entry_id
stringlengths 33
33
| published
stringlengths 14
14
| title
stringlengths 17
188
| authors
sequence | primary_category
stringlengths 5
18
| categories
sequence | text
stringlengths 2
629k
|
|---|---|---|---|---|---|---|
http://arxiv.org/abs/2307.05766v3 | 20230711194705 | Rad-ReStruct: A Novel VQA Benchmark and Method for Structured Radiology Reporting | [
"Chantal Pellegrini",
"Matthias Keicher",
"Ege Özsoy",
"Nassir Navab"
] | cs.CV | [
"cs.CV",
"cs.AI"
] |
Rad-ReStruct
Pellegrini and Keicher et al.
Computer Aided Medical Procedures, Technical University Munich, Germany
Rad-ReStruct: A Novel VQA Benchmark and Method for Structured Radiology Reporting
Chantal PellegriniContributed equally. Corresponding author: [email protected] Matthias Keicher⋆ Ege Özsoy Nassir Navab
August 12, 2023
===================================================================================================================================
Radiology reporting is a crucial part of the communication between radiologists and other medical professionals, but it can be time-consuming and error-prone. One approach to alleviate this is structured reporting, which saves time and enables a more accurate evaluation than free-text reports. However, there is limited research on automating structured reporting, and no public benchmark is available for evaluating and comparing different methods. To close this gap, we introduce Rad-ReStruct, a new benchmark dataset that provides fine-grained, hierarchically ordered annotations in the form of structured reports for X-Ray images. We model the structured reporting task as hierarchical visual question answering (VQA) and propose hi-VQA, a novel method that considers prior context in the form of previously asked questions and answers for populating a structured radiology report. Our experiments show that hi-VQA achieves competitive performance to the state-of-the-art on the medical VQA benchmark VQARad while performing best among methods without domain-specific vision-language pretraining and provides a strong baseline on Rad-ReStruct. Our work represents a significant step towards the automated population of structured radiology reports and provides a valuable first benchmark for future research in this area. We will make all annotations and our code for annotation generation, model evaluation, and training publicly available upon acceptance. Our dataset and code is available at https://github.com/ChantalMP/Rad-ReStructhttps://github.com/ChantalMP/Rad-ReStruct.
§ INTRODUCTION
Radiology is a critical medical field that relies on accurate and efficient communication between radiologists and other healthcare professionals enabled through radiology reports. However, generating these reports takes a lot of time and is prone to errors, as it often relies on ambiguous natural language. One alternative to free-text reports is to use structured reporting, which is endorsed by radiology societies, saves time, and offers standardized content and terminology <cit.>.
Automated report generation can reduce radiologists' workload and support quick diagnostic decisions. Most current research focuses on generating free-text reports, which lack standardization, and still face challenges of ambiguity and difficulties in clinical correctness evaluation <cit.>. In comparison, automating structured reporting allows an accurate evaluation of clinical correctness and can enforce the prediction of detailed findings. However, for automating structured reporting, the research is limited. Some studies predict high-level abnormalities using pre-defined template sentences <cit.>, or predict location and attributes for a single disease <cit.>. Syeda-Mahmood et al. <cit.> predict fine-grained but unstructured labels to retrieve and adapt free-text reports from a database. However, none of these works predict highly-detailed and structured annotations as needed to populate an entire structured report. A significant challenge towards this goal is the lack of public benchmarks with highly detailed structured annotations. To facilitate future research, we introduce Rad-ReStruct, the first dataset of publicly available, fine-grained, and structured annotations for Chest X-Rays. To create Rad-ReStruct, we define a detailed, multi-level structured reporting template and automatically populate it by parsing and analyzing unstructured finding summaries from the IU X-ray dataset <cit.>.
Structured reports with high standardization have a structured layout and content, e.g., organized in expendable trees with drop-down menus to select answers <cit.>. A user interface for structured reporting would pose a series of questions that, dependent on the answer, lead to expandable follow-up questions. In this setup, structured reporting can be considered several classification tasks on different levels. We model this as a hierarchical visual question answering (VQA) task and propose hi-VQA, a hierarchical, autoregressive VQA method for populating structured reports by successively filling out all fields in the report while preserving consistency. Hi-VQA considers the prior context of previously asked questions and answers, allowing to exploit inter-dependencies between questions about the same image. For structured report population, this is essential, as lower-level questions directly depend on higher levels. Further, our autoregressive formulation enhances explainability, showing at which level and for which question type the model made mistakes. As backbone, we propose a simple yet effective VQA architecture relying on large pretrained image and text encoders and a transformer-based fusion module. Using VQA <cit.> allows to exploit the knowledge encoded in large language models. It has recently received attention in the medical field, mainly on small datasets, where every question is answered independently <cit.>. One previous work explicitly models question consistency for medical VQA in the loss <cit.>. Another work had promising results using an unstructured question history in a visual dialog setting <cit.>. They ask for high-level abnormalities, such as "Pneumonia?" and use a randomly sampled, fixed history of other abnormality questions. In contrast, we define a hierarchical history with detailed questions and an autoregressive model.
We demonstrate the effectiveness of our streamlined design and hierarchical framework in our experimental results, reaching competitive results to the SOTA on the medical VQA benchmark VQARad and setting a baseline for Rad-ReStruct. Overall, our work is a significant step towards automating the population of structured radiology reports and provides a valuable benchmark for future research in this area.=-1
§ METHODOLOGY
§.§ Rad-ReStruct Dataset
We propose the first benchmark dataset to enable the development and comparison of methods for the population of structured reports entailing hierarchical and fine-grained classifications for radiological images. Rad-ReStruct is based upon the IU-Xray dataset <cit.> and consists of X-Ray images paired with fine-grained radiological findings organized as a structured report. To create the dataset, we first define a detailed report template and then populate it automatically by parsing and analyzing the unstructured expert annotations of the reports in IU-Xray.=-1
Creation of Structured Report Template We build upon the semi-structured encoded findings provided for the IU-XRay data collection <cit.>. The encoded findings were provided by medical experts, who labeled the IU-XRay free-text reports using MeSH (Medical Subject Headings) <cit.> and RadLex (Radiology Lexicon) <cit.> codes. They accurately summarize all findings in the radiological images together with a detailed attribute description. They are an unstructured collection of findings, with a sequence of annotation terms representing each finding (e.g., "infiltrate/lung/upper lobe/left/patchy/mild"...). The codes use a controlled vocabulary containing 178 terms, which include anatomies, diseases, pathological signs, foreign objects, and attributes. Anatomies and diseases can be matched to broad body regions, such as the respiratory or skeletal system. Attributes include degree, descriptive, and positional attributes.
We utilize this semi-structured finding representation to construct a highly detailed report template as shown in Figure <ref>. Our report template is structured into multiple sections and employs a multi-level hierarchy of questions that delve deeper into the findings at each level. The template can be considered a large decision tree with questions at every level. The highest level asks for the general existence of findings (signs, diseases, abnormal regions, or objects), the second level asks for specific elements, such as a certain object or disease, and the lowest-level questions ask for specific attributes. Table <ref> shows how often which question type occurs. To create the template, we parse the codes of all patients and identify all occurrences of term combinations at all levels of the defined hierarchy. We then remove unseen options to produce a streamlined report template that includes only possible options for all findings. Further, we mark all questions as either single- or multi-choice and add a "no selection" option.
Overall, our structured report template provides a rigorous and comprehensive framework for classifying radiological images and mimics the style of a structured report in a clinical setting. This enables the development and comparison of methods for the population of structured reports and the prediction of fine-grained radiological findings.
Dataset and Evaluation Metrics Our dataset consists of structured reports for each patient in the IU-XRay data collection, for which finding codes and a frontal X-Ray were available. The new dataset includes 3720 images matched to 3597 structured patient reports entailing more than 180k questions. If multiple images belong to one patient, each image is considered an independent sample. We use a 80-10-10 split to create train, validation and test set. To avoid data leakage, we ensure that different images of the same patient are in the same split.
The goal of our task is to produce fine-grained finding classifications for populating a structured report given an X-Ray image of a patient. This task involves answering a series of questions about the image, gradually adding more detail. We define several evaluation metrics for the proposed benchmark. As the distribution of questions and answers is very imbalanced, we evaluate with the macro precision, recall, and F1 score over all possible paths in the question tree to encourage methods that also perform well in under-represented question-answer combinations. One path is a unique position in the report combined with a specific answer option. Further, we employ report-level accuracy to measure how many predicted reports are entirely correct. During the evaluation, we enforce consistency within the question hierarchy. For example, if the answer to a higher-level question is "no", we prohibit to answer a lower-level question positively. This ensures the generated reports are consistent and coherent, as in a real medical report. Lastly, as multiple instances of an object, sign or pathology can occur for one patient, we iteratively ask for further occurrences, when the model predicts a positive answer. (e.g., "Are there other opacities in the lung?"). We restrict the number of follow-up questions by the maximum of per-patient occurrences in the data. As the order of occurrences is ambiguous, we apply instance matching during the metric computation. We order all predicted instances such that the highest F1 score for this finding is achieved.
§.§ Hierarchical Visual Question Answering
With Rad-ReStruct, we propose a hierarchical VQA task, where lower level questions are dependent on context information. For instance, to answer the questions "What is the degree?" it is essential to know what the question is referring to. This information is given through the previous question, which could be "Is there Pneumonia in the lung? Yes". To integrate this context, we propose a hierarchical VQA framework that can effectively answer questions about medical images by considering previously asked questions. We extend the input to the model by pre-pending the current question with the history of previously asked questions and the model's answers. This extension enables interpretable and consistent results. =-1
We leverage a pretrained image encoder, EfficientNet-b5 <cit.>, and a domain-specific pretrained text encoder, RadBERT <cit.>, to extract features from the image, history, and question. The extracted features are fused by a transformer <cit.> layer, adapted to handle multi-modal input. The fused features are then used to perform a multi-label classification over all answer options. However, we only consider outputs that are valid answers to the current question. For single-choice questions, we predict a single label applying a softmax over all valid answers, while for multi-choice questions, we predict multiple labels using a sigmoid function. Figure <ref> shows an overview of the proposed framework.
Feature Encoding For fusing the image and text features, we construct a token sequence of the form <image_tokens> <history_tokens> <question_tokens>. The image tokens consist of the flattened embedding representation of the image encoder, while the history and question text is encoded jointly using RadBERT. The different parts are separated by a <SEP> token and fused by a single transformer layer. We encode the type of input in the token-type IDs with different token types for the image tokens, history questions, history answers, and the current question. Further, we use modified positional encodings to preserve the 2D spatial information of the image as well as the sequence order of the text. We combine 2D positional embeddings <cit.> for the image tokens with the 1D positional encodings used in the original transformer architecture <cit.> for the text tokens and set the non-used part of the encoding per modality to zero.
Training and Evaluation
During training, we use teacher forcing, utilizing ground truth history, allowing for efficient batch-wise processing. The model is trained end-to-end, using a weighted masked cross-entropy loss to optimize the classification performance. For every sample in a training batch, we only consider the loss for the labels corresponding to the asked question to avoid optimizing the model on currently irrelevant outputs. The evaluation is autoregressive, thus the model utilizes the previously asked questions and their predicted answers as history. In a hierarchical VQA task such as Rad-ReStruct, the inference is interrupted if the model predicts a negative answer, and sub-questions lower in the hierarchy are automatically answered as negative, enforcing consistency of the prediction. This also improves the explainability of predictions by showing at which level the model made a mistake. For non-hierarchical VQA tasks, the history is utilized solely as context information, allowing the model to exploit inter-dependencies between different questions about the same image.=-1
§ EXPERIMENTS AND RESULTS
We test our model on Rad-ReStruct, setting a baseline for this new dataset. To further validate our model design, we compare the performance of our model with previous work on the standard VQA benchmark VQARad. We train all models on a NVIDIA A40 GPU. We use pytorch-lightning 1.8.3. and the AdamW optimizer with a learning rate of 5e-5 for VQARad and 1e-5 for Rad-ReStruct. For all models, we set the number of epochs by maximizing validation set performance.
Rad-ReStruct For Rad-ReStruct, the history includes all higher-level questions on the same question path. Additionally, attribute questions asked previously about an element, are included in the history, enabling the model to provide consistent predictions. Lastly, the history includes previously predicted instances of the same element. Table <ref> and Table <ref> show the overall and question-level results of our model.
We compare hi-VQA to a visual baseline, consisting of our image encoder and a classification layer. Hi-VQA achieves better performance than the visual baseline in all metrics, indicating the benefits of targeted information retrieval using a large language model. When comparing the RadBERT text encoder, a RoBERTa model <cit.> pretrained with radiology reports, to RoBERTaBASE, which was pre-trained on general text, the RadBERT encoder is superior. This indicates that our method can benefit from better domain-specific language encoders. Using history information improves report accuracy and precision with a slightly decreased recall and a similar F1 score, showing the benefit of history integration. We emphasize, that the history is especially important for the low-level attribute questions, as these are only meaningful with context. Therefore, it will be even more impactful with improved performance for these questions.
Our labels' hierarchical, structured formulation enables a performance analysis on different topics and levels. Hi-VQA performs well in detecting the existence of sub-topics like objects, diseases, signs, and abnormalities. However, attribute prediction performance is much lower, likely due to the rarity and complexity of these questions and error propagation from higher levels. Such an analysis is precious to understand what a model learned and when it should be trusted.
VQARad is a medical VQA benchmark with 315 radiological images and 3515 questions. The task is to make a classification over 2248 possible answers. In VQARad multiple questions are asked about one image, but in previous work they are always answered separately. To make use of possible inter-dependencies between questions, we define five question levels based on question topics in VQARad, ranging from general to specific: Modality→Plane→Organ→Presence, Count, Abnormality→Color, Position, Size, Attributes, Other. For a certain question, previously asked questions from lower or the same level are included in the history. During training, we augment the history by randomly dropping and reordering questions within a level to prevent overfitting on this small dataset.
Table <ref> shows the performance of hi-VQA compared to previous methods. Amongst the methods without domain-specific joined image-text pretraining, we reach SOTA results, even without history context. When integrating history information, our model achieves competitive results with the current SOTA method, M3AE <cit.>. This result demonstrates the promise of jointly answering questions about the same image in medical VQA tasks. Lastly, we again compare using the RadBERT encoder to RoBERTaBASE <cit.>. We can observe, also on VQARad, using RadBERT improves the performance notably, again indicating that VQA tasks benefit from domain-specific text encoders.
§ DISCUSSION AND CONCLUSION
By introducing Rad-ReStruct, the first structured radiology reporting benchmark, we create a framework to develop, evaluate, and compare structured reporting methods. The structured formulation enables an accurate evaluation of clinical correctness at different levels of granularity, focusing on levels with greater clinical importance. Moreover, such a structured finding representation could then again, rule-based, be converted to a text report while maintaining clinical accuracy. To model structured reporting, we present hi-VQA, a novel, hierarchical VQA framework with a streamlined architecture that leverages history context for multi-question and multi-level tasks. The autoregressive formulation and consistent evaluation increase interpretability and mimic the workflow of structured reporting. Moreover, as each prediction takes previous answers into account, it would allow for an interactive workflow, where the model can make predictions and react to corrections while a radiologist fills out a report.
We set a first baseline for Rad-ReStruct, with particularly good performance on higher-level questions. Although our model has limited performance on the low-level attribute questions, it performed competitive to state-of-the-art on VQARad, indicating the difficulty of our new task. We see this as an opportunity to develop methods for fine-grained understanding of radiology images, rather than solely focusing on higher-level diagnoses. Further, we show the positive effect of history integration, which is crucial for hierarchical and context-dependent tasks such as structured report population. Our work represents a significant step forward in the development of automated structured radiology report population methods, while allowing an accurate and multi-level evaluation of clinical correctness and fostering fine-grained, in-depth radiological image understanding.=-1
§ ACKNOWLEDGEMENTS
The authors gratefully acknowledge the financial support by the Federal Ministry of Education and Research of Germany (BMBF) under project DIVA (FKZ 13GW0469C) and the Bavarian Research Foundation (BFS) under project PandeMIC (grant AZ-1429-20C).
splncs04
|
http://arxiv.org/abs/2307.04674v1 | 20230710162020 | Optimal Robot Path Planning In a Collaborative Human-Robot Team with Intermittent Human Availability | [
"Abhinav Dahiya",
"Stephen L. Smith"
] | cs.RO | [
"cs.RO"
] |
B-.05emi-.025em b-.08em
T-.1667em.7exE-.125emX
=4
undefined
undefined
3pt
definition
definitionDefinition
theoremTheorem
propositionProposition
problemProblem
lemma[]Lemma
*remarkRemark
="2D
5pt
2pt
-7pt
C>c<Optimal Robot Path Planning In a Collaborative Human-Robot Team
with Intermittent Human Availability
Abhinav Dahiya and Stephen L. Smith
This research is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC)Abhinav Dahiya and Stephen L. Smith are with Department of Electrical and Computer Engineering, University of Waterloo, Waterloo
(mailto:[email protected]@uwaterloo.ca, mailto:[email protected]@uwaterloo.ca)
===================================================================================================================================================================================================================================================================================================================================================================================================================
This paper presents a solution for the problem of optimal planning for a robot in a collaborative human-robot team, where the human supervisor is intermittently available to assist the robot in completing tasks more quickly. Specifically, we address the challenge of computing the fastest path between two configurations in an environment with time constraints on how long the robot can wait for assistance. To solve this problem, we propose a novel approach that utilizes the concepts of budget and critical departure times, which enables us to obtain optimal solution while scaling to larger problem instances than existing methods. We demonstrate the effectiveness of our approach by comparing it with several baseline algorithms on a city road network and analyzing the quality of the obtained solutions. Our work contributes to the field of robot planning by addressing a critical issue of incorporating human assistance and environmental restrictions, which has significant implications for real-world applications.
§ INTRODUCTION
Robots have come a long way in the past decades, with increasing levels of autonomy transforming the way they operate in various domains, from factories and warehouses to homes and public spaces <cit.>. However, navigating dynamic environments effectively continues to be a formidable challenge. Despite the significant strides made in robot autonomy, human oversight remains vital in enhancing safety, efficiency or to comply with regulatory requirements.
For example, a robot navigating through an urban environment must abide by traffic regulations and may require human assistance in busy or construction areas to ensure safety or expedite operations. Similarly, in an exploration task, robots may require replanning due to changes in the environment, while the supervisor has already committed to a supervision schedule for other robots and is only intermittently available. By considering the operator's availability and environmental restrictions, robots can plan their paths more efficiently, avoid unnecessary waiting and decide when to use human assistance. Figure <ref> shows the problem overview with an example of a robot navigating in a city. However, the presented problem can be generalized to any arbitrary task which can be completed via different sub-tasks defined using precedence and temporal constraints.
We consider the problem of robot planning with the objective of finding the fastest path between two configurations. We demonstrate our approach through an example of robot navigation in an urban environment with intermittent operator availability, varying travel speeds, and waiting limits.
Specifically, we consider a city road network where the robot can traverse through different locations either autonomously or with the assistance of a human supervisor, each taking different amounts of time. However, the supervisor is only available at certain times, and the robot has a limited amount of time to wait at a location before it must move on to its next destination.
By formulating the problem in this way, we aim to address the challenge of collaborative robot planning in real-world environments where the availability of human supervisors may be limited and thus can affect the optimal route for the robot. In this paper, we present a method to compute the fastest path from one location to another while accounting for all these constraintsThe map shown in Fig. <ref> shows the road network of the city of Waterloo, generated using https://www.qgis.org/en/site/index.htmlQGIS, https://www.openstreetmap.org/about/OpenStreetMap and https://openrouteservice.org/OpenRouteService..
The problem of robot path planning with operator allocation in dynamic networks is inspired by real-world scenarios where the availability of human assistance and thus the robot's speed of travel and its ability to traverse certain paths can change over time, e.g., <cit.>. Traditional methods, such as time-dependent adaptations of the Dijkstra's algorithm, are not designed to handle situations in dynamic environments where waiting is limited, and the task durations may not follow the first-in-first-out (FIFO) property <cit.>. This means that a robot may arrive at its target location earlier by departing later from its previous location, for example, by using human assistance.
To address these challenges, we draw on techniques from the time-dependent shortest path literature to solve the problem. Unfortunately, existing optimal solution techniques are severely limited by their computational runtime. In this paper, we propose a novel algorithm that is guaranteed to find optimal solution and runs orders of magnitude faster than existing solution techniques.
Contributions: Our main contributions are as follows:
1) We propose a novel graph search algorithm for the collaborative planning problem with intermittent human availability. The algorithm operates by intelligently selecting the times of exploration and by combining ranges of arrival times into a single search node.
2) We provide the proof that the algorithm generates optimal solutions.
3) We demonstrate the effectiveness of our approach in a city road-network, and show that it outperforms existing approaches in terms of computational time and/or solution quality.
§ BACKGROUND AND RELATED WORK
In this section, we discuss some relevant studies from the existing literature in the area of robot planning with human supervision/collaboration. We also look into how the presented problem can be solved using existing techniques from related fields.
Planning with Human Collaboration: The problem of task allocation and path planning for robots operating in collaboration with humans has been studied extensively in recent years. Researchers have proposed various approaches, such as a data-driven approach for human-robot interaction modelling that identifies the moments when human intervention is needed <cit.>, and a probabilistic framework that develops a decision support system for the human supervisors, taking into account the uncertainty in the environment <cit.>. In the context of autonomous vehicles, studies have investigated cooperative merging of vehicles at highway ramps <cit.> and proposed a scheduling algorithm for multiple robots that jointly optimize task assignments and human supervision <cit.>.
Task allocation is a common challenge in mixed human-robot teams across various applications, including manufacturing <cit.>, routing <cit.>, surveying <cit.>, and subterranean exploration <cit.>. In addition, the problem of computing the optimal path for a robot under time-varying human assistance bears similarity to queuing theory applications, such as optimal fidelity selection <cit.> and supervisory control of robots via a multi-server queue <cit.>.
These studies provide insights into allocating assistance and path planning for robots in collaborative settings, but do not address our specific problem of computing the optimal path for a robot under bounded waiting and intermittent assistance availability. Additionally, our problem differs in that the robot can operate autonomously even when assistance is available, i.e., the collaboration is optional.
Time-Dependent Shortest Paths:
The presented problem is also related to time-dependent shortest path (TDSP) problems, which aim to find the minimum cost or minimum length paths in a graph with time-varying edge durations <cit.>. Existing solution approaches include planning in graphs with time-activated edges <cit.>, implementing modified A^*<cit.>, and finding shortest paths under different waiting restrictions <cit.>. Other studies have explored related problems such as computing optimal temporal walks under waiting constraints <cit.>, and minimizing path travel time with penalties or limits on waiting <cit.>.
Many studies in TDSP literature have addressed the first-in-first-out (FIFO) graphs <cit.>, while others have explored waiting times in either completely restricted or unrestricted settings. However, the complexities arising from non-FIFO properties, bounded waiting and the need to make decisions on the mode of operation, i.e., autonomous or assisted, have not been fully addressed in the existing literature <cit.>.
The most relevant solution technique that can be used to solve our problem is presented in <cit.>, which solves a TDSP problem where the objective is to minimize the path cost constrained by the maximum arrival time at the goal vertex. This method iteratively computes the minimum cost for all vertices for increasing time constraint value. A time-expanded graph search method <cit.> is another way of solving the presented problem by creating separate edges for autonomous and assisted modes. We discuss these two methods in more detail in Sec. <ref>. As we will see, the applicability of these solution techniques to our problem is limited due to their poor scalability for large time horizons and increasing graph size.
§ PROBLEM DEFINITION
The problem can be defined as follows. We are given a directed graph G = (V,E), modelling the robot environment, where each edge e ∈ E has two travel times corresponding to the two modes of operation: an autonomous time τ(e,0) and an assisted time τ(e,1), with the assumption that τ(e,0) ≥τ(e,1). When starting to traverse an edge, the robot must select the mode of operation for the traversal that is used for the entire duration of the edge. While the autonomous mode is always available, the assisted mode can only be selected if the supervisor is available for the entire duration of the edge (under assisted mode). The supervisor's availability is represented by a binary function μ, with μ([t_1,t_2])=1 indicating availability during the time window [t_1,t_2], and 0 otherwise.
Additionally, at each vertex v ∈ V, the robot can wait for a maximum duration of w_v ≥ 0 before starting to traverse an outgoing edge.
The robot's objective is to determine how to travel from a start vertex to a goal vertex.
This can be represented as an execution path𝒫, specified as a list of edges to traverse, the amount of waiting required at intermediate vertices and the mode of operation selected for each edge.
The objective of this problem is to find an execution path (or simply path) from a start vertex s ∈ V to a goal vertex g ∈ V∖{s}, such that the arrival time at g is minimized.
Given a set 𝒫̂ of all possible paths 𝒫 of arbitrary length n, such that 𝒫⟨(v_1,t_1,w_1,m_1), (v_2,t_2,w_2,m_2), …,(v_n,t_n,w_n,m_n)⟩, we can write the problem objective as follows:
min_𝒫∈𝒫̂ t_n
s.t. v_1 = s, v_n = g
e_v_i v_i+1 ∈ E ∀ i∈[1, n-1]
t_i+1 = t_i + w_i + τ(e_v_i v_i+1, m_i) ∀ i∈[1, n-1]
w_i ≤w_v_i ∀ i∈[1,n-1]
m_i = 1 ⇒μ([t_i + w_i, t_i+1])=1 ∀ i∈[1,n-1].
The first constraint ensures that the path starts at s and ends at g. The second constraint ensures that the topological path is valid in the graph. The third constraint ensures that the path does not violate travel duration requirements at any edge. Fourth constraint ensures that the waiting restrictions are met at each vertex. Finally, the fifth condition ensures that an edge can only be assisted if the operator is available at least until the next vertex is reached.
To efficiently solve this problem, we must make three crucial decisions: selecting edges to travel, choosing the mode of operation, and determining the waiting time at each vertex. Our proposed method offers a novel approach to computing the optimal solution. However, before delving into the details of our solution, it is necessary to grasp the concept of budget and how new nodes are generated during the search process.
§ BUDGET AND NODE GENERATION
Since the robot is allowed to wait (subject to the waiting limits), it is possible to delay the robot's arrival at a vertex by waiting at one or more of the preceding vertices. Moreover, the maximum amount of time by which the arrival can be delayed at a particular vertex depends on the path taken from the start to that vertex. Our key insight is that this information about the maximum delay can be used to efficiently solve the given problem by removing the need to examine the vertices at every possible arrival time.
We achieve this by augmenting the search space into a higher dimension, using additional parameters with the vertices of the given graph.
A node in our search is defined as a triplet (x,a_x,b_x), corresponding to a vertex x ∈ V, arrival time a_x ∈ℤ_≥ 0 and a budget b_x ∈ℤ_≥ 0. The budget here defines the maximum amount of time by which the arrival at the given vertex can be delayed. Thus, the notion of budget allows a single node (x,a_x,b_x) to represent a range of arrival times from [a_x, a_x+b_x] at vertex x. Therefore, the allowed departure time from this vertex lies in the interval [a_x, a_x+b_x+w_x].
§.§ Node Generation
The proposed algorithm is similar to standard graph search algorithms, where we maintain a priority search queue, with nodes prioritized based on the earliest arrival time (plus any admissible heuristic). Nodes are then extracted from the queue, their neighbouring nodes are generated and are added to the queue based on their priority.
Since in our search a node is defined by the vertex, arrival time and budget, we must determine these parameters for the newly generated nodes when exploring a given node.
To characterize the set of nodes to be generated during the graph search in our proposed algorithm, we define the notion of direct reachability as follows.
A node (y, a_y, b_y) is said to be directly reachable from a node (x,a_x,b_x) if x and y are connected by an edge, i.e., e_xy∈ E, and it is possible to achieve all arrivals times in [a_y, a_y+b_y] at y through edge e_xy for some departure time t_D ∈ [a_x, a_x+b_x+w_x] from x and some mode of travel.
As an example, consider a node (x,10,2) with τ(e_xy,0) = 5 and w_x=3. Then the nodes (y, 15, 5), (y, 16, 4) and (y, 17, 3) are a few directly reachable nodes from (x,10,2) (corresponding to departure times 10, 11 and 12, respectively).
Like standard graph search methods, our algorithm aims to generate all nodes directly reachable from the current node during the exploration process. One approach is to generate all directly reachable nodes from the given node (x,a_x,b_x) for all possible departure times in [a_x, a_x+b_x+w_x]. However, this results in redundancy when multiple nodes can be represented collectively using a single node with a suitable budget.
As the operator availability changes, the possible arrival times at the next vertex may present themselves as separate blocks of time. A block of arrival times can be represented using a single node, and thus we only need to generate a new node for each arrival time block. To understand this, we consider the example given in Fig. <ref>, where a node (x,a_x,b_x) is being extracted from the queue, and we want to generate the nodes corresponding to a neighbouring vertex y. The arrival time range at x, [a_x, a_x+b_x] is shown as solid purple line. The possible departure window [a_x, a_x+b_x+w_x] is shown as purple dashed line.
Under autonomous operation, the edge can be traversed by departing at any time in the departure window, resulting in possible arrival time at vertex y in the interval [a_x+α, a_x+b_x+w_x+α], shown as solid orange line. Therefore, we can represent these possible arrival times using the node n_1 = (y, a_x + α, b_x+w_x), where a_x + α is the earliest arrival time at y and b_x+w_x is the new budget. Note that the new budget is increased from the previous value by an amount of w_x.
Under assisted operation, only a subset of departure window is feasible, as shown in Fig. <ref>(b). This results in separate blocks of arrival times at y, shown as solid green lines. The range of arrival times corresponding to these blocks become the budget values for the new nodes. In the given example, two nodes are generated: n_2 = (y, t_D2+β, Δ t_2) and n_3 = (y, t_D3+β, Δ t_3).
Note that the feasible departure times are limited by operator availability and t_max. The value of t_max is the minimum of b_x + w_x and α - β. The former quantity limits the departure from x to a_x+b_x+w_x, while the latter
comes from the observation that any departure time t_D > a_x + (α - β) will result in arrival at y at a time a_y > a_x + α, and a budget b_y < b_x + w_x. However, this arrival time range is already covered by the node generated under autonomous operation.
Critical departure times: Note that the earliest arrival times for each of the three new nodes correspond to unique departure times from x (t_D1, t_D2, t_D3 in Fig. <ref>). We refer to these times as critical departure times, as exploring a node only at these times is sufficient to generate nodes that cover all possible arrival times at the next vertex.
Since in the presented problem, the edge duration depends on the mode of operation selected, the set of critical departure times for a node is a subset of times when the operator availability changes, and thus can be efficiently determined.
Next, we present how these concepts are used by our proposed Budget-A^* algorithm to solve the given problem.
§ BUDGET A^* ALGORITHM
This section details the proposed Budget A^* algorithm and its three constituent functions: EXPLORE, REFINE and GET-PATH.
To recall, a node in our search is defined as a tuple (x,a_x,b_x).
A pseudo-code for the Budget-A^* algorithm is given in Alg. <ref>, and more details on the constituent functions follow.
The algorithm initializes an empty priority queue Q, a processed set S and a predecessor function ψ.
It then adds a node (s,0,0) to Q denoting an arrival time of exactly 0 at s. The algorithm iteratively extracts the node with the earliest arrival time (plus an admissible heuristic) from Q, adds it to S, and generates new candidate nodes for each of its neighbors using the EXPLORE function. The REFINE function then checks if these nodes can be added to the queue, removes redundant nodes from Q, and updates predecessor information. The algorithm continues until Q is empty or the goal vertex is reached. The GET-PATH function generates the required path using the predecessor data and waiting limits.
§.§ Exploration
The EXPLORE function takes in several input parameters: arrival time a_x, budget b_x, waiting limit w_x, edge e_xy, travel durations τ and operator availability μ. The function returns a set 𝒩 of candidate nodes of the form (y, a_i, b_i, m_i), where a_i,b_i,m_i are the arrival time, budget and the mode of operation respectively, corresponding to all critical departure times from the node (x, a_x, b_x) to vertex y. A pseudo-code is shown in Alg. <ref>.
As discussed in Sec. <ref>, the autonomous mode generates one new node, while assisted mode can generate multiple nodes depending on operator availability, node budget and task duration. The function first adds a node (y,a_x+α, b_x+w_x) corresponding to the autonomous mode to 𝒩.
For the assisted mode, it first computes the maximum useful delay in departure t_max. Next, it generates an ordered set ℱ of feasible departure times from the current node as the times in departure window when it's possible to depart under assisted mode, computed using μ and β (line <ref>).
Lines <ref>-<ref> generate a new node (y, a_y, b_y) for each critical departure time t_d, with a budget b_y = 0 and arrival time a_y = t_d + β. The budget is then incremented for each consecutive departure time in ℱ. A gap in ℱ means a gap in arrival time at y indicating that we have considered the complete arrival time range for that critical departure time. This condition is checked in line <ref>, and the node (y,a_y, b_y, 1) is added to 𝒩.
Once all departure times in ℱ are accounted for, the set 𝒩 contains all required arrival time and budget pairs (along with the mode of operation) for the given node (x,a_x,b_x) and neighbour y.
§.§ Node Refinement
The REFINE function determines which nodes to add or remove from the search queue, based on the newly generated nodes.
The function checks if the new node is redundant by comparing its vertex and arrival time window with nodes already in the queue (Alg. <ref> line <ref>). If the new node is found to be redundant, the function returns the original queue without modifications. If not, the new node is added to the queue, and if there is any node in Q with the same vertex and an arrival time range subset of the new node's range (line <ref>), it is removed. The function then returns the updated queue and predecessor function.
§.§ Path Generation
To get the execution path from start to goal, we use the predecessor data stored in function ψ, which returns the predecessor node vertex and arrival time (x,a_x) along with the mode of travel m_x, used on the edge e_xy for a given vertex-time pair (y,a).
However, we need to determine the exact arrival and departure times at each vertex based on wait limits. To achieve this, we use the GET-PATH function shown in Alg. <ref>. The function backtracks from the goal to the start, calculating the exact departure time from the predecessor vertex based on the earliest arrival time at the current vertex and the mode of operation (line <ref>).
The exact arrival time is then determined using the departure times and the maximum allowed waiting w (lines <ref>-<ref>). The final path is stored as a list of tuples representing a vertex, the arrival time, waiting time, and mode of operation used.
§.§ Correctness Proof
Let a vertex-time pair (y, a_y) be called directly reachable from a node (x,a_x,b_x), if vertex y can be reached by departing vertex x between a_x and a_x+b_x+w_x through edge e_xy∈ E.
Consider a node (x,a_x,b_x) extracted from Q (line <ref>), and a y ∈neighbors(x) inspected in lines <ref>-<ref>. If (y, a_y) is directly reachable from (x,a_x,b_x), then there is a node in Q with vertex y whose arrival time range contains a_y.
For a given node, the critical departure times represent the number of separate arrival time blocks. Also, as discussed earlier, a single block of arrival times can be represented by a node having the earliest arrival time in that block and budget equal to the width of the block. The EXPLORE function gets called for each neighbour of x (Alg. <ref> line <ref>) and generates new nodes corresponding to each critical departure (Alg. <ref> lines <ref>-<ref>). Therefore the resulting nodes cover all possible arrival times at every neighbouring vertex of x when departing at a time in the range [a_x, a_x + b_x + w_x].
During the refinement step, only those nodes are removed for which the arrival time range is already covered by another node (Alg. <ref> line <ref>).
Therefore, after execution of the EXPLORE and REFINE functions, there exist nodes for all achievable arrival times at the neighboring vertices corresponding to the node (x,a_x,b_x).
When a node (x,a_x,b_x) is extracted from Q, for every achievable arrival time a' < a_x at x (through any path from the start vertex), there exists at least one node with vertex x in the explored set S for which the arrival time range includes a'.
We will use proof by induction.
Base case: Consider the starting node (s,0,0) (first node extracted from Q). Since it has an arrival time of 0, and arrival times are non-negative there is no earlier achievable arrival time at vertex s, so the statement is true.
Induction step: Assume the statement is true for the first k nodes extracted and added to S. We want to show that it is also true for the next node (x,a_x,b_x) extracted from Q. We will prove this by contradiction.
Suppose there exists an achievable arrival time a'<a_x at x such that no node of vertex x in S has an arrival time range that includes a'. Let (x,a') is achieved via some path[Here, only the vertex-time pairs are used to denote a path. Wait times and mode of travel are omitted for simplicity.](s,0) (u,a_u) → (v,a_v) (x,a'), where (u,a_u) and (v,a_v) are two consecutive entries in the path. Let (v,a_v) be the first pair in the path for which a node enclosing arrival time a_v is not present in S. This can also be (x,a') itself.
Since (v,a_v) is directly reachable from (u, a_u), when exploring the node corresponding to (u,a_u), a node corresponding to arrival time a_v at v must have been inserted (or already present) in Q (Lemma <ref>). Let this node be (v,a'_v, b'_v).
We have a_v ∈ [a'_v, a'_v + b'_v]. Since b'_v ≥ 0, we get a'_v ≤ a_v. Also, a_v ≤ a' because (v,a_v) and (x,a') lie on a valid path. Since we assumed a' < a_x, we get a'_v < a_x. However, since (x, a_x, b_x) is extracted from Q first, we must have a_x ≤ a'_v. Therefore, the initial assumption must be incorrect, and the statement holds for any node extracted from Q.
Consider a vertex x, and let (x,a_x,b_x) be the first node with vertex x that is extracted from Q. Then a is the earliest achievable arrival time at x.
By Lemma <ref>, if there exists an arrival time a' < a_x at x which is achievable through any path from the start, a corresponding node must be in the explored set. Since (x,a_x,b_x) is the first node with vertex x that is extracted from Q, there is no node with vertex x in the explored set S. Therefore, a is the earliest achievable arrival time at x.
§ SIMULATIONS AND RESULTS
In this section, we present the simulation setup and discuss the performance of different solution methods.
§.§ Baseline Algorithms
In this section, we present some solution approaches that we use to compare against the proposed Budget-A^* algorithm.
§.§.§ TCSP-CWT
The TCSP-CWT algorithm (Time-varying Constrained Shortest Path with Constrained Waiting Times), presented in <cit.>, solves the shortest path problem under the constraint of a bounded total travel time. To solve the given problem, we modify the original graph by creating two copies of each vertex, one for autonomous mode and another for assisted mode. New edges are added accordingly. The search is stopped at the first time step with a finite arrival time at the goal vertex.
§.§.§ Time-expanded A^*
The Time-expanded A^* algorithm is a modified version of the A^* algorithm that can be used to solve the given problem <cit.>. It creates a separate node for each vertex at each time step, and adds new edges based on the waiting limits and operator availability.
§.§.§ Greedy (Fastest Mode) Method
One efficient method for obtaining a solution is to combine a time-dependent greedy selection with a static graph search method.
This approach is similar to an A^* search on a static graph, but takes into account the arrival time at each vertex while exploring it. To determine the edge duration to the neighboring vertices, we consider the faster of the two alternatives: traversing the edge immediately under autonomous mode or waiting for the operator to become available. Once the goal vertex is extracted from the priority queue, we can stop the search and use the predecessor data to obtain the path.
§.§ Problem instance generation
For generating the problem instances, we use the map of the city of Waterloo, Ontario, Canada (a 10km × 10km area around the city centre). Using the open source tools QGIS and OpenStreetMap, we place a given number of points at different intersections and landmarks. These points serve as vertices in our graph. Next, we use Delaunay triangulation to connect these vertices and use OpenRouteService (ORS) to compute the shortest driving distance between these vertices.
An example graph of the city is shown in Figure <ref>.
To obtain the travel durations at each edge, we first sample robot speeds from a uniform random distribution. The travel durations under the two modes
are then computed by dividing the edge length (computed using ORS) by the speed values and rounding off to the nearest integer. The travel speeds are sampled as follows: autonomous speed u^0_xy∼ U[0,40]; assisted speed u^1_xy∼ U[10 + u^0_xy,30 + u^0_xy].
The maximum waiting duration at each vertex x is sampled from a uniform random distribution as w_x ∼ U[0,15].
The operator availability function is generated by randomly sampling periods of availability and unavailability, with durations of each period sampled from the range of [10, 200].
The distance values used in our simulations are in meters, times are in minutes and speeds are in meters/minute.
We test the algorithms using the Waterloo city map with varying vertex density, by selecting 64, 100 or 225 vertices to be placed in the map. We generate 20 problem instances for each density level (varying speeds, waiting limits and operator availability), and for each instance, we solve the problem for 100 randomly selected pairs of start and goal vertices. The algorithms are compared based on solution time and the number of explored nodes. We also examine some of the solutions provided by the greedy method.
Note on implementation:
All three graph search algorithms (Budget-A^*, Greedy and Time-expanded A^*) use the same heuristic, obtained by solving a problem instance under the assumption that operator is always available. This heuristic is admissible in a time-dependent graph <cit.> and can be computed efficiently. The priority queues used in all methods are implemented as binary heaps, allowing for efficient insertion, extraction and search operations. Additionally, all the methods require computation of the feasibility set (Alg. <ref> line <ref>). This is pre-computed for all departure times and is given as input to each algorithm.
§.§ Results
Figure <ref> compares the performance of the Budget-A^* algorithm with that of the Greedy algorithm in terms of durations of the generated paths. From the figure, we observe that the Greedy algorithm is able to generate optimal or close-to-optimal solutions for a large proportion of the tested problem instances. However, for many instances, the path generated by the greedy approach is much longer than that produced by the Budget-A^* algorithm, reaching up to twice the duration.
To gain further insight into our results, we present Fig. <ref>, highlighting example instances where the greedy approach fails to generate an optimal solution. Through these examples, we demonstrate how our algorithm makes effective decisions regarding path selection, preemptive waiting, and not utilizing assistance to delay arrival at a later vertex. These decisions ultimately result in improved arrival time at the goal.
Figure <ref> compares the computation time required by different solution methods for varying number of vertices and the duration of the optimal path between the start and goal vertices. The plots demonstrate that the proposed algorithm consistently outperforms the other optimal methods in terms of computation time, with the greedy method being the fastest but providing suboptimal solutions. The computation time for all methods increases with the number of vertices. The path duration has the greatest impact on the performance of the TCSP-CWT algorithm, followed by the Time-expanded A^*, the Budget-A^* algorithm, and finally the Greedy algorithm.
Figure <ref> compares the number of nodes generated and explored by Time-expanded A^*, Budget A^*, and Greedy search algorithms. The number of nodes is a key metric to evaluate search efficiency as it reflects the number of insertions and extractions from the priority queue. The Time-expanded A^* generates nodes at a faster rate with increasing vertices, while the proposed algorithm generates an order of magnitude fewer nodes, indicating better efficiency and scalability. The Greedy search algorithm terminates after exploring the least number of nodes, indicating that it sacrifices optimality for speed. In contrast, both the Time-expanded A^* and Budget A^* algorithms guarantee optimality in their search results.
§ CONCLUSION
In this paper, we introduced Budget-A^*, a new algorithm to tackle the problem of collaborative robot planning with bounded waiting constraints and intermittent human availability. Our approach computes the optimal execution path, which specifies which path should the robot take, how much to wait at each location and when to use human assistance.
Our simulations on a city road network demonstrate that Budget-A^* outperforms existing optimal methods, in terms of both computation time and number of nodes explored.
Furthermore, we note that the greedy method performs well for the majority of test cases, which could potentially be utilized to further improve efficiency of the proposed algorithm.
For future research, the Budget-A^* algorithm can be extended to handle more complex constraints such as multiple types of human assistance, non-stationary operator availability, and dynamic task requirements. Our approach can be further optimized to handle even larger networks by incorporating better heuristics and pruning techniques. Finally, our algorithm can be adapted to other applications such as emergency response in unknown environments, where fast and online task planning is crucial.
Our approach has significant implications for real-world applications like transportation systems, logistics, and scheduling, where time constraints and limited human supervision are crucial. We believe our work will inspire further research in these areas and lead to the development of more efficient algorithms for enabling human supervision under real-world restrictions.
10 url@rmstyleroyakkers2015literature
L. Royakkers and R. van Est, “A literature review on new robotics: automation
from love to war,”International journal of social robotics, vol. 7,
pp. 549–570, 2015.
mintrom2022robots
M. Mintrom, S. Sumartojo, D. Kulić, L. Tian, P. Carreno-Medrano, and
A. Allen, “Robots in public spaces: Implications for policy design,”Policy Design and Practice, vol. 5, no. 2, pp. 123–139, 2022.
dahiya2023survey
A. Dahiya, A. M. Aroyo, K. Dautenhahn, and S. L. Smith, “A survey of
multi-agent human–robot interaction systems,”Robotics and Autonomous
Systems, vol. 161, p. 104335, 2023.
riley2021assessment
D. G. Riley and E. W. Frew, “Assessment of coordinated heterogeneous
exploration of complex environments,” in IEEE Conference on Control
Technology and Applications (CCTA), 2021, pp. 138–143.
wang2019time
Y. Wang, Y. Yuan, Y. Ma, and G. Wang, “Time-dependent graphs: Definitions,
applications, and algorithms,”Data Science and Engineering, vol. 4,
pp. 352–366, 2019.
swamy2020scaled
G. Swamy, S. Reddy, S. Levine, and A. D. Dragan, “Scaled autonomy: Enabling
human operators to control robot fleets,” in IEEE International
Conference on Robotics and Automation (ICRA), 2020, pp. 5942–5948.
dahiya2022scalable
A. Dahiya, N. Akbarzadeh, A. Mahajan, and S. L. Smith, “Scalable operator
allocation for multirobot assistance: A restless bandit approach,”IEEE Transactions on Control of Network Systems, vol. 9, no. 3, pp.
1397–1408, 2022.
hickert2021cooperation
C. Hickert, S. Li, and C. Wu, “Cooperation for scalable supervision of
autonomy in mixed traffic,”arXiv preprint arXiv:2112.07569, 2021.
cai2022scheduling
Y. Cai, A. Dahiya, N. Wilde, and S. L. Smith, “Scheduling operator assistance
for shared autonomy in multi-robot teams,” in IEEE Conference on
Decision and Control (CDC), 2022, pp. 3997–4003.
fusaro2021integrated
F. Fusaro, E. Lamon, E. De Momi, and A. Ajoudani, “An integrated dynamic
method for allocating roles and planning tasks for mixed human-robot teams,”
in IEEE International Conference on Robot & Human Interactive
Communication (RO-MAN), 2021, pp. 534–539.
hari2020approximation
S. K. K. Hari, A. Nayak, and S. Rathinam, “An approximation algorithm for a
task allocation, sequencing and scheduling problem involving a human-robot
team,”Robotics and Automation Letters, vol. 5, no. 2, pp.
2146–2153, 2020.
mau2007scheduling
S. Mau and J. Dolan, “Scheduling for humans in multirobot supervisory
control,” in 2007 IEEE/RSJ International Conference on Intelligent
Robots and Systems. 1em plus 0.5em minus 0.4em IEEE, 2007, pp.
1637–1643.
gupta2019optimal
P. Gupta and V. Srivastava, “Optimal fidelity selection for human-in-the-loop
queues using semi-markov decision processes,” in 2019 American Control
Conference (ACC), 2019, pp. 5266–5271.
powel2012multiserver
N. D. Powel and K. A. Morgansen, “Multiserver queueing for supervisory control
of autonomous vehicles,” in 2012 American Control Conference
(ACC). 1em plus 0.5em minus 0.4em IEEE, 2012, pp. 3179–3185.
dean2004algorithms
B. C. Dean, “Algorithms for minimum-cost paths in time-dependent networks with
waiting policies,”Networks: An International Journal, vol. 44,
no. 1, pp. 41–46, 2004.
zhao2008algorithm
L. Zhao, T. Ohshima, and H. Nagamochi, “A* algorithm for the time-dependent
shortest path problem,” in WAAC08: The 11th Japan-Korea Joint Workshop
on Algorithms and Computation, vol. 10, 2008.
orda1990shortest
A. Orda and R. Rom, “Shortest-path and minimum-delay algorithms in networks
with time-dependent edge-length,”Journal of the ACM (JACM), vol. 37,
no. 3, pp. 607–625, 1990.
ding2008finding
B. Ding, J. X. Yu, and L. Qin, “Finding time-dependent shortest paths over
large graphs,” in International Conference on Extending Database
Technology: Advances in Database Technology, 2008, pp. 205–216.
bentert2020efficient
M. Bentert, A.-S. Himmel, A. Nichterlein, and R. Niedermeier, “Efficient
computation of optimal temporal walks under waiting-time constraints,”Applied Network Science, vol. 5, no. 1, pp. 1–26, 2020.
he2021time
E. He, N. Boland, G. Nemhauser, and M. Savelsbergh, “Time-dependent shortest
path problems with penalties and limits on waiting,”INFORMS Journal
on Computing, vol. 33, no. 3, pp. 997–1014, 2021.
foschini2011complexity
L. Foschini, J. Hershberger, and S. Suri, “On the complexity of time-dependent
shortest paths,” in ACM-SIAM Symposium on Discrete Algorithms (SODA),
2011, pp. 327–341.
cai1997time
X. Cai, T. Kloks, and C.-K. Wong, “Time-varying shortest path problems with
constraints,”Networks: An International Journal, vol. 29, no. 3, pp.
141–150, 1997.
ford1958constructing
L. R. Ford Jr and D. R. Fulkerson, “Constructing maximal dynamic flows from
static flows,”Operations research, vol. 6, no. 3, pp. 419–433,
1958. |
http://arxiv.org/abs/2307.05743v2 | 20230710174318 | An exactly solvable dissipative spin liquid | [
"Henry Shackleton",
"Mathias S. Scheurer"
] | cond-mat.str-el | [
"cond-mat.str-el",
"cond-mat.stat-mech",
"quant-ph"
] |
Department of Physics, Harvard University, Cambridge MA 02138, USA
Institute for Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany
Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
Exactly solvable Hamiltonians with spin liquid ground states have proven to be extremely useful, not only because they unambiguously demonstrate that these phases can arise in systems of interacting spins but also as a pedagogical illustration of the concept and as a controlled starting point for further theoretical analysis. However, adding dissipative couplings to the environment—an important aspect for the realization of these phases—generically spoils the exact solvability. We here present and study a Lindbladian, describing a square-lattice spin-liquid with dissipative coupling to the environment, that admits an exact solution in terms of Majorana fermions coupled to static ℤ_2 gauge fields. This solution allows us to characterize the steady-state solutions as well as “quasiparticle” excitations within the Lindbladian spectrum. We uncover distinct types of quasiparticle excitations of the Lindbladian associated with parametrically different timescales governing the equilibration time of the expectation values of different classes of observables. Most notably, for small but non-zero dissipation, we find a separation into three different timescales associated with a three-step heating profile.
On a more general level, our exactly solvable Lindbladian is expected to provide a starting point for a better understanding of the behavior of fractionalized systems under dissipative time evolution.
An exactly solvable dissipative spin liquid
Mathias S. Scheurer
August 12, 2023
===========================================
§ INTRODUCTION
Quantum spin liquids (QSLs) are exotic phases of matter characterized by emergent anyon excitations with non-trivial braiding statistics, in conjunction with the absence of any conventional long-range order <cit.>. Further interest in these states have grown due to their potential applications for use in fault-tolerant quantum computation <cit.> through their non-local encoding of quantum information.
The interplay between QSLs and open quantum systems has been an active area of research for many years, with a primary focus on the robustness of their information storage and on approaches to detect their presence when perturbations generic to experimental realization are introduced, such as a non-zero temperature, decoherence, and more <cit.>. Rather than taking this approach of considering generic forms of decoherence, we instead consider engineering a particular form of environmental coupling to a QSL in order to realize unique non-equilibrium physics. This general approach of leveraging dissipation has been shown to be efficient at preparing quantum states <cit.> including topologically-protected edge modes <cit.>. Recent applications of this idea to spin liquids <cit.> have yielded new insights into the behavior of emergent anyon excitations in the presence of dissipation.
We study a quantum spin-3 / 2 model on a two-dimensional square lattice, which is a particular limit of the QSL studied in <cit.>, and subject it to a certain choice of Markovian open dynamics generated by the Lindblad equation. We show that in a particular limit, the Lindbladian becomes exactly solvable through a parton construction. As such, exact statements about its steady-state solutions as well as transient behavior can be made. Exactly solvable Lindbladians have been studied previously using techniques such as third quantization <cit.>, Bethe ansätze <cit.>, operator-space fragmentation <cit.>, and through parton constructions <cit.> similar to our own. From a practical perspective, this exact solvability is especially useful as the wealth of analytic tools developed to approximately study the low-energy behavior of Hermitian Hamiltonians do not immediately carry over to these non-Hermitian Lindbladians, although several methods for approximately studying the spectrum of Lindbladians have been developed <cit.>.
A particular property of our exact solution that we emphasize is the existence of distinct quasiparticle excitations of the Lindbladian when viewed as an effective non-Hermitian Hamiltonian acting on an enlarged Hilbert space. We advocate for this as a powerful tool for understanding the non-equilibrium behavior of a generic state or density matrix as it equilibrates to its steady-state solution. We show that the imaginary energy gap associated with a particular type of quasiparticle excitation in this enlarged Hilbert space can be associated with the equilibration timescale of the expectation value of a certain class of observables. These classes of observables turn out to have a close relation to excitations of the corresponding unitary spin liquid.
An expert reader might immediately want to inspect Fig. <ref> for a summary of the spectrum. Importantly, the different time scales of these classes of operators have different parametric dependence on the strength γ of the coupling to the environment, which can be found simply by diagonalizing a quadratic Hamiltonian numerically, or in some cases is derived exactly analytically. For instance, in the limit of small γ, a certain set of operators, that are not conserved by the unitary dynamics, decay rapidly on a scale set by the exchange coupling rather than γ itself. Fractionalized string-like operators that can be interpreted as pairs of emergent Majorana fermion excitations in the unitary system, however, survive up to a time-scale ∝ 1/γ. After that, also the Majorana fermions heat up and only gauge-invariant fluxes of the emergent gauge fields or Wilson-loop operators remain in their original configuration. In this sense, our model realizes a three-step and exactly solvable analogue of the “fractionalized pre-thermalization” discussed recently <cit.> for stroboscopic time-evolution in the Kitaev model.
The remainder of the paper is organized as follows. A mathematical definition of all the involved operators and of the dissipative model we study can be found in Sec. <ref>. We derive an interpret the spectrum of the Lindbladian in Sec. <ref>. A discussion of perturbations away from the exactly solvable point and a conclusion are provided in Sec. <ref> and Sec. <ref>, respectively.
§ MODEL
The time evolution of a density matrix ρ can be described in its most general form by a completely-positive and trace preserving map Φ(ρ) →ρ'.
The Lindblad equation <cit.> is the most generic continuous Markovian map satisfying these properties,
ρt = ℒ[ ρ] = - i Hρ + ∑_j ( L_j ρ L_j^† - 1/2L_j^† L_j^ρ) ,
where the quantum jump operators L_j parameterize the nature of the environmental coupling. One may express the superoperator ℒ as an operator in a “doubled” Hilbert space, namely the Hilbert space of all operators. For a choice of basis in the original Hilbert space, |ψ_i⟩, i = 1…𝒟, we can represent any operator 𝒪 = ∑_i 𝒪_ij|ψ_i⟩⟨ψ_j| as a state ‖𝒪≡∑_ij𝒪_ij|ψ_i⟩⊗|ψ_j⟩ in this doubled Hilbert space, with inner product 𝒪_1 ‖𝒪_2 = 1/𝒟( 𝒪_1^†𝒪_2 ). Within this doubled Hilbert space, the action of the Lindbladian superoperator is
i ℒ = H_eff⊗𝕀 - 𝕀⊗ H_eff^† + ∑_j i γ L_j ⊗ L_j^† ,
H_eff ≡ H - i γ/2∑_j L_j^† L_j^ .
We will take L_j to be unitary, such that H_eff = H up to an overall imaginary constant.
This doubled Hilbert space construction is a powerful tool for characterizing the behavior of mixed states; notably, it has seen recent use in diagnosing the stability of quantum information stored in mixed states <cit.>. For a quantum spin model in two dimensions, it is instructive to think of this doubled Hilbert space as corresponding to a bilayer system, where the first (second) layer corresponds to the bra (ket). In this scenario, the Lindbladian consists of two copies of the Hamiltonian ± H acting on each of the two layers, with anti-Hermitian couplings i γ∑_j L_j ⊗ L_j^† between the two layers. To better connect with intuition from unitary time evolution, we will focus on the eigenvalues of the matrix iℒ rather than ℒ and refer to iℒ as “the Lindbladian”; in this convention, the imaginary components of eigenvalues correspond to dissipation, and the non-existence of exponentially growing solutions requires the imaginary part to always be negative.
§.§ Unitary time evolution
The Hermitian dynamics that we consider is a particular limit of an exactly solvable quantum spin-3 / 2 model on a square lattice first studied in <cit.>. We define this model here and review some properties of its solution, as our results are most clearly stated within this framework. Due to the four spin polarizations per site, we may express the spin-3 / 2 degrees of freedom in terms of anticommuting Gamma matrices Γ^a, a = 1… 5, which obey Γ^aΓ^b = 2 δ^ab. In terms of the physical spin operators,
Γ^1 = 1/√(3)S^yS^z , Γ^2 = 1/√(3)S^zS^x ,
Γ^3 = 1/√(3)S^xS^y , Γ^4 = 1/√(3)[ (S^x)^2 - (S^y)^2 ] ,
Γ^5 = (S^z)^2 - 5/4 .
The Hamiltonian is defined on a square lattice as
H = ∑_j [ J_x Γ^1_j Γ^2_j + x̂ + J_y Γ^3_j Γ^4_j + ŷ]
+ ∑_j [ J_x' Γ^15_j Γ^25_j + x̂ + J_y' Γ^35_j Γ^45_j + ŷ] - J_5 ∑_j Γ^5_j
where Γ^ab_j ≡Γ^a_jΓ^b_j / 2 i. For simplicity, we will assume that the lattice has an even number of sites in both the x̂ and ŷ directions. The exact solvability of this model is a consequence of an extensive number of conserved fluxes,
W_j = Γ^13_j Γ^23_j + x̂Γ^14_j + ŷΓ^24_j + x̂ + ŷ,
and can be understood most conveniently by performing a Majorana decomposition of the Γ matrices; specifically, one employs the representation
Γ_j^μ = i c_j^μ d_j , Γ_j^μ 5 = i c_j^μ d_j' , μ = 1 , 2 , 3 , 4 ,
Γ_j^5 = i d_j d_j' ,
with the constraint - i c_j^1 c_j^2 c_j^3 c_j^4 d_j d_j' = Γ_j ^1 Γ_j^2 Γ_j^3 Γ_j^4 Γ_j^5 = -1. In this representation, the Hamiltonian can be rewritten in terms of static ℤ_2 gauge fields ŵ_j,α living on the bonds of the lattice, which come from conserved bilinears of the c_j^μ operators, coupled to two species of Majorana fermions, d_j and d_j'.
We will not give a detailed review of the various properties of this solution <cit.>, as it will not be important for our analysis. However, we will emphasize the relation between these emergent degrees of freedom and physical observables, as the results of our dissipative model concisely fit into this picture. The ℤ_2 gauge fluxes - products of closed loops of ŵ_j,α operators - correspond to the conserved fluxes W_j. Pairs of Majorana fermions coupled by a string of ℤ_2 gauge fields are given by strings of Γ matrices. For a pair of d excitations, the operator can be generated by a string of bond operators:
V_j, α = Γ^1_j Γ^2_j + x̂ α = x ,
Γ^3_j Γ^4_j + ŷ α = y .
A similar construction follows for a pair of d' fermions,
V_j ,α' = Γ^15_j Γ^25_j + x̂ α = x ,
Γ^35_j Γ^45_j + ŷ α = y ,
as well as the combination of a d and d' fermion, a special case of which is Γ^5_j = i d_j d_j'.
Note that a closed loop of either the V_j,α or V_j,α' operators is equivalent to a product of the conserved fluxes contained inside the loop.
In order to retain the exact solvability upon the inclusion of dissipation, we take J_x' = J_y' = J_5 = 0, which causes the bond operators V_j ,α' to become conserved quantities. In the Majorana fermion language, this limit quenches the dispersion of the d_j' fermions and the ground state becomes highly degenerate as pairs of d_j' may be added in at no energy cost.
§.§ Jump operators
We now introduce jump operators L_j = Γ^5_j. Note that our Lindbladian jump operators commute with the conserved flux, L_jW_k = 0. This property implies that the flux operators W_j constitute strong symmetries of the system, as defined in <cit.>, and means that an initial state with a definite flux configuration will remain in such a configuration. If we express our Hermitian model as free Majorana fermions coupled to a static ℤ_2 gauge field, the interpretation of this phenomenon is that the gauge fields will remain static under the Lindbladian time evolution while generically we expect the Majorana fermions to evolve to resemble a finite-temperature Gibbs state. One may think of this behavior as “fractionalized thermalization.” For a generic set of quantum jump operators that commute with W_j, we expect the steady-state solutions of the Lindbladian can be represented as the tensor product of a thermal Gibbs state of Majorana fermions with a pure state of ℤ_2 gauge fields. We note related work studying the separation of thermalization timescales in fractionalized excitations on the Kitaev honeycomb model <cit.> under stroboscopic time evolution, as well as more directly analogous work studying the Kitaev honeycomb model coupled to jump operators that commute with the conserved fluxes <cit.>. Apart from fluxes being exactly conserved under dissipative dynamics, we also uncover below an additional, less apparent regime of fractionalized thermalization in our exactly solvable model, which occurs in the limit of small dissipation.
The above discussion follows for any jump operator that commutes with the conserved fluxes, and remains true even away from the limit J'_x = J'_y = J_5 = 0. However, our particular model admits additional conserved quantities which render the full dissipative dynamics exactly solvable.
To see this, we use the doubled Hilbert space formalism, see GeneralBilayerModel, to express the Lindbladian superoperator as an operator acting on a bilayer spin-3/2 system, with Gamma matrices Γ^a_R , Γ^a_L for the two layers - the R , L subscript indicates that they correspond to the right and left action of the gamma matrices on the physical operator. The Lindbladian can be written as
i ℒ = H[ Γ_R ] - H[ Γ_L ] + i γ∑_j Γ_j, R^5 Γ_j,L^5 - i γ N,
where N is the number of sites.
This bilayer representation makes it clear that, in addition to the intralayer fluxes W_j, R , W_j ,L which are defined in analogy to WOperators and commute with the Lindbladian separately, we have a new set of conserved interlayer fluxes U_j,α≡ V_j, α, R' V_j,α, L' defined on the plaquettes connecting the two layers, shown in Fig. <ref>. These conserved quantities are “weak” symmetries <cit.>. In contrast to the strong symmetries generated by the flux operators W_j, the operators V_j ,α' do not commute with the jump operators L_j individually, and it is exclusively the conserved superoperator consisting of the simultaneous right and left action of V_j,α' that commutes with the Lindbladian.
§.§ Parton construction
To elucidate the exact solvability of this model, we represent the Gamma matrices in terms of six Majorana fermions,
Γ_j, R^μ = i c_j, R^μ d_j, R , Γ_j, R^μ 5 = i c_j, R^μ d_j, R' , μ = 1 , 2 , 3 , 4 ,
Γ_j, R^5 = i d_j, R d_j, R' ,
with an analogous representation for Γ^μ_L in terms of c_j, L^μ , d_j, L , d_j, L '. This enlarges our Hilbert space, which necessitates the constraint - i c_j, R^1 c_j, R^2 c_j,R^3 c_j, R^4 d_j, R d_j, R' = Γ_j, R^1 Γ_j, R^2 Γ_j, R^3 Γ_j, R^4 Γ_j, R^5 = -1 on all physical states, and likewise for the Γ_L operators.
In this representation, the Hamiltonian H[Γ_R] becomes
H[Γ_R] = ∑_j J_x ŵ_j, x, R i d_j, R d_j + x̂, R + J_y ŵ_j, y, R i d_j, R d_j + ŷ, R
where ŵ_j, x, R≡ -i c_j, R^1 c^2_j + x̂, R and ŵ_j, y, R≡ - i c_j, R^3 c^4_j + ŷ, R are conserved quantities with eigenvalue ± 1. An analogous rewriting follows for the Hamiltonian on the second layer. Observe that the Majorana fermions d_j, R' , d_j, L' drop out of the intralayer Hamiltonian entirely. As a result, the interlayer coupling also becomes quadratic in the Majorana fermions,
i γ∑_j Γ^5_j, RΓ^5_j, L = - i ∑_j d_j, R d_j, R' d_j, L d_j, L'
= - γ∑_j v̂_j d_j, R d_j, L ,
where v̂_j ≡ -i d_j, R' d_j, L' is a conserved quantity with eigenvalue ± 1. With this rewriting, our model becomes one of free fermions d_j, R , d_j, L hopping on a bilayer square lattice in the presence of a background ℤ_2 gauge field ŵ_j, α, R , ŵ_j,α, L , v̂_j living on the links. Written out explicitly,
i ℒ = ∑_ℓ = L, R∑_j s_ℓ[ J_x ŵ_j, x, ℓ j d_j, ℓ d_j + x̂, ℓ + J_y ŵ_j, y, ℓ i d_j, ℓ d_j + ŷ, ℓ] - γ∑_j v̂_̂ĵ d_j, R d_j, L - i γ N
where s_L = 1, s_R = -1. This Lindbladian possesses a local ℤ_2 gauge symmetry, given by the transformation d_j, ℓ→Λ_j, ℓ d_j, ℓ, ŵ_j, α, ℓ→Λ_j, ℓŵ_j, α, ℓΛ_j + α̂, ℓ, v̂_j →Λ_j, Lv̂_j Λ_j, R, where Λ_j, ℓ = ± 1. The gauge-invariant fluxes around a single intralayer plaquette gives the conserved quantities -W_j, R , -W_j, L, and the fluxes around an interlayer plaquette gives the conserved superoperator -U_j,α. Note the relative minus signs between the two quantities - as will be relevant later, working in a sector with U_j,α = 1, which is the sector where steady-state solutions will belong to, requires us to pick a gauge configuration such as v̂_j = (-1)^j.
In order to obtain physical states, we must project back to our physical (doubled) Hilbert space. This is obtained by the projection operator P = ∏_j, ℓ1 + D_j, ℓ/2, where D_j, ℓ = - i c^1_j, ℓ c^2_j, ℓ c^3_j, ℓ c^4_j, ℓ d_j, ℓ d'_j, ℓ. A careful analysis of this for a single-layer Hamiltonian was performed in <cit.> and our analysis proceeds along similar lines. We can write P = P'(1+D), where D ≡∏_j, ℓ D_j, ℓ and P' is a linear combination of all inequivalent gauge transformations. Since D^2 = 1, Dℒ = 0, this means that we must restrict ourselves to eigenstates with D = 1. We write
D = ∏_j, α, ℓŵ_j, α, μ∏_j v̂_j ∏_j i d_j, L d_j, R .
In order to more readily leverage the gauge constraint, we re-express the Majorana fermions d_j, R , d_j, L in terms of complex fermions. A representation that will prove to be useful for future analysis is
f_j = i^j ( d_j, L + i (-1)^j d_j, R) / 2 .
With this, 2 f_j^† f_j - 1 = (-1)^j i d_j, L d_j, R and (-1)^N_f≡ (-1)^∑_j f_j^† f_j = ∏_j i d_j, L d_j, R. Therefore, gauge invariance restricts the total fermion parity, (-1)^N_f, to equal the total “gauge parity,” ∏_j, α, μŵ_j, α, μ∏_j v̂_j.
§ SPECTRUM OF THE LINDBLADIAN
In the previous section, we have shown that our Lindbladian reduces down to one of free fermions coupled to a static ℤ_2 gauge field. As such, the full spectrum and eigenvectors can in principle be calculated - analytically for translationally-invariant gauge field configurations, and by diagonalizing a non-Hermitian single-particle Hamiltonian for more general gauge configurations. However, the interpretation of these properties must be done in terms of density matrices of our physical Hilbert space, rather than a more conventional analysis of Hermitian systems. We outline our general approach to understanding these properties below.
§.§ General remarks
The most important eigenstates of the Lindbladian are those with eigenvalue zero, which correspond to steady-state solutions. Since the eigenvalues λ_i of the Lindbladian obey [λ_i] ≤ 0, every initial density matrix will eventually evolve into some superposition of these steady-state solutions (for simplicity, we we ignore the possibility of solutions with purely real eigenvalue, i.e. density matrices that do not decay but whose phase oscillates in time, as these are not present in our spectrum). Our first task will be to find these steady-state solutions and understand their properties.
Ascertaining the properties of these steady-state solutions is a non-trivial task within the doubled Hilbert space formalism. Given a density matrix ‖ρ, the expectation value of a Hermitian operator A is given by [ A ρ] = A ‖ρ. As such, standard intuition for calculating observables of pure states in ordinary Hilbert spaces, ⟨ψ| A |ψ⟩, is not applicable here. While it is possible to develop the machinery to perform such calculations, we instead proceed with a more intuitive symmetry-based analysis. The exact solvability of our model provides an extensive number of superoperators that commute with the Lindbladian, and hence ‖ρ will be an eigenstate of them. By decomposing our Hilbert space into subspaces with definite eigenvalue under these superoperators, we can conclude that A ‖ρ must vanish unless the two have the same eigenvalue. In general, this symmetry analysis only gives us limited information about ‖ρ. However, the extensive number of conserved quantities makes this perspective especially powerful for our model, and we will find that only a small amount of additional analysis is required to fully characterize the steady-state solution.
After characterizing the steady-state solutions, we will analyze the dissipative solutions - operators with eigenvalue λ_i obeying λ_i < 0. We will be interested in eigenvalues whose imaginary components have the smallest magnitude, which defines the Liouvillian gap, and a corresponding timescale associated with the decay to the steady-state solution. As the spectrum of our Lindbladian has the interpretation of fermions coupled to a ℤ_2 gauge field, we find it insightful to define distinct types of Liouvillian gaps depending on the nature of the excitation. For example, one may inquire into the Liouvillian gap with respect to fermionic excitations, or with respect to gauge excitations (visons). This is not an arbitrary labeling, the motivation for which ties back to our symmetry-based analysis of steady-state solutions. Excitations within a given sector will have different eigenvalues under the symmetries of our Lindbladian, and hence can be characterized by distinct classes of observables that have a non-zero overlap with these excitations. The corresponding Liouvillian gap for these excitations specify a timescale which governs the rate at which the expectation values for these classes of observables asymptote to their steady-state solutions. We note that a similar hierarchy of timescales was recently studied in random local Liouvillians <cit.> and in fact observed in simulations on a quantum computer <cit.> - in this model, the separation of timescales was associated with differing spatial extents of operators.
To be more explicit with our perspective, consider a steady-state solution ‖ρ_ss and a dissipative solution ‖ a which we interpret as a quasiparticle excitation of type a. A physical density matrix can be constructed by ‖ρ_d ≡‖ρ_ss + c ‖ a, where c is some constant chosen to ensure [ ρ_d^2 ] < 1. This density matrix asymptotes to ‖ρ_ss at late times but displays transient behavior dictated by ‖ a up to a timescale t_a = -[λ_a]^-1. It is useful to characterize this operator a in terms of observables {𝒪_a} such that [ 𝒪_a a ] ≠ 0, in which case one can say that the expectation value of observables 𝒪_a relax to their steady-state values with a timescale dictated by t_a for the density matrix ‖ρ_d. Of course, a generic initial density matrix will be more complicated than ‖ρ_d; however, if ‖ a is the lowest-energy excitation that has a non-zero overlap with the observables 𝒪_a, then t_a provides an upper bound on the equilibration timescale for the expectation value of these observables.
The utility of this picture is contingent on the operators 𝒪_a having a sufficiently simple representation. As we will show, these different classes of observables are most conveniently stated in terms of fractionalized operators acting on the original Hilbert space, such as the bond operators in Eq. <ref> and Eq. <ref>. In other words, we demonstrate a close connection between fractionalized excited states in the doubled Hilbert space formalism and fractionalized operators in the physical Hilbert space, with the imaginary energy of the former defining the equilibration timescale of expectation values of the latter.
§.§ Steady-state solutions
We now study the properties of the steady-state solutions. Recall that for isolated systems with similar Hamiltonians (free fermions coupled to static ℤ_2 gauge fields), there is a theorem due to Lieb <cit.> for bipartite lattices that fixes the gauge flux sector in which the ground state resides in. In a similar spirit, we leverage general arguments given in <cit.> that allow us to deduce gauge flux sectors which support steady-state solutions.
A fact that we will use in this argument is that any dissipative eigenstate of the Lindbladian must have zero trace - if it had a non-zero trace, the dissipative nature implies that the trace would decay in time, contradicting the trace preservation of the Lindbladian time evolution. Hence, the search for steady-state solutions can be recast as a search for eigenstates with a non-zero trace. This comes with the caveat that we may miss steady-state solutions that happen to also have zero trace; however, we explicitly diagonalize the Lindbladian for a 4 × 4 lattice in each gauge sector and have found no such solutions.
We first constrain the interlayer fluxes U_j,α, which constitute weak symmetries. Recall that the superoperator U_j,α acts on density matrices as U_j,α[ρ] = V_j,α' ρ V_j,α'. An eigenstate of U_j,α with non-zero trace must have eigenvalue 1, since unitarity and Hermiticity of V_j,α' implies [ρ] = [V_j,α' ρ V_j,α']. Hence, we will constrain ourselves to the U_j,α = 1 sector.
We now turn to the “strong” symmetries W_i. A similar argument as the last paragraph implies that we must constrain ourselves to sectors where W_j ρ W_j = ρ. However, recall that in the doubled Hilbert space formulation, the right and left fluxes (W_j, R and W_j, L) are conserved separately. Hence, our analysis only constrains the eigenvalues of W_j, R and W_j, L to be the same. This is actually not a new constraint - the product of fluxes around any closed surface must be +1, so the constraint that all U_j,α = +1 automatically implies W_j, R = W_j, L. We will denote this choice of W_j, R , W_j, L eigenvalue as W_j to distinguish from the operator W_j. One can prove, as in Appendix A of <cit.>, that at least one steady-state solution exists for each choice of eigenvalue.
Translating the above statements to our gauge field representation, we fix our gauge sector to be ŵ_j, α, R = ŵ_j, α, L≡ŵ_j, α and v̂_j = (-1)^j. The complex fermion representation chosen in Eq. <ref> makes the Lindbladian in the steady-state gauge sector especially simple, as
2 (f^†_j f^_j + x̂ + f_j +x̂^† f_j^) = i d_j, R d_j + x̂, R - id_j, L d_j + x̂, L ,
2 f_j^† f_j^ = 1 - (-1)^j i d_j, R d_j, L ,
where an identical relation as in the first line but for x̂↔ŷ also holds. As a consequence, the Lindbladian takes the simple form
i ℒ = ∑_j ( J_x ŵ_j, x̂ f_j^† f_j + x̂^ + J_y ŵ_j, ŷ f_j^† f_j + ŷ^ + h.c)
- 2 iγ∑_j f_j^† f_j^,
see Fig. <ref>.
The non-Hermiticity of i ℒ is manifest as simple imaginary chemical potential, and we can immediately identify the steady-state solution as the f_j^† vacuum state. The real part of the dispersion is unaffected by the dissipation, and all excitations come with the same dissipative energy penalty 2 γ.
What are the expectation values of observables in these steady-state solutions? Recall that these solutions have eigenvalue 1 under the symmetries U_jα and W_jR W_jL. Any observable with a non-zero expectation value with respect to this steady-state must have identical eigenvalues. Phrased in terms of operators on our original Hilbert space, the requirement is that observables must commute with the flux operators W_j and the bond operators V_jα'. This is a strong constraint - the only operators that satisfy this condition are precisely products of the V_jα bond operators defined in Eq. <ref>. One can check explicitly that these operators satisfy the required constraints, and the claim that these are the only operators with such a property follows from dimension counting, worked out in Appendix <ref>. Physically, these correspond to all operators that can be expressed in terms of pairs of d_j Majorana fermions connected by strings of ℤ_2 gauge fields ŵ_j α.
We now argue that among these operators, only closed loops of V_jα operators have a non-zero expectation value - recall that these correspond to products of flux operators W_j. This is a consequence of the steady-state solution being the vacuum state of the f_j^† operators, which gives an additional set of constraints: (1 - 2 f_j^† f_j) ‖ρ = ‖ρ. We can turn this into a gauge-invariant statement by the following rewriting
‖ρ = (-1)^j (1 - 2 f_j^† f_j) v̂_j ‖ρ
= d_j, R d_j, L d_j, R' d_j, L' ‖ρ
= Γ_j,R^5 Γ_j,L^5 ‖ρ .
Hence, any non-zero observable must have eigenvalue 1 under the symmetry Γ^5_j, RΓ^5_j, L (i.e., they commute with Γ^5_j), and these are precisely closed loops of V_j, α operators. Using the fact that the steady-state solution obeys the relation W_j, L‖ρ = W_j, R‖ρ≡W_j ‖ρ, we can deduce that the expectation value of the flux operators in this steady state are given precisely by the intralayer gauge fluxes W_j.
When our model is defined on a torus, the steady states of our Lindbladian exhibit a four-fold topological degeneracy arising from the possibility of flipping non-contractible loops of ŵ_j, α operators, shown in Fig. <ref>. Physically, this implies four distinct steady-state density matrices ρ_1-4 for each local flux configuration, which are distinguishable based on the expectation values of non-contractible strings of Γ matrices. We emphasize that, while this may be thought of as a topological degeneracy - and more generally, ℤ_2 topological order - within the doubled Hilbert space formalism, it does not constitute true mixed state topological order in the sense of being able to encode logical qubits in the steady-state solutions. What may appear to be a “quantum” superposition of different topological sectors ‖ρ_1 + ‖ρ_2 within the doubled Hilbert space formalism translates to a mere classical superposition of density matrices ρ_1 + ρ_2 within our original Hilbert space (moreover, the relative phase between the superposition of the two steady-states is not freely tunable - it is fixed by the Hermiticity and positive semi-definite constraint on the physical density matrix).
§.§ Liouvillian gaps
Moving beyond steady-state solutions, we can calculate the Liouvillian gap - the energy of the next-lowest state in imaginary energy. It is useful to draw a distinction between different types of Liouvillian gaps. The three types of degrees of freedom in our Lindbladian are complex fermions f_j, interlayer gauge fields v̂_j, and intralayer gauge fields ŵ_j, α, R, ŵ_j, α, L. Excitations with respect to any of these three variables may be considered. Recall from Eq. <ref> that gauge invariance requires an even number of excitations.
* Within a gauge field configuration with a steady-state solution, we compute the fermion gap, which is the energy associated with a fermionic excitation. In accordance with the condition of gauge invariance discussed previously, any valid state must include a pair of these excitations.
* We also compute the effects of interlayer gauge excitations, which corresponds to the energy associated with flipping a single v̂_j away from the “checkerboard” sector. We call this the interlayer gauge gap.
* Finally, we analyze intralayer gauge field excitations, which come from flipping a single ŵ_j, L operator. We choose left gauge fields for concreteness - an identical calculation follows for right gauge fields.
We will study each of these excitations in turn. In addition to calculating their Liouvillian gaps, we also identify operators whose equilibration timescales can be upper bounded by these gaps. We make this identification primarily through the symmetry-based analysis outlined previously in Section <ref>. To be precise, each of these excitations will be associated with a particular flux configuration, and the excitations can therefore only have a non-zero overlap with operators whose eigenvalues under the flux superoperators are identical.
This analysis is robust and can be applied to any excitation; however, for interlayer gauge excitations, we will find that the nature of the fermionic degrees of freedom allows us to say more about the structure of the long-lived excitations.
§.§.§ Fermion gap
We first study the Liouvillian gap associated with fermionic excitations within the steady-state gauge sector. As is clear from Eq. <ref>, the fermion gap is always 2 γ, and a pair of these excitations will cost energy 4 γ. As these excitations remain in the same gauge sector, they will still have eigenvalue 1 under the symmetries U_j,α, W_j,R W_j,L. Recalling the relation between f_j and the Majorana fermions in Eq. <ref>, we see that this fermion gap of 4 γ defines the inverse timescale under which the expectation values of pairs of d_j fermions will asymptote to their steady-state value of zero. The fact that also the Hermitian part of iℒ, the first line in Eq. <ref>, is quadratic means that the (in general ŵ_j,α dependent) exact eigenstates of the Lindbladian in the steady-state gauge sector and the time-dependent phases they pick up are characterized by all possible occupation numbers of the N Bloch states of the f_j and their band structure; the associated decay rate is just given by 2γ times the number of occupied Bloch states.
§.§.§ Interlayer gauge excitation
Creating an interlayer gauge excitation at site k gives us the free fermion Lindbladian
i ℒ = ∑_j ( J_x ŵ_j, x̂ f_j^† f_j + x̂^ + J_y ŵ_j, ŷ f_j^† f_j + ŷ^ + h.c)
- 2 iγ∑_j ≠ k f_j^† f_j^ - 2 i γ (1 - f_k^† f_k^) .
The structure of the Lindbladian is the same for multiple interlayer gauge excitations - the chemical potential at each site is changed from f_k^† f_k^ to (1 - f_k^† f_k^).
A single one of these flips is not gauge-invariant; one must either flip an additional gauge degree of freedom or add in an odd number of fermions in order to recover a physical excitation. The Liouvillian gap for these excitations must be computed numerically since, as opposed to Eq. <ref>, the Hermitian and anti-Hermitian part of iℒ do not commute anymore. However, we can readily see analytically that this gap vanishes in the limit of strong dissipation, γ→∞. In this limit, we ignore the Hermitian terms in Eq. <ref> and we can obtain steady-state solutions by simply placing fermions wherever the imaginary chemical potential is negative (this automatically satisfies the gauge constraint, as we place as many fermions as we flip v̂_i's).
For general γ, the gap of interlayer gauge with fermion excitations (i.e., flipping a single v̂_k and introducing a single fermion to the vacuum) is plotted in Fig. <ref>. For this and all subsequent plots, the parameters used were J_x = J_y ≡ J = 1, and N=1600. The gap depends on the background W_j flux configuration - we present results for zero flux, W_j = +1, π-flux, W_j = -1, and a random flux configuration. Note that there are two distinct contributions to the Liouvillian gap in Eq. <ref>. The first is the overall shift of 2iγ, and the second comes from the dissipative strength of the fermion excitation with the smallest imaginary energy. For small γ, the imaginary energy of this fermion excitation is positive - in other words, adding in the single fermion excitation to the vacuum is energetically unfavorable and causes the eigenstate to decay more rapidly, but one is nevertheless forced to include it by the constraint of gauge invariance. This fermion excitation energy eventually transitions from positive to negative, asymptotically approaching -2iγ.
Depending on the background flux configuration, the fermion spectrum may exhibit an anti-𝒫𝒯-symmetry breaking transition at a critical value of γ, which causes a sharp kink in the gap. In this situation, the eigenvalues with the smallest imaginary part for small γ come in pairs, with the real parts opposite in sign. The anti-𝒫𝒯-symmetry breaking transition happens when the two meet on the imaginary axis and split off. We see that in Fig. <ref>, this happens for both the uniform flux as well as the particular random flux configuration plotted, but not for the π-flux scenario. A survey of generic random flux configurations suggest that this transition is common but not necessarily guaranteed.
What is the physical interpretation of these interlayer gauge excitations? As was the case in the steady-state gauge sector, we can proceed with a symmetry analysis of the operators in this sector. In terms of gauge-invariant fluxes, the flip of a single v̂_j away from its steady-state checkerboard configuration changes the fluxes of the four neighboring U_j,α operators to be -1. Hence, operators that have a non-zero overlap with this excitation must have identical eigenvalues under these flux operators. Recall that in the steady-state sector, the operators that satisfied the flux constraint consisted of
pairs of d_j fermion excitations connected by a string of gauge fields ŵ_j,α. An interlayer gauge excitation at site k “pins” a d_k' fermion excitation to site k, and the allowed operators are gauge-invariant string-like operators that involve a d_k' fermion at site k. Therefore, the Liouvillian gap in Fig. <ref> determines the equilibration timescale of operators given by a single d' fermion coupled to a d fermion by a ℤ_2 Wilson line.
The above argument applies to all operators in this gauge sector, regardless of their energy. In the limit γ→∞, we can also analytically understand the nature of the lowest-energy (i.e. the longest lived) operator in this sector. For an interlayer gauge excitation at site k, the steady-state solution obeys f_k^† f_k ‖ψ = 1 and f_j^† f_j ‖ψ = 0 elsewhere. By leveraging this constraint using analogous manipulations as in Eq. <ref>, we find that this is only satisfied by the operator Γ^5_k, which can be interpreted as the bound state of a d and d' fermion localized on a single site [cf. Eq. <ref>]. Hence, in the limit γ→∞, we recover steady-state excitations with definite Γ^5 eigenvalue. This is a consequence of the quantum Zeno effect; if we interpret the jump operators L_j = Γ^5_j as the environment performing measurements of Γ^5 with frequency specified by γ, our state can become frozen in a Γ^5 eigenstate for large γ.
The interpretation of the lowest-energy excitation as a Γ^5_k operator also holds approximately away from the γ→∞ limit, which is a consequence of the localization of the corresponding single-particle eigenvector of Eq. <ref> around site k. As shown in Fig. <ref>, the fermion with smallest imaginary eigenvalue is highly localized around site k even for small values of γ; hence, the operator whose equilibration time is determined by Fig. <ref> retains a large overlap with Γ^5_k. We leave a detailed analysis of the extent of eigenvector localization for future work, although we mention related work <cit.> of a similar single-particle system but with a fully disordered imaginary chemical potential, rather than our case of a chemical potential that is everywhere positive expect for a single site. For their model, numerical simulations were consistent with a localization transition for arbitrarily weak disorder strength.
The above analysis has been for a single interlayer gauge field excitation. It is natural to consider multiple gauge excitations, which correspond to symmetry sectors with multiple v̂_j gauge fields flipped away from their steady-state configuration. A physically relevant quantity to consider is the Liouvillian gap associated with the f vacuum in the sector with a pair of interlayer gauge field excitations at sites k and ℓ. This determines the equilibration timescale of an operator given by a pair of d' fermions at sites k and ℓ. This state is an exact eigenstate of the Lindbladian with imaginary energy 4γ - note that for sufficiently large γ, this energy may be reduced further by including pairs of f fermions, with a quantum Zeno effect yielding a steady-state solution at γ→∞ by adding a pair of fermions at sites k and ℓ.
§.§.§ Intralayer gauge excitations
The final types of excitation we will study are intralayer gauge excitations, when we flip a gauge field on one of the two layers such that ŵ_k, α, L = - ŵ_k, α, R for some bond (k,α). Operators associated with these excitations - i.e., operators consistent with this flux configuration - are single-site operators Γ_j^μ, μ = 1, 2, 3, 4, on the two sites adjacent to the bond (k, α). A more precise identification of these operators, including the flux configurations corresponding to operators Γ^μ 5_j and Γ^μν_j, are given in Appendix <ref>.
In this gauge sector, the Lindbladian no longer has a simple expression in terms of complex fermions f_j^†, as the intralayer gauge excitation induces pairing terms into the Lindbladian - explicitly,
2 (f_j^† f_j + x̂^† + f^_j + x̂ f^_j ) = - i (-1)^j (d_j, L d_j + x̂, L + d_j, R d_j + x̂, R)
The single-particle Lindbladian is quadratic and can thus still be easily diagonalized; we provide more details of this procedure in Appendix <ref>. However, the determination of whether the resulting ground state is physical - i.e, whether it has the odd fermion parity to not be annihilated by the projection to the physical subspace - is non-trivial due to the non-Hermiticity of the Lindbladian. We leave a full analysis of this problem as an open question and plot both the ground state energy and the energy of the first excited state in Fig. <ref>. The ground state energy gives a lower bound on the physical Liouvillian gap. However, one must be careful at large γ, since the γ→∞ limit gives a fictitious quantum Zeno effect. In this limit, the ground state approaches the f_j vacuum state, which is a steady-state solution but unphysical as its fermion parity is even. As a consequence, we also plot the first excited state, which gives a more physical lower bound for large γ.
We comment on a surprising aspect of this Liouvillian gap, which is a sudden increase when an arbitrarily small γ is turned on, with a subsequent plateau at a gap of magnitude J. For finite N, the gap smoothly evolves as a function of γ, but the slope at small γ is proportional to N, as shown in the inset of Fig. <ref>. This indicates that in the thermodynamic limit, an infinitesimally small γ causes a discontinuous jump in the Liouvillian gap to J. A possible physical explanation of this fact is that, in contrast to the fractionalized operators considered earlier which have a correspondence with coherent excitations of the closed system, the operators Γ_j^1, 2, 3, 4 have no such association, and hence deconstructive interference generated by the unitary dynamics of the closed system also contributes to the decay of the expectation values of these observables. Intuition on this phenomenon can also be gained from the fermion representation - by examining the single-particle eigenstates of the Lindbladian at γ = 0 expressed in the complex fermion representation, one can see that the act of exchanging a single hopping term with a pairing term causes strong hybridization between the delocalized particle-like and hole-like excitations, which in turn leads to an extensive γ shift in the Liouvillian gap when dissipation is turned on. This phenomenon of the decay rate approaching a non-zero value as γ→ 0 in the thermodynamic limit has been found in the Lindladian dynamics of Sachdev-Ye-Kitaev models <cit.>,
This observation demonstrates a striking feature of our model in the small-γ limit. In this regime, the expectation values of string-like operators such as V_jα as well has Γ_j^5 have an γ^-1 upper bound on their equilibration timescale, in contrast to local single-site operators such as Γ^1, 2, 3, 4_j whose timescales are bounded by J^-1.
§ PERTURBATIONS AWAY FROM EXACT SOLVABILITY
As the exact solvability of our Lindbladian requires a precise set of couplings, it is natural to consider perturbations away from this exactly solvable point. Here, we discuss different types of perturbations and their physical effects. Our Lindbladian possesses an extensive number of strong symmetries W_j and weak symmetries V_j,α. The combination of the two gives us our exact solvability, and perturbations are conveniently classified in terms of their breaking of these symmetries.
The simplest perturbations retain both the strong and weak symmetries of our system. These terms are rather artificial - the most local terms consist of either explicitly adding in the flux terms W_j to the Hamiltonian, or adding a two-site jump operator L_j,α = V_j,α. Both these choices preserve the steady-state solutions as well as the structure of the quasiparticle excitations; however, details of the Liouvillian gaps will be modified.
Perturbations that break the weak symmetries but preserve the strong symmetries of our model include the J_x', J_y', and J_5 terms in the full Hamiltonian of Eq. <ref>. In this case, our quantum jump operators still commute with the fluxes W_j, and an initial state in a definite flux sector will remain in that sector for arbitrary time. However, while we can still make statements about the steady-state solutions of the Lindbladian, the full spectrum and consequently the Liouvillian gap is no longer analytically tractable in an exact way. For future work, it would be interesting to study whether coherent quasiparticle excitations still remain in this spectrum at low energies. Recall that in the exactly solvable limit, the existence of distinct types of quasiparticle excitations led to the interpretation of distinct Liouvillian gaps which give equilibration timescales for different observables - the manner in which this picture is modified away from the exactly solvable point is an important open question.
We may also consider perturbations that break the strong symmetries but conserve the weak ones. This is accomplished by a generic choice of quantum jump operator, such as Γ^1, 2, 3, 4_j. In this scenario, we expect our system to asymptote to a unique steady-state, ρ∝𝕀. The weak symmetries cause the Lindbladian spectrum to decompose into an extensive number of symmetry sectors, with the steady-state solution residing in a particular sector. This means that one still retains the ability to discuss Liouvillian gaps with respect to the steady-state sector versus gaps of different sectors, and a careful analysis of the sectors would allow one to identify the operators that live in these sectors. In passing, we note that particular choices of quantum jump operators such as Γ^μν_j with μ, ν∈{1, 2, 3, 4} will break the local strong symmetries W_j but preserve a global strong symmetry Q ≡∏_j Γ^5_j (this is not a “new” symmetry, as it can be re-expressed as a product of W_j operators). As such, in this case we expect a pair of steady-state solutions ρ_±∝𝕀± Q.
Finally, a fully generic choice of perturbation that breaks all symmetries will give a single steady-state solution. We again stress an important open question of to what extent quasiparticle excitations of the Lindbladian are robust to these types of perturbations. With regards to the extensive number of steady-state solutions in the exactly solvable limit, one will expect that a small generic perturbation away from this point will cause all but one of these steady-states to persist for a long timescale given by the inverse strength of the perturbation. Developing an analogous theory for the excitations is a promising research direction, as it emphasizes a physical interpretation of the Lindbladian spectrum that is already familiar in the study of closed systems.
§ SUMMARY AND DISCUSSION
In this work, we analyze the Lindbladian dynamics of a quantum spin-3/2 system which admits an exact solution in terms of Majorana fermions coupled to static ℤ_2 gauge fields. This allows us to characterize the steady-state solutions as well as identify distinct classes of Liouvillian gaps, with different gaps determining the equilibration timescale of different classes of observables, as summarized in Fig. <ref>. Crucially, these timescales fall into different categories with distinct parametric dependencies on γ. While closed loops of V_jα in eq:bondOperators1—i.e., the fluxes, Fig. <ref>(a), and on a torus also the non-local Wilson loops, see part (b)—do not decay at all in the exactly solvable limit, pairs of emergent Majorana fermions, Fig. <ref>(c-e), decay with rates that scale linearly with small γ; depending on whether they exhibit a quantum Zeno effect, these rates decay to zero in the large-γ limit. Finally, operators of the last category, like Γ^1,2,3,4, see Fig. <ref>(f), which are not conserved by the Hermitian dynamics, exhibit a decay rate that is singular for small γ in the thermodynamic limit; naturally, the entire dynamics is unitary at γ = 0, however, sending γ→ 0^+ after taking the thermodynamic limit N →∞, the decay rates of these operators is of order of the exchange couplings J of the Hamiltonian (<ref>). This leads to particularly non-trivial three-step fractionalized thermalization dynamics, see Fig. <ref>(g), in the thermodynamic limit: first, at times of the order of the inverse exchange couplings 1/J, all operators of the third kind decay, which is parametrically separated from the time-scale ∝ 1/γ where (gauge invariant pairs of) the Majorana fermions d and d' decay. Then only closed loops of V_jα survive, which cannot decay unless perturbations beyond our solvable limit (cf. Sec. <ref>) are included.
One promising direction for future research is the construction of additional exactly solvable Lindbladians through this fermionization technique. For closed systems, there exists a rich literature on generalizations of the Kitaev honeycomb model to other exactly solvable models <cit.>; in these cases, the exact solvability is often geometric in nature (i.e. arising from a particular choice of lattice connectivity and hopping structure) and is unaffected if a subset of couplings become non-Hermitian. One interesting phenomenon that may arise in a certain parameter regime of these models is gapless fermionic excitations, in contrast to our model where fermion excitations have a constant gap 4 γ. This would imply algebraic, rather than exponential, decay of the expectation values of certain classes of operators <cit.>. Lindbladians with gapless excitations are not new <cit.> - the intriguing new feature of this would be the ability to cleanly separate this spectrum of gapless excitations from gapped gauge excitations, implying distinct equilibration timescales of these operators.
Generalized Lindbladian constructions may also prove useful at developing a general relation between the exactly solvable open system and the underlying Hermitian dynamics. In our model, the Hermitian dynamics was given by a QSL with two species of Majorana fermions, with the dispersion of one of the fermions tuned to zero. In this limit, a particular choice of quantum jump operators admit quasiparticle excitations of the Lindbladian which display a close relation with the excitation spectrum of the closed system. It is intriguing to ask whether, in a generic system that is rendered exactly solvable through this technique, a similar relation exists between quasiparticle excitations in the doubled Hilbert space and quasiparticle operators of the physical Hilbert space. A more robust understanding of this relation, including potential violations in certain systems, is another promising direction for future research.
Note added. Just before posting our work, a related paper appeared on arXiv <cit.>, studying exactly solvable BCS-Hubbard Lindbladians. Although the starting point of their analysis involves a distinct microscopic model of complex fermions with pairing terms, a transformation to Majorana fermions yields the same Lindbladian as ours within the π-flux sector. Due to the different microscopic models, our theory also has a non-trivial gauge invariance requirement, with non-trivial consequences. For instance, the Liouvillian gap in the π-flux sector in Fig. <ref> is larger as an additional fermion has to be included.
We thank Pavel Volkov and Hanspeter Büchler for helpful feedback. M.S.S. acknowledges funding by the European Union (ERC-2021-STG, Project 101040651—SuperCorr). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. This work was partially completed at the Center for Computational Quantum Physics in the Flatiron Institute. The Flatiron Institute is a division of the Simons Foundation. H.S acknowledges funding from the U.S. Department of Energy under Grant DE-SC0019030.
§ NON-VANISHING STEADY-STATE EXPECTATION VALUES
In the main text, we claim that any operator that has eigenvalue 1 under the superoperators U_j,α and equal eigenvalues under W_j,R and W_j,L is a product of V_j,α bond operators. One can readily verify that these operators satisfy the required constraints, but a more careful argument is required to show that these are the only operators with such a property. We do so by counting the dimension of the subspace (within the doubled Hilbert space) spanned by these operators. With a square lattice having 2N bonds, there are naively 2^2N orthogonal combination of bond operators; however, this double counts the true number of operators, as the product of all bond operators is 𝕀. So, the subspace is 2^N dimensional. The full dimension of our doubled Hilbert space is 2^4N, and we have 3N independent constraints - for each site j, we have U_j,x̂ = 1, U_j,ŷ = 1, and W_j,R = W̅_j (the constraint on W_j,L is automatically satisfied under these constraints). Each constraint halves the dimension of the allowed subspace, so we find a 2^N dimensional Hilbert space, as desired.
§ DIAGONALIZATION OF THE FREE FERMION LINDBLADIAN
In this appendix, we provide more detail on the diagonalization of the free fermion Lindbladian. For a general choice of gauge sector, we work with the Lindbladian written in terms of Majorana fermions, as in Eq. <ref>. This can be re-expressed in the form
i ℒ = d^T ·A·d - i γ N
where d is a 2N-dimensional vector containing both d_j, L and d_j, R Majorana fermion operators. We follow the procedure described in <cit.> for obtaining the spectrum of this Lindbladian, which we summarize here. As A is an antisymmetric matrix, its spectrum comes in the form {β_1 , -β_1 , β_2 , -β_2 …β_N , -β_N}, where we take β_α≥ 0. One can construct N creation/annihilation operators b_α, b_α' that obey the canonical fermionic anti-commutation relations (with the caveat that b_α' is in general not the Hermitian adjoint of b_α). With this, we can write
i ℒ = - 2 ∑_α=1^N β_α b_α' b_α - (i γ N - ∑_α=1^N β_α)
The term in parenthesis gives the dissipative strength of the state with weakest dissipation within this gauge sector. Note that this Majorana fermion representation obfuscates the constraint of gauge invariance, which is most easily enforced in terms of the complex fermions f^†_j. As such, this representation is only useful in gauge sectors where pairing terms would appear if written in the f_j^† basis, in which case a proper analysis of gauge invariance is equally difficult in either representation.
§ IDENTIFICATION OF SINGLE-SITE OPERATORS WITH FLUX CONFIGURATIONS
In the main text, we emphasize that the spectrum of our Lindbladian decomposes into an extensive number of symmetry sectors, each of which is specified by a gauge flux configuration. A Liouvillian gap for each sector can be defined, and one can identify operators - which we remind the reader should be thought of as states in this doubled Hilbert space - that are contained in these symmetry sectors, which the Liouvillian gap then defines an equilibration timescale for. Here, we catalog the flux configurations associated with the set of single-site operators.
A particular flux configuration is defined by the interlayer fluxes U_jα = V_j, α = V'_j, α, R V'_j, α, L as well as the intralayer fluxes W_j, α, R, W_j, α, L. As our Lindbladian spectrum is invariant under the transformation W_j, α, R↔ W_j, α, L, we will only identify operators based on their eigenvalues under the combined flux W_j, α, R W_j, α, L. The eigenvalues of an operator under these fluxes is simply determined by whether the operators V'_j, α and W_j commute or anti-commute with the operators. If we take as our basis of operators to be products of Γ matrices, every basis operator will either commute or anti-commute with V'_j, α and W_j.
The operators Γ_k^5 commutes with all plaquette operators W_j. It also commutes with all the bond operators V'_j, α aside from the four bonds adjacent to site k. The flux configurations associated with this operator are given precisely by the interlayer gauge excitations studied in Section <ref>.
The operators Γ_k^μ, μ = 1, 2, 3, 4, commute with all the bond operators V'_j, α except for a single one adjacent to site k which anticommutes with it. Additionally, it commutes with all but two W_j operators - these two offending plaquette operators share a bond given by the anticommuting V'_j, α operator. The flux configuration associated with these operators can be obtained starting from a steady-state gauge sector and flipping an intralayer gauge field on this bond and its spectrum is analyzed in Section <ref>.
The operators Γ_k^μ 5 have the same commutation relations with the plaquette operators as Γ_k^μ, but differ with respect to the V'_j, α operators; it now anticommutes with the three V'_j, α bond operators connected to site k that aren't the bond shared by the flux operators. This flux configuration can be obtained from the intraylayer gauge excitation studied in Section <ref> and flipping an additional interlayer gauge field v̂_k.
Finally, we identify the operators Γ_k^μν, with μ, ν = 1, 2, 3, 4 and μ≠ν. For a given site k, there are 42 = 6 different operators of this type. These operators will
anticommute with two of the four V_j, α' bond operators, and either two fluxes W_j that only share a corner at site k or all four fluxes connected to site k. These flux sectors are obtained by flipping two intralayer gauge fields connected to a site k - as expected, there are 42 = 6 ways of doing this.
The Liouvillian gap of excitations corresponding to the Γ_k^μ operators are shown in Fig. <ref>. We plot the Liouvillian gap of Γ^μ 5_k and Γ^μν_k operators in Fig. <ref> and verify that similar behavior occurs. This implies that our observation of the rapid equilibration of Γ^μ_k operators holds generically for single-site operators, with the exception of Γ^5_k due to its interpretation as the bound state of two Majorana fermion excitations, or alternatively due to the fact that Γ^5_k are precisely the quantum jump operators describing the coupling to the environment.
|
http://arxiv.org/abs/2307.05437v1 | 20230711170221 | Improving the Security of Smartwatch Payment with Deep Learning | [
"George Webber"
] | cs.CR | [
"cs.CR",
"cs.LG"
] |
Tensor gradiometry with a diamond magnetometer
G. W. Morley
August 12, 2023
==============================================
I would like to express my sincere thanks to Jack Sturgess for supervising this project, and for sharing his expertise on wearable security.
I also wish to thank my friends and my family for their unwavering support throughout the highs and lows of my time at Oxford.
Making contactless payments using a smartwatch is increasingly popular, but this payment medium lacks traditional biometric security measures such as facial or fingerprint recognition. In 2022, Sturgess et al. proposed WatchAuth, a system for authenticating smartwatch payments using the physical gesture of reaching towards a payment terminal. While effective, the system requires the user to undergo a burdensome enrolment period to achieve acceptable error levels.
In this dissertation, we explore whether applications of deep learning can reduce the number of gestures a user must provide to enrol into an authentication system for smartwatch payment. We firstly construct a deep-learned authentication system that outperforms the current state-of-the-art, including in a scenario where the target user has provided a limited number of gestures.
We then develop a regularised autoencoder model for generating synthetic user-specific gestures. We show that using these gestures in training improves classification ability for an authentication system. Through this technique we can reduce the number of gestures required to enrol a user into a WatchAuth-like system without negatively impacting its error rates.
[heading=bibintoc,title=]
|
http://arxiv.org/abs/2307.04470v1 | 20230710104044 | Test-Time Adaptation for Nighttime Color-Thermal Semantic Segmentation | [
"Yexin Liu",
"Weiming Zhang",
"Guoyang Zhao",
"Jinjing Zhu",
"Athanasios Vasilakos",
"Lin Wang"
] | cs.CV | [
"cs.CV"
] |
Journal of IEEE Transactions on Artificial Intelligence, Vol. 00, No. 0, Month 2020
First A. Author et al.: Bare Demo of IEEEtai.cls for IEEE Journals of IEEE Transactions on Artificial Intelligence
Test-Time Adaptation for Nighttime Color-Thermal Semantic Segmentation
Yexin Liu, Weiming Zhang, Guoyang Zhao, Jinjing Zhu, Athanasios Vasilakos, and Lin Wang^†
Manuscript received April 19, 2023. ^† corresponding author
Y. Liu, W. Zhang, and Jingjin Zhu are with the Artificial Intelligence Thrust, HKUST(GZ), Guangzhou, China. E-mail:[email protected], [email protected], and [email protected]
G. Zhao is with the Robotics and Autonomous Systems Thrust, HKUST(GZ), Guangzhou, China. E-mail:[email protected]
Athanasios V. Vasilakos is with the Center for AI Research (CAIR), University of Agder(UiA), Grimstad, Norway. Email: [email protected]
L. Wang is with the Artificial Intelligence Thrust, HKUST(GZ), Guangzhou, and Dept. of Computer Science and Engineering, HKUST, Hong Kong SAR, China. E-mail: [email protected]
August 12, 2023
==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
The ability to scene understanding in adverse visual conditions, , nighttime, has sparked active research for color-thermal semantic segmentation.
However, it is essentially hampered by two critical problems: 1) the day-night gap of color images is larger than that of thermal images, and 2) the class-wise performance of color images at night is not consistently higher or lower than that of thermal images.
We propose the first test-time adaptation (TTA) framework, dubbed Night-TTA, to address the problems for nighttime color-thermal semantic segmentation without access to the source (daytime) data during adaptation.
Our method enjoys three key technical parts.
Firstly, as one modality (, color) suffers from a larger domain gap than that of the other (, thermal),
Imaging Heterogeneity Refinement (IHR) employs an interaction branch on the basis of color and thermal branches to prevent cross-modal discrepancy and performance degradation.
Then, Class Aware Refinement (CAR) is introduced to obtain reliable ensemble logits based on pixel-level distribution aggregation of the three branches.
In addition, we also design a specific learning scheme for our TTA framework, which enables the ensemble logits and three student logits to collaboratively learn to improve the quality of predictions during the testing phase of our Night TTA.
Extensive experiments show that our method achieves state-of-the-art (SoTA) performance with a 13.07% boost in mIoU.
Night-time segmentation, TTA, Cross-modal learning.
Night-time segmentation is a critical task for autonomous driving under challenging visual conditions. Existing methods mostly focus on daytime segmentation with perfect illumination. This has inspired active research on color-thermal semantic segmentation as thermal cameras are less affected by illumination changes and can complement color modality.
However, thermal images suffer from a lack of large-scale labeled datasets, which are labor-intensive to obtain. TTA allows for the on-the-fly adaptation to different target domains at the testing phase while protecting data privacy. In light of this, we propose the first TTA framework that achieves SoTA nighttime color-thermal segmentation performance at the testing phase without relying on the source (daytime) data. This is practically valuable for real-world application scenarios.
The proposed method presents a robust solution for all-day scene understanding, which may hopefully inspire more research in the community.
§ INTRODUCTION
Recent years have witnessed the success of deep neural networks (DNNs) for color image semantic segmentation, which is crucial for the scene understanding, , autonomous driving <cit.>.
However, models trained in favorable lighting conditions show poor generalization ability to the nighttime data. Thus, nighttime image semantic segmentation has become a challenging problem.
Recently, increasing attention has been paid to thermal images because they are inherently robust to illumination changes and may complement semantic information to the color images (especially nighttime images). <cit.>. This has sparked research for supervised <cit.> and unsupervised <cit.> color-thermal semantic segmentation as both modalities can compensate for each other’s deficiencies.
However, existing supervised methods necessitate well-label annotations, particularly for thermal images captured during nighttime, which poses significant labor-intensive challenges.
Meanwhile, most unsupervised methods (, unsupervised domain adaptation (UDA)) entail the drawbacks of time-consuming offline domain adaptation training, and its performance is greatly affected by the domain gap, leading to limited adaptation in diverse testing environments.
Therefore, it is non-trivial as only the nighttime color-thermal data is available under a limited overhead for adaption. This motivates us to explore a suitable adaptation strategy for nighttime color-thermal semantic segmentation.
Test-Time Adaptation (TTA) <cit.> presents a practical domain adaptation approach that enables the seamless adaptation of pre-trained models to the target domain in real-time during the testing phase. TTA is different from the UDA-based semantic segmentation setting <cit.>: TTA does not need to access source data during adaptation. Moreover, the TTA framework can achieve privacy protection while allowing for on-the-fly adaptation to different target domains during the testing phase without the need for offline domain adaptation training. This is practically valuable for real-world applications. However, directly extending existing TTA methods to color-thermal semantic segmentation leads to less optimal performance, as demonstrated in Tab. <ref> in the experiments. For example, entropy minimization of TENT <cit.> generates overconfident predictions. Therefore, applying it individually to color and thermal branches aggravates the color-thermal discrepancy.
Motivation: In this paper, we, for the first time, explore a TTA framework for nighttime color-thermal semantic segmentation without access to the source (daytime) color-thermal data.
Our work addresses two challenges for nighttime color-thermal semantic segmentation arising from the modality differences during TTA, as shown in Fig. <ref>.
(1) Due to the different imaging mechanisms, the day-night domain gap, denoted as G_color, of color images is larger than that, denoted as G_T, of the thermal images (See Fig. <ref>(a)). This unbalanced difference between G_color and G_T leads to the considerable cross-modal discrepancy and performance degradation in the adaption process.
We refer to this issue as imaging heterogeneity. (2) Existing color-thermal segmentation methods, , <cit.>, apply the same weights to all classes. However, we find that the class-wise performance at night (denoted as P_color) of color images is not consistently higher or lower than that of the thermal images (denoted as P_T). Therefore, these methods might neglect the discriminative features of the modalities with smaller weights during the color-thermal nighttime segmentation ensemble process. An example is shown in Fig. <ref>(b), where the performance P_T^person on the class `person' in the thermal image is larger than P_color^person of the color image. We refer to this as class-wise prediction heterogeneity.
To address aforementioned challenges, we propose a novel nighttime TTA framework, called Night-TTA, which consists of three key technical components: (1) Imaging Heterogeneity Refinement (IHR) (Sec. <ref>) and (2) Class Aware Refinement (CAR) (Sec. <ref>) and (3) a learning scheme (Sec. <ref>), as shown in Fig. <ref>(c). For IHR, we propose an interaction branch to obtain the color-thermal cross-modal invariant feature to prevent the performance degradation in the adaptation process caused by the difference in the cross-modal domain gap (G_color>G_T).
Specifically, we first take the color-thermal image pairs as input to the interaction branch and then use the two encoders to obtain the color and thermal features that need to be fused. However, directly fusing the color and thermal features induces inconsistent noises due to the private information in the two individual branches. Therefore, we introduce a novel cross-modal shared attention (CMSA) module to aggregate the cross-modal invariant features while suppressing the noisy ones between the two modalities.
The CAR strategy employs an element-wise entropy-based fusion (EEF) module to generate reliable ensemble logits. This subtly avoids neglecting the discriminative feature information of each class in each branch. Specifically, we first evaluate Shannon entropy in the channel dimension of each student's logits. Then, we re-weight the students' logits to generate more reliable ensemble logits (, teacher) based on the pixel-level distribution of three students. By performing pixel-wise re-weight on the logits of the three branches, the performance advantages of different modalities in different classes can be utilized, and more reliable ensemble logits can be obtained.
Lastly, we present a novel learning scheme to overcome the potential problematic segmentation results during TTA. By utilizing the reliable ensemble logits generated by the EEF module as a self-supervised signal, we enable three student networks to learn from each other through online distillation <cit.> during the adaptation process. This allows our Night-TTA model to fully utilize the discriminative information in each branch, thus preventing the ensemble logits from making false predictions among the categories.
Contribution: In summary, our major contributions are four-fold: (I) We make the first attempt and propose a novel TTA framework for color-thermal semantic segmentation. (II) We propose an IHR strategy with the CMSA module, to reduce the imaging heterogeneity during TTA. We also propose the CAR strategy to take advantage of the segmentation performance of different modalities in different classes and then generate reliable ensemble logits.
(III) For cross-modal ensemble distillation of our Night-TTA framework, we propose a novel learning scheme to achieve cross-modal ensemble distillation in the testing phase. (IV) Extensive experiments demonstrate that our method significantly surpasses the baselines and prior methods (at least 3.11% mIoU improvement on the MF-1 dataset, and 2.69% mIoU improvement on the KP dataset).
§ RELATED WORK
Color-Thermal Image Semantic Segmentation.
Color-thermal segmentation methods can be divided into two main categories: supervised methods and unsupervised methods. The former includes the fusion of multi-modalities using multiple encoders with a shared decoder <cit.> and the translation between the RGB and thermal images <cit.>. MFNet <cit.> extracts features from the color and thermal images using two encoders and expands the receptive field by using the 'mini-inception' module. ABMDRNet <cit.> solves the problems of multimodal disparity and multi-scale contextual information fusion by using a bridging-then-fuse strategy to obtain more discriminative cross-modal information.
UDA-based methods, , HeatNet <cit.>, propose a teacher-student learning method <cit.> to transfer the knowledge from the daytime color image domain to the nighttime thermal image domain to avoid expensive nighttime image annotation. MS-UDA <cit.> enhances the performance of thermal segmentation by transferring knowledge from color to thermal modality.
By contrast, we propose the first color-thermal TTA framework that consists of triple student networks for nighttime image semantic segmentation without access to the source domain (daytime) data. Moreover, our TTA framework not only considers the difficulty of the domain gap faced by UDA but also proposes and solves the two novel problems based on the differences between modalities.
Test-Time Adaptation (TTA).
TTA methods enable the model to adapt quickly to the target domain, which does not require access to source domain data.<cit.>. TTA has been applied to unimodal<cit.> and cross-modal<cit.> segmentation tasks.
For the former task, the typical model Tent<cit.> presents an entropy minimization strategy to optimize affine parameters during testing.
For the Cross-modal segmentation task, xMUDA<cit.> allows the 2D and 3D modalities to learn from each other via imitation, disentangled from the segmentation objective to prevent false predictions. MM-TTA<cit.> proposes two complementary modules to obtain and select more reliable pseudo-labels (from 2D and 3D modalities) as self-learning signals during TTA.
However, directly using previous TTA methods for color-thermal semantic segmentation leads to less optimal performance. Therefore, we propose the IHR and CAR strategies to make our color-thermal TTA framework more robust and generalized, with a unique learning scheme that can perform better in both the training and testing phases.
Ensemble distillation.
Compared with the standard knowledge distillation (KD) paradigm<cit.>, online KD (ensemble distillation)<cit.> enables efficient and single-stage training via collaborative learning among the student networks. Collaborative learning relies on two main ways: students learn from each other <cit.> or generate ensemble logits to supervise their learning<cit.>.
The former methods facilitate peers' mutual learning by sharing knowledge among the student networks. For example, CLNN<cit.> allows multiple classifier heads to share intermediate-level representation for collaborative learning to reduce generalization errors.
The latter methods focus on generating ensemble logits that update each student's network based on the contributions shared by the students. In particular, <cit.> select the logits based on the cross-entropy loss of each student with the true label. However, we cannot access the labels during test time. Therefore, we propose the CAR strategy to generate reliable ensemble logits, which considers the different class-wise performance between the two modalities.
§ METHOD
Overview. In multi-modal TTA for color-thermal image semantic segmentation, we consider a source domain dataset, where each sample consists of daytime paired color images (x_s^color∈ℝ^H × W × 3), thermal images (x_s^T∈ℝ^H × W × 1), and corresponding segmentation ground truth (GT). A source model is trained on the labeled source domain dataset. Usually, the source model consists of a color encoder E_color, a thermal encoder E_T, and the decoder D utilized to generate pixel-level semantic labels. The source model can be denoted as f_θ=D(E_color(x_s^color), E_T(x_s^T)).
Typically, the performance of the source model f_θ is unsatisfactory when confronted with new test data characterized by a different distribution from the source samples. The primary objective of TTA is to enhance the prediction performance in the target domain by conducting model adaptation solely on unlabeled target data. Specifically, given a target dataset t, which comprises nighttime paired color images (x_t^color) and thermal images (x_t^T).
The model is updated using *min_θ̃ℒ(𝐱;θ),𝐱∼ t
, where θ̃⊆θ represent the model parameters that should be updated (, batch normalization layer), ℒ denotes self-supervised loss functions.
Prior research works on TTA have employed the entropy minimization for single-modality (, color image) semantic segmentation <cit.> or utilized consistency loss and pseudo-labels for cross-modal (, 2D-3D) segmentation <cit.>.
However, as discussed above, applying existing TTA methods directly to color-thermal semantic segmentation poses challenges due to two main factors: imaging heterogeneity and class-wise prediction heterogeneity.
To this end, we propose a novel TTA framework for nighttime color-thermal image semantic segmentation. Specifically, as depicted in Fig. <ref>, the proposed TTA framework consists of color, thermal, and interaction branches, representing three separate student networks.
color, thermal, and interaction branches take the x_t^color, x_t^T, and both as the input, respectively.
There are two novel technical components: IHR (Sec. <ref>) and CAR (Sec. <ref>).
To solve the problems caused by imaging heterogeneity, the IHR employs an interaction branch with a novel cross-modal shared attention (CMSA) module to generate reliable pseudo labels. The CMSA module is introduced before the decoder to aggregate the complementary features and suppress the noisy features of the color and thermal modalities.
To solve the problems caused by class-wise prediction heterogeneity, the CAR is buttressed by an element-wise entropy-based fusion (EEF) module to generate the ensemble logits by aggregating the reliable logits from three branches. We also propose a specific learning scheme that enables the three student networks to collaboratively learn to improve the quality of predictions during adaptation.
§.§ Imaging Heterogeneity Refinement (IHR)
The straightforward fusion of the color and thermal branches leads to a noticeable degradation in the segmentation performance due to the significant domain gap between the two modalities, as evidenced by the results presented in Tab. <ref>. To address this challenge, we propose the integration of an interaction branch to facilitate the extraction of cross-modal invariant features, which are crucial for generating reliable pseudo labels.
Specifically, color images provide abundant textual information that is valuable for segmentation tasks, particularly in well-illuminated daytime scenarios. However, their performance suffers greatly when confronted with adverse lighting conditions. On the contrary, thermal images exhibit robustness to illumination changes but exhibit limitations such as lower resolution and ambiguous object boundaries. Therefore, a direct fusion of color and thermal features may introduce inconsistencies caused by the individual characteristics of each modality, undermining segmentation accuracy.
To mitigate these issues, the introduction of the interaction branch aims to exploit the complementary nature of color and thermal modalities. This branch facilitates the extraction of cross-modal invariant features that are resilient to domain gaps, enabling the generation of more reliable pseudo labels. By integrating these cross-modal invariant features with the individual modalities, we can effectively capture both shared and unique information, leading to improved segmentation performance in color-thermal images.
This may cause generating unreliable pseudo labels.
For this reason, we design the CMSA module (see Fig. <ref>) to rectify the noisy features and extract the cross-modal invariant features.
For the CMSA, we first embed both color (F_color∈ℝ^H × W × C) and thermal (F_T∈ℝ^H × W × C) features into two individual channel (C) attention vectors (V_color^C∈ℝ^C) and (V_T^C∈ℝ^C). Unlike <cit.>, rectifying features by utilizing the individual vectors, we generate the shared channel attention vectors ( V_shared^C∈ℝ^C) by aggregating the vectors from the color-thermal features to maintain the shared features while suppressing the noisy features.
The channel-wise feature rectification can be described as:
F^C_color =V_shared^C ⊙ F_color +F_color,
F^C_T =V_shared^C ⊙ F_T + F_T.
Similar to the channel-wise rectification, a shared spatial (S) attention vector (V_shared^S∈ℝ^H × W) is embedded to calibrate the local information, which is formulated as follows:
F^S_color =V_shared^S ⊙ F^C_color + F^C_color,
F^S_T =V_shared^S ⊙ F^C_T + F^C_T.
F^S_color and F^S_T are the rectified features after the CMSA module, which will be aggregated to the decoder of the interaction branch. Once obtained the logits in each branch, pseudo-labels are provided for the CAR.
§.§ Class Aware Refinement (CAR)
To generate ensemble logits, previous method, , <cit.> usually assigns an image-level weight to each branch by measuring the consistency between the cross-modal branches. This may encounter class performance imbalance problems for color-thermal segmentation due to the class-wise prediction heterogeneity in cross-modalities. Take the cross-modal branches as an example (See Fig. <ref>). We assume that the weights calculated by the existing method for the color and thermal branch are 0.7 and 0.3, respectively. When generating the ensemble logits, all classes in the color branch are assigned a weight of 0.7, while those of the thermal branch are assigned 0.3. This leads to poor segmentation performance for some classes that were originally better segmented in the thermal branch (, person). To alleviate this problem, we propose the EEF module to refine the ensemble logits, as shown in Fig. <ref>.
§.§.§ Element-wise Entropy-Based Fusion (EEF)
The EEF module uses the outputs of three branches as the input, which are denoted as ỹ_1^M, ỹ_2^M, and ỹ_3^M (ỹ_1^M, ỹ_2^M, ỹ_3^M ∈ℝ^H × W × C) respectively, where M ∈{s, t}and C denotes the number of channels. To assign the weight W_i for branch i, specifically, the softmax is firstly computed along the channel dimension. Then, we calculate the Shannon entropy (H(ỹ_i^M) ∈ℝ^H × W × 1) of the logits ỹ_i^M. For each pixel (i, j) ∈ H × W, we can obtain a vector v_i,j∈ 1 × C consisting of the elements of logits at position (i, j) for all channels. Then, we calculate the Shannon entropy H(v_i,j^C) of the vector v_i,j:
H(v_i,j^C)=∑_C=1^Nsoftmax(v_i,j^C) · log softmax(v_i,j^C),
where v_i,j^C denotes the value of vector v_i,j in channel C. H(ỹ_i^M) is composed of the Shannon entropy (SE) of all vectors v_i,j. Assume that the true label at position (i, j) is k. When the value on the k-th channel becomes larger, the value on other channels diminishes. Then the cross entropy (CE) loss with the label decreases, which means the segmentation performance becomes better. The ideal probability distribution is that the prediction on the k-th channel is close to 1, while the prediction on the other channels is close to 0. In this situation, Shannon entropy will be kept to a relatively small extent. An effective way to generate teacher logits is to re-weight the student's logits based on the element-wise Shannon entropy. For each element in the teacher's logits, the smaller the Shannon entropy in the channel dimension, the greater the weight of the branch. We define the teacher's logits as the combination of all students' weighted logits. The pixel-wise weights W_i of branch i are calculated as:
W_i=e^(1-H(ỹ_i^M))/temp/∑_i=1^3 e^(1-H(ỹ_i^M))/temp,
where W_i ∈ℝ^H × W × 1, temp denotes the temperature. Finally, the teacher's logits are as follows:
ỹ^EN=∑_i=1^3 W_i*ỹ_i^M.
§.§ Learning Scheme
For TTA, we denote the updated parameters of the Batch normalization layer of color, interaction, and thermal branch as γ^color, γ^Int, and γ^T, respectively. Given paired color-thermal images, there are i classes in the image. The predictions of different branches can be denoted as P_color={P_color^1,P_color^2,..., P_color^i}, P_Int={P_Int^1,P_Int^2,..., P_Int^i}, and P_T={P_T^1,P_T^2,..., P_T^i}. During TTA, the class-wise segmentation performance of one branch is not consistently higher or lower than the other branches. For some classes, one branch can achieve the best segmentation performance while the other branch could achieve the best performance in other classes. Without loss of generality, we consider the case of three classes where the color, interaction, and thermal branch achieves the best performance on class 1, 2, and 3, respectively. The ensemble logits of traditional methods are calculated by P_EN=P_color+PInt+P_T/3. Then, the consistency loss ℒ_KL^tta, which achieves knowledge distillation from ensemble logits to student logits, is used to train the three branches. During TTA, the parameters of the batch normalization layer γ are updated by:
γ_t^color=γ_t-1^color-β·▽_γℒ_KL^tta(color,EN)
γ_t^Int=γ_t-1^Int-β·▽_γℒ_KL^tta(Int,EN)
γ_t^T=γ_t-1^T-β·▽_γℒ_KL^tta(T,EN)
Based on our assumptions, for class 1, the entropy of the color branch is smaller than the ensemble logits (SE(P_color) SE(P_EN)), whereas the entropy of the interaction and thermal branches are larger than the ensemble logits (SE(P_Int) SE(P_EN) and SE(P_T) SE(P_EN)). Therefore, although the interaction and thermal branches will improve the segmentation performance, the color branch will have performance degradation after optimization. The other two classes have similar results.
To mitigate the issues mentioned above, we propose the EEF module and a learning scheme (See Fig. <ref>).
During TTA, we consider the teacher logits as the self-training signals to update the model. We define KL loss as ℒ_KL^tta(i,EN)= KL(ỹ_i^s,ỹ^EN) to ensure collaborative learning of these three students. Moreover, to boost the performance of all three student networks, we introduce the Shannon entropy loss ℒ_i^tta= SE(ỹ_i^t), and ℒ_EN^tta= SE(ỹ^EN).
For each student network i, the final learning objective is:
ℒ^tta=∑_i=1^3 ℒ_i^tta+λ_1 ℒ_EN^tta+λ_2 ∑_i=1^3 ℒ_KL^tta(i,EN),
where λ_1 and λ_2 are hyperparameters.
Dynamic Weighting Each branch. Existing methods for multi-modal test time adaptation typically assign the same weights to all branches. However, for color-thermal segmentation, the day-night domain gap in color images is more significant than in thermal images. Consequently, utilizing identical weights for all branches can lead to instability during adaptation. To address this issue, we propose a dynamic weighting scheme for these branches, which exclusively affects the loss function without incurring additional computational overhead for model adaptation. Specifically, we introduce weights ω_i for each branch according to the adaptation extent.
Measuring the extent of adaptation typically relies on labeled samples, which presents a challenge in our problem scenario where training data is unavailable, and the test samples remain unlabeled. Consequently, quantifying the extent of adaptation becomes non-trivial. To address this issue, we propose a novel approach that leverages ensemble logits to estimate the extent of adaptation. In particular, we initially compute the distance between the student logits and the ensemble logits of each branch within a batch. This computation can be formulated as follows:
D_i=1/B∑_b=1^B 1/2(KL(ỹ^EN||ỹ^M)+KL(ỹ^M||ỹ^EN)),
Then we calculate the weights of each branch as follows:
ω_i = D_i/min{D_1, D_2,D_3}
Then, the final objective is :
ℒ^tta=∑_i=1^3 ω_i ℒ_i^tta+λ_1 ℒ_EN^tta+λ_2 ∑_i=1^3 ω_i ℒ_KL^tta(i,EN),
where λ_1 and λ_2 are hyperparameters. With the EEF module, we can generate ensemble logits with small entropy at the pixel level. Then, for each class i, we have SE(P_EN) SE(P_color), SE(P_EN) SE(P_Int), and SE(P_EN) SE(P_T), which means that we have better ensemble logit to train the three branches. Adaptation with our learning scheme can continuously improve the segmentation performance of the three student branches through ensemble distillation, so as to gradually carry out more accurate segmentation results.
§ EXPERIMENTS
§.§ Datasets
MF dataset. It contains 1569 images (784 for training, 392 for validation, and 393 for test) in which 820 daytime and 749 nighttime images are mixed in training, validation, and test sets. The resolution of images is 480×640 with annotated semantic labels for 8 classes.
To evaluate our method, we just drop out the nighttime color-thermal image pairs in the original training and validation sets and drop out the daytime color-thermal image pairs in the original test sets to form a new dataset (410 for training, 205 for validation, and 188 for test), which is denoted as MF-1. For UDA methods, under our investigation, there only exist two UDA methods (HeatNet and MS-UDA) for nighttime image semantic segmentation leveraging color and thermal images. Thus, we compare the segmentation performance with these two methods. For a fair comparison, we use the same training and testing set with MS-UDA: We reorganize the daytime and nighttime images in the MF dataset as training and testing sets (820 daytime images for training and 749 nighttime images for testing ), which is denoted as MF-2. Three categories of labels overlapping the KP dataset (, car, person, and bike) are used for evaluation.
The modified KP dataset. The KAIST Multispectral Pedestrian Detection (KP) dataset <cit.> is a color-thermal paired urban driving dataset without semantic segmentation labels. Kim <cit.> create a modified KP dataset with manually annotated 503 daytime and 447 nighttime color-thermal image pairs and the pixel-level labels of 19 classes consistent with Cityscapes <cit.>. The resolution of color-thermal image pairs is 512 × 640 × 3 and 512 × 640 × 1, respectively.
§.§ Implementation Details
The proposed method is implemented using PyTorch libraries with a single A6000 GPU.
Source model.
As the first TTA framework for nighttime color-thermal semantic segmentation, our approach adopts a three-branch network structure. Each branch utilizes an untrained encoder and decoder from FEANet <cit.> (which after the training step already reaches good performance based on a supervised manner) to obtain the logits. We utilize the encoder and decoder from FEANet as the source model without changing the network architecture.
Pre-training the source model.
In our experiment setting, we want to use daytime data for training and nighttime data for testing. However, the source model from FEANet was trained and tested on day-night mixed dataset which is a different dataset splitting scheme from ours. Therefore, we pre-train the source encoder and source decoder with the source domain dataset.
For a fair comparison, We follow the training details of FEANet apart from using the original dataset.
Test-time Adaptation Details.
We apply the source model that only uses daytime data as training to each branch and use unlabeled nighttime paired data as input for test time adaptation. Similar to previous TTA methods <cit.>, we only optimize the batch norm affine parameters for one epoch. The learning rate for three sub-networks is set to 1e^-5. The temperature is set to 2.
§.§ Comparative Studies
We evaluate the proposed framework against state-of-the-art TTA methods on the MF-1, MF-2, and modified KP datasets.
MF-1 dataset. We compare our TTA framework with uni-modal and multi-modal TTA frameworks on MF-1 dataset. The quantitative and qualitative results are shown in Tab. <ref> and Fig. <ref>. The proposed Night-TTA could bring a significant adaptation effect on nighttime color-thermal image semantic segmentation compared to the source model (increases 13.07 % mIoU). Specifically, in Tab. <ref>, we conduct a comparison of the segmentation performance among different TTA frameworks across three categories: Car, Person, and Bike. Based on the analysis of the experimental data, our TTA framework exhibits a notable improvement in the segmentation performance for all three categories. Moreover, our Night-TTA achieves a substantial performance advantage over both uni-modal TTA methods, with an improvement of over 17.34% in mIoU. It should be noted that directly applying the uni-model TTA methods would degrade the segmentation performance. Our method also surpasses multi-modal TTA methods with an improvement of over 3.01% in mIoU.
MF-2 dataset. We also compare our method with existing UDA methods. The results are shown in Tab. <ref>. In the MF-2 dataset setting, where training is conducted on daytime data and testing on nighttime data, our Night-TTA approach showcases remarkable performance superiority over UDA methods, specifically achieving a significant 6.05% improvement in comparison to MS-UDA. These results highlight the efficacy and professionalism of our Night-TTA framework in addressing the challenges of domain adaptation in the context of semantic segmentation for nighttime scenarios.
The modified KP dataset. Tab. <ref> and Fig. <ref> show the quantitative and qualitative results. We can conclude that the proposed Night-TTA performs better than existing nighttime color-thermal image semantic segmentation methods. Specifically, our Night-TTA framework achieves the best segmentation performance in most categories. In addition, our proposed learning scheme for the TTA framework improves the segmentation performance of the source model (from 36.35 % mIou to 47.77 % mIou) more significantly than other TTA methods (The highest increase to 45.08% mIou).
§.§ Ablation Studies and Analysis
1) Imaging Heterogeneity Refinement
1) Interaction Branch.
We validate the effectiveness of the proposed interaction branch on the MF-1 dataset. The results are shown in Tab. <ref>. During the assessment of single-modal nighttime semantic segmentation, our findings indicate that thermal imaging exhibits superior performance compared to color imaging, highlighting its heightened robustness and reliability in low-light environments. Compared with single-modal nighttime image semantic segmentation, multi-modal (color-thermal) achieves better performance. Besides, the dual path (without the interaction branch) worsens the segmentation performance (from 49.71% mIou to 32.16 % mIou when using EEF), demonstrating the interaction branch's effectiveness.
2) CMSA.
We conduct additional experiments to validate the efficacy of the CMSA module, comparing its performance in an interaction-only network and a complete network. The results, presented in Tab. <ref>, demonstrate the significant improvements achieved by the CMSA module in both the interaction-only network (from 35.82% mIoU to 41.26% mIoU) and the triple branches networks (from 49.71% mIoU to 52.06%).
2) Class Aware Refinement
1) EEF module.
We compare EEF module against different methods of generating the ensemble logits (as shown in Tab. <ref>).
The 'Merge' approach represents taking the mean of the logits from the three branches, while 'IE' refers to methods based on image-level entropy(<cit.>). The results demonstrate that our EEF module performs better than other strategies, with an increase of 5.64% (from 47.52% mIoU to 53.16% mIoU) for 'Merge' and 6.79%(from 46.37% mIoU to 53.16% mIoU) for 'IE' in mIoU. This highlights the superior performance of our EEF module in ensemble logits generation.
2) Learning Scheme.
In this experiment, the λ_1 and λ_2 are set to 1. Tab. <ref> shows the quantitative results. Based on our experimental data, it is evident that utilizing individual losses alone or combining any two losses leads to performance improvement in adaptation. Specifically, the three ℒ^tta, ℒ_EN^tta, and ℒ_KL^tta(i,EN) contribute similarly during TTA, while ℒ_KL^tta(i,EN) plays a slightly more important role compared with others. It should be noted that our learning scheme could significantly improve the performance of the source model (13.07% mIoU).
3) Sensitivity Analysis
1) Batch size.
We explore the impact of batch size on the semantic segmentation performance of different TTA methods (as shown in Tab. <ref>). The results indicate that a small batch size (1 or 2) leads to degraded segmentation performance, while a larger batch size (4 or 8) results in improved performance. Tab. <ref> shows that the TTA method looks very sensitive to batch size. This sensitivity can be attributed to the parameters updated by the TTA method during the test phase, primarily within the batch normalization layer. Increasing the batch size brings the testing data in a batch closer to the real data contribution during the adaptation process, thus improving the segmentation performance. The proposed method consistently performs well across different batch sizes.
It outperforms the other evaluated TTA methods in terms of mIoU, showcasing its effectiveness in semantic segmentation tasks. For example, at a batch size of 8, the proposed method achieves mIoU of 53.16, surpassing the mIoU of the other methods (ranging from 49.28 to 50.05).
2) Robustness to perturbations.
We further evaluate the robustness of our methods on the MF dataset. We conduct an ablation study to evaluate the impact of different input perturbations during the test-time adaptation. Three types of perturbations are applied: image cropping, brightness adjustment, and the addition of Gaussian noise. Specifically, we crop the image at the rate of 0.2, randomly add Gaussian noise (noise range is set to 5) to the image, or just the brightness of the images to reorganize three new test sets. Tab. <ref> shows the quantitative results of different TTA methods. shows the quantitative results of different TTA methods. We can conclude that our method is more robust to noises and image corruption.
3) Parameters updated in TTA.
We conduct an analysis of the TTA performance by examining the impact of updating specific network layers. The ablation study aims to analyze the impact of updating specific network layers during TTA in semantic segmentation. Three scenarios are considered: updating only the encoder parameters, updating only the decoder parameters, and updating both the encoder and decoder parameters. The experiment is conducted with a batch size of 8. Tab. <ref> presents the results according to updating the affine parameters in different network parts for effective TTA. When only the encoder parameters are updated during TTA, the method achieved the mIoU of 48.71. Updating only the decoder parameters result in the best performance, with a mIoU of 53.16.
§ DISCUSSION
For the IHR strategy, naively combining the individual color and thermal branches yields subpar performance due to modality gap and noise (Fig. <ref>). The proposed IHR strategy enhances prediction reliability by incorporating an interaction branch and a CMSA module. The CMSA module effectively combines cross-modal invariant features while suppressing noisy information between color and thermal modalities. Evaluating with nighttime color-thermal image pairs, we observe a performance gap between color and thermal branch logits without IHR, along with considerable noise in ensemble logits. By introducing the interaction branch and CMSA module, the discrepancy between color and thermal branch logits decreases, resulting in ensemble logits that align better with ground truth labels. This reduction in cross-modal discrepancy highlights the effectiveness of the interaction branch in mitigating the influence of image heterogeneity.
As the first TTA framework, we design three branches to generate reliable pseudo labels without considering much about the parameters and computational costs, which is typical for other cross-modal TTA methods, , <cit.>. Future work will focus more on designing tight frameworks. Moreover, while our TTA framework is specifically designed for nighttime color-thermal semantic segmentation, there is potential for its application to address other types of multi-modality data. For instance, it can be extended to handle data combinations such as color and event data or color and depth data, opening up opportunities for broader applicability.
§ CONCLUSION
In this paper, we addressed two potential problems of nighttime color-thermal image semantic segmentation to reduce the cross-modal discrepancy via test time adaptation (TTA) with cross-modal ensemble distillation. We presented a novel TTA framework, dubbed Night-TTA, with two novel refinement strategies: imaging heterogeneity refinement (IHR) and class aware refinement (CAR). In the experiments, both strategies were shown effective in achieving credible performance. The experimental results also proved the benefits of our learning scheme. Moreover, for nighttime color-thermal semantic segmentation, Night-TTA outperformed the existing methods by a considerable margin.
IEEEtran
[
< g r a p h i c s >
] Yexin Liu is a Mphil. student in the Visual Learning and Intelligent Systems Lab, Artificial Intelligence
Thrust, The Hong Kong University of Science and Technology, Guangzhou (HKUST-GZ).
His research interests include infrared- and event-based vision, and unsupervised domain adaptation.
[
< g r a p h i c s >
] Weiming Zhang
is a research assistant in the Visual Learning and Intelligent Systems Lab, Artificial Intelligence
Thrust, The Hong Kong University of Science and Technology, Guangzhou (HKUST-GZ).
His research interests include event-based vision, Deep Learning, .
[
< g r a p h i c s >
] Guoyang ZHAO is a Mphil. student in the Intelligent Autonomous Driving Center, Thrust of Robotics and Autonomous Systems, The Hong Kong University of Science and Technology, Guangzhou (HKUST-GZ). His research interests include vision-based perception systems and Deep learning.
[
< g r a p h i c s >
]
Jinjing Zhu is a Ph.D. student in the Visual Learning and Intelligent Systems Lab, Artificial Intelligence Thrust, The Hong Kong University of Science and Technology, Guangzhou (HKUST-GZ). His research interests include CV (image classification, person re-identification, action recognition, etc.), DL (especially transfer learning, knowledge distillation, multi-task learning, semi-/self-unsupervised learning, etc.), omnidirectional vision, and event-based vision.
[
< g r a p h i c s >
]
Athanasios V. Vasilakos is with the Center for AI Research (CAIR), University of Agder(UiA), Grimstad, Norway. He served or is serving as an Editor for many technical journals, such as the IEEE TRANSACTIONS ON AI, IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT; IEEE TRANSACTIONS ON CLOUD COMPUTING, IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, IEEE TRANSACTIONS ON CYBERNETICS; IEEE TRANSACTIONS ON NANOBIOSCIENCE; IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE; ACM Transactions on Autonomous and Adaptive Systems; the IEEE JOURNAL ON SELECTED AREAS IN COM-MUNICATIONS . He is WoS highly cited researcher(HC).
[
< g r a p h i c s >
]
Lin Wang (IEEE Member) is an assistant professor in the AI Thrust, HKUST-GZ, HKUST FYTRI, and an affiliate assistant professor in the Dept. of CSE, HKUST. He did his Postdoc at the Korea Advanced Institute of Science and Technology (KAIST). He got his Ph.D. (with honors) and M.S. from KAIST, Korea. He had rich cross-disciplinary research experience, covering mechanical, industrial, and computer engineering. His research interests lie in computer and robotic vision, machine learning, intelligent systems (XR, vision for HCI), etc.
|
http://arxiv.org/abs/2307.03888v1 | 20230708034255 | Spectral radius, fractional $[a,b]$-factor and ID-factor-critical graphs | [
"Ao Fan",
"Ruifang Liu",
"Guoyan Ao"
] | math.CO | [
"math.CO",
"05C50, 05C35"
] |
Spectral radius, fractional [a,b]-factor and ID-factor-critical graphs[Supported by National Natural Science Foundation of China
(Nos. 11971445 and 12171440),
Henan Natural Science Foundation (No. 202300410377) and
Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. NJZY22280).]
Ao Fan^a, Ruifang Liu^aCorresponding author.
E-mail addresses: [email protected], [email protected], [email protected]., Guoyan Ao^a, b
^a School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China
^b School of Mathematics and Physics, Hulunbuir University, Hailar, Inner Mongolia 021008, China
==========================================================================================================================================================================================================================================================================================================================================
Abstract
Let G be a graph and h: E(G)→ [0,1] be a function.
For any two positive integers a and b with a≤ b, a fractional [a,b]-factor of G with the indicator function h is
a spanning subgraph with vertex set V(G) and edge set E_h such that a≤∑_e∈ E_G(v)h(e)≤ b for any vertex v∈ V(G),
where E_h = {e∈ E(G)|h(e)>0} and E_G(v)={e∈ E(G)| e v G}.
A graph G is ID-factor-critical if for every independent set I of G whose size has the same parity as |V(G)|, G-I has a perfect matching.
In this paper, we present a tight sufficient condition based on the spectral radius for a graph to contain a fractional [a,b]-factor,
which extends the result of Wei and Zhang [Discrete Math. 346 (2023) 113269].
Furthermore, we also prove a tight sufficient condition in terms of the spectral radius for a graph with minimum degree δ to be ID-factor-critical.
Keywords: Spectral radius, Fractional [a,b]-factor, ID-factor-critical, Minimum degree
AMS Classification: 05C50; 05C35
§ INTRODUCTION
Let G be a finite, undirected and simple graph with vertex set V(G) and edge set E(G).
The order and size of G are denoted by |V(G)|=n and |E(G)|=e(G), respectively.
We denote by δ(G), i(G) and o(G) the minimum degree,
the number of isolated vertices and the number of odd components of G, respectively.
We use K_n and I_n to denote the complete graph of order n and the complement of K_n.
For a vertex subset S of G, let G[S] be the subgraph of G induced by S.
Let G_1 and G_2 be two vertex-disjoint graphs.
We denote by G_1+G_2 the disjoint union of G_1 and G_2.
The join G_1∨ G_2 is the graph obtained from G_1+G_2 by adding all possible edges between V(G_1) and V(G_2).
For undefined terms and notions, one can refer to <cit.>.
Given a graph G of order n, the adjacency matrix of G is the 0-1 matrix A(G)=(a_ij)_n× n indexed
by the vertex set V(G) of G, where a_ij=1 when v_i and v_j are adjacent and a_ij=0 otherwise.
The eigenvalues of A(G) are also called the eigenvalues of G.
Note that A(G) is a real nonnegative symmetric matrix. Hence its eigenvalues are real,
which can be arranged in non-increasing order as λ_1(G)≥λ_2(G) ≥⋯≥λ_n(G).
The largest eigenvalue of A(G), denoted by ρ(G), is called the spectral radius of G.
Let g and f be two integer-valued functions defined on V(G) such that 0≤ g(v)≤ f(v) for each vertex v in V(G).
A (g,f)-factor of G is a spanning subgraph F of G satisfying g(v)≤ d_F(v)≤ f(v) for any vertex v in V(G).
Let a and b be two positive integers with a≤ b.
A (g,f)-factor is called an [a,b]-factor if g(v)≡ a and f(v)≡ b for any v∈ V(G).
An [a,b]-factor is called a 1-factor (also called a perfect matching) if a=b=1.
Let h : E(G)→ [0,1] be a function and E_G(v)={e∈ E(G)| e v G}.
If g(v)≤∑_e∈ E_G(v)h(e)≤ f(v) holds for any vertex v∈ V(G),
then we call a subgraph F with vertex set V(G) and edge set E_h a fractional (g,f)-factor of G with indicator function h,
where E_h = {e∈ E(G)|h(e)>0}.
A fractional (g,f)-factor is called a fractional [a,b]-factor if g(v)≡ a and f(v)≡ b.
In particular, for a positive integer k, a fractional [k, k]-factor of a graph G is called a fractional k-factor of G.
A fractional 1-factor is also called a fractional perfect matching.
Note that if G contains a (g,f)-factor, then it also contains a fractional (g,f)-factor.
However, if G has a fractional (g,f)-factor, G may not have a (g,f)-factor.
We start with the following well-known fractional (g,f)-factor theorem.
Let G be a graph and g,f: V(G)→ Z^+ be two integer functions such that g(v)≤ f(v) for all v∈ V(G).
Then G has a fractional (g,f)-factor if and only if for any subset S⊆ V(G), we have
f(S)-g(T)+∑_v∈ Td_G-S(v)≥0,
where T={v|v∈ V(G)-S d_G-S(v)<g(v)}.
If g(v)≡ a and f(v)≡ b, then by Theorem <ref>, we obtain the following result.
Let G be a graph and let a, b be two positive integers with a≤ b.
Then G has a fractional [a,b]-factor if and only if for any subset S⊆ V(G), we have
b|S|-a|T|+∑_v∈ Td_G-S(v)≥0,
where T={v|v∈ V(G)-S d_G-S(v)<a}.
There are many sufficient conditions which can assure a graph to have a fractional [a,b]-factors
(see for example, <cit.>).
Cho, Hyun, O and Park <cit.> posed the spectral version conjecture for the existence of [a,b]-factors in graphs.
Fan, Lin and Lu <cit.> proved that the conjecture holds for the case n≥ 3a+b-1.
Very recently, Wei and Zhang <cit.> confirmed the full conjecture.
Let a, b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1.
If ρ(G)>ρ(K_a-1∨(K_n-a+K_1)) and na≡ 0 (mod 2) when a=b, then G has an [a,b]-factor.
It is well known that if G contains an [a,b]-factor, then it contains a fractional [a,b]-factor.
Inspired by the work of Wei and Zhang <cit.>,
we obtain a tight sufficient condition in terms of the spectral radius for a graph to contain a fractional [a,b]-factor.
Let a, b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1.
If
ρ(G)≥ρ(K_a-1∨(K_n-a+K_1))
and na≡0 (mod 2) when a=b, then G has a fractional [a,b]-factor
unless G≅ K_a-1∨(K_n-a+K_1).
Note that 4+√(32a^2+24a+5)> a+1. Our Theorem <ref> improves the following result.
Let b≥ a≥ 1 be two integers, and let G be a graph of order n≥ 4+√(32a^2+24a+5).
If ρ(G)≥ρ(K_a-1∨(K_n-a+K_1)), then G has a fractional [a,b]-factor unless G≅ K_a-1∨(K_n-a+K_1).
A graph G is independent-set-deletable factor-critical, shortly ID-factor-critical, if for every
independent set I of G whose size has the same parity as |V(G)|, G-I has a perfect matching.
Let S_n, k be the join of a clique on k vertices with an independent set of n-k vertices for n>k. That is to say, S_n, k=K_k∨ I_n-k.
A graph G has a perfect matching if and only if o(G-S)≤|S| for every S⊆ V(G).
The following theorem is a direct consequence of Tutte's Theorem.
A graph G is ID-factor-critical if and only if o(G-I-S)≤|S| for every independent set I such that |I| has the same parity as |V(G)| and every subset S⊆ V(G)-I.
Using Theorem <ref>, we prove a tight spectral condition for a graph with minimum degree δ to be ID-factor-critical.
Let G be a graph of order n with minimum degree δ≥3r+1, where r≥1 is an integer.
If n≥ max{20δ+r+8, δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2} and
ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
then G is ID-factor-critical unless G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
§ PROOF OF THEOREM <REF>
Before presenting our proof, we introduce some necessary lemmas.
Let G be a graph of order n≥3. If e(G)≥n-12+1, then G has a Hamilton path.
Although the following Lemma <ref> can be obtained directly from Theorem 2 in <cit.>,
here we can present a much simpler proof of Lemma <ref> for a fractional [a,b]-factor.
Let a and b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1 and minimum degree δ≥ a. If
e(G)≥n-12+a+1/2
and na≡0 (mod 2) when a=b, then G has a fractional [a,b]-factor.
For any two disjoint vertex subsets S and T in G, let
φ(S,T)=b|S|-a|T|+∑_v∈ Td_G-S(v).
Suppose to the contrary that G has no fractional [a,b]-factor. By Corollary <ref>,
there exist two disjoint subsets S and T of V(G) such that
φ(S,T)≤-1,
where T={v|v∈ V(G)-S d_G-S(v)<a}.
n≥ a+2 and b≥2.
Note that δ≥ a. If n=a+1, then G is a complete graph. It is well known that the complete graph contains an [a,b]-factor,
and hence G contains a fractional [a,b]-factor, a contradiction. So we have n≥ a+2.
If b=1, then a=b=1, and thus e(G)≥n-12+1.
By Lemma <ref>, G has a Hamilton path.
Note that n is even. Then G contains a 1-factor, and hence G contains a fractional 1-factor, a contradiction.
Hence b≥2.
S≠∅
Assume that S=∅.
Note that G-S=G and δ(G)≥ a. Then δ(G-S)≥ a. Recall that T={v|v∈ V(G)-S d_G-S(v)<a}.
Then T=∅, and thus
φ(∅,∅)=0,
which is contrary to (<ref>).
Next we will evaluate the value of |T|.
Case 1. 0≤|T|≤ b.
Note that δ≥ a. Then
φ(S,T) = b|S|-a|T|+∑_v∈ Td_G-S(v)
= b|S|-a|T|+∑_v∈ Td_G(v)-e_G(S,T)
≥ b|S|-a|T|+a|T|-|T||S|
= (b-|T|)|S|
≥ 0,
which contradicts (<ref>).
Case 2. |T|≥ b+1.
Since S and T are two disjoint subsets of V(G), n≥|S|+|T|≥|S|+b+1.
By the assumption e(G)≥n-12+a+1/2, there exist at most n-1-a+1/2 edges which are not in E[V(G-T-S),T]∪ E(G[T]).
Hence
∑_v∈ Td_G-S(v)≥(n-1-|S|)|T|-2(n-1-a+1/2).
Subcase 2.1. a<b.
Combining Claim <ref>, we have
φ(S,T)
= b|S|-a|T|+∑_v∈ Td_G-S(v)
≥ b|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+1/2)
= (n-1-|S|-a)|T|+b|S|-2n+a+3
≥ (n-1-|S|-a)(b+1)+b|S|-2n+a+3
= (b-2)n+n-|S|-ab-b+2
≥ (b-2)n+(|S|+b+1)-|S|-ab-b+2
= (b-2)n-ab+3
≥ (b-2)(a+2)-ab+3
= 2b-2a-1
≥ 1,
which is contrary to (<ref>).
Subcase 2.2. a=b.
Recall that n≥|S|+b+1=|S|+a+1 and na ≡0 (mod 2). If a is odd, then n is even.
By Claim <ref>, we have n≥ a+3 and a≥3.
Then
φ(S,T) = a|S|-a|T|+∑_v∈ Td_G-S(v)
≥ a|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+1/2)
= (n-1-|S|-a)|T|+a|S|-2n+a+3
≥ (n-1-|S|-a)(a+1)+a|S|-2n+a+3
= (a-2)n+n-|S|-a^2-a+2
≥ (a-2)n+(|S|+a+1)-|S|-a^2-a+2
= (a-2)n-a^2+3
≥ (a-2)(a+3)-a^2+3
= a-3.
≥ 0,
a contradiction.
Next we consider that a is even. Since e(G)≥n-12+a+1/2, we obtain that e(G)≥n-12+a+2/2,
and hence ∑_v∈ Td_G-S(v)≥(n-1-|S|)|T|-2(n-1-a+2/2).
By Claim <ref>, we have n≥ a+2. Then
φ(S,T) = a|S|-a|T|+∑_v∈ Td_G-S(v)
≥ a|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+2/2)
= (n-1-|S|-a)|T|+a|S|-2n+a+4
≥ (n-1-|S|-a)(a+1)+a|S|-2n+a+4
= (a-2)n+n-|S|-a^2-a+3
≥ (a-2)n+(|S|+a+1)-|S|-a^2-a+3
= (a-2)n-a^2+4
≥ (a-2)(a+2)-a^2+4
= 0,
which contradicts (<ref>).
Let A=(a_ij) and B=(b_ij) be two n× n matrices.
Define A≤ B if a_ij≤ b_ij for all i and j, and define A< B if A≤ B and A≠ B.
Let A=(a_ij) and B=(b_ij) be two n× n matrices with the spectral radii λ(A) and λ(B).
If 0≤ A≤ B, then λ(A)≤λ(B).
Furthermore, if B is irreducible and 0≤ A < B, then λ(A)<λ(B).
We will use the following lemma in the proof of Theorem <ref>.
Let G be a graph with minimum degree δ. Then
ρ(G)≤δ-1/2+√(2e(G)-δ n+(δ+1)^2/4).
[Hong, Shu and Fang <cit.>, Nikiforov <cit.>]
For graph G with 2e(G)≤ n(n-1), the function
f(x)=x-1/2+√(2e(G)-nx+(x+1)^2/4)
is decreasing with respect to x for 0≤ x≤ n-1.
Proof of Theorem <ref>. Let G be a graph of order n≥ a+1.
Note that the minimum degree of K_a-1∨ (K_n-a+K_1) is a-1.
Let h: E(G)→ [0,1] be a function. Then for v∈ V(K_1), we have ∑_e∈ E_G(v)h(e)≤ a-1.
By the definition of a fractional [a, b]-factor, then K_a-1∨ (K_n-a+K_1) has no fractional [a, b]-factor.
Assume that G K_a-1∨ (K_n-a+K_1) (see Fig. <ref>). It suffices to prove that G contains a fractional [a, b]-factor.
First we prove the following claim.
δ≥ a.
If δ≤ a-1, then there exists a vertex v∈ V(G) such that d(v)≤ a-1.
This means that G is a subgraph of K_a-1∨ (K_n-a+K_1).
By Lemma <ref>, we have
ρ(G)≤ρ(K_a-1∨ (K_n-a+K_1)).
By the assumption ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1)), we have G≅ K_a-1∨ (K_n-a+K_1), a contradiction.
Hence δ≥ a.
We distinguish the proof into the following two cases.
Case 1. a=1.
By the assumption, we have ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1))= ρ(K_n-1+K_1)=n-2.
By Claim <ref>, Lemma <ref> and Proposition <ref>, we obtain that
n-2≤ρ(G)≤√(2e(G)-n+1).
It follows that e(G)≥n-12+1/2, and hence e(G)≥n-12+1.
By Lemma <ref>, then G contains a fractional [a,b]-factor.
Case 2. a≥ 2.
Note that K_n-1 is a proper subgraph of K_a-1∨ (K_n-a+K_1).
By the assumption and Lemma <ref>, we have ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1))>ρ(K_n-1)=n-2.
By Claim <ref>, Lemma <ref> and Proposition <ref>, we have
n-2<ρ(G)≤a-1/2+√(2e(G)-an+(a+1)^2/4).
It follows that e(G)>n-12+a/2. That is to say, e(G)≥n-12+a+1/2.
By Lemma <ref>, then G contains a fractional [a,b]-factor.
§ PROOF OF THEOREM <REF>
By the Perron-Frobenius Theorem, ρ(G) is always a positive number (unless G is an empty graph),
and there exists an unique positive unit eigenvector corresponding to ρ(G), which is called the Perron vector of G.
Let n=∑_i=1^tn_i+s. If n_1≥ n_2≥⋯≥ n_t≥ p and n_1<n-s-p(t-1), then
ρ(K_s∨(K_n_1+ K_n_2 + ⋯ + K_n_t))<ρ(K_s∨(K_n-s-p(t-1)+ (t-1)K_p)).
Graph S_δ+r,δ∨(K_n-2δ-r-1+I_δ+1) is not ID-factor-critical.
Let G= S_δ+r,δ∨(K_n-2δ-r-1+I_δ+1) (see Fig. <ref>).
Suppose to the contrary that G is ID-factor-critical.
By the definition of an ID-factor-critical graph,
we have G-I has a perfect matching for any independent set I of G whose size has the same parity as |V(G)|.
Note that S_δ+r, δ=K_δ∨ I_r. However, if we take I=I_r and let H=G-I.
Then
H≅ K_δ∨(K_n-2δ-r-1+I_δ+1).
Note that the vertices of I_δ+1 are only adjacent to the vertices of K_δ. Hence H has no perfect matching, a contradiction.
Now, we are in a position to present the proof of Theorem <ref>.
Proof of Theorem <ref>.
Assume that G is not ID-factor-critical. According to Theorem <ref>, there exists some independent set I such that |I| has the same parity as
|V(G)|=n , we have o(G-I-S)≥ |S|+1 for some subset S⊆ V(G)-I. Let |I|=r and |S|=s. Then o(G-I-S)≥ s+1.
Since n-r is even, o(G-I-S) and s have the same parity. Hence we have o(G-I-S)≥ s+2.
It is clear that G is a spanning subgraph of G'=I_r∨ (K_s∨(K_n_1+ K_n_2+⋯+K_n_s+2))
for some odd integers n_1≥ n_2≥⋯≥ n_s+2>0 with ∑_i=1^s+2n_i=n-r-s. Then we have
ρ(G)≤ρ(G'),
where equality holds if and only if G≅ G'.
Let G”=S_s+r, s∨(K_n-2s-r-1+I_s+1). By Lemma <ref>, we obtain that
ρ(G')≤ρ(G”),
where equality holds if and only if (n_1, n_2, … ,n_s+2)=(n-2s-r-1,1,… ,1).
Case 1. s=δ.
Combining (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”)=ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
By the assumption ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
we have G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
By Lemma <ref>, S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1) is not ID-factor-critical.
Hence G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Case 2. s≥δ+1.
Recall that G”=S_s+r, s∨(K_n-2s-r-1+I_s+1).
The vertex set of G” can be divided into V(G”)=V(K_s)∪ V(I_s+1)∪ V(I_r)∪ V(K_n-2s-r-1),
where V(K_s)={u_1, u_2, … ,u_s},
V(I_s+1)={v_1, v_2, … ,v_s+1}, V(I_r)={w_1, w_2, … ,w_r}
and V(K_n-2s-r-1)={z_1, z_2, … ,z_n-2s-r-1}.
Let E_1={v_iz_j|δ+2≤ i≤ s+1, 1≤ j≤ n-2s-r-1}
∪{v_iv_j|δ+2≤ i≤ s,i+1≤ j≤ s+1}
and E_2={u_iv_j|δ+1≤ i≤ s,1≤ j≤δ+1}.
Let G^*=G”+E_1-E_2. Obviously, G^*≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Let x be the perron vector of A(G”), and let ρ” =ρ(G”).
By symmetry, x takes the same value on the vertices of V(K_s), V(I_s+1), V(I_r) and V(K_n-2s-r-1), respectively.
It is easy to see that
A(G”)=[
[ (J-I)_s× s J_s× (s+1) J_s× r J_s× (n-2s-r-1); J_(s+1)× s O_(s+1)× (s+1) J_(s+1)× r O_(s+1)× (n-2s-r-1); J_r× s J_r× (s+1) O_r× r J_r×(n-2s-r-1); J_(n-2s-r-1)× s O_(n-2s-r-1)× (s+1) J_(n-2s-r-1)× r (J-I)_(n-2s-r-1)×(n-2s-r-1) ]].
We denote the entry of x by x_1, x_2, x_3 and x_4 corresponding to the vertices in the above four vertex sets, respectively.
By A(G”)x=ρ” x, we have
ρ” x_2=sx_1+rx_3,
ρ” x_3=sx_1+(s+1)x_2+(n-2s-r-1)x_4,
ρ” x_4=sx_1+rx_3+(n-2s-r-2)x_4.
Observe that n≥2s+r+2. According to (<ref>) and (<ref>), we obtain that x_4≥ x_2.
By (<ref>) and (<ref>), we have ρ” x_3-ρ” x_4=(s+1)x_2-rx_3+x_4.
It follows that x_4=(ρ”+r)x_3-(s+1)x_2/ρ”+1≥ x_2.
Then we have x_3≥ρ”+s+2/ρ”+rx_2.
Note that s≥δ+1 and δ≥3r+1. Then ρ”+s+2≥ρ”+δ+3>ρ”+r, and hence x_3>x_2.
Combining (<ref>), we have
x_2>sx_1/ρ”-r.
Recall that G^*≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Note that G^* contains K_n-2δ-r-1 as a proper subgraph. Then ρ^*>n-2δ-r-2.
Similarly, let y be the perron vector of A(G^*), and let ρ^* =ρ(G^*).
By symmetry, y takes the same value (say y_1, y_2, y_3 and y_4) on the vertices of
V(K_δ), V(I_δ+1), V(I_r) and V(K_n-2δ-r-1).
By A(G^*)y=ρ^*y, we have
ρ^*y_2=δ y_1+ry_3,
ρ^*y_4=δ y_1+ry_3+(n-2δ-r-2)y_4.
Combining (<ref>) and (<ref>), we have
y_4=ρ^*y_2/ρ^*-(n-2δ-r-2).
Note that n≥2s+r+2. Then δ+1≤ s≤n-r-2/2. Since G” is not a complete graph, ρ”< n-1.
ρ”<ρ^*.
Suppose that ρ”≥ρ^*. By x_4≥ x_2, (<ref>) and (<ref>), we have
y^T(ρ^*-ρ”)x
= y^T(A(G^*)-A(G”))x
= ∑_i=δ+2^s+1∑_j=1^n-2s-r-1(x_v_iy_z_j+x_z_jy_v_i)+∑_i=δ+2^s∑_j=i+1^s+1(x_v_iy_v_j+x_v_jy_v_i)-∑_i=δ+1^s∑_j=1^δ+1(x_u_iy_v_j+x_v_jy_u_i)
= (n-2s-r-1)(s-δ)(x_2y_4+x_4y_4)+(s-δ-1)(s-δ)x_2y_4-(s-δ)(δ+1)(x_1y_2
+x_2y_4)
≥ (s-δ)[2(n-2s-r-1)x_2y_4+(s-δ-1)x_2y_4-(δ+1)x_2y_4-(δ+1)x_1y_2]
= (s-δ)[(2n-3s-2δ-2r-4)x_2y_4-(δ+1)x_1y_2]
> (s-δ)[(2n-3s-2δ-2r-4)·sx_1/ρ”-r·ρ^*y_2/ρ^*-(n-2δ-r-2)-(δ+1)x_1y_2]
= (s-δ)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-3s-2δ-2r-4)sρ^*-(δ+1)(ρ”-r)(ρ^* .
.-(n-2δ-r-2))]
= (s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[ρ^*(2n-2δ-3s-2r-4)·s/δ+1-(ρ”-r)(ρ^*.
.-(n-2δ-r-2))].
Note that s≥δ+1, ρ”≥ρ^* and ρ^*>δ-1≥3r. Then
y^T(ρ^*-ρ”)x
> (s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[ρ^*(2n-2δ-3s-2r-4)-ρ”ρ^*+ρ”(n-2δ-r-2).
.+rρ^*-r(n-2δ-r-2)]
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-2δ-3s-2r-4)-ρ”+ρ”/ρ^*·(n-2δ-r-2).
.+r-r/ρ^*·(n-2δ-r-2)]
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-2δ-3s-2r-4)-ρ”+(n-2δ-r-2).
.+r-1/3·(n-2δ-r-2)]
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(8/3n-10/3δ-3s-5/3r-16/3-ρ”).
Since K_s⊂ G” and δ≥3r+1, ρ”> ρ(K_s)=s-1≥δ >r.
Note that s≤n-r-2/2, ρ”< n-1, ρ^*>n-2δ-r-2 and n≥ 20δ+r+8. Then
y^T(ρ^*-ρ”)x
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(8/3n-10/3δ-3·n-r-2/2-5/3r-16/3-ρ”)
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(7/6n-10/3δ-1/6r-7/3-ρ”)
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(1/6n-10/3δ-1/6r-4/3+(n-1)-ρ”)
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))·n-20δ-r-8/6
≥ 0.
This implies that ρ^*>ρ”, which contradicts the assumption ρ”≥ρ^*.
By Claim <ref>, (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”)<ρ(G^*)=ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
which contradicts ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
Case 3. s<δ.
Recall that G'=I_r∨ (K_s∨(K_n_1+ K_n_2+⋯+K_n_s+2)).
Then d_G'(v)=n_s+2+s+r-1≥δ for v ∈ V(K_n_s+2), and hence n_s+2≥δ-s-r+1.
Let G”'=I_r∨(K_s∨(K_n-s-r-(s+1)(δ-s-r+1)+(s+1)K_δ-s-r+1)).
By Lemma <ref>, we have
ρ(G')≤ρ(G”'),
where equality holds if and only if (n_1,n_2,…,n_s+2)=(n-s-r-(s+1)(δ-s-r+1),δ-s-r+1,…,δ-s-r+1).
Let ρ”'=ρ(G”').
ρ”'< n-r-1-(s+1)(δ-s+1).
Suppose to the contrary that ρ”'≥ n-r-1-(s+1)(δ-s+1).
Let x be the perron vector of A(G”'). By symmetry, x takes the same values x_1, x_2, x_3 and x_4
on the vertices of K_s, (s+1)K_δ-s-r+1, I_r and K_n-s-r-(s+1)(δ-s-r+1), respectively.
According to A(G”')x=ρ”' x, we obtain that
ρ”'x_1=(s-1)x_1+(s+1)(δ-s-r+1)x_2+rx_3+(n-s-r-(s+1)(δ-s
-r+1))x_4,
ρ”'x_2=sx_1+(δ-s-r)x_2+rx_3,
ρ”'x_3=sx_1+(s+1)(δ-s-r+1)x_2+(n-s-r-(s+1)(δ-s-r+1))x_4,
ρ”'x_4=sx_1+rx_3+(n-s-r-1-(s+1)(δ-s-r+1))x_4.
By (<ref>) and (<ref>), we have
x_3=(ρ”'+1)x_1/ρ”'+r.
Substituting (<ref>) into (<ref>) and (<ref>), we have
x_2=sx_1+r(ρ”'+1)/ρ”'+rx_1/ρ”'-δ+s+r,
x_4=sx_1+r(ρ”'+1)/ρ”'+rx_1/ρ”'-[n-s-r-1-(s+1)(δ-s-r+1)].
Since n≥δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2, we have
ρ”'≥ n-r-1-(s+1)(δ-s+1)>δ-r+1.
Substituting (<ref>), (<ref>) and (<ref>) into (<ref>), we have
ρ”'+1 = s+(s+1)(δ-s-r+1)(s+r(ρ”'+1)/ρ”'+r)/ρ”'-δ+s+r+r(ρ”'+1)/ρ”'+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r(ρ”'+1)/ρ”'+r)/ρ”'-(n-s-r-1-(s+1)(δ-s-r+1))
≤ s+(s+1)(δ-s-r+1)(s+r)/ρ”'-δ+s+r+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r)/ρ”'-(n-s-r-1-(s+1)(δ-s-r+1))
< s+(s+1)(δ-s-r+1)(s+r)/(δ-r+1)-δ+s+r+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r)/[n-r-1-(s+1)(δ-s+1)]-(n-s-r-1-(s+1)(δ-s-r+1))
= s+(s+r)(δ-s-r+1)+r+[n-s-r-(s+1)(δ-s-r+1)](s+r)/s-sr-r
= n-r-1-(s+1)(δ-s+1)-1/sr-s+r[(sr+2r)n+(2r-1)s^3+(2r^2-2δ r
+δ+1)s^2+(r^3-δ r^2-r^2-3δ r-2r+1)s+r^3-δ r^2-3r^2-2δ r-3r].
Let f(n)=(sr+2r)n+(2r-1)s^3+(2r^2-2δ r+δ+1)s^2+(r^3-δ r^2-r^2-3δ r-2r+1)s+r^3-δ r^2-3r^2-2δ r-3r.
We assert that f(n)≥ 0. Suppose that f(n)<0. Then
n<1/sr+2r[(-2r+1)s^3+(-2r^2+2δ r-δ-1)s^2+(-r^3+δ r^2+r^2+3δ r+2r-1)s-r^3+δ r^2+3r^2+2δ r+3r].
Note that 0≤ s<δ, -2r+1<0, -2r^2+2δ r-δ-1>0 and -r^3+δ r^2+r^2+3δ r+2r-1>0. Then
n < 1/sr+2r[(-2r+1)s^3+(-2r^2+2δ r-δ-1)s^2+(-r^3+δ r^2+r^2+3δ r+2r-1)s
-r^3+δ r^2+3r^2+2δ r+3r]
< 1/2r[(-2r^2+2δ r-δ-1)δ^2+(-r^3+δ r^2+r^2+3δ r+2r-1)δ-r^3+δ r^2+3r^2
+2δ r+3r]
= 1/2r[(2r-1)δ^3+(-r^2+3r-1)δ^2+(-r^3+2r^2+4r-1)δ-r^3+3r^2+3r]
< 1/2r[2rδ^3+(-r^2+3r)δ^2+(-r^3+2r^2+4r)δ-r^3+3r^2+3r]
= δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2,
which contradicts n≥δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2. Hence f(n)≥ 0.
Then
ρ”'+1 < n-r-1-(s+1)(δ-s+1)-1/sr-s+rf(n)
< n-r-1-(s+1)(δ-s+1)
≤ ρ”',
a contradiction. Therefore, we have ρ”'<n-r-1-(s+1)(δ-s+1).
By Claim <ref> and s< δ, we obtain that
ρ”' < n-r-1-(s+1)(δ-s+1)
= n-δ-r-1-[(δ-s)s+1]
< n-δ-r-1.
Note that K_n-δ-r⊂ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1). Then
n-δ-r-1=ρ(K_n-δ-r)<ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
Combining (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”')<n-δ-r-1<ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
which contradicts the assumption, as desired.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
99
Anstee1990 R.P. Anstee, Simplified existence theorems for (g,f)-factor, Discrete Appl. Math. 27 (1990) 29–38.
Berman1979 A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.
Bondy2008 J.A. Bondy, U.S.R. Murty, Graph Theory, Grad. Texts in Math. vol. 244, Springer, New York, 2008.
Cho2021 E. Cho, J. Hyun, S. O, J. Park, Sharp conditions for the existence of an even [a,b]-factor in a graph,
Bull. Korean Math. Soc. 58 (2021) 31–46.
Fan D.D. Fan, H.Q. Lin, Spectral conditions for k-extendability and k-factors of bipartite graphs, arXiv:2211.09304.
Fan2021 D.D. Fan, S. Goryainov, X.Y. Huang, H.Q. Lin, The spanning k-tree, perfect matchings and spectral radius of graphs,
Linear Multilinear Algebra 70 (2022) 7264–7275.
Fan2022 D.D. Fan, H.Q. Lin, H.L. Lu, Spectral radius and [a,b]-factors in graphs, Discrete Math. 345 (2022) 112892.
Hong2001 Y. Hong, J.L. Shu, K.F. Fang, A sharp upper bound of the spectral radius of graph,
J. Combin. Theory, Ser. B 81 (2001) 177–183.
Horn1986 R.A. Horn, C.R. Johnson, Matrix analysis, Cambridge University Press, New York, 1986.
Liu2001 G.Z. Liu, L.J. Zhang, Fractional (g,f)-factors of graphs, Acta Math. Sci. 21 (2001) 541–545.
Liu2008 G.Z. Liu, L.J. Zhang, Toughness and the existence of fractional k-factors of graphs, Discrete Math. 308 (2008) 1741–1748.
Lu2013 H.L. Lu, Simplified existence theorems on all fractional [a,b]-factors,
Discrete Appl. Math. 161 (2013) 2075–2078.
Nikiforov2002 V. Nikiforov, Some inequalities for the largest eigenvalue of a graph,
Combin. Probab. Comput. 11 (2002) 179–189.
Tutte1947 W.T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107–111.
Wang2023 J.J. Wang, J.X. Zheng, Y.L. Chen, Spectral radius conditions for fractional [a, b]-covered graphs,
Linear Algebra Appl. 666 (2023) 1–10.
Wei2023 J. Wei, S.G. Zhang, Proof of a conjecture on the spectral radius condition for [a, b]-factors,
Discrete Math. 346 (2023) 113269.
|
http://arxiv.org/abs/2307.06054v1 | 20230712101709 | Asymmetry of 2-step Transit Probabilities in 2-Coloured Regular Graphs | [
"Ron Gray",
"J. Robert Johnson"
] | math.CO | [
"math.CO",
"math.PR"
] |
Asymmetry of 2-step Transit Probabilities in 2-Coloured Regular Graphs
Ron GraySchool of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom.
J. Robert JohnsonSchool of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK. E-mail: [email protected].
================================================================================================================================================================================================================================================================
Suppose that the vertices of a regular graph are coloured red and blue with an equal number of each (we call this a balanced colouring). Since the graph is undirected, the number of edges from a red vertex to a blue vertex is clearly the same as the number of edges from a blue vertex to a red vertex. However, if instead of edges we count walks of length 2, then this symmetry disappears. Our aim in this paper is to investigate how extreme this asymmetry can be.
Our main question is: Given a d-regular graph, for which pairs (x,y)∈[0,1]^2 is there a balanced colouring for which the probability that a random walk starting from a red vertex stays within the red class for at least 2 steps is x, and the corresponding probability for blue is y?
Our most general result is that for any d-regular graph, these pairs lie within the convex hull of the 2d points {(l/d,l^2/d^2),(l^2/d^2,l/d) :0≤ l≤ d}.
Our main focus is the torus for which we prove both sharper bounds and existence results via constructions. In particular, for the 2-dimensional torus we show that asymptotically the region in which these pairs of probabilities can lie is exactly the convex hull of:
{(0,0),(1/2,1/4),(3/4,9/16),(1/4,1/2),(9/16,3/4),(1,1)}
§ INTRODUCTION
The aim of this paper is to investigate some extremal properties of a notion of edge boundary coming from random walks on a graph.
Let G be a d-regular graph on n vertices. The edge-boundary ∂(S) of a set of vertices S⊆ V(G) is defined by
∂(S)={e∈ E(G): |e∩ S|=1}
(that is the set of edges which go between S and S^c).
The problem of minimising ∂(S) for a given |S| is the edge isoperimetric inequality problem in extremal combinatorics which has been studied for various graphs. Most notably the edge isoperimetric inequality was proved in the hypercube by Harper <cit.>, Lindsey <cit.>, Bernstein <cit.> and Hart <cit.> and in the grid by Bollobás and Leader <cit.>. See also <cit.> for extremal combinatorics background.
Suppose that we colour the vertices of G with red and blue. In other words, we partition V(G) into two colour classes R and B. Clearly ∂(R)=∂(B) as each of these counts E(R,B): the set of edges with one end in each colour class. However, as we shall see, there is a natural extension of edge boundary for which the red to blue boundary may not be the same as the blue to red boundary.
To define this, we first relate the edge boundary to probabilities in random walks on G. A random walk on G moves from vertex to vertex, at each discrete time step choosing uniformly from all the neighbours of its current location. For S⊆ V(G), let (W^S_i)_i=0^∞ be the random walk on G in which W^S_0 is a uniformly random vertex in S. We will be interested in the probability that the random walk starting in a given colour class stays within this colour class for a fixed number of steps. We define these probabilities as:
P_t(R)=(W^R_1,W^R_2,…, W^R_t∈ R), P_t(B)=(W^B_1,W^B_2,…, W^B_t∈ B).
It is easy to see that the probability we stay inside the same colour class for one step is determined by the edge boundary of the colour class.
P_1(R)=1-∂(R)/d|R|, P_1(B)=1-∂(B)/d|B|
In particular, if we have a balanced colouring (that is |R|=|B|) then P_1(R)=P_1(B) because ∂(R)=∂(B). However, for t≥ 2, it is not necessarily the case that P_t(R) and P_t(B) are the same.
Like the edge boundary, the pair (P_t(R),P_t(B)) provides some quantitative measure of how interspersed the sets R and B are. But unlike the edge boundary, this quantity is not symmetric in the two coordinates (at least when t≥ 2). Our main aim in this paper is is to investigate this asymmetry at the smallest t for which it occurs: the 2-step walks.
We write P_t(R,B) for the pair (P_t(R),P_t(B))∈[0,1]^2. Our main, and very general, question is:
Given a particular d-regular graph G with |V(G)| even and a balanced colouring (R,B), what possible values in [0,1]^2 can the pair P_2(R,B) take?
Our first main result is that for any d-regular graph, the pair P_2(R,B) must lie in the region D_d⊆[0,1]^2 defined as the convex hull of the following 2d extreme points:
{(l/d,l^2/d^2):0 ≤ l ≤ d}∪{(l^2/d^2,l/d):0 ≤ l ≤ d}.
As a simple example, we briefly describe the behaviour for some 2-regular graphs. For the cycle C_n, the set above is essentially best possible, but for a disjoint union of 4-cycles the true answer is substantially smaller.
The heart of the paper concerns the discrete torus [k]^m and in particular, bounding the possible region for P_2(R,B) asymptotically for large k. This is a 2m-regular graph so by our first result this region is contained in D_2m.
For all m we show that the extreme points of the region D_2m corresponding to l=0,m,2m-2,2m-1,m can be achieved but that those corresponding to 1≤ l≤ m-1 cannot. For the 2-dimensional torus it is possible to give a more intricate argument proving an exact result (asymptotically for large k). We show that for m=2 the possible region for P_2(R,B) is significantly smaller than D_4; it is the convex hull of the following set of points
{(0,0),(1/2,1/4),(3/4,9/16),(1,1),(1/4,1/2),(9/16,3/4)}.
We finish with some questions and open problems.
We remark that another measure of boundary related to random walks is the expectation of the escape time (the smallest time at which the random walk (W^S_i)_i=0^∞ reaches a vertex outside S). The natural extremal question here is for a given |S|, which S maximises the expected escape time? In the setting of the hypercube, Samorodnitsky <cit.> showed that the extremal sets are subcubes (exactly those sets which minimise edge boundary).
Finally, we note that while our motivation is purely mathematical, this notion of using random walks to measure spatial dispersion has been used to quantify socioeconomic segregation in various contexts. In these applications, the fact that it may be easier for a random walk to move from R to B than from B to R can be socially significant. For examples of work in this direction see <cit.>.
§ NOTATION AND PRELIMINARIES
Throughout, G=(V,E) will be a d-regular graph with n vertices and n will be even. We will consider balanced colourings of V, that is partitions V=R∪ B with |R|=|B|=n/2. For a vertex v∈ R we write d_R for the degree of v in the subgraph induced by R:
d_R(v)=|{u∈ R : (u,v)∈ E}|
We write R_i for the set of vertices in R with d_R(v)=i:
R_i={v∈ R : d_R(v)=i}
and r_i=|R_i|.
The internal degree sum I(R) of R (or twice the number of internal edges in R) is
I(R)=∑_v∈ R d_R(v)=∑_i=1^d ir_i
We similarly define B_i, b_i and I(B) for the colour class B.
As our colouring is balanced I(B)=d|B|-∂(B)=d|R|-∂(R)=I(R).
The r_i also determine the 2-step probabilities P_2(R).
If G is an n-vertex, d-regular graph then
P_2(R)=2/nd^2∑_v∈ R d^2_R(v)=2/nd^2∑_i=1^d i^2r_i
The number of 2-step walks consisting entirely of vertices in R and with middle vertex u is d_R(u)^2 (we need to choose the first and last vertex in the walk independently from among all neighbours of u in R). Hence the total number of 2-step walks consisting entirely of vertices in R is
∑_u∈ R d_R(u)^2=∑_i=1^d i^2r_i.
Since the total number of 2-step walks in G starting from a vertex in R is |R|d^2 we have
P_2(R)=∑_i=1^d i^2r_i/|R|d^2=2/nd^2∑_i=1^d i^2r_i.
If S={s_1,…,s_m} is a finite set of points in ^2 we write (S) for the convex hull of S.
If P_2(R,B)=(x,y) then swapping the colours gives a colouring with P_2(R,B)=(y,x). This means that our set of possible values for P_2(R,B) is symmetric in the coordinates and it will be useful to define a symmetrised version of the convex hull to describe it. For S⊆^2 let S={(y,x) : (x,y)∈ S}. We write (S) for the set (S∪S).
§ BOUNDING THE POSSIBLE VALUES OF P_2(R,B)
Our first main result determines a region in [0,1]^2 within which P_2(R,B) lies for any d-regular graph.
Let G be a d-regular graph with |V(G)| even and let D_d ⊂ [0,1]^2 be the set ({(l/d,l^2/d^2):0 ≤ l ≤ d}). We have P_2(R,B)∈ D_d for all balanced colourings (R,B) of G.
Define α so that ∑_v∈ R d_R(v)=αdn/2 (or equivalently and more explicity, set α=1-2∂(R)/dn). Clearly 0≤α≤ 1.
By convexity, ∑_v∈ R d^2_R(v) is maximised when the d_R(v) are all 0 or d. We then must have α n/2 vertices with d_R(v)=d and (1-α)n/2 vertices with d_R(v)=0 and so by Lemma <ref>
P_2(R)=2/nd^2∑_i=1^d i^2r_i≤2/nd^2d^2α n/2=α
Again, by convexity, ∑_v∈ R d^2_R(v) is minimised when the d_R(v) are as equal as possible. So certainly, we cannot achieve a smaller value than by taking d_R(v)=α d for all v. Hence,
P_2(R)=2/nd^2∑_i=1^d i^2r_i≥2/nd^2d^2α^2 d^2 n/2=α^2.
Now crucially we have that ∑_v∈ B d_B(v)=∑_v∈ R d_R(v)=αdn/2. So, the same bounds (with the same value of α) are valid for the other colour class B.
It follows that, if P_2(R)=x then α≥ x and P_2(B)≥α^2≥ x^2=P_2(R)^2. Similarly, if P_2(B)=y then α≥ y and P_2(R)≥α^2≥ y^2=P_2(B)^2. This gives the weaker result that the possible pairs P_2(R,B) all lie in the region bounded by the lines y=x^2 and x=y^2.
P_2(R,B)∈{(x,y) : y≥ x^2, x≥ y^2}.
In this bound we did not make use of the fact that the d_R(v) are all integers. Incorporating this in the argument minimising P_2(R) gives a stronger bound. If l≤α d<l+1, the minimum of P_2(R) is attained when the d_R(v) are all l or l+1. More precisely, we have n/2(l+1-α d) of the vertices having d_R(v)=l and n/2(α d-l) of the vertices having d_R(v)=l+1. This gives
P_2(R) =2/nd^2∑_i=1^d i^2r_i
≥2/nd^2(l^2n/2(l+1-α d)+(l+1)^2n/2(α d-l))
=1/d^2(α d(2l+1)-l(l+1))
For any fixed l this is a linear function in α. Moreover, if α=l/d or α=l+1/d then we recover the bound P_2(R)≥α^2.
Using the same argument as before to constrain the possible values for the pair P_2(R,B) we get that these must lie in the convex region defined by the extreme points
{(l/d,l^2/d^2):0 ≤ l ≤ d}∪{(l^2/d^2,l/d):0 ≤ l ≤ d}.
as required.
§ SIMPLE 2-REGULAR EXAMPLES
The set D_2 has extreme points {(0,0),(1/2,1/4),(1/4,1/2),(1,1)}.
As a simple example, consider the n-cycle, C_n. Suppose that G=C_n with V={1,…,n} and E={(i,i+1) : 1≤ i≤ n-1}∪{(n,1)}. We can construct colourings of this graph which get close to each of the four extreme points of D_2. To simplify the description we assume that n is a multiple of 4.
* R={1,…,n/2}, B={n/2+1,…,n}, P_2(R,B)=(1-3/n,1-3/n)
* R={2,4,6,…,n}, B={1,3,5,… n-1}, P_2(R,B)=(0,0)
* R={x : 1≤ x≤3n/4, x≢0 3}, B=V∖ R, P_2(R,B)=(1/4,1/2+O(1/n))
* B={x : 1≤ x≤3n/4, x≢0 3}, R=V∖ B, P_2(R,B)=(1/2+O(1/n),1/4)
Note, in the asymmetric third case each red vertex has exactly one red neighbour (so P_2(R)=(1/2)^2). With a few exceptions (vertices 3n/4,3n/4+1,n-1,n), the blue vertices come in two types:
* those in {1,…,3n/4-1} have no blue neighbours (so contribute 0 to P_2(B));
* those in {3n/4+2,…,n-1} are at distance greater than 2 from the nearest red vertex (so contribute 1 to P_2(B)).
Since there are equal numbers of these two types P_2(B)=1/2+O(1/n).
Suppose we have a colouring c_1 of C_n with P_2(R,B)=(x_1,y_1) and another colouring c_2 of C_n with P_2(R,B)=(x_2,y_2). We will colour C_2n using c_1 on the first half and c_2 on the second half. That is our colouring c is:
c(x)=
c_1(x) if 1≤ x≤ n;
c_2(x-n) if n+1≤ x≤ 2n.
Each vertex apart from 1,2,n-1,n,n+1,n+2,2n-1,2n (that is those within distance 2 of the `join' between the colourings) behaves exactly as it does in colouring c_1 or colouring c_2. Hence, this new colouring satisfies
P_2(R,B)=(1/2x_1+1/2x_2+O(1/n),1/2y_1+1/2y_2+O(1/n)).
Similarly, we can join together a copies of colouring c_1 and b copies of colouring c_2 to obtain a colouring of C_(a+b)n with
P_2(R,B)=(a/a+bx_1+b/a+bx_2+O(1/n),a/a+by_1+b/a+by_2+O(1/n)).
Taking n large enough, this operation produces colourings in which P_2(R,B) can be arbitrarily close to any point in D_2.
We see from this example that constructions matching the extreme points of D_d are key. Also it may only be possible to achieve these up to O(1/n) so we will mainly be interested in asymptotic results.
Before we leave the d=2 case, note that this kind of construction will not be possible for all 2-regular graphs. For instance, if G is a disjoint union of copies of C_4 then it is not too difficult to see that the extreme point (1/2,1/4) cannot be achieved. In fact, for this graph the set of possible pairs for P_2(R,B) is essentially ({(0,0),(1/2,2/3),(1,1)}).
§ TORUS
As we noted earlier we will mainly be interested in asymptotic results. The next definition formalises this.
Let =(G_k)_k=1^∞ be a sequence of d-regular graphs with |V(G_k)| even for all k and tending to ∞ with k. Define ()⊆ℝ^2 by (x,y)∈() if and only if for every ϵ>0 there exists some k and a balanced colouring of G_k with |P_2(R)-x|<ϵ and |P_2(B)-y|<ϵ.
For fixed m≥ 2, let T^m_k be the torus [k]^m. That is the graph with vertex set [k]^m in which vertices (x_1,…,x_m) and (y_1,…,y_m) are adjacent if x_j=y_j ± 1 m for exactly one coordinate j while x_i=y_i for all i≠j. Let ^m be the sequence (T^m_k)_k=1^∞
The set (^m) is convex.
Suppose that (x_1,y_1), (x_2,y_2)∈(^m) and 0<λ<1. We will show that λ(x_1,y_1)+(1-λ)(x_2,y_2)∈(^m).
Given ϵ>0, pick k large enough that there is both a balanced colouring C^1=(R^1,B^1) of T^m_k with |P_2(R^1)-x_1|,|P_2(B^1)-y_1|<ϵ and a balanced colouring C^2=(R^2,B^2) of T^m_k with |P_2(R^2)-x_2|,|P_2(B^2)-y_2|<ϵ. Also pick s,t∈ so that |s/t^m-λ|≤ϵ.
We construct a balanced colouring (R,B) of [kt]^m by partitioning it into t^m copies of [k]^m in the obvious way and colouring s of these copies with colouring C^1 and t^m-s of them with colouring C^2. This clearly gives a balanced colouring. We say that a point of [kt]^m is a bad point if it is close enough to the boundary of one of the partitioning copies of [k]^m that it is is possible for a 2-step walk from that point to reach a different partitioning [k]^m.The number of bad points is only O(k^m-1) so their contribution to P_2(R) will only be O(1/k).
Suppose that u∈[kt]^m is not bad. Then the probability that a 2-step random walk starting from u stays within its colour class can be found by conditioning on whether the copy of [k]^m containing it was coloured with colouring C^1 or C^2. Indeed, for the balanced colouring (R,B) of [kt]^m we have
P_2(R)=s/t^mP_2(R^1)+(1-s/t^m)P_2(R^2)+O(1/k)
So
|P_2(R)-λ x_1-(1-λ)x_2|<cϵ+O(1/k)
for some constant c. But the righthand side can be made arbitrarily small by choosing ϵ and k suitably.
Similarly
|P_2(B)-λ y_1-(1-λ)y_2|<cϵ+O(1/k).
It follows that λ(x_1,y_1)+(1-λ)(x_2,y_2)∈(^m) as required.
A consequence of this result is that determining (T^m) reduces to finding its extreme points. Since T^m_k is 2m-regular we should be comparing this with D_2m.
The proof of Theorem <ref> suggests that in order to attain the extreme points of D_2m we need a colouring in which for all but o(|V|) of the vertices, every vertex in R has either 0 or 2m neighbours in R and every vertex in B has the same number of neighbours in B (or equivalently the same number of neighbours in R) or vice versa. The following definition, which we make for a general regular graph, formalises this idea and is key to constructing colourings which achieve extreme points.
Let G be a d-regular graph and 0≤ r≤ d. A set S⊆ V(G) is an independent exact r-cover if it is an independent set in the graph G and every element of S^c is adjacent to exactly r elements of S.
By double-counting edges between S and S^c we see that the density (or asymptotic density if G is infinite) of any independent exact r-cover is r/d+r.
Although we will use our constructions for the finite torus, it is convenient to express then in the infinite grid ^m, that is the graph with vertex set ^m and two vertices adjacent if they differ by adding or subtracting 1 in exactly one coordinate.
We first show how to use these objects to construct colourings of the torus with particular P_2(R,B). The idea is to split the graph into two parts. In the first part we take red vertices forming an independent exact r-cover with the remaining vertices blue, in the second part we colour all vertices red. This will ensure that with a small number of exceptions, the red vertices will all have 0 red neighbours or m red neighbours (depending on whether they are in the first of second part) while the blue vertices (which are all in the first part) will all have 2m-r blue neighbours. The size and shape of the parts are chosen so that the colouring is balanced and the number of exceptional vertices (those near the boundary of the parts) is small.
If an independent exact r-cover of ^m exists then
((2m-r/2m)^2,2m-r/2m)∈(^m)
Let S be an independent exact r-cover of ^m.
Let a=⌈2m+r/4mk⌉. We partition [k]^m into two parts X=[k]^m-1×[a] and Y=[k]^m∖ X.
Now let R=((S∩ X)∪ Y) E and B=[k]^m∖ R=(X∖ S) E where E is a small set chosen to make |R|=k^m/2.
We can count the number of vertices in X∖ S by double counting the edges from a vertex in S∩ X to a vertex in X∖ S. For all but O(k^m-1) vertices this is determined by the exact r-cover property so:
r|X∖ S| =2m(|X|-|X∖ S|)+O(k^m-1)
(2m+r)|X∖ S| =2mak^m-1+O(k^m-1)
(2m+r)|X∖ S| =2m(2m+r)k^m/4m+O(k^m-1)
|X∖ S| =k^m/2+O(k^m-1)
So, we may take |E|<ck^m-1 for some constant c depending on m and r.
With the possible exception of vertices in or within distance 2 of E and vertices near the boundary of X (that is with final coordinate in {1,2,a-1,a}) every vertex in B has exactly 2m-r neighbours in B and moreover each of those neighbours has exactly 2m-r neighbours in B. Hence:
P_2(B)=(2m-r/2m)^2+O(1/k).
Similarly, proportion 2(1-a/k)+o(1)=2m-r/2m+o(1) of the vertices in R are at distance greater than 2 from the nearest vertex in B and
proportion r/2m+o(1) of the vertices in R have all of their neighbours in B. So
P_2(R)=2m-r/2m+o(1).
By choosing k large enough we get that
((2m-r/2m)^2,2m-r/2m)∈(^m)
as required.
We will give some constructions of independent exact r-covers in ^m valid for several values of r. We also show that it is easy to scale up both r and m by a constant factor, so the most interesting open cases are those when r and m are coprime.
* For any m≥ 2, an independent exact r-cover of ^m exists when r=0,1,2,m or 2m.
* If an independent exact r-cover of ^m exists then so does an independent exact λ r-cover of ^λ m for all λ∈.
* For r=0, the set S=∅ is an independent exact r-cover.
For r=1, the set
S_1={(x_1,…,x_m)∈^m : ∑_i=1^m ix_i≡ 0 (2m+1)}
is an independent exact 1-cover. Suppose that x∈ S_1 and y∈^m differs from x by changing a single coordinate by ±1. Then ∑_i=1^m ix_i-∑_i=1^m iy_i∈{±1, ±2,…, ± m} and so we cannot have y∈ S_1. It follows that S_1 is an independent set. Also, if y∈^m∖ S_1 then ∑_i=1^m iy_i≡ r (2m+1) with r≠0. If r∈{1,2,…, m} then there is exactly one neighbour of y in S_1 obtained by decreasing the rth coordinate of y by 1. If r∈{-1,-2,…, -m} then again there is exactly one neighbour of y in S_1, this time obtained by increasing the -rth coordinate of y by 1.
For r=2, the set
S_2={(x_1,…,x_m)∈^m : ∑_i=1^m ix_i≡ 0 (m+1)}
is an independent exact 1-cover. Suppose that x∈ S_2 and y∈^m differs from x by changing a single coordinate by ±1. Then ∑_i=1^m ix_i-∑_i=1^m iy_i∈{±1, ±2,…, ± m} and so we cannot have y∈ S_2. It follows that S_2 is an independent set. Also, if y∈^m∖ S_2 then ∑_i=1^m iy_i≡ r (m+1) with r≠0. Now there are exactly two neighbours of y in S_2 obtained by decreasing the rth coordinate of y by 1 or increasing the (m+1-r)th coordinate of y by 1.
For r=m, the set
S_m={(x_1,…,x_m)∈^m : ∑_i=1^m x_i≡ 0 3}
is an independent exact m-cover. It is easy to see that S_m is an independent set. If y∈^m∖ S_m then either ∑_i=1^m y_i≡ 1 3 in which case the m nighbours obtained by decreasing any coordinate by 1 are all in S_m, or ∑_i=1^m y_i≡ 1 3 in which case the m neighbours obtained by increasing any coordinate by 1 are all in S_m.
Finally, for r=2m, it is easy to see that the set
S_m={(x_1,…,x_m)∈^m : ∑_i=1^m x_i≡ 0 2}
is an independent exact m-cover.
* Suppose that S is an independent exact r-cover of ^m. Let f:^λ m→^m be defined by
f(x_1,…,x_λ m)=(s_1,…,s_m)
where s_k=∑_i=λ(k-1)+1^λ kx_i. In other words, we split the coordinates of ^λ m into m blocks of length λ and let the ith coordinate of the image of x be the sum of the coordinates in its ith block.
Now, define S'⊆^λ m by
S'=f^-1(S)={x∈^λ m : f(x)∈ S}.
It is easy to see that S' is an independent set. Now suppose that y∈^λ m∖ S'. Then f(y)∈^m∖ S and so there are exactly r neighbours of f(y) in S. Each such neighbour gives rise to λ neighbours of y in ^λ m. For instance if increasing coordinate i of f(y) by 1 yields an element of S then increasing any of the coordinates (i-1)λ+1,(i-1)λ+2,… iλ of y by 1 yields an element of S'. Moreover each neighbour of y in S arises in this way and so y has exactly λ r neighbours in S'. It follows that S' is an independent exact λ r-cover of ^λ m.
Using the constructions in Theorem <ref> and the connection between exact r-covers and P_2(R,B) in Theorem <ref> we obtain the following corollary.
Let X_2m be the symmetric convex hull:
({(l/2m,(l/2m)^2) : l=0,m,2m-2,2m-1,2m}
Then
X_2m⊆(^m)⊆ D_2m
For [k]^2 we have an exact result.
(^2)=({(0,0),(1/2,1/4),(3/4,9/16),(1,1)})
Note that the difference between the extreme points of D_4 and the set (^2) in the statement of the Theorem is that the latter does not include (1/4,1/16) or (1/16,1/4).
Recall the notation that R_i is the set of all vertices in R which have exactly i neighbours in R (and similarly for B_i). In order to attain (1/4,1/16) we would need to find a colouring in which almost all vertices in R lie in R_0∪ R_4 while almost all vertices in B lie in B_1. We will show this is impossible by showing that B_1 being large forces many vertices in R_1∪ R_2∪ R_3.
By Theorem <ref> we have
(^2)⊆({(0,0),(1/4,1/16),(1/2,1/4),(3/4,9/16),(1,1)})
By the constructions of Corollary <ref> when m=2, r=1,2 we have
({(0,0),(1/2,1/4),(3/4,9/16),(1,1)})⊆(^2)
So to prove the Theorem it suffices to show that if (1/4,y)∈(^2) then y≥1/8.
Let (R,B) be a balanced colouring of [k]^2. Suppose that v∈ B_1. Let v=(a,b) and suppose, without loss of generality, that the single neighbour of v in B is w=(a,b+1). We will split B_1 into types as follows. We say that v has:
Type 3: if w∈ B_3∪ B_4
Type 1: if (a-1,b+1),(a+1,b+1)∈ R
Type 2: otherwise
Note that if w∈ B_1 then v certainly has Type 1, but if w∈ B_2 then v could be of Type 1 or Type 2. Let
t_i=|{v∈ B_1 : v has Type i}|
We define a bipartite graph H with bipartition (X,Y) where
X ={ v∈ B_1 : v has Type 1 or Type 2}
Y =R_1∪ R_2∪ R_3
The edges of H are of two types. If v∈ X and r∈ Y then (v,r)∈ E(H) if
* v has Type 1 and (v,r) are adjacent in T_k^2
* v=(i,j) has Type 2 and r∈{(i,j±1), (i±1,j),(i±1,j±1)}
If v∈ X then _H(v)≥ 2
If v has Type 1 then we know (a-1,b+1),(a+1,b+1)∈ R. Hence both (a+1,b) and (a-1,b) have at least one neighbour in R and one neighbour in B and so are elements of Y. It follows that v is adjacent to both (a+1,b) and (a-1,b) in H and so _H(v)≥ 2.
If v has Type 2 then without loss of generality (a-1,b+1)∈ R, (a+1,b+1)∈ B (if both these vertices where in B then v would have Type 3). Hence both (a-1,b+1) and (a-1,b) have at least one neighbour in R and one neighbour in B and so are elements of Y. It follows that v is adjacent to both (a-1,b+1) and (a-1,b) in H and so _H(v)≥ 2.
If r∈ Y then _H(r)≤ 3
Suppose that r=(i,j)∈ Y. We need to show that among the 8 points {(i,j±1), (i±1,j),(i±1,j±1)} at most 3 of them are elements of X which are adjacent to (i,j) in H. Suppose that (i-1,j+1) is adjacent to (i,j) in H. We must have that (i-1,j+1)∈ B_1 and is a Type 2 vertex. But this means that (i,j+1)∉B_1 (if it were then (i-1,j+1) would be of Type 1). It follows that among the points (i-1,j+1),(i,j+1),(i+1,j+1) either
* at most one of them is adjacent to (i,j) in H
or
* (i-1,j+1) and (i+1,j+1) are adjacent to (i,j) in H
To prove the claim, suppose for a contradiction that (i,j) has at least 4 neighbours in H. The only ways this can happen subject to the above are for these neighbours to be
* (i± 1,j), (i,j± 1)
or
* (i± 1,j± 1)
In the first case we have (i,j)∈ R_0. In the second case, note that (i+1,j) must be in R; if it were in B then (i+1,j+1) would have either Type 1 (if (i+2,j)∈ R) or Type 3 (if (i+2,j)∈ B) contradicting that (i+1,j+1) is a neighbour of (i,j) in H. Similarly, (i-1,j),(i,j+1),(i,j-1)∈ R and so (i,j)∈ R_4. A contradiction since we must have (i,j)∈ Y=R_1∪ R_2∪ R_3. This establishes the second claim.
Using these two claims to double count the edges of H we get:
2(t_1+t_2)≤ E(H)≤ 3(r_1+r_2+r_3).
Let n=k^2. Suppose that P_2(R)=1/4+ then by Lemma <ref>:
1/4+=1/8n(r_1+4r_2+9r_3+16r_4)
and so
2n+8 n=r_1+4r_2+9r_3+16r_4
we also have
I(R)=r_1+2r_2+3r_3+4r_4.
Hence
2n+8 n=4I(R)-(3r_1+4r_2+3r_3)≤ 4I(R)-3(r_1+r_2+r_3)≤ 4I(R)-2(t_1+t_2)
Now, consider P_2(B). We have that
8nP_2(B)=b_1+4b_2+9b_3+16b_4
We know that b_1=t_1+t_2+t_3. We also know that t_3≤ 3b_3+4b_4 because each Type 3 vertex in B_3 is adjacent to a vertex in B_3 or B_4. Given these constraints, to minimize P_2(B) we should take b_1=t_1+t_2+t_3, b_3=t_3/3, b_4=0 and then use the fact that I(R)=I(B) to determine b_2. That is b_2=1/2(I(R)-b_1-3b_3).
8nP_2(B) ≥ (t_1+t_2+t_3)+2(I(R)-t_1-t_2-2t_3)+9t_3/3
=2I(R)-(t_1+t_2)
≥ n +4 n
by equation (<ref>). So
P_2(B)≥1/8+/2
Turning to larger m, we can give a similar but cruder non-existence argument for independent exact r-covers of ^m when m<r<2m.
For any m≥ 2. If m<r<2m there is no independent exact r-cover of ^m.
Suppose that S⊆^m, m<r<2m and that every point in ^m∖ S has exactly r neighbours in S. Let e^i∈^m be the vector with 1 in position i and 0s elsewhere. Take x,y∈^m∖ S with x=y+e_i (so x,y are adjacent in ^m and differ in coordinate i). We can certainly find such a pair since r≠2m.
Let E={e_j: j≠i}∪{-e_j : j≠i}. Let Now X={e∈ E :x+e∈ S} and Y={e∈ E :y+e∈ S}. Since x and y each have r neighbours in S we have that |X|≥ r-1 and |Y|≥ r-1 (the bound is r-1 rather than r because x+e_i and y-e_i may be in S). But |E|=2m-2<2(r-1) so by the pigeonhole principle X∩ Y≠∅. If e∈ X∩ Y then x+e,y+e are adjacent elements of S and so S cannot be an independent set.
This argument only shows the non-existence of a set exactly meeting the definition of an independent exact r-cover. However, since the argument is a very local one, it follows easily that (^m) does not contain the points (l/2m,l^2/(2m)^2) with 1≤ l≤ m-1. However, we do not know the exact set (^m) for m≥ 3.
§ QUESTIONS
The most immediate open problem is to prove an analogue of Theorem <ref> for higher dimensions.
What is (^m) for m≥ 3?
It could be that, as in the m=2 case, the answer is
({(0,0)}∪{(l/2m,l^2/(2m)^2):m ≤ l ≤ 2m}).
For m=3, Theorem <ref> shows that this set is certainly contained in (^3) but the true answer may have additional extreme points. For m≥ 4, we are missing both constructions and non-existence results.
Closely related to this is the quesion of when an independent exact r-cover of ^m exists.
For which r,m does an independent exact r-cover of ^m exist?
As we have seen, these do exist for r=0,1,2,m,2m and do not exist for r=m+1,m+2,…,2m-1. All the missing cases of 3≤ r≤ m-1 are open although we can reduce this to the cases when r and m are coprime (by the second part of Theorem <ref>). In particular, the first open case is m=4, r=3.
We remark that all the constructions in the proof of Theorem <ref> are based on a divisibility condition on a linear combination of the coordinates in . A different type of construction of an independent exact 2-cover in ^3 is to take all points in ^3 for which the 3 coordinates are either all even or all odd. Similarly, when m=2^l-1 for some l∈ we can construct an independent exact 2-cover by taking all points in ^m for which the coordinatewise reduction modulo 2 gives an element of a fixed Hamming code (the m=3 case just described corresponds to the code {000,111}).
As well as determining () for a particular sequence of graphs, we can ask is there a sequence of graphs for which () attains the maximum given by Theorem <ref>.
For all d∈ is there a sequence of d-regular graphs for which ()=D_d?
One way of constructing such a sequence would be to find a finite graph with the property that it contains an independent exact r-cover for all 0≤ r≤ d. If we can do this then this graph has colourings with P_2(R,B) achieving each of the extreme points of D_d. The sequence of d-regular graphs formed by taking an increasing number of disjoint copies of G will have ()=D_d.
A construction of such a graph for d=3 is as follows. Take
V={a_i : 1≤ i≤ 20}∪{b_i : 1≤ i≤ 20}
with the vertices a_1,a_2,…,a_20 and b_1,b_2,…,b_20 each forming a 20-cycle. We will add to this a matching between {a_i : 1≤ i≤ 20} and {b_i : 1≤ i≤ 20} in such a way that in the resulting 3-regular graph the set
S_1={a_4,a_8,a_12,a_16,a_20,b_4,b_8,b_12,b_16,b_20}
is an independent exact 1-cover and the set
S_2={a_1,a_3,a_6,a_8,a_11,a_13,a_16,a_18,b_1,b_3,b_6,b_8,b_11,b_13,b_16,b_18}
is an independent exact 2-cover. It suffices to take
F={(1,5),(2,12),(3,9),(4,18),(6,20),(7,17),(8,14),(10,16),(11,15),(13,19)}
and edges
{(a_i,b_j) : (i,j)∈ F}∪{(b_i,a_j) : (i,j)∈ F}.
Since the graph we have constructed is bipartite it also has an independent exact 3-cover given by one part of the bipartition.
It would be interesting to know if something like this can be done for d≥ 4.
For all d∈ is there a finite d-regular graph G which contains an independent exact r-cover for all 0≤ r≤ d?
Note that the density condition on independent exact r-covers forces some divisibilty conditions on the possible size of G. The number of vertices n in such a graph must be chosen so that rn/d+r is an integer for all 0≤ r≤ d.
Finally, a natural direction to consider is extending this work to t-step walks.
What are the possible values of P_t(R,B) for the torus T^m_k?
The analogue of Theorem <ref> still holds for t-step walks (with the same proof) and it is most natural to work asymptotically. Our construction based on independent exact r-covers can be analysed with respect to t-step walks. The construction asymptotically achieving the extreme point (2m-r/2m,(2m-r/2m)^2) for P_2(R,B) also asymptotically achieves (2m-r/2m,(2m-r/2m)^t) for P_t(R,B). However, we do not have an analogue of Theorem <ref> for t>2. One obstacle to proving such a result is that the method of counting 2-step walks using the squares of degrees does not generalise to t-step walks.
Is there a general bound analogous to Theorem <ref> for the possible values of P_t(R,B) in a d-regular graph?
abbrv
|
http://arxiv.org/abs/2307.05577v1 | 20230710131915 | Comment on "Nuclear Excitation by Free Muon Capture" | [
"Natalia S. Oreshkina",
"Julian C. Berengut"
] | nucl-th | [
"nucl-th",
"physics.atom-ph"
] |
Comment on “Nuclear Excitation by Free Muon Capture"
In the paper <cit.> the process of free muon capture with simultaneous excitation of a nuclear isomer has been suggested, claiming that “the effect can be detectable for selected isotopes”.
Here, we argue that this claim can not be confirmed.
Briefly, the process is far from the dominant mechanism for nuclear excitation; it excites high energy nuclear levels that will not generally decay to the isomer; the proposal assumes all incident muons will fulfil energy criteria, ignoring dominant capture paths; and nuclei excited by muons will have a shortened lifetime due to muonic capture.
Let us start by discussing an important technical point. As stressed in <cit.>, for a free muon to be captured to its ground or first excited state, it should have a well-defined energy close to the nuclear resonance. Coupled with the small size of the target orbital, such low energy, non-relativistic scattering will be dominated by the s-wave cross-section, however this is in conflict with angular momentum and parity selection rules for most of the considered transitions in Table I.
Instead these must originate from p or d-wave muons, where the rate will be suppressed. On the other hand, radiative and Auger capture via dominant channels are always available (see Fig. <ref>) and can involve any free muon energies and bound muon states.
After a muon is captured in a highly excited state with a statistically distributed angular momentum, the dominant process is cascade towards the ground state, first via Auger, then via radiative decay <cit.>. Due to the similar energy scale for muonic and nuclear states, there is strong mixing of muon-nucleus levels driven by hyperfine interaction. This so-called dynamical splitting was discussed in <cit.> and improved later in <cit.>. The correctness of the theoretical prediction was fully confirmed by the experiment <cit.>. As a result, all states of the muonic cascade include a superposition of a few low-lying nuclear states. Decays in the cascade occur spontaneously via photon emission, without the requirement of precise energy matching, and nuclear excitations are merely a side effect of this cascade in muonic atoms. This is a vastly different physical reality from that presented in <cit.>, where the muonic and nuclear degrees of freedom are considered highly separable.
A major drawback of the paper is that it compares the probability of nuclear excitation by muon capture () almost exclusively with that of nuclear excitation by electron capture (NEEC). These are two very different physical systems: NEEC occurs with no competing nuclear excitation mechanisms. On the other hand, in order to properly evaluate the experimental feasibility of the proposal, the process should be compared with other decay channels of the same system to establish the hierarchy. The dominant process for muonic atoms, namely excitation upon muon cascade, has not received enough attention in the paper <cit.>.
Even if one allows that the mechanism might occur in some systems as a sub-dominant effect, the nuclear levels that are excited by this method are relatively high-energy, and do not necessarily cascade to the metastable isomer. For instance, the suggested nuclear state of ^207Pb at 4980.5 keV is highly excited and the direct photo-excitation rate is low. However, the excited state is separated from the ground state by over 100 levels and it is not metastable, so it would uncontrollably decay in gamma cascade. Therefore, the process will not enable the preferential feeding of nuclear isomers.
Finally, excited nuclei produced by NEμC will generally be destroyed by nuclear muon capture, which is the dominant decay mechanism for muonic atoms with heavy nuclei <cit.>.
Overall, the paper <cit.> presents an interesting mechanism for manipulating nuclear states via interaction with a muon, but ignores the dominant mechanism of muon-nucleus interaction, namely the muonic dynamical-structure cascade, and gives a misleading impression that could be observed. Despite the idea's attractiveness, based on the points mentioned above, it is highly improbable that can be visible in an experiment.
N.S.O. thanks the Gordon Godfrey fund for the financial support of the visit to UNSW Sydney, Australia.
Natalia S. Oreshkina^1 and Julian C. Berengut^2
^1 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
^2 School of Physics, University of New South Wales, Sydney NSW 2052, Australia
|
http://arxiv.org/abs/2307.05573v1 | 20230710083608 | On the first bifurcation of Stokes waves | [
"Vladimir Kozlov"
] | math.AP | [
"math.AP"
] |
We consider Stokes water waves on the vorticity flow in a two-dimensional channel of finite depth. In the paper <cit.> it was proved existence of subharmonic bifurcations on a branch of Stokes waves. Such bifurcations occur near the first bifurcation in the set of Stokes waves. Moreover it is shown in that paper that the bifurcating solutions build a connected continuum containing large amplitude waves. This fact was proved under a certain assumption concerning the second eigenvalue of the Frechet derivative. In this paper we investigate this assumption and present explicit conditions when it is satisfied.
Optical-power-dependent splitting of magnetic resonance in nitrogen-vacancy centers in diamond
Kensuke Kobayashi
Received / Accepted
==============================================================================================
§ FORMULATION OF THE PROBLEM
Stokes and solitary waves were the main subject of study in the nonlinear water wave theory up to 1980.
In 1980 (see Chen <cit.> and Saffman <cit.>) it was discovered numerically and in 2000 (see <cit.>) this was supported theoretically for the ir-rotational case for a flow of infinite depth that there exist new types of periodic waves with several crests on the period (the Stokes wave has only one crest). These waves occur as a result of bifurcation on a branch of Stokes waves when they approach the wave of greatest amplitude.
In my papers <cit.> and <cit.> the existence of subharmonic bifurcations was proved on branches of Stokes waves on vorticity flow.
The main result in the latest paper <cit.> is proved under a certain assumption on the second eigenvalue of the Frechet derivative. The main goal of this paper is to study this assumption and to give an explicit conditions for its validity.
Consider steady surface waves in a two-dimensional channel bounded below by a flat,
rigid bottom and above by a free surface that does not touch the bottom. The surface tension is neglected and the water motion can be rotational.
In appropriate Cartesian coordinates (X, Y ), the bottom coincides with the
X-axis and gravity acts in the negative Y -direction. We choose the frame of reference so that the velocity field is time-independent as well as the free-surface profile
which is supposed to be the graph of Y = ξ(X), x ∈ R, where ξ is a positive and
continuous unknown function. Thus
𝒟=𝒟_ξ = {X∈ R, 0 <Y < ξ(X)}, 𝒮=𝒮_ξ={X∈ R, Y=ξ(X)}
is the water domain and the free surface respectively. We will use the stream function Ψ, which is connected with the velocity vector ( u, v) as u=-Ψ_Y and v=Ψ_X.
We assume that ξ is a positive, periodic function having period Λ>0 and that ξ is even and strongly monotonically decreasing on the interval (0,Λ/2). Since the surface tension is neglected, Ψ and
ξ after a certain scaling satisfy the following free-boundary problem (see for example <cit.>):
ΔΨ+ω(Ψ)=0 ,
1/2|∇Ψ|^2+ξ=R ,
Ψ=1 ,
Ψ=0 ,
where ω∈ C^1,α, α∈ (0,1), is a vorticity function and R is the Bernoulli constant. We assume that Ψ is even, Λ-periodic in X and
Ψ_Y>0 ,
which means that the flow is unidirectional.
The Frechet derivative for the problem is evaluated for example in <cit.>, <cit.>, and the corresponding eigenvalue problem for the Frechet derivative has the form
Δ w+ω'(Ψ)w+μ w=0 ,
∂_ν w-ρ w=0 ,
w=0 ,
where ν is the unite outward normal to Y=ξ(X) and
ρ=
ρ(X)=(1+Ψ_XΨ_XY+Ψ_YΨ_YY)/Ψ_Y(Ψ_X^2+Ψ_Y^2)^1/2|_Y=ξ(X).
The function w in (<ref>) is supposed also to be even and Λ-periodic.
Let us introduce several function spaces. Let α∈ (0,1) and k=0,1,…. The space C^k,α(𝒟) consists of bounded functions in 𝒟 such that the norms C^k,α(𝒟_a,a+1) are uniformly bounded with respect to a∈ R. Here
𝒟_a,a+1={(X,Y)∈𝒟, : a≤ x≤ a+1}.
The space C^k,α_0,Λ(𝒟) (C^k,α_0,Λ, e(𝒟)) consists of Λ-periodic (Λ-periodic and even) functions, which belong to C^k,α(𝒟) and vanish at Y=0.
Similarly we define the space C^k,α_Λ( R) (C^k,α_Λ, e( R)) consisting of functions in C^k,α( R), which are Λ-periodic (Λ-periodic and even).
We will consider a branch of Stokes water waves depending on a parameter t≥ 0, i.e.
ξ=ξ(X,t), ψ=ψ(X,Y;t), Λ=Λ(t).
For each t the functions ξ∈ C^2,α_Λ,e( R) and Ψ∈ C^3,α_Λ,e(𝒟). This branch starts from a uniform stream solution for t=0.
The dependence on t is analytic in the sense explained in Sect. <ref>. The definition of uniform stream solution together with the dispersion equation which is required for existence of the branch of the Stokes waves (<ref>) is given in the next section <ref>. Existence of such branches was a subject of many papers. In the case of non-zero vorticity we note a fundamental work <cit.>, where a bifurcation branches for the flow with vorticity was constructed for the first time. In the case with variable period we refer to the papers <cit.> and <cit.>.
The first (lowest eigenvalue of the problem (<ref>)] is always negative and simple and the second one we denote by μ(t).
Assume that
Assumption There exists t_0>0 such that μ(t)≥ 0 for t∈ (0,t_0) and μ(t)<0 for t∈ (t_0,t_0+ϵ) for a certain positive ϵ.
This assumption describes the first bifurcation point t_0 on the branch (<ref>) in the class of Stokes waves of period Λ(t). It is convenient to separate two types of bifurcations of branches of Stokes waves:
(i) in the class of Λ(t)-periodic solutions (Stokes bifurcation);
(ii) in the class of MΛ(t)-periodic solutions (M-subharmonic bifurcation);
Then the following theorem is proved in <cit.>.
Let Assumption be fulfilled. Then there exists an integer M_0 and pairs (t_M,M), where M is integer M>M_0 and t_M>t_0, satisfying
t_M→ t_0 M→∞,
such that t_M is M- subharmonic bifurcation point. There are no subharmonic bifurcations for t<t_0.
Moreover in Theorem 9.2, <cit.>, a structure of the set of bifurcating solutions is given. In particular it was shown that the bifurcating solutions build a connected continuum containing large amplitude waves.
The main aim of this paper is to give explicit conditions for validity of Assumption. Our analysis consists of two parts:
(i) analysis of behaviour of μ(t) for small t;
(ii) analysis of μ(t) for large positive t.
For t=0, Λ(0)=Λ_0 and μ(0)=0. Our first goal is to study the functions Λ(t) and μ(t) for small t. One of the results is the following. It's quite straightforward to show that these functions has the following asymptotic representations
μ (t)=μ_2t^2+0(t^3) Λ(t)=Λ_0+Λ_2t^2+O(t^3),
where Λ_0=Λ(0).
It is proved that
μ_2=CΛ_2
with a positive constant C to be evaluated later.
To prove formula (<ref>), first we study the function
λ(t)=Λ_0/Λ(t)=1+λ_2t^2+O(t^3)
and established the relation
-4λ_2τ_*^2∫_0^dγ(Y;τ_*)^2dY=μ_2∫_0^d γ(Y;τ_*)^2dY/Ψ_Y,
where γ(Y;τ) solves the problem (<ref>).
Since Λ_2=-λ_2Λ_0, the last relation implies (<ref>) with a positive constant C. Thus the sign of μ_2 is the same as of Λ_2 and opposite to the sign of λ_2.
In the irrotational case, i.e. ω=0, we study the dependence of μ_2 on the parameter θ>1 connected with the Froude number F=d_-^-3/2 by[It follows from (<ref>)]
θ=(1+√(1+8F^-2)/4)^3F^4=(F+√(F^2+8)/4)^3F,
where the right-hand side is monotone with respect to F.
We prove that
μ_2(θ)>0 θ_0≈ 2.48.
In terms of the Froude number the eigenvalue μ(t) is positive when
F<F_0, F_0≈ 1,511.
This give a condition for validity of the first part in Assumption.
Let us turn to the second part of the above assumption. It is enough to show an appearance of negative eigenvalues of the Frechet derivative when t→∞. According to Corollary 2.2, <cit.>, there exists a sequence
{t_j}, j=1,…, such that
a). ξ(0,t_j) tends to R when j→∞ (extreme wave) or
b). ξ(0,t_j) tends to a solitary wave as j→∞
In the case a) the limit configuration is the extreme wave with the angle 120^∘ at the crest (see <cit.>, <cit.>, <cit.> and <cit.>) and the appearance of negative eigenvalues follow from Theorem 3.1, <cit.> and <cit.>.
To show that the option b) is impossible we choose parameters of the problem such that solitary waves are excluded. We will do this by using known upper estimates for the Froude number of solitary waves.
The best known upper estimate for the Froude number of solitary wave, which follows from <cit.> (see also <cit.> and Introduction of <cit.>) is the following
F<√(2).
This means that if F>√(2) there are no solitary waves with such Froude number. Hence every global branch of Stokes waves must approach a Stokes waves of maximal amplitude which have the angle 120^∘ at the crest. According to Theorem 3.1 <cit.> this fact implies appearance of infinitely many negative eigenvalues of the Frechet derivative when t→∞. This implies the validity of the second part of Assumption. Therefore
1,414<F<1,511 .
Another upper estimate for the Froude number obtained numerically (see <cit.>, <cit.>, <cit.> and Introduction in <cit.>) is F<1,29. Hence
1,29<F<1,511 .
This estimate is supported now by numerics only, but we present it because it can be used for numerical study of subharmonic bifurcations.
§.§ Uniform stream solution, dispersion equation
The uniform stream solution Ψ=U(Y) with the constant depth η =d satisfies the problem
U^”+ω(U)=0 ,
U(0)=0, U(d)=1,
1/2U'(d)^2+d=R.
In order to find solutions to this problem we introduce a parameter s=U'(0). We assume that
s>s_0:=2max_τ∈ [0,1]Ω(τ), where
Ω(τ)=∫_0^τω(p)dp.
Then the problem (<ref>) has a solution (U,d) with a strongly monotone function U for
R=ℛ(s):=1/2s^2+d(s)-Ω(1).
The solution is given by
Y=∫_0^Udτ/√(s^2-2Ω(τ)), d=d(s)=∫_0^1dτ/√(s^2-2Ω(τ)).
If we consider (<ref>) as the equation with respect to s then it is solvable if R≥ R_c, where
R_c=min_s≥ s_0ℛ(s),
and it has two solutions if
R∈ (R_c,R_0), where
R_0=ℛ(s_0).
We denote by s_c the point where the minimum in (<ref>) is attained.
Existence of small amplitude Stokes waves is determined by the dispersion equation (see, for example, <cit.>). It is defined as follows.
The strong monotonicity of U guarantees that the problem
γ^”+ω'(U)γ-τ^2γ=0, γ(0,τ)=0, γ(d,τ)=1
has a unique solution γ=γ(y,τ) for each τ∈ R, which is even with respect to τ and depends analytically on τ.
Introduce the function
σ(τ)=κγ'(d,τ)-κ^-1+ω(1), κ=U'(d).
It depends also analytically on τ and it is strongly increasing with respect to τ>0. Moreover it is an even function.
The dispersion equation (see, for example <cit.>) is the following
σ(τ)=0.
It has a positive solution if
σ(0)<0.
By <cit.> this is equivalent to s+d'(s)<0 or what is the same
1<∫_0^ddY/U'^2(Y).
The right-hand side here is equal to 1/F^2 where F is the Froude number (see <cit.> and <cit.>). Therefore (<ref>) means that F<1, which is well-known condition for existence of Stokes waves of small amplitude.
Another equivalent formulation is given by requirement (see, for example <cit.>)
s∈ (s_0,s_c).
The existence of such s is guaranteed by R∈ (R_c,R_0). One more formula for the froude number is the following
1/F^2(s)=d'(s)/s,
where the Froude number F(s) corresponds to the uniform stream solution (U(Y;s),d(s)) and R=ℛ(s). One can verified directly from
(<ref>) that
(d'(s)/s)'>0.
Therefore
ℛ'(s)=s(1-F^-2(s))
and
1-F^-2(s)=1-d'(s)/s=(d'(s)/s)'(s_0-s)+O(s_0-s).
The value σ(0) admits the following representation (see [DispEqv]):
σ(0)=-3/2κℛ'(s)/d'(s)=3(F^2(s)-1)/2κ.
The function σ has the following asymptotic representation
σ(τ)=κτ +O(1)
and equation (<ref>) has a unique positive root, which will be denoted by τ_*. It is connected with Λ_0 by the relation
τ_*=2π/Λ_0.
To give another representation of the function σ we introduce
ρ_0=1+U'(d)U^”(d)/U'(d)^2
and note that
1+U'(d)U^”(d)/U'(d)^2=κ^-2-ω(1)/κ.
Hence another form for (<ref>) is
σ(τ)=κγ'(d,τ)-κρ_0.
The following problem will be used in asymptotic analysis of the branch (<ref>) for small t:
v^”+ω'(U)v-τ^2v=f ,
v'(d)-ρ_0v(d)=g v(0)=0.
Let τ≥ 0 and τ≠τ_*. Let also f∈ C^1,α([0,d]) and g be a constant. Then the problem
(<ref>) has a unique solution v∈ C^3,α.
If τ=τ_* then the problem (<ref>) has the one dimensional kernel which consists of function
cγ(Y;τ_*).
§ A CONNECTION BETWEEN THE FUNCTIONS Μ(T) AND Λ(T) FOR SMALL T
In this section we prove formula (<ref>). It appears that the partial hodograph transform is very useful for this purpose.
§.§ Partial hodograph transform
In what follows we will study branches of Stokes waves (Ψ(X,Y;t),ξ(X;t)) of period Λ(t), t≥ 0, started from the uniform stream at t=0. The existence of such branches is established in [ConStr] with fixed period but variable R and in [KL] for variable Λ and fixed R. In our case of variable Λ
it is convenient to make the following change of variables
x=λ X, y=Y, λ=Λ_0/Λ(t)
in order to deal with the problem with a fixed period. Here as before
Λ_0=Λ(0)=2π/τ_*,
where τ_* is the root of the equation (<ref>).
As the result we get
(λ^2∂_x^2+∂_y^2)ψ+ω(ψ)=0 ,
1/2(λ^2ψ_x^2+ψ_y^2)+η=R ,
ψ=1 ,
ψ=0 ,
where
ψ(x,y;t)=Ψ(λ^-1x,y;t) η(x;t)=ξ(λ^-1 x;t).
Here all functions have the same period Λ_0:=Λ(0), D_η and B_η are the domain and the free surface after the change of variables (<ref>).
From (<ref>) it follows that
ψ_y>0 .
Using the change of variables
q=x, p=ψ,
we get
q_x=1, q_y=0, p_x=ψ_x, p_y=ψ_y,
and
ψ_x=-h_q/h_p, ψ_y=1/h_p, dxdy=h_pdqdp.
System (<ref>) in the new variables takes the form
(1+λ^2h_q^2/2h_p^2+Ω(p))_p-λ^2(h_q/h_p)_q=0 ,
1+λ^2h_q^2/2h_p^2+h=R ,
h=0 .
Here
Q={(q,p) : q∈ R , p∈ (0,1)}.
The uniform stream solution corresponding to the solution U of (<ref>) is
H(p)=∫_0^pdτ/√(s^2-2Ω(τ)), s=U'(0)=H_p^-1(0).
One can check that
H_pp-H_p^3ω(p)=0
or equivalently
(1/2H_p^2)_p+ω(p)=0.
Moreover it satisfies the boundary conditions
1/2H_p^2(1)+H(1)=R, H(0)=0.
The Froude number in new variables can be written as
1/F^2=∫_0^1H_p^3dp.
Then according to Theorem 2.1, <cit.> there exists a branch of solutions to (<ref>)
h=h(q,p;t):[0,∞)→ C^2,γ_pe(Q), λ=λ(t):[0,∞)→ (0,∞),
which has a real analytic reparametrization locally around each t≥ 0.
§.§ Bifurcation equation
In order to find bifurcation points and bifuracating solutions we put h+w instead of h in (<ref>) and introduce the operators
ℱ(w;t)=(1+λ^2(h_q+w_q)^2/2(h_p+w_p)^2)_p
-(1+λ^2h_q^2/2h_p^2)_p
-λ^2(h_q+w_q/h_p+w_p)_q+λ^2(h_q/h_p)_q
and
𝒢(w;t)=1+λ^2(h_q+w_q)^2/2(h_p+w_p)^2-1+λ^2h_q^2/2h_p^2+w
acting on Λ_0-periodic, even functions w defined in Q. After some cancelations we get
ℱ=𝒥_p+ℐ_q, 𝒢=𝒥+w,
where
𝒥=𝒥(w;t)=λ^2h_p^2(2h_q+w_q)w_q-(2h_p+w_p)(1+λ^2h_q^2)w_p/2h_p^2(h_p+w_p)^2
and
ℐ=ℐ(w;t)=-λ^2h_pw_q-h_qw_p/h_p(h_p+w_p).
Both these functions are well defined for small w_p.
Then the problem for finding solutions close to h is the following
ℱ(w;t)=0
𝒢(w;t)=0
w=0 .
Furthermore, the Frechet derivative (the linear approximation of the functions ℱ and 𝒢) is the following
Aw=A(t)w=(λ^2h_qw_q/h_p^2-(1+λ^2h_q^2)w_p/h_p^3)_p-λ^2(w_q/h_p-h_qw_p/h_p^2)_q
and
𝒩w=𝒩(t)w=(N w-w)|_p=1,
where
N w=N(t)w=(-λ^2h_qw_q/h_p^2+(1+λ^2h_q^2)w_p/h_p^3)|_p=1.
The eigenvalue problem for the Frechet derivative, which is important for the analysis of bifurcations of the problem
(<ref>), is the following
A(t)w=μ w ,
𝒩(t)w=0 ,
w=0 .
For t=0 and μ=0 this problem becomes
A_0w:=-(w_p/H_p^3)_p-(w_q/H_p)_q=0 ,
B_0w:=-w_p/H_p^3+w=0 ,
w=0 .
Since the function H depends only on p this problem admits the separation of variables and its solutions are among the functions
v(q,p)=α(p)cos (τ q), τ=kτ_*, k=0,1,….
According to <cit.> the function (<ref>) solves (<ref>) if and only if
α(p)=γ(H(p);τ)H_p,
where the function γ(Y;τ) solves the euation (<ref>) and σ(τ)=0. Therefore if τ≠τ_* then the problem (<ref>) has no non-trivial solutions. If τ=τ_* then
the kernel of the above operator is one dimensional in the class of Λ_*:=2π/τ_* periodic, even function and it is given by
v=α(p)cos(τ_*q), α(p)=γ(H(p);τ_*)H_p.
We will need also the problem
-(u_p/H_p^3)_p+τ^2u/H_p=F
u(0)=0, -u_p/H_p^3+u=c ,
where F∈ C^0,α([0,1]) and c is a constant.
Clearly this problem is
elliptic and uniquely solvable for all τ≥ 0, τ≠τ_*, the problem (<ref>) has a unique solution in C^2,α([0,1]). This solution is given by
u(p)=v(H(p))H_p(p),
where v(Y) solves the problem (<ref>) with f=F(H(y)) and g=c.
§.§ Stokes waves for small t
Here we consider asymptotics of solutions of (<ref>) for small t. For this purpose we take
h=H(p)
and represent the solution in the form
H(p)+w(q,p,t), w=tv,
where
v(q,p;t)=v_0(q,p)+tv_1(q,p)+t^2v_2(q,p)+⋯
The function λ=λ(t) is sought in the form
λ(t)=1+λ_2t^2+O(t^4).
The coefficients λ_1 and λ_3 in the above formula are zero as one can easily see from the forthcoming calculations.
Our aim is to find Stokes waves close to H. Since the functions w, v and λ analytically depend on t it is sufficient to find coefficients v_j and λ_j.
In this case
𝒥=A_1(1+w_p/H_p)^-2+A_2(1+w_p/H_p)^-2,
where
A_1=-w_p/H_p^3
and
A_2=λ^2w_q^2/2H_p^2-w_p^2/2H_p^4.
Therefore
𝒥=𝒥_1+𝒥_2+𝒥_3+O(t^4),
where
𝒥_1=A_1,
𝒥_2=A_2-2w_p/H_pA_1=λ^2w_q^2/2H_p^2+3/2w_p^2/H_p^4
and
𝒥_3=3w_p^2/H_p^2A_1-2w_p/H_pA_2=-2w_p^3/H_p^5-w_pw_q^2/H_p^3.
Furthermore
ℐ=-λ^2w_q/H_p(1+w_p/H_p)^-1
=ℐ_1+ℐ_2+ℐ_3+O(t^4).
Here
ℐ_1=-λ^2w_q/H_p, ℐ_2=λ^2w_qw_p/H^2_p, ℐ_3(w)=-λ^2w_qw_p^2/H^3_p.
Inserting (<ref>) and (<ref>) into (<ref>) and
equating terms of the same power with respect to t, we get
Av_0:=-(v_0p/H_p^3)_p-(v_0q/H_p)_q=0 ,
Bv_0:=-v_0p/H_p^3+v_0=0 ,
v_0=0 .
As we have shown in previous section the kernel of the above operator is one dimensional and is generated by the function
v_0=α_0(p)cos(τ_*q), α_0=γ(H(p);τ_*)H_p.
The next term in the asymptotics satisfies the boundary value problem
Av_1+(v_0q^2/2H_p^2+3/2v_0p^2/H_p^4)_p+(v_0qv_0p/H^2_p)_q=0 ,
Bv_1+v_0q^2/2H_p^2+3/2v_0p^2/H_p^4=0 ,
v_1=0 .
The solution of this problem, orthogonal to v_0 in L^2, is given by
v_1=α_1(p)+β_1(p)cos(2τ_* q),
where α_1 and β_1 satisfy the problem (<ref>9 with τ=0 and τ=2τ_* respectively with certain right-hand sides.
Further, the term v_2 is fond from the following problem
Av_2+(v_0qv_1q/H_p^2+3v_0pv_1p/H_p^4+𝒥_3(v_0))_p
+(v_1qv_0p+v_0qv_1p/H^2_p+ℐ_3(v_0))_q=2λ_2(v_0q/H_p)_q ,
Bv_2+v_0qv_1q/H_p^2+3v_0pv_1p/H_p^4+𝒥_3(v_0)=0
v_2(q,0)=0.
The solvability condition for the last problem has the form
2λ_2∫_Ωv_0q^2/H_pdqdp-∫_Ω((v_0qv_1q/H_p^2+3v_0pv_1q/H_p^4)v_0p+v_0qv_1p+v_1qv_0p/H^2_pv_0q)dqdp
+∫_Ω((2v_0p^3/H_p^5+v_0pv_0q^2/H_p^3)v_0p+v_0p^2v_0q/H_p^3v_0q)dqdp=0.
This relation can be used to find λ_2. It is quite difficult to find the sign of λ_2 from this relation but it implies a continuity of λ_2 on R and ω. The function v_2 has the form
v_2=α_2(p)cos(τ_* q)+β_2(p)cos(3τ_* q),
where α_2 and β_2 satisfy the problem (<ref>) with τ=τ_* and τ=3τ_* respectively with certain right-hand sides.
Thus we have shown that λ and v have the form (<ref>) and (<ref>) respectively. More exactly v_0 is given by (<ref>), v_1 is represented as (<ref>) and v_2 by (<ref>).
§.§ Formula for λ_2 and the proof of the relation (<ref>)
Using the representation (<ref>), (<ref>) with h=H+w, where w is evaluated in the previous section, we can write the Frechet derivative of the operators 𝒥 U and ℐ U in the form
d𝒥(U)=-U_p/H_p^3+(w_qU_q/H_p^2+3w_pU_p/H_p^4)-6w_p^2U_p/H_p^5
-w_q^2U_p+2w_pw_qU_q/H_p^3+O(t^3)
and
dℐ(U)=-λ^2U_q/H_p+w_pU_q+w_qU_p/H_p^2-w_p^2U_q+2w_qw_pU_p/H_p^3+O(t^3).
The eigenvalue problem is described by the boundary value problem
(d𝒥(U))_p+(dℐ(U))_q=(μ_2t^2+O(t^3))U
d𝒥(U)+U=0
U=0
We are looking for the eigenfunction U in the form
U=U(q,;pt)=U_0(q,p)+tU_1(q,p)+t^2U_2(q,p)+O(t^3), U_0=v_0.
Equating terms of the same order with respect to t, we get
AU_1+(v_0qU_0q/H_p^2+3v_0pU_0p/H_p^4)_p+(w_0pU_0q+v_0qU_0p/H_p^2)_q=0 ,
BU_1+(v_0qU_0q/H_p^2+3v_0pU_0p/H_p^4)=0 ,
U_1=0 .
Comparing this problem with (<ref>) and using that U_0=v_0, we conclude that U_1=2v_1.
Next, we write the equation for U_2
-2λ_2(U_0q/H_p)_q+AU_2+(v_1qU_0q+v_0qU_1q/H_p^2+3v_1pU_0p+v_0pU_1p/H_p^4)_p
+(v_1pU_0q+v_1qU_0p+v_0pU_1q+v_0qU_1p/H_p^2)_q
-(6v_0p^2U_0p/H_p^5
+v_0q^2U_0p+2v_0pv_0qU_0q/H_p^3)_p-(v_0p^2U_0q+2v_0qv_0pU_0p/H_p^3)_q=μ_2U_0
and the boundary equations U_2=0 for p=0 and
BU_2+(v_1qU_0q+v_0qU_1q/H_p^2+3v_1pU_0p+v_0pU_1p/H_p^4)
-(6v_0p^2U_0p/H_p^5+v_0q^2U_0p+2v_0pv_0qU_0q/H_p^3)=0
Since U_0=v_0 and U_1=2v_1, the solvability condition for (<ref>) has the form
2λ_2∫_Q_pv_0q^2/H_pdqdp-3∫_Q_p(v_1qv_0q/H_p^2+3v_0pv_1q/H_p^4)v_0pdqdp
-3∫_Q_pv_1pv_0q+v_1qv_0p/H_p^2v_0qdqdp+3(∫_Q_p(2v_0p^3/H_p^5+v_0q^2v_0p/H_p^3)v_0p+v_0p^2v_0q^2/H_p^3)dqdp
=μ_2∫_Q_p v_0^2dqdp.
Taking the sum of (<ref>) and (<ref>) with the factor -3, we get
-4λ_2∫_Ωv_0q^2/H_pdqdp=μ_2∫_Ω v_0^2dqdp,
which coincides with (<ref>).
§ THE COEFFICIENT Λ_2 FOR THE IRROTATIONAL FLOW
In this section we evaluate the coefficient λ_2
in the case ω=0. The problem (<ref>) is solvable if R≥ R_c, where R_c=3/2. If R>R_c then the equation
1/d^2+2d=2R
has exactly two solutions 0<d_-<1<d_+ which are called supercritical and subcritical, respectively. The Stokes branches appears only for the stream solutions- (Y/d_+,d_+).
We will make the following change of variables
X=x/d_+, Y=y/d_+-1, ξ(X)=η(x)/d_+-1, Ψ(X,Y)=ψ(x,y).
Then the problem (<ref>) takes the form
Δ_x,yψ=0 ,
|∇_x,yψ|^2+2θη=1 ,
ψ=1 ,
ψ=0 ,
where
θ=d_+^3.
So θ is the only parameter in the problem and θ∈ (1,∞).
In the irrotational case one can derive an explicit equation for λ_2(θ). This derivation is based on the application of the integral Byatt-Smith equation, see <cit.>.
§.§ Hodograph transformation
Let x + iy→ϕ + iψ be a conformal mapping of
D={(x,y) : x∈ R, -1<y<η(x)}
onto R × (0, 1).
Now we apply the hodograph transform, that is, use the imaginary part y(ϕ,ψ)
of the inverse conformal mapping as the unknown function instead of the stream
function ψ and the potential ϕ. From problem (<ref>) we get the following one:
y_ϕϕ+y_ψψ=0, (ϕ,ψ)∈ R×(0,1);
y=-1, ψ=0,ϕ∈ R;
y=η, ψ=1,ϕ∈ R;
(y_ϕ^2+y_ψ^2)^-1+2θ y=1, ψ=1, ϕ∈ R.
Let us eliminate y in order to obtain an equation that contains only η. It is clear
that relations (24) and (25) yield
y_ψ(ϕ,1)=(1/1-2θη(φ)-η_ϕ^2(ϕ))^1/2
Here and below we write η(ϕ) instead of η(x(ϕ, 1)) and hope that this will not cause
confusion. The Dirichlet-to-Neumann operator in the left-hand side of formula (26)
can be expressed by virtue of the Fourier transform
y(τ,ψ)=∫_-∞^∞ y(ϕ,ψ)e^iτϕdϕ.
In order to solve
the Dirichlet problem (22)–(24) we define the operator N by
Nf(ξ)=ν(ξ) f(ξ), ν(ξ)=ξξ .
The important property of this operator is
N(cos(τϕ))=ν(τ)cos(τϕ) .
Let
ℱ(u,v)=v^2H_1(u,v)-u^2H_0(u),
where
H_0(u)=2θ^2[2+S(u)]/S(u)[1+S(u)]^2, H_1(u,v)=S(u)/1+√(1-v^2S^2(u))
and S(u)=√(1-2θ u). Then equation for η=η(ϕ) has the form
(θ I-N)η=ℱ(η,η_ϕ).
Here I is the identity operator. This equation coincides with that of Byatt-Smith <cit.> up to some algebraic manipulations. It also used in
<cit.> and <cit.>, where various properties of this equation can be found.
Equation (<ref>) is valid for all solutions with arbitrary period. To fix period we make the change of the variable
φ=λϕ, λ=Λ_0/Λ.
Then equation (<ref>) becomes
(θ I-λ N)η=ℱ(η,λη_ϕ).
We are looking for a solution to (<ref>) in the form
η(φ)=t(η_1+tη_2+t^2η_3+…)
=t(cos(τ_*φ)+t(a_0+a_1cos(2τ_*φ))+t^2(a_2cos(τ_*φ)+...)+⋯)
and
λ=1+λ_2t^2+⋯,
where τ_* is the root of the equation
ττ =θ.
Using that
H_0(u)=θ^2/2(3+5θ u)+O(u^2) H_1(u,v)=1/2(1-θ u)+O(u^2+v^2)
we can solve (<ref>) asymptotically
(θ I- N)η_2=1/2η_1φ^2-3θ^2/2η_1^2
and
(θ I-N)η_3-λ_2Nη_1=1/22η_1φη_2φ-3θ^2/22η_1η_2-
θ^2/25θη_1^3-1/2θη_1η_1φ^2.
From (<ref>) it follows
(θ-1)a_0=τ_*^2-3θ^2/4, (ν(2τ_*)-θ)a_1=3θ^2+τ_*^2/4.
Using the relations
cos Acos B=1/2(cos(A+B)+cos(A-B)), sin Asin B=1/2(cos(A-B)-cos(A+B)),
and
equating in (<ref>) coefficients in cos(τ_*φ), we obtain
-λ_2ν(τ_*)=τ_*^2a_1-3θ^2a_0-3θ^2/2a_1-15θ^3/8-τ_*^2θ/8=:f(θ).
§.§ Sign of λ_2
Since ν(τ_*)=θ we have τ_*<θ. One can check that the function ν(ξ) is convex and hence
θ=ν(τ_*)<1/2(1+ν(2τ_*)) θ-1<ν(2τ_*)-θ.
In the case θ≫ 1 we have
τ_*≈θ, a_0≈-θ/2, a_1≈θ, ν(τ_*)=θ, ν(2τ_*)≈ 2τ_*,
and hence
λ_2≈θ^2.
If we assume that θ=1+ϵ where ϵ is a small positive number then we get
ν(τ)=1+τ^2/2+⋯, τ_*=√(2ϵ), ν(2τ_*)=1+4ϵ,
a_0=-3/4ϵ, a_1=3/16ϵ
and
λ_2=-9/4ϵ7/8.
Evaluating the root θ_0 of the equation f(θ)=0 we get
θ_0≈ 2.479.
Therefore if θ∈ (1,θ_0 then
Λ_2>0.
According to (<ref>) and (<ref>), we conclude that
μ_2>0 .
§.§ Upper estimates of the Froude number
The following relation connected d_-and d_+ can be found in Sect. 2.1, <cit.> (see the formula (14) there):
d_+/d_-=1+√(1+8d_-^3)/4d_-^3,
which implies
d_+=1+√(1+8d_-^3)/4d_-^2.
A necessary condition for existence of solitary wave is the lower estimate F>1, Therefore the depth d=d_- corresponds to solitary waves and the corresponding Froude number is
F=d_-^-3/2.
The best known upper estimate for the Froude number can be derived from <cit.> as it is explained in Introduction of <cit.> and it is given by (<ref>).
Since the function
x→1+√(1+8x^3)/4x^2
is strongly decreasing we conclude that the condition F^2=d_-^-3>2 which implies non-existence of solitary waves, is equivalent to θ=d_+^3>1,745.
Another numerical estimate F<1,29 is obtained in <cit.>. Both these estimates together with (<ref>) lead to relations (<ref>) and (<ref>).
§.§ On the validity of Assumption A
As before we assume here that ω=0. Consider the branch (<ref>) of Stokes waves which starts from a uniform stream solution. According to <cit.> the limit behaviour of this branch is reduced to one of the following options: the branch approches a solitary wave or it approches an extreme wave. If we assume that F>√(2) then the first option is impossible due to the estimate (<ref>). Therefore in this case the bransh is approaching an extreme wave, which has the angle 120^∘ at the crest. By <cit.> and Theorem 3.1, <cit.>
the number of negative eigenvalues of the Frechet derivative becomes more and more when t approaches infinity. As a result we arrive at (<ref>). Similarly if the numerical estimate F<1,29 is excepted then we arrive at the interval (<ref>) where the Assumption A is valid. Certainly both conditions (<ref>) and (<ref>) are sufficient for the validity of the Assumption and this problem requires further research.
§ ACKNOWLEDGMENTS
I want to thank M. Wheeler for fruitful discussions on the estimates of the Froude number.
§ REFERENCES
20
Am J Amick, Bounds for water waves,
rch. Ration. Mech. Anal., 99, pp. 91–114 1987.
T2 CJ Amick, LE Fraenkel, JF Toland, On the Stokes conjecture for the wave of extreme form,
Acta Mathematica 148 (1), 1982.
BS J.G.B. Byatt-Smith, An exact integral equation for steady surface waves, Proc. Roy. Soc. Lond. A 315 (1970) 405–418.
BDT1 B Buffoni, EN Dancer, JF Toland, The Regularity and Local Bifurcation of Steady Periodic Water Waves,
Archive for rational mechanics and analysis 152 (3), 207-240, 2000.
BDT2 B Buffoni, EN Dancer, JF Toland, The sub-harmonic bifurcation of Stokes waves,
Archive for rational mechanics and analysis 152 (3), 241-271, 2000.
Che Chen, B. and Saffman, P.G. Numerical evidence for the existence of new types of gravity
waves on deep water. Stud. Appl. Math. 62, 1980.
CSst A Constantin, W Strauss, Exact steady periodic water waves with vorticity,
Communications on Pure and Applied Mathematics 57 (4), 481-527, 2004.
HVB83 J. K. Hunter and Jean-Marc Vanden-Broeck. Accurate computations for steep solitary
waves. Journal of fluid Mechanics, 136:63–71, 1983.
KP74 G. Keady and W. G. Pritchard. Bounds for surface solitary waves. Proc. Cambridge Philos. Soc.,
76:345–358, 1974.
Koz1 V. Kozlov, The subharmonic bifurcation of Stokes waves on vorticity flow, JDE, 2023, arXiv:2204.10699.
Koz1a V.Kozlov, On first subharmonic bifurcations in a branch of Stokes waves, arXiv:2303.11440, 2023.
KN2008 V Kozlov, N Kuznetsov, On behaviour of free-surface profiles for bounded steady water waves,
Journal de mathématiques pures et appliquées 90 (1), 1-14, 2008.
KN14 V Kozlov, N Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents,
Archive for Rational Mechanics and Analysis 214 (3), 971-1018, 2014.
KN11a V Kozlov, N Kuznetsov, The Benjamin–Lighthill conjecture for near-critical values of Bernoulli’s constant,
Archive for rational mechanics and analysis 197, 433-488, 2010.
KL1 V Kozlov, E Lokharu, Global bifurcation and highest waves on water of finite depth,
arXiv preprint arXiv:2010.14156, 2020.
KL2 V Kozlov, E Lokharu, On negative eigenvalues of the spectral problem for water waves of highest amplitude,
Journal of Differential Equations, 342, 239-281, 2023.
KL3 V Kozlov, E Lokharu, On Rotational Waves of Limit Amplitude,
Functional Analysis and Its Applications 55 (2), 165-169, 2021.
KLW V Kozlov, E Lokharu, MH Wheeler, Nonexistence of subcritical solitary waves,
Archive for Rational Mechanics and Analysis, 241 (1), 535-552, 2021.
LHF74 M. S. Longuet-Higgins and J. D. Fenton. On the mass, momentum, energy and circulation
of a solitary wave. II. Proc. Roy. Soc. (London) Ser. A, 340:471–493, 1974.
McL J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Studies in Applied
Mathematics, 98 (1997), pp. 311-333.
Mil80 John W. Miles. Solitary waves. In Annual review of fluid mechanics, Vol. 12, pages
11–43. Annual Reviews, Palo Alto, Calif., 1980.
P2 PI Plotnikov, A proof of the Stokes conjecture in the theory of surface waves,
Studies in Applied Mathematics, 108 (2), 2002.
Sa Saffman, P.G. Long wavelength bifurcation of gravity waves on deep water J. Fluid Mech.
101, 1980.
Star Victor P. Starr. Momentum and energy integrals for gravity waves of finite height. J. Mar. Res., 6:175–
193, 1947.
arXiv preprint arXiv:2204.10071.
VW1 E Varvaruca, GS Weiss, A geometric approach to generalized Stokes conjectures,
Acta mathematica 206 (2), 363-403, 2011.
We M. Wheeler, The Froude number for solitary water waves with vorticity,
Journal of Fluid Mechanics 768, 91-112, 2015.
§.§ Small τ_*
Here we assume that 0<t≪τ_*≪ 1. The the relation (<ref>) for finding v_1 has the form
Av_1+3/2(v_0p^2/H_p^4)_p=O(τ_*^2) ,
Bv_1+3/2v_0p^2/H_p^4=O(τ_*^2) ,
v_1=0 .
We are looking for the solution in the form
v_1(q,p)=a_1(p)cos^2(τ_*q).
Then a_1 satisfies the equation
a_1p=3/2α_0p^2/H_p+H_p^3c_1,
where c_1 is a constant. Integrating this relation from 0 to p we get
a_1(p)=∫_0^p(3/2α_0p^2/H_s+H_s^3c_1)ds.
From the boundry condition for p=1 we get
a_1(1)-c_1=0.
Therefore
c_1(1-∫_0^1H_s^3ds)=∫_0^13/2α_0p^2/H_sds.
Since F<1 and the left hand side is equal to c_1(1-F^-2) the coefficient c_1 is negative and
c_1=(1-F^-2)^-1∫_0^13/2α_0p^2/H_sds.
Now we turn to the next term v_2. The problem for v_2 is the following
Av_2+(3v_0pv_1p/H_p^4+𝒥_3(v_0))_p=2λ_2(v_0q/H_p)_q+O(τ_*^2) ,
Bv_2+3v_0pv_1p/H_p^4+𝒥_3(v_0)=O(τ_*^2)
v_2(q,0)=0.
It is solvable if
2λ_2τ_*^2∫_Q_pv_0^2/H_pdqdp=∫_Q_p(3v_0pv_1p/H_p^4+𝒥_3(v_0))v_0pdqdp.
Since ??? we have
2λ_2τ_*^2∫_0^1α_0^2/H_pdp=-c_1∫_0^13α_0p^2/H_pdp+O(1).
|
http://arxiv.org/abs/2307.05917v1 | 20230712050541 | Early dark energy induced by non-linear electrodynamics | [
"H. B. Benaoum",
"Luz Ángela García",
"Leonardo Castañeda"
] | gr-qc | [
"gr-qc",
"astro-ph.CO",
"hep-ph",
"hep-th"
] |
Early dark energy as non-linear electrodynamics
e1e-mail: [email protected]
e2e-mail: [email protected]
e3e-mail: [email protected]
Department of Applied Physics and Astronomy, University of Sharjah, United Arab Emirates
Universidad ECCI, Cra. 19 No. 49-20, Bogotá, Colombia, Código Postal 111311
Observatorio Astronómico Nacional, Universidad Nacional de Colombia
Early dark energy induced by non-linear electrodynamics
H. B. Benaoume1,addr1
Luz Ángela Garcíae2,addr2
Leonardo Castañedae3,addr3
Received: date / Accepted: date
===============================================================================================
In this work, we introduce a parametrization of early dark energy that mimics radiation at early times and governs the present acceleration of the Universe. We show that such parametrization models non-linear electrodynamics in the early Universe and investigate the cosmological viability of the model. In our scenario, the early dark energy is encoded in the non-linearity of the electromagnetic fields through a parameter β that changes the Lagrangian of the system, and the parameters γ_s and α, that define the departure from the standard model constant equation of state. We use a Bayesian method and the modular software CosmoSIS to find the best values for the model's free parameters with precomputed likelihoods from Planck 2018, primordial nucleosynthesis data, inferred distances from different wide galaxy surveys and luminosity distances of SNIa from Pantheon and SH0ES, such that γ_s = 0.468 ± 0.026 and α = -0.947 ± 0.032, as opposed to ΛCDM where γ_s = β = 0 and there is no equivalence for the α parameter. Our results predict an earlier formation of the structure and a shorter age of the Universe compared with the canonical cosmological model. One of the main findings of our work is that this kind of dark energy alleviates the ongoing tensions in cosmology, the Hubble tension and the so-called σ_8 tension, which predicted values by our model are H_o = 70.2 ± 0.9 km/s/Mpc and σ_8 = 0.798 ± 0.007. The reported values lie between the inferred values inferred from early and late (local) Universe observations. Future observations will shed light on the nature of the dark energy, its impact on the structure formation, and its dynamics.
§ INTRODUCTION
Despite its simplicity, the standard ΛCDM model has successfully explained many cosmological observations <cit.>. However, with the improvement of the observational data, significant tensions between ΛCDM and some data might indicate new physics beyond the ΛCDM concordance model.
Based on canonical cosmic model, the observed present value of the Hubble parameter is H_o = 67.4 ± 0.5 km/s/Mpc 68% CL <cit.> from the Planck 2018 Cosmic Microwave Background (CMB) which has 5 σ tension discrepancy with a model-independent local measurement H_o = 73.04 ± 1.04 km/s/Mpc at 68% CL <cit.> from the Hubble Space Telescope (HST) observations of variable Cepheids. In addition to the observed present value of the Hubble parameter H_o disagreement, a particular important tension, related to the amplitude of the density fluctuations quantified in terms of the parameter S_8, between CMB data and the cosmic shear and redshift space distortions (RSD). A lower S_8 value ranging between 0.703 and 0.782 have been inferred from redshift surveys data and weak lensing measurements compared to the one estimated from Planck data, S_8 =0.834 ± 0.016 <cit.>. From ACT+WMAP analysis, S_8 = 0.840 ± 0.30 <cit.>, assuming ΛCDM model. Although the S_8 tension could be related to systematic errors, it might hint at a possible new physics beyond the standard ΛCDM model. Many possible alternative new physics solutions have been proposed to solve these tensions, which require modifications in the early Universe into the pre-recombination era and the late history of the Universe.
Early dark energy models have caught great interest in the community in the past few years. Early-time solutions aim to suppress the value of the sound horizon by injecting a fraction of energy before recombination without spoiling the fit to the CMB and Baryon Acoustic Oscillations (BAO) data. In particular, if 10% of the total energy density is allowed in the Hubble parameter during the radiation era (3500 < z < 5000), the Hubble parameter prediction from the early and late Universe significantly drops <cit.>. Thus, as alternatives to the cosmological constant Λ, early dark energy models are a compelling method to treat the ongoing tensions and introduce dynamics in stages when dark energy has been assumed to not play a role in the Universe. There are currently different models to describe early dark energy, among them: <cit.> introduce additional degrees of freedom in Hubble parameter through sterile neutrinos; a modified Chaplygin gas that simultaneously describes the dynamics of dark matter and dark energy <cit.>; effective parameterizations that evolve from a non-negligible contribution during the radiation-domination epoch to the current accelerated expansion of the Universe <cit.>. An extensive summary of models that compete with the cosmological constant Λ effect in the Universe´s current expansion can be found in <cit.>. Another alternative to model dark energy is non-linear electrodynamics, replacing the original Maxwell Lagrangian with a non-linear electrodynamics (NLED) Lagrangian.
Non-linear electrodynamics <cit.> is a generalization of Maxwell's electromagnetic theory and, when coupled to gravity, produces a negative pressure that tends to accelerate the expansion of the Universe at early and/or late stages. In recent years, non-linear electrodynamics has been the object of a significant amount of interest in cosmology and astrophysics <cit.>.
The outline of the paper is as follows. In section <ref>, we propose a more general parametrization of an early dark energy model that mimics radiation at early times and has an accelerated expansion at late times. Such parametrization incorporates a possible transition of the equation of state during the evolution of the Universe. In section <ref>, we present our phenomenological model based on non-linear electrodynamics that links the non-linearity of the electromagnetic fields to the early dark energy. To our knowledge, this work is the first to address the generation of early dark energy from the non-linearity of electromagnetic fields. In section <ref>, we extensively discuss the method to find the best values for the model's free parameters and present the results with Bayesian statistical inference. Section <ref> shows some cosmological tests we submit our model with the best-fits calculated in the previous section. Finally, we summarize our findings and conclusions in section <ref>.
§ PARAMETRIZATION OF EARLY DARK ENERGY
We propose a novel parametrization of dark energy. It is particularly suited to describe a small but non-negligible amount of dark energy at early stages and an accelerated expansion in the late phases of the Universe. In this section, we use the modified Chaplygin gas model <cit.> approach to parametrize the dark energy and investigate the effect of adding a small fraction of dark energy in the Universe's early evolution.
The Chaplygin gas was first introduced by S. Chaplygin <cit.> at the beginning of the 20th century to describe the lifting force experienced by the winds of an airplane when it is in the air. The gas creates a negative pressure, generating an effective opposite effect to the gravitational force.The idea is brought back in the early 2000s with the advent of dark energy by <cit.>. The author presented a model that unifies dark matter and energy in a generalized Chaplygin gas. Later on, <cit.> presented a modified Chaplygin gas that is particularly suited to describe a small but non-negligible amount of dark energy at early stages and an accelerated expansion at late phases of the Universe.
Our model is characterized by three parameters: the present-day value of dark energy density ρ_de,o≃ 1.7 × 10^-119 M_p^4 where M_p = 2.435 × 10^18GeV is the reduced Planck mass, the transition scale factor and a third parameter that controls the rapidity of this transition. In this scenario, the dark energy mimics radiation in the early Universe and accelerates its expansion at late times.
To motivate our investigation, we consider a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry described by the line element,
d s^2 = - dt^2 + a^2 (t) (dx^2 + dy^2 + dz^2 ),
where a (t) is the expansion scale factor. The expansion dynamics are governed by Friedmann's equations,
H^2 = 8 π G/3 ρ,
Ḣ = - 4 π G ( ρ + p ).
Here H = ȧ/a is the Hubble rate parameter, ρ and p are the total energy and pressure of the fluid species filling the Universe. The total cosmic fluid obeys the continuity equation,
ρ̇ + 3 H ( ρ + p ) = 0.
Throughout the present work, we assume that the Universe is composed of radiation, pressure-less dark matter, and dark energy. The total energy density and the total pressure are just the sum of the contributions of all species,
ρ = ρ_r + ρ_m + ρ_de,
p = p_r + p_de,
where the sub-index r,m and de stands for radiation, matter and dark energy. The energy densities of the radiation and matter are given by:
ρ_r = ρ_r,o a^-4
ρ_m = ρ_m,o a^-3
The dark energy is assumed to evolve independently, and its energy density is expressed by the continuity equation:
ρ̇_de + 3 H (ρ_de + p_de) = 0.
From this, it is straightforward to determine the dark energy equation of state (EoS) ω_de = p_de/ρ_de as:
ω_de = -1 - a/3d lnρ_de/d a.
It is possible to express the early dark energy ρ_de (a) as:
ρ_de (a) = ρ_de,o G (a)^1/1+α
with α > -1. The term ρ_de,0 is the energy density at present time (i.e., a=1) and G (a) is a generic function which we parametrize as having the following functional form,
G (a) = 1 - γ_s + γ_s a^- 4 (1+α).
Note that if γ_s = 0, G (a) = 1, and the model reduces to the standard ΛCDM model. This condition defines the lower limit for γ_s prior.
The solution to the dark energy EoS (<ref>) can be written in terms of the scale factor as:
ω_de = -1 + 4/31/1+ 1-γ_s/γ_s a^4 (1+ α)
Equations (<ref>), (<ref>) and (<ref>) yield several qualitative and interesting insights into the behavior of ρ_de and ω_de[It is worth noting that the parametrization we present in this document is motivated in a completely different scenario than the model presented in <cit.>. However, an unmistakable resemblance exists between (<ref>) and the early dark energy model described by <cit.>. The close similarity in the functional forms of both parametrizations is explained by the fact that both models assume that the equation of state transitions between the value for radiation in the early Universe to -1 in the late epochs when dark energy is dominant.]. First, note that in our parametrization, the EoS of dark energy converges to ω_de→ -1 at late times, which implies that dark energy is the dominant component. Also, the condition α > - 1 must be fulfilled to have the usual cosmic domination eras in place. The energy density must be subdominant at BBN/CMB scale to satisfy the constraints ρ_de/ρ_r ≤ 0.086 <cit.>. Thus one should expect that the ratio ρ_de/ρ_r at radiation era to be,
ρ_de/ρ_r|_rad≃ρ_de,o/ρ_r,oγ_s^1/α+1 < 1
One can see that this ratio depends on the parameters γ_s and α as ρ_de,o and ρ_r,o are fixed by the cosmological data.
In Figures <ref> to <ref>, we show the exact behavior of Ω_de=ρ_de/ρ_cr and ω_de as a function of the scale factor a for different values of α and γ_s. An example of the evolution of the density parameter Ω_de for α = 0.25 and γ_s = 10^-3 is shown in Figure <ref>. In Figures <ref> and <ref>, we show the behavior of Ω_de by fixing one of the parameters and varying the other one. One can see that the evolution of Ω is extremely sensitive to the values of γ_s and α.
We highlight that there is a scale factor a_t in the matter domination era that determines the transition of the gas from the radiation epoch to the current expansion, given by:
a_t = ( γ_s/1- γ_s)^1/4 (1+ α)
The way in which ρ_de and ω_de evolve with the scale factor is extremely sensitive to A_s and α where the transition scale factor a_t controls the time of transition and α defines its duration so that a small value for α corresponds to a shorter transition period. The dark energy equation of state ω_de follows peculiar behaviours, starting with ω_de= 1/3 for a << a_t during radiation era, passing by ω_de=0 for a= (γ_s/3 (1 - γ_s))^1/4 (1+ α) and ω_de = -1 in recent epochs (see Figure <ref>).
One of the important ways to check the causality of the universe to persists is the adiabatic squared speed of the sound, which is given by:
c_s^2 = δ p_de/δρ_de = ω_de - ω^'_de/3 (1+ ω_de)
= 1/3 + 4 α/31/1 + (a_t/a)^4 (1+ α) .
where primes denotes derivatives with respect to ln (a).
For the causality condition, the speed of the sound must be less than the local light speed c_s^2 ≤ 1. A positive value of c_s^2 represents a stable model, whereas a negative value of c_s^2 indicates the instability of the model. It follows from our model that c_s^2 ≈1/3 for a << a_t and c_s^2 ≈1/3 (1 + 4 α) for a >> a_t which shows that the causality and classical stability is satisfied for α <0.5.
In Figure <ref>, we plot the behavior of the square speed of sound as a function of the number of ln(a/a_t) for different values of α. As shown in the figure, one sees clearly for which values of α the causality and classical stability are satisfied (i.e., α > 0.5 breaks the model stability).
Finally, we define the dimensionless Hubble parameter E(z) as function of the redshift z, as follows:
E (z)^2 =H (z)^2/H_0^2
= Ω_o,r (1+z)^4 + (Ω_o,b +Ω_o,c) (1+z)^3
+Ω_de,o(1 - γ_s + γ_s (1+z)^4 (1+α))^1/1+α,
where Ω_o,r, Ω_o,b, Ω_c,o, Ω_de,o=1- Ω_o,r-Ω_o,b-Ω_c,o are the radiation, baryons, cold dark matter, and dark energy density fractions at present time, respectively. Also, we have assumed here the relation between the scale factor a and the redshift z, given by the equality a = 1/1+z.
§ NON-LINEAR ELECTRODYNAMICS AS AN EARLY DARK ENERGY
Here we discuss one possible avenue to describe the nature of the early dark energy. Non-linear electrodynamics is expected to play a crucial role in the evolution of the Universe. For this purpose, we propose the following non-linear electrodynamics Lagrangian <cit.>,
L_nled ≡ - F f ( F),
where F = 1/4 F_μν F^μν= 1/2 (B^2 - E^2) and f ≡ f ( F) is a functional that depends on the field strength F.
The energy-momentum tensor for this Lagrangian density L, is given by:
T^μν = H^μλ F^ν_λ - g^μν L_nled,
where H^μλ is given by:
H^μλ = ∂ L_nled/∂ F_μλ = ∂ L_nled/∂ F F^μλ.
For the Lagrangian density of our model, the energy-momentum tensor becomes:
T^μν = - ( f + Fd f/d F) F^μλ F^ν_λ +
g^μν F f,
where the energy density ρ and pressure p can be expressed as:
ρ = F f - E^2 ( f + Fd f/d F),
p = - F f + 2 B^2 - E^2/3(f + Fd f/d F).
Here we assume that the wavelength of the electromagnetic waves is typically smaller than the space-time curvature. Thus, we can use the spatial average of fields proposed by <cit.>, which defines the volumetric spatial average of a quantity Y as,
⟨ Y ⟩ = lim_V → V_0∫ Y √(-g) d^3 x
where g denotes the denotes the determinant of the metric tensor, V = ∫√(-g) d^3 x and V_0 stands for the time dependent large spatial volume.
In this procedure, the means values of the electric and magnetic fields are given by the Tolman relations, as
⟨ E_i ⟩ = 0, ⟨ B_i ⟩ =0, ⟨
E_iB_j⟩ =0,
⟨ E_iE_j⟩ =1/3E^2g_ij,⟨
B_iB_j⟩ =1/3B^2g_ij.
The latter procedure does not break the isotropy of the Friedmann-Roberson-Walker metric. In what follows, the brackets ⟨ ⟩ will be omitted for simplicity.
In the present work, we are interested in the case where the electric field is switched off. Thus, our non-linear Lagrangian is purely magnetic where we consider f( F) as a functional depending on two real parameters α and β given by:
f ( F) = ( β F^- (1+α) + 1 )^1/1+α,
where F = 1/2 B^2, β F^-(1+α) is dimensionless and for β=0, we have f( F)=1 (i.e. L = - F) which is the usual electrodynamics Lagrangian.
In this purely magnetic case, the energy density and pressure become,
ρ = F( β F^-(1+α) + 1 )^1/1+α,
p = - F( β F^-(1+α) + 1 )^1/1+α
+ 4/3 F( β F^- (1+α) + 1 )^-1+1/1+α.
The equation of state satisfied by the above nonlinear electrodynamics Lagrangian density is:
p = 1/3ρ(1 - 4 β/ρ^1+ α)
which is obviously the modified Chaplygin gas introduced in <cit.>, and discussed in detail in the previous section. It is easy to see that when the non-linearity is switched off (i.e., β =0), the above equation of state reduces to the Maxwell radiation EoS.
The evolution of the magnetic field is governed by the continuity equation:
ρ̇ + 3 H ( ρ + p ) = 0
By using equation (<ref>), a general solution of the above equation in terms of the scale factor is obtained as:
ρ = ρ_0 ( β/ρ_0^1+ α + (1 - β/ρ_0^1+ α) a^-4 (1+ α))^1/1+ α
where ρ_0 = ρ (a=1) is the present energy density. By comparing equations (<ref>) and (<ref>),
we find the relation between the non-linearity β and the parameter γ_s which is given by:
β = (1 - γ_s) ρ_de,o^1+α = (1 - γ_s) (Ω_de,oρ_crit)^1+α
with ρ_crit = 1.88×10^-29
h_o^2 · g · cm^-3.
§ STATISTICAL ANALYSIS
In order to study the cosmic evolution of this type of dark energy and compute model predictions for the free parameters, we use the software CosmoSIS[<https://bitbucket.org/joezuntz/cosmosis/wiki/Home>] <cit.>. The code provides a Bayesian framework that allows the user to implement different modules within a pipeline that combines a set of observational detections and samplers to evaluate the likelihood function and calculate the best estimates of the model.CosmoSIS computes the pipeline in discrete steps, performed by independent modules. Among these modules, we include: Consistency, which checks for cohesion and consistency among the defined cosmological parameters and assesses that set of criteria is not under or overdetermined; Camb, a Boltzmann code that calculates the underlying background quantities and computes the linear matter power spectrum <cit.>; Growth evaluates the linear growth factor and growth rate within a Universe with a state equation (<ref>). We run Camb from z = 5000 down to redshift 0 to closely follow the evolution of the components in the domination eras that are affected by the introduction of our model. The 500000 steps in redshift are dynamic to cover evenly the redshift range.
On the other hand, we choose the Emcee sampler[<http://dan.iel.fm/emcee/>] <cit.>, a Monte-Carlo Markov Chain that explores the parameter space with a collection of walkers. Each walker takes a random path and jumps to the next point after calculating a Metropolis acceptance rate. The final acceptance fraction should be in the range of 0.2-0.5 to reach the maximum posterior distribution <cit.>. The total number of samples in the chain comes from the product between the number of walkers and samples. We set 256 samples, 512 walkers, 20 steps before calculating the accuracy rate, and a random start for all walkers. The latter condition delays the convergence to the chain; thus, we remove the first 20000 realizations (burn-in stage).
The choice of the hyperparameters of the Bayesian estimator shown above obeys three criteria: i) a high success rate in the convergence of the chain; ii) posterior probability distributions of the physical parameters are unimodal; and iii) probabilities densities are not poorly populated.
The uniform prior distributions for the dark energy and the cosmological parameters imposed in CosmoSIS are presented in Table <ref>. Although the parameter α can take values up to 10, we limit the upper value for its prior distribution to be 0.5 to ensure the classical stability condition discussed in Section <ref>.
The predictions of the model are calculated with precomputed likelihoods available in CosmoSIS: CMB data with Planck 2018 (TTTEEE + lensing) <cit.>, primordial nucleosynthesis (or BBN) <cit.>, distances inferred from BAO datasets: 6dF Galaxy Survey -or 6dFGS- <cit.>, WiggleZ <cit.>, SDSS DR7 (the main galaxy survey) <cit.>, and luminosity distances from SNIa from Pan-STARRS1 <cit.> -Pantheon- and the SH0ES survey <cit.>.
We run CosmoSIS in the following combined sets of observations:Set 1: Planck 2018 (TTTEEE + lensing). Set 2: Planck 2018 (TTTEEE + lensing) + BBN + SDSS DR7 (main galaxy survey) + 6dFGS + WiggleZ + Pantheon + SH0ES.
The former setting sets strong constraints for the model at high redshift, whilst the second one acknowledges the cosmological evolution with this form of dark energy at different stages of the Universe. We perform additional tests with other combined datasets but skip presenting them here because the results are not informative; thus, we cannot set tight constraints to our free parameters. In particular, combined analysis with BAO and SNIa datasets are prior-dominated, and the parameter space is loosely explored in these cases.
The best fits for our model parameters are presented in Table <ref>. We constrain the value of γ_s and α that define the evolution of the dark energy model, three cosmological parameters {Ω_m,h_o,σ_8} and m, the nuisance parameter that accounts for the SN1a absolute magnitude. Based on the concordance model, we recover the best-fit value for Ω_de,o following the condition Ω_m + Ω_de,o = 1[We remind the reader that a spatially flat Universe is assumed throughout the document.].
The statistical analysis and visualization of the samples are performed with GetDist[<https://getdist.readthedocs.io/en/latest/plots.html>] <cit.>. Figure <ref> shows the posterior distribution of the model and cosmological parameters with the combined datasets described above.
Figure <ref> shows the evolution of our model's equation of state ω_de and dark energy density fraction Ω_de. In the upper panel, the equation of state exhibits a transition from the radiation domination epoch (ω_de∼1/3 at ln a << -3) to the De-Sitter era (ln a ∼ 0).Conversely, the dark energy density fraction as a function of the scale factor a is displayed in the bottom panel of Figure <ref>, according to the expression:
Ω_de=ρ_o/ρ_cr=Ω_de,oG (a)/E^2(a),
where E(a) is defined in equation (<ref>), and G(a) in the expression (<ref>).
The bottom panel reveals that today's dark energy density fraction value is ∼ 0.7, increasing rapidly in the future (i.e., a > 1). Even more interesting, the energy fraction of the dark energy in this model is always above this quantity associated with Λ. A natural consequence of this trend is that the matter-dark energy equality occurs earlier, so the structure formation than in the standard model case. In addition, if the amount of dark energy increases at all times with respect to other components of the matter-energy content of the Universe, the position and height of the acoustic peaks would shift with respect to the standard model prediction.
We calculate the numerical value of β = 0.52 ×ρ_crit^1+α = 0.015 ± 0.008, the parameter that quantifies the non-linearity of the electrodynamics. Considering the small (and negative) value for α derived from our analysis, the exponent accompanying the factor ρ_crit is significantly small; thus, β has a numerically non-zero value but still is a perturbative parameter. This result indicates a departure from Maxwell's classical electrodynamics in the early Universe that could explain the origin of this type of dark energy with a non-negligible contribution during the radiation domination epoch.
Additionally, we compute the value z_t -corresponding to the scale factor a_t defined in equation (<ref>)- with our best fits free parameters and find a value for z_t = 0.47. We can read the latter result as an earlier transition to the accelerated expansion epoch than in the ΛCDM model if the latter would have had dynamics associated to Λ.
Furthermore, we evaluate S_8=σ_8√(Ω_m/0.3). Plugging the best fits in our model, we obtain a value of S_8= 0.758 ± 0.007, fairly consistent with results reported by KiDS-450 and KiDS-450+2dFLenS <cit.>. However, we find that this quantity, which measures the clustering of the structure, is not the best observable to set tight constraints in this kind of cosmology. Our reports show no tension between the low and high Universe for σ_8.
Finally, it is important to stress that we do not calculate the matter perturbations due to the dark energy's early contribution to the cosmic plasma. Instead, we derive the parameter σ_8 as a product of the computations made by Camb. The complete treatment of the density field beyond the background level is outside the scope of this work.
§ COSMOLOGICAL TESTS
Based on the best fits parameters of the dark energy model presented best-fits presented in Table <ref> and Figure <ref>, we calculate the age of the Universe, following equation (19) in <cit.>:
t_o =2/31/H_o √(1 - Ω_m)ln(√(1/Ω_m - 1) + √(1/Ω_m)).
When H_o is expressed in Gyr^-1, the age of the Universe in this model is t_o = 13.4 ± 0.2 Gyr. Instead, the time of the Universe inferred from Planck 2020 <cit.> is t_o = 13.797 ± 0.023 Gyr. However, early dark energy models exhibit shorter times (see results from <cit.>), leading to an earlier structure formation in the Universe when compared with the ΛCDM model. This result is consistent with our calculation for z_t in the previous section.
Recently, <cit.> presented robust estimates of the absolute age of the globular cluster M92. Their analysis shows this cluster's age of 13.80 ± 0.75 Gyr. Evidently, no structure could have been formed before the Universe itself; hence, their main finding rules out certain cosmological models in which the age of the Universe is outside the age range for M92. Our model passes the test, lying in the lower limit defined by <cit.>.
There is an additional proxy that we submit our model, assuming the lookback time (t_lb) as a function of the scale factor (a), first explored in <cit.>:
t_lb(a) =2/31/H_o √(1 - Ω_m)arcsinh(√(1/Ω_m - 1))
-2/31/H_o √(1 - Ω_m)arcsinh(√(1/Ω_m - 1)a^3/2).
Using the expression (<ref>), we calculate the time of the completion of the Epoch of Reionization (EoR), under the assumption that 6 < z_EoR < 10 <cit.>. The lookback time predicted by our model to the duration of the cosmic Reionization is 12.37 >t_lb, EoR/Gyr> 12.93.
Moreover, we present the lookback time evolution with z for our model's best parameters in Figure <ref> and the same function for the ΛCDM model. We include a conservative duration for the EoR: 6 < z_EoR < 10, the highest redshift galaxy detected by HST: GNz-11 <cit.> and the highest z galaxy candidates observed with the JWST, by the GLASS collaboration <cit.>. We remind the reader that the calculation of the lookback time for these galaxies is model-dependant because the observable of these objects is the redshift; therefore, we need to assume a cosmology to compute the time that their light has been traveling before reaching the telescope. In this case, we consider the set of parameters by <cit.> as the fiducial cosmology for these candidates at high redshift.
Lastly, we determine the CMB shift parameter R_CMB, which measures the shift of the first acoustic peak in the power spectrum of the CMB anisotropies. We numerically calculate this observable using the dimensionless Hubble parameter E(z), equation (<ref>), which can be expressed as:
R_CMB = Ω_m^1/2∫_0^1089dz/E(z).
With the best fit parameters reported in Section <ref>, this quantity has a value of R_CMB = 1.61^+0.23_-0.04. This estimate is lower than the value inferred from the different Planck campaigns and is consistent with a dark energy model that produces an early structure evolution. Therefore, it shifts slightly between the first and second acoustic peaks.
We close this section with an important remark: introducing a form of dark energy originated by a departure from Maxwell's electrodynamics causes matter structure to form more promptly than in the standard cosmological model. This variation is mainly regulated by a non-zero γ_s parameter (or equivalently, β≠ 0). Nonetheless, if the γ_s parameter is switched off, γ_s = 0, then the ΛCDM cosmology and the classical Maxwell electrodynamics are fully recovered.
§ CONCLUSIONS
We propose an early dark energy model that mimics radiation at the early stages of the Universe and exhibits an accelerated expansion in the late evolution of the Universe. We show that such an effective parametrization can be nicely modeled within the non-linear electrodynamics framework where the non-linearly plays important during the evolution of the Universe. The model assumes three free parameters {γ_s,α,Ω_de,o}, and a parameter β that relates to the non-linearity of the electromagnetic tensor, heavily dependent on the value of γ_s.
We find the best-fit parameters for this model, combining different sets of cosmological observations: CMB, BBN, BAO, and SNIa distances, and their likelihoods in the modular code CosmoSIS. The best values for the free parameters are: γ_s = 0.468 ± 0.026, α = -0.947 ± 0.032, and Ω_de,o = 0.729 ± 0.017 (derived parameter from the MCMC best fit for Ω_m and the assumption of the Concordance model). The departure from the Maxwell electrodynamics is quantified with the derived parameter β= 0.015 ± 0.008. The non-null value of β indicates that contributions from non-linear electrodynamics in the early Universe could originate this form of dark energy. In addition, we set constraints for Ω_m= 0.271 ± 0.017, h_o = 0.702 ± 0.009, and σ_8 = 0.798 ± 0.007.
Based on the Bayesian analysis presented in this document, we demonstrate that the Universe experiences a faster expansion rate during the radiation domination epoch, induced by additional degrees of freedom in the Hubble parameter. Our Universe subjected to this type of dark energy enters the matter-domination epoch earlier than predicted by the standard model. Thus, our model foresees an earlier structure formation. This result is consistent with findings from recent works <cit.> that explore the possibility that early dark energy models explain the current Hubble tension, which has reached the 5.7σ discrepancy between Planck 2018 + ΛCDM and late time measurements of H_o <cit.>. As presented in this document, a scenario of the Universe with an early dark energy contribution alleviates the tension from comparing early and late Universe datasets <cit.>. The value for H_o reported here sits between estimates from Planck 2018 and SH0ES. The latter result is particularly encouraging to pursue future studies with this form of dark energy.
Also, we offer an estimate of the S_8 parameter, frequently associated with a tight correlation between σ_8 and Ω_m. The prediction from our model is S_8= 0.758 ± 0.007, consistent with the value reported by KiDS and the 2dFLenS. However, we stress it is outside of the scope of the paper to study the evolution of the matter perturbations within this model.
Finally, our model is a compelling formalism to narrow the current cosmological tensions between different datasets. Future generations of missions, such as the Nancy Grace Roman Space Telescope (WFIRST) and EUCLID, will shed light on the nature of dark energy and its dynamics in the early Universe.
H.B.Benaoum gratefully acknowledges the financial support from the University of Sharjah. L.A. García thanks the valuable contribution from Joe Zuntz for his insightful remarks to implement the software CosmoSIS in private communications. L. Castañeda was supported by Patrimonio Autónomo - Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas (MINCIENCIAS - COLOMBIA) Grant No.110685269447 RC-80740-465-2020, projects 69723.
caldwell2003phantom Caldwell, R. R., Kamionkowski, M., Weinberg, N., Physical review letters, 91, 7, 071301, 2003, APS.
ade2016planck Ade, P., Aghanim, N, Arnaud, M., and others. Astronomy & Astrophysics, 594, A13, 2016.
planck2020 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, AAP, 641, A6. doi:10.1051/0004-6361/201833910
riess2022 Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, APJL, 934, L7. doi:10.3847/2041-8213/ac5c5b
aiola2020atacama Aiola, S., Calabrese, E., Maurin, L., and others, Journal of Cosmology and Astroparticle Physics, 2020, 12, 047, 2020.
smith2021 Smith, T. L., Poulin, V., Bernal, J. L., et al. 2021, PRD, 103, 123542. doi:10.1103/PhysRevD.103.123542
klypin2021 Klypin, A., Poulin, V., Prada, F., et al. 2021, MNRAS, 504, 769. doi:10.1093/mnras/stab769
garcia2011 García, L., J. Tejeiro, and Castañeda L. Proceedings of the International School of Physics “Enrico Fermi”. Volume 178: From the Big Bang to the Nucleosynthesis, 309 - 316 doi:10.3254/978-1-60750-974-5-309.
bento2002 Bento, M. C., Bertolami, O., Sen, A. A. 2002, astro-ph/0210375
garcia2021 García, L. Á., Castañeda, L., Tejeiro, J. M. 2021, New Astronomy, 84, 101503. doi:10.1016/j.newast.2020.101503
schoneberg2021 Schöneberg, N., Abellán, G. F., Sánchez, A. P., et al. 2022, PhysRev, 984, 1. doi:10.1016/j.physrep.2022.07.001
poulin2023 Poulin, V., Smith, T. L., & Karwal, T. 2023, arXiv:2302.09032. doi:10.48550/arXiv.2302.09032
born1934foundations Born, M., Infeld, L., 1934, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 144,852, 425–451.
vollick2003anisotropic Vollick, Dan N., 2003, General Relativity and Gravitation, 35, 8, 1511–1516.
moniz2002quintessence Moniz, P Vargas 2002, Physical Review D, 66, 10, 103501.
garcia2000born Garcia-Salcedo, R. and Breton, N., 2000, International Journal of Modern Physics A, 15, 27, 4341–4353.
camara2004nonsingular Camara, C.S., de Garcia Maia, M.R., Carvalho, J.C., Lima, J.A.S., 2004, Physical Review D, 69, 12, 123504.
elizalde2003born Elizalde, E., Lidsey, J.E., Nojiri, S., Odintsov, S. D., 2003, Physics Letters B, 574, 1-2, 1–7.
novello2004nonlinear Novello, M., Bergliaffa, S.E., Salim, J. 2004, Physical Review D, 69, 12, 127301
kruglov2015universe Kruglov, S.I., 2015, Physical Review D, 92, 12, 123523.
ovgun2018falsifying Övgün, A., Leon, G., Magaña, J., Jusufi, K., 2018, The European Physical Journal C, 78, 6, 1–20.
ovgun2017inflation Övgün, A., 2017, The European Physical Journal C, 77, 2, 1–6.
benaoum2021 Benaoum, H.B., Övgün, A., 2021, Classical and Quantum Gravity, 38, 13, 135019.
benaoum2002 Benaoum, H. B., 2022, Universe 8, 340 .
benaoum2012 Benaoum, H. B., 2012, Advances in High Energy Physics, 2012, 357802 .
benaoum2019extensions Benaoum, Hachemi B., Luongo, Orlando and Quevedo, Hernando, 2019, The European Physical Journal C, 79, 1–9 .
chaplygin1904 Chaplygin, S. 1904, Sci. Mem. Moscow Univ. Math. Phys. 21, 1.
artymowski2021emergent Artymowski, Michał, Ben-Dayan, Ido and Kumar, Utkarsh, 2021 Phys. Rev. D, 103, 8, L121303
benaoum2023inflation Benaoum, H.B., Leon, Genly , Övgün, A. and Quevedo, H., 2023 The European Physical Journal C, 83, 367.
tolman1930 Tolman, R. and Ehrenfest, P., 1930, Phys. Rev., 36, 12, 1791–1798, doi:10.1103/PhysRev.36.1791
cosmosis Zuntz, J., Paterno, M., Jennings, E., et al. 2015, Astronomy and Computing, 12, 45. doi:10.1016/j.ascom.2015.05.005
lewis2000 Lewis, A., Challinor, A., Lasenby, A. 2000, APJ, 538, 473. doi:10.1086/309179
howlett2012 Howlett, C., Lewis, A., Hall, A., et al. 2012, JCAP, 2012, 027. doi:10.1088/1475-7516/2012/04/027
emcee Foreman-Mackey, D., Hogg, D. W., Lang, D., et al. 2013, PASP, 125, 306. doi:10.1086/670067
gelman1996 Gelman, A. and Roberts, G. O. and Gilks, W. R., Bayesian Statistics, Oxford University Press, Oxford, 1996, 599-608.
beringer2012 Beringer, J. et al. Particle Data Group. 2012, Phys. Rev. D. Vol 86, Issue 1, 1528. doi: 10.1103/PhysRevD.86.010001
cooke2016 Cooke, R. & Pettini, M. 2016, MNRAS, 455, 1512. doi:10.1093/mnras/stv2343
pitrou2021 Pitrou, C. 2021, Tools for High Energy Physics and Cosmology, 25
beutler2011 Beutler, F., Blake, C., Colless, M., et al. 2011, MNRAS, 416, 3017. doi:10.1111/j.1365-2966.2011.19250.x
beutler2012 Beutler, F., Blake, C., Colless, M., et al. 2012, MNRAS, 423, 3430. doi:10.1111/j.1365-2966.2012.21136.x
kazin2014 Kazin, E. A., Koda, J., Blake, C., et al. 2014, MNRAS, 441, 3524. doi:10.1093/mnras/stu778
ross2015 Ross, A. J. & SDSS-III Collaboration 2015, AAS
scolnic2018 Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, APJ, 859, 101. doi:10.3847/1538-4357/aab9bb
lewis2019 Lewis, A. 2019, arXiv:1910.13970
conti2017 Fenech Conti, I., Herbonnet, R., Hoekstra, H., et al. 2017, MNRAS, 467, 1627. doi:10.1093/mnras/stx200
joudaki2017 Joudaki, S., Mead, A., Blake, C., et al. 2017, MNRAS, 471, 1259. doi:10.1093/mnras/stx998
joudaki2018 Joudaki, S., Blake, C., Johnson, A., et al. 2018, MNRAS, 474, 4894. doi:10.1093/mnras/stx2820
boylan2021 Boylan-Kolchin, M. Weisz, D. R. 2021, MNRAS, 505, 2764. doi:10.1093/mnras/stab1521
jiaqi2023 Jiaqi, Ying, B. Chaboyer, E. M. Boudreaux, C. Slaughter, M. Boylan-Kolchin and D. Weisz, arXiv:2306.02180.
madau2014 Madau, P. Dickinson, M. 2014, ARAA, 52, 415. doi:10.1146/annurev-astro-081811-125615
stark2016 Stark, D. V., Kannappan, S. J., Eckert, K. D., et al. 2016, APJ, 832, 126. doi:10.3847/0004-637X/832/2/126
greig2017 Greig, B. Mesinger, A. 2017, MNRAS, 465, 4838. doi:10.1093/mnras/stw3026
garcia2017 García, L. A., Tescari, E., Ryan-Weber, E. V., et al. 2017, MNRAS, 470, 2494. doi:10.1093/mnras/stx1371
jiang2021 Jiang, L., Kashikawa, N., Wang, S., et al. 2021, Nature Astronomy, 5, 256. doi:10.1038/s41550-020-01275-y
yan2022 Yan, H., Ma, Z., Ling, C., et al. 2022, arXiv:2207.11558
review2022 Kamionkowski, M. & Riess, A. G. 2022, arXiv:2211.04492. doi:10.48550/arXiv.2211.04492
cocoa2022 García, L. A. and Castañeda, L. 2022, Ciencia en Desarrollo, 1, 2E, 1–10. doi:10.19053/01217488.v1.n2E.2022.15216
gomez2021 Gómez-Valent, A., Zheng, Z., Amendola, L., et al. 2021, PRD, 104, 083536. doi:10.1103/PhysRevD.104.083536
|
http://arxiv.org/abs/2307.04945v1 | 20230711002121 | What do LLMs need to Synthesize Correct Router Configurations? | [
"Rajdeep Mondal",
"Alan Tang",
"Ryan Beckett",
"Todd Millstein",
"George Varghese"
] | cs.NI | [
"cs.NI",
"cs.PL"
] |
Submitted to HotNets 2023
What do LLMs need to Synthesize Correct Router Configurations?
Rajdeep Mondal
mailto:[email protected]@ucla.edu
Alan Tang
mailto:[email protected]@cs.ucla.edu
Ryan Beckett
mailto:[email protected]@microsoft.com
Todd Millstein
mailto:[email protected]@cs.ucla.edu
George Varghese
mailto:[email protected]@cs.ucla.edu
October 2023
=================================================================================================================================================================================================================================================================================================================================================================
We investigate whether Large Language Models (e.g., GPT-4) can synthesize correct router configurations with reduced manual effort. We find GPT-4 works very badly by itself, producing promising draft configurations but with egregious errors in topology, syntax, and semantics. Our strategy, that we call Verified Prompt Programming, is to combine GPT-4 with verifiers, and use localized feedback from the verifier to automatically correct errors.
Verification requires a specification and actionable localized feedback to be effective. We show results for two use cases: translating from Cisco to Juniper configurations on a single router, and implementing no-transit policy on multiple routers.
While human input is still required, if we define the leverage as the number of automated prompts to the number of human prompts, our experiments show a leverage of 10X for Juniper translation, and 6X for implementing no-transit policy, ending with
verified configurations.
§ INTRODUCTION
The limitations of his knowledge were as startling as its profundity – Hardy on Ramanujam <cit.>
While GPT-4 and other language models have shown great success in some domains (e.g., writing poems, passing the LSAT) they have been shown to have issues in other domains (e.g,. math, word puzzles) <cit.>. Language models have had some success in helping users write sequential programs in systems like CoPilot <cit.> and Jigsaw <cit.>. Our paper examines GPT-4's ability to write router configuration files, traditionally written by humans, that help tune routes and forwarding decisions and are critical for network operation. Our early experiments show that GPT-4 by itself is an
“idiot-savant", capable of brilliance but also making simple errors that an operator would be fired for making.
Critics have derided large language models (LLMs) as mere “stochastic parrots” <cit.>,because they produce text (say of a program) syntactically by predicting the next word based on a statistical model derived by training on a vast corpus of text found on the Internet. Our broader goal beyond synthesizing configs is to see whether LLMs can be combined with other programs (via APIs) to come closer to a “stochastic owl” that understands program semantics.
A plausible way to introduce semantics is to pair a LLM with an automatic verifier such as a SAT solver or a model checker. But verification is not a panacea. First, a verifier cannot prove correctness without a specification. In practice,
specifications are incomplete, so not all solutions are in fact acceptable to the user.
Second, for the verifier to automatically (with minimal human aid) interact with the LLM, the verifier must provide actionable feedback. We found it was easier for the LLM to correct itself using feedback from modular verification of components of a network (individual routers <cit.> or even route maps within a router <cit.>), rather than the network as a whole.
Figure <ref> shows the traditional method of pair programming (PP), embodied in systems like GitHub CoPilot <cit.>, where a human and an AI work together to author a program. In pair programming, the AI and the human form a tuple (A, H) and the human H manually checks for correctness of the output of the AI A and then manually issues correction prompts to A as shown in the figure. Such manual initial prompting and subsequent manual correction is often called prompt engineering.
Figure <ref> shows our alternate vision. In what we call Verified Prompt Programming (VPP), the AI, the human, and a verification suite (V) form a triple (A, H, V). The verification suite checks for correctness and automatically issues localized corrections.
V may abandon automatic correction after some number of trials, and the human must still correct manually. However, our hypothesis is that human effort is reduced as the output grows “closer” to a correct program.
Notice that there is a fast inner loop between V and A, where verifier results are automatically fed back to GPT-4. Since verifier feedback is often cryptic, we use simple code that we call a humanizer that converts the feedback to natural language prompts that are given to GPT-4. When V either determines the final configuration is correct or a time bound elapses, V sends the output back to the user as part of the slow manual loop. We examine a “reduced work hypothesis": that the work in the manual loop in Figure <ref> is significantly less than then the manual work in Figure <ref>
To quantify reduced human effort we introduce a simple measure that may be useful in other VPP contexts.
Define leverage as the ratio L of the number of automated prompts in Figure <ref> to the number of human prompts. Leverage measures the effect of the verifier suite, the
potential improvement in going from (A, H) to (A, V, H), keeping the language model A and the human H the same.
The reader may think the real leverage is the improvement from H to (A, H), or from H to (A, V, H). But this depends on the capability of the human H and is hard to make uniform or repeatable. Given how error-prone
(A, H) is for configurations, we find it more natural to measure the improvement caused by VPP. If a future LLM, say GPT-6, produces near-perfect configurations, leverage will decrease as there is less need
for automatic correction. Our definition also assumes every automatic correction in Figure <ref>
would otherwise be be done by a human in Figure <ref>.
The reduced work hypothesis is that the leverage L > 1 is high. Even if the leverage is low (say 1), since it is crucial that router configurations be correct, combining with a verifier seems critical. We were happy to find that in both use cases we did end with verified configurations via GPT-4: this was not obvious at the outset.
This vision and hypothesis extends beyond synthesizing configs to more general programs. Prompt programming (as opposed to prompt engineering) also reflects the use of APIs and automatically generated feedback prompts that may be more generally useful. However, network configs are a simple enough domain to experiment with. Further, there exist config verifiers (e.g., Campion <cit.> and Lightyear <cit.>) that provide actionable localized feedback.
For the rest of this paper, we examine the reduced manual work hypothesis and measure leverage for two use cases: translating a config on a single router from Cisco to Juniper syntax, and implementing a simple policy (“no transit") on a network of 7 routers. Section <ref> describes the system organization of a potential system we call . Section <ref> describes experiments with Cisco to Juniper translation, while Section <ref> describes implementing no-transit on multiple routers. Section <ref> compares our ideas to previous work and Section <ref> describes lessons learned.
§ SYSTEM ORGANIZATION
Figure <ref> is a refinement of the more general Verified Pair Programming (VPP) vision of Figure <ref> that we call . We emphasize we have not built . While we use GPT-4 we have not been able to access the APIs, and so manually simulated the API calls with prompts to ChatGPT. Our goal is not to demonstrate a working system but instead to explore GPT-4's ability to author configurations, as in the “Sparks of AGI" paper <cit.>.
The verification suite shown in Figure <ref> consists minimally of two verifiers, a syntax verifier (we used Batfish <cit.>) and a semantics verifier (we used different ones depending on the use case). For our second use case, we used a third verifier, a topology verifier (that we wrote in Python) as we found that GPT-4 sometimes missed announcing routes to neighbors. The user provides a precise natural language description of the context (topology, routers, interfaces) and the desired task (e.g. the Cisco config and a request to translate it to Juniper). GPT-4 output is
fed first to Batfish to check for syntax errors. sends GPT-4 feedback about erroneous lines, “humanized" in natural language (see Table <ref> for examples). The boxes labelled H in Figure <ref> correspond to the humanizer in Figure <ref>.
If all syntax errors are corrected (if too many syntax correction attempts occur, punts to the user), the output is passed to the semantics verifier. For our first use case, we use Campion <cit.> as a verifier. For our second use case we use Batfish's symbolic route map analysis as the verifier, asking it to verify local policies that together ensure the desired global policy, as in Lightyear <cit.>. Once again, the semantic verifier feedback is passed back, suitably humanized, to GPT-4. We found that GPT-4 would sometimes correct a semantic error while introducing a new syntax error, in which case we had to return to the syntax verifier. When the semantic verifier attests to a correct config or too many correction attempts transpire, returns to the human.
When works with multiple routers, we used another module called a “Modularizer" (Figure <ref>). For network configs, the idea is that we start with a precise machine readable (we use JSON) description of the “modules" which in our case is the topology and the connections. The Modularizer outputs a sequence of Natural Language Prompts that describes the topology to GPT-4 (e.g.,. Router R1 is connected to Router R2 via interface I1 at R1 and I2 at R2). The Modularizer can also take a general specification of local policies (e.g. edge routers add a specific community on ingress) and output a specific local specification for each router for the semantic verifier. The Composer puts back the
pieces (in our case in a folder for Batfish).
The modularizer follows the prompt engineering paradigm "Give the Model Time to
Think" <cit.>, which suggests breaking a complex prompt into simpler sub-prompts. Exploiting modularity is a way to do so for program synthesis. A second technique we find useful is what
is called single shot prompting <cit.>. We start each chat with a set of initial instruction
prompts (IIP) (Figure <ref>) loaded from a database for avoiding common mistakes. The IIP database can be built and added by experts over time. The I/O examples in Jigsaw <cit.> are an IIP, but our IIP contains instructions not examples.
§ CISCO TO JUNIPER TRANSLATION
We translate a Cisco configuration into an equivalent Juniper one using Verified Pair Programming. Batfish <cit.> is used to identify syntax errors. Campion <cit.> is used to detect and localize semantic differences that are used to refine the result. We show examples of the issues encountered, and discuss success and limitations of the approach.
§.§ Method
First, we provide the Cisco configuration, and the prompt: "Translate the configuration into an equivalent Juniper configuration." GPT-4 will produce a translation into Junos format that typically contains several errors and differences. We try to rectify these errors iteratively. First, we use Batfish <cit.> and Campion <cit.>to detect any errors or differences, and then use a simple script to produce a prompt for GPT-4 (the humanizer H in Figure <ref>) to try and fix this error. After GPT-4 attempts to resolve the issue, we ask it to print the entire configuration and check the result using verification tools again. For our experiment we focus exclusively on behavior related to routing and forwarding, ignoring potentially important features such as NTP servers.
To design the humanizer (i.e., automatically generate a prompt informing GPT-4 of the errors present), we distinguish four classes of configuration errors:
Syntax errors: Batfish produces parse warnings identifying relevant lines that do not use valid Juniper syntax.
Structural mismatch: This is when a component, connection, or named policy is present in the original configuration but not in the translation (or is present in the translation but not the original). For example, if the original configuration defined a BGP neighbor but there is no corresponding neighbor in the translation, there would be a mismatch in the routing connections. Similarly, if there are corresponding BGP neighbor definitions in both configurations, but one configuration has an import policy defined while the other does not, that would be a mismatch in the named policies present. Campion is able to detect this, and identify the missing or extra items.
Attribute differences: This is when a numerical attribute has a different value between the two configurations. An example is OSPF link cost difference between two corresponding interfaces. Campion detects these and prints the attributes for corresponding components.
Policy behavior differences: This is when a route map or access control list has a semantic difference. Route maps are used to filter incoming or outgoing route advertisements, so a difference would mean that that there are some route advertisements that are allowed by one router but not allowed by the other. Campion is able to detect these and output the relevant policy names, prefixes, and lines for these differences.
The distinction among errors helps for two reasons. First, syntax errors and structural mismatches have to be handled earlier since they can mask attribute differences and policy behavior differences. Second, different types of errors require different humanized prompts, while errors of the same type can reuse similar prompts. Each type of error can be summarized with a formulaic prompt with some fields inserted based on the error reported by Batfish or Campion.
Table <ref> shows the formulas and examples of generated prompts. Batfish parse errors and warnings can be reused as prompts for syntax errors. Prompts for structural mismatches and attribute differences are easily generated from the relevant components and attributes. Policy behavior differences are more difficult since it is not always clear how to describe the affected input space that is treated differently. We opt for the approach of giving an example prefix.
§.§ Experience and Results
We tried translating a Cisco configuration from the Batfish examples <cit.> into Juniper format. This configuration was short enough to fit within GPT-4 text input limits, but used non-trivial features including BGP, OSPF, prefix lists, and route maps. GPT-4's synthesized Juniper router configuration had several errors. In many cases, when an automatically generated prompt, similar to those in Table <ref>, is provided to GPT-4, it will produce a response fixing the issue. In some cases, GPT-4 is unable to edit the translation correctly, either applying no change or applying an erroneous one. This often requires manual intervention via more specific prompts in order to fix. Another problem is that GPT-4 can fix one error, but introduce new errors that were not previously there. Sometimes it even reintroduces errors that were previously fixed! However, we were able to reach a point where Campion and Batfish no longer produced errors.
Leverage: The entire cycle of prompts was around 2 human prompts and 20 automated prompts, for a leverage of 10X. Some of the 20 automatic prompt correction cycles included minor cycles for syntax correction not just at the start but also
after correcting semantic errors. To be clear, we “simulated” each API call by feeding our automatically generated prompts manually to GPT-4.
Table <ref> shows errors in the translation at some point and whether GPT-4 was able to fix them using an automatically generated prompt. In more detail:
Missing BGP local-as attribute: The translated BGP neighbor declarations did not include a local AS attribute. We label this a syntax error since it produces a parse warning.
Missing/extra BGP routing policy: An import or export policy is used for a BGP neighbor in only one configuration.
Different OSPF link attributes: OSPF links have a number of attributes, and the translation sometimes contains differences in link cost or passive interface settings.
Setting wrong BGP MED value: The translation of one BGP routing policy did not update the BGP MED value. This was caused by an error in translating one of the route map clauses from the original Cisco configuration.
Different Redistribution behavior into BGP: Cisco and Juniper formats handle route redistribution into BGP differently. Juniper typically does this using the same routing policies that control importing and exporting BGP routes while Cisco configurations set a separate route map for route redistribution. In our case, Campion detected that the Juniper configuration was redistributing some routes that the Cisco configuration did not. This could be fixed by adding a "from bgp" condition to a number of locations in the policy. Unlike the previously described errors, GPT-4 was unable to fix this when given the automatically generated prompt. Instead it usually does nothing when asked to fix the error. However, it was able to fix the problem when asked more directly to add "from bgp" conditions to routing policies.
BGP prefix list issues: Another subtle issue occurred when translating prefix lists. In the original Cisco configuration, a prefix list was defined to match prefixes with length 24 or greater where the first 24 bits matched 1.2.3.4. In Cisco this is done with the command:
ip prefix-list our-networks seq 5 permit 1.2.3.0/24 ge 24
and it was applied with the definition:
route-map from_customer deny 100
match ip address prefix-list private-ips
The noteworthy part is the "ge 24" which says to match prefixes with length 24 or greater. There is no equivalent of this in defining prefix lists in Juniper, but for our use case, there are at least two methods of getting similar behavior in Juniper with different syntax. When GPT-4 is asked to translate the configuration, it often does not translate the "ge 24" part correctly, often just omitting it, so the space of prefixes matched will differ in the translation. When asked to fix this problem, it sometimes generates configurations with incorrect syntax. For example, it can output the following:
prefix-list our-networks { 1.2.3.0/24-32; }
which is not valid Juniper syntax. However, after informing it of the error, it does eventually find a correct translation.
§ GLOBAL POLICIES VIA LOCAL SYNTHESIS
Next, we used GPT-4 to generate router configs for a given network topology based on local policies for each router, inspired by Lightyear <cit.>, which does control plane verification by
verifying local invariants. We limited our scope to BGP.
For semantic correctness, we use two new modules. The first is a 'topology' verifier which checks whether the config of a particular router follows the defined topology. It checks whether GPT-4 sets up all interfaces, declares BGP neighbors and announces networks correctly. Second, we run Batfish to check local policies defined in the prompts; the outputs are used to refine the result.
§.§ Method
We begin by specifying the task to GPT in an initial prompt using a couple of sentences. The intention is to influence the LLM to start `thinking' in a certain fashion. Our goal is to make the network follow the no-transit policy, under which no two ISP's should be able to reach other. However, all ISPs should be able to reach the CUSTOMER and vice versa.
It is difficult to write a natural language description of the topology, a task prone to human error.
We wrote an automated script that generates text given the topology as input. In our experiments, we limited our scope to star networks where one router would be attached to a CUSTOMER IP, while the other routers are connected to different ISPs (Figure <ref>). All the ISP routers are directly connected to the first router. The "network generator" therefore only needs the number of routers as input. It has two outputs: 1) a textual description and 2) a JSON dictionary for the entire network topology. The textual description is used as a prompt, while the JSON dictionary is used later to check whether the generated configs match the topology.
Local versus Global Policy Prompts? We tried specifying to GPT-4 the global no-transit policy
at once. GPT-4 generated two innovative strategies: filtering routes using AS path regular expressions, and denying ISP prefixes from being advertised to other routers from the customer router. Unfortunately, we found after correcting topology and syntax errors, when we provided feedback in terms of a counterexample packet (as would be provided by a “global" network verifier like Minesweeper), GPT-4 was confused and kept oscillating between incorrect strategies. We found that specifying local policies as in Lightyear <cit.> gave us better results because it allowed us to localize verification errors to specific routers and specific route maps within those routers.
We asked GPT-4 to generate configs for each router using a new prompt each time, specifying the local policy for each router. Specifically, the policy is that R1 should add a specific community at the ingress to each ISP and then drop routes based on those communities
at the egress to each ISP. The generated errors fell into three categories:
Syntax errors: GPT-4 generates a configuration with invalid Cisco syntax including errors in which certain config lines are misplaced. Batfish produces parse warnings identifying these errors.
Topology errors: GPT-4 incorrectly declares or misses some BGP neighbors or forgets to announce certain networks. For this, we use an automated "topology verifier" that compares the config against the previously specified JSON dictionary and outputs inconsistencies.
Semantic errors / Policy errors: GPT-4 produces configs that do not follow the intended local policy. We use Batfish "Search Route Policies" for verification in this step. In case there is a semantic error, Batfish produces an example where the local policy is not followed. This examples is then fed to GPT-4 in a fresh prompt.
Classifying into separate categories allowed us to use different tools to address each one. Table <ref> lists examples of the rectifying prompts. Once all the errors are rectified, we simulate the entire BGP communication using Batfish as a final step, in order to ensure that the global policy is satisfied, though the proof technique of Lightyear <cit.> could instead be used to ensure that the local policies imply the global one.
§.§ Experience and Results
Since some GPT-4 errors were more common, we supplied it an IIP (the Inital Instruction Prompt) as follows:
CLI prompts: GPT-4 would often generate commands to enter on the Cisco command line interface, which is undesirable. Thus we specifically asked it to generate the .cfg files.
Wrong keywords: While generating the configs, it would often use certain keywords such as `exit', `end', `configure terminal', `ip routing', `write', `hostname' and `conf t'. It had a tendency to place some of them in the wrong locations.
Hence, we directed it not to use these keywords. Any extra required commands for setup were prepended to the final config files, before running them on Batfish.
Match Community:
When trying to match against a community, it sometimes generates syntax like:
route-map FILTER_ROUTES permit 10
match community 100:1
This is incorrect. The correct way to match against a community in a route-map is to first declare a community list that contains the community as in:
ip community-list 1 permit 100:1
and then while matching, make a call to a community list:
route-map FILTERROUTES permit 10
match community 1
Thus we included another IIP to define a community list and then in a route-map, match using only this list.
Adding Communities: While adding communities using a route-map, GPT-4 tends to generate syntax similar to:
route-map ADDCOUMMUNITY permit 10
set community 100:1
We observed that this happens even when we explicitly ask it to `add' a community to the route. The above route-map replaces all the communities that are already present in the route with the community 100:1.
So we added an initial prompt saying that it should always use the 'additive' keyword when adding a community to the route.
These initial prompts along with the syntax rectification scheme of Table <ref> are able to eliminate common syntax errors produced by GPT-4. Despite this, we found two egregious cases where human intervention is needed:
Placing neighbor commands in the wrong location: In a config file for BGP, all network and neighbor commands must be placed under the "router bgp" block. For example, the neighbor command is used to attach a route-map to the ingress or egress of an interface. We found that in rare situations, GPT-4 defines a route-map and then associates it with an interface outside the "router bgp" block. Batfish is able to catch this syntax error. However, the output is not informative enough for GPT-4 to be able to fix the issue.
AND/OR Semantics in match statements: GPT-4 does not understand the semantic difference between placing multiple match conditions under a single route-map stanza versus placing them in different stanzas. For no-transit, we had asked GPT-4 to generate a config for R1 that would add a different community to every route incoming from R2-R6 (Figure <ref>). We also asked it to filter routes containing any such community on the egress of the interface connecting R1 to R2-R6. GPT-4 added the correct communities at the ingress, but
at the egress at R1 it incorrectly used AND semantics for filtering routes as in the following route-map for the R1-R2 interface:
[]
ip community-list 1 permit 100:1
ip community-list 2 permit 101:1
ip community-list 3 permit 102:1
ip community-list 4 permit 103:1
ip community-list 5 permit 104:1
route-map FILTER_COMM_OUT_R2 deny 10
match community 2
match community 3
match community 4
match community 5
route-map FILTER_COMM_OUT_R2 permit 20
Community 100:1 is associated with routes incoming from R2, 101:2 with those coming from R3 and so on. We desire routes incoming from R3-R6 to be filtered out at the egress to R2. The above config will only filter out routes that have all the communities 101:1, 102:1, 103:1 and 104:1, not any one of them. When we asked Batfish whether the above route-map filters routes that have the community 101:1, it produced a counterexample but this counterexample to GPT-4 failed to rectify the issue. A human prompt was needed to ask GPT-4 to declare each match statement in a separate route-map stanza. Our attempts to help GPT-4 distinguish between AND and OR semantics using an example in the IIP also failed.
Leverage: The entire cycle took 2 human prompts and 12 automated prompts, for a leverage of 6X.
Note that the AND-OR problem required a final correction prompt.
§ PREVIOUS WORK
Jigsaw <cit.> and Copilot <cit.> use large language models for program synthesis. While they concentrate on
sequential programs, the deeper difference is that they do not pair the synthesizer with verifiers. Jigsaw <cit.> does ask users to provide test cases and tests (but does not verify) the synthesized program. Jigsaw also does some form of automatic syntax correction doing AST-to-AST transformations. CoPilot <cit.> can suggest invariants
but does not attempt an axiomatic proof. Jigsaw and Copilot do not address two questions we do: how to use
a specification, and how to provide localized feedback.
The use of ChatGPT with the Kani Rust verifier <cit.> comes closest to our vision; the Kani blog post finesses the specification question (as we do for Cisco to Juniper) by focusing on program transformations (in their case optimization) for which the source program is the specification. They also do not use modularity or local specifications. More fundamentally the Kani <cit.> use case does not do prompt programming: the user always manually switches between the verifier and the LLM, precluding possible leverage.
§ CONCLUSIONS
Our experiments are very preliminary but suggest:
1. Ramanujam Effect: As with the mathematician Ramanujam, some of whose conjectures were incorrect and needed Hardy's help <cit.> for proofs, GPT-4 by itself is not ready for use without a verifier, making elementary errors that can
bring networks down.
2. Verified Prompt Programming: Using a verifier and automated corrections via a humanizer,
GPT-4 can synthesize reasonable but not completely correct configurations for simple use cases, but the leverage in
reduced human effort can be high (5X to 10X). Modular verification seems crucial.
4. Local versus Global Specifications: Modular synthesis is the dual to modular verification. The search space for the LLM is large, which increases the chance that it will not be able to correctly complete a synthesis task based on a global specification. Instead the user needs to decide and describe the "roles" each node plays in satisfying the global spec.
Much further testing in more complex use cases is needed. Can GPT-4 add a new policy incrementally without interfering with existing verified policy? While our paper is set in the context of network configuration, the vision, definitions (e.g., leverage) and lessons (e.g., the need for actionable local feedback, modularity, humanizers and IIPs) seem more generally useful to synthesize other programs.
abbrv
|
http://arxiv.org/abs/2307.04767v1 | 20230710175940 | Semantic-SAM: Segment and Recognize Anything at Any Granularity | [
"Feng Li",
"Hao Zhang",
"Peize Sun",
"Xueyan Zou",
"Shilong Liu",
"Jianwei Yang",
"Chunyuan Li",
"Lei Zhang",
"Jianfeng Gao"
] | cs.CV | [
"cs.CV"
] |
Empirically Constraining the Spectra of a Star's Heterogeneities From Its Rotation Lightcurve
[
Received 24 May 2023 / Accepted 30 June 2023
=============================================================================================
In this paper, we introduce , a universal image segmentation model to enable segment and recognize anything at any desired granularity. Our model offers two key advantages: semantic-awareness and granularity-abundance. To achieve semantic-awareness, we consolidate multiple datasets across granularities and train on decoupled objects and parts classification. This allows our model to facilitate knowledge transfer among rich semantic information. For the multi-granularity capability, we propose a multi-choice learning scheme, enabling each click point to generate masks at multiple levels that correspond to multiple ground-truth masks. Notably, this work represents the first attempt to jointly train a model on SA-1B, generic, and part segmentation datasets. Experimental results and visualizations demonstrate that our model successfully achieves semantic-awareness and granularity-abundance. Furthermore, combining SA-1B training with other segmentation tasks, such as panoptic and part segmentation, leads to performance improvements. We will provide code and a demo for further exploration and evaluation at <https://github.com/UX-Decoder/Semantic-SAM>.
§ INTRODUCTION
The universal and interactive AI systems that follow human intents have shown their potential in natural language processing <cit.> and controllable image generation <cit.>. However, such a universal system for pixel-level image understanding remains less explored.
We argue that a universal segmentation model should possess the following important properties: universal representation,
semantic-awareness, and granularity-abundance. Regardless of the specific image domain or prompt context, the model is capable of acquiring a versatile representation, predicting segmentation masks in multi-granularity, and understanding the semantic meaning behind each segmented region.
Previous works <cit.> attempted to investigate these properties, but only achieved part of the goals. The main obstacles impeding the progress of such a universal image segmentation model can be attributed to limitations in both model architecture flexibility and training data availability.
* Model Architecture. The existing image segmentation model architectures are dominated by the single-input-single-output pipeline that discards any ambiguity. While this pipeline is prevalent in both anchor-based CNN architectures <cit.> and query-based Transformer architectures <cit.>, and has demonstrated remarkable performance in semantic, instance, and panoptic segmentation tasks <cit.>, it inherently restricts the model to predict multi-granularity segmentation masks in an end-to-end manner. Although clustering postprocessing techniques <cit.> can produce multiple masks for a single object query, they are neither efficient nor effective solutions for a granularity-aware segmentation model.
* Training Data. Scaling up segmentation datasets that possess both semantic-awareness and granularity-awareness is a costly endeavor. Existing generic object and segmentation datasets such as MSCOCO <cit.> and Objects365 <cit.> offer large amounts of data and rich semantic information, but only at the object level. On the other hand, part segmentation datasets such as Pascal Part <cit.>, PartImageNet <cit.>, and PACO <cit.> provide more fine-grained semantic annotations, but their data volumes are limited. Recently, SAM <cit.> has successfully scale up the multi-granularity mask data to millions of images, but it does not include semantic annotations. In order to achieve the dual objectives of semantic-awareness and granularity-abundance, there is a pressing need to unify segmentation training on various data formats to facilitate knowledge transfer. However, the inherent differences in semantics and granularity across different datasets pose a significant challenge to joint training efforts.
In this paper, we introduce Semantic-SAM, a universal image segmentation model designed to enable segmenting and recognizing objects at any desired granularity. Given one click point from a user, our model addresses the spatial ambiguity by predicting masks in multiple granularities, accompanied by semantic labels at both the object and part levels. As shown in Figure <ref>, our model generates multi-level segmentation masks ranging from the person head to the whole truck.
The multi-granularity capability is achieved through a multi-choice learning design <cit.> incorporated into the decoder architecture. Each click is represented with multiple queries, each containing a different level of embedding. These queries are trained to learn from all available ground-truth masks representing different granularities. To establish a correspondence between multiple masks and ground-truths, we employ a many-to-many matching scheme to ensure that a single click point could generate high-quality masks in multiple granularities.
To accomplish semantic-awareness with a generalized capability, we introduce a decoupled classification approach for objects and parts, leveraging a shared text encoder to encode both objects and parts independently. This allows us to perform object and part segmentation separately, while adapting the loss function based on the data type. For instance, generic segmentation data lacks part classification loss, whereas SAM data does not include classification loss.
To enrich semantics and granularity within our model, we consolidate seven datasets on three types of granularities, including generic segmentation of MSCOCO <cit.>, Objects365 <cit.>, ADE20k <cit.>, part segmentation of PASCAL Part <cit.>, PACO <cit.>, PartImagenet <cit.>, and SA-1B <cit.>. Their data formats are reorganized to match our training objectives accordingly. After joint training, our model obtains a strong performance across a variety of datasets. Notably, we find that learning from interactive segmentation could improve generic and part segmentation. For example, by jointly training SA-1B promptable segmentation and COCO panoptic segmentation, we achieve a gain of 2.3 box AP and a gain of 1.2 mask AP. In addition, through comprehensive experiments, we demonstrate that our granularity completeness is better than SAM with more than 3.4 1-IoU.
§ DATA UNIFICATION: SEMANTICS AND GRANULARITY
In order for multi-level semantics, we include seven datasets that contain different granularity-level masks. The datasets are SA-1B, COCO panoptic, ADE20k panoptic, PASCAL part, PACO, PartImageNet, and Objects365. Within them, COCO and ADE20k panoptic datasets contain object-level masks and class labels. PASCAL part, PACO, and PartImageNet contain part-level masks and class labels. SA-1B contains up to 6-level masks without labels, while Objects365 contains abundant class labels for object-level instances. The details of these datasets are shown in Table <ref>. We further visualize the data distribution of different data type in Fig <ref>.
[b]0.56
0.99!
2*Type 2*Data 2*#Images 2c|Semantic Concept 2cGranularity Level
Part Object Part Whole
Class-agnostic SA-1B 11B
2*Object-level
Objects365 1.7M 365
COCO 110K 133
ADE20K 20K 150
3*Part-level
PACO-LVIS 45K 201 75
PartImageNet 16K 13 11
Pascal Part 5K 30 20
tableThe data statistics in .
[b]0.43
< g r a p h i c s >
figureSemantics-Granularity 2D chart.
§
§.§ Model
Our follows <cit.> to exploit a query-based mask decoder to produce semantic-aware and multi-granularity masks. In addition to the generic queries, it supports two types of prompts including point and box, similar to SAM <cit.>. The overall pipeline is shown in Fig. <ref>.
We represent both click and box prompts into anchor boxes as a unified format. In particular, we convert user click point (x, y) into an anchor box (x, y, w, h) with small width w and height h, so that the anchor box can closely approximate the point. To capture different granularities of masks, each click is first encoded to position prompt and combined with K different content prompts, where each content prompt is represented as a trainable embedding vector for a given granularity level. Here we empirically choose K=6, considering there are at most 6 levels of masks per user click for the majority of images in SA-1B <cit.>.
More specifically, a click/box 𝐛=(x, y, w, h) is encoded into K content embeddings and one position embedding, respectively. We represent its content embeddings as a set of query vectors 𝐐 = ( _1, ⋯, _K). For the i-th query,
𝐪_i=𝐪^_i+𝐪^_i,
where
* 𝐪^ is the embedding for granularity level i,
* 𝐪^ distinguishes the query type, chosen from either the click or the box embeddings.
The position embedding of 𝐜 is implemented via sine encoding. Assuming that the output image feature from vision encoder is 𝐅, the mask decoder of the proposed represents the click on the input image as:
𝐎= (𝐐,, 𝐅 ) with O=(_1, ⋯, _K),
where (·,·,·) is a deformable decoder that takes query feature, reference box, and image features as input to output queried features. _i is the model output for the ith input query _i. Each _i=(_i, _i) consists of the predicted semantic category _i and mask _i, which are used to construct the concept recognition loss and mask prediction loss, respectively.
§.§ Training
r0.53
0.53
< g r a p h i c s >
figureDecoupled object and part classification.
Recognize Anything.
As we train with various types of data with different semantic annotations, in which some contain object-level annotations (COCO), some contain both object and part-level annotations (Pascal Part), and SA-1B has no semantic annotations but contains masks of all semantic levels. Note that a large number of part concepts are shared across different objects, for example, head for all animals. We aim to transfer the part concept knowledge across objects trained with only object-level annotations in our joint training.
To address this discrepancy between semantic annotations and better transfer semantics of different granularity, we propose to decouple object and part recognition. As shown in Fig <ref>, we utilize a shared text encoder to encode objects and parts, which are used to perform object and part segmentation separately.
Importantly, while all types of segmentation data share a unified format, the loss varies for different data types. We summarize the loss items to construct the training objective in in Table <ref>. It is the part-level data that bridges the gap to recognize semantic concepts between part and object levels, and it is the use of SAM data in Hungarian matching that bridges the gap to segment masks at any granularity.
Segment at any granularity.
To endow the model with a multi-granularity segmentation ability, we propose a many-to-many matching method during training. We found that SAM fails in providing good multi-level segmentation results with a single click because SAM uses many-to-one matching during training. In other words, the three SAM-predicted masks for each click only match with one GT mask. This causes that points located in masks of small levels cannot predict large masks with high quality according to our observation. In contrast, to enable multi-level mask prediction with a single click, we fully leverage the structures in both data and algorithm. First, we re-organize the data by clustering multiple GT masks of different levels sharing the same click. To allow multiple predictions of the same click to match with the GT masks, we employ the Hungarian algorithm to enable the many-to-many matching. The similarity matrix and scores vary based on the availability of different segmentation data components.
For box input and generic segmentation, we follow existing methods. Specifically, to generate a mask from an input box, we follow a similar idea as in denoising training (DN) <cit.>. We add noises to ground-truth boxes to simulate inaccurate box inputs from users, and these noised boxes serve as spatial prompts for the decoder. The model is trained to reconstruct the original boxes and masks given noised boxes. For the content part of box prompts, we adopt a learnable token as a general prompt. Note that this is the only difference from DN, as DN uses ground-truth label embedding as the content prompts.
For generic segmentation, we follow the same pipeline as in Mask DINO <cit.>.
Discussion.
As shown in Fig. <ref>, compared with previous interactive segmentation models, differs from previous segmentation models in two aspects. Firstly, we train the model to output all the possible segmentation masks with one click. Secondly, our output granularities are richer to generate diverse output masks.
§ EXPERIMENTS
§.§ Experimental Setup
Implementation Details.
In our experiments, we jointly train on three types of data, as shown in Table <ref>.
We implement our model based on Mask DINO <cit.> . Mask DINO is a unified detection and segmentation framework which simultaneously predicts box and mask. We follow <cit.> to use 300 latent queries and nine decoder layers for all segmentation tasks. For the visual backbone, we adopt pre-trained Swin-T/L <cit.> by default. For the language backbone, we adopt the pre-trained base model in UniCL <cit.>.
As SA-1B <cit.> dominates the data, during training, we first train on only SA-1B data. Then, we add object and part-level data to jointly train the three types of data. During training, the image resolution is 1024× 1024 for all data. We use AdamW <cit.> as the optimizer. We use large-scale jittering for object and part-level data and did not use data augmentations for SA-1B data, as SA-1B images are abundant. We set the learning rate to 0.0001, which is decayed at 0.9 and 0.95 fractions of the total number of steps by 10.
Evaluation. We mainly evaluate two datasets, including COCO Val2017 and a subset of SA-1B <cit.> with 1000 images. For evaluation metrics, we evaluate PQ and AP for generic and part segmentation datasets. For single-granularity interactive segmentation, we report Point (Max) and Point (Oracle). Max denotes we select the output mask with the maximum confidence score. Oracle
denotes we select the output mask with the max IoU by calculating the IoU between the prediction and target mask. For multi-granularity interactive segmentation, we report 1-IoU@All Granularity that matches all the possible ground-truth masks for a single click to the multi-granularity predictions and then calculate the average IoU of all granularities.
§.§ Semantic Segmentation of Anything
Generic Segmentation
As shown in Table <ref>, to validate the compatibility of multi-granularity interactive segmentation and generic segmentation, we jointly train with SA-1B <cit.> (1/10 data) and COCO panoptic segmentation. The result indicates that interactive segmentation with SAM can significantly help the instance-level detection and segmentation with a performance improvement of +2.2 AP on the box and +1.3 AP on the mask. Notably, OpenSeed <cit.> and are both based on Mask DINO <cit.>. Our joint training with SA-1B even outperforms OpenSeed which is trained with Object365 <cit.>. In addition, adding SA-1B mainly improves small object detection (APs and APm), as there are a large number of small objects in SA-1B.
Part Segmentation
We also validate the compatibility of joint training SA-1B (1/10 data) and part segmentation. As shown in Table <ref>, adding SA-1B brings a decent performance improvement on Pascal Part <cit.>.
Single-granularity Interactive Segmentation
In Table <ref>, we evaluate the 1-click mIoU (denoted as 1-IoU) for SAM and our model on COCO Val2017. Our model outperforms SAM under the same settings.
Multi-granularity Interactive Segmentation
In Table <ref>, we compare SAM <cit.> and our model on the output granularities for a single click. We adopt a Hungarian Matching to match all the possible target masks with the predicted masks for the click and calculate the average IoU score. As SAM has only three prompts, we also sample two clicks from a single mask to produce six output masks for a fair comparison. Notably, SAM has been trained on this validation set while we did not.
§.§ Abaltions
Match Strategy
As shown in Table <ref>, we compare different match strategies in our model. When using many-to-many matching to match all the possible ground-truth masks for each click, the 1-IoU@All Granularity performance is significantly improved. This validates our matching strategy is effective to learn complete granularities.
Box Interactive Evaluation
We also evaluate the 1-IoU given boxes in Table <ref>. We achieve better performance compared with object-level interactive segmentation model SEEM <cit.> and multi-granularity model SAM <cit.>.
Increasing SA-1B Training data
In Table <ref>, we show the performance improvement on COCO Val 2017 when training with more SA-1B data. The performance is saturated after using more than 15% of the total data. It indicates that we do not need
to train with the whole SA-1B data to get a good zero-shot performance.
§.§ Visualization
We compare our model with SAM to show that our model can output more levels of high-quality masks, as shown in Fig. <ref>.
Multi-Level Masks Our model outputs more meaningful granularities of masks. SAM outputs three masks at most and different levels of outputs are sometimes duplications, While, the output masks of our model are more diverse.
Mask Qualities
It is also proved that our model output masks with higher quality. SAM sometimes outputs masks with artifacts such as holes or islands especially for large masks when the click is within a small-scale mask, while our model output high-quality masks for all levels.
Compare with SA-1B Ground-truth Granularity
We output more meaningful granularity on SAM data compared with the original annotation.
Query semantics
We also find that each point content prompt embeddings learns to correspond to a fixed granularity. As shown in Fig. <ref>, when we visualize masks in a specific order of the corresponding content embeddings, the masks follow the order from small to large in each row consistently. This proves that each content embedding represents a semantic granularity level in our model.
§ RELATED WORKS
§.§ Generic Segmentation
Segmenting visual concepts is well-documented within the expansive field of computer vision <cit.>. Broad segmentation methodologies comprise several subdivisions, such as instance segmentation, semantic segmentation, and panoptic segmentation <cit.>, each catering to a unique semantic degree. For example, semantic segmentation's goal is to detect and assign a label to each pixel in an image according to its corresponding semantic class <cit.>. Conversely, instance segmentation seeks to cluster pixels associated with the same semantic class into distinct object instances <cit.>. Panoptic segmentation is the hybrid of these two tasks.
Recently, Transformer-based methods <cit.> have contributed to significant progress in segmentation tasks <cit.>.
Generic object detection and segmentation have led to the development of abundant datasets, such as MSCOCO <cit.>, LVIS <cit.>, Objects365 <cit.>, PASCAL <cit.>,CityScapes <cit.>,ADE20k <cit.>, etc.
§.§ Part Segmentation
Beyond generic segmentation, part segmentation aims to more fine-grained visual understanding.
Most early works were bottom-up methods by grouping super-pixels into parts and then objects <cit.>. Later, based on high-performance object detection networks <cit.>, top-down methods were developed by firstly
detecting an object and then parsing it to part segmentation <cit.>. To segment the scene in multi-granularity, part-aware panoptic segmentation <cit.> is introduced. PPS <cit.> establishes the baseline through assembling panoptic and part segmentation models. JPPF <cit.> simplifies the model by a shared image encoder for both panoptic segmentation and part segmentation. By representing thing, stuffs, and parts as object queries, Panoptic-PartFormer <cit.> proposes a unified architecture based on Transformer. While part segmentation data is much expensive than
object detection and segmentation data, a number of public datasets are available. Datasets for specific domains include cars <cit.>, birds <cit.>, and fashion <cit.>. General objects include Pascal-Part <cit.>, PartImageNet <cit.>, ADE20K <cit.>, Cityscapes-Panoptic-Parts <cit.>, and PACO <cit.>. More recently, SAM <cit.> provides a large-scale multi-granularity class-agnostic segmentation dataset. Our work is jointly trained on these datasets and contributes to a multi-granularity segmentation model.
§.§ Open-Vocabulary Segmentation
While generic segmentation and part segmentation have made remarkable progress, they can only segment the image in a close-set vocabulary. To expand the vocabulary size, recent works leverage the visual-semantic knowledge from large-scale foundation models like CLIP <cit.>, ALIGN <cit.> and Diffusion models <cit.> to various segmentation tasks. LSeg <cit.>, OpenSeg <cit.>, GroupViT <cit.> achieves open-vocabulary semantic segmentation ability on ADE20K and PASCAL. DenseCLIP <cit.> and MaskCLIP <cit.> achieves open-vocabulary instance and panoptic segmentation on COCO dataset. More recently, X-Decoder <cit.> proposes a unified approach to tackle various segmentation and vision-language tasks for open-vocabulary segmentation, OpenSeeD <cit.> proposes to use a large amount of detection data and a joint training method to improve segmentation. To segment open-vocabulary masks in part-level, VLPart <cit.> leverages three part segmentation datasets and learns from the dense correspondence <cit.> between base objects and novel objects. Our work unifies these tasks into one architecture and builds up open-vocabulary segmentation in multi-granularity.
§.§ Interactive Segmentation
Interactive segmentation refers to the process of separating objects by actively integrating user inputs. This enduring challenge has seen notable advancements <cit.>. Previous works only focus on a small set of data or semantic-agnostic instance masks. Recently, SAM <cit.> enlarges the training data from 0.12M COCO images to 10M SAM fine-grained images. And SEEM <cit.> enriches the modality to language and function to both generic and grounded segmentation with an impressive compositionality.
§ CONCLUSION
In this paper, we have presented , which can segment and recognize anything at any desired granularity. Apart from performing generic open-vocabulary segmentation, demonstrates the advantages of semantic awareness and granularity abundance. To achieve such advantages, we have proposed improvements on data, model, and training where we utilized datasets from multiple granularity and semantic levels, multi-choice learning for training, and a universal framework for modeling. Comprehensive experiments and visualizations have verified the semantic awareness and granularity abundance of our model. Further, is the first successful attempt to jointly train on SA-1B and other classic segmentation datasets. Experimental results also show that training with SA-1B improves other tasks such as panoptic and part segmentation.
ieee_fullname
|
http://arxiv.org/abs/2307.04019v3 | 20230708173320 | GP-guided MPPI for Efficient Navigation in Complex Unknown Cluttered Environments | [
"Ihab S. Mohamed",
"Mahmoud Ali",
"Lantao Liu"
] | cs.RO | [
"cs.RO",
"cs.AI",
"cs.SY",
"eess.SY"
] |
Explicit a posteriori error representation for variational problems and application to TV-minimization
[
August 12, 2023
========================================================================================================
@topnum0
@botnum0
empty
empty
Robotic navigation in unknown, cluttered environments with limited sensing capabilities poses significant challenges in robotics. Local trajectory optimization methods, such as Model Predictive Path Intergal (MPPI), are a promising solution to this challenge. However, global guidance is required to ensure effective navigation, especially when encountering challenging environmental conditions or navigating beyond the planning horizon.
This study presents the GP-MPPI, an online learning-based control strategy that integrates MPPI with a local perception model based on Sparse Gaussian Process (SGP).
The key idea is to leverage the learning capability of SGP to construct a variance (uncertainty) surface, which enables the robot to learn about the navigable space surrounding it, identify a set of suggested subgoals, and ultimately recommend the optimal subgoal that minimizes a predefined cost function to the local MPPI planner.
Afterward, MPPI computes the optimal control sequence that satisfies
the robot and collision avoidance constraints.
Such an approach eliminates the necessity of a global map of the environment or an offline training process.
We validate the efficiency and robustness of our proposed control strategy through both simulated and real-world experiments of 2D autonomous navigation tasks in complex unknown environments, demonstrating its superiority in guiding the robot safely towards its desired goal while avoiding obstacles and escaping entrapment in local minima. The GPU implementation of GP-MPPI, including the supplementary video, is available at <https://github.com/IhabMohamed/GP-MPPI>.
Autonomous vehicle navigation, MPPI, sparse Gaussian process (SGP), occupancy grid map path planning.
§ INTRODUCTION AND RELATED WORK
Autonomous navigation of mobile robots
in unknown, cluttered, and unpredictable environments with limited sensor capabilities is a challenging task owing to the inherent uncertainty and complexity of such environments.
To tackle this challenge, a receding-horizon strategy such as Model Predictive Control (MPC) is commonly employed.
The MPC control framework allows the robot to simultaneously plan a short trajectory (sequence of actions), following which the robot executes the immediate action while planning a subsequent trajectory.
To successfully achieve receding-horizon planning, the robot must consider both safety and persistent feasibility, where safety is achieved by avoiding collisions with any obstacles while executing a planned trajectory, and persistent feasibility is maintained by always
generating a safe trajectory that does not result in dead-ends or local minima while progressing towards the desired goal.
One of the significant challenges in robot motion planning is that the desired goal is often situated beyond the planning horizon, which requires the use of local subgoals or cost-to-go heuristics
for motion safety and persistent feasibility.
A common strategy is to rely on single-query motion planning algorithms, such as A^* and RRT^X, to identify feasible paths that direct the local planner towards its desired goal <cit.>.
For instance, the RRT^X algorithm, introduced in <cit.>, incorporates replanning techniques from Dynamic
Rapidly-exploring Random Trees (DRRT) and Rapid-exploring Random Trees (RRT^*) algorithms to adjust the path during exploration based on environmental changes. However, due to its high computational demands, implementing this algorithm in real-time on a robot can be challenging.
One alternative method to achieve efficient solutions for motion planning problems
is the integration of MPC with data-driven methods, also known as learning-based MPC <cit.>.
To name a few, a subgoal planning policy using Deep Reinforcement Learning (DRL) is recently proposed to guide the local MPC planner to navigate in crowded surroundings <cit.>.
Similarly, RL was utilized to choose the next subgoal from a set of predefined possibilities <cit.>, which guides the robot through challenging environments with dead-end corridors while also prevents the MPC planner from getting trapped in local minima.
Another related work that combines learning with MPC is POLO which aims to enhance MPC performance by learning a global value function <cit.>.
Most of these approaches typically rely on either offline training or having access to the global map of the environment.
In addition, many recent studies have suggested combining Gaussian Process (GP) with MPC to learn system dynamics, leading to better control performance and robustness to uncertainty <cit.>.
Another research avenue employed gap-based techniques that identify gaps as free spaces between obstacles, enabling a robot to move through them while avoiding local minima and obstacles.
The first developed method was the Nearness Diagram (ND) <cit.>, but many of its variants exhibited undesired oscillatory motion.
To overcome these limitations, robotics researchers have developed techniques that rely on the geometry of the gap. One such technique is the Follow-the-Gap Method (FGM), which selects a gap based on its area and computes the robot's heading using the gap center's direction relative to both the robot and the final goal <cit.>.
Another approach is the sub-goal seeking method, which assigns a cost to each sub-goal based on the goal heading error with respect to the robot and the gap heading, and then selects the sub-goal with the lowest cost (error) <cit.>.
The Admissible Gap (AG) method <cit.>, an iterative algorithm that takes into account the exact shape and kinematic constraints of the robot, identifies possible admissible gaps, and selects the nearest gap as the goal.
Different from all these strategies, our proposed framework leverages a Sparse variant of Gaussian Process (SGP) which is a new perception model by “abstracting” local perception data so that the local sub-goal for navigation can be naturally extracted.
Specifically, we introduce the GP-MPPI control strategy, which enhances the state-of-the-art sampling-based MPC, Model Predictive Path Integral (MPPI) <cit.>, by incorporating the GP-subgoal recommender policy. Such a policy takes advantage of the SGP occupancy model to learn about the navigable space surrounding the robot, identifies a set of suggested subgoals, and ultimately recommends the optimal subgoal that minimizes a predefined cost function to the MPPI local planner, as demonstrated in Fig. <ref>.
Subsequently, MPPI computes the optimal control sequence that satisfies the robot and collision avoidance constraints while moving towards the recommended subgoal, followed by executing the first optimal control 𝐮_0 to the robot.
In summary, the contributions of this work can be summarized as follows:
* We propose an online learning-based control strategy that recommends subgoals solely based on local sensory information, ensuring safety and persistent feasibility; such an approach eliminates the need for a global map of the environment or an offline training process as in RL techniques, resulting in a more flexible and agile control framework that can be easily deployed in different unexplored environments, as revealed in Section <ref>.
* To the best of the authors' knowledge, this is the first attempt to utilize the SGP occupancy model in conjunction with sampling-based trajectory optimization methods, specifically MPPI, to efficiently explore the navigable space surrounding the robot.
* In Sections <ref> and <ref>, we validate our GP-MPPI control strategy for collision-free navigation in complex and unknown cluttered environments, using both simulation and experimental demonstrations; by comparing it with two baseline sampling-based approaches (namely, MPPI <cit.>, and log-MPPI <cit.>), we show its effectiveness in overcoming local minima that may arise when the sampled trajectories of MPPI are concentrated in high-cost regions or due to challenging environmental conditions.
§ PRELIMINARIES
To provide the necessary background for our proposed work, in this section, we formulate the optimal control problem and present a concise overview of the MPPI control strategy that can be utilized to address this problem, along with a brief introduction to the Sparse Gaussian Process (SGP) which is the backbone of our GP-subgoal recommender policy.
§.§ Problem Formulation
Consider a nonlinear discrete-time stochastic dynamical system
𝐱_k+1=f(𝐱_k,𝐮_k+δ𝐮_k),
with 𝐱_k ∈ℝ^n_x and 𝐮_k ∈ℝ^n_u representing the state of the system and its control input, respectively. The disturbance introduced into the control input, δ𝐮_k, is modeled as a zero-mean Gaussian noise with co-variance Σ_𝐮.
Given a finite time-horizon N, we define the control sequence 𝐔 as 𝐔 = [𝐮_0, 𝐮_1, …,𝐮_N-1]^⊤∈ℝ^n_u N and the resulting state trajectory of the system being controlled as 𝐗 = [𝐱_0, 𝐱_1, …, 𝐱_N]^⊤∈ℝ^n_x (N+1).
Furthermore, 𝒳^d is used to represent the d-dimensional space with 𝒳_rob(𝐱_k) ⊂𝒳^d and 𝒳_o b s⊂𝒳^d representing the robot's occupied area and obstacles' area, respectively.
Let 𝐱_s and 𝐱_f denote the initial and desired (goal) state of the robot, respectively.
Given 𝒳_rob(𝐱_k), 𝒳_o b s, 𝐱_s, and 𝐱_f, we aim to find the optimal control sequence, 𝐔, that allows the robot to safely and efficiently navigate from its initial state, 𝐱_s, to the desired state, 𝐱_f, by avoiding both getting stuck in local minima and collisions with obstacles, while minimizing a cost function J.
The optimization problem at hand can be approached utilizing the classical MPPI control strategy described in <cit.>.
This optimization can be mathematically expressed as in (<ref>), with the objective of minimizing the cost function, J, which is comprised of the expectation of a combination of state terminal cost ϕ(𝐱_N), running cost q(𝐱_k), and control inputs 𝐮_k, weighted by the positive-definite matrix R∈ℝ^n_u × n_u, taking into consideration the system dynamics outlined in (<ref>) and constraints such as collision avoidance and control constraints as stated in (<ref>).
min _𝐔 J = 𝔼[ϕ(𝐱_N)+∑_k=0^N-1(q(𝐱_k)+1/2𝐮_k^⊤ R 𝐮_k)],
s.t. 𝐱_k+1=f(𝐱_k, 𝐮_k+δ𝐮_k), δ𝐮_k∼𝒩(0, Σ_𝐮),
𝒳_rob(𝐱_k) ∩𝒳_obs=∅, 𝐡(𝐱_k, 𝐮_k) ≤ 0,
𝐱_0 = 𝐱_s, 𝐮_k∈𝕌, 𝐱_k∈𝕏.
§.§ Overview of MPPI Control Strategy
In order to solve the optimization control problem defined in (<ref>), MPPI leverages Monte Carlo simulation to generate a significant number of real-time simulated trajectories by propagating them from the underlying system dynamics. It then evaluates the cost-to-go of each trajectory based on a predefined cost function and updates the optimal control sequence by considering a weighted average cost from all of the simulated trajectories. More details are given in <cit.>.
Subsequently, each trajectory τ_i in the time-horizon N can have its cost-to-go evaluated as given in (<ref>), where the cost-to-go S̃(τ_i) is calculated as the sum of the terminal state cost ϕ(𝐱_N) and the instantaneous running cost q̃(𝐱_k, 𝐮_k, δ𝐮_k,i) over all time steps.
The instantaneous running cost, q̃, expressed in (<ref>), is comprised of the state-dependent running cost q(𝐱_k) and the quadratic control cost q(𝐮_k, δ𝐮_k), where γ_𝐮 = ν -1/2ν and the aggressiveness in exploring the state-space is determined by the parameter ν∈ℝ^+. Specifically,
S̃(τ_i ) =ϕ(𝐱_N) + ∑_k=0^N-1q̃(𝐱_k, 𝐮_k, δ𝐮_k,i) ∀ i ∈{0, ⋯, M-1},
q̃=
q(𝐱_k)_State-dep.+
γ_𝐮δ𝐮_k,i^⊤ R δ𝐮_k,i+ 𝐮_k^⊤ R δ𝐮_k,i+ 1/2𝐮_k^⊤ R 𝐮_k_q(𝐮_k, δ𝐮_k): Quadratic Control Cost.
As outlined in (<ref>) from <cit.>, the optimal control sequence {𝐮_k}_k=0^N-1 in the vanilla MPPI algorithm is iteratively updated by taking a weighted average cost from all simulated trajectories, where S̃(τ_m) represents the cost-to-go of the m^th trajectory, and λ∈ℝ^+ denotes the “inverse temperature”, which regulates the selectiveness of the weighted average of the trajectories.
After smoothing the resulting control sequence with a Savitzky-Galoy filter <cit.>, the first control 𝐮_0 is executed in the system, with the remaining sequence utilized as a warm-start for the next optimization step. Formally,
𝐮_k←𝐮_k +∑_m=0^M-1exp( -1/λS̃(τ_m) ) δ𝐮_k, m/∑_m=0^M-1exp( -1/λS̃(τ_m) ).
§.§ Sparse Gaussian Process
Gaussian Process (GP) is a well-established non-parametric model described by a mean function m(z) and a co-variance function k(z, z^')
(also referred to as kernel function), where z∈ℝ^n_g is the input to the GP <cit.>; it can be mathematically expressed as
f(𝐳) ∼𝒢 𝒫(m(𝐳), k(𝐳, 𝐳^')).
Let 𝒟 = {(𝐳_i, y_i)}_i=1^n denote a dataset consisting of n input-output pairs, where each output y_i ∈ℝ is assumed to be the sum of an unknown underlying function f(𝐳_i) and Gaussian noise ϵ_i with a zero-mean and variance σ^2, i.e., ϵ_i ∼𝒩(0, σ^2).
In the context of GP regression, to estimate the output y^* for a given new input z^*, the following GP prediction equation is employed
p(y^* | y) = 𝒩(y^* | m_y(z^*), k_y(z^*,z^*) + σ^2),
m_𝐲(𝐳) =K_𝐳 n(σ^2 I+K_n n)^-1𝐲,
k_𝐲(𝐳, 𝐳^') =k(𝐳, 𝐳^')-K_𝐳 n(σ^2 I+K_n n)^-1 K_n 𝐳^',
where m_𝐲(𝐳) and k_𝐲(z,z^') are the GP posterior mean and co-variance functions, respectively, while K_nn∈ℝ^n × n refers to the n × n co-variance matrix of the training inputs and K_𝐳n∈ℝ^n is n-dimensional row vector of kernel function values between 𝐳 and the training inputs, with K_n𝐳 = K_𝐳n^⊤.
Achieving a more accurate GP prediction requires the optimization of hyper-parameters, such as kernel parameters Θ and noise variance σ^2, by maximizing the log marginal likelihood
log p(𝐲)=log[𝒩(𝐲|0, σ^2 I+K_n n)].
The standard GP can be computationally intensive due to its complexity of 𝒪(n^3), where n represents the number of training instances. To mitigate this issue, various approximation methods, collectively known as Sparse Gaussian Process (SGP), have been developed as an alternative approach. Instead of using the complete training data, SGP employs a smaller set of m_s training points, called inducing points Z_m_s, resulting in a more efficient process and a lower computation complexity of 𝒪(n m_s^2) <cit.>.
Our present work leverages the variational SGP method, proposed in <cit.>, to approximate the true posterior of a GP p(f|𝐲) using an approximated variational posterior distribution q(f,f_m_s), where f_m_s are the values of the underlying function f at the inducing points Z_m_s.
This approximation is done by augmenting the true posterior with the variable f_m_s such as p(f,f_m_s|𝐲) = p(f|f_m_s) p(f_m_s|y).
Then, the approximated variational distribution q(f,f_m_s) can be factorized in the same manner as the augmented true posterior, as follows
q(f,f_m_s) = p(f|f_m_s)ϕ(f_m_s),
where ϕ(f_m_s) is an unconstrained variational distribution over f
_m_s and p(f|f_m_s) is the conditional GP prior.
By minimizing the Kullback-Leibler (KL) divergence between the approximated and true posteriors, 𝕂𝕃[q(f, f_m_s)||p(f|𝐲)], the variational SGP obtains estimates of the inducing inputs Z_m_s and hyperparameters (Θ, σ^2).
§ GP-MPPI CONTROL STRATEGY
The goal of our present research, as outlined in (<ref>), is to determine the optimal control sequence 𝐔={𝐮_k}_k=0^N-1 that enables safe and efficient navigation of the mobile robots through complex and unknown cluttered environments, while avoiding collisions with obstacles and getting trapped in local minima.
Although the MPPI control framework, as summarized in <cit.>, has many positive attributes, it is prone to generating infeasible control sequences or trajectories, particularly when the distribution of all sampled trajectories are concentrated within high-cost regions.
To mitigate this issue, new sampling strategies proposed in <cit.> have enabled more efficient exploration of the state-space, allowing the algorithm to find better solutions and potentially reduce the risk of trapping in local minima.
Nevertheless, for specific tasks such as the one depicted in Fig. <ref>, eliminating the local minima remains a potential challenge that needs to be tackled.
One solution could be incorporating MPPI with a global planner, such as the solution presented in <cit.>, which utilizes the RRT algorithm to guide MPPI.
Instead, we introduce the GP-MPPI control strategy, a new online navigation technique
that leverages the SGP occupancy model to learn about the navigable space surrounding the robot. Specifically, we introduce the GP-subgoal recommender policy, which identifies a set of recommended subgoals and subsequently suggests the optimal subgoal that minimizes a predefined cost function to the MPPI local planner, as depicted in Fig. <ref> and explained in detail in Section <ref>.
Unlike conventional methods, a distinctive aspect of the proposed control strategy is that it does not require either a global map for long-term planning or an offline training process.
§.§ SGP Occupancy Surface Representation
Our proposed GP-subgoal recommendation policy relies on our earlier work presented in <cit.>, where we transformed pointcloud data into an occupancy surface and modeled it using a Sparse Gaussian Process (SGP).
Within this approach, the occupancy surface takes the form of a 2D circular surface centered around the sensor origin and has a predefined radius of r_oc.
This surface serves as the projection space for all observed points, which are represented in spherical coordinates (θ_i, α_i, r_i), where (θ_i, α_i, r_i) correspond to the azimuth, elevation, and radius values of each observed point, respectively.
Each point 𝐳_i on the occupancy surface is defined by two attributes: the azimuth and elevation angles 𝐳_i= (θ_i, α_i), and assigned an occupancy value f(𝐳_i) that is a function of the point radius r_i, such as f(𝐳_i)=r_oc-r_i. Afterward, the probability of occupancy f(𝐳) over the occupancy surface is modeled by an SGP occupancy model, as follows
f(𝐳) ∼𝒮𝒢𝒫(m(𝐳), k(𝐳, 𝐳^')),
k(𝐳, 𝐳^') =σ_f^2(1+(𝐳-𝐳^')^2/2 αℓ^2)^-α,
where σ_f^2 is the signal variance, l is the length-scale, and α is the relative weighting factor that manipulates large and small scale variations.
In our SGP model, the point's occupancy to radius relation is encoded as a zero-mean function, m(𝐳)=0, where the occupancy value of the non-observed points is set to zero.
The Rational Quadratic (RQ) kernel, k(𝐳, 𝐳^'), is selected as the SGP kernel due to its ability to model functions that vary across different length-scale <cit.>. This characteristic makes the RQ kernel well-suited for modeling the occupancy surface.
In Fig. <ref>, we present a concrete example of the SGP occupancy model applied to our Jackal robot, which is equipped with a Velodyne VLP-16 LiDAR and located in an unknown cluttered environment, as depicted in Fig <ref>. The figure also illustrates the raw pointcloud generated by the onboard sensor (Fig <ref>), as well as the original occupancy surface, which represents the projection of the point clouds onto the 2D circular surface with radius r_oc, where warmer colors indicate areas of lower occupancy (Fig <ref>). Furthermore, Fig <ref> exhibits the SGP occupancy surface reconstructed by the SGP occupancy model, as previously expressed in (<ref>).
The precision of the SGP occupancy model is intensively evaluated in our previous work <cit.>, where the results showed that an SGP occupancy model comprising of 400 inducing points generates a reconstructed point cloud with an average error of approximately 12.
§.§ GP-Subgoal Recommender Policy
The primary advantage of GP and its variants, compared to other modeling techniques, is their ability to provide a measure of variance, which indicates the level of uncertainty, along with a function estimate (i.e., mean).
More precisely, in the context of the occupancy surface, the SGP occupancy model prediction, as defined in (<ref>), provides both mean μ_oc_i and variance σ_oc_i values for each point on the surface, where the mean represents the expected occupancy while the variance reflects the uncertainty associated with the predicted occupancy.
Consequently, constructing the SGP occupancy surface is accompanied by an SGP variance surface that captures the uncertainty in the occupancy estimate, as depicted in Fig. <ref>.
Within this research, we have opened up a new avenue for effectively utilizing the SGP variance surface as a reliable indicator for distinguishing between occupied and free spaces around the robot, where regions with variances higher than a certain threshold V_th correspond to free space, while low-variance regions indicate occupied space.
In fact, the variance surface changes across observations due to variations in the number and distribution of observed points employed in the training of the SGP model.
As a result, the variance threshold V_th is considered to be a variable that relies on the distribution of the variance across the surface and can be calculated as V_th=K_m v_m, where K_m ∈ℝ^+ is a tuning parameter
and v_m represents the mean of the variance distribution.
To identify free navigable spaces, we define a Gaussian Process frontier (namely, GP frontier) as the centroid point (θ_i, α_i) of each high variance region. These GP frontiers {f_i}_i=1^ℱ serve as local recommended subgoals (see colored circles in Fig. <ref>).
Unlike the well-known frontier concept introduced in <cit.>, it is worth noting that our GP frontier does not rely on a global occupancy map; instead, it is extracted from the uncertainty of the current observation.
Following the identification of the GP frontiers by the SGP model, a cost function J_gp is utilized to determine the optimal GP frontier f^* that guides the local planner (in our case, MPPI) towards the desired state 𝐱_f.
Our cost function J_gp, given in (<ref>), has been established with two distinct terms. The first term, as introduced in <cit.>, calculates the distance d_fs between a GP frontier f_i and the desired state 𝐱_f. This distance criterion is used to identify the GP frontier closest to 𝐱_f. The second term, inspired by the direction criterion proposed in <cit.>, evaluates the direction θ_f_i of a GP frontier with respect to the robot heading. This criterion prioritizes a GP frontier that aligns better with the robot heading.
J_gp(f_i) = k_dst d_fs + k_dirθ_fi^2 ,
f^* =argmin _f_i∈ℱ(J_gp(f_i)),
where k_dst, k_dir are weighting factors. The GP frontier direction θ_f_i is squared to indicate the absolute direction.
Finally, the local planner receives the optimal subgoal g^*, obtained by acquiring the Cartesian coordinate of the optimal GP frontier f^*, which leads the robot to its desired state 𝐱_f.
§.§ Real-Time GP-MPPI Control Algorithm
Algorithm <ref> summarizes the real-time control cycle of the GP-MPPI algorithm, which includes two primary components: the local MPPI motion planner (described earlier in Section <ref>) and the GP-subgoal recommender (explained in Section <ref>).
Each time-step Δ t, the GP policy recommends the optimal subgoal g^*, the current state is estimated, and a M × N random control variations δ𝐮 are generated (lines 2:4). Then,
M trajectories are simulated in parallel, propagated from the system dynamics defined in (<ref>), and evaluated using (<ref>) (lines 5:13). It is noteworthy that the minimum sampled cost trajectory, denoted as S̃_min, among all simulated trajectories prevents numerical overflow or underflow without affecting the optimality of the algorithm <cit.>.
After that, the optimal control sequence {𝐮_k}_k=0^N-1 is updated, smoothed with a Savitzky-Galoy filter, and the first control 𝐮_0 is applied to the system (lines 14:18), while the remaining sequence of length N - 1 is slid down to be utilized at next time-step (lines 19:22).
In lines 25 to 38, the function known as GP-SubgoalRecommender is described, which takes a pointcloud input (PCL) and returns the optimal subgoal g^* for the local planner.
To optimize the hyper-parameters Θ and inducing points Z_m_s of the SGP occupancy model, the pointcloud data is transformed into training data 𝒟 (lines 26:29).
The mean occupancy μ_oc and variance σ_oc are then estimated over the surface Z^*, and the GP frontiers are defined as those with σ_oc > V_th, where the centroids of these frontiers are converted to Cartesian coordinates (lines 30:34). Finally, the cost function J_gp in (<ref>) is used to select the optimal subgoal g^* (lines 35:37).
In this study, we introduce two operating modes for the GP-MPPI algorithm: the simple mode (SM) and the recovery mode (RM). Under the simple mode, MPPI consistently leverages the optimal subgoal 𝐠^* suggested by the GP policy. In contrast, in the recovery mode, MPPI generates the optimal control sequence that steers the robot towards its desired state 𝐱_f, adhering to the recommended subgoal only when the robot is at risk of encountering local minima. Such local minima occur when the robot's linear velocity is zero (v=0) and its current state 𝐱_k does not match 𝐱_f (i.e., 𝐱_k ≠𝐱_f).
Thanks to the optimal control sequence {𝐮_k}_k=0^N-1 obtained by MPPI, we can efficiently anticipate the occurrence of local minima by imposing a condition on the mean of the predicted linear velocities over the time-horizon N, expressed as follows:
μ_𝐮 = 1/N∑_i=0^N-1 |v_i| < 𝐮_th,
where 𝐮_th∈ℝ^+ represents a control switching threshold set based on N.
If this condition is fulfilled, then MPPI will follow the subgoal recommended by the GP rather than navigating directly towards its desired state 𝐱_f.
§ SIMULATION-BASED EVALUATION
In this section, the effectiveness of our proposed control strategy is assessed and compared with both vanilla MPPI and log-MPPI control strategies in a goal-oriented autonomous ground vehicle (AGV) navigation task conducted in 2D cluttered environments of unknown nature.
§.§ Simulation Setup:
In this study, we consider the kinematics model of a differential wheeled robot presented in <cit.>, specifically the fully autonomous ClearPath Jackal robot, where the robot's position and orientation in the world frame are given by 𝐱 = [x, y, θ]^⊤∈ℝ^3, and the control input 𝐮 = [v,ω]^⊤∈ℝ^2 denotes the robot's linear and angular velocities.
Our autonomous AGV platform is equipped with a 16-beam Velodyne LiDAR sensor utilized for two key functions: (i) constructing the SGP variance surface, and (ii) generating the local costmap.
The simulations for all proposed control schemes were conducted with the following parameters: a prediction time of 6, a control frequency of 30 (i.e., N=180), sampling 2528 rollouts per time-step Δ t, and an exploration variance ν of 1200. Additionally, a control weighting matrix R, expressed as λΣ_n^-1/2, is utilized.
In the case of MPPI and GP-MPPI, the inverse temperature λ and the control noise co-variance Σ_𝐮 = Σ_n = Diag(σ_v^2, σ_w^2) are both set to 0.572 and Diag(0.023, 0.028), respectively. However, for log-MPPI, different values of 0.169 and Diag(0.017, 0.019) are used for these parameters, along with a normal distribution that has a co-variance of Σ_n = Diag(0.002, 0.0022) (For more details, refer to <cit.>).
The Savitzky-Galoy (SG) convolutional filter is utilized with a quadratic polynomial function, i.e., n_sg=2, and a window length l_sg of 51.
The occupancy surface was constructed with an occupancy radius r_oc of 5 meters, a full azimuth range of -180^o to 180^o,
and elevation height of 0^o to 15^o. The SGP occupancy model was designed with 400 inducing points (Z_m = 400), where the GP frontiers were identified based on a variance threshold of V_th= K_m v_m, where K_m was set to 0.4.
For the distance and direction factors K_dst and K_dir of the cost function J_gp, we assigned weighting factors of 5 and 4, respectively.
To enable the recovery mode of the GP-MPPI, we have set the control threshold, 𝐮_th, to 0.55[].
All the proposed control schemes, which are written in Python and integrated with the Robot Operating System (ROS) framework, are executed in real-time on an NVIDIA GeForce GTX 1660 Ti laptop GPU, with the GP-subgoal recommender built on GPflow<cit.>.
To accomplish the 2D navigation task, we adopt a state-dependent cost function described in (<ref>), which comprises two terms. The first term, with Q = Diag(2.5,2.5,5), aims to steer the robot towards its desired state, whereas the second term incorporates a Boolean variable 𝕀_crash to heavily penalizes collisions with obstacles.
q(𝐱_k)= (𝐱_k-𝐱_f)^⊤ Q (𝐱_k-𝐱_f) + 10^3 𝕀_crash.
Since the robot is operating in unknown environments, it relies on a 2D costmap to maintain a record of obstacles in its vicinity. This costmap is generated by analyzing sensor data from the environment and constructing a 2D occupancy grid, with each cell typically categorized as occupied, free, or unknown <cit.>.
The generated occupancy grid is subsequently employed as a 2D local costmap, feeding directly into the sampling-based MPC algorithm, enabling safe and collision-free navigation.
The robot-centered 2D local costmap, which is built by the on-board Velodyne VLP-16 LiDAR sensor, has a size of 200×200 and a grid resolution of 0.05/.
Finally, throughout the simulations, the maximum linear velocity v_max of the robot is set to 1.5/.
§.§ Simulation Scenarios and Performance Metrics:
The benchmark evaluation utilizes two types of Gazebo simulation environments, as depicted in Fig. <ref>. The first type, referred to as Forest #1, is a 50×50 forest-like environment characterized by tree-shaped obstacles with a density of 0.2/□;
The other type, named Maze #1, is a 20×20 maze-like environment with three U-shaped rooms (i.e., U_1, U_2, and U_3), as well as various other obstacles (highlighted in red in Fig. <ref>)[To evaluate the local planner's obstacle avoidance capability, the red obstacles are intentionally made undetectable as occupied space by the GP-subgoal recommender, as occupancy elevation height is set to a higher value.].
In the first scenario, denoted as Forest #1, the robot is directed to navigate from an initial pose 𝐱_s = [-5,-8,0]^⊤ to a desired pose 𝐱_f = [20,20,45]^⊤ in ([], [], []).
Meanwhile, in Maze #1, we conducted two separate control missions to (i) evaluate the robustness of our proposed control strategy, and (ii) examine its performance under the two different operating modes, previously described in Section <ref>.
The first mission, MU_1, requires the robot to navigate from 𝐱_s = [-5,-8,60]^⊤ to a desired pose 𝐱_f = [4,4,45]^⊤ located inside U_1; while, in the second mission, named MU_2, the robot starts at 𝐱_s = [-6,8,0]^⊤, crosses U_2, and reaches a desired pose of 𝐱_f = [8,-8,170]^⊤.
To ensure a fair and comprehensive comparison of the three control schemes, we have established a set of performance metrics, including the task completion percentage 𝒯_c, the average distance traveled by the robot d_av to reach 𝐱_f from 𝐱_s, the average linear velocity v_av of the robot within the cluttered environment, and the percentage of assistance 𝒜_gp provided by the GP-subgoal recommender policy to MPPI when the recovery mode is utilized. The successful task completion entails the robot reaching the target position without encountering obstacles or getting trapped in local minima ℛ_lm.
§.§ Simulation Results:
We evaluated the effectiveness of the proposed control strategies in Forest #1 and Maze #1 (i.e., MU_1 & MU_2) through 10 trials each, and the resulting performance statistics are summarized in Table <ref>.
The performance results demonstrate that, as expected, the proposed GP-MPPI control strategy outperforms both the vanilla MPPI and log-MPPI as the autonomous vehicle successfully accomplished all control missions (with 𝒯_c=100%) without getting stuck in local minima or colliding with obstacles (i.e., ℛ_lm =0), despite having limited perception range and incomplete knowledge of the environment.
In contrast, in Forest #1, log-MPPI achieved a task completion rate 𝒯_c of 95.72% over 10 trials, compared to 86.87% when MPPI was utilized. Additionally, log-MPPI encountered local minima only twice, while MPPI was trapped six times.
Nevertheless, both control methods were unable to complete any of the trials in MU_1 and MU_2 due to the challenging environmental conditions (refer to the robot trajectories generated by log-MPPI in Fig. <ref>).
Additionally, our proposed approach in Forest #1 provided a shorter route towards the desired state 𝐱_f, especially when the recovery mode (RM) is activated, similar to the optimal trajectory of the baselines, with an average linear velocity v_av of 1.30/, which approaches the maximum specified velocity of 1.5/.
Concerning the two modes of GP-MPPI, it is observed that activating the recovery mode (RM) during Forest #1 and MU_1 missions improves the average distance traveled d_av by the robot. For instance, in MU_1, d_av was approximately 32.74 with RM, whereas with the simple mode (SM), which consistently relies on the subgoal recommended by GP, d_av was roughly 34.48. On the other hand, during the MU_2 mission, the RM produced a slightly longer robot trajectory than the SM since operating our proposed GP-MPPI in the RM strikes a balance between the state-dependent cost function that directs the robot to follow a direct route towards the desired state and the optimal subgoal recommended by the GP policy that forces the robot to avoid the dead-ends associated with rooms U_2 and U_3 on its way to 𝐱_f, as illustrated in Fig. <ref>.
We can also see that, due to the presence of U-shaped rooms in Maze #1, the GP provides more assistance, represented by 𝒜_gp, than in Forest #1.
In Fig. <ref>, we illustrate through an example from the conducted trials the robot trajectories generated by GP-MPPI under the two operating modes in Maze #1. We can clearly observe that our proposed control strategy successfully achieves collision-free navigation in both modes, without getting stuck in local minima.
As an example, Fig. <ref> displays the velocity profile of the robot during the MU_1 mission shown in Fig. <ref>, while using GP-MPPI with RM, along with its corresponding mean of the predicted linear velocities μ_𝐮 over the given time-horizon N (see Fig. <ref>). The mean values that fall below the switching threshold 𝐮_th, set at 0.55[], denote the intervals where the RM is active, and are visually emphasized in yellow in Fig. <ref>.
§ REAL-WORLD DEMONSTRATION
In this section, we experimentally demonstrate the applicability of our proposed control strategy in achieving a safe 2D grid-based collision-free navigation in a complex and unknown indoor cluttered environment.
§.§.§ Experimental Setup and Validation Environment:
To conduct our experimental validation, we used the simulation setup previously outlined in Section <ref>, except for (i) setting the maximum speed v_max to 1.0/ to avoid the robot localization error associated with using the RealSense camera as a source of localization, (ii) setting the occupancy radius r_oc to 3.0,
and (iii) decreasing the size of the 2D grid map to 120×120.
r0.25
< g r a p h i c s >
Panoramic photo of our L-shaped indoor environment.
We also decreased the recovery mode switching threshold 𝐮_th to 0.3/ to be compatible with the updated v_max.
Additionally, to ensure real-time execution of the GP-subgoal recommender policy, we decrease the resolution of the SGP variance surface to one-third of its original value along the azimuth axis while keeping the original resolution along the elevation axis.
We employed an L-shaped indoor corridor environment measuring 9×14 for experimental validation. The environment has a varying width between 1.8 and 2.8 and contains randomly placed boxes-like obstacles, as depicted in Fig. <ref>.
The assigned control mission of the robot is to navigate from 𝐱_s = [0,0,0]^⊤ and arrive at 𝐱_f = [7.5,13,90]^⊤.
§.§.§ Experimental Results:
The performance statistics of our proposed GP-MPPI control scheme, gathered from four trials conducted in our indoor environment, are summarized in Table <ref> for the two operating modes.
From all trials, we can conclude that both operating modes provide collision-free navigation in the cluttered environment with an average linear velocity of 0.80, without the risk of being trapped in local minima (as ℛ_lm = 0) while moving towards its desired state. This ensures the safety and consistent feasibility of the receding-horizon planning.
In contrast, it is observed that the vanilla MPPI and log-MPPI consistently failed to complete any of the trials due to being trapped in the first edge of the L-shaped environment.
However, MPPI managed to avoid such traps with the aid of the GP-subgoal recommender policy in the recovery mode (RM), which provides an average assistance percentage 𝒜_gp of roughly 31.36%.
More details about the simulation and experimental results, including the behavior of the baselines, are provided in the supplementary video: <https://youtu.be/et9t8X1wHKI>.
§ CONCLUSION
In this work, we proposed the GP-MPPI control strategy, which comprises two primary components: the GP-subgoal recommender policy and the local planner, the MPPI. First, the GP-subgoal recommender utilized the learning capacity of SGP to create a reliable SGP variance surface, which served as an indicator for differentiating between occupied and free spaces around the robot. Consequently, a set of suggested subgoals was identified, and the optimal subgoal that minimizes a predefined cost function was recommended to the local MPPI planner. Based on the recommended subgoal, MPPI computes the optimal control input that enables the robot to navigate towards the goal efficiently and safely while accounting for its dynamics and avoiding collisions.
By conducting a combination of simulated and real-world experiments, we have shown that our proposed control strategy is superior to the vanilla MPPI and log-MPPI methods in achieving efficient and safe navigation in unknown and complex environments, thereby avoiding the risk of getting stuck in local minima.
IEEEtran
|
http://arxiv.org/abs/2307.05052v1 | 20230711070329 | Towards Understanding In-Context Learning with Contrastive Demonstrations and Saliency Maps | [
"Zongxia Li",
"Paiheng Xu",
"Fuxiao Liu",
"Hyemi Song"
] | cs.CL | [
"cs.CL",
"cs.AI"
] |
Belief Revision from Probability
Jeremy Goodman
School of Philosophy
University of Southern California, USA
[email protected]
Bernhard Salow
Faculty of Philosophy
University of Oxford, UK
[email protected]
August 12, 2023
=======================================================================================================================================================================================================
§ INTRODUCTION
Large language models (LLMs) show significant ability of in-context learning (ICL) for many NLP tasks <cit.>. ICL only requires a few input-label pairs for demonstrations and does not require fine-tuning on the model parameters. However, how each part of the demonstrations used in ICL drives the prediction remains an open research question. Previous works have mixed findings. For examples, although one might assume that ground-truth labels would have a similar impact on ICL as they do on supervised learning, <cit.> finds that the ground truth input-label correspondence has little impact on the performance of end tasks.
However, <cit.> suggests that the example ordering has a strong impact.
More recently, <cit.> find that only LLMs with larger scales can learn the flipped input-label mapping.
In this work, we use XNLP methods to understand which part of the demonstration contributes to the predictions more.
We are interested in the impact of contrastive input-label demonstration pairs built in different ways, i.e., flipping the labels, changing the input, and adding complementary explanations as shown in Fig. <ref>. We then contrast the saliency maps of these contrastive demonstrations via qualitative and quantitative analysis.
Prior works <cit.> show LLMs in relatively small scale, such as all GPT-3 models <cit.> (based on categorization in <cit.>), cannot override prior knowledge from pretraining with demonstrations presented in-context, which means LLMs do not flip their predictions when the ground-truth labels are flipped in the demonstrations <cit.>.
However, <cit.> show larger models like InstructGPT <cit.> (specifically the checkpoint) and PaLM-540B <cit.> have the emergent ability to override prior knowledge in the same setting.
We partly reproduce the results from previous work <cit.> on a sentiment classification task and find that the ground-truth labels in the demonstration are less salient after label flipping.
Meanwhile, as the other important part of the demonstrations, the effect of input distribution is understudied. <cit.> change the whole input to random words and <cit.> do no investigate input distribution at all. Therefore, we investigate the impact of input distribution at a fine-grained level, where we edit the input text's different components in correspondence to task-specific purposes. In the case of sentiment analysis, we change the sentiment-indicative terms in the input text of demonstrations to sentiment-neutral ones. We find that such input perturbation (neutralization) does not have as large impact as changing ground-truth label do. We suspect the models rely on pretrained knowledge to make fairly good predictions because the averaged importance scores for neutralized terms are smaller than the ones of original sentiment-indicative terms.
Additionally, we find that complementary explanations do not necessarily benefit sentiment analysis task as they do for symbolic reasoning tasks as shown in <cit.>, even though the saliency maps suggest the explanations tokens are as salient as the original input tokens. This suggests that we need to carefully generate complementary explanations and evaluate whether the target task would benefit from them when trying to boost ICL performance with such technique.
We hope the findings of this study can help researchers better understand the mechanism of LLMs and provide insights for practitioners when curating the demonstrations. Especially with the recent popularity of ChatGPT, we hope this study can help people from various domains have a better user experience with LLMs. The code for this study is available at https://github.com/paihengxu/XICLhttps://github.com/paihengxu/XICL.
§ BACKGROUND
§.§ Understanding ICL
Large language models (LLMs) show significant ability of in-context learning (ICL) for many NLP tasks <cit.>. <cit.> show that presenting random ground truth labels in the demonstrations does not substantially affect performance. They also change other parts of the demonstrations (e.g., label space, distribution of the input text and overall sequence format) and find these factors are the key drivers for the end task performance.
<cit.> concentrates on labels by comparing LMs across different size scales with two variants that have flipped labels or semantically-unrelated labels. They find that only large LMs can flip the predictions to follow flipped demonstrations.
<cit.> and <cit.> try to understand in-context learning by training transformer-based in-context learners on small-scale synthetic datasets.
§.§ Saliency Maps
§.§.§ Gradient-based Methods
For models with parameter access, we can estimate the importance of an input token using derivative of output w.r.t that token. The most basic method assigns importance by the gradient <cit.>. However, it suffers from some known issues such as sensitivity to slight perturbations, saturated outputs, and discontinuous gradient.
SmoothGrad <cit.> reduces the noise in the importance scores by adding Gaussian noise to the original input. Integrated Gradients (IG) <cit.> computes a line integral of the vanilla saliency from a baseline point to the input in the feature space.
§.§.§ Perturbation-based Methods
An alternative approach to generating saliency maps using input perturbations can be applied to black-box models.
Instead, the process involves systematically altering the input data (i.e., words, phrases, and sentences) and observing the changes in the model's output.
We plan to start with the standard method that falls into this category, LIME <cit.>.
The process involves creating perturbed versions of an input instance, passing them through the model, training a local linear model on the perturbed inputs and their corresponding predictions, and extracting feature importances from the local model.
§ APPROACH
Despite previous efforts on understanding ICL <cit.>, we are the first attempt to understand ICL using XNLP techniques to the best of our knowledge. We build contrastive demonstrations in various ways and contrast the saliency maps of these contrastive demonstrations to the ones of the original demonstrations to better understand ICL.
We adopt the following three methods to build contrastive demonstrations: flipping labels, perturbing (neutralizing) input, and adding complementary explanations, as shown in Fig. <ref>. We follow <cit.> to flip the labels in the demonstration. <cit.> changed input text distribution to random English words, we focus on a task-specific perturbation in this study. Since we use sentiment analysis as our task and adjectives are strong and sometimes causal indicators of the prediction in this task, we neutralize adjectives in the demonstrations. <cit.> show that adding complementary explanations benefits ICL. We want to investigate how important are these explanations using saliency map methods.
We hope that comparing the saliency maps of these contrastive prompts with the ones of the original prompt would give us insights into how different parts of the demonstrations contribute to ICL predictions.
§ EXPERIMENTAL SET-UP
Dataset We choose SST-2 <cit.>, a sentiment analysis task, as our baseline task to explain ICL paradigm. Due to budget limitations and to follow <cit.>, we randomly sampled 288 examples that are not shorter than 20 tokens from the SST-2 training set as the test set. Additionally, we randomly sample 20 examples for generating saliency maps.
§.§ Demonstration Selection
Other than the 288 examples we picked from the training set, we also hand picked 4 example demonstrations to test language models' in-context-learning ability. We picked two examples with positive labels and two examples with negative labels to have an even distribution of classes in our demonstrations. The demonstrations have significant word indicators for positiveness and negativeness. As in Fig. <ref>, shows full demonstrations under four conditions: original demonstrations, label flipping, input neutralization, and adding explanation for each demonstration.
Label Flipping
We flip the binary label for each of the prompt and use them as the demonstrations during testing.
Input Neutralization
For each of the demonstration, we prompted GPT-4 with its review, and asked it to replace indicative words or phrases (positive or negative phrases) that could lead to the labels for the prompts with neutral words and phrases. After GPT-4 generates a perturbed version of the original review, we manually examine the validity of the perturbed prompts and make sure changes from the original prompts are minimal.
Complementary Explanation
We add a complementary explanation for each of the review demonstration as shown in Fig <ref>. The explanations are generated by prompting GPT-4 with Can you give an explanation to why (REVIEW) is labled positive/negative?. After each explanation was generated, we manually rephrase the explanation to make it shorter and more concise.
§.§ Baseline LMs and Metric
We first evaluate accuracy of the following models on the sampled SST-2 dataset.
Fine-tuned BERT
We used BERT fine-tuned on SST-2 dataset to get a supervised accuracy of a language model[distilbert-base-uncased-finetuned-sst-2-english] for a better comparison on other models' accuracy on the same dataset without fine-tuning.
ChatGPT-3.5-turbo
We used the openAI gpt-3.5-turbo, with maximum number response token length set to 4096 tokens, temperature set to 0 (with no randomness of the outputs), and top_n set to 1, which always gives the top 1 prediction of the model.
Instruct-GPT
We used , a member of the Instruct-GPT family, with temperature 0, maximum tokens generated equal to 50 (to keep cost in control), and top_n also set to 1.
GPT-2
We select GPT-2 [https://huggingface.co/gpt2] with 124M parameters, the smallest version of GPT-2. The GPT-2 model does not sample when generating the output (i.e., temperature set to 0).
We are not able to run larger GPT-2 models like GPT-medium, GPT-large, and GPT-XL on a single GPU available at hand.
Metric
We use the accuracy to evaluate sentiment classification. We also use T-test to verify our hypothesis on the saliency map patterns for the three contrastive demonstrations.
§.§ Saliency Map Methods
For gradient-based method, we use IG <cit.> for models with parameter access (i.e., GPT-2) with the implementation from Ecco <cit.>. For black-box models (i.e., from Instruct-GPT family), we use LIME[https://github.com/marcotcr/lime] to explain the Instruct-GPT classifier.
We use the LimeTextExplainer to explain the instance, where the num_features is 20 and the number of neighbors is 5.
We choose such a setting because of the budget limit as it requires fewer perturbations and interactions with Instruct-GPT API. It generates more “sparse” saliency maps, we discuss this in detail in Section <ref>.
The hyperparameters for GPT-2 and GPT-3 and prompts are the same ones we used for the accuracy evaluation.
We only generated saliency maps for GPT-2 and GPT-3 models due to time and compute constraints, future work could explore more models such as ChatGPT.
§ FINDINGS
§.§ Prediction Performance of the Three Contrastive Demonstrations
We tested the performance of GPT-3.5-Turbo, InstructGPT, and GPT-2 on the 288 selected test examples with four types of demonstrations, i.e., original, label flipping, input neutralization, and complementary explanations. The results are shown in Fig. <ref> and Fig. <ref> [We excluded input perturbation from Fig. <ref> because we perturbed the demonstrations differently while testing for 8 demonstrations (replace indicative words or phrases with opposite meaning words and phrases) and cannot afford to rerun the experiments in terms of time and cost.].
We find that for label-flipping demonstration, ChatGPT-Turbo-3.5 leads to the greatest degradation in performance,
from accuracy 96% to 73% when given 4 demonstrations and to 17% when given 8 demonstrations. The performance of InstructGPT drops by a smaller amount in both scenarios.
This is consistent with the findings from <cit.>, i.e., large LMs' (Instruct-GPT and ChatGPT) performance drops with increased number of the labels flipped in exemplars.
Although the model parameter sizes are the same between GPT-3.5-Turbo and InstructGPT, GPT-3.5-Turbo seems to have a much stronger in-context-learning ability than InstructGPT.
We also noticed that GPT-2 model has much lower performance when given 4 demonstrations and almost predicts negative for all test examples when given 8 demonstrations. It is thus relatively insensitive to different types of contrastive demonstrations. The fact that the performance change of label-flipping is much smaller for GPT-2 compared with ChatGPT and Instruct-GPT also verifies the findings of emergent overriding ability of large LMS from <cit.>.
On the other hand, we observe much smaller impacts for input neutralization and explanation. The small impact of input neutralization may be due to the fact that LMs are still able to make predictions using pretrained knowledge, especially for a relatively “easy” task of sentiment analysis. Adding complementary explanation to each examplar does not benefit the model performance for both four-demonstration and eight-demonstration scenarios. One of the reasons might be that the task chosen is too trivial for explanations to be useful.
These mixed results further inspire us to contrast the patterns of the saliency maps generated by smaller LLMs and large LLMs as all the language models tested are built upon transformer architecture.
§.§ Comparison of the Saliency Maps
Due to the GPT-2's poor performance and compute cost when given 8 demonstrations, we use the setting of 4 demonstrations for saliency map comparisons as shown in Fig. <ref> and Fig. <ref>.
§.§.§ Label Flipping
Hypothesis The labels in the demonstration are less important after model flipping for smaller LMs (GPT2) but more important for large LMs ().
Analysis For example as in Fig. <ref> and Fig. <ref>, the importance of the output label in the demonstration decreases from the original prompt to the label-flipped one. This suggests that the model might pay less attention to the flipped label due to its inconsistency with the input, which results in insensitivity to label flipping in the demonstrations. We expect smaller LMs (GPT2) and large LMs ( from Instruct-GPT) to have different behaviors because <cit.> show only large LMs have the ability to override prior knowledge from pertaining to the one from demonstrations, which is also supported by our results from Fig. <ref> and Fig. <ref>.
For GPT2, on average, 3.35 / 4 of the labels in the demonstration have decreased saliency scores when the demo labels are flipped. Moreover, the average saliency scores of the 4 demo labels decrease for all 20 test examples. The p-value from a T-test for comparing average saliency scores (N=20) between original and label-flipped demonstrations is < 0.001.
For InstructGPT, the average saliency scores increase for 16/20 test examples with a p-value of 0.23 from a similar T-test as above (Fig. <ref>). As InstructGPT achieves around 60% accuracy in Fig. <ref>, we expect Instruct-GPT (with 8 demonstrations) and ChatGPT to have a more significant result as it shows the ability to fully override prior pretrained knowledge.
§.§.§ Input Perturbation (Neutralization)
Hypothesis The sentiment-indicative terms in the original prompt are more important than sentiment-neutral terms in the neutralized prompt.
Analysis The hypothesis is derived from the definition and our intuition of the sentiment analysis task. Sentiment-indicative terms are important to make sentiment predictions. To validate this hypothesis, we contrast the original and neutralized prompts and manually pick different tokens with sentiment orientations. The selected tokens are highlighted in Fig. <ref> and Fig. <ref> with red boxes respectively. We then compute the average saliency scores for each of the 20 test examples.
We find that, for GPT2, the average saliency scores for sentiment-indicative terms in the original prompt are higher than their contrastive parts in the neutralized prompt for all 20 test examples with a p-value of <0.001 from a T-test. However, for Instruct-GPT, we find that the sentiment-indicative terms in the original prompt are equal or higher in 9/20 test examples with a p-value of 0.17 from a similar T-test as above. We note that, as mentioned in Section <ref>, the saliency maps for Instruct-GPT generated by LIME are sparse and have a lot of zeros as shown in Fig. <ref>. This may lead to a mixed result with a less significant T-test result.
§.§.§ Complementary Explanation
Previous work <cit.> shows complementary explanations are beneficial for symbolic reasoning tasks including Letter Concatenation, Coin Flips, and Grade School Math. However, as we show in Fig. <ref>, complementary explanations do not necessarily improve the performance for sentiment analysis which is considered as an “easier” task for LMs. From saliency maps for GPT2, we find that 16/20 test examples have larger averaged saliency scores for tokens in the explanations than the ones in the review. On average, the averaged saliency scores for review tokens are 90% of the ones for explanation tokens. So the gaps are small and explanations are just as important as the original review in this sense.
We suspect the impact of complementary explanations varies based on the tasks at hand. It may benefit tasks requiring more logical reasoning but such conclusion requires more systematical evaluation on more benchmark datasets and we leave it to future work.
§ LIMITATIONS
One potential limitation of our work is that we only select 288 samples, only 20 examples for the saliency map and only 5 neighbors in the LIME model. This is because of budge limitation and Openai daily request limit. In the future, a larger dataset is needed for the evaluation. Additionally, all the demos we used for the ICL is randomly selected, which will influence the model's accuracy performance in some cases. Therefore, a better method would be using different methods to select the demos, like similar instance retrieval. Finally, we only pick the saliency map as the explanation method. In the future, more explanation models will support our conclusion better. In addition, since we find out that adding explanations to the demonstrations leads to performance degradation, we would like to examine the role of explanation on other tasks, or with more examples being tested for the SST-2 dataset. However, due to time-constrains and budget issues, we are unable to conduct the experiment on more complex tasks, such as grade school math problems, or common sense reasoning to further examine why adding explanation leads to performance degradation on sentiment analysis task.
§ CONCLUSION
In this study, we used XNLP techniques to study how ICL works by investigating the performance of contrastive input-label demonstration pairs built in different ways, i.e., flipping the labels, changing the input, and adding complementary explanations, and contrasting their saliency maps with the original demonstration via qualitative and quantitative analysis.
We partly reproduced the results from previous work <cit.> on a sentiment classification task and found that the ground-truth labels in the demonstration are less salient after label flipping. We also found that neutralizing the sentiment-indicative terms in the input does not have as large impact as changing ground-truth label do. The models may rely on pretrained knowledge to make fairly good predictions because the averaged importance scores for neutralized terms are smaller than the ones of original sentiment-indicative terms. Additionally, complementary explanations do not necessarily benefit sentiment analysis task as they do for symbolic reasoning tasks as shown in <cit.>. We hope the findings of this study can help researchers better understand the mechanism of LLMs and provide insights for practitioners when curating the demonstrations.
Future works can experiment with more LMs and more benchmark tasks and datasets to verify the findings of this study and make them more generalizable. Moreover, this work only focused on analyzing the saliency maps of demonstrations. Future work can investigate how demonstrations interact with the query test examples by adjusting exemplars in the demonstration based on their semantic distance from the query examples.
It is also interesting to compare gradient-based saliency map methods with perturbation-based ones and see how the saliency maps differ with the same input and model.
ACM-Reference-Format
§ EXAMPLE SALIENCY MAPS FOR INSTRUCT-GPT
|
http://arxiv.org/abs/2307.07287v1 | 20230714115228 | On generators of commutative semifields | [
"Vítězslav Kala",
"Lucien Šíma"
] | math.RA | [
"math.RA",
"Primary 12K10, 20M14, secondary 05C05, 06F20, 16Y60"
] |
We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of generators and show that it grows linearly with the depth of an associated rooted forest.
INF³: Implicit Neural Feature Fusion Function for Multispectral and Hyperspectral Image Fusion
Tai-Xiang Jiang
August 12, 2023
==============================================================================================
§ INTRODUCTION
Semirings and semifields are a natural generalization of rings and fields, which found its applications in various areas of mathematics including cryptography, theoretical computer science, and tropical geometry <cit.>.
A number of recent works <cit.> focused on the study of simple semirings and semifields, in particular on the finitely generated ones that provide rich and interesting structure.
To be more precise, recall that a (commutative) semiring is a set S
equipped with two binary associative and commutative operations, addition ⊕ and multiplication ·, such that multiplication distributes over addition (in this paper, all semirings will be commutative, i.e., multiplication is always assumed to be commutative).
Semirings offer a natural extension and generalization of (commutative) rings, and, in particular, many of the structural results on rings carry over. Of particular interest for us will be generalizations of simple rings, i.e., those possessing no non-trivial ideals. Of course, every simple (commutative) ring is just a field, and ideals in rings correspond to congruences.
This correspondence no longer holds in semirings, and so one distinguishes congruence-simple and ideal-simple
semirings.
While congruence-simple semirings are quite well understood thanks to the seminal paper <cit.> (with the exception of subsemirings of the positive real numbers), ideal-simple ones remain more mysterious, and so we focus on them.
Their structure can be quite quickly be reduced to that of (para)semifields (see Theorems <ref> and <ref>):
A semiring S is a semifield if moreover there is an element 0 such that the set of non-0 elements (S∖{0},·) forms a group and 0· s=0 for each s∈ S, and a parasemifield if (S,·) is a group.
Specifically, we will be interested in finitely generated ideal-simple semirings, motivated by the folklore result that if a field is finitely generated as a ring, then it is finite.
Semiring generalizations of this result have been recently quite intensely studied <cit.>;
the first of the goals of this article is to summarize the various classification results on finitely generated ideal-simple semirings and (para)semifields, as they have been spread throughout the literature.
Our main goal is then to focus on the most interesting case of additively idempotent parasemifields. We use their combinatorial classification <cit.> in terms of rooted forests to obtain new
results on their minimal numbers of semiring generators. As the classification states that we need to study the abelian groups (^n,+) equipped with suitable partial orders coming from the graph structure of a rooted forest F on n vertices, our proofs entail mostly elementary, but tricky and non-trivial arguments.
Surprisingly, it turns out that the minimal number of generators grows linearly with the depth of F (see Theorem <ref>). However, determining the precise value seems to be very hard and it is unclear if the answer will depend on the specific structure of F, or only on its depth.
These results are interesting not only on their own, but also because additively idempotent parasemifields are term-equivalent with lattice-ordered groups (ℓ-groups for short, see Section 2 for details). The study of ℓ-groups is another rich area of great interest <cit.>, and exploiting this connection was already crucial in the previous classification results <cit.> that relied on the classification of Busaniche, Cabrer, and Mundici <cit.>. In fact, in most of the present paper we also use the ℓ-group notation.
Very notable is also the correspondence with MV-algebras and related topics in logic <cit.>.
As for the contents of this short paper, in Section 2, we summarize the classifications of ideal-simple semirings, of semifields, and of finitely-generated semifields, following <cit.> and <cit.>.
Their structures in turn depend on parasemifields, and so accordingly in Section 3, we state the classification of parasemifields that are finitely generated as semirings (Theorem <ref>)
from <cit.>:
Each such parasemifield can be associated with a rooted forest carrying an additive group of integers on each vertex. The second semiring operation ∨ is defined as a lexicographic maximum with respect to the forest structure.
Sections 4 and 5 concern the minimal number of generators needed to generate a given parasemifield using the semiring operations. We show that this number is linear in the depth of the rooted forest that represents it (Theorem <ref>). Among other results, we also give the precise minimal number of generators for the parasemifields corresponding to equipped with coordinate-wise addition and maximum (Theorem <ref>; an elementary reformulation of this surprising result was selected for the shortlist of problems for the International Mathematical Olympiad 2022).
We conclude the article with Open Question <ref> that offers a possible precise value for the number of generators.
§ ACKNOWLEDGMENTS
We thank Miroslav Korbelář and Jiří Šíma for helpful discussions and suggestions.
§ PRELIMINARIES AND BASIC CLASSIFICATIONS
Already in the Introduction we have recalled that a semiring (S,⊕,·) consists of a set S
equipped with two binary associative and commutative operations, addition ⊕ and multiplication ·, such that multiplication distributes over addition. Moreover, a semiring S is a semifield if there is an element 0 such that (S∖{0},·) forms a group and 0 s=0 for all s∈ S, and a parasemifield if (S,·) is a group; in both cases we denote the unit element 1 and the inverse ^-1.
A semiring S is finitely generated if there are elements s_1,…,s_n for some positive integer n such that the smallest subsemiring of S containing s_1,…,s_n equals S itself.
If a semifield is finitely generated as a semiring (i.e., using only the operations ⊕, ·, but not the inverse ^-1), then we will call it an fg-semifield, and similarly in the case of an fg-parasemifield.
A semiring S is additively idempotent if s⊕ s=s for all s∈ S.
Such semirings are studied in tropical mathematics
where the semiring with its operations is commonly denoted as (S,∨, +), i.e., ∨ denotes the addition and + the multiplication (we will also frequently use this notation).
Of particular interest is the case of additively idempotent parasemifields (S,∨,+), for we have a term-equivalence with lattice-ordered groups (ℓ-groups for short) (L,+,∨,∧).
Recall that an (abelian) ℓ-group (L,+,∨,∧) is an abelian group
(L,+) that is also a lattice (L,∨,∧) such that + distributes over the lattice operations ∨,∧. The term-equivalence between additively idempotent parasemifields (S,∨,+) and ℓ-groups (S,+,∨,∧) is given by a∧ b=-((-a)∨(-b)) (and the operations ∨,+ staying the same).
For more details see, e.g., <cit.>.
Convention. Let us stress that throughout the paper, all semirings, semifields, parasemifields, and ℓ-groups are commutative.
An ideal I in a semiring S is a non-empty subset such that a⊕ b, s· a∈ I for all a,b∈ I, s∈ S.
A semiring S is ideal-simple if all ideals I in S satisfy |I|≤ 1 or I=S.
Recall that for a positive integer n, we denote ^n the direct product of n copies of . We will often consider it as a group or semigroup (^n,+) when equipped with coordinate-wise addition +.
For a vector v=(v_1,…,v_n)∈^n and an integer k, we will denote
k· v (or just k v) the vector (kv_1,…,kv_n).
Now we can summarize the classification results for ideal-simple semirings and their relation to the property of being finitely generated, originally established in <cit.>. While we do not give the proofs here, they are available in the original articles, or in the thesis <cit.>.
Let S be a semiring, |S| ≥ 3. Then S is ideal-simple if and only if one of the following cases holds:
* (S,⊕ ) is isomorphic to the p-element cyclic group (_p,+) equipped with zero-multiplication for a prime p>3,
* S is a semifield,
* S is a parasemifield.
Let us further state the complete classification of semifields.
Let S be a semifield. Then one of the following cases occurs:
* S is a field.
* S is constructed from a parasemifield (T,⊕ ,·) by adding an element 0 and letting 0⊕ s=s and 0s=0 for every s ∈ S.
* S is constructed from a multiplicative abelian group (A,·) by adding an element 0 and letting s⊕ t=0 and 0s = 0 for every s,t ∈ S.
* S is constructed from a parasemifield (P,⊕ ,·) as follows: Assume that (P,·) is a multiplicative subgroup of an abelian group (A,·) and let S =A ∪{0} and 0s=0 for every s∈ S.
The addition is defined for any x,y ∈ S as follows:
x⊕ 0 = 0
if x^-1y ∉ P, then x⊕ y = 0
if x^-1y ∈ P, then x⊕ y = (x^-1y ⊕ 1)· x.
Note that in case (4) above (as well as in Theorem <ref>(4) below) we allow P to be the trivial one-element parasemifield – this gives precisely the semifields denoted as V(A) in <cit.>.
The classification above tells us that semifields arise from well-known structures (fields and groups) or from parasemifields.
By Theorem <ref>, we see that every finitely generated, ideal-simple semiring (except for the trivial case (1)) is an fg-semifield or fg-parasemifield.
Further, it is not hard to check that the structure of fg-semifields nicely corresponds to the classification from Theorem <ref>.
Let S be an fg-semifield. Then one of the following cases occurs:
* S is a finite field.
* S is constructed from an fg-parasemifield P by adding an element 0 and letting 0⊕ s=s and 0s=0 for every s ∈ S.
* S is constructed from a finitely generated multiplicative abelian group (A,·) by adding an element 0 and letting s⊕ t=0 and 0s = 0 for every s,t ∈ S.
* S is constructed from an fg-parasemifield (P,⊕ ,·) as follows. Let (P,·) be a subgroup of a finitely generated abelian group (A,·) and let S = A ∪{0} and 0s=0 for every s∈ S.
The addition is defined for any x,y ∈ S as follows:
x⊕ 0 = 0
if x^-1y ∉ P, then x⊕ y = 0
if x^-1y ∈ P, then x⊕ y = (x^-1y ⊕ 1)· x.
The preceding theorems essentially reduce the classification of finitely generated ideal-simple semiring to that of fg-parasemifields.
While not much appears to be known about parasemifields in general, the case of fg-parasemifields is much better understood, as we discuss in the next section.
§ FINITELY GENERATED PARASEMIFIELDS
To proceed to the main topic of this paper, let us
present the classification of fg-parasemifields.
First, by a non-trivial theorem of Kala and Korbelář, we can restrict ourselves to the case of additively idempotent parasemifields.
Let S be an fg-parasemifield. Then S is additively idempotent.
Therefore, it suffices to study only additively idempotent fg-semifields, which is the case considered by Kala <cit.>. Combining his results with the preceding Theorem <ref>, we will be able to state Theorem <ref> and Corollary <ref> for general fg-parasemifields without the idempotency assumption.
However, first we need to associate a parasemifield G(T,v) to a rooted tree (T,v) and to extend this notion to rooted forests.
Recall that a rooted tree (T,v) is a (finite) un-oriented graph T without cycles together with a highlighted vertex v, called the root. A rooted forest (F,R) consists of a graph F = T_1 ⊔…⊔ T_k that is the disjoint union of finitely many rooted trees (T_i,v_i) with the set of roots R = {v_1, …, v_k}. For a graph G, we denote V(G) and E(G) its sets of vertices and edges.
A vertex v in a rooted forest is a leaf if v∉R and its degree is 1 (i.e., there is precisely 1 edge containing v), or if v∈ R and its degree is 0.
Two rooted forests are isomorphic if there is a bijection between their sets of vertices that preserves the graph structure and permutes the sets of roots.
The depth of a vertex w∈ V(F) in a rooted forest (F,R) is the largest k such that there exists a path v = v_1, v_2, …, v_k = w (i.e., a sequence of distinct vertices such that (v_i,v_i+1) is an edge for each i) from some root v∈ R to the vertex w. The depth of a rooted forest is the maximum of the depths of its vertices.
For a positive integer n, we denote [n]={1,2,…,n}.
Let (T,v) be a rooted tree on n=|V(T)| vertices. We attach a copy of the set of integers _w to each vertex w ∈ V(T) and define the set
G(T,v)=∏_w∈ V(T)_w=^n
(of course, the second equality above depends on fixing a bijection of V(T) with [n], as we will usually do).
We refer to the elements of G(T,v) as integer-valued vectors from , each coordinate corresponding to a vertex in (T,v).
Let us now define semiring operations ∨,+ on G(T,v). The multiplicative group (G(T,v), +) is given by the coordinate-wise addition in the group ( ,+) (which does not depend on structure of the tree (T,v)).
To define the semiring addition ∨, let 𝐠=(g_w), 𝐡=(h_w) be two elements from G(T,v).
We define 𝐠∨𝐡=((g∨h)_w) as follows:
For a vertex w ∈ V(T), let v = v_1, v_2, …, v_k = w be the unique path from the root v to the vertex w. If g_v_i = h_v_i for all i ∈ [k], we set (g∨h)_w = g_w = h_w. Otherwise, let i be the smallest index such that g_v_i≠h_v_i and define
(g∨h)_w =
g_w if g_v_i > h_v_i
h_w if g_v_i < h_v_i.
We can naturally extend the definition to rooted forests.
Let (F,R) be a rooted forest, F = T_1 ⊔…⊔ T_k, with the set of roots R = {v_1, …, v_k}. We define the associated parasemifield (G(F,R),∨,+) as the direct product of the parasemifields (G(T_i, v_i),∨,+).
It turns out that every fg-parasemifield arises from a rooted forest in this way, as the following theorem shows.
Let (S,⊕ ,·) be an fg-parasemifield. Then there is a rooted forest (F,R) (unique up to isomorphism) such that (S,⊕ ,·) ≃ (G(F,R),∨,+).
This classification result has an immediate corollary
[<cit.>]
Let S be an fg-parasemifield. Then S is finitely generated as a multiplicative semigroup.
Theorem <ref> gives us that S ≃ G(F,R) for some rooted forest (F,R). The multiplicative group (S,·) is thus isomorphic to ( ,+) for n = |V(F)|. The corollary follows from the fact that is clearly finitely generated as an additive semigroup (for more details, see Proposition <ref>).
As a result, we obtain the following corollary, which might be quite surprising.
Let S be an ideal-simple semiring that is finitely generated. Then S is finitely generated as a multiplicative semigroup.
First, from the classification of ideal-simple semirings (Theorem <ref>), we have that S is either isomorphic to _p with zero-multiplication (which is finite and thus finitely generated as a multiplicative semigroup), or an fg-parasemifield or an fg-semifield.
Corollary <ref> states that fg-parasemifields are finitely generated as multiplicative semigroups, which also implies that fg-semifields of type (2) from Theorem <ref> are finitely generated as well. The statement also holds for fg-semifields of the three remaining types, as they are either finite (type (1)) or obtained by adding one element to a finitely generated abelian group (types (3) and (4)).
§ GENERATORS FOR ISOLATED VERTICES
In the previous section we discussed that every fg-parasemifield S corresponds to a rooted forest (F,R) (in the sense that S ≃ G(F,R)).
From now on we will thus work with the parasemifields (G(F,R),∨,+) and accordingly denote the semiring multiplication as +, inverse as -, neutral element as 0, and addition as ∨, i.e., we will use the ℓ-group notation.
We turn our interest to determining the minimal number of generators needed to generate S as a semiring, i.e., using only the multiplication + and addition ∨, but not the inverse - (nor the neutral element 0).
One of our main results will be Theorem <ref> saying that the minimal number of semiring generators of an fg-parasemifield S is linear in the depth of the corresponding rooted forest (F,R).
In this section, we start with
some preliminary observations, and then we determine the number of generators in the case of forests without any edges.
The minimal number of semiring generators of the parasemifield G(F,R) will be denoted m(F,R).
The multiplicative group of the parasemifield G(F,R) is ( ,+) (where + is the usual addition + taken coordinate-wise). Thus, it is quite useful to determine the minimal number of vectors needed to generate as an additive semigroup.
We will denote e_i∈ the vector having 1 at the ith coordinate and 0 everywhere else, and 0=(0,…,0).
For u∈, we usually denote its coordinates as u=(u_1,…,u_n). Let us call u a positive vector if u_1>0,…,u_n>0, and similarly a negative vector if u_1<0,…,u_n<0.
Let n be a positive integer and u∈ a negative vector. Then the set {e_1, …e_n, u} generates the semigroup (,+).
Let us take any vector v∈. Since u is a negative vector, we can find a positive integer k such that w =(w_1,…,w_n) = v - k ·u is a positive vector. Then v = w + k ·u = w_1 ·e_1 + … + w_n ·e_n + k ·u for positive integers k, w_1, …, w_n, as we wanted to show.
The minimal number of semigroup generators of (,+) is n+1.
Lemma <ref> gives us a set of n+1 generators, and so it suffices to show that any n vectors do not generate . For contradiction, suppose that a set V = {v_1,…,v_n } generates . In particular, we can find non-negative integers a_i,b_i such that:
a_1 ·v_1 + … + a_n ·v_n = e_1, b_1 ·v_1 + … + b_n ·v_n = -e_1.
Adding these equations, we get a non-trivial linear combination expressing the zero vector
(a_1+b_1) ·v_1 + … + (a_n+b_n) ·v_n = 0.
It is easy to see that V generates the vector space ^n over . Because V consists of n vectors and the dimension of ^n over is n, it follows that V is a basis of ^n. But we have found a non-trivial linear combination (<ref>) that expresses the zero vector, showing that V is not linearly independent, which is a contradiction.
In order to bound m(F,R), we will start with the base case when the depth of F equals 1. We thus consider rooted forests (F,R) consisting of n isolated vertices (then clearly R = F, because each tree component consists of exactly one vertex). We will denote such forests by and give the exact value of m( ) in the rest of this section.
In other words, we are looking for a minimal set X of vectors from such that X generates all elements of using addition and maximum (both applied coordinate-wise). We will start with the easy case n ≤ 2.
Let n ∈{1,2}. Then m( ) = 2.
It is clear that one generator can not be sufficient, as the sign is preserved under both operations, i.e., every coordinate would stay either positive or negative, and so all of could not be generated by just one generator.
We finish the proof by finding the generating set X of size two. If n=1, we let X = {(1),(-1)} that generate ^1 just using +.
For n=2, we define X to be {(1,-2),(-2,1)}. It suffices to generate the following three vectors {(1,0),(0,1),(-1,-1)} since they generate ^2 using coordinate-wise addition (see Lemma <ref>). We obtain the first one as follows:
(5, -2) = (1,-2) ∨ (5 · (1,-2) )
(1,0) = (5,-2) + 2 · (-2,1),
and the second one is obtained symmetrically.
Finally, (-1,-1)=(1,-2)+(-2,1).
Let us now consider the case when n ≥ 3. Surprisingly, it turns out that m( ) = 3 regardless of the value of n. In order to show that two generators do not suffice, we need to state an auxiliary lemma.
Let 1 ≤ j ≠ k ≤ n and let us take two vectors u,v∈ G()= satisfying u_j ≥ a · u_k and v_j ≥ a · v_k for some positive real number a. Then the same inequalities hold for the vectors u + v, u∨v∈ G().
The inequality for u + v is verified by an easy computation:
(u+v)_j = u_j + v_j ≥ a · u_k + a · v_k = a · (u+v)_k.
Let m = u∨v. We have m_j ≥ u_j ≥ a · u_k and that m_j ≥ v_j ≥ a · v_k. Since m_k = u_k or m_k = v_k, the conclusion follows.
Let X = {g_1, …, g_m} be a set of vectors in G()=, g_i=(g_i,1,…,g_i,n). Let 1 ≤ j ≠ k ≤ n and let a be a positive real number such that g_i,j≥ a · g_i,k for every i∈[m].
Then X does not generate G().
By the previous lemma, the inequality u_j ≥ a · u_k holds for all vectors u=(u_1,…,u_n) generated by X – but this inequality does not hold for every vector in G()=.
We are now ready to prove the following theorem.
Let n ∈, n ≥ 3. Then m( ) = 3.
We will start with showing that two generators do not suffice. For contradiction, suppose that the set X = {u, v} generates G()=.
Suppose that there is a coordinate i such that u_i, v_i ≥ 0. Both operations preserve the sign, thus we can not generate any vector that has negative ith coordinate. Similarly if u_i, v_i ≤ 0.
Therefore, for every i ∈ [n] we have u_i > 0, v_i < 0, or u_i < 0, v_i > 0. Since n ≥ 3, there are two coordinates j ≠ k such that u_j has the same sign as u_k and v_j has the same sign as v_k.
Without loss of generality, assume that u_j,u_k > 0, v_j,v_k < 0. Let us denote the positive real number u_j / u_k by a. If v_j / v_k ≤ a, then both inequalities u_j ≥ a · u_k and v_j ≥ a · v_k are satisfied. On the other hand, if v_j / v_k ≥ a, then both u_k ≥ (1/a) · u_j and v_k ≥ (1/a) · v_j are satisfied. In either case, we found an inequality satisfied by both vectors from X, and so X does not generate G() by Proposition <ref>.
We finish the proof by finding a set of three generators of G()=. We let k = n^2 + 1 and we define X={a,b,c} by setting
a_i= i, b_i = k - i^2, c_i = -1.
Note that k is chosen so that b is a positive vector.
We will start by generating n positive vectors u_1, …, u_n such that the ith coordinate of u_i is strictly largest. We define u_i to be 2i ·a + b. Then the jth coordinate of u_i is u_i,j = 2i · a_j + b_j = 2ij + k - j^2 = k + j (2i-j). It is easy to see that this expression attains maximum for j=i, which gives us that the ith coordinate of u_i is indeed maximal, i.e., u_i,i>u_i,j for all j≠ i.
For every i ∈ [n], let us then generate v_i = u_i + ((u_i,i) - 1) ·c. Note that the ith coordinate of v_i equals 1 and all the other coordinates are non-positive.
Now it is the time to apply the coordinate-wise maximum ∨. We obtain the zero vector as 0 = (v_1 ∨v_2 ∨…∨v_n) + c. Finally, we get e_i as v_i ∨0.
By Lemma <ref>, all the vectors e_i together with the negative vector c generate (,+) as a semigroup.
§ MINIMAL NUMBER OF GENERATORS
Before we study the generators of general forests, let us introduce a partial ordering ≼ on the set of rooted forests, which is compatible with the function m in the sense that (F,R) ≼ (E,S) implies m(F,R) ≤ m(E,S).
Let (F,R) and (E,S) be two rooted forests. We say that (F,R) ≼ (E,S) if (F,R) can be obtained from (E,S) by repeatedly deleting leaves (and the edges that connected them to the forest) from the forest E. Note that R ⊆ S is the set of roots from S which were not deleted.
Let ≼ be the relation on the set of rooted forests defined as above. Then (F,R) ≼ (E,S) implies that m(F,R) ≤ m(E,S).
Let (E_1,S_1) be a rooted forest obtained by deleting a leaf l from (E,S).
If X = {g_1, …, g_k} is a minimal generating set of G(E,S), then we can obtain a generating set of size k for G(E_1,S_1) by simply deleting the coordinate which corresponds to the leaf l from all generators in X.
Thus m(E_1,S_1) ≤ m(E,S).
As (F,R) is obtained by repeatedly deleting leaves from (E,S), we can repeatedly use the result of the previous paragraph to establish m(F,R) ≤ m(E,S).
We will now determine the minimal number of semiring generators for parasemifields which correspond to rooted paths, i.e., to rooted trees that have exactly one leaf.
Let P_n = ({v_1, …, v_n},v_1) be a rooted path with the root v_1. Then m(P_n) = n+1.
It follows from the definition of the operation ∨ that we have either v∨w = v or v∨w = w for every v,w∈ G(P_n). Therefore, the minimal number of semiring generators of G(P_n) equals the minimal number of semigroup generators of , which is n+1 by Proposition <ref>.
Let (F,R) be a rooted forest of depth l. Then m(F,R) ≥ l+1.
From the definition of the depth, we can find w ∈ V(F) and v ∈ R such that there is a path P from v to w consisting of l vertices. It follows that (F,R) ≽ (P,v). Combining Lemma <ref> and Proposition <ref>, we obtain that m(F,R) ≥ m(P,v) = l+1.
Let further (F,R) be a general rooted forest.
We have two parameters for measuring the size of (F,R), namely its depth and its amount of branching, captured by the number of roots and by
the degrees of vertices. Specifically, let us define the width of a rooted forest to be the maximum of the number of roots, degrees of the roots, and of the degrees of all vertices -1, i.e.,
width(F,R)=max{|R|,max_r∈ R{deg(r)},max_v∈ V(F){deg(v)-1}},
where deg(v) denotes the degree of a vertex v.
As we have just seen in Corollary <ref>, m(F,R) grows at least linearly with the depth of (F,R) but, on the other hand, rooted forests of arbitrarily large width can still have constant m(F,R): by Theorem <ref>, m( ) = 3 for any n ≥ 3 (and one could easily modify this example, e.g., to a tree with a root connected to n leaves).
Let us thus define a `universal' rooted forest of width k and depth l.
Let k,l be positive integers. We define T_kl as the unique rooted forest such that:
* there are k roots and each of them has degree k,
* every vertex that is not a leaf or a root has degree k+1,
* every leaf has depth exactly l.
To illustrate the definition, we give the following picture containing two examples of what T_kl looks like.
Observe that G(F,R) ≼ T_kl for every rooted forest (F,R) of width k and depth l, which implies that m(F,R) ≤ m(T_kl) by Lemma <ref>. In order to give an upper bound on m(F,R), we would like to estimate m(T_kl) from above. The first step is the following quite powerful theorem.
Let (F,R) be a rooted forest, k = m(F,R), and construct a rooted forest (E,S) as the disjoint union of m ≥ 2 copies of (F,R).
(a) If m=2, then m(E,S) ≤ k+1.
(b) If m ≥ 3, then m(E,S) ≤ k+2.
Let n = | V(F)| and identify the sets ^n=G(F,R) and ()^m=G(E,S). Let X = {g_1, …, g_k }∈ be a minimal set of generators of G(F,R). Throughout the proof, we are going to work with vectors from ()^m and we will denote v∈ ()^m by (v_1, …, v_n | v_n+1, …, v_2n|…| v_(m-1)n+1, …, v_mn).
For an integer a∈, we will denote a=(a,…,a).
We shall start with part (a), i.e., m=2. Let C be an integer such that C > |g_i,j| for all i ∈ [k], j ∈ [n]. Define the set H = {h_1, …, h_k+1} of k+1 vectors from ( ^ n)^2. The first k of them are defined as h_i = (g_i + 2C|g_i-4C) and we let h_k+1 = (-C| 2C).
Our goal will be to show that we are able to generate any vector from ( ^ n)^2 from the set H. By adding 2 ·h_k+1 to each h_i, we obtain (g_i |g_i). Since X generates G(F,R), we are able to obtain all vectors of the form (v|v) for any v∈, in particular, the vector (-2C| -2C).
Since C > |g_i,j| for all i,j, we have that h_i ∨h_k+1 = (g_i + 2C| 2C). Adding (-2C| -2C) to (g_i + 2C| 2C), we obtain (g_i |0) for each i ∈ [k], which suffices to generate (v|0) for each v∈.
Finally, any vector (v|w) ∈ ( ^ n)^2 can be constructed by adding (v-w|0) to (w|w), and so we are done with the first part.
For part (b), we define the set H = {h_1, …, h_k+2} of k+2 vectors from ()^m as follows:
h_i = (g_i |g_i |g_i |…|g_i), i ∈ [k]
h_k+1 = (1|2|3|…|m)
h_k+2 = (m^2+1 - 1^2|m^2+1 - 2^2|m^2+1 - 3^2|…|m^2+1 - m^2)
= (m^2|m^2-3|m^2-8|…|1).
Our goal will be to prove that H is a generating set of ()^m. As in the first part, we can use vectors h_1, …, h_k to generate (v|…|v) for any v∈, in particular, the vector c = (-1|…|-1).
Since vectors {(1,2,…,m), (m^2,m^2-3,m^2-8,…,1), (-1,-1,…, -1)} generate ^m (see the proof of Theorem <ref>), we are able to use vectors h_k+1, h_k+2, c to generate (c_1 |c_2 |…|c_m) for any integers c_1, …, c_m.
Thanks to Lemma <ref>, it suffices now to generate any vector e_ni+j for i ∈{0,…,m-1}, j ∈ [n]. Let us take such i and j. Thanks to the previous paragraph, we can generate the vector t_i = (-1|…|-1|0|-1|…|-1) such that 0 lies in the ith copy of G(F,R). We then obtain the vector u_i,j = t_i + (e_j |…|e_j) that has all n-tuples non-positive except for the ith tuple, which contains e_j. We then obtain e_ni+j as u_i,j∨0.
Theorem <ref> plays an important role in establishing the upper bounds for m(T_kl) in the following theorem.
Let k,l be positive integers. Then
(a) m(T_1l) = l+1,
(b) l+1 ≤ m(T_2l) ≤ 2l,
(c) l+1 ≤ m(T_kl) ≤ 3l for k ≥ 3.
For part (a), it is enough to observe that T_1l=P_l is actually the path of length l and we already know that m(T_1l) = m(P_l) = l+1 (Proposition <ref>).
Let us prove parts (b) and (c) together. The lower bound follows from Corollary <ref>, as the depth of T_kl is l.
We are going to prove the upper bound by induction on l. If l=1, then T_k1 is formed by k isolated vertices, i.e., T_k1=_k. We have already proved (Proposition <ref> and Theorem <ref>) that m(_2) = 2 and m(_k) = 3 for k ≥ 3, which gives the upper bound for the case l=1.
We prove the inductive step only for part (c), the other part (b) being very similar. Let us suppose that m(T_kl) ≤ 3l and we want to show that m(T_k(l+1)) ≤ 3l+3.
As shown in Figure 2, we can construct T_k(l+1) from T_kl in two steps. First, we connect all the roots of T_kl to a new root r, thus creating the rooted tree U_kl, and then we obtain T_k(l+1) as the disjoint union of k copies of U_kl.
Let n = |V(T_kl)|. We use the assumption m(T_kl) ≤ 3l to find a set X = {g_1, …, g_3l}⊆ that generates G(T_kl) as a semiring. We are going to show that m(U_kl) ≤ 3l+1 by finding its generating set H = {h_1,…,h_3l+1}⊆^n+1 consisting of 3l+1 vectors. We can assume that the first coordinate of these vectors corresponds to the root r of U_kl.
For any i ∈ [3l], we let h_i = (-1, g_i) and we let h_3l+1 = e_1. Adding the vector h_3l+1 to each h_i, we obtain (0,g_i), which we can be used to generate (0,w) for any w∈ (since X generates G(T_kl)).
Let (c,v) ∈^n+1 be an arbitrary vector, c ∈, v∈. If c ≥ 0, we can obtain (c,v) as c ·h_3l+1 + (0,v). On the other hand, if c < 0, then we generate (c,v) as (-c) ·h_1 + (0, v - c ·g_1), which finishes the proof that m(U_kl) ≤ 3l+1.
Since T_k(l+1) is constructed as the disjoint union of k copies of U_kl, the desired bound m(T_k(l+1)) ≤ 3l+3 follows from Theorem <ref> and already proven bound m(U_kl) ≤ 3l+1.
Now it only takes one last step to give the bounds on m(F,R) for a general rooted forest (F,R). We also obtain a tighter upper bound for binary forests, i.e., in the case of width 2.
Let (F,R) be a rooted forest of depth l.
(a) We have l+1 ≤ m(F,R) ≤ 3l.
(b) If (F,R) has width 2, then l+1 ≤ m(F,R) ≤ 2l.
We prove both parts together. The lower bound follows from Corollary <ref>. Let us denote the width of (F,R) by k. It is easy to see that (F,R) ≼ T_kl, and so m(F,R) ≤ m(T_kl) by Lemma <ref>. Using the bounds on T_kl from Theorem <ref>, we obtain the result.
Note that one can prove the upper bound in Theorem <ref>(a) directly from Theorem <ref>: Let us sketch the construction of a set X generating G(F,R) such that |X| ≤ 3l, where l is the depth of (F,R).
We split vertices of (F,R) into l disjoint subsets V_1, …, V_l, where V_i = {v ∈ V(F) |depth of v is exactly i }. For each V_i, we take (at most) three generators of G(_n_i), where n_i = |V_i| (see Proposition <ref> and Theorem <ref>). We set the other coordinates (corresponding to vertices that do not belong to V_i) of those generators to 0 and include the resulting vectors in X. It can be shown that such X generates G(F,R).
§ CONCLUDING REMARKS
While we have proved that the number of generators m(F,R) grows linearly with the depth of the forest,
it seems hard to determine the precise value of m(F,R) for all rooted forests (F,R). The following open question suggests a possible answer.
Let (F,R) ≠ be a rooted forest of depth l. Does m(F,R) equal l+1?
Note that m(F,R) ≥ l+1 is true by Corollary <ref>. For Question <ref> to have positive answer, it thus suffices to find a generating set of G(F,R) of size l+1. We were able to do so for several classes of rooted forests. We end the paper by presenting these partial results.
First, in Figure 3 we give a table of generating sets of parasemifields G(F,R) such that (F,R) contains less than 5 vertices and (F,R) ≠.
This shows that Question <ref> has positive answer for small rooted forests.
We are next going to look at rooted forests that are the disjoint union of several rooted paths and we will answer Question <ref> for some of them.
For positive integers k,n, let us denote by kP_n the rooted forest formed by the disjoint union of k copies of the rooted path P_n.
Let k,n be positive integers and k ≤ n+1. Then there exists a set of n+1 generators of G(kP_n). Consequently, m(kP_n) = n+1.
It suffices to prove the theorem for k = n+1. Elements from G((n+1)P_n) are vectors from ()^n+1 which will be denoted as v = (v_1, …, v_n | v_n+1, …, v_2n|…| v_n^2+1, …, v_n^2+n).
We define the set X = {g_1, …, g_n+1} of n+1 vectors from ()^n+1 as follows:
g_1 = (-2|e_1 |e_1 |…|e_1 |e_1)
g_2 = ( e_2 |-2|e_2 |…|e_2 |e_2)
⋮ ⋱
g_n = ( e_n|e_n|e_n|…|-2|e_n)
g_n+1 = ( e_1 |e_2 |e_3 |…|e_n |-2).
We are going to show that X generates G((n+1)P_n). Let us start with generating two important vectors.
-1 = g_1 + g_2 + … + g_n+1
0 = ( g_1 ∨ 2 ·g_1) + ( g_2 ∨ 2 ·g_2) + … + ( g_n+1∨ 2 ·g_n+1)
We will finish the proof by generating all the vectors e_ni+j (for i ∈{0, …, n} and j ∈ [n]) and applying Lemma <ref>. If i ≠ j, we first generate the vector v_ij as follows.
v_ij = g_i + 3 ·g_j + ∑_k ∉{i,j} (2 ·g_k)
= g_j - g_i + 2 ∑_k=1^n+1g_k
= -2 + g_j - g_i
Observe that the vector v_ij contains e_j in the ith n-tuple and all the other n-tuples contain a non-positive vector. It follows that e_ni+j = v_ij∨0.
The approach in the case i=j is similar. We generate:
w_i = g_i + 3 ·g_n+1 + ∑_k ∉{i,n+1} (2 ·g_k)
e_ni+i = w_i ∨0
and we are done.
It can also be shown that m(kP_2) = 3 for any k ∈, using the generating set of G(_2k) from Theorem <ref>. The proof is similar to the proof of Theorem <ref>, but slightly more technical.
Finally, the analogous question concerning numbers of generators of semifields remains completely open. The answer primarily hinges on the following question that offers rich opportunities for further research.
Suppose that S is an fg-semifield constructed from a parasemifield G(F,R) and an abelian group A as in Theorem <ref>(4). How does the minimal number of semiring generators of S depend on the rooted forest (F,R) and on the embedding in the abelian group A?
B
AF
M. Anderson, T. Feil,
Lattice-Ordered Groups,
Reidel Texts in the Mathematical Sciences, 1988.
BHJK
R. El Bashir, J. Hurt, A. Jančařík, T. Kepka,
Simple commutative semirings,
J. Algebra 236 (2001), 277–306.
BDN-F
L. P. Belluce, A. Di Nola,
Yosida type representation for perfect MV-algebras,
Math. Logic Quart. 42 (1996), 551–563.
BDNF-B
L. P. Belluce, A. Di Nola, A. R. Ferraioli,
MV-semirings and their sheaf representations,
Order 30 (2013), 165–179.
BDN-E
L. P. Belluce, A. Di Nola, G. Georgescu,
Perfect MV-algebras and l-rings,
J. Appl. Non-Classical Logics 9 (1999), 159–172.
BCM
M. Busaniche, L. Cabrer, D. Mundici,
Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups,
Forum Math. 24 (2012), 253–271.
DNG
A. Di Nola, B. Gerla,
Algebras of Lukasiewicz's logic and their semiring reducts,
Contemp. Math. 377 (2005), 131–144.
droste
M. Droste, W. Kuich, H. Vogler (eds.), Handbook of Weighted Automata, Springer, 2009.
GRS
B. Gerla, C. Russo, L. Spada,
Representation of perfect and local MV-algebras,
Math. Slovaca 61 (2011), 327–340.
GH
A. M. W. Glass, W. C. Holland,
Lattice-Ordered Groups,
Kluwer Academic Publishers, 1989.
G
J. S. Golan,
Semirings and Their Applications,
Kluwer Academic, Dordrecht, 1999.
IKN
S. N. Il'in, Y. Katsov, T. G. Nam, Toward homological structure theory of semimodules: on semirings all of whose cyclic semimodules are projective,
J. Algebra 476 (2017), 238–266.
ims
I. Itenberg, G. Mikhalkin, E. Shustin, Tropical algebraic geometry (2nd ed.), Birkhäuser, Basel, 2009.
ir
Z. Izhakian, L. Rowen, Congruences and coordinate semirings of tropical varieties, Bull. Sci. Math. 140(3) (2016), 231–259.
JK
J. Ježek, T. Kepka,
Finitely generated commutative division semirings,
Acta Univ. Carolin. Math. Phys. 51 (2010), 3–27.
JKK
J. Ježek, V. Kala, T. Kepka,
Finitely generated algebraic structures with various divisibility conditions,
Forum Math. 24 (2012), 379–397.
Kala15
V. Kala, Lattice-ordered abelian groups finitely generated as semirings,
J. Commut. Alg. 9 (2017), 387–412.
KK
V. Kala, T. Kepka,
A note on finitely generated ideal-simple commutative semirings,
Comment. Math. Univ. Carol. 49 (2008), 1–9.
KKK
V. Kala, T. Kepka, M. Korbelář,
Notes on commutative parasemifields,
Comment. Math. Univ. Carolin. 50 (2009), 521–533.
KaKo1
V. Kala, M. Korbelář,
Congruence simple subsemirings of ℚ^+,
Semigroup Forum 81 (2010), 286–296.
KalKor
V. Kala, M. Korbelář, Idempotence of finitely generated commutative
semifields, Forum Math. 30 (2018), 1461–1474.
KNZ
Y. Katsov, T. G. Nam, J. Zumbrägel, On simpleness of semirings and complete semirings, J. Algebra Appl. 13(6) (2014), 29 pp.
KL
M. Korbelář, G. Landsmann, One-generated semirings and additive divisibility, J. Algebra Appl. 16 (2017), 1750038, 22 pp.
Le
E. Leichtnam, A classification of the commutative Banach perfect semi-fields of characteristic 1. Applications, Math. Ann. 369(1–2), 653–703.
litvinov
G. L. Litvinov,
The Maslov dequantization, idempotent and tropical mathematics: a brief introduction,
Idempotent mathematics and mathematical physics, Contemp. Math. 377 (2005), Amer. Math. Soc., pp. 1–17. Extended version at arXiv:math/0507014.
mmr
G. Maze, C. Monico, J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun. 1(4) (2007), 489–507.
monico
C. J. Monico,
Semirings and semigroup actions in public-key cryptography,
PhD Thesis, University of Notre Dame, USA, 2002, vi+61 pp.
M
D. Mundici,
Interpretation of AF C^*-algebras in Łukasiewicz sentential calculus,
J. Funct. Anal. 65 (1986), 15–63.
NL
A. Di Nola and A. Lettieri,
Perfect MV-Algebras are Categorically Equivalent to Abelian ℓ-Groups,
Studia Logica 53 (1994), 417–432.
Sima
L. Šíma, Finitely generated semirings and semifields, Master's thesis, Charles University, Czech Republic, 2021, iii+31 pp.
W
H. J. Weinert,
Über Halbringe und Halbkörper. I.,
Acta Math. Acad. Sci. Hungar. 13 (1962), 365–378.
WW
H. J. Weinert, R. Wiegandt,
On the structure of semifields and lattice-ordered groups,
Period. Math. Hungar. 32 (1996), 147–162.
Yang
Y. Yang,
ℓ-Groups and Bézout Domains,
PhD Thesis, Universität Stuttgart, Germany, 2006, viii+116 pp.
zumbragel
J. Zumbrägel,
Public-key cryptography based on simple semirings,
PhD Thesis, Universität Zürich, Switzerland, 2008, x+99 pp.
|
http://arxiv.org/abs/2307.15071v1 | 20230711113508 | Writer adaptation for offline text recognition: An exploration of neural network-based methods | [
"Tobias van der Werff",
"Maruf A. Dhali",
"Lambert Schomaker"
] | cs.CV | [
"cs.CV",
"cs.CL"
] |
CAT-ViL: Co-Attention Gated Vision-Language Embedding for Visual Question Localized-Answering in Robotic Surgery
Long Bai1 ⋆
Mobarakol Islam2
Long Bai and Mobarakol Islam are co-first authors.
Hongliang Ren1,3
Corresponding author.
August 12, 2023
==============================================================================================================================
Handwriting recognition has seen significant success with the use of deep learning. However, a persistent shortcoming of neural networks is that they are not well-equipped to deal with shifting data distributions. In the field of handwritten text recognition (HTR), this shows itself in poor recognition accuracy for writers that are not similar to those seen during training. An ideal HTR model should be adaptive to new writing styles in order to handle the vast amount of possible writing styles. In this paper, we explore how HTR models can be made writer adaptive by using only a handful of examples from a new writer (e.g., 16 examples) for adaptation. Two HTR architectures are used as base models, using a ResNet backbone along with either an LSTM or Transformer sequence decoder. Using these base models, two methods are considered to make them writer adaptive: 1) model-agnostic meta-learning (MAML), an algorithm commonly used for tasks such as few-shot classification, and 2) writer codes, an idea originating from automatic speech recognition. Results show that an HTR-specific version of MAML known as MetaHTR improves performance compared to the baseline with a 1.4 to 2.0 improvement in word error rate (WER). The improvement due to writer adaptation is between 0.2 and 0.7 WER, where a deeper model seems to lend itself better to adaptation using MetaHTR than a shallower model. However, applying MetaHTR to larger HTR models or sentence-level HTR may become prohibitive due to its high computational and memory requirements. Lastly, writer codes based on learned features or Hinge statistical features did not lead to improved recognition performance. [Code used for this research can be found at <https://github.com/tobiasvanderwerff/master-thesis>
^* Corresponding author]
§ INTRODUCTION
Handwriting recognition has seen major successes using deep learning, manifested in domains like handwritten text recognition <cit.>, writer identification <cit.>, binarization <cit.>, and word spotting <cit.>. However, neural networks are often still lacking when it comes to adapting to novel environments <cit.>. Arguably, much of the modern success of deep learning can be attributed to collecting massive amounts of data to cover as many parts of the underlying data distribution as possible, combined with a proportional increase in computing power and model size <cit.>. However, such a brute-force approach to learning is often not practical for handwriting recognition tasks. Large, high-quality corpora of annotated handwritten texts are often scarce, especially for historical handwriting. In this case, more efficient use of data and reusability of previously learned representations becomes important.
In this paper, we focus on improving one of the most common handwriting recognition tasks: handwritten text recognition (HTR), which refers to the process of automatically turning images of handwritten text into letter codes. HTR remains a challenging problem, mainly due to the large number of possible handwriting variations (Fig. <ref>). In this research, we attempt to make modern HTR models writer adaptive, referring to the idea that when a trained HTR model is presented with a novel writing style, it is able to modify its internal representations in such a way as to improve recognition performance for that style. We focus on cases with limited data available for adaptation (10-20 samples), as this represents a realistic scenario for real-time adaptation. In a practical setting, a user of an HTR system could be asked to supply a handful of handwriting examples in order to improve recognition performance on their writing style. How to perform writer-specific adaptation effectively remains an open problem. A popular approach for adapting existing deep learning models is transfer learning, where previously learned model parameters are reused for a new but related task that has only a modest amount of training data, leading to notable successes in fields such as natural language processing <cit.> and computer vision <cit.>.
It is important to note that the potential benefit of including writer identity as a conditional variable cannot easily be decoupled from architectural choice. For example, Hidden Markov Models <cit.> have been a common choice for HTR in the past, and methods have been developed to include writer identity in such models. However, these methods are often not usable for modern approaches to HTR using deep neural networks, which use powerful hierarchical representations that outperform past methods. In this sense, a relevant question is whether state-of-the-art deep learning approaches to HTR can benefit from explicit writer information in the first place. We will show that this benefit is not obvious, providing at best modest improvements compared to a writer-unaware baseline.
There are several problems at hand. In order to adapt effectively based on style information, there is a clear need to identify what exactly a deep learning model has not learned yet. The question can be formulated as “What novelty does this new writer introduce that is not effectively handled by the neural network?”. Another relevant question is what signal source can be provided to allow for adaptation, and the non-trivial question of effectively including such information into an HTR model. We draw inspiration from a recently published paper by <cit.> which employs meta-learning to flexibly adapt HTR models to different writers, seemingly with great success. Meta-learning (also known as learning-to-learn) is currently an active area of research <cit.>.
Meta-learning is concerned with improving the learning algorithm itself. Often, the idea is to adapt a learning algorithm to a new task based on a small number of task-specific examples. The aim is to learn underlying meta-knowledge that can be transferred to various tasks, even those unseen during training. The paper by <cit.> makes use of a modified form of model-agnostic meta-learning (MAML) <cit.>, which they call MetaHTR. As this is one of the more promising ideas for writer-aware adaptation, we explore several versions of the MAML approach and will test its ability to perform writer-specific adaptation.
Additionally, we experiment with another approach, based on writer codes: Compact vector representations of individual writers that are supposed to capture the most relevant information about a writer to allow for effective adaptation. Writer codes can be learned or explicitly given as part of the model input. The codes are inserted into a trained HTR model by adjusting the parameters of batch normalization layers. We experiment with several approaches to creating such a writer code: One based on learned feature vectors and one based on traditional handcrafted features used for writer identification. Although this approach is conceptually appealing, our version of writer codes does not yield concrete benefits for adaptation.
We summarize the contributions in this paper as follows:
* We show that MAML-based methods applied to a trained HTR model can lead to
improved data efficiency, showing an improvement between 1.4 and 2.0 word
error rate compared to a naive fine-tuning baseline;
* We test the capability of MetaHTR to perform writer-specific adaptation, finding
that it leads to an improvement of 0.7 word error rate for a deep HTR model,
but shows no significant effect for smaller models;
* We analyze how a trained HTR model can be effectively adapted based on
writer-specific vector representations, finding that fine-tuning batch
normalization scale and bias parameters can be an effective way to obtain
additional performance gains, even without writer-specific information;
* We show that writer codes based on learned features or Hinge statistical features do not lead to improved recognition performance.
This paper is structured as follows. In Section <ref>, we provide related works. In Section <ref>, we propose several techniques for writer-adaptive HTR and experiments to verify their performance. In Section <ref>, we outline our experimental setup. In Section <ref>, we show results for the proposed methods, and finally, in Section <ref> and Section <ref>, we discuss the results and future work.
§ RELATED WORKS
Handwritten text recognition:
Early approaches to HTR often employed Hidden Markov Models <cit.> (HMM). More recently, the field of HTR has progressed from HMM-based methods to end-to-end trainable neural networks with many layers. Recurrent neural networks (RNN), and in particular Multi-dimensional Long Short-Term Memory (MDLSTM), networks <cit.> have been commonly used sequence modeling architectures for HTR models <cit.>. The MDLSTM architecture, in combination with the Connectionist Temporal Classification <cit.> loss (CTC), served as a replacement for Hidden Markov Model-based methods <cit.>. Whereas standard RNN architectures process data along a one-dimensional axis – e.g., a time axis –, the MDLSTM architecture allows recurrence across multi-dimensional sequences, such as images. In more recent years, it has been observed that the expensive recurrence of the MDLSTM could be replaced by a CNN + bidirectional LSTM architecture <cit.>. The CNN-RNN hybrid + CTC has been a commonly used architecture (e.g., <cit.>). For example, in <cit.>, a spatial transformer network, residual convolutional blocks (ResNet-18), stacked BiLSTMs, and a CTC layer are used.
Although CTC has been a common decoding method, some of its downsides – such as the inability to consider linguistic dependency across tokens – have led to architectures that replace CTC in favor of attention modules <cit.>. Attention-based encoder-decoder architectures have reached state-of-the-art performance in recent years <cit.>. Attention alleviates constraints on input image sizes and the need for segmentation or image rectification <cit.> for irregular images. This thus allows for simplification in the design of HTR architectures. In <cit.>, a ResNet-31 is combined with an LSTM-based encoder-decoder along with a 2-dimensional attention module for irregular text recognition in natural scene images.
A trend in recent years has been to replace the linear recurrence of RNNs with the more parallelizable Transformer architecture and attention-based approaches more broadly. In a recent work <cit.>, various architectures for universal text line recognition are studied, using various encoder and decoder families. The authors find that a CNN backbone for extracting visual features, coupled with a Transformer encoder, a CTC decoder, and an explicit language model, is the most effective approach for recognizing line strips. Building on top of the idea <cit.> of using Transformer-only architectures for vision tasks, <cit.> explore an end-to-end Transformer encoder-decoder architecture for text recognition, initialized with a pretrained vision Transformer for extracting visual features and a pretrained RoBERTa <cit.> Transformer for sequence decoding. After initialization, the model is pretrained on large-scale synthetic handwritten images and fine-tuned on a human-labeled dataset.
Meta-learning:
Meta-learning, or learning-to-learn, is an alternative paradigm to traditional neural network training, which aims to improve the learning algorithm itself <cit.>. By learning shared knowledge across various tasks over multiple learning episodes, the aim is to improve future learning performance. The main meta-learning method we focus on here is Model-Agnostic Meta-Learning <cit.> (MAML). MAML aims to find a parameter initialization such that a small number of gradient updates using a handful of labeled samples produces a classifier that works well on validation data. MAML is related to transfer learning, in the sense that finding good initialization parameters for a model to facilitate adaptation to various tasks plays a central role. Due to its model-agnostic nature, MAML can be applied to various application domains without significant modifications.
Due to the inner/outer-loop optimization process, MAML has great flexibility in terms of the kinds of parameters that can be learned in the inner loop, e.g., parameterized loss functions <cit.>, learning rates <cit.>, and attenuation weights <cit.>. Meta-learning has been applied to various areas such as reinforcement learning and few-shot classification, but, notably, also to speech recognition, in the form of accent adaptation <cit.> and speaker adaptation <cit.>. MetaSGD <cit.> is a modification of MAML and involves learning the update direction and learning rate along with the parameter initialization. MAML++ <cit.> addresses the training instability of MAML that is commonly observed. MAML has also been used in combination with other types of meta-learning. For example, in <cit.>, the authors combine MAML with model-based meta-learning, using a latent generative representation of model parameters and applying MAML in this lower-dimensional latent space.
Writer adaptation:
Many early approaches for writer adaptation are proposed for HMMs using Gaussian Mixture Models. For example, <cit.> use linear transformations between original parameters and re-estimated parameters for adjusting GMM parameters using maximum likelihood linear regression. More recently, there have been several attempts at adaptation in the space of HTR using neural networks. In <cit.>, the authors perform simple fine-tuning on a new handwriting collection, showing that this can lead to efficient transfer between datasets using a limited amount of fine-tuning data. In <cit.>, the authors cluster writers by style and train a classifier for each cluster, using a mixture-of-experts setup for choosing the best combination of classifiers. For a new writer, the combination of classifiers is based on classification confidence for that writer. In <cit.>, the authors learn a linear writer-specific feature transformation in order to create a style-invariant classifier, which they call Style Transfer Mapping (STM). Whereas the original approach was not used in the context of neural networks, a later approach <cit.> uses STM for neural networks in the context of Chinese character recognition. In <cit.>, the authors employ writer codes for writer-specific Chinese handwritten text recognition using a CNN-HMM hybrid model. They feed a writer code into adaptation layers tied to individual convolution layers. The result is added element-wise to the intermediate CNN feature maps. At train time, writer codes are jointly learned with the adaptation layers. At test time, codes for new writers are randomly initialized and optimized using one to three gradient steps. Recently, <cit.> used a style extractor network trained on a writer identification task to extract a writer code, used to adapt a writer-independent recognizer. Specifically, the writer code is added to the convolutional layer output after being fed through a fully-connected layer.
The writer adaptation problem has also been formulated as a domain adaptation problem <cit.>. In <cit.>, a gated attention similarity unit is used to find character-level writer-invariant features. In <cit.>, the authors employ an adversarial learning approach using synthetic data. A generic HTR model is initially trained using synthetic data and adapted to new writers using a domain discriminator network.
§ METHODOLOGY
Overview:
An HTR model f_θ – corresponding to a deep neural network –, is trained to maximize the probability p(Y|; θ) of the correct transcription given an input image and ground truth character sequence Y = (y_1, y_2,…,y_L), where each y_i is picked from a vocabulary V (e.g., ASCII characters). A training dataset = (_1, Y_1), (_2, Y_2), …, (_N, Y_N) consists of tuples containing an image ℐ_i and the corresponding character sequence Y_i. The cost function is derived from cross-entropy, which, for a single example, is of the following form:
ℒ(ℐ, Y; θ) =
- 1/L∑_t=1^Llog p(Y_t = y_t|y_<t, ; θ).
§.§ Base models
We make use of two base models: FPHTR <cit.> and SAR <cit.>. FPHTR builds on the Transformer architecture, and SAR on the LSTM architecture. In Fig. <ref>, we show a high-level overview of both models to highlight their overall structure and similarity. For both models, we use two versions: a smaller version using an 18-layer ResNet backbone and a larger version with a 31-layer ResNet backbone (see Appendix <ref> for parameter counts). The base models are standard HTR models that do not make use of explicit writer information, chosen based on their competitive performance on common benchmarks. Their performance serves as a baseline for “writer-unaware” HTR models.
§.§.§ SAR
The SAR model <cit.> is based on the Long Short-Term Memory (LSTM) architecture <cit.>. It consists of a ResNet image processing backbone, LSTM encoder, LSTM decoder, and a 2-dimensional attention module. The CNN backbone consists of a modified ResNet <cit.>, which outputs a 2-dimensional feature map V. This is used by the consecutive LSTM encoder to extract a holistic feature vector for the whole image and also serves as context for the 2D attention network. The final encoder hidden state h_W is fed as the initial input to the LSTM decoder. A special start-of-sequence token (SOS) is fed as input to the decoder. At each timestep of the LSTM, a new character is sampled autoregressively. Each input at the timesteps that follow is either 1) the previous character from the ground truth character sequence (also known as teacher forcing), or 2) the sampled character from the previous timestep (at test time). If the latter is the case, the end of the sampling procedure is signified by sampling a special end-of-sequence token (EOS). All token inputs are fed in as vector representations, followed by a linear transformation, ψ(.). After being fed through an LSTM cell along with the previous hidden state, the timestep prediction is then calculated as y_t = ϕ(h'_t, g_t) = softmax(W_o[h'_t; g_t]), where h'_t is the current hidden state and g_t is the output of the attention module. W_o is a linear transformation, which maps the features to a vector whose size is equal to the number of character classes. The attention module is a modification of the standard 1D attention module for dealing with a 2D spatial layout. It takes into account neighborhood information in the 2D plane:
e_ij = tanh(W_v v_ij + ∑_p,q∈𝒩_ijW̃_p-i,q-j·v_pq + W_h h'_t)
α_ij = softmax(w_e^T ·e_ij)
g_t = ∑_i, jα_ijv_ij, i=1,…,H, j=1,…,W
Explanation of the symbols: v_ij is the local feature vector at position (i, j) in V; 𝒩_ij is the eight-neighborhood around this position; W_v, W_h, W̃ are learned linear transformations; α_ij is the attention weight at location (i, j); and g_t is the weighted sum of local features, also known as a glimpse. The difference with a traditional attention module is the addition of the ∑_p,q∈𝒩_ijW̃_p-i,q-j·v_pq term when weighing v_ij.
§.§.§ FPHTR
FPHTR <cit.> is a Transformer-based architecture, consisting of a CNN backbone combined with a Transformer <cit.> module for decoding the visual feature map into a character sequence. The architecture was originally proposed for full-document HTR, but due to its generic nature, it can easily be applied to both word and line images without any real modifications. The CNN takes an image as input and produces a 2D feature map with hidden size d_model as output. A 2D position encoding based on sinusoidal functions is added, and the feature map is flattened into a 1D sequence of feature vectors – each representing a position in the image –, that can be processed by the Transformer decoder. The Transformer decoder is a standard Transformer architecture <cit.> with non-causal attention to the encoder output (it can attend to the entire output of the encoder) and causal self-attention (it can only attend to past positions of its character input). Input vectors are enhanced with 1D position encodings. Sampling is done autoregressively, in the same way as the SAR model.
§.§ Meta-learning
Our first attempt to make HTR models writer adaptive involves meta-learning <cit.>. Adaptation occurs by providing the model with labeled examples of a writer that it should adapt to, after which the weights of the model are updated using the model-agnostic meta-learning algorithm. We first provide a brief overview of model-agnostic meta-learning in Section <ref>, then turn to the MetaHTR approach in Section <ref>. The explanation of these methods will be brief; for a more detailed explanation, we refer the reader to the original papers.
§.§.§ Model-agnostic meta-learning
Model-agnostic meta-learning (MAML) <cit.> is an approach to meta-learning aimed at finding initial parameters that facilitate rapid adaptation. Let p() be a distribution over tasks to which a model should be able to adapt. During meta-training, a batch of tasks _i ∼ p() is sampled, where samples from each task are split up in a support set D^tr of size K for adaptation (where typically K is relatively small, e.g., K ≤ 16), and a query set D^val for testing the task-specific performance after adaptation. Training is done using stochastic gradient descent (SGD), where the model parameters θ are adapted to a task as follows:
θ_i' = θ - α∇_θℒ^inner(D_i^tr; θ).
This is referred to as the inner loop, using an inner loop learning rate α. After inner loop adaptation, the adapted parameters θ_i^' are evaluated on the query set, and the original parameters are updated by aggregating the loss over the sampled tasks, using an outer loop learning rate β:
θ←θ - β∇_θ∑__i ∼
p()ℒ^outer(D^val_i; θ_i').
Whereas the inner loop optimizes for task-specific performance, the outer loop optimizes for a parameter set θ so that the task-specific training is more efficient, aiming to achieve good generalization across various tasks.
§.§.§ MetaHTR
MetaHTR is a modification of the MAML algorithm optimized for text recognition. Within the MetaHTR framework, each task instance 𝒯_i corresponds to a different writer. The full training process is summarized in Algorithm <ref>. Once MetaHTR is trained, it can be used to rapidly adapt to specific writers at inference time. This is shown in Algorithm <ref>. With respect to MAML, MetaHTR introduces two modifications: character instance-specific weights, and learnable layer-wise learning rates.
Character instance-specific weights:
Instance-specific weight values are added to the inner loop loss such that the model can adapt better with respect to characters having a high discrepancy. Given a ground truth character sequence Y = {y_1,y_2,…,y_L} and image , the inner loop loss now adds a value γ_t for each time-step t:
ℒ^inner = - 1/L∑_t=1^Lγ_t log p(y_t|; θ),
which is a modified version of cross-entropy, including γ_t values inside the summation. In order to calculate γ_t, gradient information from the final classification layer is used. The idea is that the gradients provide information related to disagreement, i.e., what knowledge is missing from the model that still needs to be learned. Specifically, let the weights of the final classification be denoted as ϕ. The gradients of the t'th instance loss with respect to the weights of the final classification layer are used, denoted as ∇_ϕℒ^t, in combination with the gradients of the mean loss (Eq. <ref>), denoted as ∇_ϕℒ. Both inputs are concatenated and fed as input to a network g_ψ, leading to character instance-specific weight γ_t, where
γ_t = g_ψ( [∇_ϕℒ^t; ∇_ϕℒ] ).
g_ψ takes the form of a 3-layer MLP with parameters ψ, followed by a sigmoid layer to produce a scalar output value in the range [0, 1].
Learnable layer-wise learning rates:
The inner loop learning rate used in MAML is replaced by a learnable one <cit.>. Specifying a learnable learning rate for every model parameter allows the model to express differences between what parameters should be updated more or less. However, using a learning rate for every parameter also doubles the parameter count, which is prohibitive. Therefore, learning rates are used for individual layers in the model, which are trained along with all the other parameters. This is also shown in Algorithm <ref>.
§.§.§ Meta-learning evaluation
We evaluate several variants of the MAML/MetaHTR approach. One downside of the MAML approach and MetaHTR, in particular, is that it leads to a notable increase in memory and computational requirements. We, therefore, analyze variations of the MAML-based approach to investigate to what degree it can be simplified. Concretely, we experiment with three different models: MAML, MAML + llr, and MetaHTR.
* MAML: The original MAML algorithm, as proposed in <cit.>, using the sequence-based cross-entropy loss function shown in <ref>.
* MAML + llr: The MAML algorithm is complemented with learnable inner loop learning rates (Section <ref>). This alleviates the need to manually set the inner loop learning rate, at the cost of only a few hundred additional parameters (see Appendix <ref>)
* MetaHTR: The full MetaHTR model is explained in Section <ref>. A downside of the MetaHTR approach is the additional complexity that it introduces. Next to the calculation of higher-order derivatives as part of the MAML algorithm, MetaHTR also requires an additional backward pass in order to calculate the instance-specific weights. This makes the approach expensive both in terms of computation and in terms of memory usage, therefore making it challenging to scale to larger contexts such as sentence-level HTR.
§.§ Writer codes
Our second attempt to include writer information into the base HTR models is based on the idea of representing style or writer information as a compact feature vector. In speech recognition, such a code is known as a speaker code <cit.>. We take a similar approach by trying to model writers or styles using a small feature vector, which is used to adapt the weights of an existing HTR model. We will refer to such vectors as writer codes. A writer code is a dense feature vector 𝐱∈^M, where M is set based on the desired representational capacity. A relevant property of writer codes is that they should be able to obtain them even for writers that are not part of the initial training set. Writer codes have certain properties that make them appealing as a method for writer-adaptive HTR: they are efficient to compute and often require minimal changes to a base architecture.
§.§.§ Code insertion
First, we address the question of how the codes should be inserted into the base model for effective adaptation. A comprehensive evaluation of possible methods for code insertion is beyond the scope of this paper, but we note here that, based on various experiments, naive insertion of codes into the base models can easily deteriorate base-level performance. Notably, naively modifying batch normalization (batch norm) parameters can lead to catastrophic forgetting. Furthermore, we found that adapting only certain key layers of the network, such as the last layers of the ResNet backbone, was not sufficient to allow for effective adaptation. Instead, an effective form of vector-based adaptation comes from fine-tuning the normalization layers of the model. This approach is inspired by work on generative models, such as conditional GANs <cit.> and methods for style transfer <cit.>. Previous work in the field of style transfer suggests that in order to adapt features to a particular style, it can be sufficient to specialize scaling and shifting parameters after normalization layers, conditioned on style information <cit.>. We adopt a similar approach, where we update the learnable weights of the normalization layers in our network, conditioned on a specific writer code. Specifically, we focus on batch normalization layers, which are present in the ResNet backbone[It is worth noting that for the FPHTR model, layer normalization is used in addition to batch normalization. However, we found no concrete benefit in adjusting these normalization layers.]. Given a minibatch of activations B = x_1,…,m, batch normalization layers are of the following form:
y_i = x_i - μ_B/√(σ^2_B + ϵ)·γ + β,
where γ and β are learnable parameter vectors of size equal to the number of channels in the input. The ϵ parameter is a small constant added for numerical stability. The normalization statistics are calculated along the batch dimension:
μ_B = 1/m∑_i=1^m x_i, σ^2_B = 1/m∑_i=1^m (x_i - μ_B)^2.
For inserting writer codes into the neural network, we modify the β and γ parameters based on an input code (corresponding to an approach called conditional batch normalization <cit.>). Given pretrained parameters β_c and γ_c, changes in these parameters are predicted based on an input code e and a two-layer MLP:
Δβ = ϕ_1(e), Δγ = ϕ_2(e),
where ϕ_1 and ϕ_2 are MLPs. The predicted deltas are then added to the original β_c and γ_c parameters:
β̂_c = β_c + Δβ_c, γ̂_c = γ_c + Δγ_c,
where β̂_c and γ̂_c replace the batch norm parameters for the current forward pass. All other parameters are frozen during training, including β and γ. By changing the γ and β affine parameters that follow normalization, there is great flexibility in changing the intermediate feature maps according to the specifics of a particular code, while the risk of catastrophic forgetting is mitigated by keeping the original batch normalization weights largely intact.
§.§.§ Code creation
Given the conditional batch normalization method for inserting writer codes into an HTR model, we turn to the question of how we create writer codes. An important criterion is that writer codes are not created under a closed writer set assumption; we should be able to instantiate them for novel writers as well. We experiment with two kinds of writer codes: learned codes, and codes based on statistical writer information (Hinge codes and style codes).
Learned codes:
Learned writer codes are obtained by training them in the same way as the weights of the network. A similar idea is commonly seen in NLP (e.g., <cit.>), where for each token in a predefined vocabulary, an associated vector representation is learned (often called an “embedding”) that is more expressive than a one-hot vector indicating the identity of the token. Note that this approach implies a fixed set of writer codes initialized at the start of training – one for each writer in the training set. In the case when a new writer is presented that is unseen during training, we follow <cit.> by randomly initializing a new writer code, followed by one or several gradient steps on the newly initialized code, using a small batch of labeled writer-specific data.
Hinge codes:
When it comes to capturing writer individuality, there exists a rich literature on this topic in the field of writer identification <cit.>. In contrast to the learned features discussed in the previous section, features for writer identification are often handcrafted or statistical in nature. One of the more successful features for writer identification is the Hinge feature <cit.>, which uses a probability distribution of the angle combination of two hinged edge fragments to characterize writer individuality. The assumption here is that these features can lead to a meaningful clustering of writers based on their style differences. These writer codes are attractive because they are easy to calculate and do not require additional adaptation data at inference time.
Style codes:
We also focus on generic style clusters in feature space, rather than features that are highly writer-specific. For example, style clusters could point to high-level writing styles such as cursive or mixed cursive. We perform k-means clustering on Hinge codes to obtain generic style clusters. For each style cluster, we train a writer code using backpropagation. Thus, given an image input, we find the closest style cluster based on the Hinge features and map the style cluster identity to a learned writer code that is updated using gradient descent.
§ EXPERIMENTS
§.§ Dataset
We use the IAM dataset <cit.> for evaluation, using word-level images. The dataset consists of English handwritten texts contributed by 657 writers, making a total of 1,539 handwritten pages consisting of 115,320 segmented words. The data is labeled at the sentence, line, and word level. Examples of word images are shown in Fig. <ref>. For splitting the data into a training, validation, and test set, we use the widely used Aachen splits <cit.>. An important property of these splits is that the writer sets are disjoint, i.e., writers seen during training are not seen during testing. The Aachen splits contain 500 writers making up a total of 75,476 images.
§.§ Implementation details
Base models:
Character error rate (CER) and word error rate (WER) are used for evaluation, with the best model chosen based on the lowest WER. We use a character-level vocabulary, converting all characters to lowercase. No linguistic post-processing on word predictions is used. We report average performance over five random seeds, along with standard deviations for all results. For training of the base models, the Adam optimizer is used <cit.>, with β_1 = 0.9 and β_2 = 0.999. We use gradient clipping to avoid exploding gradients based on the L2-norm of the gradient vector. All models are implemented using PyTorch <cit.>, using a single Nvidia V100 GPU with 32GB of memory. See appendix Table <ref> for full details about hyperparameter settings. We use random image rotation, scaling, brightness, contrast adjustment, and Gaussian noise to increase image diversity. We reduce the resolution by 50% to reduce memory footprint while keeping the text legible.
Meta-learning:
Given the K-shot N-way meta-learning formulation, we use K = 16 and N = 8, following <cit.>. This means that during adaptation, a batch of K = 16 writer-specific examples are used to adapt the model to a specific writer, and outer loop gradients are averaged over N = 8 writers (see Eq. <ref>). During training, we randomly sample writer-specific batches of size 2K, split into a support and query set of size K. At test time, we use all examples for a given writer: Given the j'th writer with N_j total examples, we randomly split the data into a support batch (adaptation batch) of size K, and use the remaining N_j - K examples for evaluation of the adapted model. Performance per writer is averaged over ten runs. For all models, we use dropout in the outer loop. Batch norm statistics are fixed to their running values and not updated during training, as this led to more stable performance (see Appendix <ref> for a more extensive discussion concerning the particulars of using batch normalization in combination with MAML). We use the learn2learn library <cit.> for implementing all meta-learning methods. Full hyperparameter settings are shown in the Appendix (Table <ref>).
Writer codes:
For the learned writer codes discussed in Section <ref>, we require adaptation data at test time to initialize codes for novel writers. Splitting of writer data is done in the same way as for meta-learning. During training, the weights of the trained HTR model are frozen, and only the writer code values and the parameters of the conditional batch norm MLPs are updated. We use a code size of 64 and an adaptation batch size of 16. For style codes, we use k-means clustering with k = 3, based on validation set performance. Complete hyperparameters are shown in Table <ref> in the Appendix.
§ RESULTS
§.§ Base models
The results for the base models on the IAM validation and test set are shown in Table <ref>. We report average performance as well as the performance of the best run. From the results in Table <ref>, we can see that the Transformer-based model (FPHTR) outperforms the LSTM-based model (SAR) on validation and test, both for the smaller 18-layer case (15-18M weights) and the larger 31-layer case (52-58M weights). This difference is significant in the case of the larger 31-layer models, with FPHTR outperforming SAR on the test with a difference of 4.1 WER and 4.8 CER. For the smaller 18-layer models, FPHTR outperforms SAR by a difference of 0.5 WER and 0.7 CER.
§.§ Meta-learning
Results for meta-learning are shown in Table <ref>. It should be noted that since all models presented here make use of additional adaptation data at test time, a direct comparison between the base models in Table <ref> is not directly meaningful. In other words, the MAML-based models have access to parts of the test data as part of their adaptation procedure. Therefore, we devise a different baseline, by evaluating the base models after performing fine-tuning on the same adaptation data that is made available to the MAML-based models. Specifically, we fine-tune the final classification layer of a base model using the adaptation data. We use the Adam <cit.> optimizer with a learning rate of 1e-3 for 3 optimization steps. Due to persistent out-of-memory errors for the SAR-31 MetaHTR model[Another performance-related issue worth mentioning is that MetaHTR requires calculation of instance-specific gradients, which, at the time of running the experiments, is something that is not supported in batch form in the PyTorch library. Therefore, this required a manual calculation of instance-specific gradients using a for-loop, which made the MetaHTR training procedure considerably slower than MAML. This problem is something that can be fixed using additional software, but the additional complexity of MetaHTR due to the extra backward pass remains.], we only include FPHTR-31 in addition to the smaller 18-layer variants. From these results, we can see that MetaHTR performs best, improving upon the baseline by 1.4, 2.0, and 1.7 WER for FPHTR-18, SAR-18, and FPHTR-31, respectively.
We plot the learned inner loop learning rates in Fig. <ref>, to get an idea of the relative weight assigned to each layer in the adaptation process. We show learned inner loop learning rates for two randomly chosen runs of the FPHTR-18 and FPHTR-31 models using MAML + llr (we include the figure for FPHTR-18 in the appendix, Fig. <ref>). Looking at these plots, we see a relatively high weight assigned to the ResNet layers, decreasing towards the head of the network. For the Transformer module, we observe an increasing trend in the learning rates across layers. This is an indication that the lower layers of the Transformer network require relatively fewer adaptation than layers closer to the output, with the final classification layer requiring the most adaptation.
It is worth noting here that the performance improvements for MetaHTR (between 1.4 to 2.0 WER compared to the baseline) are much smaller than reported in the original paper <cit.>, where MetaHTR improved upon the SAR baseline by a difference of 7.1 WER, and 6.8 after naive fine-tuning on the adaptation data. In email correspondence with the authors of the MetaHTR paper, we were not able to resolve the cause of this discrepancy. Furthermore, due to the lack of published code by the MetaHTR authors, it is difficult to cross-verify the MetaHTR results.
§.§.§ Testing the adaptation premise of MetaHTR
An important question concerning the efficacy of MetaHTR is to what degree it truly adapts based on a set of writer-specific images at test time. This is an important premise, since the additional computational overhead of MetaHTR as well as the increased complexity compared to regular neural network training is supposedly warranted by a clear goal: An ability to adapt in a flexible way to various writers leading to a performance improvement compared to a writer-unaware model. In the words of the authors, the goal of MetaHTR is to offer a “adapt to my writing button” <cit.>, where one is asked to write a specific sentence in order to make recognition performance of that handwriting more accurate.
Note that because the MetaHTR objective function and training procedure are different from the training procedure used for the baseline, it is not clear that the improved performance of MetaHTR is due to writer adaptation. The MetaHTR objective function is designed for writer-specific adaptation, but it may simply be a more effective way to train the neural network, regardless of whether writer adaptation is performed or not. The writer adaptation performed at test time is what is supposed to make MetaHTR writer adaptive. Therefore, if it is writer adaptive, it should perform better than MetaHTR without writer adaptation at test time. In order to test this, we leave out the writer-specific adaptation. More concretely, we train MetaHTR the same way as done before but evaluate it without performing inner loop adaptation on a support batch of K images. Results are shown in Table <ref>. The additional benefit of adaptation is 0.2 WER for FPHTR-18, 0.7 WER for SAR-18, and 0.7 WER for FPHTR-31. We use a two-sample t-test to measure the statistical significance of the difference in results. Using a significance level α = 0.05, we observe that the difference in results is not significant for FPHTR-18 (p = 0.4143) and SAR-18 (p = 0.0832), but is significant for FPHTR-31 (p = 0.0001). In other words, adaptation only shows a significant effect for the larger FPHTR-31 model, but not for the smaller 18-layer variants.
§.§ Writer codes
We show results for all writer codes in Table <ref>. From the table, it can be seen that the learned codes do not improve upon the performance of the baseline. The fact that writer codes at test time are created by random initialization followed by only a small number of gradient steps is a potential factor here – codes trained in this way seem to hurt performance rather than improve it.
Next, we consider Hinge and style codes. Both methods outperform the baseline. For the Hinge code, this is a difference of 1.7 and 1.6 WER for FPHTR and SAR, respectively. A similar performance improvement can be seen for the style code, obtained by clustering Hinge codes with a single learned code per style cluster. In this case, the difference is 1.8 and 1.7 WER for FPHTR and SAR, respectively.
Although these results show improvement compared to the baselines, they do not provide adequate insight into the efficacy of the codes themselves. Recall from Section <ref> that conditional batch normalization uses a 3-layer MLP with the writer codes as input to predict changes to the original batch norm weights. It is possible that the MLP learns effective bias vectors that improve performance regardless of the writer code input, i.e., the writer code could simply be ignored (e.g., assigned zero weights). To test this, we replace the writer codes with a zero code that contains no writer information whatsoever, i.e., a vector with only zero values. As seen from Table <ref>, this leads to almost identical performance compared to both the Hinge and style code. This is a strong indication that writer information is not the direct cause of the increase in performance, but rather, conditional batch normalization seems to be an effective way to fine-tune the batch norm weights, even without the presence of conditional information. Although this may be an interesting way to perform general fine-tuning, it does not rely on writer-specific information to make it possible.
§ DISCUSSIONS
§.§ Meta-learning
An appealing aspect of the meta-learning approach is that there is a great deal of flexibility in the way the model can adapt to a writer by differentially updating the layers of the model (e.g., as demonstrated in Fig. <ref>). Nevertheless, the added benefit of writer adaptation using MetaHTR is not obvious, as shown in Section <ref>. Even without using any adaptation data at test time, the MetaHTR model still improves upon the baseline performance. This indicates that more effective representations play a role in the additional performance gains, rather than rapid adaptability of the model parameters, a phenomenon observed before in the literature on meta-learning <cit.>. This makes MetaHTR interesting for improving overall model performance, but not necessarily for writer-specific adaptation. Another downside of the MetaHTR approach is the additional complexity that it introduces. Next to the calculation of higher-order derivatives as part of the MAML algorithm, MetaHTR requires an additional backward pass to calculate the instance-specific weights (Section <ref>). This makes the approach expensive both in terms of computation and memory usage and makes it challenging to scale to larger contexts such as sentence-level HTR. This is exemplified by the fact that we were not able to train MetaHTR in combination with the SAR-31 base model on a 32GB GPU due to persistent out-of-memory errors. This is somewhat problematic given our finding that a deeper model lends itself better to adaptation using MetaHTR than a shallower one. Another example of additional complexity is the difficulty caused by the interaction of MAML with batch normalization (see Appendix <ref> for a more extensive discussion on this topic).
Moreover, training of MetaHTR requires a good deal of fine-tuning of various hyperparameters to make it work well, which is something that has also been observed for MAML more broadly <cit.>. Given the modest benefits for writer adaptation (0.7 WER in the best case), combined with the increased model complexity, it can be argued that MetaHTR is perhaps not worth the extra investment for writer adaptation. This is especially true given that when more labeled examples are available, a simpler method, such as transfer learning, may be more cost and time effective.
§.§ Writer codes
The results in Table <ref> show the limited effectiveness of the writer code idea. We showed that statistical features for characterizing writer identity do not show a benefit over a constant zero vector. The fact that the Hinge feature is designed to be independent of the textual context of the handwriting samples may play a role here <cit.>. An option for future work would be to explore features that lend themselves better to characterize the most relevant writer characteristics, such as idiosyncratic letter shapes that are difficult to classify. For example, a Fraglet approach based on shape codebooks <cit.> may capture the individual shape features of a particular handwriting more appropriately (see Fig. <ref>). A histogram can be compiled by matching codebook prototypes with the character shapes observed for an individual writer, counting the matched codebook entries. The normalized histogram can subsequently be used as a vector representation.
One factor which may play an important role here is data volume. For example, consider automatic speech recognition, where the notion of “speaker adaptation” appears to be more common. One facet in which speech and text recognition diverge is the availability of large-scale labeled datasets. Whereas collecting and labeling handwriting samples can be cumbersome and labor-intensive, speech transcriptions are generally easier to obtain. Thus, if data volume is the critical bottleneck for learning robust representations that lend themselves well to adaptation, methods used in speech recognition relying on large-scale datasets may not transfer as well to HTR. Indeed, as shown by recent work on large language models <cit.>, scale may be a major enabling factor for effective few-shot adaptation.
§ CONCLUSION
In this paper, we studied various methods for making neural network-based HTR models writer adaptive. Meta-learning showed the most promising results, with both MAML and MetaHTR leading to improved performance compared to baseline models. However, we showed that only a relatively small portion of these improvements (between 14-39%, or 0.2-0.7 WER) can be attributed to writer adaptation, with most of the improvements coming from changes in the way the neural network is trained. It remains to be seen whether MetaHTR could be used to handle more radical domain shifts, as seen, for example, in historical handwriting. Given the observation that writer adaptation using MetaHTR may work better for deeper models, potential future work may focus on scaling up MetaHTR to deeper models. However, memory and/or computational requirements may become prohibitive in this case. Lastly, results show that writer code-based adaptation using learned features or statistical Hinge features does not lead to increased performance. However, updating batch normalization weights may be an effective way to perform general fine-tuning.
unsrtnat
69
urlstyle
[Michael et al.(2019)Michael, Labahn, Grüning, and
Zöllner]michael2019evaluating
Johannes Michael, Roger Labahn, Tobias Grüning, and Jochen Zöllner.
Evaluating sequence-to-sequence models for handwritten text
recognition.
In 2019 International Conference on Document Analysis and
Recognition (ICDAR), pages 1286–1293. IEEE, 2019.
[Ameryan and Schomaker(2021)]ameryan2021limited
Mahya Ameryan and Lambert Schomaker.
A limited-size ensemble of homogeneous cnn/lstms for high-performance
word classification.
Neural Computing and Applications, 330 (14):0
8615–8634, 2021.
[Yang et al.(2016)Yang, Jin, and Liu]yang2016deepwriterid
Weixin Yang, Lianwen Jin, and Manfei Liu.
Deepwriterid: An end-to-end online text-independent writer
identification system.
IEEE Intelligent Systems, 310 (2):0 45–53,
2016.
[He and Schomaker(2020)]he2020fragnet
Sheng He and Lambert Schomaker.
Fragnet: Writer identification using deep fragment networks.
IEEE Transactions on Information Forensics and Security,
15:0 3013–3022, 2020.
[Dhali et al.(2019)Dhali, de Wit, and Schomaker]dhali2019binet
Maruf A Dhali, Jan Willem de Wit, and Lambert Schomaker.
Binet: Degraded-manuscript binarization in diverse document textures
and layouts using deep encoder-decoder networks.
arXiv preprint arXiv:1911.07930, 2019.
[Chanda et al.(2018)Chanda, Baas, Haitink, Hamel, Stutzmann, and
Schomaker]chanda2018zero
Sukalpa Chanda, Jochem Baas, Daniel Haitink, Sébastien Hamel, Dominique
Stutzmann, and Lambert Schomaker.
Zero-shot learning based approach for medieval word recognition using
deep-learned features.
In 2018 16th International Conference on Frontiers in
Handwriting Recognition (ICFHR), pages 345–350. IEEE, 2018.
[Kouw and Loog(2019)]kouw2019review
Wouter M Kouw and Marco Loog.
A review of domain adaptation without target labels.
IEEE transactions on pattern analysis and machine
intelligence, 430 (3):0 766–785, 2019.
[Kaplan et al.(2020)Kaplan, McCandlish, Henighan, Brown, Chess, Child,
Gray, Radford, Wu, and Amodei]kaplan2020scaling
Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon
Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei.
Scaling laws for neural language models.
arXiv preprint arXiv:2001.08361, 2020.
[Devlin et al.(2018)Devlin, Chang, Lee, and Toutanova]devlin2018bert
Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.
Bert: Pre-training of deep bidirectional transformers for language
understanding.
arXiv preprint arXiv:1810.04805, 2018.
[Oquab et al.(2014)Oquab, Bottou, Laptev, and Sivic]oquab2014learning
Maxime Oquab, Leon Bottou, Ivan Laptev, and Josef Sivic.
Learning and transferring mid-level image representations using
convolutional neural networks.
In Proceedings of the IEEE conference on computer vision and
pattern recognition, pages 1717–1724, 2014.
[Schomaker(2002)]305ba571ed2f4e7ea01992ae48e26ac9
Lambert Schomaker.
Patronen en symbolen: een wereld door het oog van de machine.
s.n., 2002.
[Baum and Petrie(1966)]baum1966hmm
Leonard E Baum and Ted Petrie.
Statistical inference for probabilistic functions of finite state
markov chains.
The annals of mathematical statistics, 370
(6):0 1554–1563, 1966.
[Bhunia et al.(2021)Bhunia, Ghose, Kumar, Chowdhury, Sain, and
Song]bhunia2021metahtr
Ayan Kumar Bhunia, Shuvozit Ghose, Amandeep Kumar, Pinaki Nath Chowdhury,
Aneeshan Sain, and Yi-Zhe Song.
Metahtr: Towards writer-adaptive handwritten text recognition.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pages 15830–15839, 2021.
[Hospedales et al.(2020)Hospedales, Antoniou, Micaelli, and
Storkey]hospedales2020metasurvey
Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey.
Meta-learning in neural networks: A survey.
arXiv preprint arXiv:2004.05439, 2020.
[Finn et al.(2017)Finn, Abbeel, and Levine]finn2017maml
Chelsea Finn, Pieter Abbeel, and Sergey Levine.
Model-agnostic meta-learning for fast adaptation of deep networks.
In International Conference on Machine Learning, pages
1126–1135. PMLR, 2017.
[Bianne-Bernard et al.(2011)Bianne-Bernard, Menasri, Mohamad, Mokbel,
Kermorvant, and Likforman-Sulem]bianne2011hmm
Anne-Laure Bianne-Bernard, Fares Menasri, Rami Al-Hajj Mohamad, Chafic Mokbel,
Christopher Kermorvant, and Laurence Likforman-Sulem.
Dynamic and contextual information in hmm modeling for handwritten
word recognition.
IEEE transactions on pattern analysis and machine
intelligence, 330 (10):0 2066–2080, 2011.
[Graves et al.(2007)Graves, Fernández, and
Schmidhuber]graves2007multi
Alex Graves, Santiago Fernández, and Jürgen Schmidhuber.
Multi-dimensional recurrent neural networks.
In International conference on artificial neural networks,
pages 549–558. Springer, 2007.
[Puigcerver(2017)]puigcerver2017htr
Joan Puigcerver.
Are multidimensional recurrent layers really necessary for
handwritten text recognition?
In 2017 14th IAPR International Conference on Document Analysis
and Recognition (ICDAR), volume 01, pages 67–72, 2017.
10.1109/ICDAR.2017.20.
[Graves et al.(2006)Graves, Fernández, Gomez, and
Schmidhuber]graves2006ctc
Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen
Schmidhuber.
Connectionist temporal classification: labelling unsegmented sequence
data with recurrent neural networks.
In Proceedings of the 23rd international conference on Machine
learning, pages 369–376, 2006.
[Graves and Schmidhuber(2008)]graves2008offline
Alex Graves and Jürgen Schmidhuber.
Offline handwriting recognition with multidimensional recurrent
neural networks.
Advances in neural information processing systems,
21:0 545–552, 2008.
[Shi et al.(2016)Shi, Bai, and Yao]shi2016endtoendhtr
Baoguang Shi, Xiang Bai, and Cong Yao.
An end-to-end trainable neural network for image-based sequence
recognition and its application to scene text recognition.
IEEE transactions on pattern analysis and machine
intelligence, 390 (11):0 2298–2304, 2016.
[Dutta et al.(2018)Dutta, Krishnan, Mathew, and
Jawahar]dutta2018cnnrnn
Kartik Dutta, Praveen Krishnan, Minesh Mathew, and C.V. Jawahar.
Improving cnn-rnn hybrid networks for handwriting recognition.
In 2018 16th International Conference on Frontiers in
Handwriting Recognition (ICFHR), pages 80–85, 2018.
10.1109/ICFHR-2018.2018.00023.
[Sueiras et al.(2018)Sueiras, Ruiz, Sanchez, and
Velez]sueiras2018offline
Jorge Sueiras, Victoria Ruiz, Angel Sanchez, and Jose F Velez.
Offline continuous handwriting recognition using sequence to sequence
neural networks.
Neurocomputing, 289:0 119–128, 2018.
[Wigington et al.(2017)Wigington, Stewart, Davis, Barrett, Price, and
Cohen]wigington2017data
Curtis Wigington, Seth Stewart, Brian Davis, Bill Barrett, Brian Price, and
Scott Cohen.
Data augmentation for recognition of handwritten words and lines
using a cnn-lstm network.
In 2017 14th IAPR International Conference on Document Analysis
and Recognition (ICDAR), volume 1, pages 639–645. IEEE, 2017.
[Bahdanau et al.(2014)Bahdanau, Cho, and Bengio]bahdanau2014attention
Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio.
Neural machine translation by jointly learning to align and
translate.
arXiv preprint arXiv:1409.0473, 2014.
[Jaderberg et al.(2015)Jaderberg, Simonyan, Zisserman,
et al.]jaderberg2015stn
Max Jaderberg, Karen Simonyan, Andrew Zisserman, et al.
Spatial transformer networks.
Advances in neural information processing systems, 28, 2015.
[Li et al.(2019)Li, Wang, Shen, and Zhang]li2019show
Hui Li, Peng Wang, Chunhua Shen, and Guyu Zhang.
Show, attend and read: A simple and strong baseline for irregular
text recognition.
In Proceedings of the AAAI Conference on Artificial
Intelligence, volume 33, pages 8610–8617, 2019.
[Diaz et al.(2021)Diaz, Qin, Ingle, Fujii, and
Bissacco]diaz2021rethinking
Daniel Hernandez Diaz, Siyang Qin, Reeve Ingle, Yasuhisa Fujii, and Alessandro
Bissacco.
Rethinking text line recognition models.
arXiv preprint arXiv:2104.07787, 2021.
[Dosovitskiy et al.(2020)Dosovitskiy, Beyer, Kolesnikov, Weissenborn,
Zhai, Unterthiner, Dehghani, Minderer, Heigold, Gelly,
et al.]dosovitskiy2020vit
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn,
Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg
Heigold, Sylvain Gelly, et al.
An image is worth 16x16 words: Transformers for image recognition at
scale.
arXiv preprint arXiv:2010.11929, 2020.
[Li et al.(2021)Li, Lv, Cui, Lu, Florencio, Zhang, Li, and
Wei]li2021trocr
Minghao Li, Tengchao Lv, Lei Cui, Yijuan Lu, Dinei Florencio, Cha Zhang,
Zhoujun Li, and Furu Wei.
Trocr: Transformer-based optical character recognition with
pre-trained models.
arXiv preprint arXiv:2109.10282, 2021.
[Liu et al.(2019)Liu, Ott, Goyal, Du, Joshi, Chen, Levy, Lewis,
Zettlemoyer, and Stoyanov]liu2019roberta
Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer
Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.
Roberta: A robustly optimized bert pretraining approach.
arXiv preprint arXiv:1907.11692, 2019.
[Bechtle et al.(2021)Bechtle, Molchanov, Chebotar, Grefenstette,
Righetti, Sukhatme, and Meier]bechtle2021metaloss
Sarah Bechtle, Artem Molchanov, Yevgen Chebotar, Edward Grefenstette, Ludovic
Righetti, Gaurav Sukhatme, and Franziska Meier.
Meta learning via learned loss.
In 2020 25th International Conference on Pattern Recognition
(ICPR), pages 4161–4168. IEEE, 2021.
[Li et al.(2017)Li, Zhou, Chen, and Li]li2017metasgd
Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li.
Meta-sgd: Learning to learn quickly for few-shot learning.
arXiv preprint arXiv:1707.09835, 2017.
[Baik et al.(2020)Baik, Hong, and Lee]baik2020l2f
Sungyong Baik, Seokil Hong, and Kyoung Mu Lee.
Learning to forget for meta-learning.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pages 2379–2387, 2020.
[Winata et al.(2020)Winata, Cahyawijaya, Liu, Lin, Madotto, Xu, and
Fung]winata2020learning
Genta Indra Winata, Samuel Cahyawijaya, Zihan Liu, Zhaojiang Lin, Andrea
Madotto, Peng Xu, and Pascale Fung.
Learning fast adaptation on cross-accented speech recognition.
arXiv preprint arXiv:2003.01901, 2020.
[Klejch et al.(2018)Klejch, Fainberg, and Bell]klejch2018learning
Ondřej Klejch, Joachim Fainberg, and Peter Bell.
Learning to adapt: a meta-learning approach for speaker adaptation.
arXiv preprint arXiv:1808.10239, 2018.
[Antoniou et al.(2018)Antoniou, Edwards, and
Storkey]antoniou2018mamlplusplus
Antreas Antoniou, Harrison Edwards, and Amos Storkey.
How to train your maml.
In International Conference on Learning Representations, 2018.
[Rusu et al.(2018)Rusu, Rao, Sygnowski, Vinyals, Pascanu, Osindero, and
Hadsell]rusu2018leo
Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu,
Simon Osindero, and Raia Hadsell.
Meta-learning with latent embedding optimization.
In International Conference on Learning Representations, 2018.
[Vinciarelli and Bengio(2002)]vinciarelli2002writer
Alessandro Vinciarelli and Samy Bengio.
Writer adaptation techniques in hmm based off-line cursive script
recognition.
Pattern Recognition Letters, 230 (8):0
905–916, 2002.
[Nair et al.(2018)Nair, Sankaran, Kota, Tulyakov, Setlur, and
Govindaraju]nair2018knowledge
Rathin Radhakrishnan Nair, Nishant Sankaran, Bharagava Urala Kota, Sergey
Tulyakov, Srirangaraj Setlur, and Venu Govindaraju.
Knowledge transfer using neural network based approach for
handwritten text recognition.
In 2018 13th IAPR International Workshop on Document Analysis
Systems (DAS), pages 441–446. IEEE, 2018.
[Szummer and Bishop(2006)]szummer2006discriminative
Martin Szummer and Christopher M Bishop.
Discriminative writer adaptation.
In Tenth International Workshop on Frontiers in Handwriting
Recognition. Suvisoft, 2006.
[Zhang and Liu(2012)]zhang2012writer
Xu-Yao Zhang and Cheng-Lin Liu.
Writer adaptation with style transfer mapping.
IEEE transactions on pattern analysis and machine
intelligence, 350 (7):0 1773–1787, 2012.
[Zhang et al.(2017)Zhang, Bengio, and Liu]zhang2017online
Xu-Yao Zhang, Yoshua Bengio, and Cheng-Lin Liu.
Online and offline handwritten chinese character recognition: A
comprehensive study and new benchmark.
Pattern Recognition, 61:0 348–360, 2017.
[Wang et al.(2020)Wang, Du, and Wang]wang2020writer
Zi-Rui Wang, Jun Du, and Jia-Ming Wang.
Writer-aware cnn for parsimonious hmm-based offline handwritten
chinese text recognition.
Pattern Recognition, 100:0 107102, 2020.
[Wang and Du(2022)]wang2022fast
Zi-Rui Wang and Jun Du.
Fast writer adaptation with style extractor network for handwritten
text recognition.
Neural Networks, 147:0 42–52, 2022.
[Zhang et al.(2019)Zhang, Nie, Liu, Xu, Zhang, and
Shen]zhang2019sequence
Yaping Zhang, Shuai Nie, Wenju Liu, Xing Xu, Dongxiang Zhang, and Heng Tao
Shen.
Sequence-to-sequence domain adaptation network for robust text image
recognition.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pages 2740–2749, 2019.
[Kang et al.(2020)Kang, Rusinol, Fornés, Riba, and
Villegas]kang2020unsupervised
Lei Kang, Marçal Rusinol, Alicia Fornés, Pau Riba, and Mauricio
Villegas.
Unsupervised writer adaptation for synthetic-to-real handwritten word
recognition.
In Proceedings of the IEEE/CVF Winter Conference on
Applications of Computer Vision, pages 3502–3511, 2020.
[Yang et al.(2018)Yang, Zhang, Yin, Sun, and Liu]yang2018deep
Hong-Ming Yang, Xu-Yao Zhang, Fei Yin, Jun Sun, and Cheng-Lin Liu.
Deep transfer mapping for unsupervised writer adaptation.
In 2018 16th International Conference on Frontiers in
Handwriting Recognition (ICFHR), pages 151–156. IEEE, 2018.
[Singh and Karayev(2021)]singh2021fphtr
Sumeet S Singh and Sergey Karayev.
Full page handwriting recognition via image to sequence extraction.
In International Conference on Document Analysis and
Recognition, pages 55–69. Springer, 2021.
[Hochreiter and Schmidhuber(1997)]hochreiter1997lstm
Sepp Hochreiter and Jürgen Schmidhuber.
Long short-term memory.
Neural computation, 90 (8):0 1735–1780, 1997.
[He et al.(2016)He, Zhang, Ren, and Sun]he2016resnet
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
Deep residual learning for image recognition.
In Proceedings of the IEEE conference on computer vision and
pattern recognition, pages 770–778, 2016.
[Vaswani et al.(2017)Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez,
Kaiser, and Polosukhin]vaswani2017attention
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones,
Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.
Attention is all you need.
In Advances in neural information processing systems, pages
5998–6008, 2017.
[Abdel-Hamid and Jiang(2013)]abdel2013speakeradapt3
Ossama Abdel-Hamid and Hui Jiang.
Fast speaker adaptation of hybrid nn/hmm model for speech recognition
based on discriminative learning of speaker code.
In 2013 IEEE International Conference on Acoustics, Speech and
Signal Processing, pages 7942–7946. IEEE, 2013.
[Karras et al.(2019)Karras, Laine, and Aila]karras2019stylegan
Tero Karras, Samuli Laine, and Timo Aila.
A style-based generator architecture for generative adversarial
networks.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pages 4401–4410, 2019.
[Zhang and Schomaker(2022)]zhang2022divergan
Zhenxing Zhang and Lambert Schomaker.
Divergan: An efficient and effective single-stage framework for
diverse text-to-image generation.
Neurocomputing, 473:0 182–198, 2022.
[Dumoulin et al.(2016)Dumoulin, Shlens, and
Kudlur]dumoulin2016learned
Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur.
A learned representation for artistic style.
arXiv preprint arXiv:1610.07629, 2016.
[Ulyanov et al.(2016)Ulyanov, Vedaldi, and Lempitsky]ulyanov2016adain
Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky.
Instance normalization: The missing ingredient for fast stylization.
arXiv preprint arXiv:1607.08022, 2016.
[De Vries et al.(2017)De Vries, Strub, Mary, Larochelle, Pietquin, and
Courville]devries2017condbn
Harm De Vries, Florian Strub, Jérémie Mary, Hugo Larochelle, Olivier
Pietquin, and Aaron C Courville.
Modulating early visual processing by language.
Advances in Neural Information Processing Systems, 30, 2017.
[Schomaker(2007)]schomaker2007advances
Lambert Schomaker.
Advances in writer identification and verification.
In Ninth International Conference on Document Analysis and
Recognition (ICDAR 2007), volume 2, pages 1268–1273. IEEE, 2007.
[Bulacu and Schomaker(2007)]bulacu2007hinge
Marius Bulacu and Lambert Schomaker.
Text-independent writer identification and verification using
textural and allographic features.
IEEE transactions on pattern analysis and machine
intelligence, 290 (4):0 701–717, 2007.
[Marti and Bunke(2002)]marti2002iam
U-V Marti and Horst Bunke.
The iam-database: an english sentence database for offline
handwriting recognition.
International Journal on Document Analysis and Recognition,
50 (1):0 39–46, 2002.
[SLR(2023)]aachensplits
Open SLR.
Aachen data splits (train, test, val) for the iam dataset.
https://www.openslr.org/56/, 2023.
[Kingma and Ba(2014)]kingma2014adam
Diederik P Kingma and Jimmy Ba.
Adam: A method for stochastic optimization.
arXiv preprint arXiv:1412.6980, 2014.
[Paszke et al.(2019)Paszke, Gross, Massa, Lerer, Bradbury, Chanan,
Killeen, Lin, Gimelshein, Antiga, Desmaison, Kopf, Yang, DeVito, Raison,
Tejani, Chilamkurthy, Steiner, Fang, Bai, and Chintala]pytorch
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban
Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan
Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith
Chintala.
Pytorch: An imperative style, high-performance deep learning library.
In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural
Information Processing Systems 32, pages 8024–8035. Curran Associates,
Inc., 2019.
URL
<http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf>.
[Arnold et al.(2020)Arnold, Mahajan, Datta, Bunner, and
Zarkias]learn2learn
Sébastien M R Arnold, Praateek Mahajan, Debajyoti Datta, Ian Bunner, and
Konstantinos Saitas Zarkias.
learn2learn: A library for meta-learning research.
arXiv, aug 2020.
URL <http://arxiv.org/abs/2008.12284>.
[Raghu et al.(2020)Raghu, Raghu, Bengio, and Vinyals]raghu2019anil
Aniruddh Raghu, Maithra Raghu, Samy Bengio, and Oriol Vinyals.
Rapid learning or feature reuse? towards understanding the
effectiveness of maml.
In International Conference on Learning Representations, 2020.
URL <https://openreview.net/forum?id=rkgMkCEtPB>.
[Brown et al.(2020)Brown, Mann, Ryder, Subbiah, Kaplan, Dhariwal,
Neelakantan, Shyam, Sastry, Askell, et al.]brown2020language
Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla
Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell,
et al.
Language models are few-shot learners.
Advances in neural information processing systems,
33:0 1877–1901, 2020.
[Ioffe and Szegedy(2015)]ioffe2015batchnorm
Sergey Ioffe and Christian Szegedy.
Batch normalization: Accelerating deep network training by reducing
internal covariate shift.
In International conference on machine learning, pages
448–456. PMLR, 2015.
[Ba et al.(2016)Ba, Kiros, and Hinton]ba2016layernorm
Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton.
Layer normalization.
arXiv preprint arXiv:1607.06450, 2016.
§ BATCH NORMALIZATION IN MAML
In this section, we discuss the role of batch normalization in the MAML-based models. For MAML and MetaHTR models, using batch normalization <cit.> in the right way was generally crucial to obtain good performance, and would often determine whether a model would work at all. Although the current discussion is not directly relevant to the main narrative of the paper, we include it here for the sake of completeness, as it may be useful for future researchers using MAML-based methods.
It has been reported in <cit.> that the implementation from the original MAML paper <cit.> makes use of batch statistics to normalize the activations in batch normalization layers and that <cit.> discovered through experimentation that standard batch normalization using stored statistics does not work well. There is a seemingly plausible explanation for why batch normalization could be problematic when training on radically different tasks. During normal neural network training, batches of data are randomly sampled, which, if large enough, have statistics that are close to the dataset statistics. This implies that the batch statistics will remain relatively stable during training. However, introducing task-specific batches of data can potentially lead to large shifts in activation statistics during training, since batches of data are now task-specific, i.e., one batch contains a single task. Especially as the number of inner loop optimization steps is increased, the deviation from the global mean and variance will tend to grow.
Nevertheless, based on our experiments, we found the opposite to hold true for our HTR models. Using batch statistics degraded performance, and depending on the base model, it would lead to consistently inferior performance. Numerous setups have been tried out in this regard, based on what was proposed in <cit.>, e.g., fixing the γ parameter in the batch normalization layers, or only using batch statistics in the inner loop, but none of these setups yielded good results.
The explanation for this discrepancy may lie in the nature of the tasks used in MAML. In traditional MAML setups such as few-shot image classification, introducing a new task implies introducing one or several new image classes. The image distribution may therefore change radically, along with the distribution of the intermediate layer activations, and the previously stored statistics may not work well anymore. By contrast, in the HTR setting, different handwriting styles may be similar enough that shared statistics can still be used for normalization.
Notably, the effect of batch normalization was much stronger for the LSTM-based model (SAR). For the SAR base model, using batch statistics for normalization would lead to a significant drop in performance to about 40% WER. For the FPHTR model, performance was generally worse than with stored statistics, but only by a margin of a few points. Note that the only place where batch normalization takes place is in the ResNet backbone (which FPHTR and SAR both use). Therefore, the LSTM model seems to be more sensitive to the changes in normalization statistics expressed in the ResNet output. Recall that the structure of SAR is such that the output of the ResNet encoder is passed through an initial encoder LSTM processing image strips, followed by a decoder LSTM for language decoding using 2D attention. One difference between the FPHTR and SAR models is that FPHTR uses layer normalization <cit.> following the multi-head attention modules. By contrast, SAR uses no normalization layers after the ResNet encoder. Possibly, this could result in a larger sensitivity to changes in the ResNet output distribution, since the additional variability does not get normalized along the way.
§ HYPERPARAMETERS
In this section, we include all relevant hyperparameters used to train the models in Section <ref>. We show hyperparameters for the base models in Table <ref>, hyperparameters for writer code models in Table <ref>, and meta-learning hyperparameters in Table <ref>.
§ NUMBER OF PARAMETERS PER MODEL
We indicate learnable parameter counts for all models below. Base model parameters are shown in Table <ref>, whereas additional parameters required for each approach in Chapter <ref> are shown in Tables <ref> and <ref>.
§ ADDITIONAL FIGURES
|
http://arxiv.org/abs/2307.04437v2 | 20230710092701 | HORTENSIA, a program package for the simulation of nonadiabatic autoionization dynamics in molecules | [
"Kevin Issler",
"Roland Mitrić",
"Jens Petersen"
] | physics.chem-ph | [
"physics.chem-ph",
"physics.comp-ph"
] |
AIP/123-QED
HORTENSIA]HORTENSIA, a program package for the simulation of nonadiabatic autoionization dynamics in molecules
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
[email protected]
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
[email protected]
Julius-Maximilians-Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
We present a program package for the simulation of ultrafast vibration-induced autoionization dynamics in molecular anions in the manifold of the adiabatic anionic states and the discretized ionization continuum. This program, called HORTENSIA (Hopping real-time trajectories for electron-ejection by nonadiabatic self-ionization in anions), is based on the nonadiabatic surface-hopping methodology, wherein nuclei are propagated as an ensemble along classical trajectories in the quantum-mechanical potential created by the electronic density of the molecular system. The electronic Schrödinger equation is numerically integrated along the trajectory, providing the time evolution of electronic state coefficients, from which switching probabilities into discrete electronic states are determined. In the case of a discretized continuum state, this hopping event is interpreted as the ejection on an electron. The derived diabatic and nonadiabatic couplings in the time-dependent electronic Schrödinger equation are calculated from anionic and neutral wavefunctions obtained from quantum chemical calculations with commercially available program packages interfaced with our program.
Based on this methodology, we demonstrate the simulation of autoionization electron kinetic energy spectra that are both time- and angle-resolved. In addition, the program yields data that can be interpreted easily with respect to geometric characteristics such as bonding distances and angles, which facilitates the detection of molecular configurations important for the autoionization process.
Moreover, useful extensions are included, namely generation tools for initial conditions and input files as well as for the evaluation of output files both through console commands and a graphical user interface.
[
Jens Petersen
August 12, 2023
===================
For submission:
Repository link: <https://github.com/mitric-lab/HORTENSIA_LATEST.git>
Licensing: MIT
Language: Python ≥ 3.8
§ INTRODUCTION
After generation of a temporary molecular anion through electron attachment, there are three possible competing relaxation mechanisms.<cit.>
These are a) radiative deactivation, assuming that there is a lower-lying anion state that is stable with respect to ionization, b) dissociative electron attachment, in which the captured electron induces geometric change in the molecule resulting in fragmentation into more stable products, a neutral and an anionic subsystem.
And lastly, c) autoionization, in which after a finite period of time the metastable state decays via electron ejection.
The process of dissociative electron attachment is observed for example in DNA, where capture of low-energy electrons leads to single and double strand breaks<cit.>, or in a variety of substances in nanoscale thin films<cit.>.
Prominent examples for autoionization include excited dipole- and quadrupole-bound anions with binding energies slightly below the ionization threshold<cit.>, intermolecular Coulombic decay at the FADH^- cofactor involved in DNA-photolesion repair<cit.> and autoionization induced by vibrational excitation in organic molecules<cit.>.
Generally the finite lifetime of a metastable state with respect to autoionization can vary strongly from only a few femtoseconds<cit.> up to milliseconds<cit.>.
Recently, several experiments have provided insights into the dynamics of such processes in dipole- and quadrupole-bound organic anions on a (sub-)picosecond timescale.<cit.>
Although the process of autoionization is well-known and -observed experimentally by a multitude of methods, as can be seen in the references given above, the theoretical description of autoionizing systems is challenging<cit.>, especially if one is interested in the mechanistic details of the intricate ultrafast relaxation dynamics.
Autoionization processes can follow different general mechanisms, depending on how energy is redistributed among the system's degrees of freedom. Besides a purely electronic variant, where already the electronic energy of the system lies above the ionization threshold and electron ejection may proceed via tunneling, there is also the possibility of a nonadiabatic mechanism in which rotational or vibrational energy of the nuclei is transformed into the kinetic energy of the ejected electron.
In the following, we focus on the case of vibrational autoionization. This process can thus be viewed as a nonadiabatic transition between a vibrationally excited bound N-electron system and continuum electronic states consisting of an N-1 electron molecular core and a free electron. Early theoretical treatments have focused on the computation of ionization rates<cit.> as well as on establishing propensity rules for the ionization transitions<cit.>. While a full dynamical treatment of vibrational autoionization is highly desirable, an entirely quantum-dynamical approach is computationally prohibitive. As an alternative, a mixed quantum-classical ansatz can be considered, further motivated by the success of this type of methodology in the description of bound-state nonadiabatic processes and the simulation of time-resolved spectroscopic signals.<cit.> Although to date there have been several implementations of mixed quantum-classical dynamics simulations for bound-state problems made publicly available<cit.>, no program addressing the simulation of vibration-induced autoionization processes has been published so far.
Therefore, in this work we present the program package implementing our approach to describe vibrational autoionization through quantum-classical dynamics in the framework of the surface-hopping methodology in the manifold of bound and continuum electronic states as described recently<cit.>.
Therein, nuclear motion is considered classically, while the electronic system is treated quantum-mechanically.
Nonadiabatic transitions between electronic states accompanied by change of the classical vibrational energy of the molecule describe the energy exchange between the two subsystems.
With this program package and the underlying methodology, one is able to gain insight into the geometric and electronic evolution in the course of the autoionization process as well as to calculate the time-, energy- and angle-distribution of the generated free electrons, which serve as experimental observables for monitoring autoionization dynamics.
We illustrate our program on the example of the 2-cyanopyrrolide anion, which bears a dipole-bound excited state slightly below the electron detachment threshold while the vibrationally excited states are metastable and decay via autoionization.<cit.>
In the following section a brief theoretical description of the method is given. In section <ref> an overview of the actual implementation is provided. The subsequent section <ref> details performance-related issues, namely quality of approximations in the theory and runtime and memory optimization within the program, as well as a dynamics simulation example for the 2-cyanopyrrolide anion. Finally in section <ref> a conclusion and outlook are given.
§ THEORY
Our methodological framework is based on the surface-hopping procedure as proposed by Tully<cit.>, in which the coupled electron-nuclear dynamics of molecular systems is approached in a quantum-classical fashion.
Specifically, the nuclei are propagated classically according to Newton's equations of motion,
MR̈(t)
=
𝐅_i(𝐑[t])
≡
-∇_R E_i(R[t]),
where the force 𝐅_i(𝐑[t]) is obtained as the negative gradient of the electronic potential energy surface (PES) E_i(R[t]). In the above equation, M denotes a diagonal matrix containing the nuclear masses.
For an ensemble of initial conditions, this leads to trajectories R(t) moving on the given PES.
Simultaneously, the electronic time-dependent Schrödinger equation
iħΨ̇(r;R[t])
=
Ĥ_elΨ(r;R[t])
,
with the electronic Hamiltonian Ĥ_el is solved.
The electronic wavefunction can be expanded into a set of orthonormal basis states, which in the case of autoionization includes bound states Φ_m' (denoted with a primed index) as well as continuum states Φ̃_m” (denoted with a double-primed index):
Ψ(r,R[t],t)
= ∑_m' c_m'(t) Φ_m'(r,R[t])
+
∑_m”∫ d^3k c̃_m”(k,t) Φ̃_m”(k,r,R[t]),
where k denotes the continuously varying wave vector of the free electron, while m” is the quantum number of the remaining neutral state.
We assume the wavefunctions Φ_m' and Φ̃_m” to be single Slater determinants (ground state) or an expansion of singly excited Slater determinants (excited state).
In the frame of the presented methodology we discretize the continuum states, leading to
∫ d^3k c̃_m”(k,t)
Φ̃_m”(k,r,R[t])
≈∑_i
(Δ V_k)^1/2c̃_m”(k_i,t)
(Δ V_k)^1/2Φ̃_m”(k_i,r,R[t])
≈∑_i
c_m”(k_i,t)
Φ_m”(k_i,r,R[t]),
where Δ V_k denotes the volume element in k-space and the discretized and continuum state expansion coefficients are related according to c_m”(k_i,t)=(Δ V_k)^1/2c̃_m”(k_i,t). The actual determination of the wave vectors and the implementation of the discretization procedure are explained in detail in the next chapter.
Insertion of Eq. (<ref>) into the time-dependent Schrödinger equation (<ref>), multiplication from the left by an eigenstate ⟨Φ_n| and evaluation of the arising terms leads to a set of coupled differential equations for the electronic state coefficients c_n:
ċ_n(t)
=
∑_j
[
-i/ħ H_nm(R[t]) - D_nm (R[t])
]
c_m(t),
with the matrix elements of the electronic Hamiltonian H_nm = ⟨Φ_n | H_el | Φ_m|$⟩ and the nonadiabatic couplingsD_nm = ⟨Φ_n | Φ̇_m|=⟩ Ṙ·⟨Φ_n | ∇_R | Φ_m|$⟩.
These can be divided into separate expressions for the bound and continuum states, resulting in the diabatic and nonadiabatic couplings between two bound anion states,
H_n'm' =
⟨Φ_n' | Ĥ | Φ_m'|⟩hij
D_n'm' =
⟨Φ_n' | Φ̇_m'|,⟩
and between a bound and a discretized continuum state,
H_n”m'(k_i)
=
(Δ V_k)^1/2⟨Φ̃_n”(k_i) | Ĥ | Φ_m'|⟩hik
D_n”m'(k_i)
=
⟨Φ_n”(k_i) | Φ̇_m'|=⟩
(Δ V_k)^1/2⟨Φ̃_n”(k_i) | Φ̇_m'|.⟩
In the above equations, the approximation to neglect the coupling terms between the continuum states has been introduced.
The discretized continuum states consist of an antisymmetrized product of a bound N-1 electron neutral state and a molecular scattering state of the free electron
Φ̃_n”(k_i)
=
A(
Φ^(n)_n”·ψ(k_i)
).
The simplest approximation to the free electron states in the presence of a neutral molecular core are plane waves
ψ(k_i)≈ Ne^ik_i·r
with a normalization constant N = (2π)^-3/2 to satisfy the orthonormality demanded in Eq. (<ref>).
Since this function would be completely independent on the electronic and nuclear configuration of the molecular core, which is a strong simplification, the plane waves are orthogonalized with respect to the anion's molecular orbitals (MOs) ϕ_m to include (at least to a certain degree) dependence on the molecular structure according to
ψ̃(k_i)
=
(2π)^-3/2 N_ortho(
e^ik_i·r
-
∑_m^occ⟨ϕ_m | e^ik_i·r|ϕ⟩_m
)
=
N_ortho(
ψ(k_i)
-
∑_m^occ⟨ϕ_m | ψ(k_i)| ⟩ϕ_m
),
with the normalization constant
N_ortho
=
(
1
-
∑_m^occ|
⟨ϕ_m | ψ(k_i)|⟩|^2
)^-1/2
arising from the orthogonalization.
Notably, the summation over m includes the occupied MOs in all 'relevant' Slater determinants of all considered electronic states, that is, we considered all determinants which are needed to sufficiently represent the ground state and full CIS wavefunction of the excited state.
Beginning from the highest contribution to a wavefunction, determinants are included until a specific percentage or a user-adjusted maximum number of configurations per electronic state is reached (95 % / 10 configurations in the case of vinylidene<cit.>).
Considering for now the special case where only the anion's ground state is included, the used MOs are simply the energetically lowest ones up to the highest-occupied molecular orbital (HOMO).
The overlap integral between a plane wave and an MO present in Eq. (<ref>), ⟨ϕ_m | ψ(k_i)|$⟩, can be computed analytically by expanding the MO into the Gaussian atomic orbital (AO) basis, with the integral involving a single AO|ν⟩given by
⟨ν | ψ(k)|=⟩
(2π)^-3/2∫ d^3𝐫 e^ik·rφ_ν(r)
=
(2α_ν)^-3/2exp(ik·A_ν -k^2/4α_ν)
×∏_j=x,y,z
(-i√(4α_ν))^-n_ν,j
H_n_ν,j(
k_j/√(4α_ν))
,
where theH_n_ν,jare the Hermite polynomials of ordern_ν,j.
§.§ Electronic coupling terms
There are anionic systems, for example the vinylidene anion<cit.>, that do not support a bound excited state, in which case the consideration of only the ground state and the continuum in the process of autoionization is sufficient.
Besides that, for example in molecules exhibiting dipole-bound excited states <cit.>, several bound anionic states and the interaction among them are relevant as well.
Nonetheless, to keep the formalism concise, if not noted otherwise we discuss in the following the electronic coupling terms for the special case of both anion and neutral molecule being in their respective electronic ground states, which in turn are represented by a single Slater determinant.
The generalization to excited states and/or multideterminantal wavefunctions is straightforward.<cit.>
We denote the bound anionic ground state wavefunction by|Φ_0⟩and the continuum wavefunctions by|Φ_i⟩, the latter being constructed as an antisymmetrized product of the neutral ground state and a free electron state function with wave vectork_i, similar to Eq. (<ref>).
§.§.§ Diabatic couplings
In the case of two adiabatic bound anion states, the coupling matrix elementsH_n'm'given in Eq. (<ref>) yield zero for alln' ≠m'since these states are orthonormal eigenstates of the electronic Hamiltonian.
On the other hand, since in our methodology the bound and continuum state wavefunctions are constructed using separate quantum-chemical calculations for the anion and neutral, and the free electron wavefunction is taken as a plane wave, the continuum state functions are crude approximations to the actual adiabatic eigenfunctions of the electronic Hamiltonian for theN-electron system and therefore, diabatic couplings between the bound and continuum electronic states arise.
As elaborated in detail in Ref. aid, according to Eq. (<ref>) and definingV_i0^dia(k_i)as
H_i0(k_i)
≡⟨Φ_i | Ĥ | Φ_0|≡⟩(Δ V_k)^1/2
V^dia_i0(k_i),
the diabatic coupling between a bound and a continuum state can be written in terms of the AO basis as
V^dia_i0(k_i)
= ∑_λμν[
A_λμν(
⟨𝐤_i λ || μν|-⟩∑_σ
B_σ⟨σλ || μν|⟩) +
A̅_λμν(
⟨𝐤_i λ | μν|-⟩∑_σ
B_σ⟨σλ | μν|⟩)
].
In this formula the Greek letters denote the AO basis functions,⟨𝐤_i λ| μν|$⟩ is an electron-electron repulsion integral and ⟨𝐤_i λ || μν|=⟩⟨k_i λ | μν|-⟩⟨k_i λ | νμ|$⟩ its antisymmetrized variant.
The prefactorsA_λμν,A̅_λμνandB_σcomprise AO expansion coefficients and overlap integrals and are defined as follows (assuming that the extra electron of the anion hasαspin):
A_λμν
= ∑_n^occ,α∑_q,p<q^occ,α
(-1)^n+p+q-1det 𝐒_in,pq
×(
c_λ^(n)
-
∑_u^occ,α
c_λ^(u)
S_nu)
c_μ^(p) c_ν^(q)
A̅_λμν
= ∑_n̅^occ,β∑_p^occ,α∑_q̅^occ,β
(-1)^n̅+p+q̅-1det 𝐒_in̅,pq̅
×(
c_λ^(n̅)
-
∑_u̅^occ,β
c_λ^(u̅)
S_n̅u̅)
c_μ^(p) c_ν^(q̅)
B_σ
= ∑_r^occ,α∑_ρ
c_σ^(r)
c_ρ^(r)⟨k_i | ρ|,⟩
where the indices (including their variants with an overbar)p,q,rrefer to anion MOs,n,uto neutral MOs, anddet 𝐒_in,pqdenotes the minor determinant of the overlap matrix between continuum and bound state orbitals where the rows of the free electron orbitalψ̃(𝐤_i)and neutral orbitalχ_nas well as the columns of anion orbitalsϕ_pandϕ_qhave been deleted.
For the full derivation of these equations the reader is referred to Ref. aid.
§.§.§ Nonadiabatic couplings
The nonadiabatic coupling terms as defined in Eqs. (<ref>) and (<ref>) are calculated using the finite-difference approximation for the time derivative, which leads to
D_i0(t)
=
⟨Φ_i(t) | d/dtΦ_0(t)|
⟩ ≈1/2Δ t(
⟨Φ_i(t-Δ t) | Φ_0(t)|-⟩⟨Φ_i(t) | Φ_0(t-Δ t)|⟩)
In the case of two anionic bound states, these terms are evaluated according to Refs. mitric2008, werner2008, werner2010.
One can simplify the arising terms by integrating over all but one electron coordinate. For the first term of Eq. (<ref>) this yields
⟨Φ_i(t') | Φ_0(t)|=⟩
N^-1/2⟨ψ̃(k_i,t') | ψ^D(t',t)|,⟩
where we have abbreviatedt'=t-Δtand have defined the one-electron functionψ^D(t',t), which is an analog to a molecular Dyson orbital with theN- andN-1- wavefunctions taken at different time steps and geometries.
Using Eqs. (<ref>) and (<ref>) the resulting nonadiabatic coupling terms read
D_i0(k_i,t)
=
(Δ V_k)^1/2 N_ortho/2 √(N)Δ t[
⟨ψ(k_i) | ψ^D(t',t)|-⟩⟨ψ(k_i) | ψ^D(t,t')| ⟩-
∑_n
⟨ψ(k_i) | ϕ_n(t)| ⟩⟨ϕ_n(t') | ψ^D(t',t)|⟩
+
∑_n
⟨ψ(k_i) | ϕ_n(t)| ⟩⟨ϕ_n(t) | ψ^D(t,t')|⟩].
§.§ Adiabatic ionization and electronic decay
The main focus of the above presented methodology lies on describing the nonadiabatic process of vibrational autoionization. However, in the course of the molecule's dynamical evolution instances can occur where the occupied anionic state becomes unbound as the result of changes in nuclear geometry.
In this case, ionization is possible as an exclusively adiabatic electronic process without coupling to the nuclear motion.
This process can be included approximately in our method by simulating the temporal spread of the ejected electron as a wavepacket evolving freely in space. As a quantitative measure, the electronic spatial extent, i.e., the expectation value of𝐫̂^2, is calculated as a function of time.
Specifically, once a time step is reached where the VDE has become negative, the highest-occupied orbital of the last bound geometry,ϕ(r, t_0), is used as the initial free electronic wavepacket.
In the case where one only considers the anionic ground state, this corresponds to the HOMO.
If also an excited state is involved, natural transition orbitals (NTOs)<cit.> are calculated and the highest-occupied and lowest-unoccupied NTO (HONTO and LUNTO) are used for the anionic ground and excited state, respectively.
Such an electronic wavepacket is then propagated in time and its spatial extent is evaluated according to
⟨𝐫̂^2|(⟩t)
=
⟨ϕ(𝐫,t) |𝐫̂^2 |ϕ(𝐫,t)|⟩
=
∑_μν c_μ c_ν⟨φ_μ(𝐫,t) | 𝐫̂^2 | φ_ν (𝐫,t)|.⟩
Hereφ_μ, νdenote the Gaussian atomic basis functions freely propagated in time:
φ_μ(𝐫,t) = ∫ d^3𝐫' K(𝐫,𝐫',t,0) φ_μ (𝐫',0)
with the free electron propagator
K(𝐫,𝐫',t,0)
=
𝐫 | e^-i𝐩̂^2 t/2m_eħ|𝐫'.
Using Cartesian Gaussian basis functions ofs,panddtype one obtains the following analytic expression for the electronic wavepacket:
φ_μ(𝐫,t)
=
N_l_xl_yl_ze^-α/1+iβ tr^2[
-Λiβ t/2α
(1+iβ t)^-5/2
+.
.
(1+iβ t)^-3/2 - ∑_j l_j∏_j=x,y,z (r_j - A_j)^l_j],
whereAis the spatial center of the respective basis function,l_idenotes the angular momentum quantum number for thei'th spatial direction andΛis a constant that is unity if one of thel_i = 2and zero if alll_i<2.
The AO integrals in Eq. (<ref>) are calculated with an implementation of the McMurchie-Davidson scheme<cit.>.
To relate the spatial extent in a simple way to the lifetime of the unbound state, an auxiliary spherically symmetric electron distribution is considered which within the initially determined radiusr_0=√(⟨r^2|(⟩t_0))contains a probability of 99%. Subsequently, with⟨r^2|$⟩ increasing with time, the probability within r_0 decreases, giving rise to a population decay curve which can be related to a time constant τ.
The latter is incorporated into the propagation of the electronic wavefunction given by Eq. (<ref>) by adding an imaginary component to the electronic state energy,
E^(a)→
E^(a)-iħ/2τ,
which leads to an exponential population decay due to adiabatic ionization in regions where the VDE is negative for the given electronic state.
§.§ Surface-hopping procedure
Solution of the set of Eqs. (<ref>) along a nuclear trajectory yields the time-dependent electronic state coefficients c_n(t).
Within the surface-hopping methodology, a switch from the occupied bound electronic state n to any other state m is determined by the hopping probability depending on the electronic state populations ρ_nn = |c_n|^2, which is
P_n→ m
=
-ρ̇_nn/ρ_nnρ̇_mm/∑_k ρ̇_kkΔ t
for ρ̇_nn < 0 and ρ̇_mm > 0 and zero in any other instance. In the above expression, the sum over k includes all states with ρ̇_kk>0.
In case a surface hop occurs, to ensure energy conservation the nuclear velocities are rescaled such that for kinetic energies T and electronic potential energies E_n of anion (a) and neutral (n) the following conditions are fulfilled:
T'^(a)
=
T^(a) +
E_n^(a) -
E_m^(a)
for a hop between anionic bound states and
T'^(n)
=
E_n^(a) +
T^(a) -
E_m^(n) -
E_el(k_i)
for a hop into the continuum (i.e. autoionization).
For a more detailed description of the hopping procedure the reader is referred to Ref. domckebook.
§ PROGRAM IMPLEMENTATION
In the following chapter a detailed account of how the theory is actually implemented in the program package will be provided.
For an easier understanding, in Fig. <ref> the program flow is displayed schematically, with a color code indicating the module handling the respective task.
Starting from the generation of an ensemble of nuclear coordinates R(t) and velocities Ṙ(t) at the time t = t_initial using the module in the folder (red), a first quantum-chemical calculation is performed by an external quantum-chemistry program - to date these include Gaussian09/Gaussian16 <cit.> and QChem <cit.> (blue) - which yields the forces from which the accelerations R̈(t) of the nuclei are computed.
The nuclei are then propagated by integration of Newton's equations of motion for one nuclear time step using the module (orange).
With the new nuclear coordinates R(t + Δ t), a new set of quantum-chemical calculations can be performed, yielding the new energy gradients necessary for the evaluation of the velocities Ṙ(t + Δ t).
With the quantum-chemical calculations at t and t + Δ t, one is now able to construct the electronic continuum states as well as the coupling matrices of the diabatic and nonadiabatic couplings using the module (green).
From this point, the electronic state coefficients c(t) are propagated in parallel to the nuclear dynamics by integrating the electronic Schrödinger equation, yielding c(t + Δ t).
These are utilized to compute hopping probabilities from the occupied bound state to all other (bound and continuum) states.
The switching between the states is induced stochastically according to the respective hopping probabilities given in Eq. (<ref>).
After writing the results into the various output files time is shifted to t = t + Δ t, thereby completing one time step.
To make this initial overview more specific, in the following the underlying algorithms are explained in more detail.
§.§ Electronic structure calculation
All electronic structure and energy gradient calculations can be performed by using any Kohn-Sham (TD)-DFT level of theory provided within the Gaussian09, Gaussian16 or QChem program packages.
The AO basis set needs to be defined explicitly in a separate input file, thus also allowing for additional augmentation of basis sets, which is of utmost importance when describing molecular anions.<cit.>
The and modules provide an interface to the external programs by creating input files and calling the respective programs. The and modules contain classes that parse the external output files and organize the data into the form needed in the program.
§.§ Generation of initial conditions
The initial nuclear coordinates and velocities are determined by stochastic sampling of an appropriate probability distribution function for the harmonic normal modes of the system.
These can be computed from the electronic Hessian matrix at an optimized geometry of the studied molecule.
For molecules in the vibrational ground state as well as for a thermal ensemble of molecules, the Wigner function
ρ_W({Q_i,P_i})=1/(πħ)^N∏_i=1^N α_i(T) exp(-α_i(T)/ħω_i(P_i^2+ω_i^2Q_i^2))
with
α_i(T)
=
tanh(ħω_i/2k_BT)
is employed, where {Q_i,P_i} denote the normal coordinates and momenta, ω_i is the angular frequency of normal mode ν_i and T the thermodynamic temperature.
Besides these cases, in experiments investigating vibration-induced autoionization another type of initial conditions is often important in which one or more normal vibrations of the system are excited by laser irradiation.
In principle, the respective initial conditions could be also generated by using a Wigner function. However, Wigner functions for excited vibrational states can assume negative values and can thus not be directly identified with a probability distribution.
A possible approach might be to regard the positive and negative parts of the Wigner function separately as probability distributions and to run a "positive" and a "negative" ensemble of initial conditions, the final properties of the system then being obtained by appropriate averaging.
As a more efficient alternative, which gets on with only one single ensemble, we employ a positive definite probability distribution constructed from the excited-vibrational state wavefunctions in position and momentum space,
ρ^(i)_υ(Q_i,P_i)=|χ^(i)_υ(Q_i)|^2|χ̃^(i)_υ(P_i)|^2,
where χ^(i)_υ(Q_i) and χ̃^(i)_υ(P_i) are the harmonic oscillator wavefunctions for quantum state υ of normal mode ν_i in position and momentum space, respectively.
§.§ Nuclear dynamics
Given Newton's equations of motion (<ref>), the nuclei are propagated by numerical solution using the velocity Verlet algorithm <cit.> for a user-defined time step.
Within this algorithm, the nuclear coordinates at t+Δ t are obtained from a Taylor series expansion around the coordinates at t:
R(t + Δ t)
≈R(t) +
Ṙ(t)Δ t +
1/2 M^-1F(t) Δ t^2,
where in the last term the acceleration has been formulated using the force F given by the electronic potential energy gradient (cf. Eq. (<ref>)).
With the new nuclear coordinates, the force at t + Δ t can be evaluated, giving rise to the new nuclear velocities
Ṙ(t + Δ t)
=
Ṙ(t) +
Δ t/2 M^-1[
F(t) + F(t + Δ t)
]
.
Due to the approximative nature of the algorithm above and the accuracy of the calculated energy gradients, it is possible that the velocities develop small overall translational or rotational components although the initial conditions were determined with these degrees of freedom set at rest.
These numerical inaccuracies are detected, in the case of translational velocity by the shift of the center of mass away from the origin of the coordinate system, in the case of rotation by the calculation of the angular velocity according to
ω_rot = I^ -1L
with the moment of inertia I and the angular momentum L.
The translational and rotational portions of the nuclear velocities are then subtracted from the total velocity and the remaining vibrational part is rescaled to ensure energy conservation.
After each nuclear dynamics step, the new nuclear coordinates and velocities are written into separate output files, the coordinates in a format of consecutive xyz files which can be visualized easily by external software (for example with the VMD program package <cit.>, which is warmly recommended).
§.§ Electronic dynamics
Since the evaluation of electronic coupling terms in Eq. (<ref>) is, apart from the external quantum-chemistry calculations, the computationally most expensive step in the dynamics, several approximations need to be implemented, which will be discussed in the following
§.§.§ Calculation of coupling terms
Before calculating the coupling terms, the discretization procedure for the generation of wave vectors needed to construct the continuum state wavefunctions will be discussed.
To uniformly discretize angular orientation and kinetic energy of ejected electrons, it is natural to discretize angular and energetic distribution separately.
Since the kinetic energy of a plane wave is
E_kin(k_i) = ħ^2 |k_i|^2/2 m_e
and therefore proportional to the length of the wave vector squared, this length is discretized such that the desired energy range is covered evenly.
For a given energy, the vector orientations are approximately evenly distributed according to the Fibonacci sphere algorithm <cit.>.
The volume elements Δ V_k needed for calculating the bound-continuum couplings in Eqs. (<ref>) and (<ref>) are constructed as the difference of spherical caps around the corresponding wave vectors with the base diameter as an average over the six nearest points on the sphere surrounding the vector.
In the diabatic coupling terms in the AO basis (Eq. (<ref>)) two types of four-center integrals are present: (i) such involving four Gaussian-type atomic orbitals (GTOs), ⟨σλ | μν|$⟩. These are evaluated by using the library <cit.> within the PySCF program package <cit.>.
(ii) integrals involving a plane wave of wave vector𝐤_iand three GTOs,⟨𝐤_i λ| μν|$⟩.
These terms can in principle be calculated analytically as outlined, e.g., in Ref. colle1987, but this is computationally unfeasible for the present purpose since an immense number of plane waves has to be included for a proper discretization of the ionization continuum. Instead, the plane waves are approximated by their Taylor expansion around the center of basis function |μ⟩, R_μ.
As will be discussed in the Performance Section later on, for sufficient accuracy in the approximation it is necessary to include not only the zero'th order term (assuming the plane wave to be constant in the vicinity of the molecule), but also the first-order term, resulting in the approximation
e^i k·r =
e^i k·R_μe^i k· (r - R_μ)
≈e^i k·R_μ[
1 + i k· (r - R_μ)
].
This leads to two terms for the two-electron integrals as follows:
⟨𝐤_i λ | μν|≈⟩e^i k·R_μ[
⟨λ | μν|+⟩
i k⟨λ | μ̃ν|⟩].
In the above expression, |μ̃⟩ is an AO basis function with an angular momentum quantum number by one higher than |μ⟩ while having the same Gaussian exponent.
This heavily reduces the amount of two-electron integrals to be computed from n_AO^3 n_PW to n_AO^2 [n_AO + n'_AO], with n_AO being the total number of AO basis functions, n'_AO the total number of basis functions with increased quantum number and n_PW the total number of plane waves. For instance, in the case of vinylidene in Ref. aid, this amounts to a reduction by a factor of ∼30000.
These terms are again evaluated using the PySCF module.
The prefactors A, A̅ and B present in Eq. (<ref>) are straightforwardly implemented in Python according to Eqs. (<ref>), (<ref>) and (<ref>).
Evaluation of the Dyson orbitals needed for the calculation of the nonadiabatic couplings is implemented as described before in Ref. humeniuk2013 for arbitrary basis sets for the anion and the neutral molecule.
After construction of the Dyson orbitals from all bound anionic states to the neutral ground state the nonadiabatic coupling terms are then calculated according to Eq. (<ref>). To ensure that the wavefunctions of bound states do not switch their arbitrary signs (which can happen, since the external quantum-chemistry calculations are independent of each other), the overlap of electronic wavefunctions of all bound states are tracked throughout the trajectories and accounted for in all formulae involving the respective states.
§.§.§ Calculation of electronic state coefficients
The electronic degrees of freedom are propagated by solving the time-dependent Schrödinger equation (<ref>) in the manifold of all considered bound anion and continuum electronic states using Adams' method as implemented in the class of Python's module <cit.> with a user-defined integration time step. For increased computational stability the equations are beforehand transformed into the interaction picture, introducing the new electronic state coefficients
a_n(t)
=
c_n(t) e^i/ħ H_nn t.
Inserting this into Eq. (<ref>) leads to the actually implemented electronic equation of motion
ȧ_n(t)
=
∑_m
[
-i/ħH̃_nm - D_nm]
a_m(t)
e^-i/ħ (H_mm - H_nn) t
where H̃_nm denotes the Hamiltonian matrix of the system with zeros on the diagonal.
§.§.§ Hopping procedure
Hopping probabilities are directly evaluated according to Eq. (<ref>) from the state coefficients: A random number between 0 and 1 is generated using the function in the module and hopping is conducted accordingly.
Once a trajectory hops into a continuum state, it could in principle be straightforwardly continued on the potential energy surface of the neutral molecule.
Although this can be quite insightful if one is interested in the subsequent geometric changes of the ionized system, we follow a different approach and stop the trajectories after electron detachment since our focus is set on the actual autoionization process.
This allows us to implement a modification of the surface-hopping procedure that leads to a great improvement of the hopping statistics. The idea is to divide a single trajectory into 'sub-trajectories', i.e. to evaluate if a trajectory hops a number n_subtraj of times (see Fig. <ref>).
Every time a sub-trajectory hops into the continuum, n_subtraj is reduced by one and once it reaches zero, the underlying nuclear dynamics is stopped.
It has to be noted that this procedure is only followed for hops into the continuum, while for hops between bound anionic states only a single hopping event per trajectory and time step is possible due to the need to continue the nuclear dynamics on an unambiguously determined potential energy surface.
§.§ Graphical user interface
Our program package comes with a graphical user interface (GUI) for the input generation as well as an analysis tool for trajectories.
An example of the former is displayed in Fig. <ref>.
In the input generator, which is started with
[language=bash]
hortensia –gui
in addition to all relevant settings for the actual simulation, the user may find options for the generation of a complete folder structure for the trajectories as well as bash submit scripts to be used with the Slurm Workload Manager<cit.>.
Furthermore, the above mentioned Wigner ensemble scripts can be used and initial conditions can be generated. Therefore it is highly recommended to use the GUI feature.
Additionally, through the command
hortensia –analysis
one can open the analysis tool which is able to read output files and visualize them in a sub-window using the program package <cit.>.
§.§ Installation
The most convenient way to install the program package is downloading or cloning the https://github.com/mitric-lab/HORTENSIA_LATEST.gitrepository on our Github page<cit.>. In the main folder, execute
[language=bash]
python cysetup.py build_ext –inplace pip install .
to first compile the Cython modules and then install the program. The program package requires (and will automatically pip install)
*
* - for faster summation of large arrays, mainly in the calculation of the two-center integrals in Eqs. (<ref>) and (<ref>)
* - mainly in the integration of the electronic Schrödinger equation as outlined in subsection <ref>
* - for the calculation of the two-electron integrals in Eqs. (<ref>) and (<ref>)
* - for the parallelization of diabatic couplings
* - for the plots in the sub-window of the analysis tool described before
and all dependencies thereof.
Using the command
[language=bash]
pip uninstall hortensia_latest
will uninstall the program package.
§ DISCUSSION
In this section we will quantify aspects of the program related to overall performance.
This includes the quality of approximations within the methodology as well as optimization of time consumption and computational resources.
Moreover the exemplary autoionization dynamics of the 2-cyanopyrrolide anion is discussed.
§.§ Accuracy of k-vector discretization and integral approximations
The accuracy of the Fibonacci sphere algorithm for angular discretization in k-space is illustrated in Fig. <ref> by the covered surface area of a unit sphere using a given number of distributed points.
The total surface area (orange graph) is presented with the relative error|A_fib-A_sphere|/A_sphere(green graph) to the exact surface area4π≈ 12.566(blue line).
The approximated area rapidly converges to a value of∼12.243, which corresponds to a relative error of∼2.575 %.
Since in the coverage of k-vector lengths no additional approximation is introduced and for their respective volume elements the k-space is divided energetically evenly (thus covered exactly with respect to vector length), the error in the surface area for specific vector lengths equates to the overall error of the volume elements.
Therefore the sum of these volume elements results in a total volume that deviates by less than 3 % from the actual sphere for arbitrary numbers of vector orientationsn_s ≥ 30and lengthsn_E(giving a total number of wave vectorsn_k = n_E · n_s). |
http://arxiv.org/abs/2307.04582v1 | 20230710141955 | NANOGrav spectral index $γ=3$ from melting domain walls | [
"E. Babichev",
"D. Gorbunov",
"S. Ramazanov",
"R. Samanta",
"A. Vikman"
] | hep-ph | [
"hep-ph",
"astro-ph.CO",
"hep-th"
] |
NANOGrav spectral index γ=3 from
melting domain walls
E. Babichev^a, D. Gorbunov^b,c, S. Ramazanov^d, R. Samanta^d, A. Vikman^d
^aUniversité Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
^bInstitute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia
^cMoscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
^dCEICO, FZU-Institute of Physics of the Czech Academy of Sciences,
Na Slovance 1999/2, 182 00 Prague 8, Czech Republic
=========================================================================================================================================================================================================================================================================================================================================================================================================================================================
We discuss cosmic domain walls described by a tension red-shifting with the expansion of the Universe. These melting domain walls emit gravitational waves (GW) with the low-frequency spectral shape Ω_gw∝ f^2 corresponding to the spectral index γ=3 favoured by the recent NANOGrav 15 yrs data. We discuss a concrete high-energy physics scenario proposed in Refs. <cit.> which leads to such a melting domain wall network in the early Universe. This scenario involves a feebly coupled scalar field χ, which can serve as a promising dark matter candidate. We identify parameters of the model matching the GW characteristics observed in the NANOGrav data. The dark matter mass is pushed to the ultra-light range below 10^-11-10^-12 eV which is accessible through planned observations thanks to the effects of superradiance of rotating black holes.
§ INTRODUCTION AND SUMMARY
Recently several pulsar timing arrays (PTAs) such as NANOGrav <cit.>, EPTA (including InPTA) <cit.>, PPTA <cit.>, and CPTA <cit.> reported evidence for a common-spectrum signal in each dataset, with inter-pulsar angular correlations described by the Hellings-Downs (HD) curve <cit.>, pointing to a breakthrough discovery of nHz stochastic gravitational wave (GW) background. Although signals from all the PTAs are in a good agreement, in this article, we shall focus on the NANOGrav 15yrs data, because they are more stringent and with the largest statistical significance. Though no clear hints on the origin of the observed signal have been presented, the NANOGrav 15 yrs data disfavor simple GW-driven models of supermassive black hole binaries (SMBHBs) that predict Ω_gw∝ f^2/3 at 2σ CL <cit.>. Nonetheless, statistical and environmental effects may lead to different predictions, consistent with the data <cit.>. On the contrary, investigating various GW sources of cosmological origin, NANOGrav reports that a power law signal Ω_gw∝ f^1.2-2.4 is preferred at 1σ as a better fit to the data <cit.> [It is important to note that in 2020, the NANOGrav reported similar common spectrum process in their 12.5 yrs dataset but without any evidence of HD correlation. Compared to the old data, which are better fitted with a nearly scale-invariant spectrum: Ω_gw∝ f^-1.5-0.5 at 1σ, the 15 yrs data predict a much steeper spectrum, ruling out stable cosmic strings–one of the most anticipated primordial GW sources for PTAs <cit.>.]. Motivated by this, we explore the second possibility in this article and focus on GWs from cosmic domain walls <cit.>.
Compared to previous works, which fit constant tension domain walls to the NANOGrav 15 yrs signal <cit.>, we consider so-called melting domain walls characterized by a time-dependent tension, which drops as a cube of the Universe temperature <cit.>. Such domain walls are cosmology friendly, as their energy density redshifts fast enough not to overclose the Universe. They naturally arise in a well-motivated renormalizable particle physics scenario involving feebly coupled scalar field (Section 2) <cit.>.
These melting domain walls
serve as a source of GWs, which spectrum differs from the spectrum provided by constant tension domain walls (and other known sources). Most notably, the low-frequency GW spectrum from melting domain walls behaves as Ω_gw∝ f^2 <cit.>, which should be compared with Ω_gw∝ f^3 <cit.> in the case of constant tension walls.
The larger signal at small frequencies stems from the fact that the network of melting domain walls efficiently emits GWs over an extended period of time: while the most energetic GWs are produced at the network formation, later emission from somewhat melted domain walls feeds into the low energy tail of the spectrum. This contrasts sharply with the constant tension case, where GWs are mainly emitted at the end of wall evolution right before dissolving, e.g., due to slight breaking of Z_2-symmetry. Note that, there is no contradiction with causality considerations <cit.>, which typically lead to Ω_gw∝ f^3. Indeed, the standard steeper shape assumes a finite operation of the GW source, typically shorter than the Hubble time. In contrast, in the scenario <cit.> we discuss here, GWs are efficiently produced by the time-extended source over many Hubble time intervals.
Remarkably, the behaviour Ω_gw∝ f^2 better fits NANOGrav 15 yrs data compared to[At the same time, low-frequency GW emission from melting cosmic strings has a shape Ω_gw∝ f^4 <cit.>, which conflicts with NANOGrav data.] Ω_gw∝ f^3. It is conventional to parameterise the PTA GW signal as
Ω_ GW(f)=Ω_yr(f/f_ yr)^5-γ,
with γ being the spectral index and f_yr= 1 yr^-1≃ 32 nHz. The NANOGrav best-fit value of the spectral index reads γ =3.2± 0.6. In Section 3, we also identify the values of the model parameters yielding the best-fit value Ω_ yr=5.8× 10^-8. To accomplish this, one should assume
f_peak≃ f_yr, where f_peak is the predicted peak frequency of GWs, so that Ω_yr≃Ω_gw, peak. Using the relations
between f_peak and Ω_gw, peak and the model coupling constants,
we can pinpoint the particle physics scenario underlying melting domain walls.
In particular, the scalar field constituting domain walls should be extremely
weakly coupled. Given that the model constants are confined to a rather narrow range already with the current NANOGrav sensitivity, a future increase of PTA sensitivity to GWs will allow one to make decisive conclusions regarding the melting domain wall interpretation of the signal. Note also that feeble couplings involved in the interpretation make the scalar field comprising domain walls a suitable dark matter candidate, provided that its mass is confined to the ultra-light range. For such low masses superradiance <cit.> plays an important role by triggering instability of rotating black holes with astrophysical masses <cit.>.
This leads to potentially observable spin-down of rotating black holes and to stochastic GW background due to gravitational radiation of the bosonic condensate forming around black holes, see e.g. Ref. <cit.>.
§ BRIEF OVERVIEW OF MELTING DOMAIN WALLS
We start with the Z_2-symmetric model of real scalar field χ,
which interacts through the portal coupling with a scalar multiplet ϕ from the primordial thermal bath:
S=∫ d^4 x √(-g)[(∂_μχ)^2/2 -M^2_χχ^2/2-λ_χχ^4/4 +g^2 χ^2 |ϕ|^2/2] ,
where M_χ, λ_χ, and g^2 are the bare mass, quartic self-interaction constant of the field χ, and portal coupling constant, respectively <cit.>. We assume that particles ϕ are relativistic at the times of interest, which fixes the variance of the field ϕ to be
⟨ |ϕ|^2 ⟩ =N T^2/12 ,
where N counts the number of degrees of freedom associated with ϕ.
Let us fix the sign of the portal coupling constant g^2 as
g^2>0 .
It induces instability in the two-field system, which is tamed, provided that the following condition is obeyed:
β≡λ_χ/g^4≥1/λ_ϕ ,
where λ_ϕ is the quartic self-interaction constant of the multiplet ϕ. We will often use the constant β instead of self-interaction constant λ_χ in what follows. Consequently, the effective potential characterizing the field χ exhibits spontaneous symmetry breaking leading to the non-zero temperature-dependent expectation value:
⟨χ⟩ =±√(Ng^2 T^2/12λ_χ -M^2_χ/λ_χ) .
In the expanding Universe, this temperature-dependence induces time-dependence, which is crucial for our further discussions. At some (lower) temperature the bare mass term becomes relevant, and the symmetry restores with ⟨χ⟩=0, i.e., the inverse phase transition happens. However, at most times of interest we assume the bare mass M_χ negligible; it will be included only when considering dark matter implications of the model.
Spontaneous breaking of Z_2-symmetry leads to the formation of domain walls, provided that the background field χ is set to zero, i.e., χ=0, prior to falling into the minima of symmetry breaking potential.
This condition can be achieved e.g. by the non-minimal coupling to gravity ∼ξχ^2 R leading to the super-Hubble mass during inflation for ξ≳ 1; at the same time R ≈ 0 during the radiation-dominated stage, and does not affect dynamics of the system, see Ref. <cit.> for details. Domain walls are often unwelcome in cosmology because they quickly begin to dominate the evolution of the Universe, in contradiction with observational data. This problem is absent in our case, exactly due to the time dependence of the expectation value ⟨χ⟩, as it will become clear shortly.
The Universe temperature at the time of domain wall formation is defined by the balance of the Hubble friction and
the tachyonic thermal mass; it is estimated as
T_i ≃√(N) gM_Pl/√(B g_* (T_i)) ,
where g_* (T) counts the number of relativistic degrees of freedom at the temperature T, and M_Pl≈ 2.44 · 10^18 is the reduced Planck mass. The constant B here takes into account the finite duration of the field χ roll to the minimum of its potential; B ≃ 1 for the infinitely fast roll,
but generically it takes values in the range 1 ≲ B ≲ 10^3, see Ref. <cit.>.
The domain wall tension (mass per unit area) is given by
σ =2√(2 λ_χ)·⟨χ⟩^3 /3 .
The energy density of domain walls in the scaling regime with one (a few) domain wall(s) per horizon volume characterized by the size H^-1,
where H is the Hubble rate, is estimated as
ρ_wall∼σ H .
Using Eq. (<ref>), where we neglect the bare mass, from Eqs. (<ref>) and (<ref>) one finds that the energy density of melting domain walls redshifts as ρ_walls∝ 1/a^5 at the radiation-dominated stage, which is in contrast to the scenario with constant tension domain walls yielding ρ∝ 1/a^2. Hence, the energy density of melting walls drops faster than the energy density of radiation, and there is no domain wall problem in the Universe.
§ GWS FROM MELTING DOMAIN WALLS
Numerical estimation of GW emission by a network of domain walls
has been performed in Ref. <cit.>. This has been done in the case of constant
tension domain walls, but we can readily use some of the results
obtained there to the case of melting domain walls. Despite
strong differences, in both cases, most energetic GWs
are emitted within a short time interval: close to the moment of the domain
wall formation in the case of melting domain walls and near the time of
the network dissolution for constant tension domain walls. The properties
of GWs at the spectrum peak are defined by the wall tension and Hubble rate
in this short time interval. In particular, the peak frequency of emission
is estimated by F ≃ H <cit.>. Consequently, the present-day peak frequency is estimated as <cit.>
f_peak≡ f_peak (t_0) ≃ H_i ·a_i/a_0∝ T_i ,
which gives upon substituting in Eq. (<ref>):
f_peak≃ 6 √(N/B)·g/10^-18·(100/g_* (T_i))^1/3 .
Interestingly, the numerical simulations of Ref. <cit.> have shown that the Einstein quadrupole formula well captures the peak energy of GWs.
Including numerical corrections, one can write then for the fractional energy density
at the emission time
Ω_gw, peak (t_i) ≈λ_χϵ_gw A^2 ⟨χ⟩^6_i/27π H^2_i M^4_Pl∝ T^2_i ,
where the coefficients ϵ_gw and A account for the efficiency
of GW emission and scaling, correspondingly; one has ϵ_gw A^2 ≈ 0.5 <cit.>. Hence, at present, the energy density of GWs is given by
Ω_gw, peak h^2_0 ≈ 1.34 · 10^-5×(100/g_* (T_i))^1/3Ω_gw, peak (t_i) ,
where h_0=0.67 is the reduced Hubble constant <cit.>. Combining Eqs. (<ref>), (<ref>), (<ref>), and Eq. (<ref>), and using definition (<ref>), we obtain <cit.>
Ω_gw, peak h^2_0 ≃4 · 10^-14· N^4/B·β^2·(100/g_* (T_i))^7/3 .
To discriminate between GWs emitted by melting domain walls and other sources, one should consider the GW spectrum. While this requires numerical simulations, we can estimate the low-frequency part of the spectrum and show that it is distinct from the spectrum of constant tension walls. For this purpose, we observe that the low-frequency part of the spectrum is sourced by GW emission at the late times t>t_i. This is evident from Eq. (<ref>),
where one should replace T_i by T(t)<T_i.
The peak energy density can be estimated from Eq. (<ref>), where one again replaces T_i with T(t). We conclude that <cit.>
Ω_gw h^2_0 (f<f_peak) =Ω_gw, peak h^2_0 ·( T(t)/T_i)^2= Ω_gw, peak h^2_0 ·( f/f_peak)^2 .
This is in contrast to the result obtained in the case of
constant tension domain walls and many other sources,
i.e., first-order phase transitions and cosmic strings, giving Ω_gwh^2_0≃ f^3. Note that the causality is not violated in our case: indeed, according
to the discussion above, low-frequency GW emission still fulfills F ∼ H(t) and hence follows from on-horizon
dynamics of melting domain walls. Note also that the causality argument suggests the low-frequency tail of GW emission produced around the time of domain wall formation t ≃ t_i is steep enough and does not affect our estimate (<ref>). In this work, we are not much concerned about the high-frequency part of the spectrum, assuming that it is outside of the domain probed by NANOGrav (see below). We assume that it is not different from the case of constant tension walls, i.e., there is a power law decrease Ω_gw∝ 1/f at f>f_peak, which should be followed by the exponential suppression at frequencies corresponding to the inverse width of domain walls <cit.>.
Figure <ref> demonstrates that the predicted GW signal is compatible with the NANOGrav signal for the set of theoretically acceptable values of model parameters.
Below we explain the notations used and the assumed choice of model constants. GW spectral energy density
associated with the NANOGrav, or more generally, with the PTA signal is conventionally expressed as
Ω_gw(f)=2π^2/3 H_0^2f^2h^2_c(f) ,
where h_c(f) is the characteristic strain parameterised as
h_c(f)=A(f/f_yr)^(3-γ)/2,
with A and γ being the amplitude and the spectral index, respectively; recall that f_ yr= 1 yr^-1≃ 32 nHz. Note that combining Eqs. (<ref>) and (<ref>), one gets Eq. (<ref>), where Ω_yr=2π^2/3 H_0^2A^2f_yr^2. The best fit to the NANOGrav signal is provided by the values A≃ 6.4^+4.2_-2.7× 10^-15 and γ=3.2 ± 0.6. The latter agrees well with the model prediction (<ref>).
To achieve the best fit value of A, which corresponds to rather large GW energy density Ω_yr≃ 5.8 × 10^-8, we first set f_peak≃ f_yr, so that Ω_gw, peak≃Ω_yr. The reason for this choice will become clear a posteriori. Using Eqs. (<ref>) and (<ref>), one can relate the temperature at domain wall formation to the peak frequency:
T_i ≃ 1.2 ·(f_peak/f_yr) ·(100/g_* (T_i))^1/6 .
Hence, for f_gw, peak≃ f_yr, one has T_i ≃ 1.2 and g_* (T_i) ≃ 75. Now we fix the constants entering the GW energy density (<ref>):
β = 1 , B = 1 , N=24 .
Finally, using this and Eq. (<ref>), we can fix the constant g, i.e.,
g = 10^-18 .
This implies a tiny portal coupling g^2=10^-36,
while β=1 translates into the self-interaction constant λ_χ=10^-72. Such tiny coupling constants are not unfamiliar in physics, and they are characteristic for axion-like particles. Note also
that the constants β and B are chosen to be close to minimally allowed ones, see Eq. (<ref>), to achieve the observed value Ω_yr. This also explains the choice f_peak≃ f_yr, because for f_peak≫ f_yr, one would need to assume too large Ω_gw, peak≫Ω_yr. It is important to stress that one can accommodate larger values of parameters β and B by a moderate increase of the number of degrees of freedom N. Indeed, increasing β by factor ξ requires only an increase of N by smaller factor ξ^1/2. On the other hand, a change of parameter B by factor ζ requires a corresponding increase of N by a much smaller factor ζ^1/4.
We have assumed that the field ϕ is relativistic at the times when relevant GWs are emitted, which are sufficiently close to the BBN epoch. Thus, if the field ϕ is still relativistic at the temperatures T ≲ 1, one runs the risk of spoiling a well-established picture of light element primordial abundance. There are two ways of avoiding this. One is to assume that the
particles ϕ decoupled from primordial plasma at very early times,
and thus contribute insignificantly to the effective number of neutrino species N_eff. In that case, however, the effective temperature T_ϕ describing the system of particles ϕ is lower than the Universe temperature. This tends to decrease GW energy density according to Eq. (<ref>), but the decrease can be (partially) compensated by the sharp change of degrees of freedom number g_* (T) around QCD phase transition. Another way to handle the problem is to assume that the particles ϕ have mass m_ϕ in the MeV range, i.e.,
1 ≪ m_ϕ≪ 1 .
That is, the particles ϕ become non-relativistic sometime before BBN and then decay into SM species in one or another way. In that case, one can also consider the scenario with the effective temperature T_ϕ higher than the Universe temperature T.
§ IMPLICATIONS FOR DARK MATTER
The field χ being very feebly coupled to the primordial thermal bath is a proper dark matter candidate. This is despite the fact that for the
portal constant g^2 ≃ 10^-36, neither freeze-out nor freeze-out production mechanisms are operating. Yet it is possible to generate the right dark matter abundance even with this tiny coupling constant. We briefly comment on two production mechanisms below and identify the mass M_χ as a function of GW parameters assuming that the field χ constitutes all dark matter.
* Dark matter production via the direct phase transition. Oscillations of the field χ around the minima of its potential naturally feed into dark matter. These oscillations start at the times t ≃ t_i, when the domain wall network is created, and continue till present unless the particles χ are unstable. In that case, the observed dark matter abundance is achieved for extremely small M_χ:
M_χ≃ 6.5 · 10^-17 ·(f_peak/30 ) ·(g_* (T_i)/100)^1/6·√(10^-8/Ω_gw, peak· h^2_0) .
* Dark matter via inverse phase transition <cit.>, cf. Refs. <cit.>. Dark matter production
occurs also in the case, when there is an efficient decay channel for the aforementioned oscillations, and the field χ settles to the minimum of its potential. Yet coherent oscillations are produced at the inverse phase transition because symmetry restoration is a non-adiabatic process.
In that case, one has
M_χ ≃ 10^-12 · B^9/20·(g_* (T_sym)/100)^1/5·(g_* (T_i)/100)^1/20·(m_ϕ/10 )^1/2×
(f_peak/30 )^6/5·(10^-8/Ω_gw, peak h^2_0)^3/20 ,
where T_sym is the Universe temperature at the inverse phase transition.
We observe that in both cases GW parameters favoured by NANOGrav data imply ultra-light dark matter masses M_χ. Notably, with these values of M_χ, our scenario predicts superradiance instability of rotating black holes with astrophysical masses <cit.>. This suggests a complementary way of testing the model, in particular, the future LISA observations will probe the masses of dark matter particles corresponding to the direct phase transition, while the LIGO data may be used to test the masses involved in the inverse phase transition <cit.>.
§ DISCUSSIONS
We have shown that the properties of GWs emitted by the network of
melting domain walls are consistent with the signal detected at PTAs.
Keeping in mind that melting domain walls do not overclose the Universe and
the constituent field χ serves as a suitable dark matter candidate,
this makes them interesting objects deserving further investigation. Perhaps the most important prospect for future studies is the numerical study of melting domain wall evolution and eventually more precise determination
of GW parameters, i.e., peak frequency, energy density,
and the spectral shape including the high-frequency range. In particular, the formation of melting walls and settling them into the scaling regime are yet to be better understood. This is important given that the most energetic GWs signals are coming from the earliest stages of the wall network evolution. With the current estimates of GW parameters, the NANOGrav signal
is fitted in a very narrow range of model constants. Therefore, with more detailed information on the signal/improved predictions of GW properties, one will have a chance to rule out the proposed interpretation of the GW signal or establish it on firmer grounds.
On a more theoretical side, it is interesting to embed the field ϕ, with masses in a phenomenologically interesting range (<ref>), into a realistic particle physics scenario. While in the present work, we assumed that ϕ is in equilibrium with primordial plasma, it is worth investigating situations, where ϕ decouples from plasma prior to domain wall formation or has never reached thermal equilibrium.
§ ACKNOWLEDGMENTS
EB acknowledges support of ANR grant StronG (ANR-22-CE31-0015-01).
DG acknowledges support of the scientific program of the National Center for Physics and Mathematics, section 5 "Particle Physics and Cosmology", stage 2023-2025. SR acknowledges the European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS -CZ.02.1.01/0.0/0.0/15003/0000437). RS acknowledges the project MSCA-IF IV FZU - CZ.02.2.69/0.0/0.0/20 079/0017754, European Structural and Investment Fund, and the Czech Ministry of Education, Youth and Sports. AV was supported by the Czech Science Foundation (GAČR), project 20-28525S and is thankful to Enrico Barausse for discussions.
99
Babichev:2021uvl
E. Babichev, D. Gorbunov, S. Ramazanov and A. Vikman,
JCAP 04 (2022) no.04, 028
[arXiv:2112.12608].
Ramazanov:2021eya
S. Ramazanov, E. Babichev, D. Gorbunov and A. Vikman,
Phys. Rev. D 105 (2022) no.6, 063530;
[arXiv:2104.13722].
ngr1
G. Agazie et al. [NANOGrav],
Astrophys. J. Lett. 951, no.1, L8 (2023);
[arXiv:2306.16213].
ngr2
G. Agazie et al. [NANOGrav],
Astrophys. J. Lett. 951, no.1, L9 (2023);
[arXiv:2306.16217].
epta1
J. Antoniadis, P. Arumugam, S. Arumugam, S. Babak, M. Bagchi, A. S. B. Nielsen, C. G. Bassa, A. Bathula, A. Berthereau and M. Bonetti, et al.
[arXiv:2306.16214].
epta2
J. Antoniadis, S. Babak, A. S. Bak Nielsen, C. G. Bassa, A. Berthereau, M. Bonetti, E. Bortolas, P. R. Brook, M. Burgay and R. N. Caballero, et al.;
[arXiv:2306.16224].
epta3
J. Antoniadis, P. Arumugam, S. Arumugam, P. Auclair, S. Babak, M. Bagchi, A. S. B. Nielsen, E. Barausse, C. G. Bassa and A. Bathula, et al.;
[arXiv:2306.16227].
ppta1
D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Kapur, A. F. Rogers, E. Thrane and J. Askew, et al.
Astrophys. J. Lett. 951, no.1, L6 (2023);
[arXiv:2306.16215].
ppta2
A. Zic, D. J. Reardon, A. Kapur, G. Hobbs, R. Mandow, M. Curyło, R. M. Shannon, J. Askew, M. Bailes and N. D. R. Bhat, et al.;
[arXiv:2306.16230].
cpta
H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu, Z. Xue, R. N. Caballero, J. Yuan and Y. Xu, et al.
Res. Astron. Astrophys. 23, no.7, 075024 (2023);
[arXiv:2306.16216].
hd
R. w. Hellings and G. s. Downs,
Astrophys. J. Lett. 265, L39-L42 (1983).
ngr3
G. Agazie et al. [NANOGrav],
[arXiv:2306.16220].
ngr4
A. Afzal et al. [NANOGrav],
Astrophys. J. Lett. 951, no.1, L11 (2023);
[arXiv:2306.16219].
bhn1
A. Sesana, A. Vecchio, and C. N. Colacino, Mon. Not.
Roy. Astron. Soc. 390, 192 (2008), [arXiv:0804.4476].
bhn2
B. Kocsis and A. Sesana, [arXiv:1002.0584].
cs1
S. Blasi, V. Brdar and K. Schmitz,
Phys. Rev. Lett. 126, no.4, 041305 (2021);
[arXiv:2009.06607].
cs2
J. Ellis and M. Lewicki,
Phys. Rev. Lett. 126, no.4, 041304 (2021);
[arXiv:2009.06555].
cs3
R. Samanta and S. Datta,
JHEP 05, 211 (2021);
[arXiv:2009.13452].
Zeldovich:1974uw
Y. B. Zeldovich, I. Y. Kobzarev and L. B. Okun,
Zh. Eksp. Teor. Fiz. 67 (1974), 3-11
SLAC-TRANS-0165.
dom1
Y. Gouttenoire and E. Vitagliano,
[arXiv:2306.17841].
dom2
S. Blasi, A. Mariotti, A. Rase and A. Sevrin,
[arXiv:2306.17830].
dom3
X. F. Li,
[arXiv:2307.03163].
dom4
Y. M. Wu, Z. C. Chen and Q. G. Huang,
[arXiv:2307.03141].
dom5
X. K. Du, M. X. Huang, F. Wang and Y. K. Zhang,
[arXiv:2307.02938].
dom6
B. Q. Lu and C. W. Chiang,
[arXiv:2307.00746].
dom7
B. Barman, D. Borah, S. Jyoti Das and I. Saha,
[arXiv:2307.00656].
dom8
Y. Bai, T. K. Chen and M. Korwar,
[arXiv:2306.17160].
dom9
N. Kitajima, J. Lee, K. Murai, F. Takahashi and W. Yin,
[arXiv:2306.17146].
dom10
L. Bian, S. Ge, J. Shu, B. Wang, X. Y. Yang and J. Zong,
[arXiv:2307.02376].
Vilenkin:1981zs
A. Vilenkin,
Phys. Rev. D 23 (1981), 852-857.
Hiramatsu:2013qaa
T. Hiramatsu, M. Kawasaki and K. Saikawa,
JCAP 02 (2014), 031;
[arXiv:1309.5001].
Cai:2019cdl
R. G. Cai, S. Pi and M. Sasaki,
Phys. Rev. D 102 (2020) no.8, 083528;
[arXiv:1909.13728].
Hook:2020phx
A. Hook, G. Marques-Tavares and D. Racco,
JHEP 02 (2021), 117;
[arXiv:2010.03568].
Durrer:2003ja
R. Durrer and C. Caprini,
JCAP 11 (2003), 010;
[arXiv:astro-ph/0305059].
Franciolini:2023wjm
G. Franciolini, D. Racco and F. Rompineve,
[arXiv:2306.17136].
Planck:2018vyg
N. Aghanim et al. [Planck],
Astron. Astrophys. 641 (2020), A6
[arXiv:1807.06209].
Emond:2021vts
W. T. Emond, S. Ramazanov and R. Samanta,
JCAP 01 (2022) no.01, 057;
[arXiv:2108.05377].
Phinney:2001di
E. S. Phinney,
[arXiv:0108028].
Babichev:2020xeg
E. Babichev, D. Gorbunov and S. Ramazanov,
JCAP 08 (2020), 047;
[arXiv:2004.03410].
Ramazanov:2020ajq
S. Ramazanov, F. R. Urban and A. Vikman,
JCAP 02 (2021), 011;
[arXiv:2010.03383].
zeldovich1
Y. B. Zel'dovich Pis'ma Zh. Eksp. Teor. Fiz. 14 (1971) 270 [JETP
Lett. 14, 180 (1971)].
zeldovich2
Y. B. Zel'dovich Zh. Eksp. Teor. Fiz 62 (1972) 2076 [Sov.Phys.
JETP 35, 1085 (1972)].
Starobinsky:1973aij
A. A. Starobinsky,
Sov. Phys. JETP 37 (1973) no.1, 28-32.
Arvanitaki:2009fg
A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell,
Phys. Rev. D 81 (2010), 123530;
[arXiv:0905.4720].
Brito:2015oca
R. Brito, V. Cardoso and P. Pani,
Lect. Notes Phys. 906 (2015), pp.1-237
2020,
[arXiv:1501.06570].
ska
A. Weltman, P. Bull, S. Camera, K. Kelley, H. Padmanabhan, J. Pritchard, A. Raccanelli, S. Riemer-Sørensen, L. Shao and S. Andrianomena, et al.
Publ. Astron. Soc. Austral. 37, e002 (2020);
[arXiv:1810.02680].
gaia
J. Garcia-Bellido, H. Murayama and G. White,
JCAP 12, no.12, 023 (2021);
[arXiv:2104.04778].
mras
A. Sesana, N. Korsakova, M. A. Sedda, V. Baibhav, E. Barausse, S. Barke, E. Berti, M. Bonetti, P. R. Capelo and C. Caprini, et al.
Exper. Astron. 51, no.3, 1333-1383 (2021);
[arXiv:1908.11391].
lisa
P. Amaro-Seoane et al. [LISA],
[arXiv:1702.00786].
decigo
S. Kawamura, T. Nakamura, M. Ando, N. Seto, K. Tsubono, K. Numata, R. Takahashi, S. Nagano, T. Ishikawa and M. Musha, et al.
Class. Quant. Grav. 23, S125-S132 (2006).
bbo
K. Yagi and N. Seto,
Phys. Rev. D 83, 044011 (2011)
[erratum: Phys. Rev. D 95, no.10, 109901 (2017)];
[arXiv:1101.3940].
Brito:2017zvb
R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin, A. Klein and P. Pani,
Phys. Rev. D 96 (2017) no.6, 064050;
[arXiv:1706.06311].
|
http://arxiv.org/abs/2307.06023v1 | 20230712090507 | On the Uplink Distributed Detection in UAV-enabled Aerial Cell-Free mMIMO Systems | [
"Xuesong Pan",
"Zhong Zheng",
"Xueqing Huang",
"Zesong Fei"
] | cs.IT | [
"cs.IT",
"math.IT"
] |
Submitted to XXXXXXXX
On the Uplink Distributed Detection in UAV-enabled Aerial Cell-Free mMIMO Systems
Xuesong Pan, Zhong Zheng, Member, IEEE, Xueqing Huang, Member, IEEE, Zesong Fei, Senior Member, IEEE
X. Pan, Z. Zheng and Z. Fei are with the School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China (e-mail: {xs.pan, zhong.zheng, feizesong}@bit.edu.cn).
X. Huang is with the Department of Computer Science, New York Institute of Technology, NY 11568, USA (e-mail: [email protected]).
June 2023
=================================================================================================================================================================================================================================================================================================================================================================================================================================================
In this paper, we investigate the uplink signal detection approaches in the cell-free massive MIMO systems with unmanned aerial vehicles (UAVs) serving as aerial access points (APs). The ground users are equipped with multiple antennas and the ground-to-air propagation channels are subject to correlated Rician fading. To overcome huge signaling overhead in the fully-centralized detection, we propose a two-layer distributed uplink detection scheme, where the uplink signals are first detected in the AP-UAVs by using the minimum mean-squared error (MMSE) detector depending on local channel state information (CSI), and then collected and weighted combined at the CPU-UAV to obtain the refined detection. By using the operator-valued free probability theory, the asymptotic expressions of the combining weights are obtained, which only depend on the statistical CSI and show excellent accuracy. Based on the proposed distributed scheme, we further investigate the impacts of different distributed deployments on the achieved spectral efficiency (SE). Numerical results show that in urban and dense urban environments, it is more beneficial to deploy more AP-UAVs to achieve higher SE. On the other hand, in suburban environment, an optimal ratio between the number of deployed UAVs and the number of antennas per UAV exists to maximize the SE.
Unmanned aerial vehicle, cell-free massive MIMO, Rician channel, operator-valued free probability.
§ INTRODUCTION
Unmanned aerial vehicles (UAVs) have been found in a wide range of applications in wireless communication during the past few decades, especially serving as aerial base stations to expand the network coverage and enhance communication quality <cit.>. Nevertheless, due to the limited carrying capability, the UAV-based stations can be only equipped with a few antennas and it's difficult to achieve satisfactory communication quality with a single UAV-based station. Therefore, it is interesting to investigate the deployment of a swarm of UAV-based stations, which collaborate in signal transmission and processing <cit.>.
Parallel to the research on UAV communications, a novel network architecture called the cell-free massive multi-input multi-output (CF mMIMO) system has been proposed to fulfill the demand on the rapid growth of data throughput <cit.>. In the CF mMIMO system, a large number of access points (APs) are deployed and connected to a central processing unit (CPU) via the fronthaul link. Instead of creating several autonomous cells, the geographically distributed APs jointly serve the UEs by coherent transmission and reception. By exploiting the increased spatial degree of freedom due to the cooperation, CF mMIMO increases the number of simultaneously served users as well as improves the achieved sum rate, compared to the conventional noncooperative cellular systems <cit.>. Meanwhile, the CF mMIMO can provide significant performance improvement compared to the small-cell system <cit.> and more uniform communication quality to the edge UEs. Due to these advantages, it is of great interest to empower the UAV swarm with CF mMIMO capability to compensate for the limited onboard hardware resources, which could enable high-speed communications towards massive numbers of ground UEs. Moreover, benefiting from the higher altitude and flexible deployment, the UAV-based stations are more likely to establish Line-of-Sight (LoS) links with ground UEs and provide on-demand services <cit.>. As shown in Fig. <ref>, the UAVs are divided into the access point UAVs (AP-UAVs) for local signal transmission and reception, as well as the CPU-UAV for joint signal processing. The AP-UAVs and the CPU-UAV are inter-connected via wireless fronthaul links.
The integration of UAV communications and CF mMIMO was first proposed by Carmen et al. <cit.>, where UAVs are treated as aerial UEs and APs are deployed on the ground. The aerial CF mMIMO system with AP-UAVs was studied by Carles et al., where the UAVs serve as aerial APs and are equipped with a single antenna <cit.>. In <cit.>, the downlink communication between the AP-UAVs and ground UEs was studied and the location of AP-UAVs was optimized to maximize the sum rate. Carles et al. aimed to increase the spectral efficiency (SE) by maximizing the minimum local-average signal-to-interference-plus-noise ratio (SINR) in uplink CF mMIMO system, where UAVs are deployed as aerial APs <cit.>. However, the above literature all considers that the UAVs are equipped with a single antenna, which can't exploit the spatial diversity of multiple antennas.
Despite the potential performance enhancement of the UAV-based CF mMIMO systems, this network architecture requires the network-wide CSI available at the CPU-UAV to optimally design the transceivers at the AP-UAVs. However, the interaction of CSI and data between AP-UAVs and CPU-UAV results in immense interactive overhead and unpredictable delay, which is especially critical in UAV-based networks due to wireless fronthaul.
§.§ Related Works
In order to decrease the interactive overhead, network-centric and user-centric approaches are proposed in <cit.> and <cit.>, respectively, both of which divide the UEs or APs into several subsets. The interactive overhead is significantly reduced by limiting the interaction within the subsets, but leading to notable performance loss <cit.>. To utilize the macro-diversity as well as decrease the interaction overhead, the large-scale fading decoding (LSFD) strategy was first proposed by Ansuman et al. to reduce interference in cellular system <cit.> and then applied to CF mMIMO systems in <cit.>. In the LSFD strategy, the APs first detect the received signals using the local channel estimates and then forward the detected data to the CPU, where the final joint detection is performed. The key idea of LSFD is that the final detection only depends on the large-scale fading coefficients, which does not need the instantaneous interaction between CPU and APs.
Based on the same structure, a plethora of papers on LSFD have been published in recent years. Elina et al. investigated the uplink performance of CF mMIMO systems with LSFD receiver, where single-antenna APs first estimate the signal from the single-antenna UEs using the matched filter (MF) with local CSI, and the CPU computes the LSFD based on large-scale fading coefficients <cit.>. Zhang et al. considered the local zero-forcing (ZF) detector with LSFD over Rayleigh fading channels, where different variants of ZF were proposed to achieve high SE or reduce computational complexity <cit.>. Furthermore, Emil et al. proposed four different levels of cooperation in CF mMIMO systems, including fully-centralized processing, local processing with LSFD, local processing with simple centralized decoding, and small-cell network <cit.>. The local processing in <cit.> is implemented in APs with MMSE detector, which trade-offs between signal improvement and interference suppression. In the above papers, the ground UEs are assumed to be equipped with a single antenna, while the contemporary UEs have already equipped with multiple antennas.
With the emergence of new applications in 6G wireless networks, it's necessary to equip the UEs with multi-antennas to improve system performance. Trang C et al. investigated the uplink and downlink SE of CF mMIMO systems with multi-antennas UEs over uncorrelated Rayleigh fading channels, where the closed-form expressions of both uplink and downlink SE are derived <cit.>. Based on <cit.>, Wang et al. studied the uplink performance of CF mMIMO system with multi-antennas UEs, where four levels of cooperation are considered over correlated Rayleigh fading channels <cit.>. In addition, based on the same system structure, uplink precoding of multi-antennas UEs was investigated to improve uplink SE <cit.>.
However, most existing works, e.g. <cit.>, only considered the Non-Line-of-Sight (NLoS) connection between the APs and UEs and ignored the existence of the LoS path, which is the dominant channel component in the UAV-based CF mMIMO systems. In addition, the calculation of the combining weights in LSFD is given in a semi-closed form manner and requires time-consuming numerical simulations, which is difficult to practical application <cit.>.
§.§ Contributions
To address the above issues, a general UAV-based CF mMIMO system is considered with multi-antennas equipped at both APs and ground UEs, while communicating over generalized Rician fading channels. The main contributions of this paper are summarized as follows:
* We consider a CF mMIMO system with UAVs being the aerial APs and multi-antennas UEs on the ground. In addition, we adopt the Rician fading channel with Kronecker’s correlation model and investigate the pilot contamination over the Rician MIMO channel. By taking the LoS path into account, the considered channel model is more practical and suitable for the UAV-based CF mMIMO system.
* We design a two-layer detection scheme consisting of local MMSE detection at AP-UAVs and one-shot combining at the CPU-UAV. By using the operator-valued free probability theory, the asymptotically optimal combining weights are derived for the large dimensional MIMO systems, which only depend on the channel statistics and are asymptotically accurate. The free probability theory takes full advantage of the invariance properties of the propagation channels and is a convenient tool to study mutual information, information plus noise models, etc <cit.>.
* We investigate the impact of the network configuration options on the achieved sum rate in various propagation environments <cit.>. Specifically, given a fixed total number of antennas across AP-UAVs, a.k.a. fixing the total transceivers’ capability at the network side, the tradeoff between the deployed number of AP-UAVs (L) and the number of equipped antennas per each AP-UAV (M) is shown by numerical simulations. Results show that there exists an optimal ratio between L and M in suburban area due to higher LoS probability; whereas, it tends to deploy more AP-UAVs in the urban and dense urban areas.
The rest of this article is organized as follows. The channel model, the channel estimation, and the signal model are introduced in Section <ref>. In Section <ref>, the cell-free detection methods including the fully-centralized detection and the two-layer distributed detection scheme are introduced. In Section <ref>, the asymptotic combining weights of the two-layer distributed detection are calculated by using the operator-valued free probability theory. Numerical results are given in Section <ref>. Section <ref> concludes the main findings of this article.
Notations. Throughout the paper, vectors are represented by lower-case bold-face letters, and matrices are represented by upper-case bold-face letters. We use 0_n and I_n to represent an n× n all-zero matrix and an n× n identity matrix, respectively. The notation diag(A_i)_1≤ i≤ n or diag(A_1,…,A_n) denotes the diagonal block matrix consisting of square matrices A_1,…,A_n. The notation [A]_mn denotes the (m,n)-th entry of matrix A. The superscript (·)^⋆, (·)^T, and (·)^† are denoted as the conjugate, transpose, and conjugate transpose operations, respectively. We denote Tr(A) and vec(A) as the trace and vectorization of n× n matrix A, respectively. The notation E[·] denotes the expectation. The notation ⊗ is denoted as the Kronecker product.
§ SYSTEM MODEL
Consider an uplink aerial cell-free communication system consisting of L UAV-mounted APs, one CPU-UAV, and K ground UEs, as illustrated in Fig. <ref>. The users' signals are transmitted to the AP-UAVs, and then are wirelessly fronthauled to the CPU-UAV for joint signal detection. Each AP-UAV is equipped with M receiving antennas and each UE is equipped with N transmitting antennas. Denote the total number of receiving and transmitting antennas as M_tot=L× M and N_tot=N× K, respectively. We consider the more practical Rician fading channel, whose model is given in Section <ref>. In addition, we assume that the considered communication system adopts the time-division duplex (TDD) protocol and that the channel responses are reciprocal between AP-UAVs and UEs. The channel responses are assumed to be constant and frequency flat in each coherence block of τ_c-length. Furthermore, each τ_c-length coherence block consists of the τ_p-length channel estimation phase and τ_c-τ_p-length data transmission phase. The two phases are described below.
§.§ Channel Model
The ground-to-air channel coefficients between AP-UAV l and UE k are denoted as H_kl∈C^M× N, which follows the Rician MIMO model with Kronecker's correlation structure as
H_kl=H_kl+R_kl^1/2W_klT_kl^1/2,
where H_kl is the deterministic components representing the LoS propagation component and W_kl denotes the random scattering components consisting of independent identically distributed (i.i.d.) Gaussian random variables, i.e., [W_kl]_mn∈CN(0,1). The matrices R_kl∈C^M× M and T_kl∈C^N× N are the correlation matrices at the receiver and the transmitter sides, respectively. Note that vec(H_kl)∈CN(vec(H_kl), C_kl), where C_kl=E[vec(H_kl-H_kl)vec(H_kl-H_kl)^†]=T_kl^ T⊗R_kl is the full correlation matrix <cit.>.
In addition, we denote the distance-dependent pathloss between the AP-UAV l and UE k as β_kl = E{‖H_kl‖^2_F}/M given by
E{‖H_kl‖^2_F} = 1/NTr(R_kl)Tr(T_kl)+Tr(H_klH_kl^†)
Following the standard conventions <cit.>, the channel statistics H_kl, R_kl and T_kl are normalized such that
Tr(𝐑_kl)=1/κ+1β_klM,
Tr(𝐓_kl)=N,
Tr(𝐇_kl𝐇_kl^†)=κ/κ+1β_klM,
where κ is the Rician factor.
§.§ Channel Estimation
In the channel estimation phase, τ_p mutually orthogonal pilot sequences are randomly assigned to UEs. In specific, the UE k is assigned with pilot matrix Φ_k∈C^τ_p× N, where Φ_k^†Φ_i=τ_pI_N, if k=i and 0 otherwise.
For the uplink communications with massive multi-antenna transmitters, the total number of transmitting antennas can be easily larger than the available number of pilots, i.e., KN≥τ_p. In this setting, more than one UE has to be assigned with the same pilot matrix. We define the set U_k as the subset of UEs, which use the same pilot as UE k, and t_k∈{1,…, τ_p} as the index of pilot matrix assigned to UE k, respectively. Thus, the received signal Y_l^tr∈C^M×τ_p at AP-UAV l in channel estimation phase is given by
Y_l^tr = ∑_i=1^KH_ilF_iΦ_i^T+N_l^tr,
where F_i∈C^N× N denotes the precoder of UE i and N_l^tr∈C^M×τ_p is the additive noise at AP-UAV l with i.i.d CN(0,σ^2) entries, where σ^2 is the noise power. Note that the precoder F_i follows the power constrain Tr(F_iF_i^†)≤ p_i, where p_i is maximum transmitting power of UE i.
Then, the AP-UAV l correlates the received signals with the associated pilot Φ_k^⋆ as follow
Y_kl^tr = Y_l^trΦ_k^⋆ = ∑_i=1^KH_ilF_iΦ_i^TΦ_k^⋆+N_l^trΦ_k^⋆,
= τ_p ∑_i∈U_kH_ilF_i + N_l^tr,
where N_l^tr = N_l^trΦ_k^⋆. By vectorizing the received signal Y_kl^tr, the received signal can be rewritten as
y_kl^tr = vec(Y_kl^tr) =τ_p ∑_i∈U_k F_i h_il + vec( N_l^tr),
where F_i = F_i^T⊗I_M and h_il=vec(H_il). By using the MMSE estimator <cit.>, the estimated channel H_kl is given by
h_kl = vec( H_kl) = vec(H_kl)+vec(H_kl)
= h_kl + C_kl F_k^†Π_kl^-1(y_kl- y_kl),
where y_kl = τ_p ∑_i∈U_k F_i h_il with h_il = vec( H_il), H_kl represent the random uncertain components, and Π_kl is the normalized correlation matrix of the received signal y_kl^tr, which is given by
Π_kl = 1/τ_pE{(y_kl^tr- y_kl)(y_kl^tr- y_kl)^†},
= τ_p ∑_i∈U_k F_iC_il F_i^†+σ^2I_M_lN.
Considering the orthogonality property of MMSE estimation <cit.>, the estimation error h̆_kl=vec(H̆)=h_kl- h_kl∈CN(0, C_kl), where C_kl=C_kl - C_kl with C_kl=τ_pC_kl F_k^†Π_kl^-1 F_kC_kl being the correlation matrix of the estimation h_kl.
To describe the correlation of the channel coefficients and facilitate the derivation in Section <ref>, we define the one-sided correlation function of the equivalent channel H̀_kl=L H_klR parameterized by a Hermitian matrix D∈C^N× N as η_kl(L,D,R)=E[H̀_klDH̀_kl^†], where L∈C^M× M and R∈C^N× N are arbitrary matrices. The (m,n)-th element of η_kl(L,D,R) is given by
[η_kl(L,D,R)]_mn =∑_i=1^N∑_j=1^N⟨⟨R⟩⟩_jD⟨⟨R⟩⟩_i^†⟨⟨L⟩⟩_m C_kl^ji⟨⟨L⟩⟩^†_n,
where the notation ⟨⟨A⟩⟩_i denotes the i-th row of the matrix A. The matrix C_kl^ji=E[⟨ H_kl⟩_j⟨ H_kl⟩_i^†] denotes the (j,i)-th sub-matrix of C_kl, with ⟨ H_kl⟩_i and ⟨ H_kl⟩_j being the i-th and j-th column of H_kl, respectively.
The other parameterized one-sided correlation function η_kl(L, D,R)=E[H̀_kl^† DH̀_kl] parameterized by D∈C^M× M, whose (m,n)-th element is given by
[η_kl(L, D,R)]_mn =∑_i=1^M∑_j=1^M⟨L⟩_j^† D⟨L⟩_i⟨R⟩_m^†Υ_kl^ji⟨R⟩_n,
where the N× N matrix Υ_kl^ji=E[⟨⟨ H_kl⟩⟩_j^†⟨⟨ H_kl⟩⟩_i], whose (m,n)-th element is denoted as [Υ_kl^ji]_mn=[ C_kl^nm]_ij.
§.§ Uplink Data Transmission
In the data transmission phase, the received signal y_l∈C^M at the AP-UAV l is given by
y_l = ∑_k=1^KH_klP_k x_k + n_l,
where x_k∈C^N is the signal transmitted from UE k and x_k∈CN(0, I_N). The precoding matrix P_k at the UE k satisfies the power constraint as Tr(P_kP_k^†)≤ p_k. The additive noise n_l∈CN(0,σ^2I_M) is an i.i.d. complex Gaussian vector.
Based on the Rician fading channel model, in the following sections, we will investigate the decentralized signal detection scheme and derive the closed-form expressions of one-shot combining weights by utilizing the channel statistics.
§ DECENTRALIZED SIGNAL DETECTION WITH ONE-SHOT COMBINING
As illustrated in Fig. <ref>, several AP-UAVs are connected to a CPU-UAV via the wireless fronthaul, which has limited capacity and uncertain latency. In addition, the computational capability and service range of AP-UAVs are limited by their lightweight fuselage. To achieve joint reception in the aerial CF mMIMO system, a common method called the fully-centralized scheme, where AP-UAVs are treated as remote antennas forwarding the received symbols to the CPU-UAV transparently, is introduced in Section <ref> and provided as a benchmark. Although the highest SE can be achieved in the fully-centralized scheme, the huge interaction overhead and resulting latency prevent it from being used in practice. Therefore, to take advantage of macro-diversity and reduce the interactive latency, a decentralized reception framework consisting of the local detection at AP-UAVs and the one-shot combining at the CPU-UAV, is proposed in Section <ref>.
§.§ Fully-centralized Scheme
In the fully-centralized scheme, the received symbols are centrally detected in the CPU-UAV, which collects channel estimates and the received raw symbols from AP-UAVs. The AP-UAVs act as geographically distributed antennas, which only forward data to the CPU-UAV, and have no computation burden. The fully-centralized scheme can take full advantage of diversity gain and expand coverage to ensure communication quality.
The collected raw signals in the CPU-UAV can be expressed as
[ [ 𝐲_1; ⋮; 𝐲_L ]]_≜𝐲=[ [ 𝐇_1; ⋮; 𝐇_L ]]_≜𝐇[ [ 𝐏_1 ; ⋱; 𝐏_K; ]]_≜𝐏[ [ x_1; ⋮; x_K ]]_≜𝐱+
[ [ 𝐧_1; ⋮; 𝐧_L ]]_≜ n,
In a more compact form, y is rewritten as
y = HPx+n,
where H_l=[H_1l,H_2l,…,H_Kl]∈C^M× N_tot is the aggregative channel of AP-UAV l.
The CPU-UAV collects the channel estimates and recovers the received signals with the global MMSE detector, which minimizes the conditional MSE between the original signal x and the global estimated signal x=U^†y given the channel estimation H as follows
M_FC = Umin 𝔼{‖x - x ‖ ^2| H},
where H∈C^M_tot× N_tot is the aggregative channel estimates, and has the same structure as H in (<ref>). The global MMSE detector U∈C^M_tot× N_tot is given by
U = ( HPP^† H^†+C^'+σ^2I_M_tot)^-1 HP,
where C^'=diag(C_l^')_1≤ l≤ L∈C^M_tot× M_tot. The Hermitian matrix C_l^' = ∑_k=1^KC_kl^'∈C^M× M, with C_kl^' =E(H̆_klP_kP^†_kH̆_kl^†), whose (m,n)-th element is given by
[C_kl^']_mn=∑_n_1=1^N∑_n_2=1^N[P_kP_k^†]_n_2n_1[ C_kl^n_2n_1]_mn,
where C_kl^n_2n_1=E[⟨H̆_kl⟩_n_2⟨H̆_kl⟩_n_1^†] is the (n_2,n_1)-th sub-matrix of C_kl.
Based on (<ref>), the detected signals of UE k can be written as
x_k = U_k^†y
= U_k^†H_kP_kx_k+∑_i=1\ k^KU_k^†H_iP_ix_i + U_k^†n,
where U_k = ( HPP^† H^†+C^'+σ^2I_M_tot)^-1 H_kP_k. The channel coefficient H_k=[H_k1^†,…,H_kL^†]^†∈C^M_tot× N and the channel estimates H_k have the same structure as H_k.
Based on (<ref>), the achievable SE of UE k using per-user-basis MMSE-SIC detector <cit.> is given by
SE_k^FC=(1-τ_p/τ_c)E{log_2|I_N+SINR_k^FC|},
where
SINR_k^FC=P_k^† H_k^†U_kU_k^† H_kP_k/U_k^†(∑_i=1\ k^K H_iP_iP_i^† H_i^†+C^'+σ^2I_M)U_k.
The fully-centralized scheme with the global MMSE detector not only minimizes the MSE between the global estimated signals x and the original signals x, but also maximizes the SE <cit.>. However, the aggregation of instantaneous channel coefficients and raw signals at the CPU-UAV requires large transmission bandwidth, which incurs unpredictable latency. In order to overcome this drawback, we propose the distributed MMSE detection with one-shot combining scheme (MMSE-OneShot), which is a two-layer signal detection scheme, including the local MMSE detection in AP-UAVs and one-shot combining in the CPU-UAV.
§.§ Distributed MMSE Detection with One-Shot Combining
In this subsection, we will introduce the two-layer detection scheme. The main difference between the fully-centralized scheme and the MMSE-OneShot scheme is the interaction of information. In the fully-centralized scheme, the instantaneous channel coefficients and raw symbols require to be forwarded from AP-UAVs to the CPU-UAV in every coherence block. Nevertheless, in the MMSE-OneShot scheme, the CPU-UAV requires the local detected signals from each AP-UAV, as well as the long-term channel statistics between AP-UAVs and UEs, to obtain the jointly detected signals. Therefore, only the vector-sized signals need to be forwarded to the CPU-UAV in every coherence block, while the channel statistics, changing much slower than the instantaneous channel coefficients[The channel statistics are independent of frequency and about 40 times slower than the instantaneous channel <cit.>] only need to be forwarded in every several consecutive blocks. The details of the proposed MMSE-OneShot scheme are described as follows.
In the first layer of the MMSE-OneShot scheme, AP-UAVs individually recover the received signals with the local MMSE detector based on the local instantaneous channel coefficients. The detected signal at AP-UAV l is given by
x_l = U_l^†y_l,
where the local MMSE detector U_l∈C^M× N_tot minimizes the MSE between the original signals x and the detected signals x_l, which is given by
U_l = (H_lPP^† H_l^†+C_l^'+σ^2I_M)^-1H_lP,
where H_l=[H_1l,H_2l,…,H_Kl]∈C^M× N_tot is the aggregative channel estimates at AP-UAV l. Note that the detector U_l only depends on the local channel estimates, thus there is no inter-UAV interaction required in the first layer.
In the second layer, after detecting the received signal using the local detector {U_l}_1≤ l≤ L, AP-UAVs forward the detected signals { x_l}_1≤ l≤ L to the CPU-UAV. Then, the CPU-UAV collects and weighted combines the detected signals as follows
x_dist = ∑_l=1^Lω_l x_l,
where ω=[ω_1,…,ω_L]^T∈C^L is the assigned combining weights and x_dist∈C^N_tot is the final refined signal in the CPU-UAV. The optimal combining weights ω can be obtained by minimizing the MSE between the original signal x and the refined signal x_dist, which is given as follows
M_dist = ωmin 𝔼‖x_dist - x ‖ ^2,
where the expectation is taken with respect to the unknown noise. The closed-form expressions of the optimal combining weights and the corresponding MSE are given in the following proposition.
The optimal combining weights and the corresponding MSE are given by
ω = (A+Y)^-1v,
M_dist = ‖ x ‖^2 - v^† ( A+ Y)^-1v,
where v∈C^L, A∈C^L× L and Y∈C^L× L, whose entries are given by
[v]_l =x^†Q_lx,
[A]_lm = x^†Q_lQ_mx,
[Y]_ll = Tr(U_l^†(C_l^'+σ^2I_M) U_l),
and
Q_l= P^†H_l^†(H_lPP^† H_l^†+C_l^'+σ^2I_M)^-1H_lP,
The proof follows similar steps as the proof of <cit.>, and is therefore omitted.
As described in the above proposition, the calculation of combining weights requires the unknown original signal x, which is unrealistic in the implementation. In addition, the calculation of Q_l and U_l in the CPU-UAV still needs the instantaneous channel estimates, which are unavailable in the CPU-UAV and need to be transmitted from distributed AP-UAVs. Instead, when the numbers of receiving antennas per AP-UAV M and transmitting antennas per UE N grow to infinity with a fixed ratio, we can utilize the free probability theory to obtain the asymptotic expressions of the combining weights, which only depend on the long-term statistical CSI. In this way, we can still exploit the diversity gain due to multi-UAV joint detection, while expanding the range of wireless coverage. More practically, the amount of information interaction between the CPU-UAV and AP-UAVs is much reduced, compared to the fully centralized scheme.
We first approximate the expressions of v, A, and Y in (<ref>) in terms of the Cauchy transform using the large dimensional random matrix theory. Then, we derive the explicit expressions of the Cauchy transform to obtain the asymptotic expressions of the combining weights via operator-valued free probability theory, which will be introduced in Section <ref>.
According to the trace lemma <cit.>, the entries in v and A can be rewritten as
[v]_l =x^†Q_lx→_a.s.Tr(Q_l),
[A]_lm = x^†Q_lQ_mx→_a.s.Tr(Q_lQ_m),
where a.s. denotes almost surely, which represents that the left-hand-side (LHS) will converge to right-hand-side (RHS) almost surely as N_tot→∞.
Based on the matrix inversion lemma <cit.>, the matrix Q_l can be rewritten as
Q_l = I_N_tot - (P^†H_l^†(C_l^'+σ^2I_M)^-1H_lP+I_N_tot)^-1.
In addition, we define the Cauchy transform of B_l=P^†H_l^†(C_l^'+σ^2I_M)^-1H_lP∈C^N_tot× N_tot as
G_B_l(z) = 1/N_totTr∘E{(zI_N_tot - B_l)^-1}.
Then, we can rewrite the expressions of v and A in terms of G_B_l(z). The l-th element in v can be then expressed as
v_l= N_tot+N_totG_B_l(z),
Similarly, the (l,m)-th entry in A is given by
[A]_lm =
N_tot+N_totG_B_l(z)+N_totG_B_m(z)+N_tot^2Tr(B_lB_m),
l ≠ m,
N_tot+2N_totG_B_l(z)+ N_totG^'_B_l(z),
l = m,
where z=-1 in the above two formulas and G^'_B_l(z)=1/N_totE{Tr(zI_N_tot-B_l)^-2} is the derivative of the Cauchy transform G_B_l(z). The last item of [A]_lm,l≠ m can be obtained by deriving the approximation of B_l,l=1,…,L, which will be detailed in Section <ref>.
Besides, the l-th diagonal element in Y can be also denoted as
[Y]_ll =Tr(( H_lPP^† H_l^†+S_l)^-1S_l)
-Tr(( H_lPP^† H_l^†+S_l)^-2S_lS_l^†),
where S_l = C_l^'+σ^2I_M.
Define the Cauchy transform of B_l= H_lPP^† H_l^†+C_l^'∈C^M× M as
G_B_l,Ξ=1/MTr∘E{(zI_M- B_l)^-1Ξ},
where Ξ∈C^M× M is a nonnegative definite matrix with the uniformly bounded spectral norm. Then, the l-th diagonal element in Y can be rewritten as
[Y]_ll=-MG_B_l,S_l(z)-MG^'_B_l, S_lS_l^†(z),
where z=-σ^2 and G^'_B_l, S_lS_l^†(z)=1/ME{Tr((zI_M- B_l)^-2S_lS_l^†)} is the derivative of the Cauchy transform G_B_l, S_lS_l^†(z).
Therefore, the approximation of the combining weights ω is amount to finding the Cauchy transform G_B_l(z) and G_ B_l(z).
Based on the proposed MMSE-OneShot scheme, the detected signals of UE k in (<ref>) can be rewritten as
x_k= ∑_l=1^Lω_l x_kl
= ∑_l=1^Lω_l(U_kl^†H_klP_k x_k + ∑_i=1\ k^KU_kl^†H_ilP_i x_i + U_kl^†n_l),
where U_kl = (H_lPP^† H_l^†+C_l^'+σ^2I_M)^-1H_klP_k ∈C^M× N. Since the CPU-UAV is only aware of the statistical CSI, the achievable SE using MMSE-SIC detector is given in the following corollary <cit.>.
The achievable SE of UE k with MMSE-SIC detector is given by
SE_k = (1-τ_p/τ_c)log_2|I_N+S_k^†Γ_k^-1S_k|,
where S_k=∑_l=1^Lω_lE(U^†_klH_kl)P_k and Γ_k=∑_i=1^K∑_l=1^L∑_m=1^Lω_lE(U_kl^†H_ilP_iP_i^†H_im^†U_km)ω_m^⋆-S_kS_k^†+σ^2∑_l=1^Lω_lE(U_kl^†U_kl)ω_l^⋆.
The proof follows similar steps as the proof of <cit.>, and is therefore omitted.
In this section, we introduce the fully-centralized scheme as a benchmark and propose the MMSE-OneShot scheme with the optimal closed-form expressions of the combining weights. To obtain the theoretical expressions, we then represent the optimal one-shot combining weights in terms of Cauchy transforms G_ B_l(z) and G_ B_l(z) using the large dimensional random matrix theory, whose explicit expressions will be detailed in the following section.
§ ASYMPTOTIC ONE-SHOT COMBINING WEIGHTS VIA OPERATOR-VALUED FREE PROBABILITY THEORY
In this section, we will resort to the linearization trick and the operator-valued free probability theory to obtain the explicit expressions of G_ B_l(z) and G_ B_l(z). In the following discussion, we will focus on the l-th AP-UAV and thus omit the subscript l of B_l and B_l for brevity.
In the considered formulations, where B=P^†H_l^†(C_l^'+σ^2I_M)^-1H_lP and B = H_lPP^† H_l^†+C_l^', the equivalent channels in B and B (which will be defined in the following subsections) are non-central and with non-trivial spatial correlations, and thus not free over the complex algebra <cit.>.
To address this issue, the Anderson’s linearization trick is adopted <cit.>. The linearization trick transforms a complicated distribution of an arbitrary self-adjoint polynomial, such as p∈C(X_1,…,X_n)∈A with A being a complex and unital algebra, into the operator-valued distribution of a linear polynomial L_p=b_0⊗ 1+b_1⊗X_1+…+b_n⊗X_n, with b_0,…,b_n∈M_N(C), where M_N(C) is the algebra of complex N× N matrices. For any complex number z∈C, the following conditions are equivalent
* The operator zI_N-P with P:=p(X_1,…,X_n) is invertible in A.
* The operator Λ(z)-L_P is invertible in M_N(C)⊗A, where
Λ(z)=
[[ zI_N 0; 0 0; ]].
Then, for some z∈C that fulfills the above conditions, we have <cit.>
{(Λ(z)-L_P)^-1}^(1,1)=(z-P)^-1.
where {·}^(1,1) denotes the upper-left N× N matrix block.
Therefore, by using the linearization trick, we can first rewrite G_ B(z) and G_ B(z) as their operator-valued Cauchy transform counterparts, whose expressions can be derived by the operator-valued free probability theory <cit.>. Then we can obtain the expressions of Cauchy transforms G_ B(z) and G_ B(z) based on (<ref>).
§.§ Cauchy Transform G_ B(z)
Recall the Cauchy transform G_B(z)=1/N_totE{Tr(zI_N_tot-B)^-1} with B=P^†H_l^†(C_l^'+σ^2I_M)^-1H_lP, we define the equivalent channel as G=( C_l^'+σ^2I_M)^-1/2 H_lP∈C^M× N_tot and then we have B=G^†G.
By using the Anderson’s linearization trick, we can construct the block matrix L_B∈M_n, where M_n = M_n(C) denotes the algebra of n× n complex random matrices with n = N_tot+M, as follows
L_B=
[ 0_N_tot G^†; G -I_M ],
We define the sub-algebra D_n⊂M_n as the n× n block diagonal matrix. For each K_n∈D_n, we have
K_n=
[ D_1 0_N_tot× M; 0_M × N_tot D_2 ],
where the N_tot× N_tot block diagonal matrix D_1=blkdiag(D_11,…, D_1K) with {D_1k}_1≤ k≤ K being N× N sub-matrices and D_2 is a M× M sub-matrix.
The D_n-valued Cauchy transform is then defined as
G_L_B^D_n(Λ_n(z)) = E_D_n[(Λ_n(z)-L_B)^-1],
where the expectation E_D_n[X] has the same structure as K_n and is a linear function of X∈M_n, which is defined as
E_D_n[X] =
[ E[X_D_1] 0_N_tot× M; 0_M × N_tot E[X_D_2] ],
and the expectation E[X_D_1]=blkdiag(E[X_D_11],…, E[X_D_1K]). In addition, the n× n diagonal matrix Λ_n(z) in (<ref>) is given by
Λ_n(z) = [ zI_N_tot 0_N_tot× M; 0_M× N_tot 0_M ].
Based on the block matrix inversion lemma, we have
G_L_B^D_n(Λ_n(z)) = E_D_n[ (zI_N_tot-G^†G)^-1 G^†(zI_M-GG^†)^-1; (zI_M-GG^†)^-1G z(zI_M-GG^†)^-1 ] ,
Obviously, the upper-left N_tot× N_tot matrix block of G_L_B^D_n(Λ_n(z)) is equivalent to (zI_N_tot - B)^-1, thus the Cauchy transform of B is amount to
G_B(z) = 1/N_totTr({G_L_B^D_n(Λ_n(z))}^(1,1)),
where {·}^(1,1) denotes the upper-left N_tot× N_tot matrix block. Henceforth, the Cauchy transform G_B can be obtained by the operator-valued Cauchy transform. Then, in order to obtain the expression of G_L_B^D_n(Λ_n(z)), we first separate L_B into L_B and L_B, which are defined as
L_B =
[ 0_N_tot G^†; G -I_M ], L_B =
[ 0_N_tot G^†; G 0_M ],
where L_B =L_B+ L_B, G =( C_l^'+σ^2I_M)^-1/2 H_lP, and G =( C_l^'+σ^2I_M)^-1/2 H_lP.
The random variable L_B is semicircular and free from L_B over D_n.
The proof of Proposition <ref> is given in Appendix <ref>.
Based on Proposition <ref>, the limiting spectral distribution of L_B can be determined by the operator-valued free additive convolution of L_B and L_B. Thus, the operator-valued Cauchy transform G_L_B^D_n(Λ_n(z)) can be obtained by using the subordination formula <cit.>, which is given as follows
G_L_B^D_n(Λ_n(z)) = G_L_B^D(Λ_n(z) - R_L_B^D_n(G_L_B^D_n(Λ_n(z)))),
= E_D_n[(Λ_n(z) - R_L_B^D_n(G_L_B^D_n(Λ_n(z))) - L_B)^-1],
where R_L_B^D_n(·) denotes the D_n-valued R-transform of L_B. Therefore, based on the identity (<ref>), the expression of the Cauchy transform G_B(z) can be obtain through the expression of G_L_B^D_n(Λ_n(z)), which is given by the following proposition.
The expression of Cauchy transform G_B(z) is given by
G_B(z) = 1/N_totTr(G_D_1(z)),
where G_D_1(z) satisfies the following matrix-valued fix-point equations
G_D_1(z) = (Ψ(z)- G^†Ψ^-1(z) G)^-1,
G_D_2(z) = (Ψ(z)- GΨ^-1(z) G^†)^-1,
The matrix-valued function Ψ(z) and Ψ(z) are respectively denoted as
Ψ(z) =zI_N_tot-δ(G_D_2(z)),
Ψ(z) =I_M-∑_i=1^Kδ_i(G_D_1i(z)).
The functions of one-sided correlation matrices are given by
δ_i(G_D_1i(z)) =η_il(( C_l^'+σ^2I_M)^-1/2, G_D_1i(z),P_i),
δ(G_D_2(z)) =blkdiag(η_kl(( C_l^'+σ^2I_M)^-1/2, G_D_2(z), P_k))_1≤ k ≤ K.
where {G_D_1i(z)}_1≤ i ≤ K is the i-th diagonal matrix block of G_D_1(z) of dimension N× N.
The proof of Proposition <ref> is given in Appendix <ref>.
Therefore, the Cauchy transform G_B(z) can be obtained by (<ref>) and (<ref>). In addition, according to <cit.>, the entries of the matrix (zI_N_tot-B_l)^-1 can be approximated by the entries of G_D_1(z) in (<ref>) and the expression of the matrices B_l,l=1,…,L in (<ref>) can be then obtained.
§.§ Cauchy transform G_ B,Ξ(z)
Recall that G_B,Ξ =1/ME{Tr(zI_M- B)^-1Ξ} with B= H_lPP^† H_l^†+C_l^', we adopt the similar procedure as the previous subsection. We will first resort to the linearization trick and then the operator-valued free additive convolution, which will be detailed as follows.
Based on the Anderson’s linearization trick, we can construct the block matrix L_ B∈M_ n, where M_ n denotes the algebra of n× n complex random matrices and n = N_tot+2 × M, as follows
L_ B =
[[ L_ B^(1,1) L_ B^(1,2); L_ B^(2,1) L_ B^(2,2); ]]
=
[[ 0_M (C_l^')^1/2 H_lP; (C_l^')^1/2 -I_M 0_ M× N_tot; P^†H^†_l 0_N_tot× M -I_N_tot ]],
where the matrix blocks L_ B^(i,j) corresponds to the partitions shown on the RHS of (<ref>).
In addition, we define the sub-algebra D_ n⊂M_ n as the n× n block diagonal matrix. For each K_ n∈D_ n, it is defined as
K_ n=
[ D 0_M 0_M × N_tot; 0_M 0_M 0_M× N_tot; 0_N_tot× M 0_N_tot× M D ],
where D is a M× M sub-matrix and the N_tot× N_tot block diagonal matrix D is defined as D=blkdiag(D_1,…,D_K) with {D_i}_1≤ i≤ K being N× N sub-matrices.
The D_ n-valued Cauchy transform is defined as
G_L_B^D_ n(Λ_ n(z))= E_D_ n[(Λ_ n(z) - L_B)^-1],
where E_D_ n[X] has the same structure as K_ n with X∈M_ n, which is defined as
E_D_ n =
[ E[ D] 0_M 0_M × N_tot; 0_M 0_M 0_M× N_tot; 0_N_tot× M 0_N_tot× M E[D] ],
where E[D] = blkdiag(E[D_1],…,E[D_K]). In addition, Λ_ n(z) denotes the n× n matrix as
Λ(z) = [ zI_M 0_M 0_M × N_tot; 0_M 0_M 0_M× N_tot; 0_N_tot× M 0_N_tot× M 0_N_tot ].
By using the block matrix inversion lemma, we can obtain
G_L_ B^D_ n(Λ_ n(z)) = E_D_ n[ (z I_M + L_ B^(1,2)(L_ B^(2,2))^-1L_ B^(2,1))^-1 L_ B^(1,2)(-zL_ B^(2,2) - L_ B^(2,1)L_ B^(1,2))^-1; (zL_ B^(2,2) - L_ B^(2,1)L_ B^(1,2))^-1L_ B^(2,1) -(L_ B^(2,2) + z^-1L_ B^(2,1)L_ B^(1,2))^-1 ].
In particular, the upper-left block of (<ref>) can be explicitly written as
(z I_M + L_ B^(1,2)(L_ B^(2,2))^-1L_ B^(2,1))^-1 = (z I_M - H_lPP^† H_l^†-C_l^')^-1.
Thus, the Cauchy transform of B is given by
G_B,Ξ =1/Mtr({G_L_ B^D_ n(Λ_ n(z))}^(1,1)Ξ),
where {·}^(1,1) denotes the upper-left M× M matrix block.
Following the similar procedure, we separate L_ B into L_ B and L_ B, which are defined as
L =
[[ 0_M (C_l^')^1/2 H_lP; (C_l^')^1/2 -I_M 0_M× N_tot; P^†H_l^† 0_N_tot× M -I_N_tot ]] , L =
[[ 0_M 0_M H_lP; 0_M 0_M 0_M× N_tot; P^†H_l^† 0_N_tot× M 0_N_tot ]],
where L_ B= L_ B+ L_ B.
The random variable L_ B is semicircular and free from L_ B over D_ n.
The proof of Proposition <ref> is given in Appendix <ref>.
Based on Proposition <ref>, the limiting spectral distribution of L_ B can be determined by the operator-valued free additive convolution of L_ B and L_ B. Therefore, the operator-valued Cauchy transform G_L_ B^D_ n(Λ_ n(z)) can be obtained by using the subordination formula, which is given as follows
G_L_ B^D_ n(Λ_ n(z))= E_D_ n[(Λ_ n(z) - R_L_ B^D_ n(G_L_ B^D_ n(Λ_ n(z))) - L_ B)^-1],
where R_L_ B^D_ n(·) denotes the D_ n-valued R-transform of L_ B. Then, the expression of the Cauchy transform G_B,Ξ can be obtain by the following proposition.
The expression of Cauchy transform G_ B,Ξ(z) is given by
G_B,Ξ =1/Mtr(G_ D(z)Ξ),
where
G_ D(z) = ( Φ(z)- C_l^' - H_lPΦ^-1(z)P^†H_l^†)^-1,
G_D(z) =( Φ(z)-P^†H_l^†Φ^-1(z)H_lP-Υ(z))^-1,
The matrix-valued function Φ(z), Φ(z) and Υ(z) are respectively denoted as
Φ(z) =zI_M-∑_i=1^Kζ_i(G_D_i(z)),
Φ(z) =I_N_tot-ζ(G_ D(z)),
Υ(z) = P^†H_l^†Φ^-1(z)(C_l^')^1/2( I_M-(C_l^')^1/2Φ^-1(z)(C_l^')^1/2)^-1(C_l^')^1/2Φ^-1(z)H_lP,
where the functions of one-sided correlation matrices are given by
ζ(G_ D(z)) =blkdiag(η_1l(I_M, G_ D(z), P_1),…, η_Kl(I_M,G_ D(z),P_K)),
ζ_i(G_D_i(z)) =η_il(I_M,G_D_i(z),P_i),
and {G_D_i(z)}_1≤ i ≤ K is the i-th diagonal matrix block of G_D(z) of dimension N× N.
The proof of Proposition <ref> is given in Appendix <ref>.
Based on Proposition <ref> and Proposition <ref>, the derivatives of Cauchy transform G^'_B_l(z) and G^'_B_l, S_lS_l^†(z) can be obtained by matrix calculus, which is given in Appendix <ref>. In this way, the explicit expressions of v, A and Y can be obtained and the CPU-UAV can calculate the combining weights with the channel statistics, which mitigates the interaction of information.
§ NUMERICAL RESULTS
In the above sections, we proposed the MMSE-OneShot scheme and utilized the operator-valued Cauchy transforms to obtain the asymptotic expressions of the combining weights. In this section, we numerically evaluate the performance of the proposed MMSE-OneShot scheme, where both the explicit and the asymptotic combining weights are calculated and compared. In addition, we investigate the SE of the aerial cell-free systems with various numbers of deployed AP-UAVs and numbers of equipped antennas. The fully-centralized scheme in <ref> and the small-cell network are simulated as the benchmarks. In small-cell networks, each UE selects the nearest AP-UAV to transmit its signals, and the detection is only implemented in the selected AP-UAV <cit.>.
§.§ Simulation Setup
We assume that the AP-UAVs are regularly deployed in a square area of 1× 1 km^2, and UEs are randomly distributed within this area. The transmitting power of each UE is set as p_k=23dBm and the noise power in each AP-UAV is σ^2=-94dBm. The precoder of each UE is assumed to be a unit matrix, which means that the transmitting power is divided equally between each antenna, i.e. F_k=P_k=√(p_k/N)I_N. The AP-UAVs are assumed to hover at the altitude of h=100m, unless otherwise stated, and the height of UE is set as 1.5m. We consider that each AP-UAV has the same number of antennas, and each coherence block contains τ_c = 200 channel uses and τ_p = KN, unless otherwise stated.
We consider that the air-to-ground channels consist of LoS and NLoS links with different probability <cit.>. In specific, the probability of having an LoS connection between the UE k and AP-UAV l is given by
P^LoS_kl=1/1+aexp(-b(180/πtan^-1(h/d_kl)-a)),
where a and b are constants depending on the environments (suburban, urban, and dense urban) <cit.>, and d_kl denotes the distance between AP-UAV l and UE k. The probability of NLoS is then denoted by P^NLoS_kl=1-P^LoS_kl.
The path loss model for LoS and NLoS links in dB are respectively
PL_kl^LoS = FSPL_kl+ ϱ_LoS,
PL_kl^NLoS = FSPL_kl+ ϱ_NLoS,
where FSPL_kl=10log(β_kl) is the free space path loss between AP-UAV l and UE k with β_kl=ρ/d_kl^α, where ρ=-55dB is the path loss constant, and α=3 is the path loss exponent. The excessive path loss ϱ_LoS and ϱ_NLoS depend on the propagation group (LoS and NLoS) and are given in <cit.>. Therefore, the averaged path loss between AP-UAV l and UE k is
PL_kl = P^LoS_kl×PL_kl^LoS + P^NLoS_kl×PL_kl^NLoS.
In addition, according to <cit.>, the spatial correlation matrices R_kl and T_kl are generated from a uniform linear array with half wavelength spacing, whose (m,n)-th element are given as follows
[T (or 𝐑)_kl]_mn=∫_-180^180dϕ/√(2πξ_kl^2)e^jπ(m-n)sin(πϕ/180)-(ϕ-θ_kl)^2/2ξ_kl^2,
where θ_kl and ξ_kl are the mean angle and angular standard deviation (ASD), respectively. The LoS component is generated according to H_kl=a_R,l(θ^R_kl)a_T,k(θ^T_kl)^†, where
a_R,l(θ^R_kl)
=[ 1 e^jπsin(θ_kl^R/180π) … e^jπ(M-1)sin(θ_kl^R/180π) ]^ T,
a_T,k(θ^T_kl)
=[ 1 e^jπsin(θ_kl^T/180π) … e^jπ(N-1)sin(θ_kl^T/180π) ]^ T,
where θ^R_kl and θ^T_kl denote the angles of LoS path at AP-UAV l and UE k, respectively.
§.§ Accuracy Analysis
We first investigate the accuracy of our derived asymptotic expression of the combining weights with L=8 and K=4, as illustrated in Fig. <ref>. The Monte Carlo simulation curves are obtained by averaging the sum SE over 10^5 channel realizations and plotted as markers. As observed in Fig. <ref>, the sum SE monotonously increases with the number of antennas per UE and AP-UAV. The theoretical result fits the Monte Carlo simulation curve accurately, even when UEs and AP-UAVs are equipped with a small number of antennas.
§.§ Effects of Distributed Deployment
In practice, the distributed deployment of AP-UAVs, including the location, the number of AP-UAVs, and the number of antennas per AP-UAV, etc, has a significant influence on the system performance. In order to investigate how to determine the number of deployed AP-UAVs and the number of antennas per AP-UAV, we simulate the performance of the proposed MMSE-OneShot scheme in distributed systems under a given total number of receiving antennas. In Fig. <ref>, we plot the sum SE as a function of the ratio of the number of AP-UAVs to the number of antennas per AP-UAV for a fixed total number of receiving antennas M_tot=144. Note that only one AP-UAV is deployed when the ratio L/M=1/144 and the number of AP-UAVs gets larger as the ratio gets larger, while the number of antennas per AP-UAV gets smaller. We notice that the sum SE can reach a maximum value when L/M=1 in the suburban environment. Whereas in urban and dense urban environments, the sum SE continuously increases with the number of AP-UAVs getting larger as long as the number of antennas per AP-UAV is greater than one. This is due to the fact that there is a tradeoff between the LoS probability and spatial diversity. In specific, the LoS probability can be increased by deploying more distributed AP-UAVs, which also achieves macro-diversity. Nevertheless, deploying more antennas in every AP-UAV can take full advantage of the spatial diversity. In addition, the LoS probability in the suburban environment is greater than that in urban and dense urban environments <cit.>. Therefore, it is more effective to increase the LoS probability (i.e., increase the number of AP-UAVs) in urban and dense urban environments, while system performance can be improved by increasing the spatial diversity (i.e., increasing the number of antennas per AP-UAV) in the suburban environment.
The deployment strategy is further illustrated in Fig. <ref>, where the total number of receiving antennas is fixed as M_tot=180. In the suburban environment, the sum SE significantly decreases when the number of antennas per AP-UAV is smaller than 9 (when L>20). However, in urban and dense urban environments, the sum SE continuously increases until L=180, i.e., every AP-UAV is equipped with a single antenna. This phenomenon also indicates the tradeoff between the LoS probability and spatial diversity and that we should allocate the number of AP-UAVs and the number of receiving antennas per AP-UAV according to the physical propagation environment.
§.§ Effects of the Height of AP-UAV in different environments
As shown in Fig. <ref> and Fig. <ref>, the performance achieved by deploying more AP-UAVs is always better than that by deploying only one AP-UAV with multiple antennas, and it's necessary to establish a distributed system by deploying multiple AP-UAVs. Thus, we further discuss the different types of distributed systems in the following.
The effect of the height of AP-UAV in different urban environments is investigated in Fig. <ref> with L=8, K=4, M=8, and N=4. The fully-centralized scheme achieves the highest sum SE in all considered environments, while the small-cell scheme gets the worst performance. The proposed MMSE-OneShot scheme can obtain similar performance as the fully-centralized scheme, especially in urban and dense urban environments. In addition, we notice that as the height of AP-UAVs gets higher, the sum SE initially increases to reach a maximum value and then starts to decrease in urban and dense urban environments. The reason for this phenomenon is that, as the height is relatively small and becomes larger, the gain from increasing LoS probability outweighs the deterioration of the path loss caused by increasing distance. When the height exceeds 100 meters, the gain obtained from the LoS probability becomes saturated and the large path loss degrades the system performance.
§.§ Effects of the Number of UEs
In Fig. <ref>, we show the sum SE as a function of the number of UEs K with L=8, M=16, and N=4. Note that the sum SE can reach a maximum value in a specific K in any case. This phenomenon is due to the pre-log factor 1-τ_p / τ_c and the inter-UE interference. With the increase of K, the pre-log factor in SE expressions decreases, which means that most channel uses of coherence block are used for channel estimation and the ratio of signal transmission is reduced. In addition, the inter-UE interference becomes large as the number of UEs increases, which is eventually harmful to the sum SE. Therefore, as shown in Fig. <ref>, the optimal sum SE can be obtained by setting the active number of UEs as 20 to 25 in the considered system settings.
§.§ Effects of the Length of Pilots
To investigate the effects of the length of pilots of the proposed MMSE-OneShot scheme, we set different pilot-reuse factor ϑ in Fig. <ref>, where the number of pilot τ_p = KN/ϑ. The pilot reuse factor ϑ means that every ϑ UEs are allocated to the same pilot matrix and ϑ>1 leads to the pilot contamination. The other parameters are set as K=10, L=12, and M=16. We observe that there is a maximum sum SE when ϑ=1 and ϑ=2 at N=3 and N=5, respectively. When the pilot-reuse factor is small (higher τ_p), increasing the number of transmitting antennas requires a large number of pilots, which decreases the signal transmission efficiency and sum SE. On the other hand, when the pilot-reuse factor is large (lower τ_p), a smaller number of pilots are needed and the pilot contamination gets worse. However, the gain from the increasing of transmitting antennas exceeds the loss of pilot contamination and thus the system performance monotonically increases as the transmitting antennas get large. Therefore, there is a trade-off between the pilot-reuse factor and the number of transmitting antennas and it is necessary to set the pilot-reuse factor according to the number of antennas per UE.
§ CONCLUSION
Limited by the lightweight fuselage, several AP-UAVs with few antennas compose the CF mMIMO system, where all AP-UAVs forward their channel coefficients to the CPU-UAV. However, due to the limited capacity of the wireless fronthaul link, the huge interaction overhead prevents the aerial CF mMIMO system from being deployed in practice.
To address the above challenges, we propose a two-layer distributed uplink signal detection scheme for the CF mMIMO system. Specifically, the uplink signals are first recovered in AP-UAVs with local MMSE detector and then forwarded to the CPU-UAV, which weighted combines the signals to obtain the refined signals. By using the operator-valued free probability theory, the asymptotic expressions of the combining weights are obtained explicitly for the large dimensional antenna configurations, while the results are also verified to be accurate in realistic system configurations. Numerical results show that the proposed MMSE-OneShot scheme not only achieves similar system performance as the fully-centralized scheme in various environments, but also significantly alleviates the interactive overhead between the AP-UAVs and the CPU-UAV. In addition, we investigate the impact of the number of deployed AP-UAVs as well as the number of receive antennas per AP-UAV on the achieved sum SE. In the suburban environment, where the LoS probability is relatively large, the number of antennas per AP-UAV should be increased to achieve spatial diversity. While in urban and dense urban environments, more AP-UAVs should be deployed to increase the LoS probability.
§ PROOF OF PROPOSITION <REF>
According to <cit.>, a random variable L∈M is said to be D-valued semicircular if the free cumulant
κ_m^D(L b_1, L b_2, …, L b_m-1, L) = 0,
for all m≠ 2, and all b_1,…,b_m-1∈D. The free cumulant κ_m^D is a mapping from M_m to D.
We first expend L_B into a summation of n× n sub-matrices, such that
L_B =∑_k=1^K L_B,k,
where the sub-matrices L_B,k are given by
L_B,k =
[ 0_N_tot Σ_k^†; Σ_k 0_M ] ,
and Σ_k = [0_M× N, …, ( C_l^'+σ^2I_M)^-1/2 H_klP_k,…,0_M× N]∈C^M× N_tot.
Recall that C_kl is the correlation matrix of h_kl and h_kl=vec( H_kl), thus we have
H_kl = (Č_kl)^1/2X_kl,
where Č_kl =1/N∑_i=1^NC_kl^ii∈C^M× M with C_kl^ii being the i-th diagonal sub-matrix of C_kl, and the random matrix X_kl are i.i.d. complex Gaussian distributed with entries having zero mean and unit variance.
Therefore, the sub-matrices L_B,k can be rewritten as
L_B,k = A_kX_kA_k^†,
where X_k has the same structure as the block matrix L_B,k in (<ref>), while replacing the matrix ( C_l^'+σ^2I_M)^-1/2 H_klP_k with X_kl. The block matrix A_k is given by
A_k =
[ P_k^† 0_N_tot× M; 0_M × N_tot ( C_l^'+σ^2I_M)^-1/2Č_kl^1/2; ],
where P_k = blkdiag(0_N,…,P_k,…, 0_N) with P_k in its k-th diagonal block.
Since {X_k}_1≤ k≤ K are Wigner matrices and independent from each other, they are semicircular and free over the sub-algebra D_n. Following the same arguments as in <cit.>, { L_B,k}_1≤ k≤ K are semicircular and free over the subalgebra D_n. Therefore, the summation of { L_B,k} is also semicircular over the subalgebra D_n and is free from any deterministic matrix from M_n.
§ PROOF OF PROPOSITION <REF>
As previously stated, the limiting spectral distribution of L_B is the free convolution of L_B and L_B. The D_n-valued Cauchy transform G_L_B^D_n(Λ_n(z)) can be obtained by using the subordination formula (<ref>). Recall that the R-transform R_L_B^D_n(·) is the free cumulant generating function of L_B with the following formal power series expansion:
R_L_B^D_n(K_n) = κ_1^D_n(K_n) + κ_2^D_n(L_BK_n,L_B) + κ_3^D_n(L_BK_n,L_BK_n,L_B) + ⋯,
where κ_i^D_n denotes the i-th free cumulant of L_B over D_n. In addition, since L_n is semicircular over D_n, all its cumulants in (<ref>) except κ_2^D_n are zero.
Therefore, the R-transform reduces to the covariance function of L_B over D_n parameterized by K_n, i.e.,
R_L_B^D_n(K_n) = E_D_n[L_BK_nL_B],
=
[ δ(D_2) 0; 0 ∑_i=1^Kδ_i(D_1i) ],
where δ(D_2)=blkdiag(η_1l(( C_l^'+σ^2I_M)^-1/2, D_2, P_1),…,η_Kl(( C_l^'+σ^2I_M)^-1/2,D_2,P_K)) and δ_i(D_1i)=η_il(( C_l^'+σ^2I_M)^-1/2,D_1i,P_i).
We divide G_L_B^D_n(Λ_n(z))∈D_n into the same partition as (<ref>), G_L_B^D_n(Λ_n(z)) can be rewritten as
G_L_B^D_n(Λ_n(z)) =
[ G_D_1(z) 0; 0 G_D_2(z) ],
where G_D_1(z)=blkdiag(G_D_11(z), …, G_D_1K(z))∈C^N_tot× N_tot with G_D_1k(z) being a N× N sub-matrix and G_D_2(z) is a M× M matrix. By replacing K_n in (<ref>) with G_L_B^D_n(Λ_n(z)) and substituting R_L_B^D_n(G_L_B^D_n(Λ_n(z))) and L_B in the subordination formula (<ref>), we have
G_L_B^D_n(Λ_n(z)) = E_D_n[ (Ψ(z)- G^†Ψ^-1(z) G)^-1 Ψ^-1(z) G^†(Ψ- GΨ^-1(z) G^†)^-1; (Ψ- GΨ^-1(z) G^†)^-1 GΨ^-1(z) (Ψ(z)- GΨ^-1(z) G^†)^-1; ],
where Ψ(z)=zI_N_tot-δ(G_D_2(z)) and Ψ=I_M-∑_i=1^Kδ_i(G_D_1i(z)). Thus, based on the partition (<ref>), the expressions of G_D_1(z) and G_D_2(z) can be then obtained.
§ PROOF OF PROPOSITION <REF>
Following the similar procedure as in Appendix <ref>, we expend L_ B into a summation of n× n sub-matrices, such that
L_ B = ∑_k=1^K L_ B,k,
where the sub-matrices L_ B,k are given by
L_ B,k =
[[ 0_M 0_M Σ_k; 0_M 0_M 0_M× N_tot; Σ_k^† 0_N_tot× M 0_N_tot ]],
and the embedded matrix Σ_k=[0_M× N,…, H_klP_k,…, 0_M× N]∈C^M× N_tot.
We rewrite L_ B,k as follows
L_ B,k = X_k A_k X_k^†,
where X_k has the same structure as L_ B,k in (<ref>), while replacing the matrix H_klP_k with X_kl. In addition, the block matrix A_k is expressed as
A_k =
[ Č_kl 0_M 0_M× N_tot; 0_M 0_M 0_M× N_tot; 0_N_tot× M 0_N_tot× M P_k; ]
where P_k = blkdiag(0_N,…,P_k,…, 0_N).
Therefore, following the same arguments as in Appendix <ref>, the random variable L_ B is semicircular and free from L_ B over D_ n.
§ PROOF OF PROPOSITION <REF>
Based on Proposition <ref>, the random variable L_ B is semicircular and free from L_ B over D_ n. Thus, according to Appendix <ref>, the R-transform of L_ B reduces to the covariance function of L_ B over D parameterized by K_ n, i.e.,
R_L_ B^ D(K_ n) = E_D_ n[L_ BK_ nL_ B]
=
[ ∑_i=1^Kζ_i(D_i) 0_M 0_M× N_tot ; 0_M 0_M 0_M× N_tot ; 0_N_tot× M 0_N_tot× M ζ( D); ],
where ζ( D)=blkdiag(η_1l(I_M, D, P_1),…, η_Kl(I_M, D,P_K)) and ζ_i(D_i)=η_il(I_M,D_i,P_i).
By the same matrix partitioning as in (<ref>), G_L^D(Λ(z)) is partitioned into
G_L^D(Λ(z))=
[ G_ D(z) 0_M 0_M× N_tot; 0_M 0_M 0_M× N_tot ; 0_N_tot× M 0_N_tot× M G_D(z) ; ],
where G_ D(z) is a M× M matrix and G_D(z) is a N_tot× N_tot diagonal block matrix with {G_D_i(z)}_1≤ i ≤ K on the diagonal.
By replacing K in (<ref>) with G_L^D(Λ(z)) in (<ref>) and according to the subordination formula (<ref>), we have
G_L^D(Λ(z)) =E_D[[ Φ(z) -(C_l^')^1/2 -H_lP ; -(C_l^')^1/2 I_M 0 ; -P^†H_l^† 0 Φ(z) ]]^-1,
where Φ(z)=zI_M-∑_i=1^Kζ_i(G_D_i(z)) and Φ(z)=I_N_tot-ζ(G_ D(z)).
By applying the block matrix inversion lemma to (<ref>), we can obtain the expressions of G_ D(z) and G_D(z) as in (<ref>) and (<ref>).
§ DERIVATIVE OF CAUCHY TRANSFORM
Recall that the Cauchy transform G^'_B_l(z)=1/N_tE{Tr(zI_N_t-B_l)^-2} in (<ref>) denotes the derivative of G_B_l(z) and according to Proposition <ref>, we have
G^'_B_l(z) = 1/N_tTr(G^'_D_1(z)),
where G^'_D_1(z) is the derivative of G_D_1(z), which satisfies the following matrix-valued fix-point equations
G^'_D_1(z) = G_D_1(z)[G^-1_D_1(z)]^'G_D_1(z),
G^'_D_2(z) = G_D_2(z)[G^-1_D_2(z)]^'G_D_2(z),
where [G^-1_D_1(z)]^' and [G^-1_D_2(z)]^' are respectively the derivatives of G^-1_D_1(z) and G^-1_D_2(z). And some additional parameters are given as follows
[G^-1_D_1(z)]^' = Ψ^'(z)-G_l^†[Ψ^-1(z)]^'G_l,
[G^-1_D_2(z)]^' = Ψ^'(z)- G_l[Ψ^-1(z)]^' G_l^†,
[Ψ^-1(z)]^' = -Ψ^-1(z)(I_N_t - δ^'(G_D_2(z)))Ψ^-1(z),
[Ψ^-1(z)]^' = Ψ^-1(z)(∑_i=1^Kδ^'_i(G_D_1i(z)))Ψ^-1(z),
δ^'(G_D_2(z)) = δ(G^'_D_2(z)), δ^'_i(G_D_1i(z)) =
δ_i(G^'_D_1i(z)).
As for the Cauchy transform G^'_B_l, Ξ(z)=1/M_lE{Tr((zI_M_l- B_l)^-2Ξ)} in (<ref>), we have
G^'_B_l, Ξ(z) = 1/M_lTr(G^'_ D(z)Ξ),
where G^'_ D(z) is the derivative of G_ D(z), which satisfies the following matrix-valued fix-point equations
G^'_ D(z) = G_ D(z)[G^-1_ D(z)]^'G_ D(z),
G^'_D(z) = G_D(z) [G^-1_D(z)]^'G_D(z),
where [G^-1_ D(z)]^' and [G^-1_D(z)]^' are respectively the derivatives of G^-1_ D(z) and G^-1_D(z). And some additional parameters are given as follows
[G^-1_ D(z)]^' = Φ^'(z) - H_lP[Φ^-1(z)]^'P^†H_l^†,
[G^-1_D(z)]^' = Φ^'(z)-P^†H_l^†[Φ^-1(z)]^'H_lP-Υ^'(z),
[Φ^-1(z)]^' = -Φ^-1(z)(I_M_l-∑_i=1^Kζ^'_i(G_D_i(z)))Φ^-1(z),
[Φ^-1(z)]^' = Φ^-1(z)(ζ^'(G_ D(z)))Φ^-1(z),
ζ^'_i(G_D_i(z)) = ζ_i(G^'_D_i(z)), ζ^'(G_ D(z)) = ζ(G^'_ D(z)).
In addition, Υ^'(z) is the derivative of Υ(z), which can be easily obtained by the matrix calculus.
IEEEtran
00
mozaffari2021toward
M. Mozaffari, X. Lin, and S. Hayes, “Toward 6g with connected sky: Uavs and
beyond,” IEEE Commun. Mag., vol. 59, no. 12, pp. 74–80, 2021.
geraci2022will
G. Geraci, A. Garcia-Rodriguez, M. M. Azari, A. Lozano, M. Mezzavilla,
S. Chatzinotas, Y. Chen, S. Rangan, and M. Di Renzo, “What will the future
of uav cellular communications be? a flight from 5g to 6g,” IEEE
Commun. Surveys Tuts., vol. 24, no. 3, pp. 1304–1335, 2022.
ngo2017cell
H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta,
“Cell-free massive mimo versus small cells,” IEEE Trans. Wireless
Commun., vol. 16, no. 3, pp. 1834–1850, 2017.
bjornson2020scalable
E. Björnson and L. Sanguinetti, “Scalable cell-free massive mimo
systems,” IEEE Trans. Commun., vol. 68, no. 7, pp. 4247–4261, 2020.
nayebi2017precoding
E. Nayebi, A. Ashikhmin, T. L. Marzetta, H. Yang, and B. D. Rao, “Precoding
and power optimization in cell-free massive mimo systems,” IEEE Trans.
Wireless Commun., vol. 16, no. 7, pp. 4445–4459, 2017.
zheng2021uav
J. Zheng, J. Zhang, and B. Ai, “Uav communications with wpt-aided cell-free
massive mimo systems,” IEEE J. Sel. Areas Commun., vol. 39, no. 10,
pp. 3114–3128, 2021.
d2019cell
C. D'Andrea, A. Garcia-Rodriguez, G. Geraci, L. G. Giordano, and S. Buzzi,
“Cell-free massive mimo for uav communications,” in IEEE Int. Conf.
Commun. Workshops (ICC Workshops), May 2019, pp. 1–6.
diaz2022cell
C. Diaz-Vilor, A. Lozano, and H. Jafarkhani, “Cell-free uav networks:
Asymptotic analysis and deployment optimization,” IEEE Trans. Wireless
Commun., 2022.
wang2022cell
L. Wang and Q. Zhang, “Cell-free massive mimo with uav access points: Uav
location optimization,” in Proc. 2022 IEEE/CIC Int. Conf. Commun.
China (ICCC), Aug. 2022, pp. 262–267.
diaz2021deployment
C. Diaz-Vilor, A. Lozano, and H. Jafarkhani, “On the deployment problem in
cell-free uav networks,” in Proc. IEEE Global Commun. Conf.
(GLOBECOM), Dec. 2021, pp. 1–6.
zhang2009networked
J. Zhang, R. Chen, J. G. Andrews, A. Ghosh, and R. W. Heath, “Networked mimo
with clustered linear precoding,” IEEE Trans. Wireless Commun.,
vol. 8, no. 4, pp. 1910–1921, 2009.
bjornson2011optimality
E. Bjornson, N. Jalden, M. Bengtsson, and B. Ottersten, “Optimality
properties, distributed strategies, and measurement-based evaluation of
coordinated multicell ofdma transmission,” IEEE Trans. Signal
Process., vol. 59, no. 12, pp. 6086–6101, 2011.
garcia2010dynamic
I. D. Garcia, N. Kusashima, K. Sakaguchi, and K. Araki, “Dynamic cooperation
set clustering on base station cooperation cellular networks,” in
Proc. 21st Annu. IEEE Int. Symp. Pers., Indoor Mobile Radio Commun.,
Sep. 2010, pp. 2127–2132.
adhikary2017uplink
A. Adhikary, A. Ashikhmin, and T. L. Marzetta, “Uplink interference reduction
in large-scale antenna systems,” IEEE Trans. Commun., vol. 65, no. 5,
pp. 2194–2206, 2017.
bjornson2019making
E. Björnson and L. Sanguinetti, “Making cell-free massive mimo competitive
with mmse processing and centralized implementation,” IEEE Trans.
Wireless Commun., vol. 19, no. 1, pp. 77–90, 2019.
nayebi2016performance
E. Nayebi, A. Ashikhmin, T. L. Marzetta, and B. D. Rao, “Performance of
cell-free massive mimo systems with mmse and lsfd receivers,” in Proc.
50th Asilomar Conf. Signals, Syst. Comput., Nov. 2016, pp. 203–207.
zhang2021local
J. Zhang, J. Zhang, E. Björnson, and B. Ai, “Local partial zero-forcing
combining for cell-free massive mimo systems,” IEEE Trans. Commun.,
vol. 69, no. 12, pp. 8459–8473, 2021.
mai2019uplink
T. C. Mai, H. Q. Ngo, and T. Q. Duong, “Uplink spectral efficiency of
cell-free massive mimo with multi-antenna users,” in Proc. 3rd Int.
Conf. Recent Adv. Signal Process., Telecommun. Comput. (SigTelCom), Mar.
2019, pp. 126–129.
mai2020downlink
——, “Downlink spectral efficiency of cell-free massive mimo systems with
multi-antenna users,” IEEE Trans. Commun., vol. 68, no. 8, pp.
4803–4815, 2020.
wang2022uplink
Z. Wang, J. Zhang, B. Ai, C. Yuen, and M. Debbah, “Uplink performance of
cell-free massive mimo with multi-antenna users over jointly-correlated
rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 21,
no. 9, pp. 7391–7406, 2022.
wang2022iteratively
Z. Wang, J. Zhang, H. Q. Ngo, B. Ai, and M. Debbah, “Iteratively weighted mmse
uplink precoding for cell-free massive mimo,” in Proc. IEEE Int. Conf.
Commun. (ICC), May 2022, pp. 231–236.
couillet2011random
R. Couillet and M. Debbah, Random matrix methods for wireless
communications.1em plus 0.5em minus 0.4emCambridge University
Press, 2011.
alzenad20173
M. Alzenad, A. El-Keyi, F. Lagum, and H. Yanikomeroglu, “3-d placement of an
unmanned aerial vehicle base station (uav-bs) for energy-efficient maximal
coverage,” IEEE Wireless Commun. Lett., vol. 6, no. 4, pp. 434–437,
2017.
yu2002models
K. Yu and B. Ottersten, “Models for mimo propagation channels: a review,”
Wireless Commun. Mobile Comput., vol. 2, no. 7, pp. 653–666, 2002.
zhang2013capacity
J. Zhang, C.-K. Wen, S. Jin, X. Gao, and K.-K. Wong, “On capacity of
large-scale mimo multiple access channels with distributed sets of correlated
antennas,” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 133–148,
2013.
bjornson2009framework
E. Bjornson and B. Ottersten, “A framework for training-based estimation in
arbitrarily correlated rician mimo channels with rician disturbance,”
IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1807–1820, 2009.
li2016massive
X. Li, E. Björnson, S. Zhou, and J. Wang, “Massive mimo with multi-antenna
users: When are additional user antennas beneficial?” in Proc. 23rd
Int. Conf. Telecommun. (ICT), May 2016, pp. 1–6.
pan2023design
X. Pan, Z. Zheng, Z. Fei, and Z. Zhang, “On the design of optimal one-shot
distributed combining for cooperative multi-uav systems,” IEEE Trans.
Veh. Technol., vol. 72, no. 5, pp. 6827–6832, 2023.
lu2016free
A.-A. Lu, X. Gao, and C. Xiao, “Free deterministic equivalents for the
analysis of mimo multiple access channel,” IEEE Trans. Inf. Theory,
vol. 62, no. 8, pp. 4604–4629, 2016.
belinschi2017analytic
S. T. Belinschi, T. Mai, and R. Speicher, “Analytic subordination theory of
operator-valued free additive convolution and the solution of a general
random matrix problem,” J. Reine Angew. Math. (Crelles J.), vol.
2017, no. 732, pp. 21–53, 2017.
voiculescu1995operations
D. Voiculescu, “Operations on certain non-commutative operator-valued random
variables,” Astérisque, vol. 232, no. 1, pp. 243–275, 1995.
dobriban2020wonder
E. Dobriban and Y. Sheng, “Wonder: weighted one-shot distributed ridge
regression in high dimensions,” J. Mach. Learn. Res., vol. 21, no. 1,
pp. 2483–2534, 2020.
al2014optimal
A. Al-Hourani, S. Kandeepan, and S. Lardner, “Optimal lap altitude for maximum
coverage,” IEEE Wireless Commun. Lett., vol. 3, no. 6, pp. 569–572,
2014.
mingo2017free
J. A. Mingo and R. Speicher, Free probability and random matrices.1em plus 0.5em minus 0.4emNew York: Springer, 2017, vol. 35.
|
http://arxiv.org/abs/2307.06278v1 | 20230712162118 | Disorder operators and magnetic vortices in SU(N) lattice gauge theory | [
"Manu Mathur",
"Atul Rathor"
] | hep-lat | [
"hep-lat"
] |
ALPGEN
EVTGEN
PYTHIA
⟨⟨||
||⟩⟩
#1|#1⟩
#1⟨#1|
⟨[|2⟩]⟨⟩#1 |#2
|
http://arxiv.org/abs/2307.04369v1 | 20230710065440 | Exact generalized Turán number for $K_3$ versus suspension of $P_4$ | [
"Sayan Mukherjee"
] | math.CO | [
"math.CO",
"05C35"
] |
New results on the dynamics of critical collapse
Cheng-Gang Shao2
August 12, 2023
================================================
Let P_4 denote the path graph on 4 vertices.
The suspension of P_4, denoted by P_4, is the graph obtained via adding an extra vertex and joining it to all four vertices of P_4.
In this note, we demonstrate that for n≥ 8, the maximum number of triangles in any n-vertex graph not containing P_4 is ⌊ n^2/8⌋.
Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.
Keywords: generalized Turán problem, suspension of a graph, computer programming.
2020 Mathematics Subject Classification: 05C35.
§ INTRODUCTION
The generalized Turán number (n, T, H) is defined as the maximum number of copies of T in an n-vertex graph not containing H as a (not necessarily induced) subgraph.
When T=K_2, this is the Turán number (n,H) of the graph.
The first systematic study of (n, T, H) for T≠ K_2 was carried out by Alon and Shikhelman <cit.>.
In more recent years, several researchers have studied the asymptotic behavior of (n, K_3, H) for the case T=K_3 (see, for example <cit.>).
It is known that when χ(H)>3, (n,K_3,H)∼χ(H)-13/(χ(H)-1)^2· n^2, where χ(H) denotes the chromatic number of H <cit.>.
Alon and Shikhelman <cit.> extensively study the case when χ(H)=2.
Mubayi and the author <cit.> initiated the study of (n, K_3, H) for a simple family of graphs H with χ(H)=3.
For any graph G, they denoted the suspension G as the graph obtained from G by adding a new vertex v and joining it with all vertices of G.
They proceeded to analyze the asymptotic behavior of (n,K_3,G) for different bipartite graphs G.
One of the several bipartite graphs they consider is the path P_4 on four vertices. It was shown that for any n≥ 4,
n^2/8-O(1)≤(n, K_3, P_4) < n^2/8+3n.
An exact result for sufficiently large n was given by Gerbner <cit.> using the technique of progressive induction. In particular, they prove that for a number K≤ 1575 and n≥ 525+4K,
(n,K_3,P_4) = ⌊ n^2/8⌋.
They mention that a proof of the upper bound of (<ref>) for n=8,9,10,11 together with induction would suffice to prove (<ref>) for every n≥ 8.
In this note, we leverage this idea to determine the exact value of (n, K_3, P_4) for every n≥ 4, thus closing the gap in the literature for this extremal problem.
For n≥ 8, (n, K_3, P_4) = ⌊ n^2/8⌋. For n=4,5,6,7 the values of (n, K_3,P_4) are 4,4,5,8 respectively.
The lower bound constructions for Theorem <ref> are different for the cases n∈{4,5,6,7} and n≥ 8.
Figure <ref> illustrates graphs on n vertices for n∈{4,5,6,7} that achieve the maximum number of triangles.
In fact, we shall see later in Section <ref> that these constructions are unique up to isomorphism.
The general lower bound construction considered in <cit.> (for n≥ 8) was the complete bipartite graph K_⌊ n/2⌋, ⌈ n/2⌉ with a matching in any of the even parts.
A short case analysis shows that the total number of triangles in these graphs is given by ⌊ n^2/8⌋, hence proving the lower bound in Theorem <ref> for general n.
Thus, the main goal of this manuscript is to prove that these lower bounds on (n,K_3,P_4) are tight.
This work is organized as follows.
We present some preliminaries in Section <ref>.
Then, we show the upper bound of Theorem <ref> for n≥ 5 in Section <ref>.
Finally, we make some concluding remarks regarding uniqueness of the lower bound constructions in Section <ref>.
§ PRELIMINARIES
Throughout the rest of this paper, we assume without loss of generality that all graphs are edge-minimal.
This implies that every edge of the graphs considered must lie in a triangle, as we can simply delete edges that do not help forming a triangle.
We also assume that the vertex set of any n-vertex graph in the rest of this section is {0,…,n-1}, and abuse notation to represent a K_3 on vertex subset {a,b,c} as simply abc.
Let n(G), e(G) and t(G) denote the number of vertices, edges and triangles in G, respectively.
Now we recall some definitions and state a two important lemmas from <cit.> and <cit.> which are instrumental in our proof.
For a graph G, two edges e and e' are said to be triangle-connected if there is a sequence of triangles {T_1,…,T_k} of G such that e∈ T_1, e'∈ T_k, and T_i and T_i+1 share a common edge for every 1≤ i ≤ k-1.
A subgraph H⊆ G is triangle-connected if e and e' are triangle-connected for every edges e and e' of H.
A subgraph H⊆ G is a triangle block (or simply a block) if it is edge-maximally triangle-connected.
By definition, the triangle blocks of any graph G are edge-disjoint.
Let B_s denote the book graph on (s+2) vertices, consisting of s triangles all sharing a common edge. Let this common edge be called the base of the B_s. The following lemma characterizes the triangle blocks of any P_4-free graph G.
Every triangle block of a P_4-free graph G is isomorphic to a K_4 or a B_s for some s≥ 1.
Let H⊆ G be an arbitrary triangle block.
If H contains only one or two triangles, it is isomorphic to B_1 or B_2.
Suppose H contains at least three triangles.
Let two of them be abx_1 and abx_2 (see Figure <ref>).
If another triangle is of the form ax_1y for some y∈ V(H), then there are two possible cases. If y≠ x_2, then N_H(a) contains the 4-path x_2bx_1y, a contradiction. Otherwise if y=x_2, then the vertices a,b,x_1,x_2 create a K_4, and this K_4 is a triangle block by itself.
Similarly, if a triangle contained any of the edges bx_1, ax_2, bx_2, we would end up with a K_4-block, and this block cannot be extended any further.
Therefore all triangles in H would intersect the edge ab, implying H≅ B_s for some s≥ 1.
Suppose G is an n-vertex P_4-free graph containing no K_4. Then, we have
t(G)≤⌊ n^2/8⌋.
By Lemma <ref>, all triangle blocks of G are isomorphic to B_s for some s≥ 1.
Let G' be obtained from G by deleting the base edges of each of the books (if s=1, delete any arbitrary edge).
As each triangle of G contains two distinct edges from G', we have t(G)=e(G')/2.
By Mantel's theorem, e(G')≤⌊ n^2/4⌋, implying t(G)≤1/2⌊ n^2/4⌋, i.e. t(G)≤⌊ n^2/8⌋.
§ UPPER BOUNDS
In order to prove that (n,K_3,P_4)≤ K for some fixed n and K, we need to show that any n-vertex graph containing at least K+1 triangles contains a copy of P_4.
§.§ The cases 5≤ n≤ 8: brute force
While a case-by-case analysis is tractable by hand for n=5 for example, we quickly run into several possible configurations while trying to prove (8,K_3,P_4)=8.
This is where we turn to a computer-generated check.
For example, to prove that all 8-vertex graphs with more than 9 triangles is P_4-free, we can assume that 012 and 013 are two triangles in some 8-vertex graph G containing 9 triangles.
Then triangles that have an edge from the set {02, 03, 12, 13} and have a node from {4,5,6,7} are excluded from G since any of these patterns form a P_4.
This excludes 16 triangles.
Hence the plausible triangles that G may contain other than 012 and 013 are 83 - 18 = 38 in number.
We generate 387≈ 1.26× 10^7 possible graphs, filter out the ones that have exactly 9 triangles, and check for P_4's in each of them.
Our program is available at the Github repository in <cit.>.
We run |triangle_count.ipynb|.
Our computation shows that ex(n,K_3,P_4)=4,5,8,8 for n=5,6,7,8, respectively.
The total computation time required for (n,t)=(8,9) on 7 threads of an laptop processor running at 1.80GHz was around 18 minutes.
§.§ The cases 9≤ n ≤ 11: identifying K_4
The main idea behind these cases is to follow the steps of the proof in <cit.>, Section 5.2.
Suppose (n,t)∈{(9,11), (10,13), (11, 16)}, and G is an (edge-minimal) n-vertex graph with t triangles.
Then G must contain a P_4.
For the sake of contradiction, assume that G was P_4-free.
If G was also K_4-free, then by Lemma <ref>, t(G)≤⌊ n^2/8⌋ = 10, 12, 15 for n=9,10,11, contradicting our initial assumption on t(G).
Therefore G must contain a K_4.
Let this K_4 be induced by vertex subset S={u_0,u_1,u_2,u_3}⊂ V(G).
Define X_i := N(u_i) - S for 0≤ i ≤ 3.
As G[S] is a triangle block, X_i∩ X_j=∅ for every i≠ j.
Further, ∑_i=0^3|X_i|≤ n-4.
Without loss of generality assume |X_0|≤⋯≤ |X_3|.
Now we consider each case separately.
* Case 1. (n,t)=(9,11):
In this case, ∑_i=0^3|X_i|≤ 5.
If |X_1|>0, by edge-minimality we would have |X_1|≥ 2, implying |X_1|+|X_2|+|X_3|≥ 6, a contradiction.
Thus, |X_0| = |X_1| = 0, and by a similar argument, |X_2|≤ 2.
This means the vertex u_2 lies in at most one triangle outside of G[S].
Let G' be obtained by deleting {u_0,u_1,u_2} from G. Clearly n(G')=6 and t(G')≥ t(G)-5 = 6.
As (6,K_3,P_4)=5 by the discussion in Section <ref>, G' has a P_4, a contradiction.
* Case 2. (n,t)=(10,13):
Here, ∑_i=0^3 |X_i|≤ 6.
By a similar analysis as before, we can infer that |X_0|=0 and |X_1|≤ 2.
If |X_1|=0, we could consider G'=G-{u_0,u_1}, which would have n(G')=8 and t(G')=13-4=9, which would lead us to a P_4 since (8,K_3,P_4)=8 by the calculation in Section <ref>.
Thus, we have |X_0|=0, |X_1|=2, and hence |X_2|=|X_3|=2.
Now, if we consider G”=G-S, we have n(G”)=6 and t(G”) = 13-4-3=6, again implying that G” has a P_4.
* Case 3. (n,t)=(11,16):
For this pair of (n,t), we have ∑_i=0^3|X_i|≤ 7, implying |X_0|=0 again.
Since u_0 lies in exactly three triangles of G[S], G'=G-{u_0} has n(G')=10 and t(G')=13, leading us to the previous case.
In either of the three cases, we obtain a contradiction, finishing the proof for these cases.
§.§ The case n≥ 12: identifying K_4
Now that we have proved (n,K_3,P_4) = ⌊ n^2/8⌋ for 8≤ n ≤ 11, we are now ready to handle the general case using induction on n.
Our proof follows the idea of <cit.> with a more careful analysis to obtain the desired bound.
Let us assume that (k,K_3,P_4)=⌊ k^2/8⌋ for all 8≤ k ≤ n-1.
We note that a simple case analysis leads to
⌊ n^2/8⌋ - ⌊ (n-1)^2/8⌋ ≥⌊ n/4⌋
⌊ n^2/8⌋ - ⌊ (n-4)^2/8⌋ = n-2.
For the sake of contradiction, suppose G is an n-vertex P_4-free graph with t(G)≥⌊ n^2/8⌋ +1.
For a subset U⊂ V(G), let us denote by t(U) the number of triangles containing at least one vertex from U.
By (<ref>), we may assume that
|U|=1 t(U) ≥⌊ n/4⌋ +1,
|U|=4 t(U) ≥ n-1.
Now, notice that by Lemma <ref>, G must contain a K_4.
As in the previous section, let S={u_0,u_1,u_2,u_3} induce this K_4, and denote X_i=N(u_i)-S for 0≤ i ≤ 3. Again, |X_i∩ X_j|=∅ for every i≠ j.
Observe that t(S)= ∑_i=0^3 e(X_i)+4, and so by (<ref>),
∑_i=0^3 e(X_i)≥ n-5.
On the other hand, since each X_i is P_4-free, we have ∑_i=0^3 e(X_i)≤∑_i=0^3 |X_i| ≤ n-4.
Hence,
∑_i=0^3 e(X_i)∈{n-5, n-4}
This implies that e(X_i)=|X_i| for at least three u_i∈ S.
Assume that e(X_i)=|X_i| for 0≤ i ≤ 2 and e(X_3)∈{|X_3|-1, |X_3|}.
This also means that G[X_i] are vertex-disjoint unions of triangles for 0≤ i≤ 2,
and X_3 is a union of triangles and a star on r vertices for some r≥ 0.
Further, (<ref>) gives us the bound
|X_i|≥⌊ n/4⌋ -2 .
We now continue with a more detailed analysis of the neighborhoods of vertices in G.
In what follows, let x_i denote the size of X_i.
For a subset A⊂ V(G), let 𝒯(A) denote the set of triangles in G[A].
We now consider two cases.
Case 1: ∑_i=0^4x_i=n-5.
In this case, note that since ∑_i=0^3 e(X_i) = n-5, we have e(X_3)=x_3.
Thus, the subgraphs G[X_i] are all disjoint unions of triangles, and there is exactly one vertex y in V(G)-⋃_i X_i ∪ S,
and thus 3| n-5, implying n≡ 2 3.
Moreover, (<ref>) implies x_i≥ 3, and hence n≥ 17.
Now, observe that for G'=G-{y},
∑_v∈ V(G) v = ∑_i=0^3∑_vwz∈𝒯(X_i)(_G'v+_G'w+_G'z) + ∑_v∈ S v + 2 y.
We proceed by upper bounding each term of (<ref>) separately.
* Let vwz∈𝒯(X_0).
For any j≠ 0, as N(v)-X_0-S-{y} cannot contain two adjacent vertices from the same X_j, v can only be adjacent to at most one vertex from each triangle of X_j.
Finally, v is adjacent to exactly three nodes from X_0∪ S, leading to
_G' v + _G' w + _G' z ≤ 3(x_1/3 + x_2/3 + x_3/3) + 9 = (x_1+x_2+x_3)+9.
By repeating the same argument over all x_i/3 triangles from 𝒯(X_i), we have
∑_vwz∈𝒯(X_i)(_G'v+_G'w+_G'z) ≤x_i/3∑_j≠ ix_j + 3x_i.
* As y is not adjacent to any vertex of S, we have
∑_v∈ S v = (x_0+x_1+x_2+x_3) + 12 = n+7.
* For each i, N(y)∩ X_i has at most x_i/3 vertices, as otherwise by the pigeonhole principle we would have v,w∈ N(y)∩ X_i that are adjacent, leading to a triangle yvw sharing an edge with the K_4 containing u_i, v and w.
Further, y does not have a neighbor in S.
Thus,
y ≤x_0+x_1+x_2+x_3/3 = n-5/3.
Putting these inequalities together and noting that 3t(G)≤∑_v∈ V(G) v, (<ref>) gives us
3⌊ n^2/8⌋ + 3 ≤ 3t(G) ≤2/3∑_i<jx_ix_j + 3(x_0+x_1+x_2+x_3)+(n+7) + 2/3(n-5)
= 1/3(n-5)^2 - 1/3∑_i=0^3x_i^2 + 14n-34/3.
On the other hand, we note that by the Cauchy-Schwarz inequality, ∑_i=0^3x_i^2 ≥1/4(n-5)^2.
Therefore,
3⌊ n^2/8⌋ + 3 ≤1/4(n-5)^2+14n-34/3 = 1/12(3 n^2 + 26 n - 61),
A contradiction to n≥ 17. This completes the proof in this case.
▪
Case 2: ∑_i=0^4x_i=n-4.
In this case, recall that G[X_i] are disjoint unions of triangles for 0≤ i≤ 2, and X_3 is a union of triangles and a star on r≥ 0 vertices.
Let us denote this star as S^∗ = {c,ℓ_1,…, ℓ_r-1} where c is the center and ℓ_j the leaves.
We now continue with the exact same analysis of the neighborhoods of vertices in G as in the previous case.
For a subset A⊂ V(G), let 𝒯(A) denote the set of triangles in G[A].
First, we note that
∑_v∈ V(G) v = ∑_i=0^2∑_vwz∈𝒯(X_i)( v+ w+ z) + ∑_v∈ X_3 v + ∑_v∈ S v.
Let us now upper bound each term in (<ref>) separately.
* Let vwz∈𝒯(X_0).
Clearly N(v)-X_0-S cannot contain two adjacent vertices from the same X_j, j≠ 0.
Therefore, v can only be adjacent with at most one vertex from each triangle of X_j for j≠ 0.
Moreover, N(v)∩ S^∗, N(w)∩ S^∗ and N(z)∩ S^∗ are disjoint, implying
v + w + z ≤ 3(x_1/3 + x_2/3 + x_3-r/3) + r + 9 = (x_1+x_2+x_3) + 9.
Similar inequalities hold for each of the x_i/3 triangles in 𝒯(X_i), 0≤ i≤ 2.
In particular, we have
∑_vwz∈𝒯(X_i)( v+ w+ z) ≤x_i/3∑_j≠ ix_j + 3x_i.
* Let v∈ X_3.
Then, N(v)-X_3-S can have at most one vertex from each triangle of X_i.
Thus,
v ≤{[ 1/3(x_0+x_1+x_2) + 3, v∉S^∗,; 1/3(x_0+x_1+x_2) + r, v = c,; 1/3(x_0+x_1+x_2) + 2, v ∈ S^∗-{c}. ].
Thus, if r≥ 1,
∑_v∈ X_3 v ≤x_3(x_0+x_1+x_2)/3 + 3(x_3-r) + r + 2(r-1) = x_3(x_0+x_1+x_2)/3 + 3x_3 - 2,
and if r=0,
∑_v∈ X_3 v ≤x_3(x_0+x_1+x_2)/3 + 3x_3.
We use the latter inequality as it holds for any value of r.
* Finally, we have
∑_v∈ S v = (x_0+x_1+x_2+x_3)+12 = n+8.
Therefore, (<ref>) along with 3t(G)≤∑_v∈ V(G) v, gives us
3t(G) ≤2/3∑_i<jx_ix_j + 3(x_0+x_1+x_2+x_3) + n + 8.
= 1/3(n-4)^2 - 1/3∑_i=0^3 x_i^2 + 4n - 4
Observe that by Cauchy-Schwarz, ∑_i=0^3x_i^2≥1/4(n-4)^2.
Hence, (<ref>) implies,
3t(G)≤1/4(n-4)^2 + 4n-4 t(G)≤1/12 n(n+8).
By t(G)≥⌊ n^2/8⌋ + 1, this implies n≤ 14.
Note that as n-4 = ∑_i=0^3x_i ≥ 9+x_3, we would have x_3≤ 1.
By (<ref>), this would mean x_3 = 1.
However, this contradicts edge-minimality of G, as the edge between u_3 and the only vertex of X_3 would not be incident to any triangle in G, again leading to a contradiction in this case.
▪
This completes the proof of the induction step, implying (n,K_3,P_4)≤⌊ n^2/8⌋ for all n≥ 12.
§ CONCLUDING REMARKS: UNIQUENESS
For n≥ 8, one may ask whether the lower bound construction of K_⌊ n/2⌋, ⌈ n/2⌉ with a matching in any of the even parts is unique or not.
In particular, our proof of Theorem <ref> implies that if the extremal construction contained a K_4, then ⌊ n^2/8⌋≤1/12n(n+8).
This implies n≤ 16, and indeed, setting x_i=3 for every i leads us to an equality case in Case 2.
Our proof therefore gives us the following construction from Figure <ref> for n=16 consisting entirely of K_4-blocks: consider a K_4 given by S={u_0,u_1,u_2,u_3}.
For 0≤ i≤ 3, let N(u_i)-S consist of the triangles b_io_ir_i, where the b_i's are colored blue, o_i's olive and r_i's red.
Suppose the blue, red and olive vertices each form a K_4 (the diagonal edges are omitted in Figure <ref> for clarity).
Clearly each vertex neighborhood has 6 edges, leading to a total of 16· 6/3=32 triangles, and hence this graph is a valid extremal configuration for n=16.
It seems many extremal constructions are possible for smaller values of n whenever divisibility and structural constraints are satisfied.
For example, when n=8, we enumerate in our repository <cit.> all extremal constructions with 8 triangles programmatically, and these constructions are comprised of either two edge-disjoint K_4's, or only books.
However, our proof of Theorem <ref> provides uniqueness of the extremal configuration for n≥ 17.
§ ACKNOWLEDGMENTS
This work was supported by the Center of Innovations for Sustainable Quantum AI (JST Grant Number JPMJPF2221).
plain
|
http://arxiv.org/abs/2307.04201v1 | 20230709151149 | Bayesian estimation of the Kullback-Leibler divergence for categorical sytems using mixtures of Dirichlet priors | [
"Francesco Camaglia",
"Ilya Nemenman",
"Thierry Mora",
"Aleksandra M. Walczak"
] | stat.ME | [
"stat.ME",
"cs.IT",
"math.IT",
"physics.data-an"
] |
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
Department of Physics, Department of Biology, and
Initiative for Theory and Modeling of Living Systems, Emory University, Atlanta, Georgia, USA
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
Laboratoire de physique de l'École normale supérieure,
CNRS, PSL University, Sorbonne Université and Université de
Paris, 75005 Paris, France
In many applications in biology, engineering and economics, identifying similarities and differences between distributions of data from complex processes requires comparing finite categorical samples of discrete counts. Statistical divergences quantify the difference between two distributions. However their estimation is very difficult and empirical methods often fail, especially when the samples are small. We develop a Bayesian estimator of the Kullback-Leibler divergence between two probability distributions that makes use of a mixture of Dirichlet priors on the distributions being compared. We study the properties of the estimator on two examples: probabilities drawn from Dirichlet distributions, and random strings of letters drawn from Markov chains. We extend the approach to the squared Hellinger divergence. Both estimators outperform other estimation techniques, with better results for data with a large number of categories and for higher values of divergences.
Aleksandra M. Walczak
August 12, 2023
=========================
§ INTRODUCTION
Understanding of the structure and function of a large number of biological systems requires comparison between two probability distributions of their states or activities, generated under different conditions. For example, one may be interested in how the distribution of neural firing patterns underlying typical vocalizations in a song bird is different from patterns used to drive atypical, exploratory vocal behaviors <cit.>. One can similarly ask how different are the distributions of stimuli encoded by two different firing patterns; the difference then can be viewed as a measure of semantic similarity between these patterns <cit.>. In the context of immunology, one is often interested in information theoretic quantities in order to quantify diversity or to assess differences between distributions of immune receptors <cit.>. In these and similar examples, the Kullback-Leibler (KL) divergence , also known as relative entropy, is often used as a measure of dissimilarity. It is a non-symmetric measure of the difference between two probability distributions with a wide range of applications in information theory <cit.>. While not a distance in the mathematical sense, it is often the choice measure of dissimilarity since it can be applied to categorical (non-ordinal) data, when the usual statistical moments such as the mean and variance are not well defined. Indeed, like other “information theoretic quantities”, the KL divergence is not associated to the category itself, but rather to the underlying probability distribution <cit.>.
Estimation of information theoretic quantities is a hard problem, with a lot of attempts in the recent literature. Most of these have focused on the entropy and mutual information, but estimation of the KL divergence has also been investigated <cit.>. When faced with data without any knowledge of the true underlying distribution, empirical approaches (typically referred to as “naive” <cit.> or “plugin” <cit.>) are often used. These methods approximate the true probabilities of events with their empirical frequencies, with an optional pseudocount. These types of estimators have been investigated thoroughly. The consensus is that, for all entropic quantities, these estimates are typically strongly biased <cit.>. To overcome this limitation, other approaches have been proposed to estimate the Shannon entropy (or the mutual information) of categorical data. These techniques include Bayesian methods <cit.>, coverage adjusted methods <cit.> and bias corrected methods <cit.>. In the case of the KL divergence, the cross-entropy term, which diverges due to contributions where one distribution has samples and the other does not, makes it difficult to extend these methods in the absence of information about the joint distribution. The bias-corrected “Z-estimator” <cit.>, proposed for KL divergence estimation, tackles these issues. However, it has a strong dependence on the sample size.
Here we propose a Bayesian estimator of the DKL for systems with finite number of categories using a mixture of symmetric Dirichlet priors (Dirichlet Prior Mixture, or DPM). This approach is the generalization of the main idea from <cit.> that, to produce unbiased estimators, one needs to start with Bayesian priors that are (nearly) uniform not on the space of probability distributions, but directly on the quantity being estimated. Here we extend this idea beyond the estimation of entropy, for which it was first developed. We check that, for data distributed according to a Dirichlet prior, our new approach for estimation of the KL divergence consistently converges faster to the true value than other methods. We provide an algebraically equivalent expression for the Z-estimator (following <cit.>), which makes it applicable to large sample sizes. We also test the DPM technique on sequences generated by Markov chains, which are not typical within the DPM prior, obtaining better performance for datasets with many categories. We then focus our analysis on another measure of similarity between categorical distributions, the Hellinger divergence <cit.>, which, unlike the DKL, is a well defined bounded distance between distributions. To show the generality of our approach, we also develop a DPM estimator for the squared Hellinger divergence. In computational tests, we show the DPM approach to be accurate for this quantity as well. Since no estimation method can be guaranteed to estimate entropic quantities without a bias for an arbitrary underlying probability distribution, we finish by discussing the method's reliability when applied to real experimental data, where the true values of the divergences are not known a priori.
§ RESULTS
§.§ Bayesian framework for the estimation of the divergence
Our goal is to derive an estimate of the Kullback-Leibler divergence between the distributions of categorical variables t⃗ and q⃗, (q⃗‖t⃗).
We consider a discrete set of K categories labeled with i=1,…,K.
Examples of categorical variables include “words" defined as
sequences of neuron firing patterns (spike counts in time windows),
sets of coexisting ecological or molecular species or a sequence of
amino acids or nucleotides. Each category i has a certain (unknown)
probability q_i in the first condition, and t_i in the second
condition. We observe this category n_i times in an experiment done
in the first condition, and collect the data in the histogram
n⃗={ n_i}_i=1^K. An experiment in the second
condition returns the counts m⃗={ m_i}_i=1^K. We
want to estimate the Kullback-Leibler divergence between t⃗ and
q⃗ <cit.>, defined as:
(q⃗‖t⃗) = H(q⃗‖t⃗)-S(q⃗) =∑_i=1^K q_i logq_i/t_i,
where we defined the cross
entropy between t⃗, and q⃗,
H(q⃗‖t⃗) = - ∑_i=1^K q_i log t_i, and the Shannon
entropy,
S(q⃗) = - ∑_i=1^K q_i log q_i <cit.>.
Taking inspiration from Nemenman et al. <cit.>,
we choose to estimate the in a Bayesian framework.
The approach is summarized in Fig. <ref>. We do
not have access to the true probability distributions t⃗ and
q⃗, only to the empirical histograms n⃗ and m⃗.
The simple method consisting in approximating
q_i≈ n_i/∑_jn_j and likewise for t⃗ into
Eq. <ref> is known to work very
poorly <cit.>. The issue
comes from the presence of categories never observed in one sample,
while they are present in the other, resulting in divergence of the
logarithmic term. To go beyond that, we construct a prior of the
true distributions P_ prior (q⃗, t⃗) and weight the
estimate of the divergence by posterior over q⃗ and t⃗:
⟨ (q⃗, t⃗) |n⃗,m⃗⟩
=
∫dq⃗dt⃗ P_ post (q⃗, t⃗) (q⃗‖t⃗)
,
where
P_ post (q⃗, t⃗)
=
1/ZP_ prior (q⃗, t⃗) P(n⃗, m⃗|q⃗, t⃗)
,
with Z=P(n⃗,m⃗)= ∫dq⃗dt⃗ P_ prior (q⃗,t⃗) P(n⃗, m⃗|q⃗, t⃗) a normalization.
The empirical observations n⃗ and t⃗ are assumed to be independent samples of q⃗ and t⃗ respectively, and are thus distributed according to a multinomial distribution:
P( n⃗, m⃗|q⃗,t⃗ )
=
nqmt,
with
nq=
N !/∏_i=1^K n_i !∏_i=1^K q_i ^n_i
=
1/B(n⃗+1⃗)∏_i=1^K q_i ^n_i,
where N=∑_i=1^K n_i, and B(x⃗) is the multivariate Beta function: B(x⃗)=∏_i=1^K Γ (x_i)/Γ(∑_i=1^K x_i), where Γ(x) is the gamma function.
A natural choice for the prior on q⃗ and t⃗ is the Dirichlet distribution, which is the conjugate prior of the multinomial distribution, and is defined as
qα = δ(∑_i=1^Kq_i-1)/B(α⃗)∏_i=1^K q_i ^α-1,
where α∈ (0,∞) is the “concentration parameter”,
α⃗={α}_i=1^K and δ(x) is
the Dirac's delta function imposing normalization. Rank plots
associated to qα are shown in
Fig. <ref>A.
For α→∞, the prior tends to a uniform distribution q_i=1/K.
For small concentration parameters α, the distribution is peaked with weights given to just a few categories.
As noted in Ref. <cit.>, entropies of
distributions drawn from a Dirichlet with the same α all have
similar entropies, strongly biasing the Shannon entropy estimate,
especially in the large K limit. To reduce the bias, one then
uses a mixture of Dirichlet distributions at different α,
allowing substantially different values of the entropy a
priori. For a certain choice of the mixture distribution (the
prior over α, ρ(α)), one can achieve a
nearly-uniform a priori distribution of entropies and,
consequently, a much smaller estimation bias
<cit.>.
We expect also to have very similar values for all
distributions generated from the Dirichlet priors with fixed α
and β. We then expect that a good estimator may be produced by
using a mixture of Dirichlet
distribution that allows to span different values of the expected
:
P_ prior( q⃗,t⃗)=
∫_0^∞∫_0^∞dαdβ ρ(α,β) qαtβ
,
where ρ(α,β) is a “hyper-prior”, i. e., a prior over the hyper parameters α and β.
Plugging this prior into Eq. <ref> and <ref> gives:
⟨|n⃗,m⃗⟩
=
=
1/Z∫dαdβ
P( n⃗|α)P(m⃗|β)ρ(α,β)
⟨|n⃗,m⃗ ; α,β⟩
,
with
P( n⃗|α)
= ∫dq⃗ nq qα
=
B(n⃗+α⃗)/B(α⃗)B(n⃗+1⃗)
and likewise for P(m⃗|β).
The normalization now reads
Z = ∫dαdβ
P( n⃗, m⃗|α, β)
.
The expected value of the inside the integral may be computed analytically (see <ref>):
⟨|n⃗,m⃗ ; α,β⟩
=
=
∫dq⃗dt⃗ P(q⃗,t⃗|m⃗,n⃗,α,β)(q⃗‖t⃗)
=
∑_in_i+α/N+Kα{Δψ(M+Kβ , m_i+β)
-Δψ(N+Kα+1 , n_i+α+1) }
.
where Δψ(z_1,z_2)=ψ(z_1)-ψ(z_2) is the difference of digamma functions ψ (i. e., polygamma function of order 0, see Eq. <ref>).
Similarly we can calculate
⟨^2 |n⃗,m⃗⟩, which we can use to
compute the posterior standard deviation of our method
( <ref>). For a given choice of
ρ(α,β), the DPM estimate for in
Eq. <ref> can be computed numerically (same for
^2 in Eq. <ref>), as described in
detail in <ref>.
The code is available on github as specified in <ref>.
We expect that, in the limit of large data (N,M≫ K), the integral
of Eq. <ref> will be dominated by the values of
α and β that maximize the likelihoods
P(n⃗|α) and P(m⃗|β), regardless of
the
hyper-prior ρ(α,β).
The dominant role of the likelihood P(n⃗|α) for
increasing N was equivalently observed for the NSB entropy
estimator <cit.>. By contrast, we
expect the prior ρ(α,β) to play a role in the
low-sampling regime, as can be seen from Fig. <ref>.
A simplified approach for the estimation of the would then
be to provide a choice for the concentration parameters that
maximizes the likelihoods P(n⃗|α) and
P(m⃗|β) (see Eq. <ref>). We refer to the
application of Eq. <ref> with the
maximum-likelihood values of α and β
as the Dirichlet Prior (DP) estimator.
§.§ Choosing the hyper-prior
To finalize the estimation, we need to choose a functional form for the hyper-prior ρ(α,β) in such a way that the resulting ensemble has an evenly distributed .
In the limit of large numbers of categories (K≫ 1), both contributions of the , S(q⃗) and H(q⃗‖ t) are very peaked around their mean values, which can be computed analytically ( <ref>):
𝒜(α)≡⟨ S |α⟩=Δψ(Kα+1,α+1)≤log K,
and
ℬ(β) ≡⟨ H |α,β⟩=Δψ(Kβ,β) ≥log K
(which only depends on β), at fixed concentration parameters.
These mean values are shown in Fig. <ref>B, and
the corresponding =H-S in Fig. <ref>C as a
function of α and β.
We are interested in finding a hyper-prior such that the
resulting prior over is not peaked. This results in the
following inverse problem for finding the hyper-prior ρ_z(z),
where we denote by z:
ρ_z(z) ≈∫_0^∞dα∫_0^∞dβ ρ(α,β) δ( ℬ(β) - 𝒜(α)- z ),
with the choice ρ_z(z) to be made. Because we have a
one-dimensional target distribution ρ_z(z), but a 2-dimensional
hyper-prior ρ(α,β), there are infinitely many solutions
to this inverse problem. Without losing generality, we can make the
change of variable from α and β to 𝒜 and
ℬ:
ρ_z(z) ≈∫_0^log Kd𝒜∫_log K^+∞dℬ ρ_AB(𝒜,ℬ) δ( ℬ - 𝒜- z ),
with
ρ(α,β)=|∂_α𝒜||∂_βℬ|ρ_AB(𝒜(α),ℬ(β)).
Then a natural choice is to pick the Ansatz imposing that all values of 𝒜 and ℬ with the same are equiprobable:
ρ_AB(𝒜,ℬ)=ϕ(ℬ-𝒜).
Then ϕ(z) satisfies:
ρ (z) =
ϕ(z) ∫_0^log Kd𝒜 θ( z + 𝒜 -log K )
=
ϕ(z) { z θ(log K - z) + log K θ(z - log K) }
,
where θ(x)=1 if x≥ 1 and 0 otherwise (Heaviside function), or after inversion:
ϕ(z) =
ρ(z) z^-1 z < log K
ρ(z) 1log K otherwise.
Eqs. <ref>, <ref>, and <ref> give us
the final form of the hyper-prior ρ(α,β). We are
left with the choice of the distribution of the , ρ(z).
We pick a log-uniform (also known as “reciprocal”)
distribution,
ρ(z)∝ z^-1 <cit.>, allowing
to evenly span over different orders of magnitude of the .
The resulting hyper-prior is represented in
Fig. <ref>D.
§.§ Tests on synthetic Dirichlet samples
To assess the properties of the DPM estimator, we test it on data
generated from distributions drawn from Dirichlets
q⃗∼qα, t⃗∼qβ
(Eq. <ref>), for various values of α and
β. Having in mind applications to polypeptide sequences, we
perform our tests for three different numbers of categories K=20^2,
20^3, and 20^4, the numbers of all possible 2-mers, 3-mers
and 4-mers that can be produced with an alphabet of 20 letters
(e. g., amino acids). For each choice of q⃗ and t⃗,
samples n⃗ and m⃗ are generated from these distributions.
This application may be viewed as a the consistency check for te
estimator, since the estimator relied on the Dirichlet hypothesis,
which is satisfied by the data.
We know that standard Bayesian consistency applies, ensuring that
DPM (and DP) estimators converge to the true value in the limit of
large samples. To understand how DPM estimator converges to the
true value, we extract subsamples of increasing sizes N=M from a
larger sample of size 2· 10^7.
Fig. <ref> compares our
estimate to several state-of-the-art estimators: the additive
smoothing method with different values of the pseudo-count (see below
for details), the Z estimator, and the simplified version of our
method, the DP estimator, obtained by fixing α and β to
their maximum-likelihood values.
Additive smoothing estimators are defined as: (q⃗̂⃗‖t⃗̂⃗), with q̂_i=(n_i+a)/(N+Ka), and t̂_i=(m_i+b)/(M+Kb).
We use 4 choices for the pseudo-counts a and b, summarized in Table <ref>.
To avoid infinite values, in the case b=0 we set to zero the terms for which m_i=0.
It has been shown that naive estimators converge to the true
value in the limit of large samples, but have an infinite bias due
to low-probability categories <cit.>. The
“Z-estimator” <cit.> was introduced to
remove this bias asymptotically. Although its original definition was
given as a series, one can show following <cit.>
that its expression reduces to (Appendix <ref>):
D_ KL^(Z)
=
∑_i=1^K n_i/N[
Δψ (M+1, m_i+1)-Δψ (N, n_i)
],
where the first term in the sum corresponds to an estimator of
H(q⃗‖ t), and the second term is the classic
Schurmann-Grassberger estimator of the entropy
S(q⃗) <cit.>. In Appendix <ref> we
observe that
⟨|n⃗, m⃗, α, β⟩→D_ KL^(Z) in the limit α→ 0,
β→ 1, N ≫ K and M ≫ K.
Comparing the convergence of the different estimators to the true
value as a function of the subsample size N/K for
α=β=1 and K=20^2
(Fig <ref>A), we see that the DPM
performs better than other estimators. To assess how performance
depends on the concentration parameters, we repeated this convergence
analysis for different values of α and β. We measure
convergence through N^∗, defined as the sample size where the
estimator get within 5% of the true value
(Fig <ref>B). This measure
of accuracy has the advantage to be applicable to all considered
estimation methods.
Our estimator consistently performs well and compares favorably
to other methods when data was generated from distributions drawn
from symmetric Dirichlet. In most cases, the proposed DPM estimator
converges faster than all other considered methods
(Fig <ref>C).
The better performance is striking also for larger numbers of
categories, K=20^3 and 20^4
(Fig. <ref>).
§.§ Tests on synthetic Markov chain sequences
To test the performance of DPM on a different synthetic system that
does not satisfy the Dirichlet assumption, we generated L-long
sequences (or “L-grams”) from a Markov chain described by the
transition matrix Ŵ∈ℳ_20 with 20 states
μ=1,⋯,20. We choose each transition probability
P(μ→ν) from a uniform distribution in (0,1) and then impose
that the transition matrix is a right stochastic matrix,
P(μ→ν)=W_νμ by normalizing to 1 each column of the
transition matrix. An illustration where the states are the 20
amino acids is shown in Fig. <ref>A.
With this choice for the Markov transition matrix, all states
communicate and are non absorbing.
We verify there exists a stationary probability vector π⃗ = {π_μ}_μ=1^L that satisfies
π⃗ = Ŵπ⃗. The number of categories is K=20^L
and each category i corresponds to the L-gram
(x_1,⋯,x_L) with the stationary probability q_i equal to
q_i=π_x_1 W_x_2 x_1⋯ W_x_L x_L-1.
We analytically compute the entropy associated to the stationary distribution q⃗ of L-grams to get:
S^(L)(q⃗)
=
S(π⃗)
-
(L-1) ∑_μν W_νμπ_μlogW_νμ.
Typical values for the Shannon entropy of L-grams are shown in Fig. <ref>B along with the convergence curve of the NSB estimator.
We assume that the L-grams of a second system are generated by a similar Markov process but with a transition matrix V̂ and stationary probabilities of the σ⃗={σ_μ} states.
The cross-entropy between the t⃗ and q⃗ distributions reads:
H^(L)(q⃗‖t⃗)
=
H(π⃗‖σ⃗)
-
(L-1) ∑_μν W_νμπ_μlogV_νμ.
Similarly to the analysis in the previous section, we generate a large sample of L-grams from each distribution, with N=M=2·10^8.
We subsample this dataset at different sample sizes and estimate the and its standard deviation for L=2,3,4.
To study the average behavior, we divide the estimate by the expected result (Eq. <ref>) and we average over 30 simulations.
We observe that, in the case of small numbers of categories (K=20^2,
Fig. <ref>C top panel), DPM (and DP) perform
quite similar to the best alternative (Jeffreys), but with different
sign biases. However, the DPM estimator performance greatly
improves for larger K (Fig. <ref>C middle and
bottom panels). In all cases, the standard deviation associated to
the DPM estimator (red bars in Fig. <ref>C) captures
the spread across the different repetitions of the convergence curve
(red shade in Fig. <ref>C).
§.§ Estimator for the Hellinger divergence
Lastly, we extend the DPM method to estimate the Hellinger divergence between the discrete distributions q⃗ and
t⃗ <cit.>.
The Hellinger divergence is a symmetric statistical distance that satisfies the triangular inequality, making it a true distance in the mathematical sense <cit.>:
(q⃗,t⃗)^2
=
1/2∑_i=1^K(√(q_i)-√(t_i))^2=1-∑_i=1^K√(q_it_i).
Following the same approach as for the Kullback-Leibler divergence (details in <ref>), we obtain the DPM estimator for ^2:
⟨^2 |n⃗,m⃗⟩
=
=
1/Z∫dαdβ ρ_ H (α,β) P ( n⃗|α) P ( m⃗|β)
⟨^2 |n⃗, m⃗;α,β⟩,
with
⟨^2 |n⃗, m⃗;α,β⟩
=
=
1-∑_i=1^KB(1/2, N + Kα)/B(1/2,n_i + α)B(1/2, M + Kβ)/B(1/2,m_i + β),
where Z = ∫dαdβ ρ_ H (α,β) P ( n⃗|α) P ( m⃗|β) and B(x_1, x_2) = Γ(x_1)Γ(x_1)/ Γ(x_1+x_2) is the two-dimensional Beta function.
We test the Hellinger divergence estimator on the same synthetic datasets as in Fig. <ref>B (Fig. <ref>).
For datasets generated with Dirichlet-multinomial distributed samples, the DPM outperforms all considered plugin estimators
1 - ∑_i=1^K √(q̂_it̂_i), with q̂_i and t̂_i defined as before with pseudo-counts
a,b chosen according to Table <ref> (Fig. <ref>A).
As for the case of KL divergence, the performance improves for larger categories (Fig. <ref>).
Tests on the synthetic Markovian L-grams (see previous paragraph) show the DPM estimator performs better for larger numbers of categories K, with comparable performance to the best alternative (Jeffreys) for K=20^2 (Fig. <ref>B).
§ DISCUSSION
Correctly estimating statistical divergences between two distributions is an open problem in the analysis of categorical systems. Alongside the entropy, divergences such as the Kullback-Leibler and the Hellinger distance, are an important tool in the analysis of categorical data <cit.>.
We focused on categorical distributions with finite numbers of categories K (bounded domain), where K is a known quantity. We proposed a way (DPM) to extend the approach of Nemenman et al. <cit.> developed for Shannon entropy estimation, to Kullback-Leibler estimation. DPM introduces a
mixture of symmetric Dirichlet priors with a log-uniform a priori expected divergence distribution (Eq. <ref>).
We restricted our analysis to the case of the two finite samples of the same size N, although the method works for different sample sizes.
We also propose a simplified estimator (DP), which assumes a Dirichlet prior with concentration parameter fixed to the maximum value of the likelihood.
This estimator is faster to compute as it does not require to integrate over the concentration parameters.
We showed that the DPM method outperforms the tested empirical plugin techniques
in terms of estimation for synthetic data sampled from a Dirichlet-multinomial distribution with fixed concentration parameters. The estimation task gets harder for distributions with larger concentration parameters, i. e., closer to a uniform distribution, but easier for large numbers of categories K.
These convergence trends were confirmed by tests on sequences of L states generated by Markov chains.
In this case, DPM compares well to the best plugin estimator in the low sample size regime of K=20^2 and outperforms it for K≥ 20^3.
Similar results were obtained for the DPM estimate of the Hellinger divergence for both Dirichlet-multinomial and Markov chain datasets.
To our knowledge, DPM estimator of the Hellinger divergence is the first attempt to extend the ideas of Ref. <cit.> and to build a uniform prior estimator for a non-entropy-like quantity.
Our tests were restricted to categorical systems with rank distributions having exponentially decaying tails.
As previously discussed for the case of the NSB entropy estimator, the Dirichlet prior has major limitations in capturing the Shannon entropy if the system rank distribution is not decaying fast <cit.>.
Many real systems exhibit long-tailed rank distributions that decay as power-laws <cit.>, which are not well captured by a Dirichlet prior.
Preliminary (unpublished) tests of the DPM method for such systems show poor performance.
Similarly to the case of entropy estimates, we speculate that the limitations of this method are related to issues with the poor representation of long tails by Dirichlet priors.
Introducing a Pitman-Yor prior <cit.> could overcome this problem, as has been shown for entropy estimation by Archer et al. <cit.>, and offers a direction to generalize the applicability of the DPM method.
Extending the Pitman-Yor prior to the case of statistical divergences would require to compute expected values over the probabilities of both systems, but to the best of our knowledge this is not possible because of the lack of an analytical expression for the Pitman-Yor distribution.
Another difficulty may lie in the difficulty to encode correlations between the ranks of categories in the two distributions.
Our priors assume that the two unknown distributions q⃗ and t⃗ are drawn independently. However in real data they are generally correlated, which could have an impact on the quality of estimators when the distribution of frequencies becomes very broad.
In view of these complications, it is important to have practical criteria to ascertain if the output of the DPM estimator can be trusted for a specific dataset, or if it remains biased. Similar questions exist for estimation of many quantities, and specifically of entropic quantities, on categorial data since no estimator can be universally unbiased for them, and the decay of the bias with the sample size may be excruciatingly slow <cit.>. For entropy and mutual information, the standard approach is to observe if the empirical output drifts systematically as the sample size changes. One then declares the estimator trustworthy if the bias does not drift by more than the posterior standard deviation over about an order of magnitude change in the amount of data <cit.>. We expect this approach to transfer nearly verbatim to the and the Hellinger divergence context, easily detecting whether the DPM approach can be used for a specific dataset, or if other analysis methods should be sought.
§ ACKNOWLEDGEMENTS
We thank Antonio C. Costa for helpful discussions.
This work was partially supported by the
European Research Council Consolidator Grant n. 724208 and ANR-19-CE45-0018 “RESP-REP” from the Agence Nationale de la Recherche.
I.N. was supported in part by the Simons Foundation Investigator grant and by the U. S. NSF grant No. 2209996.
unsrt
10
hernandez_unsupervised_2022
Damián G Hernández, Samuel J Sober, and Ilya Nemenman.
Unsupervised Bayesian Ising Approximation for decoding neural
activity and other biological dictionaries.
eLife, 11:e68192, March 2022.
Publisher: eLife Sciences Publications, Ltd.
ganmor_thesaurus_2015
Elad Ganmor, Ronen Segev, and Elad Schneidman.
A thesaurus for a neural population code.
eLife, 4:e06134, September 2015.
Publisher: eLife Sciences Publications, Ltd.
mora_quantifying_2019
Thierry Mora and Aleksandra Walczak.
Quantifying lymphocyte receptor diversity.
In Jayajit Das and Ciriyam Jayaprakash, editors, Systems
Immunology, An Introduction to Modeling Methods for Scientist.
CRC Press, 2019.
camaglia_quantifying_2023
Francesco Camaglia, Arie Ryvkin, Erez Greenstein, Shlomit Reich-Zeliger, Benny
Chain, Thierry Mora, Aleksandra M Walczak, and Nir Friedman.
Quantifying changes in the T cell receptor repertoire during thymic
development.
eLife, 12:e81622, January 2023.
Publisher: eLife Sciences Publications, Ltd.
kullback_information_1951
Solomon Kullback and Richard A. Leibler.
On Information and Sufficiency.
The Annals of Mathematical Statistics, 22(1):79–86, March
1951.
Publisher: Institute of Mathematical Statistics.
zhang_entropic_2022
Jialin Zhang.
Entropic Statistics: Concept, Estimation, and Application in
Machine Learning and Knowledge Extraction.
Machine Learning and Knowledge Extraction, 4(4):865–887,
December 2022.
Number: 4 Publisher: Multidisciplinary Digital Publishing Institute.
zhang_nonparametric_2014
Zhiyi Zhang and Michael Grabchak.
Nonparametric Estimation of Küllback-Leibler Divergence.
Neural Computation, 26(11):2570–2593, November 2014.
strong_entropy_1998
Steven P. Strong, Roland Koberle, Rob R. de Ruyter van Steveninck, and William
Bialek.
Entropy and Information in Neural Spike Trains.
Physical Review Letters, 80(1):197–200, January 1998.
Publisher: American Physical Society.
antos_convergence_2001
András Antos and Ioannis Kontoyiannis.
Convergence properties of functional estimates for discrete
distributions.
Random Structures & Algorithms, 19(3-4):163–193, 2001.
paninski_estimation_2003
Liam Paninski.
Estimation of Entropy and Mutual Information.
Neural Computation, 15(6):1191–1253, June 2003.
zhang_entropy_2012
Zhiyi Zhang.
Entropy Estimation in Turing's Perspective.
Neural Computation, 24(5):1368–1389, May 2012.
jiao_maximum_2015
Jiantao Jiao, Kartik Venkat, Yanjun Han, and Tsachy Weissman.
Maximum Likelihood Estimation of information measures.
In 2015 IEEE International Symposium on Information
Theory (ISIT), pages 839–843, Hong Kong, Hong Kong, June 2015. IEEE.
nemenman_entropy_2001
Ilya Nemenman, Fariel Shafee, and William Bialek.
Entropy and Inference, Revisited.
In Advances in Neural Information Processing Systems,
volume 14. MIT Press, 2001.
archer_bayesian_2014
Evan Archer, Il Memming Park, and Jonathan W. Pillow.
Bayesian Entropy Estimation for Countable Discrete
Distributions.
Journal of Machine Learning Research, 15(81):2833–2868, 2014.
chao_nonparametric_2003
Anne Chao and Tsung-Jen Shen.
Nonparametric estimation of Shannon's index of diversity when there
are unseen species in sample.
Environmental and Ecological Statistics, 10(4):429–443, 2003.
schurmann_bias_2004
Thomas Schürmann.
Bias analysis in entropy estimation.
Journal of Physics A: Mathematical and General, 37(27):L295,
June 2004.
schurmann_note_2015
Thomas Schürmann.
A Note on Entropy Estimation.
Neural Computation, 27(10):2097–2106, October 2015.
hellinger_neue_1909
Ernst Hellinger.
Neue begründung der theorie quadratischer formen von unendlichvielen
veränderlichen.
Journal für die reine und angewandte Mathematik,
1909(136):210–271, 1909.
shannon_mathematical_1948
Claude E. Shannon.
A Mathematical Theory of Communication.
Bell System Technical Journal, 27(3):379–423, 1948.
hernandez_low-probability_2023
Damián G. Hernández, Ahmed Roman, and Ilya Nemenman.
Low-probability states, data statistics, and entropy estimation.
Physical Review E, 108(1):014101, July 2023.
Publisher: American Physical Society.
nemenman_coincidences_2011
Ilya Nemenman.
Coincidences and Estimation of Entropies of Random Variables
with Large Cardinalities.
Entropy, 13(12):2013–2023, December 2011.
hamming_distribution_1970
Richard W. Hamming.
On the distribution of numbers.
The Bell System Technical Journal, 49(8):1609–1625, October
1970.
Conference Name: The Bell System Technical Journal.
jeffreys_invariant_1946
Harold Jeffreys.
An invariant form for the prior probability in estimation problems.
Proceedings of the Royal Society of London. Series A.
Mathematical and Physical Sciences, 186(1007):453–461, September 1946.
Publisher: Royal Society.
trybula_problems_1958
Stanislaw Trybula.
Some Problems of Simultaneous Minimax Estimation.
The Annals of Mathematical Statistics, 29(1):245–253, March
1958.
Publisher: Institute of Mathematical Statistics.
perks_observations_1947
Wilfred Perks.
Some observations on inverse probability including a new indifference
rule.
Journal of the Institute of Actuaries, 73(2):285–334, December
1947.
Publisher: Cambridge University Press.
hausser_entropy_2009
Jean Hausser and Korbinian Strimmer.
Entropy Inference and the James-Stein Estimator, with
Application to Nonlinear Gene Association Networks.
Journal of Machine Learning Research, 10(50):1469–1484, 2009.
liese_statistical_2008
Friedrich Liese and Klaus J. Miescke.
Statistical Models.
In Statistical Decision Theory: Estimation, Testing, and
Selection, Springer Series in Statistics, pages 1–74. Springer, New
York, NY, 2008.
zipf_human_1949
George Kingsley Zipf.
Human behavior and the principle of least effort.
Human behavior and the principle of least effort. Addison-Wesley
Press, Oxford, England, 1949.
Pages: xi, 573.
pitman_two_parameter_1997
Jim Pitman and Marc Yor.
The two-parameter Poisson-Dirichlet distribution derived from a
stable subordinator.
The Annals of Probability, 25(2):855 – 900, 1997.
holmes2019estimation
Caroline M Holmes and Ilya Nemenman.
Estimation of mutual information for real-valued data with error bars
and controlled bias.
Physical Review E, 100(2):022404, 2019.
wolpert_estimating_1995
David H. Wolpert and David R. Wolf.
Estimating functions of probability distributions from a finite set
of samples.
Physical Review E, 52(6):6841–6854, December 1995.
Publisher: American Physical Society.
bhattacharyya_on_1943
Anil K. Bhattacharyya.
On a measure of divergence between two statistical populations
defined by their probability distributions.
Bulletin of the Calcutta Mathematical Society, 35:99–109,
1943.
simpson_measurement_1949
Edward H. Simpson.
Measurement of Diversity.
Nature, 163(4148):688–688, April 1949.
Number: 4148 Publisher: Nature Publishing Group.
*
§ SUPPLEMENTARY INFORMATION
§.§ Mathematical relations
We first introduce mathematical relations and notations that are used for the computation of the DP and DPM estimators for and ^2.
§.§.§ Wolpert-Wolf integrals
Given a vector x⃗={ x_i }_i=1^K, where x_i ∈ (0,∞) for all i=1,…, K, where K is a finite number of categories, the Wolpert-Wolf <cit.> integral is a multivariate Beta function B:ℝ_+^K→ℝ_+ in x⃗:
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K q_j ^x_j-1
= ∏_j=1^K Γ (x_j)Γ(X) = B(x⃗),
where X=∑_i x_i and the function Γ is the Euler Gamma function Γ(x) = ∫_0^∞dt e^-tt^x-1.
All Bayesian calculations with multinomial likelihoods and multivariate Dirichlet priors involve the integral:
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K f_j(q_j)
=
=
L^-1[ ∏_j=1^K L[ f_j(q) ](s)](q'=1),
where the f_i are regular functions, L is the Laplace transform in q (which is a function of s) and L^-1 is the inverse Laplace transform (which is a function of q').
§.§.§ Partial derivative operation
The “partial derivative operator” for the i-th dimension ∂_i = ∂/∂ x_i applied to the Beta function B returns
( ∂_i B ) (x⃗ )
=
∫dq⃗ δ(‖q⃗‖_1-1) ∏_j=1^K q_j ^x_j -1log q_i
= B(x⃗) [ ψ(x_i)-ψ(X) ],
where the function ψ is the polygamma function of order 0. The polygamma function of order ℓ is defined as
ψ_ℓ(x) = .d^ℓ/d y^ℓlogΓ(y)|_y=x.
In order to simplify the calculations, we define the following quantities related to the partial derivative operation (Eq. <ref>):
Λ_i(x⃗)
=
( ∂_i B ) (x⃗ )/B(x⃗)
=
Δψ(x_i,X) ,
where we make use of the contraction Δψ(z_1,z_2) = ψ(z_1)-ψ(z_2). Iterating this derivation on the function B, one can express the double derivatives as follows:
Λ_ij(x⃗)
=
(∂_ij B) ( x⃗)/B( x⃗)
=
(∂_i( B Λ_j))( x⃗)/B( x⃗)
=
Λ_i( x⃗)Λ_j( x⃗) + ( ∂_iΛ_j) ( x⃗)
,
where the derivative ∂_iΛ_j( x⃗) = δ_ijψ_1( x_i) -ψ_1( X) is a consequence of Eq. <ref>.
§.§.§ Shift Operation
The “shift operator” e^λ∂_i of parameter λ∈ℝ for the i-th dimension acts on the function B as follows:
( e^λ∂_iB )( x⃗)
=
B( x⃗+λi)
=
∫dq⃗ δ( ∑_i=1^K q_i-1) ∏_j=1^K q_j ^x_j-1q_i^λ
=
B( x⃗) B ( λ , X)B(λ, x_i)
,
where i = {δ_ij}_j=1^K indicates the i-th versor in the K-dimensional space of categories, with the condition x_i+λ>0. The function B( z_1,z_2) is the regular (two-dimensional) Beta function :
B( z_1,z_2)
=
Γ( z_1) Γ( z_2) Γ( z_1+z_2)
.
When λ =n ∈ℕ_+, the shift simplifies to
( e^n∂_i B ) ( x⃗)
=
B( x⃗) ∏ ^n-1_n'=0x_i+n'X+n'
as an immediate consequence of the recurrence relation Γ (x + 1) = x Γ(x).
Similarly to the case of partial derivatives, we introduce a class of functions to deal with the shift:
Ω _i( x⃗)
=
(e^∂_iB )( x⃗)/B ( x⃗)
=
x_iX
,
from which we compute two-dimensional shifts
Ω _ij( x⃗)
=
( e^∂_ie^∂_jB ) ( x⃗)/B( x⃗)
=
e^∂_i( B Ω_j) ( x⃗)/B( x⃗)
=
Ω_i( x⃗) Ω_j( x⃗ + i)
=
x_iX( x_j +δ _ij) /X+1.
§.§.§ Composed operations
For the sake of this work, another useful class of functions are the first order derivatives of the functions Ω defined as
∂_jΩ_i/Ω_i
=
∂_jlogΩ_i
=
δ_ij/x_j - 1/X
,
and, for the two-dimensional shift,
∂_kΩ_ij/Ω_ij
=
∂_klogΩ_ij
=
δ_ijk( 1/x_k + 1/x_k+1)
+ (δ_ik + δ_jk ) (1-δ_ij)1/x_k
+
- 1/X - 1/X+1
.
Similarly for the second order derivatives:
∂_jkΩ_i/Ω_i
=
∂_j∂_klogΩ_i + (∂_jlogΩ_i) (∂_klogΩ_i)
=
2/X^2 - δ_ij+δ_ik/x_i X
,
and
∂_khΩ_ij/Ω_ij
=
∂_k (Ω_ij∂_hlogΩ_ij)/Ω_ij
=
1/X^2 + 1/(X+1)^2
- δ_ijkh( 1/x_k^2 + 1/(x_k+1)^2)
+
-(δ_ikh + δ_jkh ) (1-δ_ij)1/x_h^2
+ (∂_klogΩ_ij) (∂_hlogΩ_ij).
Using all these definitions, we compute the following quantities:
e^∂_i∂ _j B/B
=
Ω_iΛ_j + ∂_jΩ_i,
e^∂_i e^∂_j∂ _k B/B
=
Ω_ijΛ_k + ∂_kΩ_ij,
e^∂_i∂ _jk B/B
=
Ω_i Λ_jk + (∂_j Ω_i) Λ_k
+
(∂_k Ω_i) Λ_j + ∂_jkΩ_i
and
e^∂_i e^∂_j∂ _kh B/B
=
Ω_ijΛ_kh + (∂_k Ω_ij) Λ_h
+
(∂_h Ω_ij) Λ_k + ∂_khΩ_ij.
which are used to reconstruct all estimators presented in this work.
§.§.§ A priori and a posteriori expected values
The operations presented in the previous sections are used compute the posterior expected values ⟨ F(q⃗,t⃗) |n⃗,m⃗;α, β⟩ for all the functions that can be expressed as :
F(q⃗,t⃗)
=
∑_i=1^K f_i(q⃗) g_i (t⃗).
Since the concentration parameters α, β are independent, for fixed concentration parameters the expected value of F factorizes:
⟨ F(q⃗,t⃗) |n⃗,m⃗;α, β⟩
=
∑_i=1^K ⟨ f_i |n⃗;α⟩⟨ g_i |m⃗;β⟩,
with
⟨ f_i |n⃗;α⟩B(n⃗+α⃗)/B(α⃗) B(n⃗+1⃗)
=
∫d q⃗ δ( ∑_i=1^K q_i-1)
qαnq
f_i(q⃗)
=
∫d q⃗ δ( ∑_i=1^K q_i-1)
∏_j^K q_j ^n_i+α-1/ B(α⃗) B(n⃗+1⃗)
f_i (q⃗).
For all functions f_i that can be expressed in terms of partial derivative (Eq. <ref>) and/or shift operators (Eq. <ref>),
a factor B(n⃗+α⃗) appears and the expected value is obtained explicitly simplifying the constant factors.
Specifically:
⟨ q_i |n⃗;α⟩
=
(e^∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨log q_i |n⃗;α⟩
=
(∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i log q_i |n⃗;α⟩
=
(e^∂_i∂_i B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i q_j |n⃗;α⟩
=
(e^∂_i e^∂_j B )(n⃗+α⃗)/B(n⃗+α⃗),
⟨ q_i q_j log q_i |n⃗;α⟩
=
(e^∂_i e^∂_j∂_i B )(n⃗+α⃗)/B(n⃗+α⃗)
and
⟨ q_i q_j log q_i log q_j |n⃗;α⟩
=
(e^∂_i e^∂_j∂_i ∂_j B )(n⃗+ α⃗) /B(n⃗+α⃗)
.
The a priori expected values are computed in the same way, noticing that ⟨ f_j |α⟩ = ⟨ f_j |n⃗=0⃗;α⟩.
§.§.§ KL divergence estimation
We can use the previous results to compute the a posteriori expected value for the .
We start by computing the a posteriori expected value for the crossentropy H which is given by
⟨∑_i^K
H(q‖t)
|n,m, α, β⟩
=
∑_i^K⟨ q_i |n, α⟩⟨log t_i |m, β⟩
=
∑_i^K e^∂_i B(n+α) /B(n+α)∂_i B(m+β) /B(m+β)
=
∑_i^Kn_i+α/N+KαΔψ (M+Kβ, m_i+β)
,
where we took advantage of independence and used the relations Eq. <ref> and <ref> to obtain the explicit expressions in Eq. <ref> and <ref>.
Subtracting the a posteriori expected Shannon entropy ⟨ S |n,m, α, β⟩ = ∑_i n_i+αN+KαΔψ(N+Kα+1, n_i +α+1), we finally obtain the KL expected value in Eq. <ref>:
⟨|n⃗,m⃗ ; α,β⟩
=
∑_in_i+α/N+Kα{Δψ(M+Kβ , m_i+β)
- Δψ(N+Kα+1 , n_i+α+1) }.
§.§.§ Squared KL divergence estimation
In order to compute the posterior standard deviation of the Kullback-Leibler divergence estimator, we calculate the expected value of the squared KL divergence:
⟨^2 |n⃗,m⃗⟩
=
∫dαdβ
P(n⃗,α)P(m⃗|,β)ρ(α,β)
⟨^2 |n⃗,m⃗;α,β⟩
.
Similarly to the case of ,
we can compute explicitly
⟨^2 |n⃗,m⃗;α,β⟩
=
∑_ij⟨ q_i q_j logq_i/t_ilogq_j/t_j|n⃗,m⃗;α,β⟩
,
which requires to rewrite
q_iq_jlogq_it_ilogq_jt_j
=
q_iq_jlog q_ilog q_j
- 2 q_iq_jlog q_ilog t_j
+ q_iq_jlog t_ilog t_j
.
The explicit expression computed using Wolpert-Wolf properties (Eqs. <ref>, <ref>, <ref>, <ref>, <ref>) is:
⟨ q_i q_j logq_i/t_ilogq_j/t_j|n⃗,m⃗;α,β⟩
=
x_i(x_j+δ_ij)/X(X+1){δ_ijψ_1( x_i+2 ) -ψ_1( X+2 )
+ Δψ( x_i+1+ δ_ij,X+2) · (i↔ j)
- 2Δψ( x_i+1+δ_ij,X+2)
Δψ( y_j,Y)
+δ_ijψ _1(y_i) - ψ _1 (Y)
+
Δψ( y_i,Y)·(i↔ j)
}
,
where we have introduced the following notation to contract the expression: x⃗=n⃗+α⃗, X=N+Kα, y⃗=m⃗+β⃗ and Y=M+Kβ.
The factor (i↔ j) means taking the term that it multiplies with inverted indexes i and j.
§.§ Zhang-Grabchak divergence estimator
In Ref. <cit.> Zhang and Grabchak proposed an estimator for the Kullback-Leibler divergence, defined as:
D_ KL^(z)
=
∑_i=1^K n_i/N{ ∑_v=1^M-m_i1/v∏_s=1^v(1 - m_i/M-s+1)
-
∑_v=1^N-n_i1/v∏_s=1^v(1 - n_i-1/N-s)
}
,
where v and s are dummy variables.
§.§.§ Expression of the Z-estimator
Schurmann <cit.> has shown that, in the entropy term of Eq. <ref>, the summation in v of the i-th element can actually be expressed in a more concise way as
∑_v=1^N-n_i1/v∏_s=1^v(1 - n_i-1/N-s)
=
Δψ (N, n_i)
,
times a factor n_i/N.
The sum of these terms returns the Shurman-Grassberger entropy estimator S_ SG = ∑_i=1n_i/NΔψ (N, n_i) <cit.>.
If we simply plug N=M+1 and n_i = m_i+1 in Eq. <ref>, we can show that the analogous i-th crossentropy term in Eq. <ref> is equal to the following:
∑_v=1^M-m_i1/v∏_s=1^v(1 - m_i/M-s+1)
=
Δψ (M+1, m_i+1)
.
Finally, if we substitute Eq. <ref> and <ref> in Eq. <ref>, we obtain:
D_ KL^(z)
=
∑_i=1^K n_i/N[
Δψ (M+1, m_i+1)-Δψ (N, n_i)
]
.
which is the expression in Eq. <ref> of the main text.
§.§.§ Relation between the DP and the Z estimator
The Z-estimator can be expressed as an a posteriori expected value of the at α=0 and β=1, up to an additive constant.
We start by showing the following relation
lim_α→ 0⟨ S |n, α⟩
=
∑_i=1^K n_i/NΔψ (N+1, n_i+1)
=
1-K/N
+
∑_i=1^K n_i/NΔψ (N, n_i)
,
which makes use of the fact that ψ(x+1)=ψ(x)+1/x.
Considering now the crossentropy term with β=1, and performing the same limit as before, we observe that
lim_α→ 0⟨ H |n, m, α, β
=1 ⟩
=
∑_i=1^K n_i/NΔψ (M+K, m_i+1)
=
Δψ (M+K, M+1) +∑_i=1^K n_i/NΔψ (M+1, m_i+1)
,
where we used the fact that Δψ(x, x) = ψ(x)-ψ(x)=0 to add the term Δψ (M+1,M+1) in the sum.
Recognizing the two terms in Eq. <ref> we subtract Eq. <ref> and <ref> to obtain that
lim_α→ 0⟨|n, m, α, β=1 ⟩
=
Δψ (M+K, M+1)
+
K-1/N
+
D_ KL^(z)
.
§.§ The DPM squared Hellinger divergence estimator
We compute the DPM estimator for the squared Hellinger divergence ^2 (Eq. <ref>).
We do so by starting from the Bhattacharyya coefficient <cit.>
(q⃗,t⃗)
=
∑_i=1^K√(q_i)√(t_i)
=
1 - ^2(q⃗,t⃗)
.
Its a priori expected value under the assumption of the prior
P_ prior(q⃗,t⃗) =p( q⃗,t⃗|α,β)
=
qαtβ
is equal to :
⟨|α , β⟩
=
∑_i=1^K(e^1/2∂_iB)(α⃗)/B(α⃗)(e^1/2∂_iB)(β⃗)/B(β⃗)
=
K B(1/2, Kα)/B(1/2,α)B(1/2, Kβ)/B(1/2,β)
,
where we used the shift property (Eq. <ref>) with parameter λ=1/2. Following the derivation of the in the main text, we choose a metaprior ρ_ H (α,β) to control the a priori squared Hellinger divergence ⟨^2 |α , β⟩ = 1 - ⟨|α , β⟩:
ρ_ H (z) =
∫dαdβ ρ_ H (α,β) δ( ⟨^2 |α , β⟩ -z ).
We define g(x)= √(K) B(1/2, Kx) / B(1/2,x), which is a function g : ℝ→ [0,1).
Using a similar Ansatz of the one in the main text, we obtain ρ_ H (α,β) = |∂_α g (α) ||∂_β g (β) |ϕ( ⟨^2 |α , β⟩ ), where the condition in Eq. <ref> imposes
ϕ ( z ) = ρ_ H (z) (1-z)^2/z(2-z).
We choose ρ_ H (z) to be log-uniform.
Finally, knowing that the calculation for the posterior expected squared Hellinger divergence is analogous to the a priori expectation, we obtain the DPM squared Hellinger estimator in Eq. <ref> and <ref>.
§.§ Numerical implementation
§.§.§ Computations with multiplicities
In the low sampling regime (sparse data), there is a limited number of values the counts can take, which means that many categories will see the same pairs of values x⃗_⃗i⃗=(n_i,m_i).
To reduce the computational cost associated to summation over the K categories, we introduce a set of “multiplicities” <cit.> contained in the vector ν_x⃗, where each entry is the number of instances that appear n times in the first sample and m in the second.
Since by construction the dimension of ν_x⃗≤ K, we expressed all summation in terms of the multiplicities vector. Given a function of the two counts f, the sum over all categories is:
∑_i=1^K f(x⃗_⃗i⃗) = ∑_x⃗ν_x⃗ f(x⃗),
where the last sum runs over the ensemble of distinct pairs of observed counts. In the case of double sums (e.g. for ^2), one needs to re-express the function as:
f(x⃗_⃗i⃗, x⃗_⃗j⃗)
=
δ_ij f_∥(x⃗_⃗i⃗) + (1-δ_ij)f_(x⃗_⃗i⃗,x⃗_⃗j⃗),
where f_∥ and f_ is the function f for i=j and i≠ j.
The summation over the terms in δ_ij is calculated as before, and the double summation is
∑_i,j f_(x⃗_⃗i⃗, x⃗_⃗j⃗)
=
∑_x⃗, x⃗'⃗ν_x⃗ν_x⃗'⃗ f_(x⃗, x⃗'⃗).
These formulas allow us to exploit vectorial expressions in the numerical calculations.
§.§.§ Numerical integrations
Similarly to Ref. <cit.>, to compute numerically the quantities ⟨|n⃗,m⃗⟩ (Eq. <ref>) and ⟨^2 |n⃗,m⃗⟩ (Eq. <ref>), we first seek for the maximum (α_*,β_*) of the quantity ℒ(α,β) (see Fig. <ref> for further details).
For accuracy, we perform this computation in logarithmic space of logα and logβ.
Rescaling ℒ(α,β) by its maximum, integrands are 𝒪(1) for (α,β) ∼ (α_*,β_*).
To find the maximum of the log-evidence (minimum of the opposite), we use the “Limited-memory BFGS” optimization algorithm as coded in the function “minimize”, module optimize of the Python package scipy (version 1.7.3).
We evaluate the integrals using the trapezoidal rule. From the
Hessian at the maximum of the log-evidence, we compute the standard
deviation in the α and the β-direction as if the
posterior was Gaussian. We use this standard deviation to pick a
range of parameters spanning 3 standard deviations on both sides of
the maximum. We heuristically chose the number of bins within
the ranges for the integration, to be equal to
10(K/N)^2 for α
(10(K/M)^2 for β).
§.§ Code availability
The software for the DP, DPM and alternative estimators of the Kullback-Leibler and the Hellinger divergence presented in this article are collected in a Python package which can be found in the repository at <https://github.com/statbiophys/catede>.
In addition, the package provides a Python version for the NSB entropy estimator <cit.>, and a NSB estimator for the Simpson index <cit.>.
§ SUPPLEMENTARY FIGURES
|
http://arxiv.org/abs/2307.07423v1 | 20230712124918 | Enhancing ECG Analysis of Implantable Cardiac Monitor Data: An Efficient Pipeline for Multi-Label Classification | [
"Amnon Bleich",
"Antje Linnemann",
"Benjamin Jaidi",
"Björn H Diem",
"Tim OF Conrad"
] | eess.SP | [
"eess.SP",
"cs.LG",
"68U10 (Primary) 68T10, 62H30 (Secondary)",
"J.3; I.5.3"
] |
Implantable Cardiac Monitor (ICM) devices are demonstrating as of today, the fastest-growing market for implantable cardiac devices. As such, they are becoming increasingly common in patients for measuring heart electrical activity. ICMs constantly monitor and record a patient's heart rhythm and when triggered - send it to a secure server where health care professionals (denote HCPs from here on) can review it. These devices employ a relatively simplistic rule-based algorithm (due to energy consumption constraints) to alert for abnormal heart rhythms. This algorithm is usually parameterized to an over-sensitive mode in order to not miss a case (resulting in relatively high false-positive rate) and this, combined with the device's nature of constantly monitoring the heart rhythm and its growing popularity, results in HCPs having to analyze and diagnose an increasingly growing amount of data. In order to reduce the load on the latter, automated methods for ECG analysis are nowadays becoming a great tool to assist HCPs in their analysis. While state-of-the-art algorithms are data-driven rather than rule-based, training data for ICMs often consist of specific characteristics which make its analysis unique and particularly challenging. This study presents the challenges and solutions in automatically analyzing ICM data and introduces a method for its classification that outperforms existing methods on such data. As such, it could be used in numerous ways such as aiding HCPs in the analysis of ECGs originating from ICMs by e.g. suggesting a rhythm type.
ECG, ICM, Classification, Semi-Supervised-Learning.
Enhancing ECG Analysis of Implantable Cardiac Monitor Data: An Efficient Pipeline for Multi-Label Classification
Amnon Bleich,
Antje Linnemann, Benjamin Jaidi, Björn H Diem,
and Tim OF Conrad
================================================================================================================
§ INTRODUCTION
§ INTRODUCTION
Automated classification of ECG data has been a field of active research over the past years, and various methods have been proposed to solve this task (see e.g. <cit.>). However, most of these approaches require that not only the ECG data itself has very good quality, but also that the data labels for each episode are available and correct. These requirements seem to be fulfilled in most clinical research setups (with professional ECG devices and trained staff), but are rarely met in the world of implantable devices, such as with Implantable Cardiac Monitors (ICMs).
ICMs are an important part of implantable cardiac devices and as such play a large role in medical diagnostics. They are becoming more and more common as a diagnostic tool for patients with heart irregularities. These devices are characterized by a relatively simple classification algorithm that allows detecting and sending of suspicious ECG episodes to HCPs for further inspection. Because this algorithm prioritizes sensitivity over specificity (as it can't afford to miss heart rhythm irregularities) many of the detections are false positives, i.e. the recorded episodes are often not a real indication of a problem. The outcome is an increasing amount of ECG data that is sent regularly to HCPs for manual diagnoses. As a result, the current bottleneck of the ICM in treating patients is not in the device itself, but rather in the scalability in terms of manual analysis done by HCPs. In other words, due to the rapid growth in the number of sent episodes, it is expected that in the near future HCPs will not be able to give the proper attention needed for analyzing it <cit.>. Therefore, to reduce the load on HCPs that analyze episodes recorded by ICMs, an automated method to assist HCPs in annotating and/or suggesting probable arrhythmia in ICM episodes is becoming essential.
Furthermore, most available methods and existing studies for ECG analysis focus on the relatively high quality of at least 2-lead ECG signals and contain high-quality manually assigned labels. In contrast to that, ICMs produce only 1-(variant) lead ECG data – or more correctly 1 subcutaneous ECG (sECG) data – with variant morphology due to variable implantation sites and are of comparably low resolution (128 Hz). In addition, manually labeled datasets often only provide inaccurate labels. Thus, available analysis methods for the better quality data (2-or-more leads) are hardly suitable for the analysis of the lower quality ICM data (see also section <ref> for a performance evaluation of these methods on ICM data).
In this paper, we address the above-mentioned problems and present a novel method that allows de-novo labeling and re-labeling for ECG episodes acquired from ICM devices. More specifically, the method allows assigning labels to segments of ECG episodes. Moreover, the method is applicable to low-resolution data-sets with small sample size, class imbalance, and inaccurate and missing labels. We compare our method to two other commonly used methods on such (or similar enough) data and show the superiority of our method (especially on minority classes). Although the overall performance has room for improvement, it can nevertheless be seen as a milestone on the way to automatically classify such data or e.g., serve as an arrhythmia recommender for HCPs.
This paper proposes a novel data analysis pipeline for ICM data. The highlights of this pipeline are:
* The novel method allows for de-novo labeling and re-labeling of ECG episodes acquired from Implantable Cardiac Monitors (ICMs). The main goal is to reduce the load on HCPs by assisting in annotating and suggesting probable arrhythmia in ICM episodes.
* The method is applicable to low-resolution data-sets with small sample size, class imbalance, and inaccurate and missing labels. This is a significant contribution as most available methods and existing studies for ECG analysis focus on high-quality manually assigned labels for at least 2-lead ECG signals, which are not suitable for the analysis of low-quality ICM data.
* The results of our experiments suggest that our new approach is superior to two other commonly used methods on ICM data, especially on minority classes. Although the overall performance has room for improvement, it can be seen as a milestone on the way to assisting HCPs as an arrhythmia recommender.
Before we dive into the details of our new method (section <ref>), we will first provide background information about ICM devices and the acquired data in the next section. We will then present our experiments and results (sections <ref> and <ref>), including comparison and evaluation with other established methods. Finally, we discuss and conclude our findings (sections <ref> and <ref>).
§ IMPLANTABLE CARDIAC MONITORS
Implantable Cardiac Monitors (ICMs) continuously monitor the heart rhythm (see <cit.> for a detailed overview). The device, also known as implantable loop recorder (ILR), continuously records a patient's subcutaneous electrocardiogram (sECG). Once triggered, it stores the preceding 50 seconds of the recording up to the triggering event, plus 10 seconds after the triggering event. (Note that there are cases where the recorded signal is less than 60 seconds due to compression artifacts). The result is a sECG which is (up to) 60 seconds long and in which we expect to see the onset and/or offset of an arrhythmia (i.e. the transition from normal sECG to abnormal one or vice versa). The stored episodes are sent to a secure remote server once per day.
Possible detection types include: atrial fibrillation (AF), high ventricular rate (HVR), asystole, bradycardia, and sudden rate drop.
Detections (or triggers) are based on simple rules and thresholds, namely: (i) the variability of the R-R interval exceeding a certain threshold for a set amount of time for AF, (ii) a predefined number of beats exceeding a certain threshold of beats-per-minute for HVR, (iii) mean heart rate below a certain threshold for a set time for bradycardia, and (iv) pause lasting more than a set amount of time for asystole.
In summary, data is retrieved daily from the device through a wireless receiver for long-distance telemetry. The receiver forwards the data to a unique service center by connecting to the GSM (Global System for Mobile Communication) network. The Service Center decodes, analyzes and provides data on a secure website, with a complete overview for the attending hospital staff. Remote daily transmissions include daily recordings of detected arrhythmia. Transmitted alerts were reviewed on all working days by a HCP who is trained in cardiac implantable electronic devices.
§.§ Challenges with ICM Data
Data acquired from an ICM device has several quite unique characteristics, compared to data acquired by more commonly used ECG devices, e.g. in a clinical setting. Some of the key challenges include:
Variant ECG morphologies: Unlike e.g., classical Holter ECG recording device, the ICM device allows for a variant implant site relative to the heart, which results in a variant ECG morphology. This variance in ECG morphology introduces an artifact that makes it difficult for machine learning models to compensate for.
No “normal” control group: Due to the nature of ICMs, a record is only sent when an abnormal event occurs. This means that the available data consist of - almost only - ECG episodes that contain an unusual event - which triggered the sending of the episode.
Inaccurate labels: The human factor in the manual labeling of thousands of episodes, each of which requires specialized attention, contributes to a degree of inaccuracy in the labels assigned to the samples. Furthermore, the varying definitions of labels among HCPs contribute to this issue. This is further increased by the fact that many annotation platforms allow the assignment of global labels only. This means that an annotation doesn't have a start and end time, but rather refers to the entire episode. This is unsuitable for the typical data-set produced by ICMs as these consist in the majority of cases of 60 seconds long episodes, in which the onset of a particular arrhythmia is expected - i.e. by definition a single heart rhythm (viz. label) cannot relate to an entire episode.
Class imbalance: The false detection ratio of the device results in the vast majority of episodes being Sinus episodes with some artifact that triggered the storing of the episode.
Small multi-label training-sets: Most supervised learning models of ECG data require relatively large data-sets of manually annotated ECG episodes. However, since it involves highly proficient personnel to thoroughly review every sample in the training data-set (especially in the case of multi-class labeling), these tend to be relatively small.
Low resolution: ICMs compress data before storing it, resulting in a low sampling rate (128 Hz compared to a minimum of 300 Hz in comparable applications).
Noise: Movements of patients might result in noisy amplitude which could trigger a device recording - this could result in the irregular class distribution in the data-set, especially regarding noise.
§ RELATED WORK
Various methods have been published in recent years for the automatic diagnosis of ECG data. In <cit.>, Stracina and co-workers give a good overview of the developments and possibilities in ECG recordings that have been made since the first successful recording of the electrical activity of the human heart in 1887. This review also includes recent developments based on deep learning. Most of the available analysis methods aim at the classification of ECG episodes, e.g., to determine whether an ECG signal indicates Atrial Fibrillation (AF) or any other non-normal heart rhythm.
Most state-of-the-art solutions for ECG classification use either feature-based models such as bagged/boosted trees <cit.>, SVM, k-nearest neighbor, etc. <cit.> or Deep Learning models such as convolutional neural networks (CNN) <cit.>. While Deep Learning approaches usually use the raw 1d ECG signal <cit.>, feature-based ECG arrhythmia classification models involve the extraction of various features from ECG signals, such as time-domain and frequency-domain features.
Spectral features are derived from the frequency components of the ECG signal and are extracted e.g. using Wavelet transform, Wavelet decomposition, and power spectral density analysis <cit.>.
Time domain features, on the other hand, are based on the P-waves, QRS complexes, and T-waves, including their regularity, frequency, peaks, and onsets/offsets <cit.>. Wesselius and co-workers <cit.> have evaluated important time-domain features, such as Poincaré plots or turning point ratios (TPR), as well as other ECG wave detection algorithms (such as P- and F-waves) and spectral-domain features, including Wavelet analysis, phase space analysis, and Lyapunov exponents. These time domain features usually require the detection of the respective waves, with the majority of algorithms focusing on the detection of QRS complexes due to their importance in arrhythmia classification and the fact that they are relatively easy to detect. For instance, a few of the most important features for ECG signal classification are based on RR intervals, including the time between two consecutive R peaks, with R-peaks commonly detected via the detection of QRS complexes <cit.>. Popular QRS detectors that are used by e.g. <cit.> are jqrs (<cit.>) which consists of a window-based peak energy detector, gqrs <cit.> which consists of a QRS matched filter with a custom-built set of heuristics (such as search back) and wqrs<cit.> which consists of a low-pass filter, a nonlinearly scaled curve length transformation, and specifically designed decision rules.
§.§ PhysioNet/CinC 2017 Challenge
We were also interested in methods that have been shown to be successful in applied settings, such as challenges or real-world comparisons. For this, we searched the literature for recent surveys. We used PubMed in January 2023 with the query “single lead ECG machine learning review” OR “single lead ecg AI review” and restricted the time-frame to 2020 and newer. The results were manually inspected and led us to a challenge that took place in 2017 and was related to our approach.
The PhysioNet/Computing in Cardiology 2017 Challenge <cit.> was an organized competition by the PhysioNet initiative - a research resource that offers open-access to databases of physiological signals and time series, and the Computing in Cardiology (CinC) - an annual conference that focuses on the application of computer science techniques to problems in cardiovascular medicine and physiology. In 2017 the two collaborated to conduct the PhysioNet/CinC 2017 Challenge, which was focused on the detection of atrial fibrillation (AF) from short ECG recordings. This challenge was intended to encourage the development of novel algorithms and techniques and attracted experts from diverse fields, including computer science, cardiology, and signal processing, thereby facilitating a multidisciplinary approach to the problem. The results of the competition demonstrated the potential of data-centered techniques, such as deep learning and ensemble methods, for enhancing the accuracy of AF detection.
The best-performing approaches include solutions based on deep learning and decision trees. For example, the winning solution of the challenge by Datta and co-workers <cit.> approaches multi-label classification via cascaded binary Adaboost <cit.> classifiers along with several pre-processing steps including noise detection via signal filters, PQRST points detection and 150 features that are derived from them.
Another example is the approach by one of the top-scoring teams that proposes a deep learning based solution<cit.>, consisting of a 24-layer CNN and a convolutional recurrent neural network.
Two of the top-scoring algorithms of this challenge, for which source code was publicly available, are: (1) a feature-based approach that extracts 169 features for each episode and uses an ensemble of bagged trees as well as a multilayer perceptron for classification and (2) a deep learning-based method using a 34-layer ResNet. Both approaches have been compared in a paper by Andreotti and co-workers <cit.> and are still considered state-of-the-art approaches for short single-lead ECG data classification. We will use these two approaches as a baseline to compare to our approach (see section <ref> for more details).
§ METHOD DETAILS
In this section, we provide a detailed description of our pipeline, which we developed to analyze and classify 60-second episodes of single-lead subcutaneous ECG (sECG) data. The main steps include segmentation, noise detection, embedding, and clustering (see Figure <ref>). The resulting clustering is then used to construct the final classifier, which allows for the prediction of labels for new episodes. Each of these pipeline steps will be discussed in detail in the following subsections.
§.§ Input Data
The input to the pipeline is a data set consisting of multiple 60-second sECG episodes for which at least one label has been assigned by medical experts. Every episode potentially contains the onset and/or offset of arrhythmia (see section <ref>) and thus, by definition, more than one rhythm type per episode. The annotation platform used for our dataset allows assigning labels to episodes in a manner of selection out of a bank of 38 possible labels. However, due to the relatively small data-set, in order to reduce the number of classes and thus increase the number of episodes belonging to each class, the various labels are grouped into 5 categories: normal, pause, tachycardia, atrial fibrillation (denote aFib) and noise.
§.§ Training
The training pipeline consists of several steps, which are shown in Fig.<ref>. The following paragraphs explain the building blocks of this pipeline in more detail.
startstop = [rectangle, rounded corners,
minimum width=3cm,
minimum height=1cm,
text centered,
draw=black,
fill=red!30]
io = [trapezium,
trapezium stretches=true,
trapezium left angle=70,
trapezium right angle=110,
minimum width=3cm,
minimum height=1cm, text centered,
draw=black, fill=blue!30]
process = [rectangle,
minimum width=3cm,
minimum height=1cm,
text centered,
text width=3cm,
draw=black,
fill=orange!30]
decision = [diamond,
minimum width=3cm,
minimum height=1cm,
text centered,
draw=black,
fill=green!30]
arrow = [thick,->,>=stealth]
§.§.§ Segmentation
In order to have homogeneous episodes, i.e., one heart rhythm per episode (which correlates to one label per episode), we divide the 60-second episodes into six 10-second, non-overlapping sub-episodes. Each sub-episode is assigned the label of its parent episode. Note that this label is presumed to be inaccurate, as it could have been chosen due to a rhythm that appears in another sub-episode of that episode and the later steps in the pipeline will correct for this.
§.§.§ Noise Detection
ECG data - and in particular data from ICM devices - typically contains noise, such as baseline wander, high/low frequency interruptions and the like (<cit.>). Although certain technologies in ICM are aimed at reducing this effect (<cit.>), filtering out noise is essential and improves downstream processes, such as detection of the QRS complex. Several approaches have been adjusted from the general field of signal processing, such as extended Kalman filter, median, low or high-pass filters or Wavelet transform <cit.>. In the proposed method, we suggest using an algorithm based on empirical mode decomposition (EMD) <cit.>, which has been shown to deal well with high-frequency noise and baseline wander. The pseudocode in Algorithm <ref> describes a step-by-step description of the CEEMD (Complete Ensemble Empirical Mode Decomposition <cit.>) - based algorithm used in order to detect noisy ECG signals. It first decomposes the signal into its IMFs (intrinsic mode function <cit.>) using CEEMD - where the first IMFs account for high-frequency signal and the last IMFs account for low frequency (e.g. baseline wander, see figure <ref>). It then sums up the first 3 IMFs (the components accounting for the high-frequency signal - denoted by Hn) and checks over a sliding window whether Hn in that window contains a value greater than a certain threshold and if it has more than a set number of zero-crossings. The result of this is a binary vector with 1s where both conditions are met and 0s otherwise (denote Gn - see figure <ref>). Finally, if a certain gate (a sequence of 1s in the Gn vector) is longer than a set threshold, the corresponding part of the original ECG signal is marked as noise. All the parameters (namely threshold, window size and gate length) were selected empirically using Jaccard index to quantify the overlap between the detected noise and the segments annotated as noise in the test dataset, as the objective function. To conclude, in order to see the relevance of the noise detection algorithm to our data, we marked all the noisy segments in it and compared with the assigned labels. Results of this step can be found in the result section (section <ref> and in particular Fig. <ref>).
§.§.§ Embedding
To be able to deal with the high-dimensional ECG data and prevent overfitting (due to the curse-of-dimensionality problem) as much as possible, we perform a projection of the data (approximately 7600D for an episode or 1300D for a sub-episode) into a low-dimensional (2D) space. We suggest the following steps for the dimensionality reduction (see Figure <ref>). First, we (1) perform QRS complex detection on the ECG signals, to identify the R-peaks (Figure <ref>). Based on the R-peaks, we can (2) transform the ECG signals into a Lorenz plot (Figure <ref>), which we (3) discretize into a 2D histogram, using a N× N grid (Figure <ref>). Then, we (4) flatten the 2D histogram into a 1D vector with N · N elements. The last step is (5) to use t-SNE to project the resulting vectors into the 2D space to get the final embedding. In more details:
(1) R-peak Detection: The R-peak is a component of the QRS complex which represents the heart-beats in an ECG signal. There are several algorithms for QRS detection such as gqrs <cit.>, Pan-Tompkins <cit.>, maxima search <cit.> and matched filtering. In our pipeline, we use an aggregation of these four approaches via a kernel density estimation based voting system.
(2) Lorenz Plot Transformation: In the context of ECG, a Lorenz plot (also called Poincaré plot) <cit.>, is the plotting of dRR(i) vs. dRR(i+1) for each R peak i in an ECG signal, with dRR(i)=RR(i+1)-RR(i), where RR(i) is the i'th R-R interval (the time between two consecutive R peaks). This visualization is often used by HCPs to detect arrhythmia.
(3) Discretization: In order to produce a 2D histogram of the Lorenz plot, the plot is divided into equal sized 25 bins by overlaying it with a 5-by-5 grid. The histogram is thus a count of points (viz. dRR(i)/dRR(i+1) intervals) in each bin. The result is a 5-by-5 matrix that is flattened to form a 25 dimensional vector.
(4) t-SNE Embedding: t-SNE is an algorithm for dimension reduction <cit.>. It is primarily meant for visualization (i.e. projection of multi-dimensional data on a 2D or 3D plane) and since it does not preserve distance between points it is not suitable for distance-based clustering (such as DBSCAN - the algorithm used in our work). However, since it maximizes the likelihood that contiguous data points will not be in the same cluster, it performs better than alternative embedding methods, such as LDA <cit.> and PCA <cit.>.
§.§.§ Clustering
There are various clustering approaches described in the literature, with popular ones including k-means, spectral clustering or DBSCAN <cit.>. In our pipeline, we chose the DBSCAN algorithm since it outperformed other approaches we tried on our datasets. DBSCAN stands for Density-Based Spatial Clustering of Applications with Noise and as its name suggests it clusters together areas with high density of points separated by areas with low density while leaving points in sparse areas as outliers.
In order to select the maximal distance parameter, we created a k-distance plot with k=2 × number of features, which in the case of 2D t-SNE is k=4. The reasoning for this is that the distance threshold should be proportional to the number of dimensions to account for sparsity resulting from it, with 2 × k being a common rule of thumb. Once the minimal point number is set, the k-distance plot is a sorted representation of the average distance of each point in the data to its k nearest neighbors. The area after the elbow would therefore represent points in the data that could be considered outliers in this sense (as finding k nearest neighbors would require searching increasingly further) so this could mark a reasonable maximal distance to look for neighbors. The latter resulted in a distance of eps=1.5. We then explored the neighborhood of that point interchangeably with the minimal number of points. For a minimal number of points, we simply tried different values on a scale from a conservative lower to an upper bounds (2 to 30) until we found a visually satisfying results, with the number of outliers (points not belonging to any cluster) not exceeding 10% of the number of sub-episodes. The optimal values for our case were 0.75 for maximal distance and 15 as the minimal number of points.
§.§.§ Building the classifier
From the clustering results from the previous paragraph, a classifier that enables label prediction of previously unseen episodes can be created. In order to classify a sub-episode that belongs to a certain cluster (denote cluster c) we compute the p-value of the label proportions for all labels in that cluster, given the label proportion in the dataset. The p-value is computed using the cumulative binomial distribution defined as:
for cluster c, ∀ l ∈labels,
F(k_cl,n_c,p_l):=Pr(X < k_cl)=∑_i=0^k_cl-1n_ci p_l^i (1-p_l)^n-i
i.e. the probability for a maximum of k_cl successes in n_c trials with probability p_l for a success in a single trial.
In our case, k_cl denotes the observed number of sub-episodes with label l in cluster c, n_c the size of cluster c and p_l the proportion of label l in the dataset (defined as p_l:=d_l/d with d_l=label count in the dataset and d=dataset size ).
Formula <ref> is used to assign p-values of P(l,c)=Pr(X ≥ k_cl)=1-Pr(X<k_cl) to each label l in cluster c and the label with the lowest p-value is assigned to the sub-episode. Or in words - for a given label and cluster with label count of k_cl in that cluster, the p-value computed is the probability to observe at least k_cl sub-episodes of that label in that cluster given the label's proportion in the dataset and the cluster size. The lower the probability, the higher the significance of that cluster being labeled with that label and therefore the label with the lowest p-value in a cluster is set to be the label of this cluster and all sub-episodes clustered to it will be labeled as such.
To use the classifier for predicting the labels of new, unseen episodes, the given ECG episode is segmented into 10 seconds sub-episodes and each is projected in the same way as described above. Then, we perform a k-nearest neighbor search (with k=1) on each sub-episode. We assign the cluster of the nearest neighbor of each sub-episode as the cluster of that sub-episode and assign the label accordingly.
Note: Several approaches could be taken for the choice of label based on the p-value, according to the use case. For example, if one wants to ensure high sensitivity for several classes at the expense of specificity for, say, pause rhythm, a threshold could be set such that if the p-value for pause is lower than that threshold, the cluster's label will be “pause” regardless of other classes with possibly lower p-values. More on that in section <ref>
§ EXPERIMENTAL SETUP
In the following sections, we introduce the two algorithms selected as comparative benchmarks for our novel method, and detail the dataset used for this study.
§.§ Baseline Methods
Based on our literature search (Section <ref>), we selected two algorithms as baseline methods that we use to compare our new method to:
(1) a feature-based approach that extracts 169 features for each episode and uses an ensemble of bagged trees as well as a multilayer perceptron for classification. (2) A deep learning-based method using a 34-layer ResNet. The two methods were used as proposed by the authors, with the following adjustments made to match our study:
* The feature-based approach uses Butterworth band-pass filters that were adjusted for lower frequency episodes (as the original solution deals with a frequency 300 Hz and our data has 128 Hz).
* Class imbalance - The original method deals with this problem by adding episodes from minority classes (namely AF and noise) from additional ECG datasets. Since we wanted to evaluate the case where the data is imbalanced, we skipped this step.
* In the deep learning approach, padding is performed for episodes shorter than the selected size, in our case 60 seconds. Due to the nature of labeling in our test dataset and the fact that heart rhythms tend to last few seconds rather than a minute, the test samples must be significantly shorter than 60 seconds (10 seconds in our method) so every sub-episode was padded by 0 values to complete a 60-second episode.
§.§ Data
The dataset used for this study was produced by the BIOTRONIK SE & Co KG BIOMONITOR III and BIOMONITOR IIIm <cit.> and anonymized for this analysis.
The data produced by the device is characterized by a single-lead, 60 seconds long ECG episodes with a sample rate of 128 Hz (7680 data points per 60 seconds episode).
Out of the 60 seconds long stored episodes, 3607 were labeled by up to three HCPs, each assigning a global label (a label is given to an entire episode albeit more than one label can be given in case the HCP recognizes more than one heart rhythm in an episode) and 4710 episodes with similar characteristics but where the labels were assigned with start and end time (i.e. per segment as opposed to a global label). In this study, the former was used for learning (denote training-set) and the latter for evaluation (denote test-set).
It should be noted that although the test-set labels resolve the primary problem with the training dataset - that a label is assigned to the entire episode - they might still have the additional inaccuracies, and thus the test set is still not fully trustworthy as ground truth labels.
The labels of the training- and test-sets are assigned as follows: The training-set has global labels (i.e. one or multiple labels for the entire episode without start/end time - see section <ref> for details) and therefore the labels of all sub-episodes are derived from that of the parent episode and if more than one label is present the episode is duplicated. The test-set has non-overlapping segmented labels assigned by a different annotation platform that allows for such annotations (i.e., each label has start and end time - see section <ref> for details) and so a sub-episode is assigned the label that has a start and end time that overlaps at least half of it (5 or more seconds in case of 10 seconds segmentation). If there was no such label, the sub-episode is removed from the dataset. The label distribution of the datasets can be seen in figure <ref>. The figure highlights the class imbalance where e.g. in the training set 2795 episodes represent normal episodes, 546 pause episodes, 467 tachycardia episodes, 985 aFib episodes, and 932 noise episodes. (Note that multiple labels per episode are counted once for each label, and therefore the number of labels sums up to more than the number of episodes).
The reasoning for the selection of test and train sets is that the training-set presents the difficulties referred to in section <ref> and the test-set contains similar difficulties but due to the different labeling system, the labels are assumed to be more accurate.
§ RESULTS
The following section presents the results of our experiments and compares the performance of our proposed method for classifying cardiac rhythms to the described baseline methods. Our prediction pipeline involves a series of steps that enable the generation of sECG labels, as described above. The following sections describe the application of our pipeline to sECG data and the achieved results, starting with the noise detection step.
§.§ Noise Detection: Only Limited Effects
Our proposed pipeline begins with segmentation of the given episodes, followed by the noise detection step. To understand the influence of this noise detection step, we performed two experiments: (A) We performed the full pipeline without the noise detection (and removal) step. (B) We performed the full pipeline, including the noise detection step. Here, 1444 sub-episodes (out of 25186) were identified as noise and were removed from further processing. We then looked at the ground-truth labels of the sub-episodes that were labeled as “noise” by the noise identification step. As shown in Fig. <ref>, the majority of the sub-episodes detected were indeed noisy parts of the ECG data.
Afterwards, we compared the overall performance of pipeline A (“without noise detection step”) and pipeline B (“with noise detection step”). The comparison reveals that removing the noisy sub-episodes before the training does have only a limited effect overall (see Table <ref> for details). The only class with a significant change in performance is the class “noise” with an increased F1-score from 0.24 to 0.30 and, more significantly, an increased sensitivity from 0.34 to 0.61.
§.§ Embedding
Applied to our 10s segmented combined test- and training dataset, the pipeline described in section <ref> resulted in a 2D sub-episode projection that is illustrated in Figure <ref> with the class distribution of the training sub-episodes highlighted in green. The figure visualizes the effectiveness of the embedding process, where the different classes tend to concentrate in different areas of our 2D space.
§.§ Clustering
We used the DBSCAN algorithm to cluster the 2D projection of the embedded points from the previous step. Fig. <ref> shows the results of this clustering step applied to the full dataset. One can see that the clustering algorithm manages to separate the dataset in an organic manner via areas of lower density of data points (as opposed to e.g. arbitrary pentagonal clusters, or many small/few large clusters). In addition, the visualization of the clusters combined with the class distribution depicted in figure <ref> can suggest probable labels that will be given to certain clusters as part of the labeling procedure (for clusters with high density of a certain class).
§.§ Overall Performance: Better label prediction and lower runtime
The overall performance of our new method compared to the two selected competitive methods is summarized in table <ref>. The results show that our method's prediction outperforms the other two methods in almost all classes, based on the F1 score, especially in the minority classes. As can be seen in the table, there is quite a difference in performance between the various heart rhythm types. The class “normal rhythm” seems to be easiest to classify. This might be due to the large number of training samples of type “normal” in the training-set - almost half (49.1%) of all samples have this label. This is the only class where the baseline method outperforms our method, by a difference of 0.06 in the F1 score. Moreover, this is arguably the rhythm we are the least concerned about, and even in this rhythm our method significantly outperforms the baseline in specificity with 0.84 without noise (Our A) cleanup and 0.89 with noise cleanup (Our B) compared to 0.35 and 0.62 for the feature-based and deep learning methods respectively. In the case of Normal heart rhythm, in most use-cases (such as alerting patients of possible heart issues they might have), specificity is preferred over sensitivity - i.e. we require to correctly reject Normal labels for non-normal episodes even at the expense of incorrectly rejecting Normal labels.
Furthermore, it is apparent that our method deals better with class imbalance, with the performance of minority classes significantly better than the baseline. In addition, our method has a significantly lower runtime: while our method needs about 10 minutes on the machine used for this study the baseline methods need several hours to finish computation on the same machine. Also, our method has the ability to 'fix' inaccurate training labels as well as label shorter segments of the episodes via semi-supervised learning.
§ DISCUSSION
The proposed method aims to reduce clinician workload by recommending and emphasizing certain insights, but not with a focus on making autonomous decisions. The performance of the method supports this tendency, as while it outperforms existing methods for the use case for which it is intended, it is not as accurate as a professional's manual diagnosis. In this situation, the use-case can affect model parameters, which could have a significant effect on the model's efficacy. According to the authors of this work, one of the strengths of this method is its adaptability to various scenarios through small modifications. In order to decrease false-negatives despite an increase in false-positives (falsely assigning normal episodes with non-normal labels), one could call the non-normal episode even if it has a higher p-value than the normal label. Another option is to set distinct thresholds for various labels, as required by the use case. Such modifications could even be made to a point where the false-negative rate is close to zero, thereby reducing the number of episodes sent to clinicians. In such a case, additional research should be conducted to determine whether its use as a filter on top of the ICM device's algorithm reduces the number of false-positives without increasing the number of false-negatives. As opposed to the Lorenz plot histogram, larger modifications are possible, such as using a 2D projection of a distinct set of characteristics. Such modifications, however, necessitate additional research.
Another noteworthy aspect of the method is its potential to be incorporated into a larger decision support system, which could combine the strengths of various techniques or algorithms. In such a framework, the proposed method could serve as a pre-processing phase or supplementary tool to improve the system's overall performance. This integration could facilitate stronger decision-making and reduce the workload of clinicians further.
When implementing the proposed method in a real-world clinical setting, user-centered design principles should also be considered. Usability, interpretability, and user acceptability will be vital to the successful adoption of the method by healthcare professionals. Future research could investigate methods for enhancing the method's user interface and providing meaningful feedback to clinicians, thereby ensuring that the system's recommendations are in line with their requirements and expectations.
Finally, the method's ethical implications must be exhaustively examined. As the method is intended to reduce clinician burden rather than replace human decision-making, it is crucial to ensure that its implementation does not inadvertently absolve healthcare professionals of responsibility or liability. This may involve developing explicit guidelines for the application of the method, establishing robust monitoring mechanisms, and promoting transparency and accountability in the design and implementation of the system.
In conclusion, the proposed method contains great potential for reducing clinician workload and enhancing the effectiveness of medical diagnosis. However, additional research is required to investigate its adaptability to diverse scenarios, optimize its performance for different use cases, and integrate it into a larger framework for decision support. In order to ensure its successful adoption in clinical practice, it is also essential to address the human factors and ethical considerations associated with its implementation.
§ CONCLUSION & OUTLOOK
The research presented in this paper demonstrates that automatic classification of data derived from ICM devices can be applied effectively for a variety of practical applications, demonstrating superior performance compared to existing state-of-the-art techniques.
Despite inherent complexities, our study demonstrates that achieving a satisfactory level of automatic diagnosis with ICM device data is feasible. This includes the use of ECG clustering as a supplementary tool for medical professionals managing increasing volumes of ECG episodes and the implementation of clustering as a preparatory step for segment adaptive parameter optimization during post-processing. Notably, our proposed method excels in dealing with underrepresented categories in ICM sECG data sets.
As unlabeled ICM data rapidly becomes one of the most prevalent types of ECG data, its anticipated volumetric growth emphasizes the importance of precise labeling. Unfortunately, the lack of specific labels diminishes its practical utility. This study reveals that contemporary classification algorithms struggle with such datasets. Nonetheless, the method proposed in this study provides a promising foundation for annotating such data, which could pave the way for large ECG episode databases with labels. This could circumvent existing restrictions and unlock a variety of applications.
The present study pioneers a flexible methodology that enables users to develop customized extensions and modifications. Several possibilities within the proposed pipeline have already been investigated, but future research may consider additional refinements to optimize overall performance based on the data and devices in use. Limiting label prediction to high-confidence sub-episodes (as indicated by low p-values) may be an effective strategy for evaluating automated classification, while leaving more complex sub-episodes for manual diagnosis. As an alternative to static 10-second sub-episodes, dynamic episode segmentation may be utilized. This strategy would entail the identification of specific instances during an episode in which the cardiac rhythm shifts, enabling the segmentation of episodes based on this criterion. This would result in sub-episodes with similar cardiac rhythm types. In addition to the high-frequency noise detector presented in this study, non-high-frequency noise detection techniques, such as clipping or missing data, should be considered for comprehensive noise reduction.
In conclusion, this study provides not only a powerful new tool for the automatic classification of ICM device data, but also a foundation upon which further advancements can be built.
§ ACKNOWLEDGMENTS
§ ACKNOWLEDGMENT
This work was supported by the German Ministry for Education and Research (BMBF) within the Berlin Institute for the Foundations of Learning and Data - BIFOLD (project grants 01IS18025A and 01IS18037I) and the Forschungscampus MODAL (project grant 3FO18501).
amsplain
|
http://arxiv.org/abs/2307.07622v1 | 20230714204608 | Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime | [
"Conrado da Costa",
"Jonathon Peterson",
"Yongjia Xie"
] | math.PR | [
"math.PR",
"60K37"
] |
§ INTRODUCTION
Random walks in cooling random environments (RWCRE) are a model of random walks in dynamic random environments introduced by Avena and den Hollander in <cit.>.
This is a model for random motion in an inhomogeneous environment which experiences “shocks" at certain prescribed times when the entire environment is resampled.
By adjusting the sequence of times when the environment is resampled, the RWCRE model interpolates between that of random walk in a random environment (RWRE)
where the environment is “frozen" and never resampled
and that of a simple random walk.[if the environment is resampled on each step, then it is easy to see that the annealed distribution of the RWCRE is the same as that of a simple random walk.]
The adjective “cooling" was attached to the model because if the gaps between successive resamplings of the environment increases without bound then the environment is becoming more and more “frozen" as time goes on, and in this case we might expect the walk to retain some of the strange asymptotic behaviors of a RWRE.
Naturally, the behavior of a RWCRE depends on both the distribution μ that the environments are sampled from and the sequence of times τ = {τ(n)}_n≥ 1 (called the cooling map) at which the environment is resampled. Previous results <cit.> have shown that not only can one retain some of the characteristics of a RWRE by choosing a rapidly growing cooling sequence, but that the model of RWCRE can exhibit new limiting distributions (such as tempered stable distributions or sums of independent copies of Kesten-Sinai random variables) which do not occur in either the RWRE model or simple random walks .
The study of limiting distributions of RWCRE has been divided according to the type of the limiting distribution for the RWRE model with distribution μ on environments. The case where the corresponding RWRE model is recurrent was studied in <cit.> and <cit.>.
The limiting distributions for transient RWRE are characterized by a parameter κ > 0 which depends on the distribution μ on environments <cit.>.
The limiting distributions for the cases κ > 2 (the diffusive, Gaussian regime) and κ∈ (1,2) (the ballistic, stable regime) were studied in <cit.> and <cit.>, respectively.
In the present paper, we give the limiting distributions for RWCRE in the cases κ∈ (0,1) (the sub-ballistic, non-stable regime) and κ = 2 (the Gaussian, super-diffusive regime). The only remaining case (κ=1) is left for a future work.
Our main result in the case κ∈ (0,1) gives examples of new limiting distributions (sums of independent Mittag-Leffler distributions), while our results in the case κ=2 show that the limiting distribution is always Gaussian but with a non-trivial scaling factor that depends very delicately on the specifics of the cooling map.
In the remainder of the introduction we will briefly recall the model of one-dimensional RWRE as well as some of the relevant results that we will use.
Then we will introduce the model of RWCRE and state our main results on the limiting distributions in the cases κ∈ (0,1) and κ=2. In the process of proving the limiting distributions for the RWCRE we also obtain some new results for RWRE which may be of independent interest. We state some of these new RWRE results in the introduction as well.
§.§ RWRE
Here we will give a brief overview of the model of one-dimensional RWRE. The interested reader can see the lecture notes of Zeitouni <cit.> or the various references below for more details.
Throughout the paper we use the notation
N_0 := N∪{0} for the set of non-negative integers.
The classical one-dimensional (static) RWRE model is defined as
follows. Let ω=(ω(x),x∈Z) ∈ [0,1]^Z be a
one dimensional environment. The random walk in environment
ω starting from z ∈Z is the probability law
P^ω_z on the space of trajectories Z^N_0 which
corresponds to the Markov chain Z = (Z_n)_n∈N_0 on Z
with initial condition z and transition probabilities
P^ω_z(Z_n+1 = x + e | Z_n = x)
= {[ ω(x) e = 1,; 1 - ω(x) e = - 1, ].
n ∈N_0.
Let G be the sigma algebra on the space of trajectories
Z^N_0. By the monotone class theorem, one can verify the
measurability of the map ω↦ P_z^ω (G) for any
G ∈G and z ∈Z. Thus, for any probability measure μ on the space [0,1]^Z of environments we can define the
probability measure P^μ_x on Z^N_0 as the semi-direct product
P^μ_x(·):= μ⋉ P_z^ω(·) = ∫ P_z^ω(·) μ (ω).
The stochastic process Z = (Z_n)_n∈N_0 is called a RWRE and the distributions P^ω_x(·) and P^μ_x(·) are referred to as the quenched and annealed laws of the RWRE, respectively.
A standard assumption on the distribution μ on environments, which we will also make here, is that the environments are i.i.d.
That is,
μ = α^Z,
for some probability distribution α on [0,1].
We write ⟨·⟩ to denote the expectation w.r.t. α.
A crucial quantity to characterize the asymptotic properties of RWRE
is the ratio of the transition probabilities to the left and to the
right at the origin
ρ_0 = 1 - ω_0/ω_0.
For the remainder of the paper, we assume that
⟨logρ_0⟩ <0,
which, as shown by Solomon <cit.>, guarantees right
transience for the RWRE; that is, P^μ_0( lim_n→∞ Z_n = ∞ ) = 1.
In addition to (<ref>) and (<ref>), we will also assume the conditions on the distribution μ given by the following definition.
We say that a measure μ on environments is κ-regular for some κ>0 if it satisfies (<ref>), (<ref>), the distribution of logρ_0 is non-lattice,
⟨ρ_0^κ⟩ = 1,
and ⟨ρ_0^κ+⟩ < ∞ for some >0.
Since the moment generating function M(t) = ⟨ e^t logρ_0⟩ = ⟨ρ_0^t ⟩ is convex, if (<ref>) holds
then under mild additional assumptions
then there is a (unique) κ>0 such that
(<ref>) holds.
The additional technical condition ⟨ρ_0^κ+⟩ < ∞ can be seen as a sort of mild ellipticity condition.
For some results that we will use, such as the limiting distributions from <cit.>, a weaker ellipticity condition ⟨ρ_0^κ (logρ_0)_+ ⟩ < ∞ is sufficient. However, the stronger ellipticity condition ⟨ρ_0^κ+⟩ < ∞ is needed for the precise large deviation results from <cit.> that will be instrumental in our analysis (see (<ref>) below).
The parameter κ given by (<ref>) characterizes a number of aspects of the asymptotic behavior of the RWRE.
For instance,
since the convexity of t ↦⟨ρ_0^t ⟩ implies that κ > 1 if and only if ⟨ρ_0 ⟩ < 1, Solomon's LLN for RWRE in <cit.> can be written as
lim_n Z_n/n = v =
0 if κ≤ 1
1 - ⟨ρ_0⟩/1 + ⟨ρ_0⟩ if κ > 1,
P^μ_0-a.s.
That is, the RWRE is sub-ballistic (v=0) if κ∈ (0,1] and ballistic (v>0) if κ > 1.
The parameter κ also appears in many other results for one-dimensional RWRE, including the characterization of the limiting distributions for transient RWRE <cit.> and identifying the correct sub-exponential rate of decay for certain large deviation probabilities <cit.>. In this paper we will be concerned only with the cases when κ∈ (0,1) and κ=2, so next we will recall some of the limiting distribution and large deviation results that are known for these cases.
§.§.§ The sub-ballistic, non-stable case: kappa in (0,1)
The following theorem states the limiting distribution proved in <cit.> for RWRE in the regime κ∈ (0,1).
We refer to this as the sub-ballistic, non-stable regime because the walk has limiting speed v=0 by (<ref>) and the limiting distribution is non-stable – in contrast to the case κ=1 where the walk is sub-ballistic and the limiting distribution is a 1-stable distribution.
We note that here and throughout the paper we will use ⟹ to denote convergence in distribution.
Let (Z_n)_n ∈N_0 be a RWRE with distribution μ on environments that is κ-regular with κ∈ (0,1).
There is a constant b>0 such that
under the annealed law
Z_n/n^κn→∞⟹,
where is a non-negative random variable with Laplace transform
E[ e^-λ ] = ∑_n=0^∞(-bλ)^n/Γ(1+κ n), λ > 0.
The characterization of the limiting distribution in (<ref>)
is quite different from what is given in <cit.>. Indeed, in <cit.> the limiting distribution is
of the form (S)^-κ where S is a κ-stable which has Laplace transform E[e^-λ S] = e^-c λ^κ for some c>0.
However, as can be seen from <cit.>[Section XIII.8] these two characterizations are equivalent.
Since the Laplace transform in (<ref>) can be written as ℰ_κ(-bλ) where
ℰ_κ(z) = ∑_n=0^∞z^n/Γ(1+κ n) is the Mittag-Leffler function with parameter κ,
we say that is a Mittag-Leffler random variable.
There is another family of non-negative random variables which also bear the name “Mittag-Leffler." These are non-negative random variables Y_κ with cumulative distribution functions given by P(Y_κ≤ x) = 1 - ℰ_κ(-x^κ), for x≥ 0, and have Laplace transform E[e^-λ Y_κ] = 1/1+λ^κ.
To distinguish these two families, the random variables Y_κ are said to have Mittag-Leffler distribution of the first kind, whereas are said to have Mittag-Leffler distribution of the second kind.
Since we will only be concerned with the Mittag-Leffler distributions of the second kind in this paper, we will omit the descriptor “of the second kind" when referring to for the remainder of the paper.
§.§.§ The Gaussian, non-diffusive case: kappa equals 2
The limiting distribution results in <cit.> show that RWRE with κ-regular distributions μ have Gaussian limiting distributions only when the parameter κ≥ 2. However, the limiting distribution has diffusive √(n) scaling only when κ>2 whereas the case κ=2 has non-diffusive scaling √(nlog n).
The following theorem collects this limiting distribution result and some large deviation results that we will use in the remainder of the paper.
Let (Z_n)_n ∈N_0 be a RWRE with distribution μ on environments that is κ-regular with κ = 2. Then, the following results hold.
Limiting distribution <cit.> There is a constant b>0 such that under the annealed law
Z_n - nv/b√(nlog n)n→∞⟹Φ,
where Φ is a standard Gaussian random variable.
Large deviations <cit.> The sequence of random variables {Z_n/n}_n ≥ 1 satisfies a large deviation principle with speed n and good, convex rate function I_μ(x) with
the property that I_μ(x)>0 x ∉ [0,v].
In particular, for any ϵ>0 there is a constant C_ϵ>0 such that
P^μ_0( Z_n - nv > ϵ n ) ≤ e^-C_ϵ n and
P^μ_0( Z_n < -ϵ n) ≤ e^-C_ϵ n,
for all n large enough.
Moderate and large deviation slowdowns <cit.> There is a constant K_0>0 such that
lim_n→∞sup_√(n)log^3 n ≤ x ≤ nv - log n | P^μ_0(Z_n - nv < -x)/(nv-x)x^-2 - K_0 | = 0.
§.§ RWCRE
A random walk in a cooling random environment (RWCRE) is a random walk in a
space-time random
environment built by partitioning _0 into a sequence of intervals,
and assigning independently to each interval an environment sampled
from μ. Formally, let (T_k)_k∈ be an increment sequence
such that T_k∈. We will refer to this sequence as cooling
increment sequence. We denote further by
τ(k):=∑_i=1^kT_i the k-th cooling time, i.e. the time
at which a new environment is freshly sampled from μ. We will
refer to τ as the cooling map. For convenience of notation we will let τ(0) = 0 so that T_k = τ(k)-τ(k-1) for all k≥ 1.
The RWCRE (X_n)_n∈_0 is defined as follows.
Let ω̅ = (ω^(k))_k≥ 1= ( (ω^(k)(x))_x
∈Z)_k≥ 1 be an i.i.d. sequence of environments with
ω^(k)∼μ. The RWCRE X then starts at X_0=0 and evolves on each interval [τ(k-1),τ(k)) as a random walk in the environment ω^(k). More precisely, given a sequence of environments ω̅ and the cooling sequence τ we define the quenched law P^ω̅,τ(·) of the RWCRE as that of a (time inhomogeneous) Markov chain with transition probabilities given by
P^ω̅,τ(X_n+1 = x + e | X_n = x)=
{[ ω^(k)(x) e = 1,; 1 - ω^(k)(x) e = - 1, ].
if τ(k-1)≤ n < τ(k).
The annealed law ^μ,τ(·) of the RWRE is then obtained by averaging the quenched law with respect to the measure μ^ on the sequence of environments ω̅. That is,
P^μ,τ(·) := μ^N⋉ P_0^ω̅,τ(·) = ∫ P_0^ω̅,τ(·) μ^N(ω̅),
Because we will always be discussing the RWCRE for a fixed distribution μ and cooling map τ, in a slight abuse of notation we will simply use in place of ^μ,τ for the annealed law of the RWCRE for the remainder of the paper.
Throughout the paper, we will use the following representation of the RWCRE. Let (Z^(k))_k≥ 1 = ( (Z^(k)_n)_n≥ 0)_k≥ 1 be a sequence nearest neighbor random walks that are i.i.d. with distribution Z^(k)∼ P^μ_0; that is each Z^(k) is an independent copy of a RWRE with distribution μ on the environment.
For a fixed n≥ 1
Let
ℓ_n := sup{ℓ : τ(ℓ) ≤ n}
be the number of resamplings of the environment by time n.
With this notation, it is easy to see that (under the annealed measure for the RWCRE)
X_n Law=∑_k = 1^ℓ_
n Z^(k)_T_k + Z^(ℓ_n + 1)_n-τ(ℓ_n), n ≥ 0.
(by convention if ℓ_n=0 then the empty sum on the right side is zero so that X_n Law= Z^(1)_n.)
At times it will be convenient to have some notation to rewrite the right side of (<ref>) as a single summation term. To this end, we can write
X_n
Law=∑_k Z^(k)_T_k,n,
where
T_k,n =
T_k if k≤ℓ_n
n-τ(ℓ_n) if k = ℓ_n + 1
0 if k > ℓ_n+1.
The main results of this paper concern the limiting distributions for the RWCRE. Naturally the limiting distribution depends both on the distribution μ on environments and the cooling map τ, but it is natural to separate the analysis according to type of the limiting distribution for the RWRE with environment distribution μ.
In this paper we will be concerned with the cases where μ is κ-regular with either κ∈ (0,1) or κ = 2. Before stating the results we obtain in these cases, however, we will first review some of the limiting distributions for RWCRE that have already been obtained for other regimes of RWRE.
The Sinai regime: “κ=0".[Since the parameter κ>0 characterizes the limiting distributions of transient RWRE, in a slight abuse of notation we will refer to the recurrent regime for RWRE (i.e., where μ is such that ⟨logρ_0 ⟩=0) as having parameter κ=0.]
For recurrent RWRE, Sinai proved the limiting distribution Z_n/(log n)^2⇒ V, where V is a non-Gaussian random variable that can be represented as a functional of a standard Brownian motion <cit.>. The limiting distributions for RWCRE for μ in the Sinai regime were studied first in <cit.> for a few special cases of cooling maps and then later in <cit.> for general cooling maps. The results of <cit.> showed that all subsequential limits
of X_n-[X_n]/√((X_n))
are either Gaussian, sums of independent copies of the random variable V, or an independent mixture of Gaussian and sums of independent copies of V (the limiting distribution can depend both on the cooling map τ and the subsequence n_j→∞).
Functional limit laws for a few special cases of cooling maps were also obtained in <cit.>.
The diffusive Gaussian regime: κ>2.
When μ is κ-regular with κ>2 then a CLT-like limiting distribution holds: Z_n-nv/b √(n)⇒Φ for some b>0, where Φ is a standard Gaussian random variable <cit.>. For μ in this regime it was shown that for any cooling map τ the limiting distribution for the RWCRE is X_n-[X_n]/√((X_n))⇒Φ.
The ballistic, stable regime: κ∈ (1,2).
When μ is κ-regular with κ∈ (1,2), the limiting distributions for RWRE are of the form Z_n-nv/n^1/κ⇒𝒮_κ, where 𝒮_κ is a κ-stable that is totally skewed to the left and has mean zero <cit.>.
Limiting distributions for RWCRE with μ in this regime were studied in <cit.> where sufficient conditions were given on the cooling map τ which lead to limiting distributions for the RWCRE which are (1) Gaussian, (2) κ-stable of the type S_κ, (3) generalized tempered κ-stable, or (4) a mixture of independent random variables of the first three types.
The main results of the paper concern the limiting distributions of the RWCRE in the cases where the distribution μ on environments is κ-regular with either κ∈ (0,1) or κ = 2.
Since both the results and the methods of proof are very different in these two cases we will state our results in each case separately.
§.§.§ Limiting distributions for the case kappa in (0,1)
The decomposition of the RWCRE in (<ref>) as a sum of increments of independent copies of a RWRE, together with the fact that limiting distributions are known for the RWRE, suggests that one might be able to approximate the distibution of X_n by an appropriate linear combination of independent copies of the limiting distribution of the RWRE (which in the case κ∈ (0,1) is a Mittag-Leffler random variable ). We will refer to this general approach to proving a limiting distribution for the RWCRE as the replacement method approach.
Previous results for RWCRE have shown that the replacement method works sometimes (e.g., for the cases when the RWRE is either recurrent or κ-regular with κ>2, <cit.>) but not always (e.g., when the RWRE is κ-regular with κ∈ (1,2), <cit.>). Our main result for the case κ∈ (0,1) is that the replacement method does indeed work for this case.
To prepare for the statement of our main results in this case, note that using (<ref>) we can rewrite the normalized RWCRE as
X_n - [X_n]/√((X_n))Law=∑_k Z^(k)_T_k,n - E_0^μ[Z_T_k,n]/√((X_n))
= ∑_kλ_τ,n(k) Z^(k)_T_k,n - E_0^μ[Z_T_k,n]/√((Z_T_k,n)),
where the coefficients in the last line are given by the vector λ_τ,n = (λ_τ,n(k))_k≥ 1 with
λ_τ,n(k)
= √((Z_T_k,n)/(X_n)).
The terms of the vector λ_τ,n reflect the relative weight that each
term in the sum in (<ref>)
contributes to the distribution of X_n. If the terms of the vector λ_τ,n converge to zero uniformly, then it is natural to expect that the limiting distribution of X_n will be Gaussian. On the other hand, if some terms of λ_τ,n remain bounded away from zero then we expect the Mittag-Leffler random variables to appear in the limiting distribution. To make this precise, and state our main results in the case κ∈ (0,1) we need to first introduce some notation.
Let ℓ^2 = {𝐱∈ℝ^ : ∑_k≥ 1 x(k)^2 < ∞} be the collection of square summable sequences, and note that λ_τ,n∈ℓ^2 since ∑_k≥ 1λ_τ,n(k)^2 = 1. For any non-negative sequence 𝐱∈ℓ^2 there exists a unique non-increasing sequence 𝐱^↓∈ℓ^2 that is a re-ordering of the terms of 𝐱.[That is 𝐱^↓ = (x^↓(k))_k≥ 1 is the unique element of ℓ^2 such that
x^↓(·) = x(π(·))
for some bijection π: → and such that x^↓(k)≥ x^↓(k+1) for all k≥ 1.]
Finally, for any random variable Z with finite variance let Z = Z-E[Z]/√((Z)) denote the normalized version of Z and for any 𝐱∈ℓ^2 let
( Z)^⊗𝐱 = ∑_k≥ 1 x(k) Z_k,
where Z_1,Z_2,… are i.i.d. copies of the random variable Z.
Having introduced the necessary notation, we are now ready to state our main result in the case κ∈ (0,1) which says that subsequential limits of RWCRE when κ∈ (0,1) are sums of independent Mittag-Leffler random variables, Gaussian random variables, or a mixture of the two.
Let X_n be a RWCRE with κ-regular distribution μ with κ∈ (0,1) and cooling map τ.
Assume that n_j →∞ is a subsequence such that
lim_j→∞λ_τ,n_j^↓(k) = λ_*(k) for all k≥ 1 for some λ_* ∈ℓ^2.
Then,
X_n_j-[X_n_j]/√((X_n_j))j→∞⟹( )^⊗λ_*
+ a(λ_*)Φ,
where a(λ_*) := ( 1 - ∑_k λ^2_*(k))^1/2∈ [0,1] and
Φ is a standard Gaussian random variable independent from
( )^⊗λ_*. Moreover, the
convergence in law also holds in L^p for all p>0.
Theorem <ref> shows that any subsequential limit of the RWCRE must be a random variable of the form in the right side of (<ref>) for some λ_*. Indeed, using a diagonalization argument it is easy to see that for any subsequence there is always a further subsequence so that λ_τ,n_j^↓(k) converges for all k (and the limiting vector λ_* must be in ℓ^2 by Fatou's Lemma).
Moreover, we give explicit examples in Section <ref> which show that that the limit can be a pure Gaussian (a(λ^*) = 1), a pure mixture of centered Mittag-Leffler random variables (a(λ_*) = 0) or a mixture of the two (a(λ_*) ∈ (0,1).
We believe by choosing an appropriate cooling map and subsequence n_j one can obtain a limit as in (<ref>) for any non-negative non-increasing λ^* ∈ℓ^2, but we do not prove this here.
Theorem <ref> as stated is quite general. However, to check the convergence of λ^↓_τ,n along some subsequence one needs control on the variance of the corresponding RWRE.
The following theorem, which is also the key to the proof of Theorem <ref>, also provides the necessary asymptotics on the variance to be able to identify the (possibly subsequential) limiting distributions for specific choices of cooling maps τ.
Let Z=(Z_n)_n≥ 0 be a RWRE with distribution μ on environments that is κ-regular with κ∈ (0,1).
Then the convergence in distribution in (<ref>) also holds in L^p for all p>0.
In particular E_0^μ[Z_n] ∼μ_𝔐 n^κ and (Z_n) ∼^2 n^2κ as n→∞, where
μ_𝔐 := E[] = b/Γ(1+κ) and ^2 := () = b^2 ( 2/Γ(1+2κ) - 1/Γ(1+κ)^2).
In the case where T_k →∞ (that is, the gaps between resampling times of the environment diverge),
Theorem <ref> implies that (X_n) ∼^2 ∑_k (T_k,n)^2 κ.
Using this it is then straightforward to obtain
the following more explicit way to check the conditions for the subsequential limiting distributions in (<ref>).
Let X_n be a RWCRE satisfying the assumptions of Theorem <ref>. If in addition the cooling map is such that lim_k→∞ T_k = ∞, then the conclusion of Theorem <ref> holds true if λ_τ,n is replaced by the vector λ̃_τ,n defined by
λ̃_τ,n(k) =
(T_k,n)^κ/V_n,
where
V_n = √(∑_j (T_j,n)^2κ),
where T_k,n is defined as in (<ref>).
§.§.§ Limiting distributions for the case kappa equals 2
The limiting distribution result for the RWCRE in the case κ=2 (Theorem <ref>) inherits some of the properties of both the case κ>2 and κ∈ (1,2).
Like the κ>2 case, the limit will be Gaussian for any cooling map.
On the other hand, like the case κ∈ (1,2) one cannot use the replacement method to prove limiting distributions and determining the proper scaling for the limiting distribution is a major difficulty.
In order to use the replacement method to prove a limiting distribution for the RWCRE, one needs to improve the limiting distribution for the RWRE to convergence in L^2 (see the discussion on the replacement method in Section <ref>).
The first main result in this section gives new asymptotics on the variance of the RWRE in the case κ=2, and as a consequence shows that one does not have L^2 convergence in this case.
Let Z=(Z_n)_n≥ 1 be a RWRE with distribution μ on environments that is κ-regular with κ =2.
Then,
lim_n→∞ E_0^μ[ ( Z_n - nv/√(nlog n))^2 ] = b^2 + K_0 v,
where v>0 is the limiting speed as in (<ref>) and the constants b and K_0 are as in (<ref>) and (<ref>), respectively.
While (<ref>) shows that the limiting distribution in (<ref>) cannot be improved to L^2-convergence,
it also implies that | Z_n-nv/√(nlog n)|^p is uniformly integrable for any p ∈ (0,2), and thus the convergence in distribution in (<ref>) can be improved to L^p convergence for all p ∈ (0,2).
In particular, this implies the following asymptotics for the mean and variance of the RWRE.
Under the same assumptions as Theorem <ref>, we have
lim_n→∞E_0^μ[Z_n]-nv/√(nlog n) = 0,
and lim_n→∞(Z_n)/nlog n = b^2 + K_0 v.
The asymptotics of the mean and variance in Corollary <ref> imply that we can restate the limiting distribution in (<ref>) as
Z_n - E_0^μ[Z_n]/β√((Z_n))n→∞⟹Φ,
where β = b/√(b^2 + K_0 v) < 1.
Of course the interesting part of the limiting distribution as stated in (<ref>) is that the constant β<1.
The fact that the RWRE must be scaled my a non-trivial multiple of the standard deviation to get a standard Gaussian limit is then reflected in our main result for the limiting distributions for RWCRE in the case κ=2 where the appropriate multiplicative constant depends very delicately on the cooling map τ and the distribution μ.
Let X_n be a RWCRE with 2-regular distribution μ and cooling map τ.
There exists a sequence of numbers β_n = β_n(μ,τ) ∈ [β,1] for n≥ 1 such that
X_n - [X_n]/β_n √((X_n))⇒Φ,
where Φ is a standard normal random variable.
The formula for the scaling constants β_n in terms of the cooling map τ and the distribution μ is given explicitly in (<ref>) below and involves certain truncated variance terms for the RWRE of the form ((Z_n - E_0^μ[Z_n])_|Z_n - E_0^μ[Z_n]|≤ x ). As part of the proof of Theorem <ref> we will also give precise asymptotics for such truncated variance terms, and thus one can compute the scaling constants β_n for certain specific choices of cooling maps. In particular, we will give examples in Section <ref> which show that the constants β_n can fill the entire range from β to 1 and that the sequence β_n can also oscillate with n.
The proof of Theorem <ref> is the most difficult and innovative part of the paper. There are two natural ways to try to prove Gaussian limits for the RWCRE, but neither works for all cooling maps.
Approach 1: The first approach is to try to apply the Lindberg-Feller CLT using the representation in (<ref>) as a sum of independent random variables. This approach will work only if the cooling map grows slowly enough so that the triangular array is uniformly asymptotically negligible, i.e.,
lim_n →∞max_k ≤ℓ_n + 1P(Z^(k)_T_k,n>√((X_n)))=0,
see also <cit.>. For instance, this approach will work for cooling maps with T_k ∼ A k^α for some A,α>0 (i.e., polynomial cooling). Even in this case, applying the Lindeberg-Feller CLT is not always straightforward as one sometimes needs to apply a truncation step first before applying the CLT for triangular arrays (e.g., polynomial cooling with α>1).
Approach 2: If the cooling map grows sufficiently fast, then the distribution of X_n is essentially controlled by the last few terms of the sum in (<ref>).
For instance, this approach will work for cooling maps with T_k ∼ A e^ck for some A,c>0 (i.e., exponential cooling).
The idea with this approach is that one first fixes m, and then notes that the sum of the largest m terms in (<ref>) converges in distribution (after appropriate centering and scaling) to a Gaussian. Then one argues that for cooling maps growing fast enough the distribution of the sum of the largest m terms in (<ref>) is not very different from the distribution of X_n if m is large enough.
Since the first approach above only works for cooling maps growing sufficiently slowly and the second approach only works for cooling maps growing sufficiently fast, it is not obvious how to prove Gaussian limits for cooling maps which are more irregular. In our proof of Theorem <ref> we show how the two approaches can be combined together to cover general cooling maps. One splits the sum in (<ref>) into the terms where T_k is “large" or “small", respectively (whether T_k is classified as “large" or “small" depends on its relative size among all the other terms in the sum) and then simultaneously applies approach 1 to the “small" terms and approach 2 to the “large" terms.
There are two main difficulties to implementing this approach for general cooling maps.
The first difficulty is finding the appropriate way to divide the “small" and “large" terms so that both approaches can be applied simultaneously.
The second difficulty is that when applying approach 1 to the “small" terms one still needs to truncate the terms before applying the Lindeberg-Feller CLT, and it is a very delicate matter to choose a truncation that works.
§.§ Notation
Before continuing on with the rest of the paper, we will introduce here some notation that we will use throughout the remainder of the paper.
In the description of the models above, in order to more clearly articulate the difference between the models for RWRE and RWCRE we have used the notation P_0^μ for the annealed law of RWRE and for the annealed law of the RWCRE. However, we could expand the measure to include copies of RWRE so that equalities in law such as (<ref>) become almost sure equalities. We will assume throughout the remainder of the paper that we have done such an expansion of and will therefore in a slight abuse of notation also use in place of P_0^μ for the annealed law of a single RWRE.
Because our main results are stated for the RWCRE centered by its mean, we will often want to use a centered version of the RWRE. Thus, we will use the notation Z̃_n = Z_n - [Z_n].
Our proofs of tail asymptotics of RWRE in Sections <ref> and <ref> will use certain facts about regeneration times for RWRE. We recall here the definition of regeneration times for a RWRE as well as some basic facts about regeneration times that we will use in the proofs.
For a transient RWRE {Z_n}_n≥ 0, the regeneration times 0<R_1<R_2<R_3<… are
defined by
R_1 =inf{n>0 max_m < n Z_m< Z_n ≤min_m>n Z_m }
and
R_k = inf{n>R_k-1max_m < n Z_m< Z_n ≤min_m>n Z_m }, for k>1.
We collect here a few properties of regeneration times that we will
use in our analysis below. Details of these facts can be found in
<cit.>, <cit.>, and the references
therein.
To state these facts, for convenience of notation we will let R_0 =
0 though this is a slight abuse of notation because R_0 is not
necessarily a regeneration time (as reflected in the first fact
below). We will be assuming that the distribution μ on
environments is κ-regular with κ∈ (0,2], though most
of these properties are true in greater generality.
* I.i.d. structure. The sequence of joint random
variables {(Z_R_k-Z_R_k-1,R_k-R_k-1) }_k≥ 1 are
independent, and for every k≥ 2 the vector
(Z_R_k-Z_R_k-1,R_k-R_k-1) has the same distribution as
(Z_R_1,R_1) under the measure
(·) = ( ·| Z_n≥ 0, ∀ n≥ 0 ).
* Regeneration distances have light tails. There are
constants C,c>0 such that
(Z_R_1 > n) ≤ C e^-cn.
Note that (Z_R_2-Z_R_1 > n ) = (Z_R_1 > n) ≤(Z_R_1 > n)/(Z_n ≥ 0, ∀ n≥ 0), so that Z_R_2-Z_R_1 also has exponential tails.
* Regeneration times
have heavy tails. There is a constant C>0 such that
(R_2-R_1 > n) = (R_1 > n) ∼ C n^-κ.
Under the
measure we have the slightly weaker control on the tail of the
first regeneration time: [R_1^γ] < ∞ for all
γ∈ (0,κ).
* Connection with the limiting speed. If κ > 1
then the limiting speed of the RWRE as defined in
(<ref>) is given by
v = [Z_R_2-Z_R_1]/[R_2-R_1] = [Z_R_1]/[R_1].
(Note that (<ref>) holds true when κ∈ (0,1] as
well in the sense that v=0 and [R_1] = ∞.)
§ RWRE RESULTS FOR THE CASE KAPPA IN (0,1)
In this Section we prove Theorems <ref> and <ref>.
We will first prove Theorem <ref> as it is the key element of the proof of Theorem <ref>.
§.§ Proof of Theorem <ref>
By Theorem 4.6.3 in <cit.>, to prove Theorem <ref>
it is enough to prove that
{|Z_n/n^κ|^p}_n≥ 1 is uniformly
integrable for any p<∞, i.e., that
lim_M→∞lim sup_n→∞∫_M^∞ p x^p-1(|Z_n| > x n^κ) x = 0.
We will obtain bounds on (|Z_n| > x n^κ), the tail
probabilities in the integral above by bounding both the left tails,
(Z_n <- x n^κ) and the right tails (Z_n > x n^κ).
Left tail bounds. It was shown in <cit.>
that
lim_n→∞log(-log(X_n<-n^κ) )/log n = κ.
Therefore, there exists an n_0 such that if n≥ n_0
and x≥ 1 we have that
(Z_n < -x n^κ) ≤(Z_n < -n^κ) ≤
e^-n^κ/2.
Also, since (Z_n < -x n^κ) = 0 for
x ≥ n^1-κ we conclude for
n≥ n_0 that
lim_n→∞sup_M ≥ 1∫_M^∞ p x^p-1(Z_n < -x n^κ) x
≤lim_n →∞sup_M ≥ 1∫_M^n^1-κ p
x^p-1 e^-n^κ/2 x
≤lim_n →∞ n^(1-κ)p e^-n^κ/2 = 0.
Right tail bounds.
For the right tail bounds we will use regeneration times.
Note that if the (m+1)-st regeneration time occurs after time n
then Z_n ≤ Z_R_m+1.
Therefore, for any m≥ 1 we have that
( Z_n > x n^κ ) ≤(R_m+1 < n) + (Z_R_m+1 > x n^κ)
≤(R_m < n) + (Z_R_1 > x n^κ/2)
+ (Z_R_m > x n^κ/2),
where in the last inequality we used that R_m+1-R_1 and
Z_R_m+1-Z_R_1 have the same distribution under as do
R_m and Z_R_m, respectively, under the measure . By
(<ref>), there are C,c >0 such that
(Z_R_1 > x n^κ/2) ≤ C e^-c x
n^κ≤ Ce^-cx.
To bound the first and third terms in (<ref>)
choose m depending on x and n as follows
m = m(x,n) = ⌊x n^κ/4 [Z_R_1] ⌋.
For the first probability in (<ref>), by the i.i.d structure
of {R_k-R_k-1}_k≥ 1 under ,
we obtain that
(R_m < n)
≤(R_k-R_k-1 < n, ∀ k≤ m )
= (R_1 ≤ n)^m
= ( 1-(R_1 ≥ n) )^m.
Now, by (<ref>) and (<ref>),
we obtain that there is n_0 ∈N and a constant c>0 such
that
(R_m(x,n) < n) ≤ e^-c x for all n >n_0, x ≥ 1.
For the third probability in (<ref>), first we use our choice of m in (<ref>) to get that (Z_R_m > x n^κ/2)
≤( Z_R_m/m > 2 [Z_R_1] ).
Since Z_R_m is the sum of m i.i.d. random variables with exponential tails and mean [Z_R_1],
by Cramér's theorem, see <cit.>,
there is a constant c>0 and n_1 ∈N such that
(Z_R_m > x n^κ/2)
≤( Z_R_m/m > 2 [Z_R_1] )
≤ e^-cm≤
e^-c x n^κ≤ e^-cx for all n>n_1, x≥ 1.
By (<ref>), (<ref>), and
(<ref>) we obtain from (<ref>) that there is n_2
∈N and c>0 such that
(Z_n > x n^κ) ≤ 3 e^-cx for all n > n_2, x≥ 1.
From this it then follows that
lim_M→∞lim_n→∞∫_M^∞ p x^p-1(Z_n > x n^κ) x = 0.
§.§ Proof of Theorem
<ref>
To prove of Theorem <ref> we follow the ideas in <cit.> which we now sketch. Essentially the idea is to use a threshold J>0 to distinguish small increments, T_k,n≤ J, from large increments, T_k,n>J, and let this threshold grow after we take n →∞ to obtain the limit statement. By the CLT for iid random variables, we may replace the small terms by independent copies of Mittag-Leffler distributions as both have the same Gaussian limit.
Next by using the convergence of (<ref>) which holds in L^2 we can show that there is a coupling of (Z^(k)_m, ^(k), k,m ∈N) for which the difference between the increments and copies of Mittag-Leffler can be neglected and we are allowed to replace the left hand side of (<ref>) by weighted sums of independent Mittag-Leffler random variables. More explicitly, recall (<ref>) and note that for any J>0 we have
𝔛_n :=
X_n- E[X_n]/√((X_n))
= ∑_k λ_τ,n(k) _T_k,n≤ J[ Z^(k)_T_k,n - E[Z^(k)_T_k,n]/√((Z^(k)_T_k,n))] +∑_k λ_τ,n(k) _T_k,n >J[ Z^(k)_T_k,n - E[Z^(k)_T_k,n]/√((Z^(k)_T_k,n))],
For the small increments, the CLT for iid random variables give us that for any J>0 and any fixed bounded continuous function f: R→R
lim_n E[f(∑_k λ_τ,n(k) _T_k,n≤ J[ Z^(k)_T_k,n - E[Z^(k)_T_k,n]/√((Z^(k)_T_k,n))]) ]
-E[f(∑_k λ_τ,n(k) _T_k,n≤ J𝔐_k^(k))] = 0.
Now, provided the coupling of the random variables ensures almost sure convergence (<ref>), we obtain
lim_J →∞lim_n→∞E[(∑_k λ_τ,n(k) _T_k,n >J[ Z^(k)_T_k,n - E[Z^(k)_T_k,n]/√((Z^(k)_T_k,n)) - ^(k)] )^2] = 0.
A combination of (<ref>) and (<ref>) allows us claim that (𝔛_n, n∈N) has the same subsequential weak limits as (()^⊗λ_n, n ∈N).
The final step in the proof of (<ref>) is to identify the subsequential weak limits of (()^⊗λ_n, n ∈N).
For ease of notation, let λ_j : = λ_τ,n_j^↓, note that ()^⊗λ_j(d)= ()^⊗λ_τ,n_j, and recall that we are assuming that lim_j→∞λ_j(k) = λ_*(k) for all k≥ 1. Now note that for any K>0
∑_k = 1^K λ_j(k) ^(k)j→∞⟹∑_k = 1^K λ_*(k) ^(k).
By (<ref>), we may take K_j →∞ slowly enough such that
∑_k = 1^K_jλ_j(k) ^(k)j→∞⟹∑_k = 1^∞λ_*(k) ^(k).
To conclude, we note that we may also take K_j →∞ slowly enough such that
a(λ_*)^2 = 1 - ∑_k=1^∞(λ_*(k))^2 = ∑_k=1^∞(λ_j(k))^2 - ∑_k = 1^∞(λ_*(k))^2
= lim_j∑_k=1^∞(λ_j(k))^2 - ∑_k = 1^K_j(λ_*(k))^2 = lim_j ∑_k=1^∞(λ_j(k))^2 - ∑_k = 1^K_j(λ_j(k))^2
= lim_j ∑_k = K_j+1^∞(λ_j(k))^2,
and using the Lindeberg condition for triangular arrays, see Theorem 3.4.10 in <cit.>, we obtain that
∑_k = K_j+1^∞λ_j(k) ^(k)j→∞⟹ a(λ_*) Φ,
where Φ_0 is a standard Gaussian random variable.
By the independence of the sequence (^(k), k ∈N)
we obtain that
∑_k = 1^∞λ_j(k) ^(k) = ∑_k = 1^K_jλ_j(k) ^(k) + ∑_k = K_j + 1^∞λ_j(k) ^(k)j→∞⟹∑_k = 1^∞λ_*(k) ^(k) + a(λ_*) Φ_0,
where the two terms on the right are independent. This
concludes the proof of (<ref>). To obtain convergence in L^p for all p<∞ it is enough to show that sup_n E[(𝔛_n)^2r] < ∞ for all r ∈N.
This can be proved by representing 𝔛_n as in (<ref>), using a binomial expansion of (𝔛_n)^2r and using the following facts: (1) the terms in the decomposition in (<ref>) are independent with zero mean, (2) for any ℓ≥ 2 we have ∑_k (λ_τ,n(k))^ℓ≤∑_k (λ_τ,n (k))^2 = 1, and (3) Theorem <ref> implies that sup_n [(Z_n - [Z_n]/√((Z_n)))^ℓ ] ≤ C_ℓ < ∞ for all ℓ <∞.
§.§ Proof of Corollary <ref>
Let V_n : = √(∑_j=1^ℓ_n (T_j)^2κ +
(n-τ(ℓ_n))^2κ).
If lim_k →∞T_k = ∞ it follows from Theorem <ref> that
lim_k (Z_T_k)/^2 (T_k)^2κ = 1.
Therefore, we also obtain that
lim_k (X_n)/V^2_n = ∑_k
(Z_T_k)/^2 ( ∑_k = 1^ℓ_n
(T_k)^2 κ + (n-τ(ℓ_n))^2κ) = 1.
Now assume that there is a subsequence (n_j, j ∈N) and
λ_* ∈ℓ^2 for which
lim_j→∞λ^↓_τ, n_j(k) =
λ_*(k) for all k ∈N.
Let λ̃_τ,n be defined by (<ref>). From (<ref>) definition (<ref>)
we obtain that (<ref>) holds true if and only if
lim_j→∞λ̃^↓_τ, n_j(k) =
λ_*(k) for all k ∈N.
This concludes the proof of Corollary <ref>.
§ RWRE RESULTS FOR THE CASE KAPPA EQUALS 2
In this section we will prove some of the new RWRE results that will be needed for the analysis of the limiting distributions of RWCRE when the distribution μ is 2-regular.
This will include the proof of Theorem <ref>, but will also include some new large and moderate deviation tail bounds as well as some asymptotics of truncated moments that will be crucial later in the proof of Theorem <ref>.
Since the moments of the RWRE can be expressed in terms of the tails of the distribution of the RWRE, we will need good tail asymptotics of the RWRE to prove Theorem <ref>. We will divide our analysis of the tails of the RWRE into the right and left tails separately since the asymptotics are very different in either case.
§.§ Right tail estimates
Our main result in this section is the following Gaussian right tail estimate for the RWRE.
If the distribution μ is 2-regular, then there exist constants C,c>0 such that for all n sufficiently large,
ℙ(Z_n-vn≥ x)≤ Ce^-cx^2/nlog n, ∀ x>0.
First of all, note that it is enough to prove the inequality in (<ref>) for x ∈ (0,(1-v)n] since the probability on the left is zero for x>(1-v)n.
To this end, recall the notation regarding regeneration times introduced at the beginning of Section <ref> and for any
x∈ (0,(1-v)n]
let m=m(x,n) be defined by
m
= ⌊vn+x/4/[Z_R_1]⌋.
We will use in several places below that
this choice of m implies that there is a constant c_1>0 such that c_1 n ≤ m ≤1/c_1 n for all
x∈ (0,(1-v)n].
For this choice of m we have that for n sufficiently large
ℙ(Z_n-vn≥ x)
≤ℙ(Z_R_1≥ x/2)+ℙ(Z_R_m+1-Z_R_1≥ vn+x/2)+ℙ(R_m+1-R_1<n)
≤ P(Z_R_1≥ x/2) + ( Z_R_m- [ Z_R_m ] ≥ x/4 ) + ( R_m - [R_m] < -x/(8v) ),
where in the last inequality we used (<ref>), (<ref>), and the i.i.d. structure of regeneration times.
let m=m(x,n) be defined by
m-1
= ⌊vn+x/4/[Z_R_1]⌋.
We will use in several places below that
this choice of m implies that there is a constant c_1>0 such that c_1 n ≤ m ≤1/c_1 n for all
x∈ (0,(1-v)n].
For this choice of m we have that for n sufficiently large
ℙ(Z_n-vn≥ x) ≤ℙ(Z_R_1≥ x/2)+ℙ(Z_R_m-Z_R_1≥ vn+x/2)+ℙ(R_m-R_1<n)
≤ P(Z_R_1≥ x/2) + ℙ( Z̃_R_m-Z̃_R_1≥ x/4 ) + ℙ(R̃_m-R̃_1< -x/(8v) ),
where in the last inequality we used (<ref>), (<ref>), and the i.i.d. structure of regeneration times.
(we're also using the Z̃_R_1 and R̃_1 notation here, so be sure we've introduced it earlier.)
We will bound the three terms in (<ref>) separately.
Since Z_R_1 has exponential tails, see (<ref>), the first term in (<ref>) can be bounded by
Ce^-cx for x>0.
For the second term in (<ref>), since
under we have that Z_R_m is a sum of m i.i.d. random variables with exponential tails,
standard large deviation results
(e.g. <cit.>) imply that
there exists a constant c>0 such that
(Z_R_m-[Z_R_m] ≥ x/4 )≤exp{-c x^2/n}, ∀ x ∈ (0,(1-v) n].
(Note that here we are using that c_1 n ≤ m ≤ n/c_1 as noted above.) To bound the last term in (<ref>) we will use the fact that R_m - [R_m] is a sum of m i.i.d. random variables which have mean zero, are bounded below, and have tails decaying like x^-2 to the right. Thus, applying Corollary <ref> we get for n large enough and x ≤ (1-v)n that
(R_m - [R_m] < -x/(8v) ) ≤ e^-c x^2/m log m≤ e^-c x^2/n log n.
(Note that to apply Corollary <ref> we are using both that x < (1-v)n and that m≥ c_1 n, while for the second inequality above we are using that m≤ n/c_1.)
Putting together the bounds for the three terms in (<ref>), we have for n large enough that
ℙ(Z_n-vn≥ x)≤ Ce^-cx+e^-cx^2/n+e^-cx^2/nlog n≤ C e^-c x^2/nlog n, ∀ x ∈ (0,(1-v)n].
Again, since the probability is zero for x>(1-v)n this completes the proof of the lemma.
An immediate consequence of (<ref>) is the following bound on the truncated right tail L^p norm of Z_n-vn.
If the distribution μ is 2-regular, then for any p>0
lim_M→∞lim sup_n→∞1/(nlog n)^p/2𝔼[|Z_n-vn|^p1_{Z_n-vn≥ M√(nlog n)}]=0.
§.§ Left tail estimates
The left tail asymptotics of Z_n-vn are much more delicate than the right tail asymptotics.
We can use the annealed large deviation principle to get good bounds on the left tail probabilities (Z_n - vn≤ -x), but only when x ≫ nv since the annealed large devation rate function is zero on [0,v] (see Theorem <ref> or <cit.>).
On the other hand, the estimates of Buraczewski and Dyszewski stated in (<ref>)
allow us to get very precise estimates for these probabilities, but only covering x ∈ [√(n)log^3 n , nv - log n].
The following is a rougher estimate but allows us to give a useful upper bound for any
x ∈ (0,nv/2].
If the distribution μ is 2-regular, then
there exists a constant C>0 such that for n large enough we have
ℙ(Z_n-vn≤ -t√(nlog n))≤C/t^2(log n) + C/t^4, for 0< t ≤v/2√(n/log n).
By taking the constant C>1, the bound in (<ref>) becomes trivial for t ≤ 1/√(log n). Thus, for the remainder of the proof we will assume that 1/√(log n)≤ t ≤v/2√(n/log n).
Letting
m = m(n,t) = 1/[R_1](n-t√(nlog n) ) we have
( Z_n - nv ≤ -t √(nlog n))
≤( R_m+1 > n ) + ( Z_R_m+1≤ nv - t√(nlog n))
≤( R_1 > t/2√(nlog n)) + ( R_m > n - t/2√(n log n))
+ ( Z_R_m≤ nv - t√(nlog n))
≤( R_1 > t/2√(nlog n)) + ( R_m - [R_m] > t/2√(nlog n))
+ ( Z_R_m - [ Z_R_m ] ≤ -( 1-v/2) t √(nlog n)).
Since
[R_1^γ] < ∞ for all γ < κ = 2, it follows that
the first term in (<ref>) is bounded by
2[R_1]/t√(nlog n)≤v[R_1]/t^2log n for all t≤v/2√(n/log n).
For the second term in (<ref>), since R_m - [R_m] is the sum of i.i.d. terms with tail decay (R_1 > x) ∼ C x^-2 and since m/n = m(n,t)/n is uniformly bounded away from 0 and ∞ for t ≤v/2√(n/log n), by applying Lemma <ref> we obtain that
( R_m - [R_m] > t/2√(nlog n))
≤( R_m - [R_m] > C t √(mlog m))
≤C/t^2 log n + C/t^4.
Finally, since Z_R_m-[Z_R_m] is the sum of i.i.d. terms with exponential tails, standard large deviation estimates (see <cit.>) again imply that for n large enough and t≤v/2√(n/log n) the probability in (<ref>) is bounded above by
exp{ -C t^2 n log n/m ∨ t √(nlog n)}≤exp{ -C t^2 log n }≤C/t^2 log n.
(We are again using here that m/n is bounded away from 0 and ∞.)
This completes the proof of the Proposition.
A consequence of the above left tail estimates for the RWRE is the following lemma which is the key to the proof of Theorem <ref>.
If the distribution μ is 2-regular, then
lim_M→∞lim sup_n→∞| 1/nlog n𝔼[(Z_n-vn)^21_{Z_n-vn≤ -M√(nlog n)}] - K_0 v | = 0,
where K_0 is the constant from (<ref>) and v is the limiting speed of the RWRE as in (<ref>).
For any fixed M, noting that since |Z_n|≤ n, we have that
1/nlog n𝔼[|Z_n-vn|^2 1_{Z_n-vn≤ -M√(nlog n)}]
= M^2 ( Z_n - vn ≤ -M√(nlog n) ) + 1/nlog n∫_M √(nlog n)^(1+v)n 2x ℙ(Z_n-vn≤-x) x.
Using Proposition <ref> we have that
M^2 ( Z_n - vn ≤ -M√(nlog n) ) ≤C/log n + C/M^2,
and therefore to prove (<ref>) it suffices to show that
lim_M→∞lim sup_n→∞|1/nlog n∫_M √(nlog n)^(1+v)n 2x ℙ(Z_n-vn≤-x) x - K_0v|=0.
Now, the integral term in (<ref>) may be split as
1/nlog n∫_M √(nlog n)^(1+v)n 2x ℙ(Z_n-vn≤-x) x
= 1/nlog n{∫_M √(nlog n)^√(n)log^3 n +∫_√(n)log^3 n^nv-log n +∫_nv-log n^(1+v)n}2x ℙ(Z_n-vn≤-x) x.
To conclude the proof of (<ref>) we need only control the three integral terms in (<ref>). To this end we will show that the main term is given by the second integral and the first and third integrals are negligible as first n→∞ and then M→∞.
To control the first integral in (<ref>) we will again use the general left tail estimates from Proposition <ref>. That is,
1/nlog n∫_M √(nlog n)^√(n)log^3 n2x ℙ(Z_n-vn≤-x) x
=∫_M^(log n)^5/2 2t ℙ( Z_n-vn≤-t√(nlog n)) t
≤∫_M^(log n)^5/2( C/t log n + C/t^3) t
≤C loglog n/log n + C/M^2,
which vanishes as we take n→∞ followed by M→∞.
For the third integral in (<ref>), for n sufficiently large and all x ≥ nv-log n we can use (<ref>) to bound
(Z_n - nv ≤ -x) ≤(Z_n ≤log n) ≤2 K_0 log n/(nv-log n)^2≤C log n/n^2,
and thus
1/nlog n∫_nv-log n^(1+v)n 2x ℙ(Z_n-vn≤-x) x
≤
C/n,
which also vanishes as n→∞.
Finally, for the second integral in (<ref>) we can apply the precise left tail estimates in (<ref>) for the probabilities in the integrand
to get that
lim_n→∞1/nlog n∫_√(n)log^3n^nv-log n 2x ℙ(Z_n-vn≤-x) x
= lim_n→∞1/nlog n∫_√(n)log^3n^nv-log n 2x K_0(nv-x)x^-2 x
= K_0 v.
This completes the proof of the lemma.
§.§ Proof of Theorem 1.11 and centered tail estimates
Having obtained the left and right tail estimates in the previous two sections, we are now ready to give the proof of Theorem <ref>.
First of all, note that for M fixed we have
| [(Z_n-nv)^2]/nlog n - (b^2+K_0 v) |
≤| [ (Z_n-nv)^2 _|Z_n-nv| < M√(nlog n)] /nlog n - b^2 E[Φ^2 _|Φ| < M] |
+ | [ (Z_n-nv)^2 _Z_n-nv ≤ -M√(nlog n)] /nlog n - K_0 v |
+ b^2 E[Φ^2 _|Φ| ≥ M]
+ [ (Z_n-nv)^2 _Z_n-nv ≥ M√(nlog n)] /nlog n.
It follows from (<ref>) and the bounded convergence theorem that the first term on the right vanishes as n→∞ for any fixed M.
Then, it follows from Corollary <ref> and Lemma <ref> that the last three terms on the right can be made arbitrarily small as n→∞ by fixing M large enough. This completes the proof of the theorem.
The left and right tail asymptotics for the RWRE proved above were for the random walk centered by its limiting speed. However, since results of Theorem <ref> are proved for the RWCRE centered by its mean, we will need some results on the RWRE centered by its mean as well.
Since it follows from Corollary <ref> that [Z_n] = nv + o(√(nlog n)), we can then easily obtain the following analogs of the above results for the RWRE centered by its mean (with different constants and sometimes slightly smaller ranges to which the tail estimates apply).
Let (Z_n)_n≥ 0 be a RWRE with a 2-regular distribution μ on environments. Then the following right and left tail estimates hold for the centered RWRE.
* Right tail estimate: There exist constants C,c such that for n sufficiently large
ℙ(Z_n-𝔼Z_n≥ x) ≤ C e^-c x^2/nlog n, ∀
x>0.
* General left tail estimate: There is a constant C such that for n sufficiently large
ℙ(Z_n-𝔼Z_n≤ -t√(n log n)) ≤C/t^2 log n + C/t^4, for 1 ≤ t ≤v/2√(n/log n).
* Precise left tail estimate: With K_0 the same constant as in (<ref>), we have
lim_n→∞sup_√(n)log^4n ≤ x ≤ nv-√(n log n)|ℙ(Z_n-𝔼Z_n≤ -x)/(nv-x)x^-2-K_0|=0.
We close this section with some consequences of the above tail estimates for certain truncated first and second moments of the RWRE which will be needed for our proof of Theorem <ref>.
For these lemmas (and throughout Section <ref>), recall from Section <ref> that Z̃_n = Z_n - [Z_n] is the notation for the centered version of the RWRE.
There exists a constant C>0 such that for n sufficiently large and a ≥√(n)log^4 n,
[ |Z̃_n| _|Z̃_n| > a] ≤Cn/a
First of all, note that since |Z̃_n| ≤ 2n the bound holds trivially for a ≥ 2n. On the other hand, one easily sees that if (<ref>) holds for all a ∈ [√(n)log^4 n, nv/2] then by changing the constant C we get that it also holds for a ∈ [nv/2,2n]. Indeed, if a ∈ [nv/2,2n] then
[ |Z̃_n| _|Z̃_n| > a]
≤[ |Z̃_n| _|Z̃_n| > nv/2]
≤ C n/nv/2≤(4C/v) n/a.
Thus, for the remainder of the proof we will assume that √(n)log^4 n ≤ a ≤ nv/2.
To bound the expectation in (<ref>) we first decompose it as [ |Z̃_n| _|Z̃_n| > a] = [ Z̃_n _Z̃_n > a] + [ (-Z̃_n) _Z̃_n < -a].
For the right truncated expectation,
using (<ref>) we get that
[ Z̃_n _Z̃_n > a]
= a (Z̃_n > a) + ∫_a^∞(Z̃_n > x ) x
≤ C a e^-ca^2/(nlog n) + ∫_a^∞ C e^-c x^2/(nlog n) x
≤ C a e^-ca^2/(nlog n) + C nlog n/a e^-c a^2/(nlog n).
Since a≥√(n)log^4 n implies that e^-ca^2/(nlog n)≤ e^-c log^7 n and since a ≤ nv/2, it follows that the above bound is less than C'n/a for some C'<∞ for all n sufficiently large and a ∈ [√(n)log^4 n, nv/2].
For the left truncated expectation,
using the assumption that √(n)log^4 n ≤ a ≤ nv/2 and the precise tail bounds in (<ref>) we obtain for n sufficiently large that
[ (-Z̃_n) _Z̃_n < -a]
= a ( Z̃_n < -a ) + ∫_a^2n( Z̃_n < -x ) x
≤ a (Z̃_n < -a ) + ∫_a^nv-√(nlog n)( Z̃_n < -x ) x
+ 2n ( Z̃_n < -(nv-√(nlog n)) )
≤2K_0 v n/a + ∫_a^nv-√(nlog n)2K_0 v n/x^2 x + 8 K_0 √(log n)/v^2√(n)
≤4K_0 v n/a + 8 K_0 √(log n)/v^2 √(n).
Finally, note that a≤ nv/2 implies that √(log n)/√(n)≤n/a for n sufficiently large.
If √(n log n)≤ a ≤√(n)log^4 n, then
[ (Z̃_n)^2 _a < |Z̃_n| ≤√(n)log^4 n] ≤ C n loglog n + C n^2 log^2 n/a^2.
We begin by noting that
[ (Z̃_n)^2 _a < |Z̃_n| ≤√(n)log^4 n]
≤ a^2 ( |Z̃_n| > a ) + ∫_a^√(n)log^4 n 2x ( |Z̃_n| > x ) x.
For √(nlog n)≤ x ≤√(n)log^4 n it follows from
(<ref>) and (<ref>) that
( |Z̃_n| > x )
= ( Z̃_n > x ) + ( Z̃_n < -x )
≤ C e^-cx^2/nlog n + Cn/x^2 + C n^2 log^2 n/x^4.
Applying this to the first term on the right in (<ref>)
we get
a^2 ( |Z̃_n| > a ) ≤ C a^2 e^-c a^2/nlog n + Cn + C n^2log^2 n/a^2≤ Cn + C n^2log^2 n/a^2,
where the last inequality is justified by noting that (by elementary calculus)
a^2 e^-ca^2/nlog n = 1/a^2(a^4 e^-ca^2/nlog n) ≤1/a^2( 4/e^2 c^2 n^2 log^2 n ).
For the integral term in (<ref>) we obtain that
∫_a^√(n)log^4 n 2x ( |Z̃_n| > x ) x
≤ C ∫_a^√(n)log^4 n{ x e^-c x^2/nlog n + n/x + n^2log^2 n/x^3} x
≤ C { nlog n e^-c a^2/nlog n + n log( √(n)log^4 n/a) + n^2 log^2 n/a^2}
≤ C { n loglog n + n^2 log^2 n/a^2},
where in the last inequality we used (<ref>) and that a≥√(nlog n) implies log( √(n)log^4 n/a) ≤7/2loglog n.
If b is the scaling constant from the limiting distribution in (<ref>), then
lim_n→∞(Z̃_n _|Z̃_n| ≤√(n)log^4 n)/nlog n =
lim_n→∞[ (Z̃_n)^2 _|Z̃_n| ≤√(n)log^4 n]/nlog n
= b^2.
First of all, note that
| (Z̃_n _|Z̃_n| ≤√(n)log^4 n)
- [ (Z̃_n)^2 _|Z̃_n| ≤√(n)log^4 n] |
= [ Z̃_n _|Z̃_n| ≤√(n)log^4 n]^2
= [ Z̃_n _|Z̃_n| > √(n)log^4 n]^2
≤C n/log^8 n,
where in the second equality we used that [Z̃_n] = 0 and in the last inequality we used Lemma <ref>.
Since this bound is o(nlog n), it is enough to only prove the second equality in (<ref>).
To this end, fix M>1 and note that
| [ (Z̃_n)^2 _|Z̃_n| ≤√(n)log^4 n]/nlog n - b^2 |
≤| [ (Z̃_n)^2 _|Z̃_n| ≤ M√(nlog n)]/nlog n - b^2 E[Φ^2 _|Φ|≤ M ] |
+ [ (Z̃_n)^2 _M√(nlog n) < |Z̃_n| ≤√(n)log^4 n]/nlog n + b^2 E[Φ^2 _|Φ|>M ].
The first term on the right vanishes by the bounded convergence theorem (together with (<ref>) and Corollary <ref>), while the second term on the right can be bounded by Cloglog n/log n + C/M^2 by Lemma <ref>. Therefore, taking first n→∞ and then M→∞ completes the proof.
§ TRANSIENT RWCRE WHEN KAPPA EQUALS 2: ARBITRARY COOLING
In this section we will give the proof of Theorem <ref>.
We will prove that the limiting distribution is Gaussian by proving that for any subsequence n_j→∞ there is a further subsequence n_j_k along which X_n_j_k - [ X_n_j_k ] /β_n_j_k s_n_j_k⇒Φ. Therefore, at several times throughout the proof we will be able to assume that some nice additional property holds by passing to a subsequence along which it is true.
Before starting the proof of Theorem <ref> we will introduce some notation and give a brief idea of the proof.
Our starting point for the analysis of the distribution of X_n is the decomposition into sums of increments of independent copies of RWRE as in (<ref>).
We will analyze this sum by separating the terms in the sum into the “large" and the “small" terms. The small terms will be those whose variance makes up a negligible fraction of the total variance of the sum, and the large terms will be the remaining terms. To be more precise, let
s_n^2 = (X_n) = ∑_k=1^ℓ_n+1σ_k,n^2,
where σ_k,n^2 = (Z_T_k,n),
and then for any δ>0 and n≥ 1
let
I_n,δ^+ = { k∈{1,2,…,ℓ_n+1}: σ_k,n^2 > δ s_n^2 },
and
I_n,δ^- = {k∈{1,2,…,ℓ_n+1}: σ_k,n^2 ≤δ s_n^2}.
Finally, letting
𝒵_n,δ^+ = ∑_k ∈ I_n,δ^+ (Z_T_k,n^(k) - [Z_T_k,n])
and 𝒵_n,δ^- = ∑_k ∈ I_n,δ^- (Z_T_k,n^(k) - [Z_T_k,n])
we can rewrite X_n - [X_n] = 𝒵_n,δ^+ + 𝒵_n,δ^-.
The main idea is then to show that, after appropriate scaling, both Z_n,δ^+ and Z_n,δ^- are approximately Gaussian (unless the set I_n,δ^+ or I_n,δ^- is empty in which case one only needs that the non-zero term is approximately Gaussian).
Unfortunately, this argument doesn't quite work for a fixed δ as our analysis of the small terms will actually require δ to vanish as n→∞.
That is, we will show that there exists a sequence δ_n → 0 such that both 𝒵_n,δ_n^+ and 𝒵_n,δ_n^- converge in distribution to Gaussians when properly scaled. Since 𝒵_n,δ_n^+ and 𝒵_n,δ_n^- are independent this will then imply that X_n - [X_n] converges to a Gaussian when properly scaled.
Finally, we note that as part of our proof we will show that the proper scaling ends up differing from s_n = √((X_n)) by a multiplicative factor which asymptotically lies in [β,1], where the constant β is defined in (<ref>).
Having outlined the general strategy, we will now begin proving some lemmas which complete the main steps of the proof.
Gaussian convergence for the large parts.
The first step is to prove the convergence of the sums over only the large cooling intervals.
We cannot simply claim that 𝒵_n,δ^+ ⇒Φ as n→∞ since it could be that I_n,δ^+ = ∅ for infinitely many n.
However, the following lemma show that 𝒵_n,δ^+ is approximately Gaussian whenever the set I_n,δ^+ is not empty.
For any δ>0 and n≥ 1 let s_n,δ,+^2 := (𝒵_n,δ^+). If n_j→∞ is a subsequence such that s_n_j,δ,+>0 (or equivalently I_n_j,δ^+ ≠∅) for all large j then
𝒵_n_j,δ^+/s_n_j,δ,+j→∞⟹βΦ.
We begin by noting that if s_n_j,δ,+>0 then we can write
𝒵^+_n_j,δ/s_n_j,δ,+
= ∑_k ∈ I_n_j,δ^+σ_k,n_j/s_n_j,δ,+Z̃_T_k,n_j^(k)/σ_k,n_j.
Since the set I_n,δ^+ can contain at most 1/δ elements,
by passing to a subsequence if needed we can assume without loss of generality that there is an integer 1≤ m ≤1/δ such that |I_n_j,δ^+| = m for all j; that is, the sum in the decomposition has exactly m terms.
It then follows from the limiting distribution for the RWRE in (<ref>)
that {Z̃^(k)_T_k,n_j / σ_k,n_j}_k ∈ I_n_j,δ^+⇒{βΦ_i }_1≤ i≤ m, where the Φ_i are i.i.d. standard Gaussian random variables.[Note here that we are using that T_k,n is large if k ∈ I_n,δ^+. Indeed, it follows from Corollary <ref> and the definition of I_n,δ^+ that k ∈ I_n,δ^+ implies that T_k,nlog (T_k,n) ≥ C δ s_n^2 for some C>0.
]
The claimed convergence in (<ref>) then follows from this, together with the observation that ∑_k ∈ I_n_j,δ^+ (σ_k,n_j/s_n_j,δ,+)^2 = 1.
There exists a sequence δ_n → 0 such that either (1) s_n,δ_n^+ = 0 for all n large, or (2) for any subsequence n_j→∞ such that s_n_j,δ_n_j,+>0 for all but finitely many j we have
𝒵^+_n_j,δ_n_j/s_n_j,δ_n_j,+j→∞⟹βΦ.
Let d be a metric on the space of Borel probability measures on ℝ that is consistent with the topology of weak convergence (e.g., the Levy metric), and in a slight abuse of notation for random variables X and Y with distributions μ_X and μ_Y, respectively, we will use d(X,Y) in place of d(μ_X,μ_Y). With this notation, for any n≥ 1 and δ>0 we can let
_n,δ =
d( 𝒵_n,δ^+/s_n,δ,+, βΦ) if s_n,δ,+>0
0 if s_n,δ,+ = 0.
It follows from Lemma <ref> that lim_n→∞_n,δ = 0, for any fixed δ>0.
Let m_0 = 0 and define integers m_j for j≥ 1 inductively as follows.
Let m_j > m_j-1 be such that _n,1/j < 1/j for all n>m_j. Let δ_n = 1 for n< m_1 and if m_j ≤ n < m_j+1 for some j≥ 1 then δ_n = 1/j.
For this sequence δ_n we have lim_n→∞_n,δ_n = 0, and this completes the proof of the corollary.
Let the sequence δ_n → 0 be chosen as in Corollary <ref>.
In general we would like to prove that 𝒵_n,δ_n^- properly scaled also converges to a centered Gaussian, but in fact this isn't necessarily true if
(𝒵_n,δ_n^-)
doesn't grow to ∞. However, the following corollary shows that in this case we already have the limiting distribution we are after because only the large parts will be relevant in the limit.
For any δ>0 let s_n,δ,- := (𝒵_n,δ^-).
If δ_n is the sequence from Corollary <ref> and n_j→∞ is a subsequence such that
lim_j→∞s_n_j,δ_n_j,-/s_n_j = 0,
then
X_n_j - [X_n_j] /β s_n_jj→∞⟹Φ.
Since s_n^2 = s_n,δ_n,-^2 + s_n,δ_n,+^2, the assumption that s_n_j,δ_n_j,-/s_n_j→ 0 implies that
lim_j→∞s_n_j,δ_n_j,+/s_n_j = 1.
Next we write the decomposition
X_n_j - [X_n_j] /β s_n_j
= 𝒵_n_j,δ_n_j^-/β s_n_j + 𝒵_n_j,δ_n_j^+/β s_n_j.
The variance of the first term on the right vanishes as j→∞ by the assumption on the subsequence n_j in the statement of the corollary, and the second term converges in distribution to a standard Gaussian by (<ref>) and Corollary <ref>.
Gaussian convergence for the small parts.
Due to Corollary <ref>,
we will only need to consider the limiting distributions X_n along subsequences n_j such that
lim inf_j→∞s_n_j,δ_n_j,-/s_n_j = θ∈ (0,1].
The following lemma shows that under this assumption 𝒵_n_j,δ_n_j^- converges to a Gaussian, but determining the proper scaling to get a standard Gaussian limit is quite delicate.
Let δ_n → 0 and assume that n_j is a subsequence such that (<ref>) holds. Then,
𝒵_n_j,δ_n_j^-/s̃_n_j,δ_n_j⇒Φ,
where for any n≥ 1 and δ>0
s̃_n,δ^2 = ( ∑_k ∈ I_n,δ^-Z̃_T_k,n^(k)_A_k,n)
and
A_k,n = { |Z̃_T_k,n^(k)|≤s_n/√(log s_n)∨√(T_k,n)log^4 T_k,n}.
For simplicity of notation, we will give the proof under the assumption that (<ref>) holds without taking a subsequence. That is,
lim inf_n→∞s_n,δ_n,-/s_n = θ∈ (0,1].
We begin by decomposing
𝒵_n,δ_n^-/s̃_n,δ_n
= 1/s̃_n,δ_n∑_k ∈ I_n,δ_n^-( Z̃_T_k,n^(k)_A_k,n - [ Z̃_T_k,n^(k)_A_k,n] )
+ 1/s̃_n,δ_n∑_k ∈ I_n,δ_n^-( Z̃_T_k,n^(k)_A_k,n^c - [ Z̃_T_k,n^(k)_A_k,n^c] ).
We will prove that (<ref>) converges in distribution to a standard Gaussian while (<ref>) converges in distribution to zero.
Proof that (<ref>) is negligible.
We will show that (<ref>) converges to zero in L^1.
To this end, note first of all that
[ | ∑_k ∈ I_n,δ_n^-( Z̃_T_k,n^(k)_A_k,n^c - [ Z̃_T_k,n^(k)_A_k,n^c] ) | ]
≤ 2 ∑_k ∈ I_n,δ_n^-[ |Z̃_T_k,n| _A_k,n^c].
Since |Z̃^(k)_T_k,n| ≤ 2T_k,n, the event A_k,n^c is empty if 2T_k,n < s_n/√(log s_n). Therefore, we need only consider the terms in the sum where T_k,n≥s_n/2√(log s_n)≥√(s_n). For these terms we can use Lemma <ref> to bound the sum by
∑_ k ∈ I_n,δ_n^- [ |Z̃_T_k,n| _A_k,n^c]
≤ C ∑_k ∈ I_n,δ_n^-
T_k,n≥√(s_n)T_k,n/( s_n/√(log s_n))
≤ C √(log s_n)/s_n∑_k ∈ I_n,δ_n^-
T_k,n≥√(s_n)σ_k,n^2/log T_k,n≤C/s_n√(log s_n)∑_k=1^ℓ_n+1σ_k,n^2
= C s_n/√(log s_n),
where in the second inequality we used that Corollary <ref> implies that there is a constant C such that T_k,nlog T_k,n≤ C σ_k,n^2, and in the third inequality we used that T_k,n≥√(s_n) implies that log T_k,n≥ (1/2) log s_n.
Thus, to complete the proof that (<ref>) converges to zero in L^1, we need only to show that lim_n→∞s_n/s̃_n,δ_n√(log s_n) = 0.
In fact, we will show that
lim inf_n→∞s̃_n,δ_n/s_n,δ_n,-≥β,
which combined with our assumption (<ref>) is enough to show that
lim_n→∞s_n/s̃_n,δ_n√(log s_n) = 0.
To prove (<ref>), note first of all that
s̃_n,δ_n^2 = ∑_k ∈ I_n,δ_n^-( Z̃_T_k,n^(k)_A_k,n)
and that
s_n,δ_n,-^2 = ∑_k ∈ I_n,δ_n^-σ_k,n^2
so that it is enough to show that
lim inf_n→∞inf_k ∈ I_n,δ_n^-( Z̃_T_k,n^(k)_A_k,n) /σ_k,n^2≥β^2.
To this end, first note that
( Z̃_T_k,n^(k)_A_k,n) = σ_k,n^2
if T_k,n≤√(s_n) since as noted above
_A_k,n≡ 1
in this case.
Thus, we need only to get a good bound on ( Z̃_T_k,n^(k)_A_k,n) when T_k,n≥√(s_n).
For this, note that
( Z̃_T_k,n^(k)_A_k,n)
= [ (Z̃_T_k,n^(k))^2 _A_k,n] - [ Z̃_T_k,n^(k)_A_k,n]^2
= [ (Z̃_T_k,n^(k))^2 _A_k,n] - [ Z̃_T_k,n^(k)_A_k,n^c]^2
≥[ (Z̃_T_k,n)^2 _|Z̃_T_k,n| ≤√(T_k,n)log^4 T_k,n] - [ |Z̃_T_k,n| _|Z̃_T_k,n| > √(T_k)log^4 T_k,n]^2
≥[ (Z̃_T_k,n)^2 _|Z̃_T_k,n| ≤√(T_k,n)log^4 T_k,n] - C T_k,n/log^8 T_k,n,
where the second equality follows from the fact that [ Z̃_T_k^(k)_A_k,n] = - [ Z̃_T_k^(k)_A_k,n^c] since [ Z̃_T_k ] = 0, and the last inequality follows from Lemma <ref>.
Recall that we only need to use the lower bound (<ref>) when T_k ≥√(s_n) and note that (<ref>) and Lemma <ref> imply that
lim_n→∞[ (Z̃_n)^2 _|Z̃_n| ≤√(n)log^4 n]/(Z_n)
= β^2.
This completes the proof of (<ref>) and thus also the proof that (<ref>) converges to zero in L^1.
Proof of convergence of (<ref>).
To prove the convergence of (<ref>) to a standard Gaussian we will use the Lindeberg-Feller CLT.
Since the normalization s̃_n,δ_n is chosen so that (<ref>) has variance 1 and since assumption (<ref>) together with (<ref>) implies that s̃_n,δ_n→∞, we need
only to check the Lindeberg condition; that is, for any >0
lim_n→∞1/s̃_n,δ_n^2∑_k ∈ I_n,δ_n^-[ ( Z̃_T_k,n^(k)_A_k,n - [ Z̃_T_k,n^(k)_A_k,n] )^2 _| Z̃_T_k,n^(k)_A_k,n - [ Z̃_T_k,n^(k)_A_k,n] | > s̃_n,δ_n] = 0.
However, by (<ref>) and (<ref>) it is enough to prove the above statement with s̃_n,δ_n replaced by s_n. That is, we need to show for any >0 that
lim_n→∞1/s_n^2∑_k ∈ I_n,δ_n^-[ ( Z̃_T_k,n^(k)_A_k,n - [ Z̃_T_k,n^(k)_A_k,n] )^2 _| Z̃_T_k,n^(k)_A_k,n - [ Z̃_T_k,n^(k)_A_k,n] | > s_n ] = 0.
To obtain a simple bound on the expectation above, note that
E[ (Y-μ)^2 _|Y-μ|>a] ≤ 2E[Y^2 _|Y|>a/2] + 2μ^2, if |μ|<a/2.
To apply this simple bound to the expectations in (<ref>) we need to check that
[ Z̃_T_k,n^(k)_A_k,n] ≤ s_n/2, ∀ k ∈ I_n,δ_n^-.
It follows from Lemma <ref> that [ Z̃_T_k,n^(k)_A_k,n] ≤ C T_k,n/s_n/√(log s_n).
Also, note that by Corollary <ref> there is a constant C such that k ∈ I_n,δ_n^- implies that T_k,nlog T_k,n≤ C σ_k,n^2 ≤ C δ_n s_n^2, and thus C T_k,n/s_n/√(log s_n)≤ s_n/2
for all n large enough and k ∈ I_n,δ_n^-.
This completes the verification of (<ref>), and thus to check (<ref>) it is enough to prove for all >0 that
lim_n→∞1/s_n^2∑_k ∈ I_n,δ_n^-{[ (Z̃_T_k,n^(k))^2 _A_k,n∩{|Z̃_T_k,n^(k)| > s_n }]
+ [ Z̃_T_k,n^(k)_A_k,n]^2
} = 0.
For the first expectation inside the sum in (<ref>), note that
for n sufficiently large (so that 1/√(log s_n) < ϵ) we have
A_k,n∩{|Z̃_T_k,n^(k)| > s_n } = { s_n < |Z̃_T_k,n^(k)| ≤s_n/√(log s_n)∨√(T_k,n)log^4 T_k,n}
= { s_n < |Z̃_T_k,n^(k)| ≤√(T_k,n)log^4 T_k,n},
and thus
[ (Z̃_T_k,n^(k))^2 _A_k,n∩{|Z̃_T_k,n^(k)| > s_n }]
≤[ Z̃_T_k,n^2 _ s_n < |Z̃_T_k,n| ≤√(T_k,n)log^4 T_k,n].
Note that this shows that the expectation above is zero unless √(T_k,n)log^4 T_k,n > s_n, and since for n large enough we have s_n ≥√(s_n)log^4 s_n, we can conclude that the above expectation is zero unless T_k,n > s_n.
To bound this truncated second moment in the case T_k > s_n we would like to use Lemma <ref>,
but to apply this we need to first check that that s_n > √(T_k log T_k).
However,
it follows from Theorem <ref> and the definition of I_n,δ_n^- that
for n sufficiently large we have
T_k,nlog T_k,n≤ C δ_n s_n^2 ≤ϵ^2 s_n^2 for all k ∈ I_n,δ_n^-. Therefore, we can conclude for n large and k ∈ I_n,δ_n^- that
[ (Z̃_T_k,n^(k))^2 _A_k,n∩{|Z̃_T_k,n^(k)| > s_n }]
≤ C ( T_k,nloglog T_k,n + T_k,n^2 log^2 T_k,n/^2 s_n^2) _T_k,n > s_n
≤ C σ_k,n^2 ( loglog T_k,n/log T_k,n + σ_k,n^2/ s_n^2) _T_k,n > s_n
≤ C σ_k,n^2 ( loglog s_n/log s_n + δ_n/).
For the second expectation in the sum in (<ref>), recall that if T_k,n≤√(s_n) then A_k,n^c = ∅ and thus
this expectation is zero.
On the other hand, since
|[ Z̃_T_k,n^(k)_A_k,n ] | = |[ Z̃_T_k,n^(k)_A_k,n^c ]| ≤[ |Z̃_T_k,n| _|Z̃_T_k,n > √(T_k,n)log^4 T_k,n ],
we can also use Lemma <ref> to obtain the bound
[ Z̃_T_k,n^(k)_A_k,n ]^2
≤C T_k,n/log^8 T_k,n_T_k,n > √(s_n)≤C σ_k,n^2/log^9 T_k,n_T_k,n > √(s_n)≤C σ_k,n^2 /log^9 s_n.
Combining the estimates in (<ref>) and (<ref>)
and noting ∑_k ∈ I_n,δ_n^-σ_k,n^2 ≤∑_k=1^ℓ_n+1σ_k,n^2 = s_n^2, one obtains (<ref>).
This completes the verification of the Lindeberg condition and completes the proof of the lemma.
Having completed the preparatory steps, we are now ready to give the proof of Theorem <ref>.
Let δ_n → 0 be as in Corollary <ref> and let
b_n^2 = β^2 s_n,δ_n,+^2 + s̃_n,δ_n^2/s_n^2.
Note that since
( Z̃_T_k,n^(k)_A_k,n)
≤[ ( Z̃_T_k,n^(k))^2_A_k,n]
≤[ ( Z̃_T_k,n^(k))^2 ] = σ_k,n^2,
we have
that s̃_n,δ_n^2 ≤ s_n,δ_n,-^2,
and because s_n^2 = s_n,δ_n,+^2 + s_n,δ_n,-^2 and β<1, it then follows that b_n ≤ 1 for all n.
Moreover, since we have shown that (<ref>) holds whenever lim inf_n→∞s_n,δ_n,-/s_n > 0, we can also conclude that
lim inf_n→∞ b_n ≥β.
Therefore, if we can prove that
X_n - [X_n]/b_n s_nn→∞⟹Φ,
then the conclusion of Theorem <ref> will hold with β_n = b_n ∨β.
We will prove
(<ref>)
by proving that every subsequence has a further subsequence that converges to a standard Gaussian.
To this end, let n_j →∞ be a fixed subsequence and consider the following cases.
Case 1: lim inf_j→∞s_n_j,δ_n_j,-/s_n_j = 0.
By passing to a further subsequence we can assume that lim_j→∞s_n_j,δ_n_j,-/s_n_j = 0. It then follows from Corollary <ref> that X_n_j - [X_n_j ]/β s_n_j⇒Φ.
Since s̃_n,δ_n≤ s_n,δ_n,- always holds, then the definition of b_n and the assumption on the subsequence in this case implies that b_n_js_n_j/β s_n_j→ 1 as j→∞, so that X_n_j - [X_n_j ]/b_n_j s_n_j⇒Φ as j→∞.
Case 2: s_n_j,δ_n_j,+ = 0 for infinitely many j≥ 1.
In this case, by passing to a further subsequence we can assume that s_n_j,δ_n_j,+ = 0 for all j, in which case
X_n_j - [ X_n_j ] = 𝒵_n_j,δ_n_j^-,
s_n_j,δ_n_j,- = s_n, and b_n_j s_n_j = s̃_n_j,δ_n_j so that Lemma <ref> implies that
X_n_j - [ X_n_j ] /b_n_j s_n_j
= 𝒵_n_j,δ_n_j^-/s̃_n_j,δ_n_jj→∞⟹Φ.
Case 3: lim inf_j→∞s_n_j,δ_n_j,-/s_n_j > 0 and s_n_j,δ_n_j,+ > 0 for all but finitely many j.
Since s_n_j,δ_n_j,+>0 for all j large enough we can decompose
X_n_j - [ X_n_j ] /b_n_j s_n_j
= β s_n_j,δ_n_j,+/b_n_j s_n_j𝒵_n_j,δ_n_j^+/β s_n_j,δ_n_j,+ + s̃_n_j,δ_n_j/b_n_j s_n_j𝒵_n_j,δ_n_j^-/s̃_n_j,δ_n_j.
Corollary <ref> implies that 𝒵_n_j,δ_n_j^+/β s_n_j,δ_n_j,+⇒Φ, and
Lemma <ref>
gives that 𝒵_n_j,δ_n_j^-/s̃_n_j,δ_n_j⇒Φ also. Also, the two terms on the right side of (<ref>) are independent random variables and the squares of the coefficients of the two terms sum to 1 by the definition of b_n. This implies that the right side of (<ref>) converges to Φ in distribution as j→∞.
A disadvantage to the above proof of Theorem <ref> is that the formula for the scaling multiple β_n depends on the choice of the sequence δ_n → 0 in Corollary <ref> which is non-explicit.
The following lemma gives another sequence that is asymptotically equivalent to the scaling constants used in the proof above, but which has the advantage of not relying on the choice of δ_n and thus which can be used to compute the scaling constants β_n for certain choices of cooling maps.
The sequence β_n in Theorem <ref> can be chosen as
β_n = b̃_n ∨β,
where b̃_n = ∑_k=1^ℓ_n+1( Z̃_T_k,n1_A_k,n) /s_n^2,
and where the set A_k,n is defined as in (<ref>).
Comparing (<ref>) with (<ref>) and (<ref>), we see that it's enough to prove that
lim_n→∞sup_k ∈ I_n,δ_n^+| ( Z̃_T_k,n^(k)1_A_k,n)/β^2 σ_k,n^2 - 1 | = 0,
where again δ_n is the sequence from Corollary <ref> (if I_n,δ_n^+ = ∅ then the supremum in the display above is by convention taken to be zero).
Note that we are always free to pick the sequence δ_n→ 0 slow enough so that δ_n ≥1/log s_n. If this is the case, then for n large enough and k ∈ I_n,δ_n^+ we have that
s_n/√(log s_n)≤√(δ_n s_n^2)≤σ_k,n≤ C √(T_k,nlog T_k,n)≤√(T_k,n)log^4 T_k,n.
Therefore, for n large enough and k ∈ I_n,δ_n^+ we have
( Z̃_T_k,n^(k)1_A_k,n)
= ( Z̃_T_k,n^(k)1_ |Z̃_T_k,n^(k)| ≤√(T_k,n)log^4 T_k,n),
and
T_k,n≥s_n^2/log^5 s_n.
From this, it follows that for n large enough we have
sup_k ∈ I_n,δ_n^+| ( Z̃_T_k,n^(k)1_A_k,n)/β^2 σ_k,n^2 - 1 |
≤sup_m≥s_n^2/log^5 s_n| ( Z̃_m _|Z̃_m| ≤√(m)log^4 m ) /β^2 (Z_m) - 1 |,
and then (<ref>) follows from this together with Lemma <ref>.
§ EXAMPLES
In this section we consider specific cooling maps that display interesting/illustrative behaviour in the study of the limit distribution of RWCRE in the subbalistic (κ∈(0,1)) and in the Gaussian critical (κ = 2) regime.
To explore the features of RWCRE, we consider both regular cooling maps (maps for which T_k = τ_k - τ_k-1 admits an asymptotic behavior) as well as some cooling maps with more irregular behavior.
§.§ Sub-balistic regime
For the examples in this subsection we will assume that the distribution μ on environments is κ-regular with κ∈ (0,1).
[Polynomial cooling when κ∈ (0,1)]
Let T_k ∼ A k^α for some constants A,α>0.
Since T_k→∞ we can use Corollary <ref> to determine the limiting distributions. It is easy to check for this example that max_k λ̃_τ,n(k) ≤C/√(ℓ_n)→ 0,
and thus we can conclude that
X_n-[X_n]/√((X_n))⇒Φ.
Moreover, we can use Theorem <ref> to replace the scaling by the standard deviation of X_n with a more explicit scaling in this case. Indeed, using (Z_n) ∼ n^2κ we have in this case that
(X_n) = ∑_k=1^ℓ_n(Z^(k)_T_k) + (Z^(ℓ_n +1)_n-τ(ℓ_n) ) ∼^2 A^2κ/2ακ+1ℓ_n^2ακ+1,
as n→∞.
Since ℓ_n ∼( α+1/A)^1/α+1 n^1/α+1 we can then conclude that
X_n-[X_n]/σ_A,α n^2ακ+1/2(α+1)n→∞⟹Φ,
where σ_A,α^2 = ^2 A^2κ/2ακ+1( α+1/A)^2ακ+1/α+1.
Note that the scaling exponent 2ακ+1/2(α+1) converges to 1/2 as α→ 0 and to κ as α→∞ (if κ=1/2 then the scaling exponent is always to 1/2 for all α>0).
[Exponential cooling when κ∈ (0,1)]
Let T_k ∼ Ce^ck for C,c>0.
Again we can use Corollary <ref> to determine the limiting distributions.
In this case one can only obtain limiting distributions along certain subsequences, but the limiting distribution is always a sum of independent (normalized) Mittag-Leffler random variables. For simplicity we will only describe the limiting distribution along the subsequence n_j = τ(j).
For this choice of n_j we have
(V_n_j)^2 = ∑_k=1^j (T_k)^2κ∼∑_k=1^j (C e^ck )^2κ∼(C e^c)^2κ/e^2cκ-1 e^2cκ j, as j→∞.
Since for k≥ 1 fixed and j large enough we have that λ̃^↓_τ,n_j(k) = (T_j-k+1)^κ/V_n_j, it follows that
lim_j→∞λ^↓_τ,n_j(k) = λ_c,*(k):= √((θ_c^-2-1)) (θ_c)^k, ∀ k≥ 1, where θ_c = e^-cκ.
Since ∑_k≥ 1 (λ_c,*(k))^2 = 1 we can conclude from Corollary <ref> that
X_n_j - [X_n_j]/√((X_n_j))j→∞⟹( )^⊗λ_c,*.
We can also obtain more explicit centering and scaling terms for this example. Indeed, it follows from Theorem <ref> that
[X_n_j] ∼μ_𝔐 (C e^c)^κ/e^cκ-1 e^cκ j and
√((X_n_j))∼ (C e^c)^κ/√(e^2cκ-1) e^cκ j,
and since n_j = τ(j) ∼C e^c/e^c-1 e^cj implies that (C e^c)^κ e^cκ j∼ (e^c-1)^κ n_j^κ we can re-write these asymptotics as
[X_n_j] ∼μ_𝔐 (e^c-1)^κ/e^cκ-1 n_j^κ and
√((X_n_j))∼ (e^c-1)^κ/√(e^2cκ-1) n_j^κ.
Note that since both the mean and standard deviation are multiples of n_j^κ we can remove the centering from the left side of (<ref>) at the cost of adding the constant μ_𝔐√(e^2cκ-1)/ (e^cκ-1) to the right side.
To simplify further note that we can re-write the limit on the right side of (<ref>) as
( )^⊗λ_c,*
=∑_k=1^∞λ_c,*(k) ^(k) - μ_𝔐/ = 1/∑_k=1^∞λ_c,*(k) ^(k) - μ_𝔐/∑_k=1^∞λ_c,*(k)
= 1/∑_k=1^∞λ_c,*(k) ^(k) - μ_𝔐/√(e^2cκ-1)/e^cκ-1,
where ^(1),^(2),… are i.i.d. copies of the random variable .
Thus, finally we obtain the simplified form of the limiting distribution where the scaling is the same as in the RWRE case
X_n_j/ n_j^κj→∞⟹(e^c-1)^κ/√(e^2cκ-1)∑_k=1^∞λ_c,*(k) ^(k)
= (e^c-1)^κ∑_k=1^∞ (θ_c)^k ^(k).
[super exponential cooling when κ∈ (0,1)]
If log(T_k) ∼ e^ck for some c>0,
then all subsequential limits are a linear combination of one or two independent Mittag-Leffler random variables.
To give a specific example consider the cooling map with τ(j) = 2^2^j for j≥ 1. Then, fix a parameter θ≥ 0 and consider the subsequence n_j = (1+θ) 2^2^j. Note that for j large enough we have τ(j) ≤ n_j < τ(j+1), so that the representation in (<ref>) becomes X_n_jLaw=∑_k=1^j Z^(k)_T_j + Z^(j+1)_θ 2^2^j.
One can show
that only the last two terms in the sum on the right survive in the limiting distribution.
Indeed, since the variance asymptotics in Theorem <ref> imply that (X_n) ∼(Z_T_j) + (Z_θ 2^2^j) ∼^2 (1+θ^2κ) n_j^2κ, we can then apply Corollary <ref> to get
X_n_j-[X_n_j] /√((X_n_j))j→∞⟹1/√(1+θ^2κ)( ^(1)-μ_𝔐/) + θ^κ/√(1+θ^2κ)( ^(2)-μ_𝔐/).
Finally, since the mean and variance asymptotics in Theorem <ref> imply that [X_n_j] ∼μ_𝔐(1+θ^κ)n_j^κ and
√((X_n_j))∼√(1+θ^2κ) n_j^κ we can rewrite the limiting distribution above as
X_n_j/n_j^κj→∞⟹^(1) + θ^κ^(2).
[Mixtures of Mittag-Leffler and Gaussian when κ∈ (0,1)]
The basic idea to build mixtures of Mittag-Leffler random variables with Gaussian is to build a cooling map by interweaving a fast growing cooling map where the limiting distribution is a sum of Mittag-Leffler distributions with a slow growing cooling map where the limiting distribution is Gaussian.
To give a specific example of this, let τ be the cooling map with cooling intervals given by
T_2^i = ⌊ 2^(i-1)/(2κ)⌋ for i≥ 1, and T_k = 1 if k ∉{2^i: i∈}.
We will compute the limiting distribution along the subsequence n_j = τ(2^j).
To this end, first note that from Theorem <ref> that
(X_n_j) = ∑_i=1^j (Z_2^(i-1)/(2κ)) + (2^j-j) (Z_1)
∼∑_i=1^j ^2 2^i-1 + 2^j (Z_1)
∼(^2 + (Z_1) ) 2^j, as j→∞.
Moreover, since the k-th largest cooling interval among the first 2^j cooling intervals is T_2^j-k+1 = 2^(j-k)/(2κ) we get that for any fixed k≥ 1,
lim_j→∞λ^↓_τ,n_j(k)
= lim_j→∞√((Z_2^(j-k)/(2κ)) /(X_n_j))
= √(^2/^2 + (Z_1)) 2^-k/2
=: λ_*(k).
Since a(λ_*) = ( (Z_1)/^2 + (Z_1))^1/2∈ (0,1),
we get in this case that the limiting distribution is a mixture of sums of independent Mittag-Leffler random variables and an independent Gaussian.
More precisely,
X_n_j-[X_n_j]/√((X_n_j))j→∞⟹( )^⊗λ_*
+ ( (Z_1)/^2 + (Z_1))^1/2Φ.
§.§ Gaussian critical regime
For the examples in this subsection we will assume that the distribution μ on environments is 2-regular. The examples below demonstrate the various properties of the sequence of scaling constants β_n in Theorem <ref> that can be obtained by changing the cooling map τ.
Recall that the scaling constants β_n can be given by the formula in (<ref>). However, for this formula to be of practical use one needs some way of approximating the truncated variance terms involved. To this end, one can use the following result which follows from the tail bounds for Z̃_n in Section <ref>.
Let (Z_n)_n≥ 0 be a RWRE with a 2-regular distribution μ on environments. Then,
lim_n→∞sup_x≥√(n)log^4 n| (Z̃_n _|Z̃_n| ≤ x )/b^2 nlog n + 2K_0 v n log( x ∧ (nv/2)/√(n)) - 1 | = 0.
The proof of Corollary <ref> is straightforward (using similar methods as in the proofs of Lemmas <ref> and <ref>), but is rather tedious.
Since Corollary <ref> is needed only for the justifying the computations of β_n in the examples below, we
give its proof in Appendix <ref>.
The first two examples give families of cooling maps which show that
we cannot change the condition in Theorem <ref> that β_n ∈ [β,1] to a smaller interval.
[Polynomial cooling when κ=2]
If T_k ∼ A k^α for some constants A,α>0, we claim that the scaling constants β_n can be chosen to be equal to a constant σ_α∈ (β,1] which depends only on α>0.
More precisely,
X_n - [X_n]/σ_α√((X_n))n→∞⟹Φ,
where σ_α^2 =
1 if α≤ 1
b^2+K_0 v/α/b^2 + K_0 v if α > 1.
(Note that σ_α∈ (β,1] for all α>0 with σ_α→β as α→∞.)
To simplify computations, we first determine the scaling constants along the subsequence τ(n) of cooling times.
It follows from the variance asymptotics in Corollary <ref> that
s_τ(n)^2 = (X_τ(n))
∼ (b^2 + K_0 v)Aα∑_k=1^n k^αlog k
∼(b^2 + K_0 v) A α/α+1 n^α+1log n.
A consequence of this is that max_k≤ n√(T_k)log^4 T_k ≤s_τ(n)/√(log s_τ(n)) for n sufficiently large, and therefore using (<ref>) we have that
β_τ(n)^2 ∼∑_k=1^n ( Z̃_T_k_|Z̃_T_k| ≤s_n/√(log s_n)) /(b^2 + K_0 v) A α/α+1 n^α+1log n .
Since it follows from (<ref>) that s_τ(n)/√(log s_τ(n))∼ C n^α+1/2 for some C, we can then use Corollary <ref> to deduce that
β_τ(n)^2
∼∑_k=1^n { b^2 T_k log T_k + 2K_0 v T_k log( n^(α+1)/2∧ T_k /√(T_k)) }/(b^2 + K_0 v) A α/α+1 n^α+1log n
∼∑_k=1^n { b^2 α k^αlog k + K_0 v k^αlog( n^α+1∧ k^2α/k^α) }/(b^2 + K_0 v) α/α+1 n^α+1log n n→∞⟶
1 if α≤ 1
b^2+K_0 v/α/b^2 + K_0 v if α > 1.
(Note that in the second line we are replacing T^k by k^α instead of A k^α inside the logarithm of the second term since the multiplicative constant A inside the logarithm doesn't change the asymptotics.)
We next divide the analysis into the cases α < 1 and α≥ 1.
Case 1: α < 1. Since s_τ(n)/√(log s_τ(n))∼ C n^α+1/2 for some C, it follows that for n large enough we have that s_τ(n)/√(log s_τ(n))≥T_k v/2 for all k≤ n. Therefore, using Corollary <ref> we have that
β_τ(n)^2
∼ (b^2 + K_0 v) ∑_k=1^n T_klog T_k /(b^2 + K_0 v) A α/α+1 n^α+1log n ∼ (b^2 + K_0 v) Aα∑_k=1^n k^αlog k /(b^2 + K_0 v) A α/α+1 n^α+1log n n→∞⟶ 1.
Case 2: α≥ 1. Again using that s_τ(n)/√(log s_τ(n))∼ C n^α+1/2 for some C, it follows from Corollary <ref> that
β_τ(n)^2
∼∑_k=1^n { b^2 T_klog T_k + 2K_0 v T_k log(n^(α+1)/2∧ T_k/√(T_k)) }/(b^2 + K_0 v) A α/α+1 n^α+1log n
∼ (b^2-K_0 v) ∑_k=1^n T_klog T_k + 2K_0 v ∑_k=1^n T_k log(n^α+1/2∧ T_k ) /(b^2 + K_0 v) A α/α+1 n^α+1log n
∼(b^2-K_0 v)Aα/α+1 n^α+1log n + K_0 v A n^α+1log n /(b^2 + K_0 v) A α/α+1 n^α+1log n
= b^2 + K_0 v/α/b^2 + K_0 v.
Thus, we have justified the formula for the scaling constant σ_α in (<ref>), but only along the subsequence τ(n).
To justify the general limiting distribution, we decompose
X_n-[X_n]/σ_α√((X_n))
= X_τ(ℓ_n) - [X_τ(ℓ_n)]/σ_α√((X_n))
+ Z_n-τ(ℓ_n)^(ℓ_n+1) - [Z_n-τ(ℓ_n)] /σ_α√((X_n)).
One can then check that (X_n) ∼(X_τ(ℓ_n)) and that (Z_n-τ(ℓ_n)) = o( (X_n)) so that the first term on the right converges to a standard Gaussian and the second term on the right converges to zero in distribution.
Note: For polynomial cooling case, I believe that we have (X_n) ∼(b^2 + K_0 v) α/α+1 nlog n. So we could write the limiting distribution in the form X_n - [X_n]/c_α√(nlog n)⇒Φ, but I don't think the asymptotics of the mean are good enough for us to replace [X_n] with n v. I'm not sure this is worth adding to the paper.
[Exponential cooling when κ=2]
Let τ be a cooling map with exponentially growing cooling intervals T_k ∼ e^rk for some r>0.
For this example the cooling intervals grow fast enough that in the decomposition of the variance
(X_n) = ∑_k=1^ℓ_n+1(Z_T_k,n),
only the “large" terms in the sum contribute to the asymptotics of the variance.
More precisely, using the notation from Section <ref> we have that lim_δ→ 0lim inf_n→∞s_n,δ,+^2/s_n^2 = 1,
or equivalently lim_δ→ 0lim sup_n→∞s_n,δ,-^2/s_n^2 = 0.
Then it follows from Corollary <ref> that for this example we have X_n - [X_n]/β√((X_n))n→∞⟹Φ.
If T_k ∼ e^r k as k→∞ for some r>0, we claim that we can let the scaling constants β_n ≡β, where β is the constant defined in (<ref>).
As with the previous example, it is most convenient to first consider limiting distributions along the subsequence τ(n).
For this, we have
s_τ(n)^2
= ∑_k=1^n (Z_T_k)
∼ (b^2 + K_0 v)r ∑_k=1^n k e^r k∼(b^2 + K_0 v)r e^r/e^r-1 n e^rn.
Note that for this example the cooling intervals T_k grow fast enough that only the largest terms in the sum contribute to the asymptotics above.
In particular for any >0 we have s_τ(n)^2 ∼∑_k=n- log n^n(Z_T_k). Similarly one can adjust the formula for the scaling constants in (<ref>) so that the numerator involves only the last log n truncated variance terms without changing the asymptotics of the scaling constant.
The benefit of this is that if one chooses >0 sufficiently small then it follows from (<ref>) that for n sufficiently large we have √(T_k)log^4 T_k > s_τ(n)/√(log s_τ(n)) for all n-log n≤ k ≤ n, and therefore
β_τ(n)^2
∼∑_k=⌈ n- log n ⌉^n ( Z̃_T_k_|Z̃_T_k| ≤√(T_k)log^4 T_k ) /∑_k=⌈ n- log n ⌉^n (Z_T_k) n→∞⟶β^2,
where we used Lemma <ref> for the last limit.
Note: For the exponential cooling case, I believe that the asymptotics of the mean and variance are good enough for us to conclude that
X_n - nv/b√(nlog n)⇒Φ.
Again, I'm not sure if we should include this.
Hi Jon, I did some computations and found the following:
b^-1(X_n) = ∑_k = 1^ℓ_n e^rk rk + T_ℓ_n,nlog (T_ℓ_n,n)
Ignoring for now the boundary term we have for n_j = τ(j)
b^-1(X_n_j) = ∑_k = 1^j e^rk rk = r/e^r -1∑_k = 1^j [e^r(j+1) - e^j] = r/e^r - 1[ j e^r(j + 1) - e^r (j + 1) - e^r/e^r - 1]
Now n_j = τ(j) = e^r(j+1) - e^r/e^r - 1∼ e^rje^r/e^r - 1. So the above (variance with no boundary) relation turn into
(X_n_j) ∼ bn_jlog n_j
Can we extend this relation to all points n ∈ [τ(j), τ(j+1)]? In general, for n ∈ [τ(j), τ(j+1)] I believe n ∼α_n e^r(j+1)/e^r - 1 with α_n ∈ [1 - e^-rj, e^r - e^-rj. So roughly, α_n ∈ [1,e^r]. Will the boundary term affect the scaling?, We compare
lim_jτ(j+1) logτ(j+1)/τ(j) logτ(j) =lim_jr(j+2)e^r(j+2)/r(j+1) e^r(j+1) = e^r
If the above limit is not 1, then can we be sure of the scaling as a function of n? Yes consider
n_α = τ(j) + α e^r(j+1)∼αe^r(j+2) + (1-α) e^r(j+1)/e^r-1 with α∈ [0,1]
We obtain that
(X_n_α = r/e^r-1[j e^r(j+1) - e^r(j+1) - e^r/e^r - 1] + α e^r(j+1) r(j+1).
Now we see that,
sup_α∈ [0,1]n_αlog n_α/(X_n_α)→ 1 as j →∞.
While the above two examples show that one cannot restrict the scaling constants to an interval smaller than [β,1], these examples are all regular enough so that the scaling constant can be a fixed constant and doesn't need to oscillate with n. The following gives an explicit example of a cooling map where one cannot obtain a limiting distribution without letting β_n depend on n.
One can give somewhat simpler examples which demonstrate this oscillation of β_n, but the example below has lim inf_n→∞β_n = β and lim sup_n→∞β_n = 1.
[Oscillating scaling constants when κ=2]
Fix A>0 and Let τ be the cooling map given by
T_j2^j = 2^j , j∈,
and
T_k = 1 if k ∉{j2^j: j∈}.
Fix t ∈ [0,1] and consider the distribution of
X_τ(n2^n + t n2^n) - [X_τ(n2^n + t n2^n)]
= ∑_j=1^n (Z^(j2^j)_2^j - [Z_2^j] ) + ∑_k≤ (t+1)n2^n
k ∉{j 2^j: j∈} (Z^(k)_1 - [Z_1]).
As n→∞, the variance of the first term on the right is asymptotic to
(b^2 + K_0 v) (log 2) ∑_j=1^n j 2^j
∼ (b^2 + K_0 v) (log 4) n 2^n,
while the variance of the second term on the right is
((t+1)n 2^n - n)(Z_1) ∼ (t+1)(Z_1) n 2^n,
so that the variance of both terms are of the order n 2^n.
On the other hand, by Example <ref> the first term converges to a standard Gaussian when scaled by β times its standard deviation whereas the second term converges to a standard Gaussian when scaled by its standard deviation.
From this we see that
X_τ(n2^n + t n2^n) - [X_τ(n2^n + t n2^n)]/α_t√(( X_τ(n2^n + t n2^n) ))n→∞⟹Φ,
where α_t^2 = b^2 (log 4) + (t+1) (Z_1)/(b^2+K_0 v) (log 4) + (t+1) (Z_1).
Thus, for this cooling map the scaling constants β_n must oscillate in the interval [α_0,α_1].
While Example <ref> gives an explicit example where the scaling constants β_n do not converge to a constant as n→∞, the interval [α_0,α_1] where they oscillate in is strictly smaller than the interval [β,1]. It is intuitively clear that one can construct a cooling map where the scaling constants oscillate throughout the entire interval [β,1], but it seems such an example would be more complicated than the one in Example <ref> and so we leave this to the interested reader to construct such an example.
[Full oscillation with super exponential cooling κ = 2]
In fact, to construct such an example, consider
T_2^2^i = 2^2^i, i≥ 1,
and
T_k = 1 if k ∉{ 2^2^i: i ∈}.
Set Δ_j := τ(2^2^j+1) - τ(2^2^j) take t ≥ 0 and let
n_j(t) = τ(2^2^j) + tτ(2^2^j)∧Δ_j
a similar computation to the one above, shows that
X_n_j(t) - [X_τ(n_j(t))]
= ∑_i=1^j (Z^(2^2^i)_2^2^i - [Z_2^2^i] ) + ∑_k≤ n_j(t)
k ∉ A (Z^(k)_1 - [Z_1]).
As n→∞, the variance of the first term on the right is asymptotic to
(b^2 + K_0 v) (log 2)∑_i=1^j 2^2^i 2^i
∼ (b^2 + K_0 v) (log 2) 2^2^j 2^j,
while the variance of the second term on the right is
(n_j(t) - j)(Z_1) ∼ n_j(t)(Z_1).
By Example <ref> the first term converges to a standard Gaussian when scaled by β times its standard deviation whereas the second term converges to a standard Gaussian when scaled by its standard deviation.
Let τ̂(j) =∑_i = 1^j (log 2)∑_i=1^j 2^2^i 2^i. From this we see that
X_n_j(t) - [X_n_j(t)]/α_t√(( X_τ(n_j)) )n→∞⟹Φ,
where α_t^2 = b^2 τ̂(j) + (n_j(t)) (Z_1)/(b^2+K_0 v) τ̂(j) + (n_j(t) ) (Z_1).
Note that
lim_j n_j(t) /τ̂(j) = t + 1
and thus we may set
α_t ∼b^2 + (t+1)(Z_1)/b^2 + K_0v + (t+1) (Z_1).
If we take instead of n_t = τ(2^2^j) + t_j Δ_j with t_j →∞, we have that we see that for this cooling map the scaling constants β_n must oscillate in the interval [α_0,1].
[Full oscillation of multiplicative scaling constant β_n]
For i≥ 1 let m_i = 2^2^i and r_i = i m_i = i 2^2^i. Then, let
τ be the cooling map given by
T_r_i = m_i, i≥ 1,
and
T_k = 1 if k ∉{ r_i: i ≥ 1 }.
For a fixed t ≥ 0 we will consider the distribution of the RWCRE along the subsequence n_j=n_j(t) = τ( r_j + t m_j log(m_j) ).
Note that for j large enough we have τ(r_j) ≤ n_j(t) < τ(r_j+1), and thus for j large enough we can decompose X_n_j(t) - [X_n_j(t)] as
X_n_j(t) - [X_n_j(t)] =
∑_i=1^j ( Z_m_i^(r_i) - [Z_m_i] )
+ ∑_ k≤ r_j + t m_j log(m_j)
k∉{r_i: i≥ 1 } (Z_1^(k) - [Z_1]).
As j→∞, the variance of the first term on the right in (<ref>) is asymptotic to
(b^2 + K_0 v)∑_i=1^j m_i log(m_i)
∼ (b^2 + K_0 v) m_j log(m_j),
while the variance of the second term on the right is
(r_j + t m_j log(m_j) - j) (Z_1) ∼
r_j (Z_1) if t=0
t m_j log(m_j) (Z_1) if t>0.
Since the second term in (<ref>) is a sum of i.i.d. random variables, it converges in distribution to a standard Gaussian when scaled by its standard deviation, while for the first term in (<ref>) we can apply Theorem <ref> to the cooling map τ' with increments T'_k = m_k = 2^2^k to get that this sum converges to a standard Gaussian when scaled by β times its standard deviation.
From this we see that
X_n_j(t) - [X_n_j(t)] /α_t √((X_n_j(t)))j→∞⟹Φ,
where α_t^2
= b^2 + t(Z_1)/b^2 + K_0 v + t (Z_1).
Finally, note that α_0 = β and α_t ↗ 1 as t↗∞. This shows that in applying Theorem <ref> to the cooling sequence τ in this example, not only does one need to let the scaling constant β_n vary with n, but also that the sequence β_n will continue to oscillate through the entire interval [β,1].
§ TECHNICAL RESULTS FOR THE TAIL ESTIMATES
In this appendix we collect some results for sums of i.i.d. random variables that are needed in Section <ref>.
Some of these results may be already known, but we include them here for completeness.
Assume ξ_1 has mean zero, is bounded below by -L for some L>0, and has right tail decay P(ξ_1>x)=O(x^-2). Then, there exists a constant C>0 that depends on the distribution of ξ_1 such that
E[e^-λξ_1]≤ e^Cλ^2|logλ|, for all λ∈ (0,1/e).
Define ξ̂_1=ξ_1+L so that ξ̂_1 is non-negative and E[ξ̂_1]=L.
We begin by noting that
e^-λ LE[e^-λξ_1]
=E[e^-λξ̂_1]
= 1 - λ L + E[e^-λξ̂_1-1+λξ̂_1]
= 1 - λ L + λ∫_0^∞ (1-e^-λ x) P( ξ̂_1 > x ) x
≤ 1 - λ L + λ∫_0^∞min{λ x, 1} P( ξ̂_1 > x ) x.
Since λ <1,
bounding the probability inside the integral by 1 when x<1 and by Kx^-2 when x≥ 1 we obtain
e^-λ LE[e^-λξ_1]
≤ 1-λ L+λ^2∫_0^1 x x +Kλ^2∫_1^λ^-1 x^-1 x +Kλ∫_λ^-1^∞ x^-2 x
= 1-λ L+1/2λ^2+Kλ^2|logλ|+Kλ^2
≤exp{-λ L+(2K+1/2)λ^2|logλ|},
where in the last inequality we also used that λ∈ (0,1/e) implies that |logλ| > 1.
Finally, the proof is completed by multiplying both sides of the above inequality by e^λ L.
Let ξ_1,ξ_2,… be i.i.d. random variables which are bounded below and have right tail decay P(ξ_1 > x) = O(x^-2). Then, for any a>0 there exist constants c,C'>0 (depending on a) such that
P( ∑_i=1^n ξ_i ≤ - x ) ≤ C' e^-c x^2/nlog n,
∀ x ∈ (0,an].
First of all, for any λ∈ (0,1/e) we have from Lemma <ref> that
P( ∑_i=1^n ξ_i ≤ - x ) ≤ e^-λ x( E[ e^-λξ_1] )^n ≤exp{ -λ x + Cn λ^2 |logλ| }.
It is not easy to find a value of λ that minimizes the upper bound on the right, but we can achieve a nearly optimal upper bound by choosing λ such that λ |logλ| = x/2Cn.
Since the function λ↦λ |logλ| achieves its maximum of 1/e at λ = 1/e, we can find such a λ∈ (0,1/e) only if x ≤2C/e n. Thus, for now we will restrict ourselves to x < 2C/e n and will extend our bound to x ≤ an later.
With this choice of λ we then have
P( ∑_i=1^n ξ_i ≤ - x )
≤exp{-λ|logλ| x + C n λ^2 |logλ|^2/|logλ|}
= exp{-x^2/4Cn |logλ|}.
Next, we claim that our choice of λ above implies that
|logλ| ≤ 2 log( 2Cn/x) .
To see this, note that t^2 |log(t^2)| = 2t (t |log t|) < 2/et < t for all t ∈ (0,1/e), and applying this with t = x/2Cn < 1/e yields that
( x/2Cn)^2 | log( x/2Cn)^2 | < x/2Cn = λ |logλ|. The monotonicity of λ |logλ| on (0,1/e) then implies that 1/e > λ > ( x/2Cn)^2 which in turn implies the claim in (<ref>).
Combining (<ref>) and (<ref>) we get
P( ∑_i=1^n ξ_i ≤ - x )
≤exp{-x^2/4Cn log( 2Cn/x) }≤exp{-x^2/4Cn log n }, for all 2C ≤ x ≤2C/en,
and then by choosing C' large enough we get P( ∑_i=1^n ξ_i ≤ - x ) ≤ C' exp{-x^2/8Cn log n } for all x ≤2C/e n.
Finally, we can extend this bound to x≤ an by changing the coefficient in the exponent. Indeed, if 2C/en < x ≤ an then
P( ∑_i=1^n ξ_i ≤ - x )
≤ P( ∑_i=1^n ξ_i ≤ - 2C/e n )
≤ C' exp{-(2Cn/e)^2/4Cn log n }
≤ C' exp{ -C/e^2a^2x^2/nlog n}.
This completes the proof of the corollary.
Assume that {ξ_i}_i≥ 1 are i.i.d., zero mean random variables such that P(|ξ_i| > x) = 𝒪(x^-2), and let S_n = ∑_i=1^n ξ_i.
Then, there exists a constant C>0 such that
P(|S_n| > t √(n log n)) ≤C/t^2 log n + C/t^4 ∀ t ≤√(n/log n).
First of all, note that by choosing C>0 large enough we can assume that t> 1/√(log n).
Next, note that
P(|S_n| > t√(nlog n))
≤ n P( |ξ_1| > t √(nlog n)) + P( | ∑_i=1^n ξ_i _|ξ_i| ≤ t √(nlog n)| > t √(nlog n)).
Since the assumption on the tail of |ξ_1| implies that the first term on the right is 𝒪( 1/t^2 log n), it remains only to bound the second probability for 1/√(log n)≤ t≤√(n/log n). To this end, first note that
| E[ ξ_i _|ξ_i| ≤ x] | =
| E[ ξ_i _|ξ_i| > x] |
≤ E[ |ξ_i| _|ξ_i| > x] = 𝒪( x^-1 ),
where the first equality is because E[ξ_i]=0 and the last equality follows from the assumptions on the tails of ξ_i. Therefore, there exists a constant L>0 such that
| E[ ∑_i=1^n ξ_i _|ξ_i| ≤ t √(nlog n)] |
≤ n | E[ ξ_i _|ξ_i| ≤ t √(nlog n)] |
≤t/2√(nlog n) for all t > L/√(log n).
Letting ξ̅_i,t,n = ξ_i _|ξ_i|≤ t √(nlog n) - E[ξ_i _|ξ_i|≤ t √(nlog n)],
we can conclude that for t>L/√(log n) we have
P( | ∑_i=1^n ξ_i _|ξ_i| ≤ t √(nlog n)| > t √(nlog n))
≤ P( | ∑_i=1^n ξ̅_i,t,n| > t√(nlog n)/2)
≤4/t^4 n^2 (log n)^2 E[ ( ∑_i=1^n ξ̅_i,t,n)^4 ]
≤C/t^4 n^2(log n)^2{ n E[ ξ_i^4 _|ξ_i| ≤ t√(nlog n)] + n^2 E[ ξ_i^2 _|ξ_i| ≤ t√(nlog n)]^2 }.
Since the tail decay assumptions imply that
E[ ξ_i^4 _|ξ_i| ≤ x] = 𝒪( x^2 )
and
E[ ξ_i^2 _|ξ_i| ≤ x] = 𝒪( log x ), we can conclude that for n large enough and t ∈ [1/√(log n), √(n/log n)] we have
P( | ∑_i=1^n ξ_i _|ξ_i| ≤ t √(nlog n)| > t √(nlog n))
≤C/t^2log n + C/t^4 (log n)^2 E[ ξ_i^2 _|ξ_i| ≤ n]^2
≤C/t^2log n + C/t^4.
§ PRECISE TRUNCATED VARIANCE ASYMPTOTICS
First of all, we claim that it is enough to prove the claimed asymptotics for the truncated second moment instead of the truncated variance. That is, we claim it is enough to prove
lim_n→∞sup_x≥√(n)log^4 n| [ (Z̃_n)^2 _|Z̃_n| ≤ x] /b^2 nlog n + 2K_0 v n log( x ∧ (nv/2)/√(n)) - 1 | = 0.
To see that it is enough to prove (<ref>), note that Lemma <ref> implies for n sufficiently large and x≥√(n)log^4 n that
| (Z̃_n _|Z̃_n| ≤ x ) - [ (Z̃_n)^2 _|Z̃_n| ≤ x] |
= [ Z̃_n _|Z̃_n| ≤ x]^2
= [ Z̃_n _|Z̃_n| > x]^2
≤C n^2/x^2.
Using this, the statement of the lemma follows easily from (<ref>).
The advantage of (<ref>) rather than the original statement in the Lemma is that the truncated second moment [ (Z̃_n)^2 _|Z̃_n| ≤ x] is monotone in x whereas the truncated variance is not.
In particular, since [ (Z̃_n)^2 _|Z̃_n| ≤ x] ≤(Z_n) ∼ (b^2 + K_0 v)nlog n then we need only to get good upper bounds on the truncated second moment when x ∈ [√(n)log^4 n, nv/2].
For such x we then have from Lemma <ref> for M fixed and n sufficiently large that
[ (Z̃_n)^2 _|Z̃_n| ≤ x]
≤[ (Z̃_n)^2 _|Z̃_n| ≤ M√(nlog n)] + [ (Z̃_n)^2 _M√(nlog n) < |Z̃_n| < √(n)log^4 n ]
+ [ (Z̃_n)^2 _√(n)log^4 n ≤ |Z̃_n| ≤ x ] .
By Lemma <ref> we can bound the second expectation in the last line by C n loglog n + Cnlog n/M^2, while the Bounded Convergence Theorem implies the first expectation is asymptotic to b^2 E[Φ^2_|Φ|≤ M] nlog n as n→∞. Thus, by first choosing M large and then n large enough we get that for any >0 there exists an n_0 = n_0() such that the two terms on the right in (<ref>) can be bounded above by (1+2)b^2 nlog n for all n≥ n_0. For the last term in (<ref>), it follows from (<ref>) and (<ref>) that for any >0 there is an n_1 = n_1() such that (|Z̃_n| > t) ≤(1+)K_0 v n/t^2 for all t ∈ [√(n)log^4 n, nv/2].
Applying this bound we get for all n large enough (depending on ) that
[ (Z̃_n)^2 _√(n)log^4 n ≤ |Z̃_n| ≤ x ]
≤ n log^8 n ( |Z̃_n| > √(n)log^4 n ) + ∫_√(n)log^4 n^x 2t (|Z̃_n| > t) t
≤ (1+)K_0 v n + 2(1+) K_0 v n log( x/√(n)log^4 n)
≤ (1+2)2 K_0 v n log( x/√(n)).
Since >0 is arbitrary this completes the proof of the needed upper bound for [(Z̃_n)^2 _|Z̃_n|≤ x ].
For the lower bound on [(Z̃_n)^2 _|Z̃_n|≤ x ] we note that this truncated second moment is non-decreasing in x and so it's enough to only give the necessary lower bounds for x ∈ [√(n)log^4 n, vn/2].
Using Lemma <ref> we can bound the truncated second moment below by [(Z̃_n)^2 _|Z̃_n|≤√(n)log^4 n ] ≥ (1-2)b^2 nlog n for any ϵ>0 and n sufficiently large.
This is a good enough lower bound for x ∈ [√(n)log^4 n, √(n)e^√(log n) ], but it
remains to get a good lower bound for x ∈ [√(n)e^√(log n), vn/2].
For such x we can begin
by noting that Lemma <ref> implies that for any >0 and n sufficiently large we have
[ (Z̃_n)^2 _|Z̃_n| ≤ x]
≥ (1-2)b^2 nlog n + [ (Z̃_n)^2 _√(n)log^4 n ≤ |Z̃_n| ≤ x ].
For the second term on the right, it follows from (<ref>) that (|Z̃_n| ≥ t) ≥(Z̃_n ≤ -t) ≥ (1-)K_0 (nv-t)t^-2 for all t ∈ [√(n)log^4 n, nv/2] and n sufficiently large.
Therefore, for n sufficiently large we have
[ (Z̃_n)^2 _√(n)log^4 n ≤ |Z̃_n| ≤ x ]
≥∫_√(n)log^4 n^x 2t (|Z̃_n| > t ) t - x^2 (|Z̃_n| > x)
≥∫_√(n)log^4 n^x 2 (1-)K_0 (nv-t)t^-1 t - (1+)K_0 nv
≥ 2(1-2)K_0 v n log( x/√(n)),
where in the last inequality we used that x ≥√(n)e^√(log n).
§ DELETE
We now examine L^p convergence for p <2r with r ∈N.
, that is, we would like to prove
lim_n E[ |ℜ_n |^p] = 0 for all p ∈ (2,2r).
Note that (<ref>) implies that
ℜ_n → 0 almost surely.
By Theorem 4.6.3 in <cit.>, to obtain L^p convergence it suffices to prove that
for any r ∈N
sup_n E[ |ℜ_n |^2r] < ∞.
Now note that since ∑_k (λ_n(k))^2 = 1, it follows that sup_k(λ_n(k))^2 ≤ 1 and therefore, for any r ∈N
∑_ℓ = 1^2r∑_k_1, …, k_ℓe_k_i≥ 2, i =1,…,
ℓ∑_e_k_1 + … +e_k_ℓ = 2r∏_i = 1^∞(λ_n(k_i))^e_k_i
≤∑_ℓ = 1^2r2r - ℓ -1ℓ -1∑_k_1, … k_ℓ∏_i =
1^∞(λ_n(k_i))^2
= ∑_ℓ = 1^2r2r-ℓ -1ℓ -1≤ 2^2r + 1 < ∞.
10
AP16
Sung Won Ahn and Jonathon Peterson, Oscillations of quenched slowdown
asymptotics for ballistic one-dimensional random walk in a random
environment, Electron. J. Probab. 21 (2016), Paper No. 16, 27.
3485358
Ash00
R.B. Ash and C.A. Doleans-Dade, Probability and measure theory, Elsevier
Science, 2000.
ACdCdH20
Luca Avena, Yuki Chino, Conrado da Costa, and Frank den Hollander,
Random walk in cooling random environment: recurrence versus
transience and mixed fluctuations, arXiv:1903.09200 (2019).
ACP21
Luca Avena, Conrado da Costa, and Jonathon Peterson, Gaussian,
stable, tempered stable and mixed limit laws for random walks in cooling
random environments, arXiv e-prints (2021), arXiv:2108.08396.
AveHol19
Luca Avena and Frank den Hollander, Random walks in cooling random
environments, Sojourns in Probability Theory and Statistical Physics - III
(Singapore) (Vladas Sidoravicius, ed.), Springer Singapore, 2019, pp. 23–42.
BD18
Dariusz Buraczewski and Piotr Dyszewski, Precise large deviations for
random walk in random environment, Electron. J. Probab. 23 (2018),
Paper no. 114, 26. 3885547
CGZ00
Francis Comets, Nina Gantert, and Ofer Zeitouni, Quenched, annealed and
functional large deviations for one-dimensional random walk in random
environment, Probability Theory and Related Fields 118 (2000),
no. 1, 65–114.
DPZ96
Amir Dembo, Yuval Peres, and Ofer Zeitouni, Tail estimates for
one-dimensional random walk in random environment, Comm. Math. Phys.
181 (1996), no. 3, 667–683. 1414305
Durret
Rick Durrett, Probability—theory and examples, Cambridge Series in
Statistical and Probabilistic Mathematics, vol. 49, Cambridge University
Press, Cambridge, 2019, Fifth edition of [ MR1068527]. 3930614
Fel71
William Feller, An introduction to probability theory and its
applications. Vol. II., Second edition, John Wiley & Sons Inc., New
York, 1971.
FGP10
Alexander Fribergh, Nina Gantert, and Serguei Popov, On slowdown and
speedup of transient random walks in random environment, Probab. Theory
Related Fields 147 (2010), no. 1-2, 43–88. 2594347
GZ98
Nina Gantert and Ofer Zeitouni, Quenched sub-exponential tail estimates
for one-dimensional random walk in random environment, Comm. Math. Phys.
194 (1998), no. 1, 177–190. 1628294
dHol00
F. Hollander, Large deviations, FIM Series, American Mathematical
Society, 2000.
KKS75
H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in
a random environment, Compositio Math. 30 (1975), 145–168.
Petrov
V. V. Petrov, Sums of independent random variables, Springer-Verlag, New
York-Heidelberg, 1975, Translated from the Russian by A. A. Brown, Ergebnisse
der Mathematik und ihrer Grenzgebiete, Band 82. 0388499
Sinai
Ya. G. Sinaĭ, The limit behavior of a one-dimensional random walk in
a random environment, Teor. Veroyatnost. i Primenen. 27 (1982),
no. 2, 247–258. 657919
S75
Fred Solomon, Random walks in a random environment, Ann. Probab.
3 (1975), no. 1, 1–31.
SZ99
Alain-Sol Sznitman and Martin Zerner, A law of large numbers for random
walks in random environment, Ann. Probab. 27 (1999), no. 4,
1851–1869. 1742891
Yon19
Yongjia Xie, Functional Weak Limit of Random Walks in Cooling Random
Environment, Electron. Commun. Probab. 25 (2020), 1–14.
Z04
O. Zeitouni, Random Walks in Random Environment, Lecture Notes in
Mathematics 1837 (2004), 1–121.
|
http://arxiv.org/abs/2307.04932v1 | 20230710225935 | Tur\' an number for bushes | [
"Zoltán Füredi",
"Alexandr Kostochka"
] | math.CO | [
"math.CO",
"05D05, 05C65, 05C05"
] |
myheadings
Füredi and Kostochka: Bushes, July 4, 2023
Turán number for bushes
Zoltán Füredi
Alfréd Rényi Institute of Mathematics, Budapest, Hungary.
E-mail: .
Research partially supported by National Research, Development and Innovation Office NKFIH grants 132696 and 133819.
Alexandr Kostochka
University of Illinois at Urbana–Champaign, Urbana, IL 61801
and Sobolev Institute of Mathematics, Novosibirsk 630090, Russia. E-mail: .
Research supported in part by NSF
grant DMS-2153507 and NSF RTG grant DMS-1937241.
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================
Let a,b ∈ Z^+,
r=a + b, and let T be a tree
with parts U = {u_1,u_2,…,u_s}
and V = {v_1,v_2,…,v_t}.
Let U_1, … ,U_s and V_1, …, V_t be disjoint sets, such that |U_i|=a and |V_j|=b for all i,j.
The (a,b)-blowup of T is the
r-uniform hypergraph with edge set
{U_i ∪ V_j :
u_iv_j ∈ E(T)}.
We use the Δ-systems method to prove the following Turán-type result.
Suppose a,b,t∈ Z^+, r=a+b≥ 3, a≥ 2, and
T is a fixed tree of diameter 4 in which the degree of the center vertex is t.
Then there exists a C=C(r,t,T)>0 such that
|ℋ|≤ (t-1)n r-1 +Cn^r-2 for every
n-vertex r-uniform hypergraph ℋ not containing
an (a,b)-blowup of T. This is asymptotically exact when t≤ |V(T)|/2.
A stability result is also presented.
Mathematics Subject Classification: 05D05, 05C65, 05C05.
Keywords: Hypergraph trees, extremal hypergraph theory, Delta-systems.
§ INTRODUCTION
§.§ Basic definitions and notation
An r-uniform hypergraph (an r-graph, for short), is a family of r-element subsets of a finite set.
We associate an r-graph with its edge set and call its vertex set V().
Often we take V()=[n], where [n]:={ 1, 2, 3,…, n}.
Given an r-graph ,
let the Turán number of , _r(n,), denote the maximum number of edges in an r-graph on n vertices that does not contain a copy of .
Since
a (graph) tree is connected and bipartite, it uniquely defines the parts in its bipartition. So, we say a tree
T is an (s,t)-tree if one part of V(T) has s vertices and the other has t vertices.
Let s,t, a,b > 0 be integers,
r=a + b, and let T = T(X,Y) be an (s,t)-tree
with parts U = {u_1,u_2,…,u_s}
and V = {v_1,v_2,…,v_t}.
Let U_1, … ,U_s and V_1, …, V_t be pairwise disjoint sets, such that |U_i|=a and |V_j|=b for all i,j.
So | ⋃ U_i∪ V_j | = as+bt.
The (a,b)-blowup of T, denoted by (T,a,b), is the
r-uniform hypergraph with edge set
(T,a,b):= {U_i ∪ V_j :
u_iv_j ∈ E(T)}.
The goal of this paper is to find the asymptotics of the Turán number for (a,b)-blowups of many trees of radius 2 using the
Δ-systems method.
Earlier, (a,b)-blowups of different classes of trees and different pairs (a,b) were considered in <cit.>. The main result in <cit.>
is the following.
Suppose r ≥ 3, s,t ≥ 2, a + b = r, b < a < r.
Then (as n→∞) any
𝒯-free n-vertex r-graph satisfies
|| ≤ (t - 1)n r-1 + o(n^r-1).
This is asymptotically sharp whenever t≤ s.
This theorem asymptotically settles about a half of possible cases, but when t<s and a≤ b, it is expected that the asymptotics is different from the one in Theorem <ref>. More is known on (a,b)-blowups of paths. Let P_ℓ denote the (graph) path with ℓ edges.
The case of P_2 was resolved asymptotically by Frankl <cit.> (for b=1) and by Frankl and Füredi <cit.> (for all 1≤ a≤ r-2 and b=r-a):
_r(n, (P_2,a,b)) = Θ( n^max{ a-1,b }).
The case of P_3 was fully solved for large n
by Füredi and Özkahya <cit.>. They showed that for fixed 1≤ a,b <r with r=a+b and for n> n_0(r),
_r(n,( P_3,a,b)) = n-1r-1.
For longer paths, the following was proved in <cit.>.
Let a+b=r, a,b≥ 1 and ℓ≥ 3. Suppose further that
(i) ℓ is odd, or
(ii) ℓ is even and a>b, or (iii) (ℓ, a,b) = (4,1,2).
Then
_r(n,(P_ℓ,a,b)) = ⌊ℓ-1/2⌋n r - 1+ o(n^r - 1).
So, the situation with blowups of P_ℓ is not resolved for the case when ℓ≥ 4 is even and a≤ b apart from
the case (ℓ, a,b) = (4,1,2).
In this paper, we first consider (a,b)-blowups of special trees of radius 2.
A graph bush B_t,h is the radius 2 tree obtained from the star K_1,t by joining each vertex of degree one to other h new vertices. So B_t,h has 1+t +th vertices. Let s=1+th. Then B_t,h is an (s,t)-tree with s>t.
Suppose that a,b,t,h are positive integers, a+b=r and t≥ 2.
By (a,b,t,h)-bush, _t,h(a,b), we will call the (a,b)-blowup of B_t,h.
This means the center vertex of B_t,h is replaced by an a-set A,
its neighbors by the b-sets B_1, …, B_t and its second
neighbors by a-sets A_i,j, i∈ [t], j∈ [s].
In particular, the (a,b)-blowup of the path P_4 is the (a,b,2,1)-bush _2,1(a,b).
Since _t,h(a,b) has t disjoint edges B_i∪ A_i,1 for i=1,…,t, the example of the r-uniform hypergraph with vertex set [n] in which every edge intersects the interval [t-1] shows that
_r(n,_t,h(a,b))≥n r-n-t+1 r∼ (t-1)n-1 r-1.
We will use the Δ-systems approach to show that this is asymptotically correct in many cases.
Recall that for a>b≥ 2 the asymptotic equality follows from Theorem <ref>.
In this paper we deal with all cases and also present a somewhat refined result by
considering shadows of hypergraphs.
Recall that for an r-graph ℋ the shadow, ∂ℋ, is the collection of (r-1)-sets that lie in some edge of ℋ.
Our first main result is the following.
Suppose that a,b,t,h are positive integers, r=a+b≥ 3.
Also suppose that in case of (a,b)= (1,r-1) we have h=1.
Then there exists a C=C(r,t,h)>0 such that the following holds.
If ℋ is an n-vertex r-uniform family not containing
a bush _t,h(a,b) then
|ℋ|≤ (t-1)|∂ℋ| +Cn^r-2.
This implies that (<ref>) is asymptotically exact in these cases as r,t,h are fixed and n→∞.
Note that for (a,b)=(1,r-1) we prove (<ref>) only for h=1.
In fact, in this case (<ref>) does not hold for h≥2.
An example is this: V()=[n], A=[t-1] and =_1∪_2, where
_1 is the set of r-subsets of [n] with exactly one vertex in A and
_2 is the Steiner system S_1(n-t+1,r,r-1) on [n]-A.
This example has asymptotically (t-1+1/r)n r-1 edges and does not contain
_t,2(1,r-1).
By increasing h, we can get examples without _t,h(1,r-1) that have even more edges.
McLennan <cit.> proved that in the graph case (a=b=1) we have
_r(n, B_t,h)= 1/2(t+th-1)n+O(1), some extremal graphs are vertex-disjoint unions of complete graphs K_t+th.
For the case t=1 the inequality (<ref>) is rather weak, _r(n, _1,h(a,b))= O(n^r-2) is known for a≥ 2. Even better bounds were proved in <cit.>. So we usually suppose that t≥ 2, r≥ 3.
Since each tree of diameter 4 with the degree of the center equal to t is a subgraph of a graph bush B_t,h for some h, Theorem <ref>
yields the following somewhat more general result.
Suppose a,b,t∈ Z^+, r=a+b≥ 3 and a≥ 2.
Let T be a fixed tree of diameter 4 in which the degree of the center vertex is t.
Then there exists a C=C(r,t,T)>0 such that the following holds.
If ℋ is an n-vertex r-graph not containing
an (a,b)-blowup of T, then
|ℋ|≤(t-1)n r-1 +Cn^r-2.
We also use the Δ-systems approach to show that for many a≥ 2 and b≥ 2 with a+b=r, the r-uniform hypergraphs without _t,h(a,b) of cardinality "close" to extremal contain vertices of "huge" degree.
Suppose that a,b,t,h are positive integers, a,b≥ 2 and r=a+b≥ 5.
Then for any C_0>0
there exist n_0>0 and C_1>0 such that the following holds.
If n>n_0, ℋ is an n-vertex r-uniform family not containing
a bush _t,h(a,b) and |ℋ|> (t-1)n-1r-1-C_0 n^r-2, then
there are t-1 vertices in [n] each of which is contained in at least n-1r-1-C_1 n^r-2 edges of .
The structure of this paper is as follows. In the next section, we discuss the Δ-system method and present a lemma
by Füredi <cit.> from 1983 that will be our main tool. In Section <ref> we describe properties of so called intersection structures. It allows us to prove the main case of Theorem <ref> (the case a≥ 2) in
Section <ref> and the case of a=1 and h=1 in
Section <ref>. Then in
Section <ref> we prove Theorem <ref>.
§ DEFINITIONS FOR THE Δ-SYSTEM METHOD AND A LEMMA
A family of sets {F_1,…,F_s} is an s-star or a Δ-system or an s-sunflower of size s with kernel A, if
F_i∩ F_j=A for all 1≤ i<j≤ s. The sets F_i∖ A are called petals.
For a member F of a family ℱ, let the intersection structure of F relative to ℱ be
ℐ(F,ℱ)={F∩ F': F'∈ℱ∖{F}}.
An r-uniform family ℱ⊆[n] r is r-partite if there exists a partition
(X_1,…,X_r) of the vertex set [n] such that |F∩ X_i|=1 for each
F∈ℱ and each i∈ [r].
For a partition (X_1,…,X_r) of [n] and a set S⊆ [n], the pattern Π(S) is the set
{i∈ [r]: S∩ X_i≠∅}. Naturally, for a family ℒ of subsets of [n],
Π(ℒ)={Π(S):S∈ℒ}⊆ 2^[r].
For any positive integers s and r, there exists a positive constant c(r,s) such that every
family ℱ⊆[n] r contains a subfamily ℱ^*⊆ℱ satisfying
1. |ℱ^*|≥ c(r,s)|ℱ|.
2. ℱ^* is r-partite, together with an r-partition (X_1,…,X_r).
3. There exists a family 𝒥 of proper subsets of [r] such that Π(ℐ(F,ℱ^*))=𝒥
holds for all F∈ℱ^*.
4. 𝒥 is closed under intersection, i.e., for all A,B∈𝒥 we have A∩ B∈𝒥, as
well.
5. For any F∈ℱ^* and each A∈ℐ(F,ℱ^*), there is an s-star in ℱ^*
containing F with kernel A.
Remark 1. The proof of Lemma <ref> in <cit.> yields that if ℱ itself is r-partite with an r-partition (X_1,…,X_r), then the r-partition in the statement can be taken the same.
Remark 2. By definition, if for some M⊂ [r] none of the members of the family 𝒥 of proper subsets of [r] in
Lemma <ref> contains M, then for any two sets F_1,F_2∈ℱ^*, their intersections with
⋃_j∈ MX_j are distinct. It follows that if |M|=m, then
|ℱ^*|≤∏_j∈ M|X_j| ≤( n-(r-m)/m)^m.
Thus, if |ℱ^*|> ( n-r+m/m)^m, then every m-element subset of [r] is contained in some
B∈𝒥.
Call a family 𝒥 of proper subsets of [r] m-covering if every m-element subset of [r] is contained in some
B∈𝒥. In these terms, Remark 2 says that
For k=0,1,…,r, define the family 𝒥^(k) of proper subsets of [r] as follows. It contains (a) the sets [r]-{i} for 1≤ i≤ k, (b) all (r-2)-element subsets of [r] containing {1,2,…,k}, and (c) all the intersections of these subsets.
By definition, each 𝒥^(k) is (r-2)-covering. Moreover,
14.5cm each (r-2)-covering family of proper subsets of [r] closed under intersections contains a subfamily isomorphic to some 𝒥^(k).
Indeed, if an (r-2)-covering family 𝒥 of proper subsets of [r] contains exactly k sets of size r-1,
then it must contain as members all (r-2)-element subsets of [r] not contained in these k sets, so properties (a) and (b) of the definition hold. Part (c) follows since 𝒥 is closed under intersections.
§ GENERAL CLAIMS ON INTERSECTION STRUCTURES.
Call a set B a (b,s)-kernel in a set system ℱ if B is the kernel of size b in a sunflower with s petals formed by members of ℱ.
If a+b=r and an r-uniform family ℋ does not contain _t,h(a,b), then there do not exist
disjoint sets A_0,B_1,B_2,…,B_t with |A_0|=a, |B_1|=… =|B_t|=b such that all B_1,…,B_t are (b,thr)-kernels in ℋ and the sets A_0∪ B_1,… ,A_0∪ B_t are edges of ℋ.
Suppose, there are such disjoint sets A_0,B_1,B_2,…,B_t. Let D_0=A_0∪⋃_j=1^tB_j. For i=1,…,t, do the following. Since B_i is a (b,thr)-kernel and |D_i-1∖ B_i|=a+(t-1)b +(i-1)ha≤ t h r -h, there exist h petals A_i,j (1≤ j≤ h) of a thr-sunflower with kernel B_i which are disjoint from D_i-1. Let D_i=D_i-1∪_j Y_i,j.
After t steps, we find a _t,h(a,b) whose edges are A_0∪ B_i and B_i∪ A_i,j for i=1,…,t, j=1,2,…,h.
Suppose a+b=r and 𝒢⊂[n] r with |𝒢|>1/c(r,thr)n^r-2
does not contain _t,h(a,b).
By Lemma <ref> and (<ref>), there is 𝒢^*⊆𝒢
satisfying the lemma such that the corresponding family 𝒥 of proper subsets of [r] is (r-2)-covering.
Let (X_1,…,X_r) be the corresponding partition.
Family 𝒥 does not contain disjoint members A and B such that |A|=a and |B|=b.
Suppose, it does. By renaming the elements of 𝒥, we may assume that A={1,…,a} and
B={a+1,…,r}. Let X={x_1,…,x_r}∈𝒢^*, where x_i∈ X_i for all i.
Since A∈𝒥, {x_1,…,x_a} is an (a,thr)-kernel in 𝒢^*. Let B_1,…,B_t be some t petals in the sunflower with kernel {x_1,…,x_a}. Since [r]-[a]=B∈𝒥, each of B_1,…,B_t
is a (b,thr)-kernel in 𝒢^*, contradicting Lemma <ref>.
If a+b=r and 2≤ a,b≤ r-2, then for each 0≤ k≤ r-2 and for k=r the family
𝒥^(k) has
disjoint members A and B such that |A|=a and |B|=b, unless (r,a,b,k)=(4,2,2,1).
If (a,b)=(1,r-1) or (a,b)=(r-1,1) and r≥ 3, then for each 1≤ k≤ r-2 and for k=r the family
𝒥^(k) has
disjoint members A and B such that |A|=a and |B|=b.
If k≥ a, then we let A=[a], B=[r]-[a], and represent them as follows:
A=⋂_k+1≤ i<i'≤ r([r]-{i,i'})∩⋂_a+1≤ i≤ k([r]-{i}),
B=⋂_1≤ i≤ a([r]-{i}).
If k≥ b, then we have a symmetric representation.
In particular, this proves the claim for (a,b)=(1,r-1) or (a,b)=(r-1,1).
If k≤ a-2, then we again let A=[a], B=[r]-[a], but represent them as follows (using that a≤ r-2 and k≤ a-2):
A=⋂_a+1≤ i<i'≤ r([r]-{i,i'}),
B=⋂_1≤ i≤ k([r]-{i})∩⋂_k+1≤ i<i'≤ a([r]-{i,i'}).
By symmetry, the only remaining case is that k=a-1=b-1. So r is even, and a=b=r/2=k+1. In this case, if r>4, then
we let A=[k-1]∪{k+1,k+2}, B=([r]-[k+2])∪{k}, and represent them as follows:
A=⋂_k+2≤ i<i'≤ r([r]-{i,i'})∩ ([r]-{k}),
B=⋂_1≤ i≤ k-1([r]-{i})∩ ([r]-{k+1,k+2}).
§ PROOF OF THE MAIN THEOREM
§.§ Basic procedure
Now we are ready to prove Theorem <ref>.
Assume that an n-vertex r-uniform family ℋ
does not contain _t,h(a,b).
Define C=C(r,t,h) := 1/c(r,thr), where c is from Lemma <ref>.
For any r-uniform family 𝒢, let 𝒢^* denote a family
satisfying Lemma <ref> and 𝒥(𝒢^*)⊂ 2^[r] denote the corresponding intersection structure.
Do the following procedure. Let ℋ_1=(ℋ)^*
and _1=𝒥(ℋ^*).
For i=1,2,…, if
|ℋ∖⋃_j=1^i ℋ_j|≤ n^r-2/c(r,thr),
then stop and let m:=i and
ℋ_0=ℋ∖⋃_j=1^i ℋ_j;
otherwise, let
ℋ_i+1:=(ℋ∖⋃_j=1^i ℋ_j)^*
and _i+1=𝒥((ℋ∖⋃_j=1^i ℋ_j)^*).
This procedure provides a partition of ℋ,
ℋ=⋃_i=0^m ℋ_i.
Let denote ⋃_i=1^m ℋ_i.
By definition,
|ℋ_0| ≤ Cn^r-2,
so we get
|| + Cn^r-2≥ | ℋ|.
We distinguish four cases, a,b≥ 2 and r≥ 5 (discussed in Subsection <ref>), (a,b)=(2,2) and r=4 (Subsection <ref>), (a,b)=(r-1,1) and r≥ 3 (Subsection <ref>), and finally
(a,b)=(1,r-1), r≥ 3 and h=1 (Section <ref>).
§.§ Case of a,b≥ 2 and r≥ 5
Here 2≤ a,b≤ r-2 and (a,b)≠ (2,2), so Lemma <ref> implies that for each 1≤ i≤ m, 𝒥(ℋ_i) has exactly r-1 (r-1)-subsets, it contains 𝒥^(r-1).
Hence for each hyperedge E∈ℋ_i⊂ (1≤ i≤ m) there exists an element c(E)∈ E such that
each proper subset of E containing c(E) is a kernel of a thr-star in ℋ_i.
Beware of the fact that although each ℋ_i is k-partite, the partitions might differ for different values of i.
This does not cause any problem in our argument, we only need the existence of the element c(E).
Define the function α on [n] r-1 as follows:
Given an (r-1)-set Y, let α(Y) be the number of edges E∈ with Y=E∖{ c(E)}.
α(Y)≤ t-1, for all (r-1)-subsets Y of [n].
Suppose to the contrary that there are t distinct E_1∈ℋ_i_1, …, E_t∈ℋ_i_t
such that E_i=Y∪{ c(E_i)}.
If i_j=i_j' for some j≠ j',
then the intersection structure of ℋ_i_j would contain the (r-1)-set Y, a contradiction.
Thus i_1,…,i_t are all distinct. By relabelling we may suppose that E_i∈ℋ_i.
We will find the t+1 disjoint sets A_0, B_1,B_2,…,B_t contradicting Lemma <ref> using induction as follows.
Let A_0⊂ Y arbitrary, |A_0|=a and let D_0:= ∪ E_i, |D_0|=t+r-1.
We define the sets E_i', D_i, B_i step by step as follows. We will have D_i:=D_0∪_j ≤ i E_j' and |D_i|= t+r-1+i(r-1-a).
For i=1, 2, …, t consider the family ℋ_i and its member E_i in it.
By the intersection structure of ℋ_i,
the set A_0∪{c(E_i)} is an (a+1,thr)-kernel in ℋ_i.
One of the thr petals of the sunflower in ℋ_i with kernel A_0∪{c(E_i)}
should be disjoint from D_i-1; let E_i' be the corresponding set in ℋ_i. Since c(E_i)∈ E_i', and homogeneity gives c(E_i')=c(E_i), the set
B_i:=E_i'-A_0 is a (b,thr)-kernel.
Claim <ref> implies
(t-1)|∂| ≥∑_∀ (r-1)-set Yα(Y)= ||.
This, together with (<ref>), completes the proof of Theorem <ref> in this case.
§.§ The case (a,b)=(2,2)
Lemma <ref> implies that for each 1≤ i≤ m, either—
𝒥^(3) is contained in 𝒥(ℋ_i), it has exactly three 3-subsets, so [4]∖{1},
[4]∖{2}, and [4]∖{3}
are in 𝒥 and 𝒥 also contains
all subsets containing the element 4 but it does not contain 123, or —
𝒥(ℋ_i) is of 𝒥^(1) type, it has a unique 3-subset, 234, and { 1, 12,
13, 14}⊂𝒥. Call ℋ_i (and its edges) type α if 𝒥(ℋ_i) has three 3-subsets.
Each of these edges E∈ℋ_i has an element c(E)∈ E such that
each proper subset of E containing c(E) is a kernel of a 4· t· h-star in ℋ_i.
The union of the families ℋ_i of type α is _α.
Call ℋ_i (and its edges) type β if 𝒥(ℋ_i) has a unique 3-subset.
Each of these edges E∈ℋ_i has an element b(E)∈ E such that
each set K⊂ E of the form {b(E), x} (x∈ E∖{ b(E)}) and the set
E∖{ b(E) } is a kernel of a 4· t· h-star in ℋ_i.
The union of the families ℋ_i of type β
is _β.
Define the function α on [n] 3 as follows:
Given an 3-set Y let α(Y) be the number of edges E∈_α with Y=E∖{ c(E)}.
Define the function β : →[n] r-1 as follows:
Given a 3-set Y, let β(Y) be the number of edges E∈_β with b(E)∈ Y⊂ E.
α(Y)+1/3β(Y)≤ t-1, for all (r-1)-subsets Y of [n].
Let E_1∈ℋ_i_1, …, E_α∈ℋ_i_α be the α(Y) distinct edges with
E∈_α and Y=E∖{ c(E)} and
let E_α+1∈ℋ_i_α+1, …, E_α+β∈ℋ_i_α+β be the β(Y) distinct edges with E∈_β and b(E)∈ Y ⊂ E.
If i_j=i_j' for some j≠ j',
then the intersection structure of ℋ_i_j would contain the 3-set Y, a contradiction.
Thus i_1, i_2,… are all distinct. By relabelling we may suppose that E_i∈ℋ_i.
Suppose to the contrary that α(Y)+1/3β(Y)> t-1, so α + ⌈β/3 ⌉≥ t.
Since |Y|=3, one can find an element y_0∈ Y such that b(E_j)=Y at least ⌈β/3 ⌉ times.
So we may suppose that there are t distinct E_i∈ℋ_i
such that the elements c(E_1), …, c(E_α) and d(E_j):= E_j∖ Y for α < j≤ t are all distinct
and E_i=Y∪{ c(E_i)}, b(E_j)=y_0.
We will find the t+1 disjoint sets A_0, B_1,B_2,…,B_t contradicting Lemma <ref> using induction as follows.
Let A_0: Y∖{ y_0}, |A_0|=2 and let D_0:= ∪ E_i, |D_0|=t+3.
We define the sets E_i', D_i, B_i step by step as follows. We will have D_i:=D_0∪_j ≤ i E_j' and |D_i|= t+3+i.
For i=1, 2, …, t, consider the family ℋ_i and its member E_i in it.
By the intersection structure of ℋ_i,
the set A_0∪{c(E_i)} (and the set A_0∪{c(E_i)}) is an (a+1,4· t· h)-kernel in ℋ_i.
One of the 4· t· h petals of the sunflower in ℋ_i with kernel A_0∪{c(E_i)}
should be disjoint from D_i-1; let E_i' be the corresponding set in ℋ_i. Since c(E_i)∈ E_i', the set
B_i:=E_i'-A_0 is an (b,4· t· h)-kernel.
Claim <ref> implies
(t-1)|∂| ≥∑_∀ 3-set Y( α(Y)+1/3β(Y))= |_α|+ |_β|.
This, together with (<ref>), completes the proof of Theorem <ref> in this case.
§.§ The case (a,b)=(r-1,1)
Call ℋ_i (as above) of type α if 𝒥(ℋ_i) has r-1 (r-1)-subsets.
Each of these edges E∈ℋ_i has an element c(E)∈ E such that
each proper subset of E containing c(E) is a kernel of a thr-star in ℋ_i.
The union of these ℋ_i families is _α.
Call ℋ_j (and its edges) type β if 𝒥(ℋ_j) has no (r-1)-subset.
Note that each element y of an edge E∈ℋ_j
is a kernel of a thr-star in ℋ_j.
The union of these ℋ_j families is _β.
The definition of α on [n] r-1 is the same as in the previous subsections,
the definition of β :→[n] r-1 is even simpler:
β(Y) is the number of edges E∈_β with Y⊂ E.
If α(Y)+β(Y)> t-1, then taking A_0:=Y and B_i:=E_i∖ Y one can see that each B_i is a kernel of a large star,
contradicting Lemma <ref>.
Finally, we complete the proof of this case as follows
(t-1)|∂| ≥∑_∀ (r-1)-set Y( α(Y)+β(Y))= |_α|+ r|_β|≥ ||.
§ HYPERGRAPHS WITHOUT A BUSH _T(1,R-1)
Call an r-graph t-normal if it has no (r-1)-tuples of vertices whose codegree is positive but less than t.
For every edge Y in an r-graph ℋ and any x∈ Y, let Q_ℋ(Y,x)={y∈ V(ℋ)-x: Y-x+y∈ℋ}.
In a
t-normal r-graph ℋ, |Q_ℋ(Y,x)|≥ t-1
for every edge
Y∈ℋ and vertex x∈ Y.
We will prove this case by induction on the number of edges. For |ℋ|<α(r,t) the claim is trivial.
Suppose ℋ is a counter-example with the fewest edges.
If our ℋ is not t-normal, then choose a (r-1)-tuple Y of vertices whose codegree is positive but less than t and
let ℋ' be obtained from ℋ the
edges containing Y. Then |ℋ'|-(t-1) |∂ℋ'|≤ |ℋ|-(t-1) |∂ℋ|, so ℋ' satisfies (<ref>) and is
_t(1,r-1)-free. This contradicts the minimality of ℋ. Thus ℋ is t-normal.
Let q^* be a huge number in terms of rt but small w.r.t. α(r,t)
By (<ref>),
there is a subfamily ℋ^* of ℋ satisfying Lemma <ref> for q=q^* with
|ℋ^*|>n^r-2. Let 𝒥 be the corresponding intersection structure.
By Lemmas <ref> and <ref>, 𝒥 contains a family isomorphic to 𝒥^(r-1), or
to 𝒥^(0). In both cases, 𝒥 contains a singleton. So, ℋ contains an element u and
disjoint (r-1)-tuples U_1,…, U_q^* such that Y_i=U_i+u is an edge in ℋ^* for each 1≤ i≤ q^*.
Since H is t-normal, for each 1≤ i≤ q^* we can choose a (t-1)-element subset
Q'_ℋ(Y_i,u) of Q_ℋ(Y_i,u). Construct the auxiliary bigraph R with parts
{U_i: 1≤ i≤ q^*} and V(ℋ) where U_iv∈ E(R) iff v∈ Q'_ℋ(Y_i,u).
Case 1. R has a matching of size (r+1)t, say M={U_1,v_1,…,U_(r+1)tv_(r+1)t}.
Since all U_i here are disjoint and all v_i are distinct, we can greedily find t disjoint sets U_i∪{v_i}:
start from M and one by one take a set U_i∪{v_i} and delete from the list all U_i'∪{v_i'} with i'≠ i such that
v_i∈ U_i' or v_i'∈ U_i'. If our disjoint sets are U_1∪{v_1},…,U_t∪{v_t}, then
we have _t(1,r-1) with the set of edges
{U_1∪{u},…,U_t∪{u},U_1∪{v_1},…,U_t∪{v_t}}
Case 2. R has no matching of size (r+1)t. Then it has a vertex cover of size less than (r+1)t.
So there is a set V⊂ V(ℋ) with |V|<(r+1)t containing Q'_ℋ(Y_i,u) for at least
q^*-(r+1)t sets Y_i.
Then some (t-1)-element set T=T(u)={z_1,…,z_t-1} serves as Q'_ℋ(Y_i,u) for
at least (q^*-(r+1)t)(r+1)t t-1^-1 sets Y_i. We choose q^* so that this number is at least t-1 and suppose
T(u) serves
for Y_1,…,Y_t-1. If for some other Y_i, Y_i∩ T=∅ and
Q'_ℋ(Y_i,u)≠ T, then we take an element z∈ Q'_ℋ(Y_i,u)- T and construct a _t(1,r-1) with the set of edges
{Y_1∪{u},…,Y_t-1∪{u},Y_i∪{u}}∪{Y_1∪{z_1},…,Y_t-1∪{z_t-1},Y_i∪{z}}.
So, T=Q'_ℋ(Y_i,u) for all 1≤ i≤ q^* such that Y_i∩ P=∅. Hence if for at least one other edge Y in ℋ containing
u and disjoint from T, Q_ℋ(Y_i,u)≠ T, then we find a _t(1,r-1) similarly to (<ref>) because q^* is large. It follows that denoting T'(u)=T(u)∪{u},
14.5cm
the set of edges of H containing u and disjoint from T'(u)-z is contained in the set of edges of H containing z_j and disjoint from T'(u)-z_j for each 1≤ j≤ t-1.
Case 2.1. 𝒥 contains a family isomorphic to 𝒥^(r-1). We may assume that 𝒥 contains all proper subsets of [r] containing 1 and that {x_1,…,x_r}∈ℋ where x_i∈ X_i for i∈ [r]. Since
{1,2}∈𝒥, there are 2t disjoint sets U_h={x_3,h,x_4,h,…,x_r,h} such that
U'_h=U_h∪{x_1,x_2}∈ℋ^* for all 1≤ h≤ 2t. At most t-1 of them intersect T. So we may assume
U'_1,…,U'_t are disjoint from T. Since [r]-{2}∈𝒥, for every 1≤ h≤ t, there are 2t elements
x_2,h,g∈ X_2 such that U_h,g=U_h∪{x_1,x_2,h,g}∈ℋ^* for all 1≤ h≤ t and
1≤ g≤ 2t. We may rename the elements x_2,h,g so that for all 1≤ h≤ t and
1≤ g≤ t, x_2,h,g∉ T∪{x_2}. After that we rename them again so that for all 1≤ h≤ t
the elements x_2,h,1 are distinct. Now, letting z_t=x_1, by (<ref>) we have a _t(1,r-1) with the set of edges {U_h∪{z_h,x_2}: h∈ [t]}∪{U_h∪{z_h,x_2,h,1}: h∈ [t]}.
Case 2.2. For each subfamily ℋ^* of ℋ satisfying Lemma <ref> for q=q^* with
|ℋ^*|>n^r-2, the intersection structure
𝒥=𝒥(ℋ^*) contains a family isomorphic to 𝒥^(0).
Do the following procedure. Let ℋ_1=(ℋ)^*. For i=1,2,…, if
|ℋ-⋃_j=1^i ℋ_j|≤ C· n^r-2, then stop, let m=i
and ℋ'=⋃_j=1^i ℋ_j and ℋ_0=ℋ-ℋ'; otherwise,
let ℋ_i+1=(ℋ-⋃_j=1^i ℋ_j)^*.
By the case, ℋ_i
contains a family isomorphic to 𝒥^(0) for each i≥ 1.
By the structure of 𝒥^(0), each v∈ V(ℋ' can serve as vertex u in the argument at the beginning of Case 2. So, let u∈ V(ℋ') and T'=T'(u) be defined as above.
By (<ref>), if for any z_j∈ T(u) and some edge Y in ℋ containing
z_j and disjoint from T'(u)-z_j, Q_ℋ(Y,z_j)≠ T'(u)-z_j, then we again find a _t(1,r-1) similarly to (<ref>) because q^* is large. It follows that
14.5cm there are disjoint t-element sets T'_1,…, of the form T'(u) covering V(ℋ') such that for
any two u,u'∈ T'_i
the set of edges of H containing u and disjoint from T_i-u equals the set of edges of H containing u' and disjoint from T_i-u' for every T_i.
Since 𝒥^(0) does not contain sets of size r-1, each (r-1)-element set Y∈∂ℋ_i
is only in one set in ℋ_i, thus |ℋ_i|≤ |∂ℋ_i|/r. Moreover, since
|ℋ_0|<α(r,t)n^r-2, by (<ref>) some (r-1)-tuple Y_0 belongs to at least r(t-1) families
ℋ_i, say to ℋ_1,…,ℋ_r(t-1).
For i∈ [r(t-1)], let y_i be the vertex such that Y_0∪{y_i}∈ℋ_i. Let Y'={y_1,…,y_r(t-1)}.
For every y∈ Y_0, |T'(y)∩ Y'|≤ t-1; thus there is y_j∈ Y'∖⋃_y∈ Y_0 T'(y), say j=1.
Let T(y_1)={z_1,…,z_t-1}. Since |Y'|>t, we may assume y_2∉ T'(y_1).
Since 𝒥(ℋ_1) contains all singletons, there are
disjoint (r-1)-tuples U_1,…, U_q^* such that U_i+y_1 is an edge in ℋ_1 for each 1≤ i≤ q^*.
Since q^* is large, we can choose among them t-1 sets, say U_1,…, U_t-1 disjoint from
Y_0∪ Y'∪ T(y_1). Then we have a _t(1,r-1) with the set of edges
{U_h∪{y_1}: h∈ [t-1]}∪{U_h∪{z_h}: h∈ [t-1]}∪{Y_0∪{y_1},Y_0∪{y_2}},
a contradiction.
§ STABILITY: PROOF OF THEOREM <REF>
Recall the Lovász form of the Kruskal-Katona Theorem:
If x is a positive real, 1≤ k<n, ⊆[n] k and ||= x k, then
|∂|≥x k-1.
Choose n_0 so that
n_0-1 r-1>3(C+C_0)n_0^r-2 n_0^r-1/(r-1)!<2n_0-1 r-1.
Let n>n_0 and ℋ be an n-vertex r-uniform family not containing
a bush _t,h(a,b) with |ℋ|> (t-1)n-1r-1-C_0 n^r-2.
Define C, m, _0,…,_m and as in Subsection <ref>. By (<ref>),
|| ≥ (t-1) n-1r-1-(C+C_0)n^r-2.
As in Subsection 4.2, for each 1≤ i≤ m, the intersection structure 𝒥(ℋ_i) contains 𝒥^(r-1). So, again
for each hyperedge E∈ℋ_i⊂ (1≤ i≤ m) there is an element c(E)∈ E such that
each proper subset of E containing c(E) is a kernel of a thr-star in ℋ_i.
For every Y∈[n] r-2, let (Y) be the set of vertices v∈ [n]-Y such that there is an edge E∈
containing Y and such that v=c(E).
Suppose Y∈[n] r-2, v,v'∈(Y), v≠ v' and edges E,E'∈ are such that v=c(E), v'=c(E') and
Y⊆ E∩ E'.
If
E∈_i and E'∈_i', then i≠ i'.
Suppose v≠ v', but i=i'.
We may assume that the partition of [n] corresponding to _i is (X_1,…,X_r) and
v,v'∈ X_r. By symmetry, we also may assume that Y⊂ X_1∪…∪ X_r-2.
Since Y⊆ E∩ E', [r-2]∈_i or [r-1]∈_i. Since 𝒥^(r-1)⊂_i and 2≤ a,b≤ r-2,
[r]-[a]∈_i and [a]∪{r}∈_i. Since _i is intersecting, [a]=([a]∪{r})∩ [r-1]=([a]∪{r})∩ [r-1]∈_i.
Together with [r]-[a]∈_i, this contradicts Lemma <ref>.
Similarly to Claim <ref>, the following holds.
For every (r-2)-subset Y of [n],
|(Y)|≤ t-1.
Suppose to the contrary that there are t distinct v_1,…,v_t∈ [n] and
distinct E_1,…,E_t∈ such that Y⊆ E_1∩…∩ E_t and v_i=c(E_i) for i=1,…,t.
Let E_1∈ℋ_i_1, …, E_t∈ℋ_i_t.
By Claim <ref>, i_1,…,i_t are all distinct. By relabelling we may suppose that E_i∈ℋ_i.
We will find the t+1 disjoint sets A_0, B_1,B_2,…,B_t contradicting Lemma <ref> using induction as follows.
Fix any subset A_0 of Y with |A_0|=a and let D_0:= ∪ E_i. Then |D_0|≤ 2t+r-2.
We define the sets E_i', D_i, B_i step by step as follows. We will have D_i:=D_0∪⋃_j ≤ i E_j' and |D_i|≤ 2t+r-2+i(r-1-a).
For i=1, 2, …, t consider the family ℋ_i and its member E_i in it.
By the intersection structure of ℋ_i,
the set A_0∪{c(E_i)} is an (a+1,thr)-kernel in ℋ_i.
One of the thr petals of the sunflower in ℋ_i with kernel A_0∪{c(E_i)}
should be disjoint from D_i-1; let E_i' be the corresponding set in ℋ_i. Since c(E_i)∈ E_i', and homogeneity gives c(E_i')=c(E_i), the set
B_i:=E_i'-A_0 is a (b,thr)-kernel.
For v∈ [n], let (v)={E∈: v=c(E)} and (v)={E-v: E∈}. Let =⋃_v∈ [n](v).
By Claim <ref>,
|(v)|=|(v)| for each v∈ [n]; in particular, by (<ref>), ||=||≥ (t-1) n^r-1-(C+C_0)n^r-2.
In these terms, Claim <ref> can be restated as follows:
For every j∈ [n], there a real x_j such that |(j)|=x_j r-1. Reorder the elements of [n] so that x_1≥ x_2≥…≥ x_n. Let b=3 (r-1)!(C+C_0).
For every 1≤ i≤ t-1,
n-x_i≤ b.
Suppose the claim fails and 1≤ i≤ t-1 is the smallest index for which (<ref>) does not hold.
Then
Since for each real
0<x<n,
x r-1/n r-1= (1-n-x/n-r+2)x r-2/n r-2,
Theorem <ref> yields that for every i≤ j≤ n,
|∂(j)|≥x_j r-2=n r-2/n r-1x_j r-1/(1-n-x_j/n-r+2)≥n r-2/n r-1|(j)|(1+b/n-r+2),
and for every 1≤ j≤ i-1,
|∂(j)|≥x_j r-2≥n r-2/n r-1x_j r-1≥n r-2/n r-1|(j)|.
Hence
∑_j=i^n |∂(j)|≥n r-2(1+b/n-r+2)/n r-1∑_j=i^n|(j)| ∑_j=1^i-1 |∂(j)|≥n r-2/n r-1∑_j=1^i-1|(j)|.
By (<ref>) and (<ref>),
(t-1)|∂|≥∑_j=1^n |∂(j)|=∑_j=1^i-1 |∂(j)|+∑_j=i^n |∂(j)|≥n r-2/n r-1(∑_j=1^n |(j)|+ b/n-r+2∑_j=i^n |(j)|
).
Since |∂|≤n r-2 and by (<ref>), ∑_j=1^n |(j)|=||=||≥ (t-1) n-1r-1-(C+C_0)n^r-2, this implies
(t-1)n r-1≥ (t-1) n-1r-1-(C+C_0)n^r-2+b/n-r+2∑_j=i^n |(j)|.
Again by (<ref>), ∑_j=i^n |(j)|≥ ||-(i-1) n-1r-1≥ (t-i) n-1r-1-(C+C_0)n^r-2. Since t-i≥ 1, plugging this into (<ref>) and rearranging we get
(C+C_0)n^r-2≥b/n-r+2( n-1r-1-(C+C_0)n^r-2).
Using the definition of b and (<ref>), inequality (<ref>) implies
(C+C_0)n^r-2≥3 (r-1)!(C+C_0)/n-r+2( n-1r-1-1/3n-1r-1),
which in turn again using (<ref>) yields
n^r-2(n-r+2)≥ 2(r-1)! n-1r-1≥ n^r-1,
a contradiction.
Now we are ready to finish the proof of the theorem. By Claim <ref>, for every 1≤ i≤ t-1,
|(i)|=x_i r-1≥n-b r-1=n r-1∏_j=0^r-2n-b-j/n-j≥n r-1(1-(r-1)b/n-r+2)
=n r-1-bn r-2≥n r-1-b/(r-1)!n^r-2.
This proves Theorem <ref> for C_1=b/(r-1)!=3(C+C_0).
99
DEF
M. Deza, P. Erdős and P. Frankl, Intersection properties of systems of finite
sets, Proc. London Math. Soc. (3) 36 (1978), 369–384.
EG P. Erdős, and T. Gallai,
On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.
Frankl1977 P. Frankl,
On families of finite sets no two of which intersect in a singleton.
Bull. Austral. Math. Soc. 17 (1977),
125–134.
FF85 P. Frankl, and Z. Füredi,
Forbidding just one intersection.
J. Combin. Th., Ser. A 39 (1985), 160–176.
FF60 P. Frankl, and Z. Füredi, Exact solution of some Turán-type problems.
J. Combin. Th., Ser. A 45 (1987), 226–262.
Furedi1 Z. Füredi, On finite set-systems whose every intersection is a kernel of a star, Discrete Math. 47 (1983), 129–132.
FJKMV5 Z. Füredi, T. Jiang, A. Kostochka, D. Mubayi, and J. Verstraëte,
Extremal problems for hypergraph blowups of trees. To appear in SIDMA, 17 pp.
FurOzk
Z. Füredi, and L. Özkahya,
Unavoidable subhypergraphs: 𝐚-clusters.
J. Combin. Th., Ser. A 118 (2011), 2246–2256.
Tree
A. McLennan,
The Erdős-Sós conjecture for trees of diameter four.
J. Graph Theory 49 (2005), no. 4, 291–301.
|
http://arxiv.org/abs/2307.07306v1 | 20230714123041 | C3: Zero-shot Text-to-SQL with ChatGPT | [
"Xuemei Dong",
"Chao Zhang",
"Yuhang Ge",
"Yuren Mao",
"Yunjun Gao",
"lu Chen",
"Jinshu Lin",
"Dongfang Lou"
] | cs.CL | [
"cs.CL",
"cs.AI"
] |
Verification of Quantum Systems using Barrier Certificates
Marco Lewis1Corresponding email: [email protected]
0000-0002-4893-7658
Paolo Zuliani1,2Currently at Università di Roma; work predominately done at Newcastle University.
0000-0001-6033-5919
Sadegh Soudjani1,30000-0003-1922-6678
August 12, 2023
================================================================================================================================================================================================================================================
This paper proposes a ChatGPT-based zero-shot Text-to-SQL method, dubbed C3, which achieves 82.3% in terms of execution accuracy on the holdout test set of Spider and becomes the state-of-the-art zero-shot Text-to-SQL method on the Spider Challenge. C3 consists of three key components: Clear Prompting (CP), Calibration with Hints (CH), and Consistent Output (CO), which are corresponding to the model input, model bias and model output respectively. It provides a systematic treatment for zero-shot Text-to-SQL. Extensive experiments have been conducted to verify the effectiveness and efficiency of our proposed method. Our code is available at https://github.com/bigbigwatermalon/C3SQLhttps://github.com/bigbigwatermalon/C3SQL
§ INTRODUCTION
Text-to-SQL, which aims to convert natural language questions into executable SQL queries, can be used to provide a user-friendly interface to relational databases. It can benefit various aspects of data management, such as accessibility to databases, flexibility of website design and so on.
Traditional Text-to-SQL methods <cit.> typically fine-tune a decoder-encoder model with an amount of training data to achieve proper Text-to-SQL performance. These fine-tuning-based methods require a training set that consists of amounts of text-SQL pairs. However, in practice, obtaining the text-SQL pairs is extremely expensive. Furthermore, this fine-tuning paradigm can cause overfitting of the training set and degenerate the transferability of the model.
To address these problems, few-shot or zero-shot Text-to-SQL methods are natural choices. However, the decoder-encoder model-based methods cannot achieve proper performance <cit.>. Recently, the emergent abilities of Large Language Models (LLMs) make few-shot <cit.> or zero-shot <cit.> Text-to-SQL possible. The few-shot Text-to-SQL method <cit.> can outperform the fine-tuning-based methods by means of few-shot in-context learning; however, it requires more than 10,000 tokens for each query, which is expensive and infeasible in practical Text-to-SQL tasks. By contrast, the zero-shot Text-to-SQL method <cit.> can save tokens; however, its performance is inferior to the fine-tuning-based methods.
To address these issues, this paper proposes a novel ChatGPT-based zero-shot Text-to-SQL method, dubbed C3, which utilizes only approximately 1,000 tokens per query and achieves a better performance than fine-tuning-based methods. C3 consists of three key components: Clear Prompting (CP), Calibration with Hints (CH) and Consistent Output (CO), which are corresponding to the model input, model bias and model output respectively. Specifically, CP is a novel prompting paradigm for zero-shot Text-to-SQL, which improves the zero-shot Text-to-SQL performance via adopting proper input. Furthermore, CH is proposed to mitigate the biases of ChatGPT in generating SQL queries, which improves the zero-shot Text-to-SQL performance via calibrating the model biases. Besides, CO is designed to keep the consistency of the generated SQL queries, which improves the zero-shot Text-to-SQL performance via overcoming the inherent randomness and uncertainty in the output of large language models.
We evaluate our proposed C3 method on the widely used Spider dataset, where C3 achieves 82.3% in terms of execution accuracy on the holdout test set of Spider and becomes the state-of-the-art zero-shot Text-to-SQL method. It outperforms state-of-the-art fine-tuning-based approaches by 2.4% execution accuracy on the holdout test set; besides, it only uses approximately 1,000 tokens per query. Furthermore, our extensive experimental analysis provides several noteworthy discoveries, which are potential to forge a new trend in GPT-based Text-to-SQL research.
§ RELATED WORK
Existing Text-to-SQL methods can be divided into several categories: rule-based methods, fine-tuning methods and ICL-based methods. The rule-based Text-to-SQL methods <cit.> use well designed templates to generate SQL queries, which have achieved good performance in specific cases. However, these methods rely heavily on manual rule design, making them difficult to apply to other domains and limiting their scalability and generalizability.
To address these limitations, researchers explore Seq2Seq models based on bidirectional LSTMs <cit.> and CNNs <cit.>. These models offer improved flexibility and effectiveness but face challenges in integrating database structural information seamlessly. To tackle these constraints, researchers utilize graph neural networks, treating the database schema as a graph, to better integrate structural information <cit.>. Besides, fine-tuning on pretrained language models like T5 has demonstrated better performance <cit.>. However, fine-tuning methods typically require a substantial volume of labeled training data specific to the target task and they often suffer from overfitting to the training data.
The emergence of large language models (LLM) has left a deep impression and points towards new directions for the Text-to-SQL task. LLMs like GPT models outperform fine-tuning models on many NLP down-stream tasks in few-shot or zero-shot setting due to the ability of in-context learning (ICL) <cit.>. The design of prompts in the ICL profoundly influences the quality of outputs <cit.> in LLM. Some work <cit.> conduct the evaluation of LLMs' ICL performance on Text-to-SQL task with different prompts. However, none of them outperform the current fine-tuning methods.
All though the standard prompting only provides a lower bound on the capabilities of LLMs in principle <cit.>, LLMs with well designed prompts have the potential to achieve better performance. Recently, a few-shot paradigm using GPT-4 has achieved SOTA performance on Text-to-SQL task <cit.>. But this method needs some handcraft demonstrations and numerous tokens, which are time-consuming and resource-intensive.
In our work, we demonstrate that ChatGPT can effectively generate SQL queries without the need for any demonstrations when provided with proper instructions. We show its capability as a skilled zero-shot SQL writer.
§ PRELIMINARIES
§.§ Problem Definition of Text-to-SQL
Given natural language question Q and database schema 𝒮, the database schema 𝒮={𝒯, 𝒞, ℛ} includes multiple tables 𝒯, columns 𝒞 and foreign key relations ℛ. The problem of Text-to-SQL parsing aims to generate the SQL query 𝒴 corresponding to the question.
§.§ Large Language Model for Text-to-SQL
The task of Text-to-SQL parsing has been formulated as a generation task in recent works <cit.>, utilizing appropriate prompts 𝒫 to guide a large language model ℳ. This model estimates a probability distribution over SQL queries 𝒴 and allows us to generate queries token by token. The generation process for the SQL query 𝒴 can be formulated as follows:
P_ℳ(𝒴 | 𝒫, 𝒮, Q) = ∏_i=1^|𝒴| P_ℳ(𝒴_i | 𝒫, 𝒮, Q, 𝒴<i)
Here, 𝒴<i is the prefix of the SQL query 𝒴 and P_ℳ(𝒴_i | ·) is the conditional probability of the i-th token in the SQL query 𝒴 given the prefix 𝒴_<i, the prompt 𝒫, the schema 𝒮 and question 𝒬.
Recent studies have demonstrated that large language models (LLMs) can learn from a few examples within a given context, known as in-context learning <cit.>. These works have shown that in-context learning effectively enables LLMs to perform a range of complex tasks. However, including additional examples tends to increase the manual cost and token cost associated with using the OpenAI API. Therefore, in this study, we specifically focus on the zero-shot prompt setting.
§ PROPOSED APPROACH
In this section, we propose the C3 method which can achieve proper zero-shot Text-to-SQL performance based on ChatGPT. The framework of the C3 method is demonstrated in Figure <ref>. C3 consists of three key components: Clear Prompting (CP), Calibration with Hints (CH), and Consistency Output (CO), which are corresponding to the model input, model bias and model output respectively. The details of each component are introduced as follows.
§.§ Clear Prompting
The Clear Prompting (CP) component aims to provide effective prompts for Text-to-SQL parsing. It consists of two parts: clear layout and clear context.
Clear Layout
In Text-to-SQL, the widely used prompt layout styles can be divided into two types. We denote them as complicated layout and clear layout respectively.
The details of each type are introduced as follows.
* Type 1: Complicated Layout: This type of prompt layout directly concatenates the instruction, question and context (database schema) together, which looks a mess. An example of this prompt layout is illustrated in Figure <ref>.
* Type 2: Clear Layout: This type of layout divides the instruction, context (database schema) and questions by adopting sharp symbols, which look clear. An example of this prompt layout is illustrated in Figure <ref>.
Intuitively, the clear layout will be easily understood by ChatGPT and achieve better performance. The experiments illustrate that the clear layout outperforms the complicated layout by 7.0% in term of execution accuracy, which verify this intuition.
The experimental setting can be found in Section <ref>). Therefore, this paper adopts a clear layout to construct our prompt template.
We also provide an interesting material that asks ChatGPT itself whether clear layout is important (please refer to Appendix <ref>). However, our experimental results show that directly using the instruction provided in Figure <ref> causes redundant columns in the generated SQL.
To solve this problem, we improve the instruction part by adding a clause and do not select extra columns that are not explicitly requested in the query.
Moreover, the context provided in Figure <ref> requires to refine. We refine it in the following section.
Clear Context
Including the entire database schema in the context part of a prompt causes two issues. Firstly, introducing too many irrelevant items in the prompt increases the likelihood of ChatGPT generating irrelevant schema items in the output SQL. Secondly, using the complete database schema results in excessive text length, leading to unnecessary API costs. To overcome these two issues, we propose that it is necessary to conduct schema linking which recalls relevant tables and columns. With the schema linking results, we can only put the linked tables and columns into the context part. This paper presents a ChatGPT-based zero-shot schema linking approach, which can be divided into the following two steps:
* Table Recall We design a zero-shot prompt to instruct ChatGPT to recall tables using three steps. Firstly, the tables should be ranked based on their relevance to the question. Secondly, the model should check if all tables have been considered. Finally, the output format is specified as a list. To ensure the stability of table recall, we employ a self-consistency method. Specifically, the model generates ten sets of retrieval results, each set containing the top four tables. The final result is determined by selecting the set that appears most frequently among the ten sets. The complete prompt is shown in Appendix <ref>.
* Column Recall Based on the table recall results, we further retrieve the columns within the candidate tables. We also design a zero-shot prompt and instructed ChatGPT to recall columns in two steps. Firstly, all columns within each candidate table are ranked based on their relevance to the question. Then, the output format is specified as a dictionary. In the prompt, we also emphasize that columns matching more with the question words or the foreign key should be placed ahead to assist in more accurate recall results. Similarly, we employ the self-consistency method. Specifically, for each table, we generate ten sets of recalled columns at once. Then we choose five columns that appear most frequently among each set as the final result. The complete prompt is shown in Appendix <ref>.
Besides the recalled tables and columns, we also add foreign key information of the recalled tables into the context part to specify the columns required for JOIN operations.
Combining the Clear Layout and Clear Context, we propose Clear Prompt, as demonstrated in Figure <ref>.
§.§ Calibration of Model Bias
Through analyzing the errors that occurred in the generated SQL queries, we find that some errors are caused by certain biases inherent in ChatGPT. As demonstrated in Figure <ref>, ChatGPT prefers to provide extra columns and extra execution results. This paper concludes them as the following two kinds of bias.
* Bias 1: ChatGPT tends to be conservative in its output and often selects columns that are relevant to the question but not necessarily required. Furthermore, we find that this tendency is particularly pronounced when it comes to issues involving quantities. For example, for the first question in Figure <ref> (left), ChatGPT chooses Year and COUNT(*) in the SELECT clause. However, the gold SQL in Spider dataset only selects Year as COUNT(*) is only needed for ordering purposes.
* Bias 2: ChatGPT tends to use LEFT JOIN, OR and IN when writing SQL queries, but often fails to use them correctly. This bias often leads to extra values in execution results. Some examples of this bias can be found in Figure <ref> (right).
To calibrate these two biases, we propose a plug-in calibration strategy, dubbed Calibration with Hints (CH). CH incorporates prior knowledge into ChatGPT by using contextual prompts which include historical conversations.
In the historical conversation, we initially regard ChatGPT as an excellent SQL writer and guide it to follow our proposed debias hints.
* Hint 1: For the first bias, we design a tip to guide ChatGPT in selecting only the necessary column. This tip is illustrated in the upper right part of Figure <ref>. It emphasizes that items like COUNT(*) should not be included in SELECT clause when it is only needed for ordering purposes.
* Hint 2: For the second bias, we design a tip to prevent ChatGPT from misusing SQL keywords. As shown in the upper right part of Figure <ref>, we straightforwardly ask ChatGPT to avoid using LEFT JOIN, IN and OR, and use JOIN and INTERSECT instead. We also instruct ChatGPT to use DISTINCT or LIMIT when appropriate to avoid repetitive execution results.
By incorporating these two hints, ChatGPT can generate SQL queries that align more closely with the desired output. As shown in Figure <ref>, using our CH prompt can effectively calibrate the model biases. More examples of CH prompt can be found in Appendix <ref>.
§.§ Consistency Output
Using the CP and CH methods, ChatGPT is able to generate higher-quality SQL. However, the output of ChatGPT is unstable due to the inherent randomness of large language models <cit.>. To find out the influence of the uncertainty output of ChatGPT, we analyze the distribution of correctness counts on the dev set across thirty independent experiments under different prompts, as shown in Figure <ref>. In this figure, ChatGPT-SQL is the method proposed in literature <cit.>; besides, CP and CP + CH denote our proposed Clear Prompt and the combination of Clear Prompt and Clear Hint method respectively.
Regardless of the method used, only less than 65% of SQL statements can consistently be written correctly.
This means that by enhancing the consistency of the output, the model has a great potential to correctly write the majority of the queries.
To enhance the consistency, we draw inspiration from the concept of self-consistency <cit.> proposed in previous work. Self-consistency method is motivated by the fact that in complex reasoning problems, there are multiple different reasoning paths to the unique right answer. It first samples multiple different reasoning paths and then selects the most consistent answer to improve the quality of the output remarkably. Text-to-SQL problem is similar to reasoning problems, where there are multiple ways to write SQL queries to represent the same meaning as shown in the figure <ref>. Therefore, we implement execution-based self-consistency to Text-to-SQL.
Specifically, we first sample multiple reasoning paths to generate diverse SQL answers. Then, execute these SQL queries on the database and collect the execution outcomes. After removing the errors from all the outcomes, we identify the most consistent SQL as the final SQL by applying a voting mechanism to these execution results. For instance, in figure <ref>, we classify the SQL queries based on the execution outcomes and represent them with different colors. Then we compare the categories to determine which one contains more SQL queries, and select one SQL from that category as the final SQL. This approach allows us to harness the collective knowledge derived from these multiple paths, resulting in more reliable and resilient results in the generation of SQL queries.
§ EXPERIMENTS
§.§ Experimental Setup
Datasets
We conduct experiments on the Spider dataset. Spider <cit.> is a large-scale complex and cross-domain Text-to-SQL dataset. It consists of 10,181 questions and 5,693 unique complex SQL queries on 200 databases with multiple tables covering 138 different domains. It is divided into 8659 training samples across 146 databases, 1034 evaluation samples across 20 databases and 2147 test samples across 34 databases. There is no overlap between the databases used in each of these sets. Meanwhile, according to their difficulty levels, these samples are classified into four categories: easy, medium, hard, and extra.
Evaluation Metrics
The most commonly used evaluation metrics are exact match (EM) and execution accuracy (EX). For EM, it considers each part of the SQL query predicted by the model after removing the values, and it is considered a correct prediction only if all parts match the corresponding parts in the standard SQL query. For EX, it evaluates the correctness of the predicted SQL query by comparing the execution results of the predicted SQL and the standard SQL in the same database. In our experiment, we only use EX as the evaluation metrics<cit.>, because we think that the SQL can be written in multiple ways to generate the same result. Moreover, ChatGPT has its own style of writing SQL queries. Thus it is more meaningful to make a direct comparison of the results in our experiment.
Implementation
We use OpenAI ChatGPT API: GPT-3.5-Turbo-0301 as our experiment model. We recall tables and columns in two steps with ChatGPT on their relevance according to the given question. Then we construct input prompt using our CP and CH methods. Lastly, we utilize ChatGPT to generate 20 SQL queries at once and select the most consistent SQL as the final result using our CO method.
Baselines
We conduct experiments on the Spider dataset and compare our method with the following baselines:
* ChatGPT-SQL <cit.> introduces a simple zero-shot method to evaluate the capability of ChatGPT on the Text2SQL task.
* RATSQL <cit.> proposes a unified framework utilizing the relation-aware self-attention to encode the relational structure of a database schema and the query.
* PICARD <cit.> introduces an incremental parsing method with constrained decoding, which checks and filters out inadmissible tokens at each generation step.
* Graphix <cit.> designs a Graphix layer to encode a combination of semantic and structural information based on the pre-trained T5 model.
* SC-Promp <cit.> decomposes the Text2SQL task into two subtasks, predicting structure and content separately, achieving high performance with fewer training samples.
* RESDSQL <cit.> proposes a ranking-enhanced encoding and skeleton-aware decoding framework to decouple the schema linking and the skeleton parsing. It is the best approach based on fine-tuning.
* DIN-SQL <cit.> decomposes the text-to-SQL task into smaller subtasks and designs different prompts for each subtask to instruct GPT-4 to complete each subtask and obtain the final SQL. It is the SOTA at the time of writing.
§.§ Overall Performance
In Table <ref>, we report the performance of our method and baseline methods on the Spider dataset. It can be observed that our method outperforms all traditional fine-tuning-based methods in terms of execution accuracy on the test set. Our method also outperforms the zero-shot setting of ChatGPT-SQL by 9.5% on the dev set. At the time of writing, our model achieved 2rd on the Spider Leaderboard. Compared with the few-shot setting of top-1 method, DIN-SQL, our method focuses on the zero-shot setting. Our approach only uses approximately 10% of the token numbers of DIN-SQL; furthermore, the cost of GPT-3.5 engine is also significantly lower than GPT-4 which is used in DIN-SQL. Therefore, our method is more budget-friendly.
§.§ Effect of Clear Prompt
In this section, we experimentally investigate the impact of the prompt layout and the content on the performance of Text-to-SQL respectively. The results are the average execution accuracy obtained on the dev set over five runs.
Clear Layout We first explore the effect of clear prompt layout by comparing the performance of the clear layout (refer to Type 2 in Section <ref>) and the complicated layout (refer to Type 1 in Section <ref>). The results are recorded in Table <ref>. From this table, we can see that clear layout outperforms complicated layout by 7.0%. It demonstrates that the clear prompt layout is important for chatGPT-based zero-shot Text-to-SQL. We suppose that this is because a clear layout allows ChatGPT to better understand the prompt, while the complicated one may blur the information. So a clear layout can help ChatGPT better follow our instructions and make the most of the information provided in the prompt.
Clear Content To investigate the effect of our proposed clear prompt content, we conduct ablation study on (1) recall of tables and columns and (2) specify foreign keys. The results are recorded in Table <ref>. From this table, we can see that the performance of C3 decreases by 2.3% when removing the operation of recalling of tables and columns; besides, the performance of C3 decreases by 2.6% when removing the operation of specifying foreign keys. It demonstrates that the operations of recalling tables and columns and specifying foreign keys, which provides clear prompt content, are necessary for C3. We further use Area Under ROC Curve (AUC) to evaluate the performance of table and column recall. As illustrated in Section <ref>, we only keep top-4 tables in the database and top-5 columns for each remained table to form a ranked schema sequence. The results are recorded in Table <ref>, which shows that the AUC of table recalling and column recalling are 0.9588 and 0.9833 respectively.
§.§ Effect of Calibration of Model Bias
To explore the effect of our proposed calibration method, Calibration with Hints (CH), we perform an ablation study on it. The experimental results are illustrated in Table <ref>. From this table, we can see that the execution accuracy decreases by 1.5% when calibration is removed. It verifies that calibration of model bias is necessary for chatGPT-based Text-to-SQL.
§.§ Effect of Self-Consistency
In this section, we discuss about the effect of self-consistency. We first conduct an experiment to show ChatGPT's high potential on Text-to-SQL task. We perform k independent experiments and record the changes of accuracy. The results are shown in Figure <ref>. For each question, if ChatGPT generates one or more correct SQL in the k experiments, we think ChatGPT can generate a accurate SQL for the question. It can be observed that when using ChatGPT-SQL, the accuracy can reach approximately 85.7% when k=30. However, the accuracy is only 72.3% when k=1, resulting in a 13.4% gap. This gap shows the enormous potential of ChatGPT. When using our two prompts, though the gaps are relatively smaller and the curves are flatter, the gap is still noticeable. However, We can find that a slight increase of k can lead to a large increase of accuracy. This is due to the unstability of ChatGPT, which may not generate the correct answer every time. Therefore, as illustrated in <ref>, using self-consistency can utilize multiple answers to get the most consistent one and mitigate this unstability.
In the experiment, we set the parameter n to 20 in GPT-3.5 API call function in order to generate twenty SQL queries in a single run. Among these SQL queries, we cluster them based on their execution results and select one SQL query from the largest cluster as the final SQL. The ablation study of self-consistency is recorded in Table <ref>, we can see that self-consistency improves performance by 1.3% and is therefore necessary to maintain the stable performance.
§.§ Error Analysis
To clearly demonstrate the effectiveness of our proposed method, we conduct an error analysis of ChatGPT-SQL and our C3 method on the dev set. We adopt the error classification method proposed in literature <cit.>. Specifically, we manually examined the errors occurred in the generated SQL answers and classified them into six categories, as shown in Figure <ref>.
Firstly, we can see that, by using our C3 method, the total number of errors decreases by approximately 34% compared to ChatGPT-SQL. The most noteworthy reduction happens in the Schema-linking category, which contains three subcategories. Among these subcategories, the subcategory Extra-columns is mostly caused by Bias 1(refer to Section <ref>). We can see that the errors belonging to this subcategory has a proportional decrease of 11%. It demonstrates the significant effectiveness of CH in calibrating the model bias.
Secondly, the subcategory Set op in category Nested demonstrates the misuse of operations like INTERSECT and EXCEPT, and the subcategory Distinct in category Other indicates repetitive execution results. Errors belonging to these subcategories are mostly caused by Bias 2 (refer to Section <ref>). Though the proportions of these categories change a little, the actual number of errors in these categories decreases. This indicates that CH can effectively calibrate Bias 2.
And finally, in the category Schema-linking and Join, we can see that the errors of choosing wrong tables and columns significantly decrease, which can be credited to the operation of table and column recall as well as specifying foreign keys in the proposed CP. As for the other categories, the number of errors is also reduced, which can be credited to the CO method for it improves the stability and maintains high performance in each experiment.
The overall reduction of errors is attributed to the combined effect of the C3 method.
§ CONCLUSION
This paper proposes a novel zero-shot Text-to-SQL method based on ChatGPT, called C3, which achieves the state-of-the-art zero-shot Text-to-SQL performance. C3 provides a systematic treatment for GPT-based Text-to-SQL from the perspective of model input, model bias, and model output. It is potential to forge a new trend in the GPT-based Text-to-SQL research.
§ ACKNOWLEDGMENTS
We thank Yu Mi and Yilin Li for their valuable assistance with the experiments.
acl_natbib
§ CLEAR PROMPTING
Large language models (LLMs) are highly sensitive to the input prompt. A clear prompt can have a substantial positive impact on the performance of downstream tasks. This claim is empirically evidenced by our C3 prompt and corresponding experimental results in the Text-to-SQL task. Furthermore, we present the following question to ChatGPT for suggestions:
1.0
*theoremQ1
Human: Does using a clear prompt layout benefit ChatGPT in performing Text-to-SQL tasks?
ChatGPT: Yes, using a structured prompt style can greatly benefit ChatGPT in performing Text-to-SQL tasks. Text-to-SQL tasks involve converting natural language queries into SQL queries that can be executed on a database. These tasks often require a clear understanding of the query intent and the underlying database schema.
Structured prompts provide a specific format or template that guides the model to generate SQL queries in a structured and predictable manner. By using a structured prompt style, you can specify the desired structure and components of the SQL query, such as the SELECT, WHERE, and JOIN clauses, as well as the table and column names. This helps to guide the model's generation process and improve the accuracy and consistency of the generated SQL queries.
Structured prompts can also help overcome some of the challenges of language models, such as ambiguity and verbosity. By providing explicit instructions and constraints in the prompt, you can guide the model to generate more precise and concise SQL queries. This reduces the chance of the model producing queries that are syntactically correct but semantically incorrect or incomplete.
However, it's important to note that while structured prompts can improve the performance of ChatGPT in Text-to-SQL tasks, they are not a complete solution. These tasks can be complex, and the model may still require additional training, fine-tuning, or post-processing techniques to achieve optimal performance. Additionally, structured prompts may limit the model's flexibility in handling novel or out-of-domain queries, so a balance between structure and flexibility should be considered when designing prompts for Text-to-SQL tasks.
§ SCHEMA LINKING
§.§ The example of table recall prompt
Given the database schema and question, perform the following actions:
1 - Rank all the tables based on the possibility of being used in the SQL according to the question from the most relevant to the least relevant, Table or its column that matches more with the question words is highly relevant and must be placed ahead.
2 - Check whether you consider all the tables.
3 - Output a list object in the order of step 2, Your output should contain all the tables. The format should be like:
[
"table_1", "table_2", ...
]
Schema:
# continents ( contid, continent )
# countries ( countryid, countryname, continent )
# car_makers ( id, maker, fullname, country )
# model_list ( moddeli, maker, model )
# car_names ( makeid, model, make )
# cars_data ( id, mpg, cylinders, edispl, horsepower, weight, accelerate, year )
Question:
### What is the name of the different car makers who produced a car in 1970?
§.§ The example of column recall prompt
Given the database tables and question, perform the following actions:
1 - Rank the columns in each table based on the possibility of being used in the SQL, Column that matches more with the question words or the foreign key is highly relevant and must be placed ahead. You should output them in the order of the most relevant to the least relevant.
Explain why you choose each column.
2 - Output a JSON object that contains all the columns in each table according to your explanation. The format should be like:
{
"table_1": ["column_1", "column_2", ......],
"table_2": ["column_1", "column_2", ......],
"table_3": ["column_1", "column_2", ......],
......
}
Schema:
# car_makers ( id, maker, fullname, country )
# model_list ( modelid, maker, model )
# car_names ( makeid, model, make )
# cars_data ( id, mpg, cylinders, edispl, horsepower, weight, accelerate, year )
Foreign keys:
# model_list.maker = car_makers.id
# car_names.model = model_list.model
# cars_data.id = car_names.makeid
Question:
### What is the name of the different car makers who produced a car in 1970?
§ THE EXAMPLES OF PROMPTS
§.§ Clear Layout
### Complete sqlite SQL query only and with no explanation
### Sqlite SQL tables, with their properties:
#
# stadium ( stadium_id, location, name, capacity, highest, lowest, average );
# singer ( singer_id, name, country, song_name, song_release_year, age, is_male );
# concert ( concert_id, concert_name, theme, stadium_id, year );
# singer_in_concert ( concert_id, singer_id ).
#
### How many singers do we have?
SELECT
§.§ Complicated Layout
Complete sqlite SQL query only and with no explanation.
How many singers do we have? Sqlite SQL tables, with their properties: singer : singer.singer_id , singer.name , singer.country , singer.age , singer.song_name , singer.song_release_year , singer.is_male | stadium : stadium.location , stadium.name , stadium.capacity , stadium.highest , stadium.lowest , stadium.average , stadium.stadium_id | concert : concert.theme , concert.year , concert.concert_id , concert.concert_name , concert.stadium_id | singer_in_concert : singer_in_concert.concert_id , singer_in_concert.singer_id
SELECT
§.§ C3 Prompt
Role: SYSTEM
Content:
You are now an excellent SQL writer, first I'll give you some tips and examples, and I need you to remember the tips, and do not make same mistakes.
Role: USER
Content:
Tips 1:
Question: Which A has most number of B?
Gold SQL: select A from B group by A order by count ( * ) desc limit 1;
Notice that the Gold SQL doesn't select COUNT(*) because the question only wants to know the A and the number should be only used in ORDER BY clause, there are many questions asks in this way, and I need you to remember this in the the following questions.
Role: ASSISTANT
Content:
Thank you for the tip! I'll keep in mind that when the question only asks for a certain field, I should not include the COUNT(*) in the SELECT statement, but instead use it in the ORDER BY clause to sort the results based on the count of that field.
Role: USER
Content:
Tips 2:
Don't use "IN", "OR", "LEFT JOIN" as it might cause extra results, use "INTERSECT" or "EXCEPT" instead, and remember to use "DISTINCT" or "LIMIT" when necessary.
For example,
Question: Who are the A who have been nominated for both B award and C award?
Gold SQL should be: select A from X where award = 'B' intersect select A from X where award = 'C';
Role: ASSISTANT
Content:
Thank you for the tip! I'll remember to use "INTERSECT" or "EXCEPT" instead of "IN", "NOT IN", or "LEFT JOIN" when I want to find records that match or don't match across two tables. Additionally, I'll make sure to use "DISTINCT" or "LIMIT" when necessary to avoid repetitive results or limit the number of results returned.
Role: USER
Content:
### Complete sqlite SQL query only and with no explanation, and do not select extra columns that are not explicitly requested in the query.
### Sqlite SQL tables, with their properties:
#
# singer ( singer_id, name, country, age )
# stadium ( capacity, highest, lowest, average )
# concert ( theme, year, concert_id, concert_name )
# singer_in_concert ( concert_id, singer_id )
# concert.stadium_id = stadium.stadium_id
# singer_in_concert.singer_id = singer.singer_id
# singer_in_concert.concert_id = concert.concert_id
#
### How many singers do we have?
SELECT
|
http://arxiv.org/abs/2307.04015v1 | 20230708164731 | Emotion-Guided Music Accompaniment Generation Based on Variational Autoencoder | [
"Qi Wang",
"Shubing Zhang",
"Li Zhou"
] | cs.SD | [
"cs.SD",
"cs.MM",
"eess.AS"
] |
Emotion-Guided Music Accompaniment Generation Based on Variational Autoencoder
Qi Wang, Shubing Zhang , Li Zhou 1
China University of Geosciences(Wuhan)
{wangqi233,zhouli}@cug.edu.cn
* Corresponding author
1 This research was funded by the Chinese Regular Projects of the Humanities and Social Sciences Fund of the Ministry of Education of Grant No.16YJAZH080.
August 12, 2023
===================================================================================================================================================================================================================================================================================================
Music accompaniment generation is a crucial aspect in the composition process. Deep neural networks have made significant strides in this field, but it remains a challenge for AI to effectively incorporate human emotions to create beautiful accompaniments. Existing models struggle to effectively characterize human emotions within neural network models while composing music. To address this issue, we propose the use of an easy-to-represent emotion flow model, the Valence/Arousal Curve, which allows for the compatibility of emotional information within the model through data transformation and enhances interpretability of emotional factors by utilizing a Variational Autoencoder as the model structure. Further, we used relative self-attention to maintain the structure of the music at music phrase level and to generate a richer accompaniment when combined with the rules of music theory.
Our experimental results indicate that the emotional flow of the music generated by our model has a strong correlation with the input emotion, demonstrating the model's strong interpretability and control of emotional flow. The generated music is also well-structured, diverse, and dynamic, outperforming the baseline models.
Music Accompaniment Generation, Emotional Flow, Variational Autoencoder, Rule constraints
§ INTRODUCTION
Music evokes emotions in listeners, making it a powerful and intuitive medium for understanding. It also serves as a driving force for musicians to create. One important aspect of composing is incorporating emotional expression into the music. Composers use their emotions along with their technical skills and knowledge to craft their compositions.
Current AI methods fall short of replicating a composer's approach. Neural networks primarily focus on combining and utilizing pre-existing knowledge of compositions, rather than incorporating emotions as high-level information. Our research aims to overcome this limitation by developing a model for generating accompaniment that takes emotions into account.
The way emotions are processed impacts every aspect of music composition and, as a result, every aspect of deep neural networks <cit.>. This puts a significant emphasis on the need for network control. While autoregressive models can effectively capture key elements of music, they lack transparency and do not guarantee internal control and interpretability of musical information. Adversarial networks <cit.> can separate elements like pitch, rhythm, and texture, but they struggle with capturing emotional information and prioritize interpretability over musicality and structure.
Additionally, many music generation models <cit.> primarily focus on identifying and evaluating the emotional aspects of music, rather than using them as a controllable variable. Therefore,
instead of using subjective and limited emotional labels<cit.>, such as "relaxed" or "nervous," we have adopted Thayer's continuous emotion model<cit.>. This model takes into account two quantitative and controllable factors: valence, which measures the level of positivity or negativity, and arousal, which measures the level of excitement or calmness. This approach provides a controlled understanding of human emotions.
Thus, we designed a system based on Variational Autoencoder, a controllable deep learning model, which incorporates emotional factors into the neural network's learning process. The user inputs valence and arousal trends, which are then encoded using our Valence Encoder and Arousal Encoder. The model then decodes and reconstructs this information to generate 2-bar piano accompaniments that match the emotional flow of the user's input.
To compose a dynamic piece of music, we take into account two key elements: tonality<cit.>, which enhances the beat and rhythm of the music by incorporating rule-based constraints in the model's decoder, and structural organization<cit.>, which improves the storytelling aspect of the music and preserves the internal structure of the piece through a self-attention mechanism.
Our data, code, and samples have been made publicly available [<https://github.com/Duoluoluos/Emotion-Guided-Music-Accompaniment-Generation>]online.
Our main contributions include:
* Emotion-Guided Composition, where the user inputs an Emotion-Flow Curve and the model generates music
that closely matches the input emotions.
* Enhanced accompaniment generation, incorporating global tonality, music phrases, and local texture for a more realistic and dynamic improvised accompaniment.
* Integration of rules and deep learning, combining the creative capabilities of deep networks with the constraints of music theory to improve the transparency of the music creation process.
§ RELATED WORKS
§.§ Accompaniment Generation
Generating musical accompaniment is essentially a specific type of music generation problem<cit.>, where the melody is used as a constraint, and the accompaniment is the generated music. In the past, accompaniment generation was approached in the same way as music generation, treating pitch and temporal values as simple data. Algorithms such as Hidden Markov Chain (HMC)<cit.>, Random Forest (RF), Support Vector Machine (SVM)<cit.> <cit.>, etc. were used to approach the problem from a regression perspective. However, with the advancement of deep learning, more accurate prediction models have been developed.
DeepBach<cit.>, a well-known music generation network based on RNN/LSTM<cit.> networks, represents Bach choral as voice lists with metadata lists and embedding representation to RNN for prediction. However, RNN/LSTM networks alone may not be sufficient for achieving the required level of long-range coherence in accompaniment. Hybrid models, such as the RNN-LSTM model in paper <cit.> and the RNN-RBM model in paper <cit.>, have been proposed to address this issue. The RNN-LSTM model learns different models in stages, while the RNN-RBM model uses several Restricted Boltman Machines (RBMs) and samples the output of the RBMs as input for the RNN, training local information and then makes autoregression for each information.
In 2018, the Music Transformer <cit.> was introduced, which shifted the focus from regression problems and note prediction to natural language processing (NLP) techniques for recognizing relationships between different segments of music and evaluating the logicality of musical phrases, similar to how NLP tasks analyze relationships and coherence in language. The Transformer model uses attention mechanisms, positional coding, and other techniques to ensure long-range coherence, making it useful for various accompaniment generation tasks such as drum and piano accompaniment. The model is similar to text completion in NLP, using a priori melodic data and key information such as drum beats to "fill in" missing features. Papers <cit.> have expanded upon this data representation and the MuMidi proposed in paper <cit.> can solve harmonic problems in a long-term context by integrating pitch, time value, and tempo. However, the generation process is not always interpretable or controllable and the randomness of notes can increase over time, resulting in non-sequential music.
To improve control over the music generation process, various methods have been employed. MuseBert <cit.> uses data corruption and fine-tuning during the inference learning process, while Music VAE <cit.> <cit.> uses decoupled feature representations such as pitch, chord, and texture, and employs interpolation, back-and-forth sampling, and temperature factors to increase accompaniment diversity. MuseGAN <cit.> treats music data as images and can generate multi-track accompaniments, but the structure of each track is not well-constrained by composition rules and the resulting music may not be as listenable. It is worth noting that the "hidden space" of the Variational Autoencoder(VAE) is better suited to the music generation problem than the image representation method used in the generative adversarial network. Unlike pass-through data, notes are affected by pitch, time, and velocity and have a high dimensionality of information. The VAE <cit.> normalizes this information to the hidden space for posterior estimation and reconstruction using an Encoder-Decoder architecture, which can be combined with a "learning from scratch" strategy and improve the model's ability to migrate and transfer. Therefore, we chose to use VAE as a controllable accompaniment generation model. Our model can generate well-structured accompaniments that conform to certain composition rules and follow an Emotion Flow.
§.§ Emotional Flow Guided Composition
Valence and Arousal are commonly used as quantitative measures of musical emotion in research. Studies<cit.> have shown that the rhythmic density of music, determined by the duration of notes in each measure, can affect a person's arousal levels independently of note velocity. Additionally, the melodic and harmonic direction of a song can affect the overall emotional direction <cit.>, referred to as valence. These factors can have a significant impact on the emotional response to a piece of music.
The objective of our research is to extract features from Emotion Flow, specifically the Valence Curve and Arousal Curve <cit.>, and then systematically associate those features with the generated accompaniment. Previous research, as shown in the paper <cit.>, used dynamic programming and template-matching methods to complete the Emotion-Flow Guided Accompaniment Generation. However, these methods can ensure the audibility of the music but do not guarantee the diversity of the accompaniment. In contrast, deep neural networks can achieve accompaniment diversity through large-scale learning, but they struggle to maintain the structure of the music compared to methods such as template matching <cit.>. Although self-similarity <cit.> can maintain some of the structure, neural network methods have difficulty ensuring the structure of the music because the music structure is strongly regulated through music phrases. Therefore, decoding music segments into "phrase" units is the key to maintain music structure. In this paper, we propose using a VAE which makes full use of structured features of the music to improve the overall structure and diversity of the accompaniment.
§ METHODS
§.§ Data Preparation
The POP909 Dataset <cit.> comprises 909 popular music tracks, which are piano-based and have a total running time of 60 hours. Each track is stored in MIDI file format and includes three separate components - melody, bridge, and piano. The bridge and piano tracks serve as an accompaniment. Additionally, the dataset includes chord and bar annotations for each song.
The POP909 dataset includes melodies that are broken down into 2-bar, 4-bar, and 6-bar fragments. The bar annotations in the dataset provide information about the structure of these fragments. The chord annotations, on the other hand, provide information about the harmony of each bar in the melodies.
To address the issue of music structure in a consistent manner, we discovered that the majority of music is composed of 2-bar segments. As a result, we carried out data cleaning, filtering out 2/4-bar segments and 2/4-bar segments with 6-bar introductory fragments. The training and testing sets were then split in an 8:2 ratio.
As sample data, we selected a subset from the Nottingham Dataset <cit.>. This dataset comprises over 1000 European and American folk songs, all of which have chord annotations. For validation purposes, we chose 2-bar and 4-bar segments from the dataset. The collated data information is presented in Table <ref>. (It is worth noting that if the user-supplied music does not have chord annotations like the sample data, we used Bi-LSTM Harmonizer <cit.> to implement the chord annotations)
To showcase the capabilities of our model, we chose two representative songs, one with high valence and the other with low valence, from the 20 songs we used. These songs were made available on a web page for users to evaluate and [<https://soundcloud.com/ko9isjyplxrb/sets/demos-of-emotion-guided-generated-accompaniment>]enjoy.
§.§ Models
§.§.§ The Conversion of Valence and Arousal
The overall architecture is illustrated in Figure <ref>.
The initial music data is represented by piano rolls. Each row of the piano roll matrix corresponds to one of the 128 pitch values and each column corresponds to a unit of time, with the duration of a 16th note used as the unit of time. The accompaniment tracks were merged and transformed to produce the accompaniment piano roll p_T^ACC, where T represents the duration of the altered accompaniment fragment. Similarly, the rhythm piano roll is represented as p_T^RHY, and the labeled chord progression is represented as c_T. According to the twelve-mean meter <cit.>, c_T is a matrix of 12 × T, where 12 is the number of notes in an octave.
Valence_T=V(c̅_̅T̅)
Where V(·) is the Valence's mapping and c̅_̅T̅ is the chord data after normalizing the root note of c_T to the C3 note. This is to ensure that the Valence is in the same key, and we set the T here to 8.
Also with respect to Arousal's mapping as A( · ), there are,
Arousal_T= A(p_T^ACC+p_T^RHY)
The operation of mapping A is to transform the multitrack music data into a tree structure <cit.>, where the nodes of the tree can more clearly characterize the density distribution of notes. Arousal is a four-dimensional matrix of size 128× T × 16 × 8, denoting the pitch-duration-density grouping, respectively.
Denote the quantization operation of Arousal and Density as | · |,
|Arousal|_T=1/5 · T∑_T∑_pitch A(p_T^ACC+p_T^RHY)
|Valence|_T = ∑_T W_V(c̅_̅T̅)
The W value in this context refers to the chroma weights of each chord and serves as a measure of the valence, or emotional assessment, of each chord. By performing a quantization-transformation operation, the emotional content of the music can be translated into a format that the composition model can understand, allowing for the user's desired Emotion Flow to be incorporated into the final output.
§.§.§ Valence/Arousal Encoder
Arousal and Valence Encoder are both dominated by LSTM as the backbone network. Arousal Encoder extracts the features of pitch-time-value information through a CNN with a (4,12) sized kernel in convolutional layer and (1,4) sized kernel in max pooling layer.
In fact, after the features are extracted by the convolutional network, the arousal information is more concise and refined [38], so that Decoder can learn better emotional features.
The layers of the LSTM network are all 1, and both are bidirectional. the dimension of the input weight of the Arousal Encoder is 256, and the dimension of the output weight is 1024. the dimension of the input weight of the Valence Encoder is 32, and the dimension of the output weight is 1024. Both are encoded to calculate the mean and variance of the probability distribution and sampled to obtain a 256-dimensional latent space variable z_Arousal or z_Valence.
§.§.§ Decoder
The Valence Decoder is introduced first, and the LSTM encoder of the decoder is roughly the same, except that the input side is fused with z_valence, and the dimension is modified to 292. The reconstructed Valence is estimated by calculating the variance and mean, and it is input to the LSTM as a token so that the decoding part of the model is completed. The probability distribution of valence is a 12-dimensional Bernoulli distribution.
PianoTree Decoder, on the other hand, refers to the design of the paper <cit.> and uses the model in this paper as a baseline. The original model is divided into two main stages, one is the time domain decoding and the other is the decoding of notes for each pitch. Since different notes may be concatenated into fragments and have some autocorrelation in the structure of the music to form the music phrases, we performed a note summary operation after the time-domain decoding operation and introduced a self-attention mechanism, which we will explain the ins and outs in detail in the next subsection.
The role of the first Pianotree-LSTM in Figure <ref> is to decode 512-dimensional latent space vectors. latent space vectors are the hidden space mapping changes of notes, and LSTM (hidden size=1024) is to summarize and summarize the results of the changes in the temporal dimension, so we call the summarized results note summary with size (1,512). After obtaining the relative self-attention, it is then decoded in the dimension of the pitch by LSTM(2) and mapped to 128 pitches through the fully connected layer. For each or each class of notes, respective temporal values are then decoded by LSTM (hidden size=16) to obtain the emotional stream/music sequence after reconstruction.
§.§.§ Relative Self-Attention
In order to maintain the structural organization in the music sequences, we introduce a self-attentive mechanism. This inspiration comes from the paper <cit.>, which does this by comparing a template music sequence fragment with a training music fragment and obtaining the correlation of the relative positions in the two sequences by one-dimensional/two-dimensional convolution, and the resulting correlation data is called self-similarity.
In this paper, self-similarity is not done by convolution operation because we do not have template fragments, but by note summary, a tensor of stacked pitch and mood information in the time domain. Similarly, since self-attention obtains the autocorrelation information inside the input by soft addressing, it is just possible to obtain the autocorrelation of note summary in the time domain and thus maintain the structured organization of the music fragments as the estimated "music phrases".
Since there is some time invariance in the relative positions of the sequences <cit.>, we also introduce offsets. Each fragment is not very informative, and to optimize the efficiency of the algorithm, we use a single-head attention mechanism. The query, key, and value tensor of relative attention are written as Q, K, and V, respectively. S^rel represents the offset matrix and the matrix element r=NS_k-NS_q, where NS_k and NS_q are the note summary query and key's position code, then the formula for relative self-attention(abbreviated as Att) is
Att = Softmax(QK^T+S^rel/√(D))V.
As for the parameter settings, we set the weight dimension of Q to 1024 and the weight dimension of K, V to D=128.
§.§.§ Rules-based Constraint
Two rules are very common in the realm of improvised accompaniment, enriching the player's accompaniment performance by changing tonality. The first principle is to add variety to the chords by making small adjustments to the chord tuning.
The second technique is to add a sense of layering between the different voices by shifting the tonality of the chords significantly at the same time.
Either way, chord arrangement is the most important thing.
If we want to use the rules in our accompaniment generator, we need to grasp the key information and build the model. Whether it's chord transposition or pitch shifting, it's essentially shifting pitch. So instead of inferring from the model, we can use the chord arrangement and transposition information directly to shift the pitch and change the generated accompaniment.
To obtain the chord transposition information, a mathematical evaluation is required. We note that the originally labeled chords of the input melody are C^pre and the chords generated by PianoTree decoding are C^gene, each chord is represented by 12 mean meters, so it is a 12-dimensional vector. The two are compared and the maximum difference is used as the criterion for transposition. Note the current bar number i, the pitch shift Δ C refers to:
Δ C = argmax(C^pre_i C^gene(T)_i /|| C^pre_i || · || C^gene(T)_i ||)
Here T denotes matrix transposition.
Each bar has a chord best transposition selection, and a number of bars with large Δ C are selected for pitch shift so that tonality adjustment is achieved by means of rules and mathematical modeling.
§.§ Training Objective
The training objective of VAE <cit.> is much the same, and its loss function mainly consists of regularization loss and reconstruction loss. To shorten the formulation, we abbreviate Valence and Arousal as V and A.
For the regularized loss, we set the prior Gaussian distributions of Valence and Texture as p(z_V) and p(z_A), and the posterior distributions after encoder are noted as p(z_V|V), p(z_A|A), respectively. To find the regularization loss of the two probability distributions, we commonly use the KL scatter [40], denoted here as KL(·).
For the reconstruction loss, we set the probability distribution of the Valence Decoder output as p(V|z_V) and the PianoTree Decoder output as p(A|z_A, z_V), and the reconstruction loss is generally found by finding the log probability expectation value. In summary, the loss function Loss(V, A) of the model is
Loss(V, A) = E_p[log p(V|z_V) + log p(A|z_V,z_A)]
+ KL(p(z_V|V) || p(z_V)) + KL(p(z_A|A) || p(z_A))
§ EXPERIMENTS
§.§ Training Details of Our Proposed Model
The experiment was run on a host with a 12th Gen Intel(R) Core(TM) i7-12700H and a single NVIDIA GeForce RTX3060 6GB.
In the section <ref>, we explain the dataset and convert the MIDI files in the dataset into a piano roll representation and a 12-measure chord representation, respectively We set the batch size to 128, so that the model is trained with a time value of 32 for each arousal fragment and 8 for the valence fragment.
When training our VAE model, we set the epoch to 6 and the learning rate to 10^-3 with an exponential decay of 0.999 and a minimum value of 10^-5. To speed up the training speed and reduce the possibility of model divergence, we use the Teacher-Forcing strategy. The Teacher-Forcing training ratio of Encoder-PianoTree Decoder , and Encoder-Valence is set to 0.6 and 0.5 respectively. The training ratio of Encoder-Valence Decoder is set to 0.5.
§.§ Baseline Models
Our baseline models are Poly-dis and M-GPT chosen from the model in the paper <cit.> <cit.>. Poly-dis, the state-of-the-art disentanglement learning-based model, decouples the characterization of harmony and texture. Unlike our rule constraint and modeling, this model achieves the adjustment of the generated accompaniment by learning prior and posterior sampling. M-GPT is the state-of-the-art piano music generation model and can harmonize the melody using auto-regression principles.
§.§ Emotional Flow Comparison Test
The experiment aims to compare the correlation between the Emotional Flow entered by the user, used as a guide, and the Emotional Flow finally generated by the system. This is an important indicator of the effectiveness of the system's control over the input Emotional Factors.
We evaluate the correlation by comparing the Pearson coefficients between the two sequences, referring to the evaluation metrics in the paper <cit.>, so as to avoid misevaluation due to misalignment of the Emotional Flow.
There are two constraints on the Emotional Flow of the user input guidelines. The first is that there cannot be more than five extreme points per flow curve, except for the start and end points. This is because the melodic data of the sample data does not exceed 90s in length, and too many extreme points mean too many melodic ups and downs, which is not in accordance with the rules of music composition. The second is that each flow curve must have a certain amount of ebb and flow, because too much flatness is not necessary for correlation. Specifically, V̅ and A̅ are the mean values of the valence and arousal curves, and the duration of the melody is set to T.
1/T∫_0^T (V-V̅)^2 dt > 0.15
1/T∫_0^T (A-A̅)^2 dt > 0.15
The data for the experiment were obtained from the "Samples" mentioned in the section <ref>, with 20 pieces of music to be validated. Four typical cases were selected to visualize the results. The criteria we chose are similar to the idea of control variables, which are the correlation of Arousal Flow in the low arousal and high arousal cases, and the correlation of Valence Flow in the Low Valence and High Valence cases, respectively. We calculated the average valence and arousal correlation values for 20 samples of music. For statistical convenience, high arousal/valence is denoted as High Input Basis (HIB) and low arousal/valence is denoted as Low Input Basis (LIB).
The visualization in Figure <ref>, a combination of a heat map and box plot, presents a comparison of the input and output Emotional Flow. The heat map illustrates the specifics of the Emotional Flow, while the box plot offers a broader statistical comparison. The results reveal that the mean values and quartiles of the Emotional Flow are similar for both the user input and the system output. This suggests that the system-generated Emotional Flow aligns with the user input statistically, regardless of the Emotional Flow's baseline.
We also compared the association values between the baseline model and our VAE model, as shown in Table <ref>. Where the baseline model is abbreviated as Poly-Dis, our model is called VA-VAE.
It can be seen that the average correlation of our model outperforms the baseline models for both valence flow and arousal flow. The correlation of our VA-VAE also outperforms the baseline model under HIB versus LIB.
§.§ Subjective Musicality test
The subjective musicality assessment was mainly a professional assessment by music experts. A total of 44 junior and senior music majors and graduate students were invited. The music experts were randomly selected from two of the eight sample groups, and each group contained two pieces of music, one with the accompaniment generated by the baseline Transformer model and the other with the accompaniment generated by the VA-VAE model. The two pieces of music were not distinguished by name; in other words, the music experts' music was selected in a completely blind manner.
The music experts evaluated the level of the accompaniment from four angles: 1) whether the overall layout of the composition was appropriate; 2) whether the chords were harmoniously chosen and connected; 3) whether the rhythmic density (articulation points) was specific to the melody; and 4) whether there was a sub-melody or passing phrase that accentuated the melody. Each evaluation angle is evaluated quantitatively using a rating value, and is assigned a score of 1 to 5. The above four perspectives are abbreviated as Q1, Q2, Q3 and Q4.
The experimental results are shown below, and the final score for each assessment perspective is based on the weighted average score.
From the experimental results shown in Fig <ref>, we can see that the weighted average score of our VA-VAE model is stronger than that of the Baseline models in terms of the overall layout of the weave (Q1), chord selection and connection (Q2), melodic counterpoint (Q3), and melodic underscoring (Q4). The overall arrangement of the accompaniment generated by our model is more reasonable, and the chord selection and connection are more fully considered, and the rhythm between the accompaniment and the melody is more organized and regular, which can also better support the melody. The musical accompaniment generated by our model has a more artistic character.
Refer to Figure <ref> for a visual representation of the music's attention structure.
The darker the color of the music phrases, the greater the weight of attention. The structure of the different "music phrases" gathered by attention mechanism is divided by dotted lines, so that the music as a whole is well organized.
§.§ Ablation Study
For the ablation study, we abbreviated the control group without relative self-attention and Rule Constraint (RC) as CG, the model after adding relative self-attention as CG+NS, and then after adding Rule Constraint as CG+NSR. We used a quantitative approach to assess the generation The quality of the accompaniment in the ablation experiment is assessed quantitatively. Quantitative metrics such as pass/fail ratios, null ratios, etc. are less applicable in our piano improvisation accompaniment generation task. The key criteria for the evaluation of the accompaniment task are the texture of the accompaniment, the harmony of the accompaniment with the melody, the contribution to the melody, etc. This way of evaluation is very similar to that of the translation task, where the harmony of the accompaniment is like the valuation of the translated utterance, the weaving arrangement is like the wording of the translation, and the contribution to the melody is like the synthesis and comparison of the information in the translation task. Therefore, we chose the MUTE evaluation index from the paper <cit.>, which is analogous to the F-Score evaluation index in the translation task, to accurately and quantitatively assess the level of the accompaniment arrangement.
In MUTE, F1 Score(FS) evaluates the "translation accuracy" of the accompaniment from the perspective of 128 pitches and is suitable for evaluating texture, while the F1 Score Pitch Class(FSPC) normalizes the pitches to 12 basic pitches and is therefore suitable for evaluating harmony.
As seen in Table <ref>, the model incorporating relative self-attention and RC outperformed the CG and CG+NS control groups in both FS and FSPC metrics. Whether it is harmony or texture, the newly incorporated relative self-attention mechanism and rule constraint can be better designed and orchestrated to create higher quality accompaniment. Further, we visualized the comparison test of the rule constraints, as shown in Figure <ref>, and found that the rule constraints did indeed shift the range of the accompaniment to better harmonize the melody.
§ CONCLUSION
In this study, we investigate the generation of musical accompaniment that is guided by emotional flow. We focus on two key aspects of the problem. First, we establish a mechanism for converting emotional streams into music information data and a VAE network architecture that is tailored to emotional quantization data, allowing us to control the network model with emotional factors. Secondly, we optimize the structural planning of accompaniment generation by introducing the Self-Similarity and relative self-attention mechanism. By using rule constraints, we further improve the local and global tonality of the music. This approach of progressing from the whole to the local, layer by layer, allows us to create an automatic accompaniment system that has excellent emotional flow control and high-quality music generation.
In the future, we plan to further improve our research. Currently, the accompaniment is generated by a single instrument and we intend to extend it to include multiple instruments to create an automated orchestra. Additionally, the representation of emotional flow is not yet clear, and we will research on better visualization methods to make the AI technology more user-friendly.
§ ACKNOWLEDGMENT
This research was funded by the Regular Projects of the Humanities and Social Sciences Fund of the Ministry of Education of Grant No.16YJAZH080.
00
b1 Wu, Yi-Chan, and Homer H. Chen. "Emotion-flow guided music accompaniment generation." 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2016.
b2 Thayer, Robert E. The biopsychology of mood and arousal. Oxford University Press, Oxford, UK, 1990, ch. 2-5.
b3 Boulanger-Lewandowski, Nicolas, Yoshua Bengio, and Pascal Vincent. "Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription." arXiv preprint arXiv:1206.6392 (2012).
b4 Choi, Keunwoo, George Fazekas, and Mark Sandler. "Text-based LSTM networks for automatic music composition." arXiv preprint arXiv:1604.05358 (2016).
b5 Dua, Mohit, et al. "An improved RNN-LSTM based novel approach for sheet music generation." Procedia Computer Science 171 (2020): 465-474.
b6 Lyu, Qi, et al. "Modelling high-dimensional sequences with lstm-rtrbm: Application to polyphonic music generation." Twenty-Fourth International Joint Conference on Artificial Intelligence. 2015.
b7 Yang, Li-Chia, Szu-Yu Chou, and Yi-Hsuan Yang. "MidiNet: A convolutional generative adversarial network for symbolic-domain music generation." arXiv preprint arXiv:1703.10847 (2017).
b8 Luo, Jing, et al. "MG-VAE: deep Chinese folk songs generation with specific regional styles." Proceedings of the 7th Conference on Sound and Music Technology (CSMT). Springer, Singapore, 2020.
b9 Lattner, Stefan, Maarten Grachten, and Gerhard Widmer. "Imposing higher-level structure in polyphonic music generation using convolutional restricted boltzmann machines and constraints." Journal of Creative Music Systems 2 (2018): 1-31.
b10 Zhao, Jingwei, and Gus Xia. "AccoMontage: Accompaniment Arrangement via Phrase Selection and Style Transfer." arXiv preprint arXiv:2108.11213 (2021).
b11 Hadjeres, Gaëtan, François Pachet, and Frank Nielsen. "Deepbach: a steerable model for bach chorales generation." International Conference on Machine Learning. PMLR, 2017.
b12 Huang, Cheng-Zhi Anna, et al. "Music transformer." arXiv preprint arXiv:1809.04281 (2018).
b13 Huang, Yu-Siang, and Yi-Hsuan Yang. "Pop music transformer: Beat-based modeling and generation of expressive pop piano compositions." Proceedings of the 28th ACM International Conference on Multimedia. 2020.
b14 Wu, Shih-Lun, and Yi-Hsuan Yang. "The Jazz Transformer on the front line: Exploring the shortcomings of AI-composed music through quantitative measures." arXiv preprint arXiv:2008.01307 (2020).
b15 Jin, Cong, et al. "A transformer generative adversarial network for multi‐track music generation." CAAI Transactions on Intelligence Technology 7.3 (2022): 369-380.
b16 Wang, Ziyu, and Gus Xia. "MuseBERT: Pre-training Music Representation for Music Understanding and Controllable Generation." ISMIR. 2021.
b17 Jiang, Junyan, et al. "Transformer VAE: A hierarchical model for structure-aware and interpretable music representation learning." ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2020.
b18 Tanaka, Keitaro, et al. "Pitch-Timbre Disentanglement Of Musical Instrument Sounds Based On Vae-Based Metric Learning." ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2021.
b19 Yang, Ruihan, et al. "Deep music analogy via latent representation disentanglement." arXiv preprint arXiv:1906.03626 (2019).
b20 Song, Kai, Xia Liang, and Junmin Wu. "ViT-based VQ-VAE Generative Network for Accompaniment Generation." 2021 4th International Conference on Algorithms, Computing and Artificial Intelligence. 2021.
b21 Liu, Weiming. "Literature survey of multi-track music generation model based on generative confrontation network in intelligent composition." The Journal of Supercomputing (2022): 1-23.
b22 Wu, Yi-Chan, and Homer H. Chen. "Emotion-flow guided music accompaniment generation." 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2016.
b23 Wallis, Isaac, et al. "A rule-based generative music system controlled by desired valence and arousal." Proceedings of 8th international sound and music computing conference (SMC). 2011.
b24 Morreale, Fabio, and Antonella De Angeli. "Collaborating with an autonomous agent to generate affective music." Computers in Entertainment (CIE) 14.3 (2016): 1-21.
b25 Miyamoto, Kana, Hiroki Tanaka, and Satoshi Nakamura. "Online EEG-Based Emotion Prediction and Music Generation for Inducing Affective States." IEICE TRANSACTIONS on Information and Systems 105.5 (2022): 1050-1063.
b26 Kaliakatsos-Papakostas, Maximos, Andreas Floros, and Michael N. Vrahatis. "Artificial intelligence methods for music generation: a review and future perspectives." Nature-Inspired Computation and Swarm Intelligence (2020): 217-245.
b27 Boulesteix, Anne‐Laure, et al. "Overview of random forest methodology and practical guidance with emphasis on computational biology and bioinformatics." Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 2.6 (2012): 493-507.
b28 Eddy, Sean R. "What is a hidden Markov model?." Nature biotechnology 22.10 (2004): 1315-1316.
b29 Hearst, Marti A., et al. "Support vector machines." IEEE Intelligent Systems and their applications 13.4 (1998): 18-28.
b30 Boulanger-Lewandowski, Nicolas, Yoshua Bengio, and Pascal Vincent. "Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription." arXiv preprint arXiv:1206.6392 (2012).
b31 Dahale, Rishabh, et al. "Generating Coherent Drum Accompaniment With Fills And Improvisations." arXiv preprint arXiv:2209.00291 (2022).
b32 Vaswani, Ashish, et al. "Attention is all you need." Advances in neural information processing systems 30 (2017).
b33 Ren, Yi, et al. "Popmag: Pop music accompaniment generation." Proceedings of the 28th ACM International Conference on Multimedia. 2020.
b34 Temperley, David. The cognition of basic musical structures. MIT press, 2004: 10-20.
b35 Wang, Ziyu, et al. "Pop909: A pop-song dataset for music arrangement generation." arXiv preprint arXiv:2008.07142 (2020).
b36 Medeot, Gabriele, et al. "StructureNet: Inducing Structure in Generated Melodies." ISMIR. 2018.
b37 Wang, Ziyu, et al. "Pianotree vae: Structured representation learning for polyphonic music." arXiv preprint arXiv:2008.07118 (2020).
b38 Sharif Razavian, Ali, et al. "CNN features off-the-shelf: an astounding baseline for recognition." Proceedings of the IEEE conference on computer vision and pattern recognition workshops. 2014.
b39 Wang, Ziyu, et al. "Pianotree vae: Structured representation learning for polyphonic music." arXiv preprint arXiv:2008.07118 (2020).
b40 An, Jinwon, and Sungzoon Cho. "Variational autoencoder based anomaly detection using reconstruction probability." Special Lecture on IE 2.1 (2015): 1-18.
b41 Wang, Ziyu, et al. "Learning interpretable representation for controllable polyphonic music generation." arXiv preprint arXiv:2008.07122 (2020).
b42 Lim, Hyungui, Seungyeon Rhyu, and Kyogu Lee. "Chord generation from symbolic melody using BLSTM networks." arXiv preprint arXiv:1712.01011 (2017).
b43 Gover, Matan, and Oded Zewi. "Music Translation: Generating Piano Arrangements in Different Playing Levels." Ismir 2022 Hybrid Conference. 2022.
|
http://arxiv.org/abs/2307.03987v1 | 20230708142557 | A Stitch in Time Saves Nine: Detecting and Mitigating Hallucinations of LLMs by Validating Low-Confidence Generation | [
"Neeraj Varshney",
"Wenlin Yao",
"Hongming Zhang",
"Jianshu Chen",
"Dong Yu"
] | cs.CL | [
"cs.CL"
] |
PCG-based Static Underground Garage Scenario Generation
Wenjin Li, Kai Li
Wenjin Li, Kai Li are with the Department of Computer Science and Technology, Southern University of Science and Technology, Shenzhen, 518055, China
August 12, 2023
============================================================================================================================================================================
Recently developed large language models have achieved remarkable success in generating fluent and coherent text. However, these models often tend to `hallucinate' which critically hampers their reliability.
In this work, we address this crucial problem and propose an approach that actively detects and mitigates hallucinations during the generation process.
Specifically,
we first identify the candidates of potential hallucination leveraging the model's logit output values, check their correctness through a validation procedure, mitigate the detected hallucinations, and then continue
with the generation process.
Through extensive experiments with the `article generation task', we first demonstrate the individual efficacy of our detection and mitigation techniques.
Specifically, the detection technique achieves a recall of ∼88% and the mitigation technique successfully mitigates 57.6% of the correctly detected hallucinations.
Importantly, our mitigation technique does not introduce new hallucinations even in the case of incorrectly detected hallucinations, i.e., false positives.
Then, we show that the proposed active detection and mitigation approach successfully reduces the hallucinations of the GPT-3 model from 47.5% to 14.5% on average.
In summary, our work contributes to improving the reliability and trustworthiness of large language models, a crucial step en route to enabling their widespread adoption in real-world applications.
§ INTRODUCTION
Recently developed large language models such as GPT-3 <cit.>, InstructGPT <cit.>, PaLM <cit.>, LLaMA <cit.>, and several others <cit.>
have achieved remarkable performance on a wide range of language understanding tasks.
Furthermore, they have been shown to possess an impressive ability to generate fluent and coherent text.
Despite all these abilities, their tendency to `hallucinate' critically hampers their reliability and limits their widespread adoption in real-world applications.
Hallucination in the context of language refers to the generation of text or responses that seem syntactically sound, fluent, and natural but are factually incorrect, nonsensical, or unfaithful to the provided source input <cit.>.
These hallucinations can lead to serious consequences such as spreading of misinformation and violation of privacy.
Thus, in this work, we focus on the crucial problem of `addressing' hallucinations of the large language models.
We propose to actively `detect' and `mitigate' hallucinations during the generation process.
This is crucial as we show that a generated sentence is hallucinated more often when the model has already hallucinated in its previously generated sentences for the input.
Thus, actively detecting and mitigating hallucinations is also important to prevent the propagation of hallucinations in the subsequently generated sentences. We divide our approach into two stages, Detection and Mitigation.
In the hallucination detection stage, we first identify the candidates of potential hallucination, i.e., the key `concepts' of the generated sentence.
Next, leveraging the logit output values of the model, we calculate model's `uncertainty' on the identified concepts.
We demonstrate that this uncertainty provides a signal for hallucination.
However, we note that this is an additional signal and not a necessary requirement for our approach.
Then, we check the correctness of the
`uncertain' concepts through a validation procedure where we:
(a) create a query that tests the correctness of the information pertaining to the concept,
(b) retrieve knowledge relevant to the validation question, (c) answer the validation question leveraging the retrieved knowledge, and verify the corresponding information in the generated sentence to detect hallucinations.
This is followed by the hallucination mitigation stage in which we
`repair' the potentially hallucinated sentence using the retrieved knowledge as evidence.
Figure <ref> illustrates the key steps of our approach.
Furthermore, we conduct a systematic and wide study exploring multiple techniques to achieve the objective of each of the steps.
We design an experimental setup where
we prompt the model to write about topics from diverse domains such as sports, politics, music, literature, etc.
Then, we annotate the correctness of the first five generated sentences for each topic.
We first demonstrate the individual efficacy of our detection and mitigation techniques.
Specifically, the detection technique achieves a recall of ∼88% and the mitigation technique successfully mitigates 57.6% of the correctly detected hallucinations.
Importantly, our mitigation technique does not introduce new hallucinations even in the case of incorrectly detected hallucinations, i.e., false positives.
Then, we show that the proposed active detection and mitigation approach successfully reduces the hallucinations of the GPT-3 model from 47.5% to 14.5% on average (Figure <ref>).
We conduct a thorough analysis that further
results in several interesting and important findings.
Lastly, we release our code and correctness annotations that will also facilitate a systematic future research in addressing hallucinations.
§ APPROACH
§.§ Overview
We propose to actively detect hallucinations and mitigate them during the generation process.
This is crucial as we show that
a generated sentence is hallucinated more often
when the model has already hallucinated in its previously generated sentences for the input (Section <ref>).
Similarly, a generated sentence is relatively less often hallucinated when the model has not hallucinated in its previously generated sentences.
Thus, actively detecting hallucinations and mitigating them is also important to prevent the propagation of further hallucinations in subsequently generated sentences.
To this end, we iteratively generate sentences through the model and actively detect and mitigate hallucinations.
Figure <ref> illustrates the key steps of our approach.
In section <ref>, we detail the steps of our hallucination detection approach, i.e., identifying the important `concepts' of the generated sentence, i.e., the candidates of potential hallucination (<ref>), calculating model's uncertainty on the concepts using the logit output values (<ref>), and checking the correctness by creating validation query (<ref>), finding relevant knowledge (<ref>), and verifying information leveraging the retrieved knowledge (<ref>).
We describe various techniques to achieve the objective of each of these steps and also elaborate on several important points such as
using a `self-inquiry' method to answer validation questions without using an external knowledge source and trade-off between executing the validation procedure in parallel for all the concepts and in sequential order based on their `uncertainty'.
For each step, we also indicate the most preferred technique with (*) and provide our justification.
In section <ref>, we detail our hallucination mitigation approach.
Specifically, we `repair' the hallucinated sentence by removing or substituting the hallucinated information leveraging the retrieved knowledge as evidence and also utilize the retrieved knowledge as context (prepended to the input) to generate the next sentence.
§.§ Hallucination Detection
§.§.§ Identify Key Concepts
In the first step, we identify the important concepts from the generated sentence.
We identify these concepts because validating the correctness of the entire sentence at once is infeasible; this is because a sentence may contain a number of different facets all of which can not be validated at once.
On the other hand, individually validating the correctness corresponding to the concepts provides opportunities for accurately detecting hallucinations.
Thus, the objective of this step is to identify the candidates of potential hallucination.
We note that a concept or keyphrase is essentially a span of text consisting of one or more words.
We study the following techniques to identify the concepts:
Entity Extraction:
Entities are usually an important part of a sentence, thus, we use an off-the-shelf entity extraction model
to identify the concepts.
A limitation of this method is that a concept need not necessarily be an entity and can be a non-entity span also.
We address this limitation with a keyword extraction model.
Keyword Extraction:
To also identify the non-entity concepts, we explore an off-the-shelf keyword extraction model[https://huggingface.co/ml6team/keyphrase-extraction-kbir-kpcrowd].
This model uses Keyphrase Boundary Infilling with Replacement (KBIR) as its base model and fine-tunes it on the KPCrowd dataset <cit.>.
*Instructing the Model*:
Since state-of-the-art language models perform remarkably well on a wide range of tasks, in this technique, we directly instruct the model to identify the important concepts from the generated sentence.
An important characteristic of this technique is that it doesn't require calling a task-specific tool (entity or keyword extraction model) for this task.
Table <ref> (in Appendix <ref>) illustrates examples of concepts identified using the three techniques.
It shows that the entity extraction model misses many important concepts while the keyword extraction model identifies a lot of insignificant concepts also.
In contrast, instruction technique successfully identifies all the important concepts.
Moreover, it doesn't require calling a task-specific tool.
Thus, we represent this technique with (*), our preferred technique for this step.
§.§.§ Calculate Model's Uncertainty
GPT-3 <cit.> and several other publicly available models also provide logit output values in their prediction response.
Thus, we study if these logit output values can be utilized to
detect hallucinations.
However, we note that this is an additional source of information and not a necessary requirement for our hallucination detection method as some models that are available only via API calls do not provide these logit output values.
Recall that a concept can consist of more than one token also (note that the model provides logit output values at the level of tokens); thus, we study three different techniques for calculating a probability score for a concept.
Consider a concept consisting of n tokens and having the maximum softmax probabilities as p_1, p_2, p_3, ..., p_n for the n token positions respectively.
We obtain these probabilities by applying the softmax function over the logit values for each token position.
We study the following techniques:
Average of Token Probabilities:
In this technique, we simply take the average of the probabilities of the tokens corresponding to the concept:
score = AVG (p_1, p_2, ..., p_n)
Normalized Product of Token Probabilities:
Here, we take a normalized product of the probabilities of the tokens:
score = (p_1 × p_2 × ... × p_n)^1/n
*Minimum of Token Probabilities*:
Here, we take the minimum of probabilities as the score.
score = MIN (p_1, p_2, ..., p_n)
This is our preferred technique for this step as the other techniques average out the effect of model's uncertainty on the tokens while low probability in even one token of the concept provides a strong evidence of the model being uncertain.
For example, if the model is uncertain on the name of the USA president then its uncertainty on the first token (`Joe') would be high but on the next token (`Biden') would be very low as the token `Joe' is frequently followed by the token `Biden'.
Thus, averaging or normalizing the probabilities will have a limited capability to capture this signal.
Through our experiments (Section <ref>), we show that this score (especially `MIN') indeed provides a signal for hallucination, i.e., the more uncertain a model is on a concept (low probability score), the more likely it is to be hallucinating about that concept.
However, we note that this score is just a signal for hallucination and in no way provides a guarantee for presence of hallucinations.
We utilize this signal and check for hallucinations with respect to the uncertain concepts using our validation procedure (<ref>-<ref>).
In the absence of logit output values:
For models that do not provide the logit output values, all or some heuristically selected concepts (depending on the computational and latency budget of the system) can be passed to the validation stage for detecting hallucinations.
§.§.§ Create Validation Question
We start the validation procedure for a concept by creating a question that tests the correctness of the information (in the generated sentence) pertaining to the concept.
We create Yes/No Questions, i.e., questions for which the answer is either a `Yes' or a `No'.
Table <ref> shows examples of validation questions.
For creating these questions, we explore the following two techniques:
Question Generation Tool:
Here, we use an off-the-shelf answer-aware question generation model.
*Instructing the Model*:
Here, we directly instruct the model to create a validation question checking the correctness of the information about the selected concept.
For the same reason as in the concept identification step, this is our preferred technique as it does not require calling a task-specific tool.
We note that instead of Yes/No questions, Wh-questions can also be used for validation.
We prefer Yes/No questions as it is relatively easier to check the answer for these questions.
We leave exploring Wh-questions for validation for future work.
§.§.§ Find Relevant Knowledge
*Web Search*:
In order to answer the validation question, we retrieve knowledge relevant to it which serves as additional context.
For generality and wide coverage, we use web search (via Bing search API) for retrieving this knowledge.
However, we note that any other search API or knowledge corpus can also be utilized for this purpose.
Self-Inquiry:
We also explore a self-inquiry technique where we directly prompt the model to answer the validation question.
In this technique, the model relies on its parametric knowledge to answer the validation question.
This technique has several drawbacks as compared to web search such as lack of a reliable strategy to extract the parametric knowledge from the model and staleness of the parametric knowledge.
§.§.§ Answer Validation Question
In this step, we prompt the model to answer the validation question (leveraging the retrieved knowledge as context) and verify its response.
If the validation procedure succeeds for all the uncertain concepts then we continue generating the next sentence; otherwise, we interrupt the generation process, mitigate the potential hallucination in the sentence, and then continue generation.
Order of Validation of Concepts:
Validation of different concepts can be done in a sequence (in ascending order of their calculated probability score) or in parallel.
However, running this in parallel would require starting multiple threads which may not be supported by all machines.
Thus, in this work we study only the sequential validation strategy but note that it can be made more efficient by running it in parallel.
We regard this sequential validation as a greedy exiting strategy as we proceed to the mitigation stage on detection of the first potential hallucination.
§.§ Hallucination Mitigation
For mitigating the hallucination in the generated sentence, we instruct the model to repair the generated sentence by either removing or substituting the hallucinated information using the retrieved knowledge as evidence.
Table <ref> shows the instructional prompts for different steps of our approach.
Note: We note that the result of the validation procedure is contingent on the retrieved knowledge and the model's ability to leverage that knowledge in answering the validation question.
Thus, a case is plausible in which the validation procedure reports hallucination even though the sentence is actually not hallucinated.
However, in Section <ref>, we show that our approach performs fairly well on this task.
Moreover, it achieves a very high recall demonstrating its efficacy at detecting hallucinations.
Moreover, in Section <ref>, we show that our mitigation approach does not introduce new hallucinations even in the case of incorrectly detected hallucinations, i.e., false positives.
§ EXPERIMENTS AND RESULTS
In this section, we first demonstrate the two findings that motivate our approach (<ref> and <ref>).
Then, we show the individual efficacy of our hallucination detection and mitigation techniques in <ref> and <ref>, respectively.
Finally, in <ref>, we show the effectiveness of the proposed active detection and mitigation approach in addressing hallucinations.
Data and Annotation:
In our experimental setup, we prompt the large language model (GPT-3: text-davinci-003) to write about various topics.
Specifically, we use a total of 150 topics from diverse domains.
Figure <ref> shows the distribution of different domains in our topic set.
In each domain, we include different kinds of topics; for instance, Sports domain consists of sports persons, administrators, teams, and games, Music consists of musicians, songs, music labels, and bands, Politics includes politicians, political parties, and elections, Film & TV includes actors, TV personalities, shows, and movies, History includes historians and events, etc.
For selecting the names of people, we use randomly sampled names from the top 20% of longest articles in WikiBio dataset <cit.> as done in <cit.>.
Similarly, for the other topics, we randomly sample from the longest Wikipedia articles.
This is done to ensure that no obscure or ambiguous concept is selected.
Equipped with the list of topics, we give the following input prompt to the model:
for each topic.
Following this, we (the authors) manually annotate the correctness of the first five sentences generated by the model for each topic.
For annotating the correctness, we look at search results from the web to find the relevant knowledge that either supports or contradicts the information present in the generated sentence.
In some cases, multiple web searches were required to check the correctness of different facets of a sentence.
Furthermore, in a small number of cases, we could not find information supporting or contradicting the information in the generated sentence, we mark it as a case of extrinsic hallucination.
We opt for this expert annotation strategy because despite our annotation task being a simple binary classification task, it requires considerable effort in checking the correctness of a given sentence which can not reliably be collected via crowdsourcing.
In addition to this sentence-level annotation, we also annotate correctness at the concept-level that we will detail in <ref>.
We release both sentence-level and concept-level hallucination annotations that will also facilitate a systematic future research in this direction.
§.§ Motivating Findings
§.§.§ Hallucination Causes Further Hallucination
Recall that we consider the first five sentences generated by the model for each topic and annotate their correctness.
Since the sentences are sequentially generated, we investigate the
relationship between `hallucination in a generated sentence' and `hallucination in the previously generated sentences' for an input.
Since there are two binary variables, there exist four possibilities in this relationship, i.e.,
a sentence is hallucinated and there was hallucination in the previously generated sentences (A), the sentence is not hallucinated and there was hallucination in the previously generated sentences (B), the sentence is hallucinated and there was no hallucination in the previously generated sentences (C), the sentence is not hallucinated and there was no hallucination in the previously generated sentences (D).
For illustration, consider a sample case for sentence 3, the two binary variables are whether sentence 3 is hallucinated and whether there was hallucination in the previously generated sentences (i.e. in sentence 1 OR sentence 2).
Figure <ref> demonstrates this relationship for sentences 2, 3, 4 and 5 aggregated over all the topics in our data.
We do not show this for sentence 1 as there is no previously generated sentence for it.
From this figure, we draw the following inferences:
(a) A > B: Cases A and B correspond to the scenario when there is hallucination in the previously generated sentences. It can be observed that A is considerably greater than B which implies that when there is hallucination in the previously generated sentences, a sentence is hallucinated more often.
Moreover, the gap keeps increasing as the sentence number increases.
(b) A > C: Cases A and C correspond to the scenario when a generated sentence is hallucinated. It can be observed that A is greater than C which implies that a generated sentence is hallucinated more when there is hallucination in the previously generated sentences as compared to when there is no previous hallucination.
(c) D > C: Cases C and D correspond to the scenario when there is no hallucination in the previously generated sentences. Here, D is greater than C which implies that when there is no hallucination in the previously generated sentences, a generated sentence is more often not hallucinated.
(d) D > B: Cases B and D correspond to the scenario when a generated sentence is not hallucinated. D is greater than B which implies that a generated sentence is not hallucinated more when there is no previous hallucination as compared to when there is previous hallucination.
This shows that hallucination in a sentence often results in further hallucinations in the subsequently generated sentences and thus actively detecting and mitigating hallucinations can not only fix the current hallucination but can also prevent its propagation in the subsequently generated sentences.
Next, we demonstrate the utility of logit output values in detecting hallucinations.
§.§.§ Logit Output Values Provide a Signal for Hallucination
In this subsection, we first show the trend of hallucination with the probability score.
Note that this score is calculated using the logit output values.
Then, we demonstrate the benefit of identifying concepts from the generated sentence in detecting hallucinations.
Finally, we compare the efficacy of different probability calculation techniques in detecting hallucinations.
Hallucination vs Probability Score:
In order to study the relationship between logit output values and hallucination, we annotate correctness at concept-level also (in addition to sentence-level annotations described earlier).
Specifically, for each identified concept, we mark whether the information about it in the generated sentence is hallucinated or not.
This can be different from sentence-level annotation as it focuses only on the correctness of the information about the concept in the sentence.
Table <ref> shows examples of both sentence-level and concept-level annotations.
Figure <ref> shows the trend of hallucination with our calculated probability scores at both sentence and concept levels.
For a sentence, we use the minimum across tokens of all its identified concepts as the probability score and for a concept, we use the minimum across all its tokens as the probability score.
It can be observed that as the probability score increases (or uncertainty decreases), tendency to hallucinate decreases.
This shows that these probability values can be utilized as a signal for hallucination, i.e., the low probability concepts in a generated sentence can be considered as candidates of potential hallucination and their correctness in the generated sentence can be validated for detecting hallucinations.
On average, we observe an absolute difference of ∼0.15 between the probabilities of concepts when the model is hallucinating vs when it is not hallucinating.
Benefit of Identifying Concepts from a Sentence:
Now, we demonstrate the benefit of identifying concepts from a sentence and leveraging the logit output values corresponding to their tokens for detecting hallucinations.
To this end, we plot precision-recall curves for the hallucination detection task corresponding to two methods that use the probabilities calculated from the logit output values.
The blue curve corresponds to the technique in which we use the minimum probability across all tokens of the sentence and the orange curve is for the technique in which we use the minimum over only the tokens of the identified concepts.
Figure <ref> shows the two curves.
The orange curve achieves higher area under the precision-recall curve implying that utilizing the probabilities of the concept tokens provides a stronger signal for hallucination as compared to the probabilities corresponding to all the tokens.
Comparing Probability Calculation Techniques:
Figure <ref> shows the Precision-Recall curves for the hallucination detection task (at concept-level) using the three probability calculation techniques, i.e., Minimum, Average, and Normalized (described in <ref>).
The `Minimum' technique achieves the highest area under the curve and hence is better at the hallucination detection task.
§.§ Hallucination Detection Performance
In this subsection, we demonstrate the hallucination detection performance of various techniques at both sentence and concept-levels.
Self-Inquiry vs Web Search:
Table <ref> and <ref>
show the hallucination detection performance of the self-inquiry and web search techniques at sentence-level and concept-level, respectively.
For sentence-level results, we predict the sentence to be hallucinated if the validation procedure fails on any identified concept.
Note that in these results, we do not leverage the uncertainty score to select concepts for validation, instead we validate all the identified concepts.
We study the relationship of recall with probability thresholds in Figure <ref> (in Appendix).
From the tables, it can be observed that the web-search technique achieve considerably high recall in detecting hallucinations.
Here, we emphasize on the high `recall' of web-search technique as we show that our mitigation approach does not introduce any new hallucinations even in the case of incorrectly detected hallucinations, i.e., false positives (<ref>).
Figure <ref> shows the recall of hallucination detection vs Probability threshold plot for Self Inquiry and web search techniques at both sentence-level and concept-level.
Web-search is consistently and considerably better than self-inquiry.
§.§ Hallucination Mitigation Performance
On sentences where our validation procedure (using Web search) reports hallucinations, we apply our mitigation technique.
We note that a sentence which is reported as hallucination can either be actually hallucinated or not hallucinated, i.e., it could also be a false positive.
Table <ref> shows the result of our method.
It successfully mitigates the hallucination on 57.6% of the correctly detected hallucinations (True Positives); we refer to this metric as `success'.
Furthermore, it achieves this at minimal `deterioration' (3.06%), i.e., it incorrectly converts a minimal 3.06% of the non-hallucinated instances to hallucinated.
§.§ Active Detection and Mitigation
The two findings in Section <ref> motivate our approach of addressing hallucinations in which we actively detect hallucinations leveraging the logit output values and mitigate them during the generation process to prevent their propagation.
Specifically, using the calculated probability scores, we identify the uncertain concepts and check their correctness using our validation procedure.
We generate one sentence at a time and when our detection method reports hallucination, we fix it using our mitigation approach and continue generating the next sentence.
We demonstrated separate detection and mitigation efficacy in <ref> and <ref>, respectively.
Figure <ref> compares the percentage of hallucination in the output of GPT-3 model and our active detection and mitigation approach.
Our approach reduces the percentage of hallucinations from 47.4% to 14.53%.
In Figure <ref>, we demonstrate this comparison for different categories of hallucination.
It shows that our approach reduces hallucinations for all categories.
§ RELATED WORK
Advancements in the field of natural language processing led to the development of models that possess an impressive ability to generate fluent and coherent text. However, these models are vulnerable to a phenomenon called text hallucination.
Prior work <cit.> has categorized text hallucinations into two classes: Intrinsic (when the generated output contradicts the source content) and Extrinsic (when the generated output cannot be verified from the source content, i.e., it that can neither be supported nor contradicted by the source).
One thread of research pertaining to hallucinations has focused on studying different causes of this phenomenon such as training data quality <cit.>, source-target divergence <cit.>, ill-suited modeling <cit.>, and randomness during inference <cit.>.
The other thread focuses on addressing the hallucination problem <cit.>.
<cit.> propose a sampling-based hallucination detection approach in which they first sample multiple responses from the model and then measure the information consistency between the different responses. They posit that when a language model knows a given concept well, the sampled responses are likely to be similar and contain consistent facts; on the other hand, for hallucinated facts, stochastically sampled responses are likely to diverge and may completely contradict one another.
Another recent work <cit.> leverage LLM's internal state to identify the truthfulness of a statement. Using an annotated dataset, they train a separate classifier that takes the LLM's activation values as input and predicts its truthfulness.
<cit.> hypothesize that the randomness of sampling is more harmful to factuality when it is used to generate the latter part of a sentence than the beginning of a sentence and propose a new sampling algorithm named factual-nucleus sampling that dynamically adapts the `nucleus' p along the generation of each sentence.
<cit.> propose an approach motivated by The Society of Mind and multi-agent settings in which multiple models individually propose and jointly debate their responses and reasoning processes to arrive at a common answer.
In our approach, we leverage the logit output values and web search to actively detect and mitigate hallucinations.
§ CONCLUSION
In this work, we proposed an approach that actively `detects' and `mitigates' hallucinations of the large language models.
Through systematic and extensive experiments, we show that our approach successfully reduces the hallucinations of the GPT-3 model from 47.5% to 14.5% on average.
We also demonstrate the individual efficacy of our detection and mitigation techniques.
Specifically, our detection technique achieves a high recall and our mitigation technique successfully mitigates majority of the correctly detected hallucinations.
Notably, the mitigation technique does not introduce new hallucinations even in the case of incorrectly detected hallucinations, i.e., false positives.
Overall, our work contributes to improving the reliability and trustworthiness of text generation systems, a crucial step en route to enabling their widespread adoption in real-world applications.
acl_natbib
§ APPENDIX
§ APPROACH
Table <ref> shows the instructional prompts used for different steps of our approach.
We note that these techniques are the preferred techniques as they do not require calling an external task-specific tool to achieve the corresponding objectives.
§.§ Identify Key Concepts
Table <ref> shows examples of concepts identified using the three methods, i.e., Entity Extraction, Keyword Extraction, and Instructing the Model.
It shows that the entity extraction model misses many important concepts while the keyword extraction model identifies a lot of insignificant concepts also.
In contract, instruction technique successfully identifies majority of the important concepts.
§.§ Create Validation Question
Table <ref> shows examples of validation questions corresponding to each concept created via instructing the model technique.
It shows examples of both the question types, i.e., Yes/No and Wh questions.
We prefer Yes/No questions as it is relatively easier to check the answer for these questions.
We leave exploring Wh-questions for validation for future work.
§ EVALUATION DATA
Table <ref> shows the statistics of the sentences generated by the GPT-3 (text-davinci-003 with temperature 0) model.
A sentence has ∼18 word on average and each sentence has ∼3.2 key concepts that are identified by our instruction technique.
Table <ref> shows examples of sentence-level and concept-level hallucination annotations.
§ RECALL OF HALLUCINATION DETECTION VS PROBABILITY THRESHOLD
Figure <ref> compares recall of hallucination detection for self-inquiry and web search techniques at different probability thresholds.
Web search considerably outperforms self-inquiry at all thresholds.
|
http://arxiv.org/abs/2307.10200v1 | 20230709023156 | Disentangling Societal Inequality from Model Biases: Gender Inequality in Divorce Court Proceedings | [
"Sujan Dutta",
"Parth Srivastava",
"Vaishnavi Solunke",
"Swaprava Nath",
"Ashiqur R. KhudaBukhsh"
] | cs.CY | [
"cs.CY",
"cs.AI",
"cs.CL",
"cs.LG"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
====================================================================================
Divorce is the legal dissolution of a marriage by a court. Since this is usually an unpleasant outcome of a marital union, each party may have reasons to call the decision to quit which is generally documented in detail in the court proceedings. Via a substantial corpus of 17,306 court proceedings, this paper investigates gender inequality through the lens of divorce court proceedings. While emerging data sources (e.g., public court records) on sensitive societal issues hold promise in aiding social science research, biases present in cutting-edge natural language processing (NLP) methods may interfere with or affect such studies. We thus require a thorough analysis of potential gaps and limitations present in extant NLP resources. In this paper, on the methodological side, we demonstrate that existing NLP resources required several non-trivial modifications to quantify societal inequalities. On the substantive side, we find that while a large number of court cases perhaps suggest changing norms in India where women are increasingly challenging patriarchy, AI-powered analyses of these court proceedings indicate striking gender inequality with women often subjected to domestic violence.
§ INTRODUCTION
The 2011 decennial census in India gave its citizens the following choices to select their marital status – never married, separated, divorced, widowed, married. Based on the census data, a study reported some startling facts <cit.>: 1.36 million of the Indian population is divorced which accounts for 0.24% of the married population, and 0.11% of the total population. More women were separated or divorced than men, and the number of separation was almost three times as high as the number of divorce.
Divorce, a historically taboo topic in India for ages <cit.>, seldom features in mainstream Indian discourse <cit.>. Recent indications of changing social acceptance of divorcees notwithstanding <cit.>, divorce in India still carries a considerable social stigma <cit.>.
How do we quantify gender inequality in Indian divorce? Surveys about divorce often have limited participation and a small sample size <cit.>, perhaps due to the social stigma attached. A vulnerable community – Indian women under conjugal distress – had limited visibility to social scientists. Via a substantial corpus of 17,306 divorce court proceedings, this paper conducts the first-ever computational analysis of gender inequality in Indian divorce based on public court records.
Even though written in English, legal texts are often domain-specific <cit.>. The considerable variation of legal jargon across countries and courts makes domain-specific analysis important. In that vein, Indian legal NLP is an emerging field <cit.>. Most NLP research on legal texts thus far has focused on building robust tools to analyze legal text. Recent research, however, on in-group bias <cit.> and sexual harassment <cit.>, and <Ref> and <Ref> suggest that automated methods to glean social insights from large-scale, legal texts merit investigation. Barring few recent lines of work <cit.>, there is surprisingly little literature on large-scale linguistic analysis of gender bias in India, let alone on legal text zeroing in on divorce.
While emerging data sources (e.g., public court records available on the web) offer opportunities for social scientists to study important and sensitive social issues that previously had limited survey data, applying cutting-edge NLP methods to newer domains requires careful evaluation of the critical question: How much of the (perceived) gender inequality as quantified by the methods truly reflects the corpus and how much of it is due to the inherent biases of the employed NLP methods?
In this paper, we show that the subtleties present in legal text present unique challenges. Unless we consider them and make non-trivial changes to existing methods, we may end up drawing inaccurate social conclusions.
We further show that sophisticated NLP methods built on top of large language models (LLMs) need scrutiny when applied to social inference tasks involving genders. We, in fact, conduct a much broader bias audit of these systems. Our audit reveals well-known LLMs often exhibit gender bias even on simple subject-verb-object sentence completion tasks. Through a corpus-specific text entailment analysis, we demonstrate that downstream applications such as natural language inference (NLI) systems also exhibit sensitivity to gender. We finally, present a novel inconsistency sampling method to mitigate this bias and present our social findings.
To summarize, our contributions are the following:
Social: We create a substantial corpus of 17,306 divorce court proceedings and conduct the first-ever analysis of gender inequality through the lens of divorce proceedings. While a large number of court cases perhaps suggest changing norms in India where women are increasingly challenging patriarchy <cit.>, our analyses reveal widespread domestic violence, dowry demands, and torture of the bride.
Methodological: We address extant gaps and limitations in multiple NLP frameworks. We propose non-trivial modifications to the framework <cit.> to make it suitable for legal text.
We demonstrate a novel application of text entailment <cit.> in quantifying gender inequality. We investigate several potential sources for model bias in NLP resources that can interfere with quantifying gender inequality. We present a novel inconsistency sampling method exploiting counterfactuals to mitigate this bias.
§ DATASET
§.§ Collection
We scrape all the publicly available court proceedings with the word between January 1, 2012 to December 31, 2021 from <https://indiankanoon.org/> (hereafter ), an Indian law search engine launched in 2008 and the largest free online repository of the court proceedings of different courts and tribunals of India. Prior computational law research <cit.> and gender focused social science studies <cit.> have used as source of data.
We download 86,911 case proceedings containing the word from using its advanced search feature. Filtering based on the keyword is a high-recall approach to obtain relevant cases with precedence in computational social science research <cit.>. However, the presence of the keyword may not always indicate a divorce court proceeding; for instance, the keyword can be used to describe the marital status of any of the litigants. It can also be used in an altogether different context (e.g., divorced from reality). We use the following heuristic to further refine our dataset. We also look for other words (e.g., , , ) and phrases (e.g., ), and check if such occurrences repeat for a minimum threshold (set to 5). On a random sample of 100 cases after we apply this cleaning method, a manual inspection reveals that 94 are divorce cases. Hence, our keyword-based filtering is reasonably precise. This pruning step retains 25,635 cases.
§.§ Data Pre-processing
To quantify gender inequality in court proceedings, we must disambiguate the legal parties – the plaintiff and the defendant – and accurately identify of the husband and the wife, who plays which role. Indian legal documents use a wide range of legal terms to denote the plaintiff (e.g., appellant, applicant, complainant, petitioner) and the defendant (e.g., respondent, nonapplicant, opponent). We observe different courts have different formats (sometimes, multiple formats) to summarize the proceedings. The documents also specify which party in marriage represents which role in several different ways (e.g., respondent/wife, respondent-wife, respondent aggrieved wife). We write a regular-expression-based pipeline and consolidate such information to identify the gender of the plaintiff and the defendant across all the states.
The names and salutations (e.g., , , , ) of the plaintiff and defendant also provide gender information. Subcultural naming conventions played a key role in assigning gender to the litigants in some of the cases. For instance, , meaning princess, is a Punjabi last name only for females <cit.>. Or , meaning sister, is solely used in many female names in Gujarat <cit.>. Dependence information of the litigants also provides gender information (e.g., , , ).[We did not find a single mention of in our dataset.]
Of the 25,635 cases, we could unambiguously assign gender to 17,306 cases. For each case, we replace each mention of the litigants as or accordingly. For example, a proceeding snippet “The plaintiff/wife has filed for a divorce. The plaintiff was married to the defendant for three years.”, will be modified to “The wife has filed for a divorce. The wife was married to the husband for three years.” This data set, _divorce, consists of 30,615,754 (30 million) tokens.
§ BRIEF OVERVIEW OF INDIAN LEGAL SYSTEM
Indian Judicial System is largely based on the English Common Law system (where, the law is developed by judges through their decisions, orders, and judgments). The nation has 28 states and 8 union territories (UT), and a total of 25 high courts (some high courts have jurisdiction of more than a state or UT). The federal structure has a supreme court coupled with the high courts that roughly handle the cases in a state or UT. The legal cases of divorce are usually handled by the family or district courts. However, some unresolved cases or sometimes fresh cases are also heard by the high courts. Since the court proceedings are public records and are digitally made available freely by , we found this dataset to be quite appropriate for a large-scale study on gender equality in court proceedings.
§ DOWRY IN DIVORCE PROCEEDINGS
The dowry system involves a transaction of financial assets between the bride's family and the bridegroom's family with the latter being the recipient of the financial assets. While legally prohibited in India since 1961 <cit.>, this practice has continued well after its legal prohibition and has a strong link to social crises such as female feticide <cit.>, domestic abuse and violence <cit.>, and dowry deaths <cit.>. In order to protect the bride from marital cruelty and domestic violence, Indian Penal Code introduced Section 498 in 1983 <cit.>.
Figure <ref> reflects relative proportions of divorce cases containing the text tokens and . For each state, we report the fraction of divorce cases that contain at least one mention of these two tokens. A higher intensity color indicates a larger proportion of such cases. We observe that overall, 24.38% of all cases and 21.86% of all cases mention and , respectively.
Jacob and Chattopadhyay, <cit.> reported that divorce in India does not follow any one-size-fits-all pattern across different states; there exists sufficient interstate variation even for the rate of divorce. We notice a considerable variation in mentions of dowry and section 498-A across different states indicating variance in reported cases of dowry or domestic violence. Among the states and the union territories, the top three entries in terms of dowry mentions are Telangana, Delhi, and Bihar while the top three entries in terms of Section 498-A mentions are Bihar, Telangana, and Andhra Pradesh. Bihar and Telangana have social science literature documenting dowry and domestic violence <cit.>. Apart from the overlap in the top three entries, the statewise dowry and 498-A mentions are moderately correlated (correlation coefficient: 0.67).
We next conduct a qualitative analysis of (alleged) dowry demands [This analysis follows the statements made by the plaintiffs]. On a random sample of 100 court proceedings where the (alleged) dowry demand is explicitly recorded, we observe that the estimated demanded amount is 393,100 ± 544,876. We observe demanded amounts as low as 5,000 to as high as 3,000,000 which explains the staggeringly high variance in our estimation. This also indicates the broad economic spectrum present in India and how far and wide the system of dowry (allegedly) persists. We further observe that cash is not always the solely demanded financial asset. Gold is the second-most commonly demanded asset. Out of the 100 cases, 34 cases report gold demands (71.2 ± 84.6 gm). When we adjust the valuation of demanded gold replacing it with the historical average gold price in India across 2012 and 2021 [Obtained from <https://www.bankbazaar.com/gold-rate/gold-rate-trend-in-india.html>], the estimated (alleged) demanded dowry is 474,798 ± 567,219.
§ METHODS OVERVIEW
We use two NLP methods to quantify gender inequality: (1) Word Embedding Association Test; and (2) a text entailment framework. A brief description follows.
§.§ Word Embedding Based Methods
The first metric is called ord mbedding ssociation est () introduced by <cit.>. To calculate the metric, the words are embedded and the vectors a and b are obtained for the words a and b respectively. The cosine similarity of these words are denoted by cos(a,b). The metric considers two sets of target words given by and , and two sets of attribute words Å and . Then, the score is defined as (, , Å, ) = (_x ∈σ(x, Å, ) - _y ∈σ(y, Å, ))/_w ∈∪σ(w, Å, ),
where, σ(w, Å, ) = _a ∈Åcos(w,a) - _b ∈cos(w,b).
Intuitively, σ(w, Å, ) measures the association of w with the attribute sets, and the score measures the differential association of the two sets of target words with the attribute sets. A positive score implies that the target words in is more associated with the attribute words in Å than and the words in is more associated with than Å.
§.§ Text Entailment Based Methods
Quantifying gender inequality relying on the distributed representation of words presents a diffused, bird's-eye view of the larger trends. Also, these methods are known to be data-hungry <cit.>. Data availability often becomes a limiting factor to conducting contrastive studies at different spatio-temporal granularity. In what follows, we present a novel application of text entailment, a natural language inference (NLI) task <cit.> that bypasses the data size requirement and equips us with a finer lens through which we can compare and contrast gender inequality with respect to individual verbs.
An NLI system take a premise 𝒫 and a hypothesis ℋ as input and outputs entailment, contradiction, or semantic irrelevance. For instance, the hypothesis some men are playing a sport is entailed by the premise a soccer game with multiple males playing <cit.>. As one can see, textual entailment is more relaxed than pure logical entailment and it can be viewed as a human reading 𝒫 would infer most likely ℋ is true. This framework has gained traction in several recent social inference tasks that include estimating media stance on policing <cit.>, aggregating social media opinion on election fairness <cit.>, and detecting COVID-19 misinformation <cit.>.
Formally, let NLI(𝒫,ℋ) takes a premise 𝒫 and a hypothesis ℋ as inputs and outputs o ∈{entailment, contradiction, neutral}. Following <cit.>, we define entailment ratio (denoted by ent(𝒟, ℋ)) for given corpus 𝒟 and a hypothesis ℋ, as the fraction of the individual sentences present in 𝒟 that entails ℋ:
ent(𝒟, ℋ) = ∑_𝒫∈𝒟I(NLI(𝒫, ℋ) = entailment)/|𝒟|,
where I is the indicator function. A larger value of ent(𝒟, ℋ) indicates greater support for ℋ in the corpus.
Consider we are interested in learning how often the husband and the wife are accused of torture (physical or emotional) in our corpus. We analyze this research question in the following way. We first construct a sub-corpus 𝒟_torture from the divorce court proceedings consisting of sentences that (1) mention or at least once; and (2) mention as a verb at least once. We next construct two hypotheses – ℋ_,torture and ℋ_,torture – using a and a as victims and perpetrators interchangeably. ℋ_,torture is A woman tortures a man and ℋ_,torture is A man tortures a woman. We next compute the entailment gap defined as
gap(𝒟_torture,torture) =
ent(𝒟_torture,ℋ_,torture) - ent(𝒟_torture,ℋ_,torture)
Effectively, this means we compute the fraction of sentences that entail A woman tortures a man in 𝒟_torture and subtract it from the fraction of sentences that entail A man tortures a woman in 𝒟_torture. An overall positive number indicates that the male has been described as the torturer more often than the female in court proceedings. A negative value would indicate the opposite way. Similar analysis can be extended to other verbs such as , , or .
§ DESIGN CONSIDERATIONS
Adapting the and entailment frameworks to quantify gender inequality in our domain requires careful consideration of several aspects described in what follows.
§.§ Verbs for Target Sets
Traditionally, score is used to quantify gender or racial stereotypes. Majority of the elements present in those attribute sets would be nouns and adjectives (e.g., criminals, terrorists, doctors, police) <cit.> and seldom verbs <cit.>.
We are interested in understanding the action space of the two parties fighting a divorce case; we want to know if the court described that one party tortured or abused the other. Hence, verbs are a natural choice for our target set.
We inspect the list of high-frequency verbs in the corpus and narrow down to the following ten verbs: _unpleasant =
{,
,
,
,
,
,
,
, , }. A small subset of these words are already present in the list of unpleasant stimuli presented in <cit.>. We further compute the average valence score of these words as per the lexicon presented in <cit.>. We find the average valence score of _unpleasant is 2.7, comparable to the average valence score (2.16) of unpleasant stimuli presented in <cit.>.
Divorce being a bitterly fought family situation, we observe a sparse presence of pleasant verbs such as , , or in our corpus. Since infrequent words in the corpus do not have reliable embeddings <cit.>, in contrast with traditional applications of score, we choose the target set to be an empty set.
§.§ The Torturer and the Tortured
The attribute sets Å and as defined in the score represents the identifiers used for the plaintiff and defendant in our data (e.g., Å consisting of , , , and consisting of , , etc.). However, notice that score is agnostic about whether the identifier is the contributor or the receptor of target words. For example, torture does not happen in isolation; it requires a torturer and one who is tortured. Unlike nouns, verbs are typically associated with two entities – the subject and the object. To disambiguate between “the husband tortured the wife” and “the wife tortured the husband”, a word embedding needs to understand this nuance. Otherwise, the embedding is likely to place both the plaintiff and defendant identifiers equidistant to the verb.
To disambiguate these two situations, we run the corpus through the POS tagger <cit.> to find out the subject and object of the sentences and whether the statements are in active or passive voice. Based on this, we classify the subjects and objects as `male perpetrator', `female perpetrator', `male victim', or `female victim', in the sentences that has the target verbs. We replace these four cases with four unique words (denoted by , ,, and , respectively) so that those words do not occur anywhere else in any of the documents. We call this new dataset _replaced.
§ WORD EMBEDDING BASED ANALYSIS
We are interested in two research questions:
RQ 1: How does gender inequality manifest in divorce court proceedings with respect to unpleasant verbs in 𝒳?
RQ 2: Is our careful disambiguation of the torturer and the tortured necessary at all?
In order to answer these two questions, we run two sets of experiments with identical training configurations. First, we run experiments on _replaced using the target and attribute sets as defined in the previous section. We train the word embedding model 10 times and calculate the scores for each of the following two cases: when both genders are (a) perpetrators, i.e., when Å={}, ={}, and (b) victims, i.e., when Å={}, ={}. We use the default parameters for training our FastText <cit.> Skip-gram embedding with the dimension set to 100 for all word-embeddings in this paper. Second, we run a baseline experiment with the original text data without replacing them with the four unique words (_divorce) and use the attribute sets as Å={} and ={}. The number of runs and the embedding method are the same in both experiments. The results are shown in <Ref>.
As already described, a negative score indicates is more associated with the target set as compared to Å. Hence, if we look from the perspective of the victim, we find that women are more associated with the unpleasant verbs than men. In contrast, when viewed from the perpetrator's perspective, a positive score implies that men are more associated with the unpleasant verbs. Hence, our results indicate that in our corpus, women are more often the victims while men are more often the perpetrators.
Our baseline experiments that do not make any distinction between the perpetrator and the victim give a score close to zero indicating near-perfect gender equality. This inaccurate result, while highly surprising from a social science perspective, is not unexpected given how the original framework functions. The two entities (husband and wife) are present around the unpleasant verbs with nearly equal frequency. If the method does not make any distinction between the roles of victim and perpetrator, will give inaccurate results. We thus carefully use the score to elicit the correct
gender bias when applied to legal texts for our social science research question.
§ SOCIETAL INEQUALITY AND MODEL BIAS
Our word embeddings are computed from scratch while our next set of experiments relies on downstream applications built on top of large language models. Large language models (LLMs) are known to have a wide range of biases due to the train data <cit.> and extant literature has examined gender bias in the form of occupational stereotypes present in NLI systems <cit.>. We thus need to disentangle societal inequalities that are potentially reflected in our corpus and model biases that are potentially present in the NLP applications.
Essentially, for a premise/hypothesis pair ⟨𝒫,ℋ⟩, the NLI system estimates the probability P(ℋ |𝒫). However, how LLMs encode the probability P(ℋ) when the hypotheses primarily consists of the two genders (male and female) and a set of verbs is understudied. A thorough investigation first reveals that the masked word prediction probability of several well-known LLMs is sensitive to gender. We next present a measure to quantify gender bias sensitivity of NLI frameworks and present mitigating strategies. Finally, we use a bias-mitigated NLI system on our corpus and report findings.
§.§ Implicit Bias in Agent and Theme in LLMs
Unlike existing literature that primarily target occupational stereotypes to quantify and analyze gender bias <cit.>, we focus on a very basic unit in a sentence – the verbs. Following <cit.>, let in a sentence X verbs Y, X represent the agent and Y represent the theme.
Many verbs imply the relative authority levels between the agent and the theme. For example, in the sentence The football coach instructed the players to play a conservative game, the agent (the football coach) has more authority than the theme (the players). In contrast, the agent has less authority than the theme in the sentence The football coach honored the players' suggestion to play a conservative game. First proposed in <cit.>, the connotation relation of power captures this notion of power differential between an agent and a theme with respect to a given verb.
While the connotation relation of power has been analyzed in the context of gender inequality in movie scripts <cit.> and follow-on research focused on editorial fixes to remove bias <cit.>, little or no literature exists that documents the implicit gender bias present towards the agent and the theme when specific verbs are considered. This research is important and has a broader impact beyond our current social inference task. For instance, if an LLM encodes that it is less likely for a woman to inspire or guide someone than a man, this bias may percolate to downstream tasks leading to erroneous social conclusions when applied to large-scale data for other social inference tasks.
We use cloze tests to evaluate this implicit bias. A brief description of cloze test follows.
Cloze test: When presented with a sentence (or a sentence stem) with a missing word, a cloze task <cit.> is essentially a fill-in-the-blank task. For instance, in the following cloze task: In the , it snows a lot, is a likely completion for the missing word. Word prediction as a test of LLM's language understanding has been explored in <cit.>.
Bias Evaluation Framework:
We describe our proposed testing framework for gender bias. Let _𝑐𝑙𝑜𝑧𝑒 (w, 𝒮) denote the completion probability of the word w with a masked cloze task 𝒮 as input. For a given verb v, we consider the following four cloze tests:
* A [MASK] v a woman (denoted by v_womanAsTheme)
* A [MASK] v a man (denoted by v_manAsTheme)
* A man v a [MASK] (denoted by v_manAsAgent)
* A woman v a [MASK] (denoted by v_womanAsAgent)
In an ideal world where the LLM treats men and women equally, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) and _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) should be equal. However, our preliminary exploratory analysis indicates that is not the case. For example, when v is set to inspire, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) is 0.20 whereas _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) is 0.16. When we set v to guide, the gap widens – _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) is 0.71 whereas _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme) is 0.36.
Again, in an ideal world where the LLM treats men and women equally, _𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsAgent) and _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsAgent) should be equal.
Let 𝒱 denote the set of all verbs listed in <cit.> where the agent has more power than the theme. Our overall measures of implicit bias are:
(a) (1/|𝒱|) ·( ∑_v ∈𝒱 (_𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsTheme) - . . _𝑐𝑙𝑜𝑧𝑒 (woman, v_manAsTheme)) ), and (b) (1/|𝒱|) ·(∑_v ∈𝒱 (_𝑐𝑙𝑜𝑧𝑒 (man, v_womanAsAgent) - . . _𝑐𝑙𝑜𝑧𝑒 (woman, v_womanAsAgent)) ).
Measure (a) quantifies bias_agent. A positive value indicates that the LLM encodes a man being in the position of agent likelier than a woman on expectation.
Measure (b) quantifies bias_theme. A positive value indicates that the LLM encodes a man being in the position of theme likelier than a woman on expectation. We investigate three well-known LLMs for this audit: <cit.>; <cit.>; and <cit.>. We consider 1,222 verbs listed in <cit.>. We also consider verbs in 𝒳_unpleasant for this study.
Table <ref> summarizes our gender bias audit of LLMs with respect to verbs implying more power to the agent than the theme. We first note that for both verb sets, bias_agent is substantially larger than bias_theme. This result indicates that men are considerably more likely to be considered as the agent when women is the theme and the verb implies that the agent has greater power than
the theme. We also note that the completions favor mildly men over women even for the theme, however, the values are closer to 0.
§.§ Implicit Bias in NLI Systems
We describe our approach to quantify model bias in our NLI framework specific to our task.
Consider we modify the sub-corpus 𝒟_torture to 𝒟_torture^flipped where the gender identifiers in each premise sentence are flipped to the equivalent identifier of the opposite gender. For instance, the premise The wife tortured the husband both mentally and physically will be modified as The husband tortured the wife both mentally and physically. Flipping gendered words to test bias through counterfactuals in the context of coreference resolution has been previously explored in <cit.>. We argue that if a premise in 𝒟_torture entails A man tortures a woman, the flipped premise in 𝒟_torture^flipped should entail A woman tortures a man instead in a gender-neutral NLI system. Hence the entailment gap for computed on 𝒟_torture should be equal in magnitude and opposite in polarity as the entailment gap computed on 𝒟_torture^flipped. The NLI system's (ℳ) overall bias score with respect to verbs present in 𝒳_unpleasant is thus computed as
NLI_bias(ℳ, 𝒳_unpleasant) = ∑_v ∈𝒳_unpleasantabs( (gap(𝒟_v, v) + gap(𝒟_v^flipped, v))/|𝒳_unpleasant|.
In simple words, for each verb, we compute the entailment gap (value_1) for the relevant sub-corpus and the flipped sub-corpus (value_2). We subtract value_2 from value_1 and take the absolute value of the sum. The bias score is the average value of this sum across all verbs: a score close to 0 indicates that the NLI system has a minimal bias, whereas larger values indicate greater bias.
Our baseline is an off-the-shelf NLI system from Allen NLP trained using (denoted by ℳ_base). We find that NLI_bias(ℳ_base, 𝒳_unpleasant) is 0.27 [We note that a bias-aware NLI variant from Allen NLP has a better starting point (bias score 0.20) than the base model. However, the bias-aware model exhibits slower convergence than the base model when we conduct our active learning steps as discussed in Section 7.3. With identical experimental setting, after iteration 3, the bias-aware model improves its bias score to 0.133.].
§.§ Bias Mitigation Via Inconsistency Sampling
Active Learning is a powerful and well-established form of supervised machine learning technique <cit.> characterized by the interaction between the learner, aka the classifier, and the teacher (oracle or annotator). Each interaction step consists of the learner requesting the teacher the label of an unlabeled instance sampled using a given sampling strategy and augmenting the data set with the newly acquired label. Next, the classifier is retrained on the augmented data set. This sequential label-requesting and re-training process continues until some halting condition is reached (e.g., exceeded annotation budget or the desired classifier performance). At this point, the algorithm outputs a classifier, and the objective for this classifier is to closely approximate the (unknown) target concept in the future. The key goal of active learning is to reach a strong performance at the cost of fewer labels.
Some of the well-known sampling methods include uncertainty sampling <cit.>, certainty sampling <cit.>, and density-based sampling <cit.>.
Beyond a static strategy, more complex strategies such as adapting strategy selection parameters based on estimated future residual error reduction or combining multiple sampling strategies to balance the label distribution in the procured data set have been explored in <cit.> and <cit.>, respectively.
Inconsistency Sampling. First introduced in Dutta et al. <cit.>, this sampling technique exploits the underlying logical structure of the ⟨ premise, hypothesis ⟩ space. For instance, a premise cannot both entail (or contradict) a given hypothesis and its negation. In our work, we extend this idea and exploit a ⟨ premise, hypothesis ⟩ space richer than Dutta et al. <cit.> for logical inconsistency.
Consider the premise/hypothesis pair Continuously her husband used to harass and torture her everyday/A man tortures a woman. We argue that if this premise entails the hypothesis (which it does), the modified premise/hypothesis pair with replacing every gendered word with the opposite gender – i.e., Continuously his wife used to harass and torture him everyday/A woman tortures a man – should also entail. If not, it signals a logical inconsistency. For each sampling iteration, we add 60 samples giving equal weightage to the verbs present in 𝒳_unpleasant.
Table <ref> summarizes our active learning results. For both models, ℳ_base and ℳ_bias-aware, we conduct three rounds of active learning using inconsistency sampling and stop when the performance improvement becomes indiscernible (≤ 0.01). All annotations are independently conducted by two annotators. Since legal documents are typically written in clear, unambiguous language, we observe a near-perfect agreement (Cohen's κ value 0.96). The remaining disagreements are resolved through a post-annotation adjudication step. Table <ref> indicates that with subsequent active learning steps, our NLI system exhibits lesser bias. Given that the maximum possible bias score is 2, we achieve substantial improvement in mitigating the bias.
Now that we are more confident that our model inferences are less sensitive to gender, we evaluate the societal bias present in our corpus.
Figure <ref> summarizes our text entailment results. Barring , for all other verbs, men are identified as perpetrators more often than women. We further note that verbs that indicate physical abuse, such as and , particularly stand out with larger values. The average entailment gap for verbs unambiguously indicating physical harm – , , , , and – is much higher (0.41) than verbs that may or may not indicate physical harm (0.19) such as , , , , and . A manual inspection of randomly sampled 200 ⟨ premise, hypothesis⟩ pairs aligns with our automated method's overall findings.
§ DISCUSSIONS AND LIMITATIONS
In this paper, we present the first-ever computational analysis (to our knowledge) of gender inequality in divorce court proceedings in India. Based on the documented allegations of parties involved in the divorce, our analyses indicate a striking gender inequality as described in these public records. While documented evidence of marital distress in India exists in social science literature, how such factors play out in divorce has limited understanding. Our study sheds light on a vulnerable and vulnerable and practically invisible community in India.
Methodologically, we identify and address several gaps and limitations of existing NLP techniques to quantify gender inequality. We believe our finding specific to legal text is new, and our method to address it is simple, effective, and intuitive. Casting the problem of quantifying gender inequality as a text entailment task is also new. Our results on text entailment results suggest that NLI can be a viable tool to computational social science researchers to analyze similar research questions (e.g., who gets the child custody can be estimated with hypotheses the husband gets the custody of the child and the wife gets the custody of the child). Moreover, our bias mitigation strategy exploiting a novel inconsistency sampling technique using counterfactuals holds promise.
Our work has the following limitations.
Sentence level processing: An important point to keep in mind, however, is that our analyses operate at the sentence level. If in a court proceeding, a sentence records that the plaintiff accuses the defendant of wrongdoing which the defendant denies in a subsequent sentence, how these two contradicting claims are resolved in the court cannot be inferred without language models that can handle document-level contexts. We believe our research will open the gates for investigation with newer-age LLMs that can handle broader contexts.
Archival limitation: The sparse presence of the North-Eastern region in our dataset is most likely due to archival limitation as some of these states record the highest rate of divorce <cit.>. Our study is also limited by the overall archival extent of .
Economic independence: Some of the court proceedings mention the litigants' occupations. We annotated randomly 100 sampled occupations for women.
While an overwhelming majority of the sampled occupations are homemakers, compared to World Bank Data on labor force participation of women in India (23%), 32% of the women are working women in our sampled occupations.
Economic independence and divorce merit a deeper exploration.
Out-of-court settlements, separation, abandonment: Finally, not all unhappy marriages end up in divorce and reach court for dissolution. Many out-of-court settlements happen. As documented in <cit.>, the number of separated women in 2011 is almost three times the number of divorced women. Since divorce is still looked at as a social stigma <cit.> and family institutions are highly valued in India, there could be many women who continue with their dysfunctional marriages while unhappy. The court does not know their stories.
§ ETHICAL STATEMENT
We work with public court records. Prior studies exist on Indian court proceedings <cit.>. We conduct aggregate analysis refraining from presenting any personally identifiable information in the paper. Hence, we do not see any ethical concern. Rather, we believe our findings and methods can be valuable to policymakers and social scientists.
A study on binary gender inequality runs the risk of oversimplifying gender, which we acknowledge lies on a spectrum. Same-sex marriage is yet not legal in India. Further nuances will be needed to extend our work to other cultures allowing same-sex marriages. We are also sensitive to previous studies that point out the potential harms of the erasure of gender and sexual minorities <cit.>.
10
jacob2016marriage
Suraj Jacob and Sreeparna Chattopadhyay.
Marriage dissolution in india: Evidence from census 2011.
Economic and Political Weekly, 51(33):25–27, 2016.
dommaraju2016divorce
Premchand Dommaraju.
Divorce and separation in india.
Population and Development Review, pages 195–223, 2016.
goode1962marital
William J. Goode.
Marital satisfaction and instability-a cross-cultural class analysis
of divorce rates.
International social science journal, 14(3):507–526, 1962.
mani2017study
A Santhosh Mani and Bhanu Priya.
A study on the recent trends of divorce in india.
ZENITH International Journal of Multidisciplinary Research,
7(8):25–32, 2017.
belliappa2013gender
Jyothsna Belliappa.
Gender, class and reflexive modernity in India.
Springer, 2013.
vasudevan2015causes
Bindhu Vasudevan, Devi M. Geetha, Anitha Bhaskar, Binu Areekal, Anupa Lucas,
et al.
Causes of divorce: a descriptive study from central kerala.
Journal of evolution of medical and dental sciences,
4(20):3418–3427, 2015.
bhattacharya2019comparative
Paheli Bhattacharya, Kaustubh Hiware, Subham Rajgaria, Nilay Pochhi,
Kripabandhu Ghosh, and Saptarshi Ghosh.
A comparative study of summarization algorithms applied to legal case
judgments.
In ECIR, pages 413–428. Springer, 2019.
kalia2022classifying
Arvind Kalia, Naveen Kumar, and Nischay Namdev.
Classifying case facts and predicting legal decisions of the indian
central information commission: a natural language processing approach.
In Advances in Deep Learning, Artificial Intelligence and
Robotics, pages 35–45. Springer, 2022.
ash2021group
Elliott Ash, Sam Asher, Aditi Bhowmick, Sandeep Bhupatiraju, Daniel Chen,
Tanaya Devi, Christoph Goessmann, Paul Novosad, and Bilal Siddiqi.
In-group bias in the Indian judiciary: Evidence from 5 million
criminal cases.
Technical report, Working paper, August, 2021.
kumar2020sexual
Anil Kumar.
Sexual harassment of women at workplace: How far is indian law
protective?
International Academic Journal of Law, 1(1):35–39, 2020.
madaan2018analyze
Nishtha Madaan, Sameep Mehta, Taneea Agrawaal, Vrinda Malhotra, Aditi Aggarwal,
Yatin Gupta, and Mayank Saxena.
Analyze, detect and remove gender stereotyping from bollywood movies.
In MAccT, pages 92–105. PMLR, 2018.
DBLP:conf/acl-trac/BhattacharyaSKB20
Shiladitya Bhattacharya, Siddharth Singh, Ritesh Kumar, Akanksha Bansal, Akash
Bhagat, Yogesh Dawer, Bornini Lahiri, and Atul Kr. Ojha.
Developing a multilingual annotated corpus of misogyny and
aggression.
In Proceedings of the Second Workshop on Trolling, Aggression
and Cyberbullying, pages 158–168, 2020.
khadilkar2021gender
Kunal Khadilkar, Ashiqur R. KhudaBukhsh, and Tom M. Mitchell.
Gender bias, social bias, and representation in Bollywood and
Hollywood.
Patterns, 3(4):100486, 2022.
rao1973dowry
R. Jaganmohan Rao.
Dowry system in India — a socio-legal approach to the problem.
Journal of the Indian Law Institute, 15(4):617–625, 1973.
ahmad2008dowry
Nehaluddin Ahmad.
Dowry deaths (bride burning) in India and abetment of suicide: a
socio-legal appraisal.
JE Asia & Int'l L., 1:275, 2008.
sonawat2001understanding
Reeta Sonawat.
Understanding families in india: A reflection of societal changes.
Psicologia: Teoria e Pesquisa, 17:177–186, 2001.
caliskan2017semantics
Aylin Caliskan, Joanna J. Bryson, and Arvind Narayanan.
Semantics derived automatically from language corpora contain
human-like biases.
Science, 356(6334):183–186, 2017.
maccartney2008modeling
Bill MacCartney and Christopher D. Manning.
Modeling semantic containment and exclusion in natural language
inference.
In COLING 2008, pages 521–528, 2008.
mandal2021unsupervised
Arpan Mandal, Kripabandhu Ghosh, Saptarshi Ghosh, and Sekhar Mandal.
Unsupervised approaches for measuring textual similarity between
legal court case reports.
Artificial Intelligence and Law, 29(3):417–451, 2021.
HaltermanKSO21Policing
Andrew Halterman, Katherine A. Keith, Sheikh Muhammad Sarwar, and Brendan
O'Connor.
Corpus-Level Evaluation for Event QA: The IndiaPoliceEvents Corpus
Covering the 2002 Gujarat Violence.
In ACL/IJCNLP 2021, volume ACL/IJCNLP 2021 of Findings
of ACL, pages 4240–4253, 2021.
DuttaPolice
Sujan Dutta, Beibei Li, Daniel S. Nagin, and Ashiqur R. KhudaBukhsh.
A murder and protests, the capitol riot, and the chauvin trial:
Estimating disparate news media stance.
In IJCAI, pages 5059–5065, 2022.
kaur2019gap
Harjnder Kaur-Aulja, Farzana Shain, and Alison Lilley.
A Gap Exposed: What is Known About Sikh Victims of Domestic Violence
Abuse (DVA) and Their Mental Health?
European Journal of Mental Health, 14(1):179–189, 2019.
mistry1982personal
PJ Mistry.
Personal names: Their structure, variation, and grammar in
Gujarati.
South Asian Review, 6(3):174–190, 1982.
ghansham2002female
Devaki Monani Ghansham.
Female foeticide and the dowry system in India.
In Townsville International Women’s Conference, James Cook
Univ., Australia, 2002.
banerjee2014dowry
Priya R. Banerjee.
Dowry in 21st-century India: the sociocultural face of
exploitation.
Trauma, Violence, & Abuse, 15(1):34–40, 2014.
rastogi2006dowry
Mudita Rastogi and Paul Therly.
Dowry and its link to violence against women in India: Feminist
psychological perspectives.
Trauma, Violence, & Abuse, 7(1):66–77, 2006.
carpenter2016protecting
Deepshikha Carpenter and Polly Vauquline.
Protecting Women from Domestic Violence in Assam, India? Evaluating
Section 498-A, The Indian Penal Code (IPC), 1983 vs the Protection of Women
from Domestic Violence Act (PWDVA), 2005.
Journal of International Women's Studies, 18(1):133–144, 2016.
babu2011dowry
Gopalan Retheesh Babu and Bontha Veerraju Babu.
Dowry deaths: a neglected public health issue in India.
International health, 3(1):35–43, 2011.
jakimow2013everyone
Tanya Jakimow.
‘everyone must give’: Explaining the spread and persistence of
bridegroom price among the poor in rural telangana, india.
Journal of Asian and African Studies, 48(2):180–194, 2013.
DBLP:conf/nips/MikolovSCCD13
Tomás Mikolov, Ilya Sutskever, Kai Chen, Gregory S. Corrado, and Jeffrey
Dean.
Distributed representations of words and phrases and their
compositionality.
In Advances in Neural Information Processing Systems, pages
3111–3119, 2013.
dagan2005pascal
Ido Dagan, Oren Glickman, and Bernardo Magnini.
The pascal recognising textual entailment challenge.
In Machine Learning Challenges Workshop, pages 177–190.
Springer, 2005.
bowman-etal-2015-large
Samuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning.
A large annotated corpus for learning natural language inference.
In EMNLP, 2015.
halterman-etal-2021-corpus
Andrew Halterman, Katherine Keith, Sheikh Sarwar, and Brendan O'Connor.
Corpus-level evaluation for event QA: The IndiaPoliceEvents
corpus covering the 2002 Gujarat violence.
In ACL-IJCNLP, pages 4240–4253, 2021.
Capitol2022
Ashiqur R. KhudaBukhsh, Rupak Sarkar, Mark S. Kamlet, and Tom M. Mitchell.
Fringe news networks: Dynamics of US news viewership following the
2020 presidential election.
In ACM WebScience, pages 269–278, 2022.
hossain-etal-2020-covidlies
Tamanna Hossain, Robert L. Logan IV, Arjuna Ugarte, Yoshitomo Matsubara, Sean
Young, and Sameer Singh.
COVIDLies: Detecting COVID-19 misinformation on social media.
In Proceedings of the 1st Workshop on NLP for COVID-19 (Part
2) at EMNLP 2020, December 2020.
DBLP:conf/naacl/ManziniLBT19
Thomas Manzini, Yao Chong Lim, Alan W. Black, and Yulia Tsvetkov.
Black is to criminal as caucasian is to police: Detecting and
removing multiclass bias in word embeddings.
In NAACL-HLT, pages 615–621, 2019.
greenwald2014malice
Anthony G. Greenwald and Thomas F. Pettigrew.
With malice toward none and charity for some: ingroup favoritism
enables discrimination.
American Psychologist, 69(7):669, 2014.
DBLP:conf/acl/HoyleWWAC19
Alexander Hoyle, Lawrence Wolf-Sonkin, Hanna M. Wallach, Isabelle Augenstein,
and Ryan Cotterell.
Unsupervised discovery of gendered language through latent-variable
modeling.
In ACL 2019, pages 1706–1716, 2019.
warriner2013norms
Amy Beth Warriner, Victor Kuperman, and Marc Brysbaert.
Norms of valence, arousal, and dominance for 13,915 english lemmas.
Behavior research methods, 45(4):1191–1207, 2013.
DBLP:conf/iclr/LampleCRDJ18
Guillaume Lample, Alexis Conneau, Marc'Aurelio Ranzato, Ludovic Denoyer, and
Hervé Jégou.
Word translation without parallel data.
In ICLR. OpenReview.net, 2018.
qi2020stanza
Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, and Christopher D. Manning.
Stanza: A Python natural language processing toolkit for many human
languages.
In ACL: System Demonstrations, 2020.
bojanowski2017enriching
Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov.
Enriching word vectors with subword information.
TACL, 5:135–146, 2017.
bender2021dangers
Emily M. Bender, Timnit Gebru, Angelina McMillan-Major, and Shmargaret
Shmitchell.
On the Dangers of Stochastic Parrots: Can Language Models Be Too
Big?
In ACM FaccT, pages 610–623, 2021.
rudinger2017social
Rachel Rudinger, Chandler May, and Benjamin Van Durme.
Social bias in elicited natural language inferences.
In Proceedings of the First ACL Workshop on Ethics in Natural
Language Processing, pages 74–79, 2017.
DBLP:journals/corr/abs-2105-05541
Shanya Sharma, Manan Dey, and Koustuv Sinha.
Evaluating gender bias in natural language inference.
CoRR, abs/2105.05541, 2021.
kumar2020nurse
Vaibhav Kumar, Tenzin Singhay Bhotia, and Tanmoy Chakraborty.
Nurse is closer to woman than surgeon? mitigating gender-biased
proximities in word embeddings.
TACL, 8:486–503, 2020.
SAPPowerAgency
Maarten Sap, Marcella Cindy Prasettio, Ari Holtzman, Hannah Rashkin, and Yejin
Choi.
Connotation frames of power and agency in modern films.
In EMNLP 2017, pages 2329–2334, 2017.
PowerTransformer
Xinyao Ma, Maarten Sap, Hannah Rashkin, and Yejin Choi.
Powertransformer: Unsupervised controllable revision for biased
language correction.
In EMNLP 2020, pages 7426–7441, 2020.
taylor1953cloze
Wilson L. Taylor.
“Cloze procedure”: A new tool for measuring readability.
Journalism quarterly, 30(4):415–433, 1953.
paperno-etal-2016-lambada
Denis Paperno, Germán Kruszewski, Angeliki Lazaridou, Ngoc Quan Pham,
Raffaella Bernardi, Sandro Pezzelle, Marco Baroni, Gemma Boleda, and Raquel
Fernández.
The LAMBADA dataset: Word prediction requiring a broad discourse
context.
In ACL 2016, pages 1525–1534, 2016.
ettinger-2020-bert
Allyson Ettinger.
What BERT is not: Lessons from a new suite of psycholinguistic
diagnostics for language models.
TACL, 8:34–48, 2020.
devlin2018bert
Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.
Bert: Pre-training of deep bidirectional transformers for language
understanding.
arXiv preprint arXiv:1810.04805, 2018.
Roberta
Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer
Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.
Roberta: A robustly optimized BERT pretraining approach.
CoRR, abs/1907.11692, 2019.
Megatron
Mohammad Shoeybi, Mostofa Patwary, Raul Puri, Patrick LeGresley, Jared Casper,
and Bryan Catanzaro.
Megatron-lm: Training multi-billion parameter language models using
model parallelism.
CoRR, abs/1909.08053, 2019.
lu2020gender
Kaiji Lu, Piotr Mardziel, Fangjing Wu, Preetam Amancharla, and Anupam Datta.
Gender bias in neural natural language processing.
In Logic, Language, and Security, pages 189–202. Springer,
2020.
settles2009active
Burr Settles.
Active learning literature survey.
Computer Sciences Technical Report 1648, University of
Wisconsin–Madison, 2009.
sindhwani2009uncertainty
Vikas Sindhwani, Prem Melville, and Richard D. Lawrence.
Uncertainty sampling and transductive experimental design for active
dual supervision.
In ICML, pages 953–960. ACM, 2009.
nguyen2004active
Hieu T. Nguyen and Arnold Smeulders.
Active learning using pre-clustering.
In ICML, page 79, 2004.
donmez2007dual
Pinar Donmez, Jaime G Carbonell, and Paul N Bennett.
Dual strategy active learning.
In Machine Learning: ECML 2007, pages 116–127. Springer, 2007.
palakodety2020voice
Shriphani Palakodety, Ashiqur R. KhudaBukhsh, and Jaime G. Carbonell.
Voice for the voiceless: Active sampling to detect comments
supporting the Rohingyas.
In AAAI 2020, volume 34-01, pages 454–462, 2020.
ArjunErasurePaper
Sunipa Dev, Masoud Monajatipoor, Anaelia Ovalle, Arjun Subramonian, Jeff M.
Phillips, and Kai-Wei Chang.
Harms of gender exclusivity and challenges in non-binary
representation in language technologies.
In EMNLP, pages 1968–1994, 2021.
|
http://arxiv.org/abs/2307.05205v1 | 20230711122030 | Entanglement and entropy in multipartite systems: a useful approach | [
"A. Bernal",
"J. A. Casas",
"J. M. Moreno"
] | quant-ph | [
"quant-ph",
"hep-th"
] |
1.1
|
http://arxiv.org/abs/2307.03892v1 | 20230708035958 | Embedding Mental Health Discourse for Community Recommendation | [
"Hy Dang",
"Bang Nguyen",
"Noah Ziems",
"Meng Jiang"
] | cs.IR | [
"cs.IR",
"cs.CL"
] |
Feature selection simultaneously preserving both class and cluster structures
Suchismita Dasmycorrespondingauthor and Nikhil R. Pal
August 12, 2023
=============================================================================
*These authors contributed equally to this work
Our paper investigates the use of discourse embedding techniques to develop a community recommendation system that focuses on mental health support groups on social media. Social media platforms provide a means for users to anonymously connect with communities that cater to their specific interests. However, with the vast number of online communities available, users may face difficulties in identifying relevant groups to address their mental health concerns. To address this challenge, we explore the integration of discourse information from various subreddit communities using embedding techniques to develop an effective recommendation system. Our approach involves the use of content-based and collaborative filtering techniques to enhance the performance of the recommendation system. Our findings indicate that the proposed approach outperforms the use of each technique separately and provides interpretability in the recommendation process.
§ INTRODUCTION
The rise of social media as a platform has allowed people all over the world to connect and communicate with one another.
Further, these communities that exist online are able to keep their members anonymous from one another, allowing new communities to form which would have a hard time existing without anonymity.
Specifically, this new and robust anonymity has allowed an explosion of online communities with a focus on giving each other advice on health issues.
While being involved in seeking peer support in a community with people that have experienced similar issues can provide a significant positive impact on someone's ability to navigate their personal problems <cit.>, finding communities with relevant discourse is not trivial.
Often, the platforms which host these communities have a very large quantity of them.
There are over 100,000 different communities on Reddit alone.
Further, some communities are not easily found due to their inherently anonymous nature, so the only way a user can decide if they fit within the community is by spending time reading through the discourse happening within the community.
For these reasons, new users seeking others who have experienced similar situations may have a very hard time finding communities that would help them the most, even if they are familiar with the platform which hosts the communities.
Recently, embedding long sequences of text has received lots of interest both from the research community and from practitioners.
A number of studies have shown embeddings can be useful for measuring the similarity both between document pairs and between question-document pairs <cit.>, allowing for retrieval of the most similar documents given a new question or document.
However, little work has been done investigating how the discourse within a community, which represents the meaning of that community, can be represented in a single embedding. The discourse of a community in this context can be all users' posts in that specific community or represented community's description.
This poses a unique challenge as discourse within these communities is often in the form of threads that, unlike documents, are not naturally represented as a single block of text.
The goal of this work is to develop a system to recommend support groups to social media users who seek help regarding mental health issues using embeddings to represent the communities and their discourse.
Specifically, we aim to leverage the text of a given user's posts along with the description and posts in each subreddit community to help recommend support groups that the user could consider joining.
Our main research questions are as follows:
* In representing online communities through discourse embeddings, what type of information can be used?
* To what degree do these representations improve the accuracy of predicting users' behaviors regarding their involvement in sharing experiences within groups or communities?
* Do different discourse embedding methods change the prediction capacity of our community recommendation model?
In exploring these research questions, we propose a hybrid recommendation approach that leverages both content-based and collaborative filtering to construct our community recommendation model. As shown in Fig. <ref>, the content-based filtering component investigates different methods of embedding discourse within a community to recommend similar communities to users. It is then combined with a matrix factorization model that learns user engagement behavior in a community to improve recommendation decisions. Utilizing users' past interactions as well as text-based information about the communities, we show that our model achieves promising accuracy while offering interpretability.
§ RELATED WORK
There are a number of studies related to our work.
<cit.> and <cit.> constructed discourse embeddings to find relations between short text segments.
While the two studies were similar in concept, they focused on short text segments where this work instead focused on constructing discourse embeddings for entire social media communities.
<cit.> showed NLP techniques could be used with electronic health records to predict mental health crises 4 weeks in advance.
While online communities were no replacement for professional medical help, this suggested many who had looming mental health problems seek help before a crisis.
<cit.> experimented on the same dataset we used with Natural Language Processing techniques such as TF-IDF and sentiment analysis to understand the effects of COVID-19 on mental health.
Although working on the same dataset, our work studies a different task: to recommend mental health-related support community to Reddit users.
<cit.> adopted a similar approach to ours in content-based filtering for recommendation.
Specifically, they mapped a Wikipedia page to each item and generate its corresponding vector representation using three feature-extraction methods - Latent Semantic Indexing, Random Indexing, and Word2Vec.
We extended this method by exploring more recent representations of text such as BERT <cit.> and OpenAI embeddings.
<cit.> recommended threads in health forums based on the topics of interest of the users.
Specifically, self-reported medical conditions and symptoms of treatments were used as additional information to help improve thread recommendations <cit.>.
While our work is also situated in the health domain, we are interested in recommending a broader support group to users rather than a specific thread.
<cit.> used sentiment and other features to automatically evaluate dialog, showing NLP techniques could be used to evaluate quality of discourse.
In doing so, they leveraged weak supervision to train a model on a large dataset without needing quality annotations.
§ PROBLEM DEFINITION
Suppose we have a Reddit's "who-posts-to-what" graph, which is denoted by G = (U, V, E) where U is the set of users, V is the set of subreddit communities, and E, a subset of U× V, is the set of edges.
The number of user nodes is m = |U| and the number of subreddit communities is n = |V|. So, U = {(u_1, P_1), (u_2, P_2) , ..., (u_m, P_m)} where P_i is the set of posts by user u_i and V = {(v_1, P^'_1), ..., (v_n, P^'_n)} where P^'_j is the set of all posts in subreddit v_j.
If a user u_i posts to subreddit v_j, there is an edge that goes from u_i to v_j, which is denoted by e_ij = e(u_i, v_j).
The problem is that given G, predict if e_ij = e(u_i, v_j) exists.
In other words, will user u_i post to subreddit v_j?
§ METHODOLOGY
Figure <ref> illustrates our recommendation pipeline, which adopts a hybrid approach by incorporating both content-based filtering (CBF) and collaborative filtering, specifically matrix factorization (MF) strategies. The CBF model recommends new subreddits based on the average of a user's previous interactions, weighted by how similar the previous subreddits are to the new ones. Meanwhile, users and subreddits are represented in a k-dimensional joint latent space in the MF model. The distance between users and subreddits in this latent space is used to provide recommendations for new subreddits. The predictions from these two components are linearly combined to obtain the final recommendation of subreddits to users.
The collaborative filtering component of our solution leverages nonnegative matrix factorization to represent our users and subreddits in lower-dimensional latent space. In this sense, we redefine the adjacency matrix 𝐀 in our problem definition so that it works with nonnegative factorization. More specifically, users' past interactions with items are represented by the adjacency matrix 𝐀∈{5, 1, 0}^m × n. A_ij = 5 if the user u_i has posted to subreddit j, A_ij = 1 if the user u_i has NOT posted to the subreddit v_j, and A_ij = 0 is the missing connection that needs predicting. Given this adjacency matrix 𝐀, the task is to predict the missing elements A_ij = 0. In the following sections, we elaborate on each component of our recommendation model and then discuss how they are combined to obtain our final solution.
§.§ Content-based Filtering
In recommending items to users based on their past interactions and preferences, content-based filtering methods represent each item with a feature vector, which can then be utilized to measure the similarity between items <cit.>. If an item is similar to another item with which a user interacted in the past, it will be recommended to that same user. Thus, in addition to the adjacency matrix 𝐀, we utilize another matrix 𝐂 of size n× n, where 𝐂_ab is the similarity between the embeddings for two subreddits with embedding vectors 𝐚 and 𝐛.
In this paper, we use cosine similarity as the similarity measure:
𝐂_ab = 𝐚·𝐛𝐚𝐛,
To predict the value of the missing element where A_ij = 0 (whether user u_i will post to subreddit v_j), we compute the average of user u_i's past interactions (which subreddits user u_i posted and did not post to), weighted by the similarity of these subreddits to subreddit v_j.
Mathematically,
A^'_ij = ∑_k=1^n A_ik C_kj/∑_k=1^n C_kj.
We can generalize the above formula to obtain the new predicted adjacency matrix using matrix-level operations:
𝐀^(CBF) = (𝐀𝐂) ⊙𝐃,
where
* 𝐃 = 1. / (𝐈·𝐂) (element-wise),
* 𝐈 is an indicator matrix such that I_ij = 1 if A_ij≠ 0, otherwise I_ij = 0,
* and ⊙ is the Hadamard product.
§.§.§ Representing Subreddit Discourse with Description and Posts
It is helpful to consider the specific domain of the application to represent each item as an embedding. In the context of our subreddit recommendation problem, we take advantage of two types of text-based information about a subreddit to construct the similarity matrix: (1) the posts within the subreddit itself and (2) the general description about the reddit provided by the subreddit moderators.
We then use a feature extraction method to obtain two embeddings of a subreddit, one based on its description and the other based on its posts. As a subreddit contains many posts, each of which has a different embedding given the same feature-extraction method, we take the average of the embeddings across all posts within a subreddit to obtain one embedding for the subreddit.
§.§.§ Feature Extraction
In this paper, we consider three feature-extraction methods: Term Frequency-Inverse Document Frequency (TF-IDF), Bidirectional Encoder Representations from Transformers (BERT) <cit.>, and OpenAI.[OpenAI API Embeddings: <https://platform.openai.com/docs/guides/embeddings>]
TF-IDF: The TF-IDF algorithm represents a document as a vector, each element of which corresponds to the TF-IDF score of a word in that document.
The TF-IDF score for each word in the document is dictated by (1) the frequency of the word in the document <cit.>, and (2) the rarity of the word in the entire text corpus <cit.>.
That is, a term is important to a document if it occurs frequently in the document but rarely in the corpus.
We use the implementation from scikit-learn <cit.> to obtain the TF-IDF representations of our subreddits.
BERT: We employ BERT to generate sentence embeddings as another feature extraction technique <cit.>.
BERT takes a sentence as input and generates a fixed-length vector representation of the sentence.
This representation is meant to capture the syntactic and semantic meaning of the input sentence in a way that can be used for various natural language processing tasks, such as sentence classification or semantic similarity comparison.
In the context of our problem, we can treat each subreddit description or each post as a sentence and feed it to a pre-trained BERT model to generate the embeddings that represent the subreddit. Long posts are truncated to fit within the context limits of pre-trained models. We experiment with 4 different variations of BERT embeddings:
* BERT base and large <cit.>
* Sentence-BERT, or SBERT <cit.>
* BERTweet <cit.>
OpenAI: Similar to BERT embeddings, OpenAI embeddings take in a string of text and output an embedding that represents the semantic meaning of the text as a dense vector.
To do this, the input string is first converted into a sequence of tokens.
The tokens are then fed to a Large Language Model (LLM), which generates a single embedding vector of fixed size.
OpenAI's text-embedding-ada-002 can take strings of up to 8191 tokens and returns a vector with 1536 dimensions.
§.§ Nonnegative Matrix Factorization for Collaborative Filtering
Matrix factorization (MF) approaches map users and items (subreddits in this case) to a joint latent factor space of a lower dimension k <cit.>. The goal of this method is to recommend to a user the subreddits that are close to them in the latent space. More formally, MF involves the construction of user matrix 𝐏 of dimension m× k and subreddit matrix 𝐐 of dimension n× k. In this sense, the resulting term, 𝐩_i^⊤𝐪_j, captures user u_i's interest in item v_j’s characteristics, thereby approximating user u_i's rating of item v_j, or denoted by A_ij.
This modeling approach learns the values in 𝐏 and 𝐐 through the optimization of the loss fuction
min_𝐏,𝐐∑_A_ij∈𝐀 ( A_ij - 𝐩_i^⊤𝐪_j )^2 + λ ( 𝐩_i ^2 + 𝐪_j ^2).
Matrix factorization offers the flexibility of accounting for various data and domain-specific biases that may have an effect on the interaction between user u_i and subreddit v_j. In this paper, we consider three types of biases: global average μ, user bias b_i^(p), and subreddit bias b_j^(q). The updated loss function is given by:
min_𝐏,𝐐∑_A_ij∈𝐀 ( A_ij - μ - b_i^(p) - b_j^(q) - 𝐩_i^⊤𝐪_j )^2 +
λ ( 𝐩_i ^2 + 𝐪_j ^2 + b_i^(p)^2 + b_j^(q)^2).
After optimization, each element in the new predicted adjacency matrix 𝐀^𝐌𝐅 is given by:
𝐀^(MF)_ij = 𝐩_i^⊤𝐪_j + μ + b_i + b_j
§.§ Final Model: Hybrid Approach
Our main model leverages insights from both content-based filtering and matrix factorization by taking a linear combination of their predicted adjacency matrix. Specifically, the new adjacency matrix is given by:
𝐀^(MF+CBF) = β𝐀^(CBF) + (1 - β) 𝐀^(MF),
where β is a hyperparameter that controls how much the CBF model (vs MF model) contributes to the final prediction.
§ DATA AND EXPERIMENTAL SETUP
For the experimental setup, we use the data from <cit.> working on Reddit platforms in mental health domains, particularly health anxiety.
§.§ Data Description
The dataset is collected from 28 mental health and non-mental health subreddits.
The dataset is suitable for studying how subreddits and social media platforms correlated with individuals' mental health and behavior.
The original data comprises 952,110 Reddit posts from 770,176 unique users across 28 subreddit communities, which include 15 mental health support groups, 2 broad mental health subreddits, and 11 non-mental health subreddits. We also manually collect descriptions of the 28 subreddits and use that information along with the posts to conduct the content similarity matrix.
§.§ Data Preprocessing
Although the original dataset has a large number of unique users, the majority of them only contribute posts to one or two different communities. This presents a challenge when evaluating our specific task. As our objective is to examine users' behavior over time and provide recommendations for engaging in suitable subreddits, we have implemented a filter to exclude users who post to fewer than three subreddits. After filtering, the remaining users and posts are 16,801 and 69,004, respectively, while the number of subreddits remains to be 28.
We also seek to understand the distribution of interactions between users and different subreddits. The detailed distribution of post frequency across subreddits is visualized in Figure <ref>.
§.§ Experimental Setup
§.§.§ Data Splits
To construct our data splits, for each user in our dataset, we choose the most recent subreddit that the user first posted to as the test example.
For example, if the user post history is [subreddit1, subreddit2, subreddit3, subreddit1, subreddit2], then subredddit3 will be used as the test example.
For each positive training example, we pair it with a negative example randomly sampled from the list of subreddits where the user has not posted to.
§.§.§ Evaluation Metrics
In assessing the performance of our recommendation method and the baseline, we use the following evaluation metrics: Recall@K and Mean Reciprocal Rank (MRR).
§.§ Results
Table <ref> presents the performance of our hybrid recommendation system as well as its individual components (MF or CBF). For CBF, we report its performance on different types of embeddings constructed using different information (posts or description) and different feature extraction methods (TF-IDF, BERT, or OpenAI). Figure <ref> visualizes the results of exemplary models in a diagram for better analysis using Recall@K.
According to Table <ref>, all variants of our recommendation method outperform the random predictor. Among all the variants, the hybrid solution using the content similarity matrix generated from OpenAI embeddings achieves the highest performance in MRR (0.4244) and average Recall@K.
For CBF, operating a feature-extraction method on subreddit posts results in higher performance than operating the same method on description. For example, the MRR for CBF - BERT base is 0.3140 when using posts and 0.3024 when using description. It can also be observed that given the same information (either posts or information), deep-learning-based feature extraction methods like OpenAI and BERT bring about better performance for CBF than TF-IDF.
As our recommendation model combines both MF and CBF, we investigate the effect of hyperparameter β, which dictates how much CBF contributes to the final prediction. Figure <ref> illustrates the performance of the hybrid models on varying β. When β = 0, the hybrid model's performance is the same as that of MF. When β = 1, the hybrid model's performance is the same as that of CBF. It can be seen from the peak of these curves that this way of linearly combining MF and CBF brings about significant improvement in MRR.
§.§ Case Studies
We perform a series of case studies to understand why certain information and methods are more helpful than others in recommending subreddits to users. We present our findings by comparing the behavior of the following models: (1) CBF models using TF-IDF and OpenAI Embedding on Subreddit Descriptions, (2) CBF models using OpenAI Embeddings on Subreddit Descriptions and Posts, and (3) MF model and Hybrid model.
§.§.§ CBF models using TF-IDF and OpenAI Embedding on Subreddit Descriptions
The objective of the first case study is to investigate the impact of different types of embedding methods on the performance of recommendations. To achieve this, we employ TF-IDF and OpenAI Embedding approaches to analyze subreddit descriptions and compare their predictions using content-based filtering (CBF) approaches, as illustrated in Figure <ref>. Specifically, we consider User A's historically interacted subreddits, which relate to depression, loneliness, and anxiety, respectively, with the ground truth of socialanxiety. For CBF models, the content similarity C between historically interacted and ground truth subreddits is crucial for accurate predictions. Hence, we evaluate the similarity scores between them. According to the result, the OpenAI Embedding technique outperforms TF-IDF in learning the representation of subreddits.
Based on the analysis of content similarity matrices of the two approaches, we observe that TF-IDF has low similarity scores among subreddits due to its bag-of-words (BOW) approach, which fails to capture semantic relationships in short texts <cit.>, such as subreddit descriptions. In contrast, OpenAI Embeddings, which can capture semantic meanings, performs better for encoding the meanings of subreddit descriptions for recommendation tasks.
§.§.§ CBF models using OpenAI Embeddings on Subreddit Descriptions and Posts
The second case study aims to investigate the impact of different types of information on the performance and recommendations of CBF models. To achieve this goal, we evaluate OpenAI Embeddings approaches on two types of information, subreddit descriptions, and posts. Figure <ref> illustrates the predictions using CBF approaches utilizing OpenAI Embeddings on posts and descriptions. Specifically, we examine User B's historical posts, which are in depression and personalfinance, and the ground truth label is legaladvice. To understand the behavior of CBF on these two types of information, we analyze the similarities between historical subreddit interactions of User B and how the ground truth label is correlated with these subreddits. Our analysis shows that using OpenAI Embeddings on subreddit posts can capture strong relationships between personalfinance and legaladvice, where many legaladvice posts are related to financial information. However, when only using subreddit descriptions of legaladvice, which is "A place to ask simple legal questions, and to have legal concepts explained.", the model fails to capture this relationship.
Furthermore, as shown in Table <ref>, the use of subreddit posts as representations for communities generally exhibits higher performance across most metrics when compared to using community descriptions. The reason is that subreddit descriptions contain less information than posts describing only the general purpose of the subreddit. In contrast, using subreddit posts can accurately learn the representations of the subreddits. Therefore, among the two types of information, using subreddit posts to represent subreddits helps models achieve better performance.
§.§.§ MF vs MF + CBF model using OpenAI Embeddings on Subreddit Discourses
The objective of the third study is to investigate the performance improvement achieved by combining MF and CBF. Specifically, we aim to explore how the use of discourse embeddings to generate content similarity matrices among subreddits can address challenges encountered by the MF approach. To this end, we evaluate the MF and MF + CBF approaches using OpenAI Embeddings on posts. The predictions generated by the two models are presented in Figure <ref>.
We further examine the construction of scores using MF for this case study. The scores values are generated using latent features P, Q, μ, b^(p), and b^(q), representing user, item features, global average, user, and item biases, respectively. However, due to the imbalance in the dataset, there are more posts in some subreddits than others, leading to a cold start problem for the MF approach to accurately learn communities with a small number of examples. In this case study, MF fails to generate correct predictions for the divorce community due to the limited number of posts available. Additionally, MF is biased towards subreddits with more posts, as reflected by the b^(q) values that have strong correlations with the number of posts in the subreddit communities, as depicted in Figure <ref>.
We demonstrate that the top three predictions generated by MF are the subreddits with the highest item biases compared to other subreddits, which are also the ones with the most posts. However, as divorce only accounts for 0.78% of the dataset, the performance of MF is limited. By utilizing OpenAI Embeddings on Subreddit Discourses to represent subreddit communities, we can integrate semantic information into the prediction process, thereby overcoming the cold start problem encountered by MF. Furthermore, this approach captures the relationships between the target recommended subreddit, historically interacted communities and semantic similarities. In this case, the most similar subreddits to personalfinance are legaladvice and divorce, while the most similar subreddits to parenting are autism and divorce.
Overall, we showcase that integrating semantic information into MF can address the cold start problem, and combining MF with CBF using discourse embeddings can make better recommendations.
§ CONCLUSION
This study aimed to investigate the effectiveness of different types of discourse embeddings when integrated into content-based filtering for recommending support groups, particularly in the mental health domain. Our findings showed that the hybrid model, which combined content-based filtering and collaborative filtering, yielded the best results. Moreover, we conducted an extensive case study to demonstrate the interpretability of our approach's predictions.
Previous studies have brought to light the use of past behaviors to make more accurate recommendations in mental health <cit.>. They also emphasize effective communication between the recommender system and the user as an essential factor for users' proper understanding of mental health in general as well as in their own journey <cit.>. Through promising prediction accuracy and interpretability, we believe that this method can serve as a valuable tool to support individuals, particularly those with mental health concerns, to share and seek help regarding their issues.
§ LIMITATIONS
In our current project, we have not taken into account the temporal information that treats the historical behavior of users as a sequence of actions. Thus, the model may not capture how user behaviors change over time. To ensure full support to users in need, we recommend that future work should address this limitation by considering users' historical behaviors as a sequence of actions. Moreover, although our pre-trained models achieved significant results without fine-tuning discourse embeddings, we suggest that fine-tuning these models can enhance performance by capturing the nuances of the datasets' distribution and contexts. Furthermore, conducting a detailed comparison of additional open-source Large Language Models (LLMs) would provide more comprehensive insights into their performance.
Additionally, in addition to analyzing the efficiency of different models, it is crucial to evaluate the cost associated with implementing these models. Therefore, future work should consider both fine-tuning and evaluating additional LLMs, while also taking into account the costs of utilizing these models.
§ ACKNOWLEDGEMENT
This work was supported by NSF IIS-2119531, IIS-2137396, IIS-2142827, CCF-1901059, and ONR N00014-22-1-2507.
style/acl_natbib
|
http://arxiv.org/abs/2307.04116v1 | 20230709081305 | Neutron scattering and muon-spin spectroscopy studies of the magnetic triangular-lattice compounds $A_2$La$_2$NiW$_2$O$_{12}$ ($A$ = Sr, Ba) | [
"B. C. Yu",
"J. Y. Yang",
"D. J. Gawryluk",
"Y. Xu",
"Q. F. Zhan",
"T. Shiroka",
"T. Shang"
] | cond-mat.str-el | [
"cond-mat.str-el",
"cond-mat.mtrl-sci"
] |
plain
Preprint: August 12, 2023,
These authors contributed equally
Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
These authors contributed equally
Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China
Spallation Neutron Source Science Center (SNSSC), Dongguan 523803, China
Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zürich, Switzerland
[Corresponding authors:
][email protected]
Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
Chongqing Key Laboratory of Precision Optics, Chongqing Institute of East China Normal University, Chongqing 401120, China
We report on the geometrically frustrated two-dimensional triangular-lattice magnets A_2La_2NiW_2O_12 (A = Sr, Ba) studied mostly by means of neutron powder diffraction (NPD) and muon-spin rotation and relaxation (µSR) techniques. The chemical pressure induced by the Ba-for-Sr substitution suppresses the ferromagnetic (FM) transition from 6.3 K in the Ba-compound to 4.8 K in the Sr-compound.
We find that the R3̅ space group reproduces the NPD patterns better than the previously reported R3̅m space group. Both compounds adopt the same magnetic structure with a propagation vector k = (0, 0, 0), in which the Ni^2+ magnetic moments are aligned ferromagnetically along the c-axis. The zero-field µSR results reveal two distinct internal
fields (0.31 and 0.10 T), caused by the long-range ferromagnetic order.
The small transverse muon-spin relaxation rates reflect the homogeneous internal field
distribution in the ordered phase and, thus, further support the simple FM arrangement of the Ni^2+ moments. The small longitudinal muon-spin relaxation
rates, in both the ferromagnetic- and paramagnetic states of A_2La_2NiW_2O_12, indicate that spin fluctuations are rather weak.
Our results demonstrate that chemical pressure indeed changes the superexchange interactions in A_2La_2NiW_2O_12 compounds, with the FM interactions being dominant.
Neutron scattering and muon-spin spectroscopy studies of the magnetic triangular-lattice compounds A_2La_2NiW_2O_12 (A = Sr, Ba)
T. Shang
Accepted 08-Jul-2023. Received 18-Jun-2023; in original form 23-May-2023
================================================================================================================================
3pt
§ INTRODUCTION
8pt
Geometric frustration occurs when a system of interacting spins is unable
to find its lowest energy state because of how the spins are arranged.
This property plays an important role at microscopic scales in solids.
In particular, in certain cases, such as in spin glasses, spin ice, and
spin liquids <cit.>, the
localized magnetic moments interact through competing exchange
interactions that cannot be simultaneously satisfied, thus giving rise
to a highly degenerate magnetic ground state.
For instance, in a spin-liquid system, the constituent spins are
highly correlated, but still strongly fluctuating down to zero
temperature <cit.>.
Such fluctuations lead to remarkable collective phenomena such as emergent gauge fields and fractional excitations <cit.>.
Most of the magnetic frustrations have a simple geometric origin <cit.>, usually occurring in materials
with a 2D triangular- or kagome lattice, or a 3D pyrochlore lattice,
etc., with the nearest-neighbor interactions being antiferromagnetic
(AFM) <cit.>.
A two-dimensional triangular lattice with antiferromagnetic interactions
provides one of the prototypes of magnetic frustration <cit.>.
The perovskite-derived compounds A_4B'B_2O_12 (A = Sr, Ba, La; B' = Mn, Co, Ni; B = Sb, Te, W, Re) represent one such system <cit.>.
Depending on the valence states of the B' and B atoms, the A site can be occupied by either a Sr^2+ (Ba^2+) or La^3+ ion, or by their combinations.
Here, the magnetic B' ions form a layered structure with a 3-fold site symmetry [see Fig. <ref>(a) for the B' = Ni^2+ case].
Since the magnetic B' layers are well separated by the nonmagnetic
A- and BO_6 layers, the former give rise to a magnetic
quasi-2D triangular lattice,
which can potentially host magnetic frustrations.
To date, different magnetic ground states have been found to occur
in the A_4B'B_2O_12 family <cit.>,
whose magnetic properties are thought to be determined mostly by the competition between the
ferromagnetic (FM-) B'-O-B-O-B' and antiferromagnetic B'-O-O-B' superexchange interactions, shown by solid- and dashed lines in Fig. <ref>(c) <cit.>. The spin state
of the magnetic B' ions plays a decisive role in the competition between the
two superexchange interactions. As a consequence, A_4CoB_2O_12
(effective spin S = 1/2 for Co^2+) and
Ba_2La_2NiW_2O_12 (S = 1 for Ni^2+) are reported to be ferromagnetic, while
Ba_2La_2MnW_2O_12 (S = 5/2 for Mn^2+) is reported to be antiferromagnetic <cit.>.
Similar superexchange interactions and their competitions have been observed in other triangular-lattice magnets, e.g., Ba_3B'Nb_2O_9 <cit.> and AAg_2B'(VO_4)_2 <cit.>.
Unsurprisingly, such closely competing interactions can be tuned by either external pressure or by chemical substitution,
each of which able to introduce lattice distortions and to modify
the bond lengths and angles <cit.>, thus, tuning the magnetic order and frustration.
For example, in A_4CoB_2O_12, the chemical pressure (i.e., the substitution of Ba with Sr and/or La, or W with Re) can tune the FM transition temperature <cit.>.
However, the effects of chemical pressure on the magnetic properties
of A_4NiB_2O_12 have not been investigated in detail.
To clarify the above issues, in this paper, we synthesized polycrystalline samples of A_2La_2NiW_2O_12 (A = Sr, Ba)
and studied their magnetic properties by means of magnetization specific heat-, neutron scattering-, and muon-spin rotation and relaxation (µSR) measurements. The chemical pressure is introduced by substituting Ba with Sr, which suppresses the
FM transition temperature from 6.3 down to 4.8 K, while the magnetic
moments of the Ni^2+ ions are ferromagnetically aligned along the c-axis in both compounds.
Our results suggest that the chemical pressure indeed changes the superexchange interactions in A_2La_2NiW_2O_12, with the B'-O-B-O-B' superexchange path dominating
the competition between the FM and AFM interactions. External
pressure on Sr_2La_2NiW_2O_12 or chemical substitution on the
Ni site may further tune the magnetic interactions and lead
to magnetic frustration.
§ EXPERIMENTAL DETAILS
8pt
The A_2La_2NiW_2O_12 (A = Sr, Ba) polycrystalline samples were prepared by
the solid-state reaction method. Stoichiometric amounts of La_2O_3,
BaCO_3, SrCO_3, NiO, and WO_3 powders were used to prepare the
materials. The La_2O_3 rare-earth oxide was annealed for
15 hours in atmosphere to remove moisture. The powders were then mixed, ground, and sintered at 1200^∘C for 24 hours. After grinding the samples again, the powders were pressed into pellets and sintered at 1200^∘C for extra 48 hours. The magnetic-susceptibility and heat-capacity measurements were performed
on a Quantum Design magnetic property measurement system (MPMS) and
physical property measurement system (PPMS), respectively.
Neutron powder diffraction (NPD) measurements were carried out at the Swiss Neutron Source SINQ of the Paul Scherrer Institute in Villigen, Switzerland. The A_2La_2NiW_2O_12 powder samples were introduced in cylindrical vanadium cans (8 mm in diameter and 50 mm high) and mounted on a helium cryostat
stick (2–300 K). High-resolution room-temperature NPD patterns were recorded at the powder diffractometer HRPT [Ge (822), λ = 1.154 Å].
To discern the magnetic diffraction peaks, high-intensity NPD patterns were collected at 1.7 K on the DMC diffractometer using a longer wavelength [pyrolitic graphite (002), λ = 2.458 Å].
The collected NPD patterns were analyzed using the Rietveld package of the FullProf suite <cit.>.
The bulk µSR measurements were carried out at the general-purpose
surface-muon instrument (GPS) of the Swiss muon source at Paul Scherrer
Institut, Villigen, Switzerland.
In this study, we performed two types of experiments: zero-field (ZF)-, and longitudinal-field (LF) µSR measurements.
In both cases, we aimed at studying the temperature evolution of the magnetically ordered phase and the spin fluctuations.
The µSR spectra were collected upon sample heating and then analyzed by the software package <cit.>.
§ RESULTS AND DISCUSSION
8pt
§.§ Magnetic susceptibility
The A_2La_2NiW_2O_12 samples were first characterized by magnetic-susceptibility measurements. Figures <ref>(a) and (d) show the temperature-dependent magnetic susceptibility χ(T) collected in an applied magnetic field of 0.1 T using a zero-field-cooling (ZFC) protocol. χ(T) shows a sharp increase close to T_c, the
temperature where the Ni^2+ moments give rise to a FM order. The
Curie temperatures T_c can be determined from the derivative
of susceptibility with respect to temperature dχ/dT [see Fig. <ref>(c) and (f)] which, in a 0.1-T applied field, provides a T_c of 6.3 and 4.8 K for
Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, respectively.
The magnetic susceptibility was also measured under various magnetic fields up to 6 T. As shown in Fig. <ref>(b) and (e),
as the magnetic field increases, the transition becomes broader and T_c moves to higher temperatures, both features typical of ferromagnetic materials.
The insets in Fig. <ref>(a) and (d) show the Curie-Weiss fits to
the inverse susceptibility (solid lines), which yield a Weiss temperature θ_p = 7.4 K
for Ba_2La_2NiW_2O_12 and θ_p = 8.4 K for Sr_2La_2NiW_2O_12. The positive θ_p values indicate that FM interactions are
dominant in both compounds.
The estimated effective moments are μ_eff = 3.17 μ_B and 3.13 μ_B for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, respectively.
Both are close to the theoretical value of spin-only
Ni^2+ ions (2.83 μ_B), i.e., assuming a
quenching of the orbital moment, typical of octahedral complexes <cit.> — such as the NiO_6 units in Fig. <ref>(a).
The FM ground state was further confirmed by field-dependent magnetization
measurements (see Fig. <ref>). For T < T_c, a small yet clear
magnetic hysteresis loop is observed. For both materials, the
magnetization starts to saturate for μ_0H > 5 T. After substituting
the Ba with Sr, the magnetism becomes softer. The coercive field
of Ba_2La_2NiW_2O_12 is about 67 mT, while,
in Sr_2La_2NiW_2O_12, it decreases to 4 mT.
Thus, in A_2La_2NiW_2O_12, the chemical pressure suppresses
both the magnetization and the T_c, hence suggesting an
enhancement of the magnetic competition. Nevertheless, the FM
interactions remain dominant also in Sr_2La_2NiW_2O_12.
§.§ Heat capacity
We measured the zero-field heat-capacity
of A_2La_2NiW_2O_12 from 2 to 300 K.
The low-T heat-capacity data were also collected under various
external fields, up to 9 T. As shown in Fig. <ref>,
in both compounds, there is a sharp λ-like transition at
low temperatures, typical of long-range magnetic order.
The C(T) data show a distinct peak at T_c = 6.1 and 4.7 K for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, which are consistent with the T_c values determined from magnetization data (see Fig. <ref>).
To extract the magnetic contribution, the normal-state (i.e., T ≫ T_c)
specific-heat data were fitted to C/T = γ + βT^2, where
γ≡ 0, due to the insulating nature of both compounds [see solid lines in Fig. <ref>(a) and (d)]. The derived β values are 0.0013 and 0.0012 J/mol-K^4 for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, which yield a Debye temperature θ_D = 142 and 145 K, respectively. After subtracting the phonon contribution (i.e, the βT^2 term), the magnetic specific heat C_m/T vs. temperature is plotted in Fig. <ref>(b) and (e) for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, respectively.
Upon increasing the magnetic field, the peak at T_c becomes broader
and moves to higher temperatures, once more confirming the FM nature of
the magnetic transition in both materials.
The zero-field magnetic entropy S_m(T) obtained by
integrating C_m(T)/T is shown in Fig. <ref>(c)
and (f) for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, respectively.
In both compounds, at temperatures close to T_c,
S_m reaches Rln(2) (corresponding to S = 1/2).
In Ba_2La_2NiW_2O_12, at temperatures above T_c,
S_m reaches Rln(3) (corresponding to S = 1), while
in Sr_2La_2NiW_2O_12, S_m is slightly smaller
than Rln(3). Such a deviation is most likely
due to an over-subtraction of the phonon contribution from
the specific-heat data. To properly subtract the phonon contribution
and estimate the magnetic entropy, heat-capacity measurements on
the non-magnetic counterparts, as e.g., A_2La_2ZnW_2O_12, are highly desirable.
§.§ Neutron diffraction
To determine the crystal- and magnetic structures of A_2La_2NiW_2O_12,
neutron powder diffraction patterns were collected at both the
paramagnetic (300 K)- and ferromagnetic states (1.7 K).
The room-temperature patterns were first analyzed by using the space group
R3̅m (No. 166), as reported in previous studies <cit.>.
With this model, the powder x-ray diffraction (XRD) patterns could be fitted reasonably well with a goodness of fit χ_r^2 ∼ 7.
However, in case of the NPD patterns, although the Bragg peaks were located at the right positions, the R3̅m space group yielded a fairly large χ_r^2 ∼ 18,
as evinced also from the clear discrepancy
between the observed- and calculated intensities. This indicates
that the space group R3̅m does not describe the crystal
structure of A_2La_2NiW_2O_12 compounds accurately and,
thus, further corrections to the structural model are required.
Considering that neutron diffraction is more sensitive to the oxygen
atoms than x-ray diffraction <cit.>, the oxygen positions are
most likely to require corrections. We found that the space group R3̅ (No. 148) reproduces the
NPD patterns quite well. In fact, both R3̅m and
R3̅ groups belong to the trigonal system, with the latter
exhibiting slightly different oxygen positions.
Figures <ref>(a) and (b) show the Rietveld refinements of NPD
at 300 K using the R3̅ space group for both compounds.
These refinements yield a significantly reduced χ_r^2 ∼ 2,
thus confirming that, in both cases, the R3̅ space group
is more appropriate than R3̅m.
With R3̅, the NiO_6 and WO_6 octahedra rotate in
opposite directions around the c-axis, which breaks the mirror symmetry.
A similar symmetry breaking has been observed also in the
Ba_2La_2NiTe_2O_12 compound <cit.>. The refined lattice parameters, atomic positions, and bond lengths/angles,
together with the goodness of fits are summarized in Table <ref> for A_2La_2NiW_2O_12 compounds.
To clarify the magnetic structure of Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, the NPD patterns were also collected in the magnetically ordered state
(i.e., 1.7 K) using long wavelength neutrons (λ = 2.458 Å).
The LeBail fits of the magnetic diffraction patterns
reveal a commensurate magnetic structure with a propagation
vector k = (0, 0, 0) for A_2La_2NiW_2O_12 compounds.
For such a magnetic vector, the little group G_k is identical to the
space group R3̅ and it includes the symmetry elements
1, 3^+, 3^-, 1̅, 3̅^+, and 3̅^- <cit.>.
The magnetic unit cell of A_2La_2NiW_2O_12 possesses a single orbit with only one site located at the Ni (0, 0, 0) position.
For k = (0, 0, 0), G_k has six different irreducible representations (irreps) τ1, τ2, τ3, τ4, τ5, and τ6, among which only τ1, τ3, and τ5 allow for a long-range magnetic order at the Ni site. Table <ref> summarizes the basis vectors of τ1, τ3, and τ5 irreps calculated with BasIreps.
For the R3̅ space group, the Ni atoms are located at the
3a site (0, 0, 0), invariant under all the symmetry operations.
As a consequence, all the allowed irreps generate a FM coupling with the
spins aligned along the c-axis for τ1, or lying within the ab-plane for τ3 and τ5 (see details in Table <ref>).
According to the Rietveld refinements of the 1.7-K NPD pattern [see Fig. <ref>(c) and (d)], the best fits were obtained by using the τ1 irrep, yielding the smallest
χ_r^2 = 1.93 and 2.77 for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12, respectively. The refined magnetic structure is shown in Fig. <ref>(b).
The magnetic moments of Ni atoms obtained from the refinements are 1.94(2) and 1.84(3) μ_B for Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12,
consistent with their saturation magnetization (see Fig. <ref>).
§.§ ZF- and LF-µSR
The large gyromagnetic ratio of muons, combined with their
availability as 100% spin-polarized beams, makes ZF-µSR a very sensitive probe for investigating magnetic materials.
Here, to study the magnetic properties of A _2La_2NiW_2O_12
at a local level, we collected a series of ZF-µSR spectra at temperatures covering both the paramagnetic- and ferromagnetic states.
Since neutron diffraction data suggest FM ground states for both Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12
(with the Ni^2+ moments aligned along the c-axis),
for our µSR measurements we focused on Ba_2La_2NiW_2O_12
due to its slightly higher T_c value.
In a magnetic material with a long-range order, the time evolution of ZF-µSR asymmetry,
A_ZF(t), encodes both the intrinsic magnetic fields and their distribution at the muon-stopping site <cit.>.
The ZF-µSR spectra of Ba_2La_2NiW_2O_12 collected at different temperatures are shown in Fig. <ref>(a).
In the paramagnetic state (T > T_c), the ZF-µSR spectra exhibit a relatively
slow muon-spin depolarization (∼0.5–1 µs^-1 at 10 K), indicating rather weak spin fluctuations.
Considering the two muon-stopping sites in Ba_2La_2NiW_2O_12, attributed to two distinct oxygen sites (see Table <ref>),
the ZF-µSR spectra in the paramagnetic state were analyzed using the following model:
A_ZF(t)= ∑_i=1^2 A_i e^-λ^L_it.
Here, λ^L_i represent the longitudinal muon-spin relaxation rates,
while A_i are the asymmetries of the two nonequivalent muon-stopping sites.
In the FM state (T < T_c), the ZF-µSR spectra are characterized by
highly-damped oscillations, typical of long-range magnetic order.
These are clearly visible in Fig. <ref>(b), where short-time
oscillations are superimposed on a long-time slow relaxation.
The ZF-µSR spectra in the FM state were, hence, analyzed using
the following model:
A_ZF(t)= ∑_i=1^2A_i[αcos(ω_it+ϕ)e^-λ^T_it + (1-α)e^-λ^L_it].
Here, α and 1–α are the oscillating (i.e., transverse) and nonoscillating (i.e., longitudinal) fractions of the µSR signal, respectively,
whose initial total asymmetry is equal to A_1 and A_2.
In polycrystalline materials with a long-range magnetic order, one expects α = 2/3, since statistically one third of the muon spins are aligned parallel to the local field direction
(i.e., S_μ∥ B_int) and, hence, do not precess;
ω_i (=γ_μ B_i^int) represents the muon-spin precession frequency,
with γ_μ= 2π×135.5 MHz/T the muon gyromagnetic ratio and B_i^int the local field sensed by muons;
λ^T_i are the transverse muon-spin relaxation rates, reflecting the internal field distributions; ϕ is a shared initial phase.
The derived fitting parameters are summarized in Fig. <ref>(c)-(e).
The B_i^int, λ^T_i, and λ^L_i
all show a distinct anomaly at T_c.
The T_c determined from ZF-µSR is consistent with the value determined from magnetic susceptibility and heat capacity (see Figs. <ref> and <ref>).
As shown in Fig. <ref>(c), below T_c, there are two distinct internal fields, here reflecting the two different muon-stopping sites.
In the FM state, the temperature evolution of B^int_i(T)
resembles the typical mean-field curve. To estimate the zero-temperature internal field,
B^int_i(T) was analyzed by means of
a phenomenological model:
B^int_i(T) = B^int_i(0) [1-(T/T_c)^γ]^δ,
where B^int_i(0) is the zero-temperature internal field,
while γ and δ represent two empirical parameters.
As shown by solid lines in Fig. <ref>(c), the above model describes the data reasonably well, yielding B^int_1(0) = 0.30 T
and B^int_2(0) = 0.10 T
for Ba_2La_2NiW_2O_12. The resulting
power exponents are γ = 5.5(2) and δ = 0.54(2) for B_1^int(T), and γ = 4.6(2) and δ = 0.26(1) for B_2^int(T), respectively.
The lack of any anomalies in
B^int_i(T) below T_c is consistent with the simple FM structure of Ba_2La_2NiW_2O_12 (see Fig. <ref>).
In fact, in some complex magnetic materials with multiple transitions,
one observes a more complex B^int(T), since changes in
magnetic structure are reflected in the local-field distribution <cit.>.
The transverse muon-spin relaxation rate λ^T reflects the static magnetic field distribution at the muon-stopping site and is also affected by dynamical effects such as spin fluctuations,
while its longitudinal counterpart λ^T is solely determined by spin fluctuations.
The λ_i^T(T) of Ba_2La_2NiW_2O_12 exhibits the typical behavior of
magnetic materials with a long-range order <cit.>, i.e., diverging at T_c and continuously decreasing well
inside the magnetic state [see Fig. <ref>(d)].
In the paramagnetic state, λ_i^T is zero, due to the lack of a magnetic moment in the absence of an external field.
The λ_i^L(T) in Fig. <ref>(e) shows a similar
behavior to the λ_i^T(T), i.e., λ_i^L(T)
diverges near T_c, followed by a significant drop at T < T_c,
indicating that spin fluctuations are the strongest close to the onset
of the FM order. Note that, the absolute values of longitudinal relaxation are much smaller than the transverse ones.
Thus, at 1.5 K, λ^L/λ^T∼ 0.097
and 0.002 for the two different muon-stopping sites.
In the paramagnetic state (i.e., T > 8 K), λ_i^L is
also very small, suggesting weak spin fluctuations in both the ferromagnetic and
paramagnetic states of Ba_2La_2NiW_2O_12.
Such weak spin fluctuations are further supported by LF-µSR measurements.
Figure <ref> shows the 2-K LF-µSR spectra
collected in a longitudinal field of 0.1 and 0.5 T. Once the external
field exceeds the internal field (here, ∼ 0.3 T), the µSR spectra become
almost flat. This suggests that, in Ba_2La_2NiW_2O_12,
muon spins are fully decoupled from the electronic magnetic moments
in a field of 0.5 T.
§ DISCUSSION
Although our comprehensive set of measurements suggest that
both Ba_2La_2NiW_2O_12 and Sr_2La_2NiW_2O_12
have FM ground states, the magnetic susceptibility and neutron diffraction
results indicate that the competition between FM- and AFM couplings is indeed tuned
by the chemical
pressure induced by the substitution of Ba- with the smaller Sr ions.
To understand this, we examine the crystal-structure parameters
of A_2La_2NiW_2O_12 (see details in Table <ref>),
including the bond lengths and angles. The latter are directly
related to the magnetic superexchange interactions and, thus, control the
magnetic properties. In A_4B'B_2O_12, the B'O_6 octahedra
share their corners with the BO_6 octahedra via oxygen atoms, thus leading to two superexchange interaction paths, i.e.,
B'-O-B-O-B' and B'-O-O-B' [see details in Fig. <ref>(c)].
According to the Goodenough-Kanamori rule,
which provides the signs of the competitive interactions that are
responsible for non-collinear spin ordering <cit.>,
the B'-O-B-O-B' superexchange interaction (with ∠O-B-O ∼ 90^∘) favors a FM coupling, while the B'-O-O-B' path (with ∠B'-O-O ∼ 120-180^∘) allows for an AFM coupling. Although the R3̅ space group implies
reduced O-B-O and B'-O-O bond angles with respect to the previously
reported R3̅m space group <cit.>, the change is such
that the FM or AFM character of the superexchange interactions is maintained.
For instance, in Ba_2La_2NiW_2O_12, R3̅m gives
∠Ni-O2-O2 = 137.2^∘ and ∠O2-W-O2 = 86.7^∘; while
in R3̅, these bond angles become 121.5^∘ and 84.5^∘.
Consequently, the B'-O-B-O-B' and B'-O-O-B' superexchange
interaction paths remain
valid also in the R3̅ space group.
The competition between these FM and AFM interactions eventually determines
the magnetic ground state of A_4B'B_2O_12.
Since Sr has a smaller atomic radius than Ba, by
replacing Ba with Sr, the lattice constants along both the
a- and c-axis are reduced by a factor of 1.14 and 2.81%,
the Ni-O bond length decreases from 2.064 Å to 2.051 Å, while
the Ni-O2-O2 bond angle increases from 121.50^∘ to 120.62^∘.
By contrast, the W-O bond length and the O2-W-O2 bond angle are
less affected, most likely because the W-O2 layer is further
away from the Ba- or Sr-layers [see Fig. <ref>(a)].
The O2-W-O2 bond angle increases slightly from 84.51^∘ to
84.53^∘. The changes of Ni-O2-O2 and O2-W-O2 bond angles induced by chemical pressure (i.e., the substitution of Ba by Sr)
tune the competition between FM- and AFM superexchange interactions in A_2La_2NiW_2O_12.
The physical pressure might further
tune the competition between the FM- and AFM interactions, and yield
magnetic frustration.
Previous studies reveal that the magnetic ground states of
A_4B'B_2O_12 can also be tuned
by chemical substitution on the B sites <cit.>.
The substitution on the B'-site of Ni may enhance the B'-O-O-B'
AFM interactions and stabilize the AFM ground state. For instance,
Ba_2La_2MnW_2O_12 shows an AFM order below 1.7 K <cit.>.
The Ni^2+ ions can also be substituted by Cu^2+ ions, but the
latter case is not yet studied, although it may represent
another interesting compound to exhibit magnetic frustration. Finally, the introduction of magnetic ions on the A site
(e.g., the substitution of Ba^2+ or Sr^2+ with Eu^2+),
whose magnetic interactions can compete with the above superexchange
interactions, may lead to exotic magnetic properties.
§ CONCLUSION
To summarize, we studied the effects of chemical pressure on the
magnetic triangular-lattice compounds A_2La_2NiW_2O_12 (A = Sr, Ba).
Their magnetic properties (due to the Ni^2+ ions) were investigated by means of magnetic susceptibility, specific heat, neutron diffraction,
and µSR spectroscopy. When replacing Ba with Sr, chemical pressure is introduced which can tune the competition between the FM- and AFM superexchange interactions. While the Curie temperature T_c is suppressed
from 6.3 K to 4.8 K, the FM interactions still persist
in Sr_2La_2NiW_2O_12. According to the refinements of neutron
diffraction patterns, in both compounds, the magnetic moments of
Ni atoms are aligned along the c-axis, with a propagation vector
k = (0, 0, 0).
By using ZF-µSR measurements, we could follow the temperature
evolution of the spin fluctuations and of the local magnetic fields.
The estimated internal fields at zero temperature for the two
different muon-stopping sites are 0.31 and 0.1 T.
The smooth transverse muon-spin relaxation rates λ_T
in the ordered phase confirm the simple FM structure of
A_2La_2NiW_2O_12. In both materials, spin fluctuations
are rather weak, reflected in a small longitudinal muon-spin relaxation rate in both the ferromagnetic- and paramagnetic states.
In the future, it could be interesting to check if the combined
physical pressure and chemical substitution on the A and B' sites can further tune the magnetic competitions in
Sr_2La_2NiW_2O_12, and eventually lead to magnetic frustration or to a quantum spin-liquid state.
This work was supported by the Natural Science Foundation of Shanghai
(Grants No. 21ZR1420500 and 21JC1402300), Natural Science
Foundation of Chongqing (Grant No. 2022NSCQ-MSX1468), and the Schweizerische
Nationalfonds zur Förderung der Wissenschaftlichen Forschung
(SNF) (Grants No. 200021_188706 and 206021_139082). Y.X. acknowledges
support from the Shanghai Pujiang Program (Grant No. 21PJ1403100) and the Natural Science Foundation of China (Grant No. 12274125).
|
http://arxiv.org/abs/2307.04064v1 | 20230709000056 | Local null controllability of a class of non-Newtonian incompressible viscous fluids | [
"Pitágoras de Carvalho",
"Juan Límaco",
"Denilson Menezes",
"Yuri Thamsten"
] | math.AP | [
"math.AP",
"math.OC",
"35K55, 76D55, 93B05, 93C10"
] |
UESPI]P. Carvalho
[email protected]
IMEUFF]J. Límaco
[email protected]
IMEUFF]D. Menezes
[email protected]
IMEUFF]Y. Thamsten
[email protected]
[UESPI]Departamento de Matemática, Universidade Estadual do Piauí, Teresina, PI, Brasil
[IMEUFF]Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, Brasil
We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.
Null controllability, shear dependent viscosity, nonlinear partial differential equations, non-Newtonian fluids.
[2010] 35K55, 76D55, 93B05, 93C10.
§ INTRODUCTION
Let us fix an integer N ∈{ 2,3 }, and let us take a non-empty, open, connected, and bounded subset Ω of ℝ^N with a smooth boundary ∂Ω, and a real number T>0. Henceforth, we write Q:= ]0,T[×Ω, and Σ := [0,T]×∂Ω. In general, we understand all of the derivatives figuring in this work in the distributional sense.
We interpret the set Ω as a region occupied by the particles of a fluid with a velocity field y. We represent its pressure by p, whereas v stands for a distributed control which acts as a forcing term through a given open set ω⋐Ω. We assume ω≠∅. The model comprising the subject of the current investigation is the following:
[ D y/Dt - ∇·𝒯(y,p) = χ_ω v, in Q,
∇· y = 0, in Q,
y = 0, on Σ,
y(0) = y_0, in Ω. ]
Above, the function χ_ω denotes the indicator function of ω, we define the material derivative as
Dy/Dt := y_t + ( y·∇) y,
the stress tensor, 𝒯, is given by
𝒯(y,p) := -p I + ν(∇ y) ∇ y, ν(∇ y) := ν_0 + ν_1 |∇ y|^r ,
in such a way that the constitutive law for the deviatoric stress tensor reads as
ν(∇ y)∇ y := ( ν_0 + ν_1 |∇ y|^r) ∇ y,
where
|∇ y| := [ ∑_i,j=1^N ( ∂_j y_i)^2 ]^1/2.
We remark that the three constants ν_0, ν_1, and r appearing above are strictly positive, typically with ν_0 ≫ν_1, although this assumption is not necessary in this work.
Therefore, we are focusing on the class of power-law shear-dependent fluids. Pioneers in the study of the system (<ref>)-(<ref>) were O. A. Ladyzhenskaya and J.-L. Lions, see <cit.>. Particularly, let us introduce the usual spaces we use in the mathematical analysis of fluid dynamics, i.e.,
H := { y ∈ L^2(Ω)^N : ∇· y = 0 in Ω, y· n = 0 on ∂Ω}
and
V := {y ∈ H^1_0(Ω)^N : ∇· y = 0 in Ω},
where n denotes the outward unit normal on ∂Ω. Then, the results <cit.> (cf. <cit.>) imply the following:
Let us suppose that
r > N/2 - 1.
as well as
y_0 ∈ H and χ_ω v ∈ L^q^'(0,T; V^'),
where
1/q + 1/q^' = 1, for q := r+2.
Then, the problem (<ref>)-(<ref>) admits a unique solution (y,p) such that
y ∈ L^r+2(0,T;V) ∩ L^∞(0,T;H) and p ∈ L^2(Q).
For r=1 and N=3, the system (<ref>)-(<ref>) is the simple turbulence model of Smagorinsky, see <cit.>. Since then, gradient-dependent (or shear-dependent) viscosity models of incompressible viscous fluids have attracted considerable attention from the mathematical, physical, and engineering communities. Some other works investigating the well-posedness for the model (<ref>)-(<ref>) under consideration are <cit.>. The paper <cit.> studies the energy dissipation for the Smagorinsky model. For the investigation of some regularity properties of solutions of (<ref>)-(<ref>), see <cit.> and the references therein.
On the one hand, the Navier-Stokes (NS) system of equations (corresponding to formally replacing ν_1 = 0 in (<ref>)) is deeply relevant, not only in mathematics, but for physics, engineering, and biology, see <cit.>. For standard well-posedness results, which are now classic, see <cit.>. However, even with a great effort of researchers, among the main longstanding open problems are the questions about global existence or finite-time blow-up of smooth solutions in dimension three of the incompressible Navier-Stokes (or else the Euler) equations. The system (<ref>)-(<ref>) is a generalization of the Navier-Stokes equations. From a practical perspective, as <cit.> points out, every fluid which solutions of NS decently models is at least as accurately described by those of (<ref>)-(<ref>).
On the other hand, for real-world problems, the advantage of considering the more general fluids of power-law type is not slight. In effect, as <cit.> describes, practitioners employed them to investigate problems in chemical engineering of colloids, suspensions, and polymeric fluids, see <cit.>, in ice mechanics and glaciology, see <cit.>, in blood-rheology, see <cit.>, and also in geology, see <cit.>, to name a few instances.
We briefly describe the physical meanings of the constants ν_0, ν_1, and r. Firstly, ν_0 stands for the kinematic viscosity of the fluid. If the physical variables are nondimensionalized, then ν_0^-1 is the Reynolds number of the fluid. Secondly, we can conceive the constants ν_1 and r in light of the kinetic theory of gases and the definition of a Stokesian fluid, see <cit.>. For instance, from the point of view of turbulence modeling, we have ν_1 = C_0ℓ^2, where C_0 is a model parameter and ℓ≪ 1 is a mixing length, see <cit.>. In the latter perspective, a possible derivation of the model stands on the Boussinesq assumption for the Reynolds stress, further stipulating that the eddy viscosity ν_t takes the particular form
ν_t = ν_1 |∇ y|^r,
see <cit.>. The term ν_t given by (<ref>) leads to a stabilizing effect by increasing the viscosity for a corresponding increase in the velocity field gradient, see the discussion in <cit.>; hence, we call these fluids shear-thickening.
From the viewpoint of control theory, <cit.> establishes the local null controllability for the Navier-Stokes equations under no-slip boundary conditions; later developments worth mentioning are, e.g, <cit.>. For the study of the Ladyzhenskaya-Smagorinsky model, see <cit.>. The paper <cit.> deals with a similar one-dimensional problem. Regarding local exact controllability properties for scalar equations having a locally nonlinear diffusion, some advances are <cit.>. However, although the diffusion coefficients can be functions of the state (in the case of <cit.> in a simplified form), the methods used in these works seem not enough to tackle the situation in which these coefficients depend on the gradient of the controlled solution. Furthermore, the assumptions they make rule out more general diffusions with power-law type nonlinearities. In the present work, we can circumvent all of these difficulties.
The notion of controllability we consider in this paper is defined as follows.
We say that (<ref>)-(<ref>) is locally null-controllable at time T>0 if there exists η>0 such that, for each y_0 ∈[H^5(Ω)∩ V]^N satisfying the compatibility conditions Ay_0,A^2y_0 ∈[H^1_0(Ω)]^N, as well as
y_0_H^5(Ω)^N < η,
we can find v ∈ L^2(]0,T[×ω)^N for which the corresponding velocity field y of (<ref>)-(<ref>) satisfies
y(T,x) = 0 for almost every x ∈Ω.
We now state the main theoretical result we establish in this paper.
Let us suppose r ∈{1,2} or r ⩾ 3. For each T>0, the system (<ref>)-(<ref>) is locally null-controllable at time T.
Although we stated Theorem <ref> in terms of weak solutions, our methodology yields smooth controls and transient trajectories for the nonlinear system (<ref>)-(<ref>). Namely, we will be able to prove that there is a control parameter v such that
ρ_4 v, (ζ v)_t, ζΔ v, ( ζ v_t )_t, ζΔ v_t, ζ D^4 v ∈ L^2(Q)^N,
with a corresponding trajectory y satisfying
ρ_6∇ y, ρ_7 y_t, ρ_7 Δ y, ρ_8∇ y_t, ρ_9y_tt, ρ_9Δ y_t, ρ_10∇ y_tt, ρ_10D^3 y_t, ρ_9 D^4 y, ρ_11y_ttt, ρ_11Δ y_tt∈ L^2(Q)^N,
ρ_6 y, ρ_7 ∇ y, ρ_8y_t, ρ_9Δ y, ρ_9 ∇ y_t, ρ_9 D^3 y, ρ_10y_tt, ρ_10Δ y_t, ρ_11∇ y_tt∈ L^∞(0,T; L^2(Ω)^N),
for appropriate time-dependent positive weights ρ_4, ρ_6, ρ_7, ρ_8, ρ_9, ρ_10, ρ_11, ζ, ζ which blow up exponentially as t↑ T. For more details and the proofs, we refer to Sections <ref> and <ref>. Of course, there is a trade-off between such regularity and our requirements on the initial datum. We will comment upon questions that are related to this relation on Section <ref>.
We will prove Theorem <ref> with the aid of a local inversion-to-the-right theorem. Namely, we will introduce Banach spaces Y and Z (we provide the details in the second subsection of Section <ref>) as well as a mapping H: Y → Z, such that a solution (y,p,v) of the equation
H(y,p,v) = (0,y_0),
for a given initial data y_0 meeting the assumptions of Theorem <ref>, is a solution of the control problem, i.e., a tuple subject to (<ref>)-(<ref>) and (<ref>). We will use the inversion theorem to guarantee the existence of a local right inverse of H. For proving that H is well-defined, as well as that it enjoys suitable regularity properties, the key steps are novel high-order weighted energy estimates for a control and the solution of the linearization of the system (<ref>)-(<ref>) around the zero trajectory.
Taking advantage of the invertibility properties of DH(0,0,0), we construct the following algorithm allowing the computation of a tuple (y,p,v) solving (<ref>)-(<ref>) and (<ref>).
The following local convergence result for Algorithm 1 holds.
There exist a small enough constant η > 0, as well as appropriate Banach spaces Y and Z,[We provide, in the second subsection of Section <ref>, the explicit definitions of both Y and Z.] such that, if y_0_H^5(Ω)^N < η, with y_0 satisfying the compatibility conditions of Definition <ref>, then it is possible to find κ∈]0,1[ with the following property: the relations (y^0,p^0,v^0) ∈ Y and
(y^0,p^0,v^0)-(y,p,v)_Y < κ,
imply the existence of θ∈]0,1[ for which
(y^n+1,p^n+1,v^n+1) - (y,p,v)_Y ⩽θ(y^n,p^n,v^n)-(y,p,v)_Y,
for all n⩾ 0. In particular, (y^n,p^n,v^n) → (y,p,v) in Y.
Here, we fix some notations that we will use throughout the whole paper. Firstly, C denotes a generic positive constant that may change from line to line within a sequence of estimates. In general, C depends on Ω, ω, T, ν_0, ν_1, and r. In case C begins to depend on some additional quantity a (or we want to emphasize some dependence), we write C=C(a). We will also write, for every integer k⩾ 0,
|D^k y| := [∑_i=1^N ∑_|α|=k(∂^α y_i)^2 ]^1/2,
where we used the standard multi-index notation above. We denote the standard norm of L^2(Ω)^N by ·. Finally, we set D^k y := | D^k y|.
We finish this introductory section outlining the structure of the remainder of the work.
* In Section 2, we study the linearization of (<ref>)-(<ref>) around the zero trajectory — it is a forced Stokes system. With the aid of a global Carleman estimate, we can to show that this system is null controllable. Assuming sufficiently regular initial data, we employ a bootstrapping argument to deduce higher regularity for the control, taking advantage of its characterization via Pontryagin's minimum principle. The higher control regularity naturally leads to higher regularity of the velocity field..
* In Section <ref>, we use a local inversion-to-the-right theorem for mappings between Banach spaces to show that the model (<ref>)-(<ref>) is locally null controllable.
* It is in Section 4 that we prove Theorem <ref>. Then, we conduct some numerical experiments to illustrate our theoretical findings.
* Finally, we conclude the work in Section <ref> with some comments and perspectives.
§ STUDY OF THE LINEARIZED PROBLEM
§.§ Some previous results
Our aim in the present Section is to establish the null controllability of the linear system:
[ Ly + ∇ p = χ_ω v + f, in Q,
∇· y = 0, in Q,
y = 0, on Σ,
y(0) = y_0, in Ω, ]
In (<ref>), we have written Ly := y_t - ν_0 Δ y. We achieve this result via a suitable Carleman inequality for the adjoint system of (<ref>); upon writing L^*φ := -φ_t - ν_0 Δφ, it reads
[ L^*φ + ∇π = g, in Q,
∇·φ = 0, in Q,
φ = 0, on Σ,
φ(T) = φ^T, in Ω. ]
In the present subsection, we fix notations that we will employ henceforth. Let us consider ω_1 ⋐ω, with ω_1 ≠∅. For the proof of the following lemma, see <cit.>.
There is a function η^0 ∈ C^2(Ω) satisfying
η^0 >0 in Ω, η^0 = 0 on ∂Ω, |∇η^0| > 0 on Ω\ω_1.
We take l ∈ C^∞([0,T]) with
l(t) ⩾ T^2/4 on [0,T/2], l(t) = t(T-t), on [T/2,T].
We define
γ(x) := e^λ(η^0(x) +mη^0_∞),
α(x) := e^5/4λ mη^0_∞ - e^λ(η^0(x) + mη^0_∞),
γ_1 := min_Ωγ, γ_2 := max_Ωγ,
α_1 := min_Ωα, α_2 := max_Ωα,
and
γ := γ/l^4, α := α/l^4.
Given C>1, m>4, there exists λ_0=λ_0(m,C)>0 such that α_2 ⩽ Cα_1, for all λ⩾λ_0.
For s,λ>0, we write
I(s,λ,φ) := s^3λ^4 ∫_Q e^-2sαγ^3|φ|^2d(t,x) + sλ^2∫_Q e^-2sαγ |∇φ|^2 d(t,x)
+ s^-1∫_Q e^-2sαγ^-1(|φ_t|^2 + |Δφ|^2 ) d(t,x).
We are ready to recall the Carleman inequality that is the key to study the null controllability of the linear system (<ref>).
There exist positive constants s, λ and C depending solely on Ω and ω for which the relations g ∈ L^2(Q)^N, φ^T ∈ H, λ⩾λ and s ⩾s(T^4 + T^8) imply
[ I(s,λ,φ) ⩽ C(1+T^2)(s^15/2λ^20∫_Q e^-4sα_1 +2sα_2(γ_2/l^4)^15/2|g|^2d(t,x); + s^16λ^40∫_0^T ∫_ω_1 e^-8sα_1 + 6sα_2(γ_2/l^4)^16|φ|^2dx dt ), ]
where φ is the solution of (<ref>) corresponding to g and φ^T.
As a consequence, we get the following Observability Inequality.
With the notations of Proposition <ref> (possibly enlarging s, λ and C, the latter now depending on T), we have
φ(0)^2 ⩽ C(s^15/2λ^20∫_Q e^-4sα_1 +2sα_2γ_2^15/2|g|^2d(t,x) + s^16λ^40∫_0^T ∫_ω_1 e^-8sα_1 + 6sα_2γ_2^16|φ|^2dx dt ).
From now on, we fix λ = λ and s=s. Moreover, in view of Remark <ref>, given γ > 0, we can take λ = λ(γ) large enough in such a way that
α_2 < (1+γ)α_1.
Whenever we need (<ref>) in subsequent estimates, for a suitable positive real number γ, we will assume it holds in all that follows.
For p,q,r ∈ℝ, we introduce the weights
μ_p,q,r(t):= exp{psα_1 l^-4(t) }exp{qsα_2 l^-4(t) } l^r(t).
Regarding these weights, it is valuable to note:
Let p,p_1,p_2,q,q_1,q_2,r,r_1,r_2 be nine real numbers.
(a) One has the equality μ_p_1,q_1,r_1μ_p_2,q_2,r_2 = μ_p_1+p_2,q_1+q_2,r_1+r_2. In particular, for integral k, μ_p,q,r^k = μ_kp,kq,kr.
(b) There exists a constant C>0 such that
|d/dtμ_p,q,r|⩽ Cμ_p,q,r-5,
|d/dt(μ_p,q,r^2) |⩽ C μ_p,q,r-5/2^2.
(c) There exists a constant C>0 such that μ_p_1,q_1,r_1⩽ Cμ_p_2,q_2,r_2 if, and only if,
p_1α_1 + q_1α_2 = p_2α_1 + q_2α_2 and r_1 ⩾ r_2,
or
p_1α_1 + q_1α_2 < p_2α_1 + q_2α_2.
We define the weights
ρ_0 := μ_0,1,6,
ρ_1 := μ_0,1,2,
ρ_2 := μ_0,1,-2,
ρ_3 := μ_2,-1,15,
and
ρ_4 := μ_4,-3,32.
With these notations, we can gather Proposition <ref> and Corollary <ref> together, resulting in the following statement.
There is a constant C=C(Ω,ω,s,λ,m,T)>0 such that the solution φ of (<ref>) corresponding to g ∈ L^2(Q)^N and φ^T ∈ H satisfies
[ φ(0)^2 + ∫_Q[ ρ_0^-2|φ|^2 + ρ_1^-2|∇φ|^2 + ρ_2^-2(|φ_t|^2 + |Δφ|^2 )]d(t,x); ⩽ C(∫_Qρ_3^-2|g|^2d(t,x) + ∫_0^T ∫_ω_1ρ_4^-2|φ|^2 dx dt ). ]
§.§ Null controllability of the linear system
We suppose y_0 ∈ H, ρ_0 f ∈ L^2(Q)^N. Then there exist controls v ∈ L^2(]0,T[×ω)^N such that the state y of (<ref>) corresponding to v, f and y_0 satisfies
∫_Q ρ_3^2|y|^2d(t,x) + ∫_0^T∫_ωρ_4^2|v|^2dx dt ⩽ Cκ_0(y_0,f),
where
κ_0(y_0,f) := y_0_H^2 + ∫_Q ρ_0^2|f|^2d(t,x).
In particular, y(T) = 0 almost everywhere in Ω.
We define P_0 := { (w,σ) ∈ C^2(Q)^N+1 : ∇· w ≡ 0, w|_Σ≡ 0, ∫_Ωσ dx = 0 }, we take χ∈ C^∞_c(ω), with 0 ⩽χ⩽ 1, χ|_ω_1≡ 1, and we consider on P_0 the continuous bilinear form
b((w,σ),(w,σ)) := ∫_Q {ρ_3^-2(L^*w +∇σ)·(L^* w + ∇σ) + χρ_4^-2w·w} d(t,x).
By Corollary <ref>, b is an inner product on P_0. Let us denote P:= P_0^b(·,·), i.e., P is the completion of P_0 under the norm induced by b(·,·). We also deduce, from the corollary we just mentioned, that that the linear form
Λ : (w,σ) ∈ P ⟼∫_Ω y_0· w(0) dx + ∫_Q f· w d(t,x) ∈ℝ
is continuous, with
|Λ(w,σ)| ⩽ Cκ_0(y_0,f)^1/2b((w,σ),(w,σ))^1/2.
The Riesz representation theorem guarantees the existence of a unique (φ,π) ∈ P for which
Λ(w,σ) = b((w,σ),(φ,π)) (for all (w,σ) ∈ P).
Upon taking (w,σ) = (φ,π) above, we get
b((φ,π),(φ,π)) = Λ(φ,π) ⩽ Cκ_0(y_0,f)^1/2b((φ,π),(φ,π))^1/2,
whence
b((φ,π),(φ,π)) ⩽ Cκ_0(y_0,f).
Let us set
y:= ρ_3^-2(L^*φ + ∇π), z:= ρ_4^-2φ, v:= -χ z.
We observe that (v) ⊆ω, that (y,v) is a solution of (<ref>) corresponding to the datum y_0 and f, and applying Corollary <ref> once more,
∫_Q ρ_3^2|y|^2d(t,x) + ∫_0^T ∫_ωρ_4^2|v|^2 dx dt ⩽ Cb((φ,π),(φ,π)) ⩽ Cκ_0(y_0,f).
This proves the theorem.
§.§ Weighted energy estimates
Along this subsection, we let y_0 ∈ H, ρ_0 f ∈ L^2(Q)^N, and let us denote by (v,y) the control-state pair constructed in the proof of Theorem <ref>.
Let us define ρ_6 := μ_1,-1/2,35/2 and ρ_7 := μ_1,-1/2,20. We have
sup_[0,T]( ∫_Ωρ_6^2 |y|^2 dx) + ∫_Q ρ_6^2 |∇ y|^2 d(t,x) ⩽ Cκ_0(y_0,f),
and, if y_0 ∈ H^1_0(Ω)^N, then
∫_Q ρ_7^2(|y_t|^2+|Δ y|^2 )d(t,x) + sup_[0,T](∫_Ωρ_7^2|∇ y|^2 dx ) ⩽ Cκ_1(y_0,f),
where
κ_1(y_0,f) := y_0_H^1_0(Ω)^N^2 + ∫_Qρ_0^2 |f|^2d(t,x).
For each n ⩾ 1, let v_n(t, ·), f_n(t, ·) and y_0,n(·) be the projections of of v(t, ·), f(t, ·) and y_0(·) in the first n eigenfunctions for the Stokes operator A: D(A) → H, respectively. Let us denote by y_n the corresponding solution for the finite dimensional approximate forced Stokes system. For simplicity, unless we state otherwise, we omit the subscript n throughout the current proof. Moreover, we emphasize that we can take all of the constants C appearing below to be independent of n.
Using ρ_6^2y as a test function in system (<ref>), and doing some integrations by parts, we derive the identity
1/2d/dt(∫_Ωρ_6^2 |y|^2dx ) + ν_0 ∫_Ωρ_6^2 |∇ y|^2 dx = ∫_ωρ_6^2 v· y dx + ∫_Ωρ_6^2 f· y dx
+ 1/2∫_Ωd/dt(ρ_6^2 ) |y|^2dx.
From (<ref>) and Remark <ref>, item (c), we have ρ_6 ⩽ Cρ_4 ⩽ Cρ_3 ⩽ Cρ_0, whence
∫_ωρ_6^2 v· y dx ⩽ C(∫_ωρ_4^2 |v|^2 dx + ∫_Ωρ_3^2 |y|^2 dx ),
and
∫_Ωρ_6^2 f· y dx ⩽ C(∫_Ωρ_0^2 |f|^2 dx + ∫_Ωρ_3^2 |y|^2 dx ).
From Remark <ref>, item (b), we have |d/dt(ρ_6^2)| ⩽ Cρ_3^2, from where it follows that
∫_Ωd/dt( ρ_6^2 )|y|^2dx ⩽ C∫_Ωρ_3^2|y|^2 dx.
Using (<ref>), (<ref>) and (<ref>) in (<ref>), and applying Gronwall's inequality together with (<ref>), we infer (<ref>).
Henceforth, we will tacitly apply (<ref>) and Remark <ref>.
Now, we use ρ_7^2(y_t-ν_0 A y) as a test function in (<ref>), from where we easily derive that
[ ∫_Ωρ_7^2(|y_t|^2 + ν_0^2|Δ y|^2 )dx + 1/2d/dt(∫_Ωρ_7^2|∇ y|^2 dx ) = ∫_ωρ_7^2 v·(y_t-ν_0A y)dx; + ∫_Ωρ_7^2 f·(y_t-ν_0A y)dx + 1/2∫_Ωd/dt( ρ_7^2 )|∇ y|^2 dx. ]
We observe that, for any ϵ>0,
∫_ωρ_7^2 v·(y_t-ν_0A y)dx ⩽C/ϵ∫_ωρ_4^2|v|^2dx + Cϵ[∫_Ωρ_7^2(|y_t|^2 + |Δ y|^2)dx ],
∫_Ωρ_7^2 f·(y_t-ν_0A y)dx ⩽C/ϵ∫_Ωρ_0^2|f|^2dx + Cϵ[∫_Ωρ_7^2(|y_t|^2 + |Δ y|^2)dx ],
∫_Ωd/dt( ρ_7^2)|∇ y|^2 dx ⩽ C∫_Ωρ_6^2 |∇ y|^2 dx.
We take ϵ sufficiently small, in such a way that that the terms involving y in (<ref>) and (<ref>) are absorbed by the left-hand side of (<ref>). Also, from (<ref>) and (<ref>), the time integral of the third term in the right-hand side of (<ref>) is bounded by Cκ_0(y_0,f). Thus, it suffices to apply Gronwall's Lemma to conclude (<ref>) for the Galerkin approximates y_n instead of the actual solution y. Employing standard limiting arguments, as n →∞, we conclude that (<ref>) does hold for the actual solution y.
(a) If ζ := μ_-1,1,0, then
ζ v ∈ L^2(0,T;H^2(ω)∩ H^1_0(ω))∩ C([0,T];V), (ζ v)_t ∈ L^2(]0,T[×ω)^N,
with the estimate
∫_0^T ∫_ω[|(ζ v)_t|^2 + |ζΔ v|^2 ]dx dt + sup_[0,T]ζ v_V^2 ⩽ Cκ_0(y_0,f).
(b) Let us also assume that y_0 ∈ H^1_0(Ω)^N. For ζ := μ_-1,1,5, we have the memberships
(ζ v_t)_t ∈ L^2(]0,T[×ω)^N, ζ v_t ∈ L^2(0,T;[H^2(ω)∩ H^1_0(ω)]^N),
ζ v ∈ L^2(0,T;[H^4(ω)∩ H^1_0(ω)]^N),
and the following inequality holds
∫_0^T ∫_ω[|(ζ v_t)_t|^2 + |ζΔ v_t|^2 + |ζΔ^2v|^2 ] dx dt ⩽ Cκ_1(y_0,f).
(a) For p,q,r ∈ℝ, we notice that
L^*(μ_p,q,rz) = -d/dt(μ_p-8,q+6,r-64)φ + μ_p-4,q+4,r-34y - μ_p-8,q+6,r-64∇π.
Choosing p=-1, q=1, and r=0, it follows that
| d/dt(μ_p-8,q+6,r-64)| ⩽ Cρ_0^-1, μ_p-4,q+4,r-34⩽ Cρ_3, μ_p-8,q+6,r-64⩽ C.
Thus, u:= ζ z and π := μ_-9,7,-64π solve the Stokes equation
Lu + ∇π = h, in Q,
∇· u = 0, in Q,
u = 0, on Σ,
u(T) = 0, in Ω,
where
h := -d/dt(μ_-9,7,-64)φ + μ_-5,5,-34y ∈ L^2(Q)^N.
By standard regularity results for solutions of the Stokes system, we can infer the stated regularity for ζ v = -χ u.
(b) As in the previous item, for p,q,r ∈ℝ, we derive
[ L^*(μ_p,q,rz_t) = -φd/dt[μ_p,q,rd/dt(μ_-8,6,-64) ] + yμ_p+4,q-2,r+64d/dt(μ_-8,6,-64); - d/dt(μ_p-8,q+6.r-64)φ_t + μ_p-4,q+4,ry_t + μ_p-8,q+6,r-64d/dt(μ_4,-2,64) y; - μ_p-8,q+6,r-64∇π_t. ]
For the choice p=-1, q=1, r=5, it is straightforward to check the inequalities
|d/dt[μ_p,q,rd/dt(μ_-8,6,-64) ] | ⩽ Cμ_p-8,q+7,r-58ρ_0^-1 = Cμ_-9,8,-53ρ_0^-1⩽ Cρ_0^-1,
|μ_p+4,q-2,r+64d/dt(μ_-8,6,-64) | ⩽ Cμ_p-14,q+6,r-152ρ_3 = Cμ_-15,7,-147ρ_3 ⩽ Cρ_3,
|d/dt(μ_p-8,q+7,r-64) | ⩽ Cμ_p-8,q+7,r-71ρ_2 = Cμ_-9,8,-66ρ_2 ⩽ Cρ_2,
μ_p-4,q+4,r = μ_p-6,q+5,r-20ρ_7 = μ_-7,6,-15ρ_7 ⩽ Cρ_7,
|μ_p-8,q+6,r-64d/dt(μ_4,-2,64) | ⩽ Cμ_p-6,q+5,r-37ρ_3 = Cμ_-7,6,-32ρ_3 ⩽ Cρ_3,
and
μ_p-8,q+6,r-64 = μ_-9,7,-59⩽ C.
We can conclude by arguing similarly as in the first two memberships and corresponding estimates. The third ones are obtained doing the same analysis for the term L^*(ζΔ z).
Let us set ρ_8 := ζ = μ_-1,1,0 and ρ_9 := μ_-1,1,5/2. Supposing y_0 ∈ H^2(Ω)^N∩ V, Ay_0 ∈[H^1_0(Ω)]^N, ρ_8f_t ∈ L^2(Q)^N, we have the following estimates:
sup_[0,T](∫_Ωρ_8^2|y_t|^2dx ) + ∫_Q ρ_8^2|∇ y_t|^2 d(t,x) ⩽ Cκ_2(y_0,f).
If furthermore y_0 ∈ H^3(Ω)^N, f(0) ∈ H^1_0(Ω)^N,
∫_Q ρ_9^2(|y_tt|^2 + |Δ y_t|^2 )d(t,x) + sup_[0,T][∫_Q ρ_9^2(|∇ y_t|^2 + |Δ y|^2 )dx ] ⩽ Cκ_3(y_0,f),
where
κ_2(y_0,f):= y_0_H^2(Ω)^N^2 + ∫_Qρ_0^2 |f|^2d(t,x) + ∫_Q ρ_8^2|f_t|^2 d(t,x)
and
κ_3(y_0,f) := y_0_H^3(Ω)^N^2 + ∫_Qρ_0^2 |f|^2d(t,x) + ∫_Q ρ_8^2|f_t|^2 d(t,x) + f(0)_H^1_0(Ω)^N^2.
We establish the current estimates by following the same approach as in the proof of Lemma <ref>. Here, we begin by differentiating the system (<ref>) with respect to time, and we use ρ_8^2 y_t as a test function:
1/2d/dt(∫_Ωρ_8^2|y_t|^2 dx ) + ν_0 ∫_Ωρ_8^2 |∇ y_t|^2 dx = ∫_ωρ_8^2 v_t· y_t dx + ∫_Ωρ_8^2 f_t· y_t dx
+ 1/2∫_Ωd/dt(ρ_8^2)|y_t|^2 dx.
We note that
| d/dt(ρ_8^2) | ⩽ Cρ_7^2, ρ_8 ⩽ Cρ_7 ⩽ Cρ_4;
hence,
∫_ωρ_8^2 v_t· y_t dx ⩽ C[∫_ω(|ρ_4 v|^2 + |(ζ v)_t|^2 )dx + ∫_Ωρ_7^2 |y_t|^2 dx ],
∫_Ωρ_8^2 f_t · y_t dx ⩽ C(∫_Ωρ_8^2 |f_t|^2 dx + ∫_Ωρ_7^2|y_t|^2 dx ).
By Lemmas <ref> and <ref>,
∫_Qρ_7^2 |y_t|^2 d(t,x) + ∫_0^T∫_ω(|ρ_4 v|^2 + |(ζ v)_t|^2 )dx dt ⩽κ_1(y_0,f),
so that by using (<ref>) and (<ref>) in (<ref>), then integrating in time and applying of Gronwall's lemma, it follows that
sup_[0,T](∫_Ωρ_8^2|y_t|^2 dx) + ∫_Q ρ_8^2|∇ y_t|^2 d(t,x) ⩽ C (y_t(0)_L^2(Q)^N^2 +∫_Q ρ_8^2|f_t|^2d(t,x) .
+ κ_1(y_0,f) ).
It is simple to infer the subsequent estimate:
y_t(0)_L^2(Q)^N^2 ⩽y_0_H^2(Ω)^N^2 + f(0)_L^2(Q)^N^2 + v(0)_L^2(]0,T[×ω)^N^2 ⩽κ_2(y_0,f).
Relations (<ref>) and (<ref>) imply (<ref>).
Next, we use ρ_9^2(y_tt-ν_0A y_t) as a test function in the system (<ref>) differentiated with respect to time to deduce
[ ∫_Ωρ_9^2 (|y_tt|^2 + ν_0^2|Δ y_t|^2 )dx + ν_0d/dt( ∫_Ωρ_9^2 |∇ y_t|^2 dx ) = ∫_ωρ_9^2 v_t· (y_tt-ν_0A y_t) dx; + ∫_Ωρ_9^2 f_t· (y_tt-ν_0A y_t)dx + ν_0∫_Ωd/dt(ρ_9^2 )|∇ y_t|^2 dx ]
We observe that
∫_Ωd/dt(ρ_9^2) |∇ y_t|^2dx ⩽ C∫_Ωρ_8^2|∇ y_t|^2 dx,
and for each ϵ > 0,
∫_ωρ_9^2 v_t· (y_tt-ν_0A y_t)dx ⩽ C[ 1/ϵ∫_ωζ^2 |v_t|^2 dx + ϵ∫_Ωρ_9^2( |y_tt|^2 + |Δ y_t|^2 ) dx ],
as well as
∫_Ωρ_9^2f_t · (y_tt-ν_0A y_t)dx ⩽ C[1/ϵ∫_Ωρ_8^2|f_t|^2 dx + ϵ∫_Ωρ_9^2(|y_tt|^2 + |Δ y_t|^2)dx ].
We fix a sufficiently small ϵ, whence the second terms within the brackets in the right-hand sides of (<ref>) and (<ref>) are absorbed by the left-hand side of (<ref>). Then, using (<ref>) in (<ref>), we infer
[ ∫_Ωρ_9^2 (|y_tt|^2 + |Δ y_t|^2 )dx + d/dt( ∫_Ωρ_9^2 |∇ y_t|^2 dx ) ⩽ C(∫_ωζ^2 |v_t|^2 dx + ∫_Ωρ_8^2 |f_t|^2dx; + ∫_Ωρ_8^2 |∇ y_t|^2 dx ). ]
Employing Gronwall's lemma in (<ref>), we obtain
∫_Q ρ_9^2 (|y_tt|^2 + |Δ y_t|^2)d(t,x) + sup_[0,T](∫_Ωρ_9^2|∇ y_t|^2 dx ) ⩽ C(∇ y_t(0)_L^2(Q)^N^2 + κ_2(y_0,f) ).
We easily establish, with the aid of item (a) of Lemma <ref>, that
∇ y_t(0)_L^2(Q)^N^2 ⩽ C(y(0)_H^3(Q)^N^2 + ∇ v(0)_L^2(Q)^N^2 + ∇ f(0)_L^2(Q)^N^2 ) ⩽ Cκ_3(y_0,f),
whence
∫_Q ρ_9^2 (|y_tt|^2 + |Δ y_t|^2)d(t,x) + sup_[0,T](∫_Ωρ_9^2|∇ y_t|^2 dx ) ⩽ Cκ_3(y_0,f).
Finally, we use ρ_9^2Δ y_t in the undifferentiated partial differential equation of system (<ref>) as a test function to get
[ ∫_Ωρ_9^2 |∇ y_t|^2 dx + ν_0/2d/dt(∫_Ωρ_9^2 |Δ y|^2 dx ); = ∫_ωρ_9^2 v·Δ y_t dx + ∫_Ωρ_9^2 f·Δ y_t dx + ν_0/2∫_Ωd/dt(ρ_9^2)|Δ y|^2 dx; ⩽ C(∫_ωρ_4^2 |v|^2 dx + ∫_Ωρ_0^2 |f|^2 dx + ∫_Ωρ_9^2 |Δ y_t|^2 dx + ∫_Ωρ_7^2|Δ y|^2 dx). ]
We use Gronwall's lemma in (<ref>), in such a way that
sup_[0,T](∫_Ωρ_9^2 |Δ y|^2 dx ) ⩽ Cκ_3(y_0,f).
From the estimates (<ref>) and (<ref>), together with the compatibility condition Ay_0 ∈[H^1_0(Ω)]^N, we derive (<ref>).
We write ρ_10:= ζ = μ_-1,1,5 and ρ_11 := μ_-1,1,15/2. Let us assume that y_0 ∈ H^4(Ω)^N∩ V, Ay_0, A^2y_0 ∈[H^1_0(Ω)]^N, ρ_9 Δ f ∈ L^2(Q)^N, ρ_10 f_tt∈ L^2(Q)^N, ρ_10f_t ∈ L^2(0,T; H^1_0(Ω)^N), f(0) ∈[H^2(Ω)∩ H^1_0(Ω)]^N and f_t(0)∈ L^2(Ω). Then, the following estimate holds
[ sup_[0,T][∫_Ωρ_10^2 (|y_tt|^2 + |Δ y_t|^2 )dx + ρ_9 y_H^3(Ω)^N^2 ]; + ∫_Q (ρ_10^2 |∇ y_tt|^2 +ρ_10^2|D^3 y_t|^2 + ρ_9^2|D^4 y|^2 ) d(t,x) ⩽ Cκ_4(y_0,f). ]
If, furthermore, y_0 ∈ H^5(Ω)^N, f(0) ∈ H^3(Ω)^N, Af(0) ∈ V, and f_t(0) ∈ H^1_0(Ω)^N, then
sup_[0,T](∫_Ωρ_11^2 |∇ y_tt|^2 dx ) + ∫_Qρ_11^2(|y_ttt|^2 + |Δ y_tt|^2 )d(t,x) ⩽ C κ_5(y_0,f).
where we have written
κ_4(y_0,f) := ∫_Q(ρ_9^2 |Δ f|^2 +ρ_10^2|∇ f_t|^2 + ρ_10^2|f_tt|^2) d(t,x)
+ y_0_H^4(Ω)^N^2 + f(0)_H^2(Ω)^N^2 + f_t(0)_L^2(Ω)^N^2 + κ_3(y_0,f),
κ_5(y_0,f) := y_0_H^5(Ω)^N^2 + f(0)_H^3(Ω)^N^2 +f_t(0)_H^1_0(Ω)^N^2 + κ_4(y_0,f).
Again, we proceed in the same framework as in the proof of Lemma <ref>. We begin by applying the Stokes operator A on the equation of system (<ref>), and then use -ρ_9^2A^2 y as a test function:
[ ν_0∫_Ωρ_9^2|Δ^2 y|^2 dx = -∫_ωρ_9^2 Δ v · A^2 y dx - ∫_ΩΔ f · A^2 y dx + ∫_Ωρ_9^2 Δ y_t · A^2 y dx; ⩽ C(∫_ωζ^2|Δ v|^2 dx + ∫_Ωρ_9^2 |Δ f|^2 dx + ∫_Ωρ_9^2 |Δ y_t|^2 dx ) + 1/2∫_Ωρ_9^2 |Δ^2 y|^2 dx, ]
We integrate (<ref>) with respect to time, whence
∫_Q ρ_9^2 |Δ^2 y|^2 d(t,x) ⩽ Cκ_4(y_0,f).
We can now easily argue that, under suitable limiting arguments (having in view the compatibility conditions we required in the statement of the present lemma), Eq. (<ref>) yields the corresponding estimate for the solution of (<ref>). We observe that the relations ρ_9A y_t ∈ L^2(Q)^N and ρ_9A^2 y ∈ L^2(Q)^N imply ρ_9 y ∈ L^∞(0,T; H^3(Ω)^N), with
sup_[0,T]ρ_9 y_H^3(Ω)^N^2 ⩽ C∫_Q ρ_9^2 ( |Δ y_t|^2 + |Δ^2 y|^2 )d(t,x) ⩽ Cκ_4(y_0,f).
In the differential equation of system (<ref>) differentiated once with respect to time, we use the test function ρ_10^2 A^2 y_t:
1/2d/dt(∫_Ωρ_10^2 |Δ y_t|^2 dx ) + ν_0∫_Ωρ_10^2 |∇Δ y_t|^2 dx
= ∫_ωρ_10^2 ∇ v_t : ∇Δ y_t dx + ∫_Ωρ_10^2 ∇ f_t : ∇Δ y_tdx
+ 1/2∫_Ωd/dt( ρ_10^2 ) |Δ y_t|^2 dx.
For ϵ > 0,
[ ∫_ωρ_10^2 ∇ v_t : ∇Δ y_tdx ⩽ C_ϵ∫_ωζ^2 |∇ v_t|^2 dx + ϵ∫_Ωρ_10^2|∇Δ y_t|^2 dx; ⩽ C_ϵ∫_ω( |(ζv_t)_t|^2 + |(ζ v)_t|^2 + ρ_4^2|v|^2)dx + ϵ∫_Ωρ_10^2 |∇Δ y_t|^2 dx, ]
[ ∫_Ωρ_10^2 ∇ f_t: ∇Δ y_t dx ⩽ C_ϵ∫_Ωρ_10^2 |f_tt|^2 dx + ϵ∫_Ωρ_10^2 |∇Δ y_t|^2 dx , ]
∫_Ωd/dt( ρ_10^2)|Δ y_t|^2 dx ⩽ C ∫_Ωρ_9^2 |Δ y_t|^2,
and
[ Δ y_t(0)^2 ⩽ C(y_0_H^4(Ω)^N^2 + Δ v(0)_L^2(]0,T[×ω)^N^2 + Δ f(0)^2 ); ⩽ Cκ_4(y_0,f). ]
Therefore, by taking ϵ sufficiently small, and using (<ref>)-(<ref>) in (<ref>), we deduce
sup_[0,T](∫_Ωρ_10^2 |Δ y_t|^2dx ) + ∫_Q ρ_10^2 |∇Δ y_t|^2 d(t,x) ⩽ Cκ_4(y_0,f).
Next, we differentiate the equation of system (<ref>) twice with respect to time and we use the test function ρ_10^2 y_tt:
1/2d/dt(∫_Ωρ_10^2 |y_tt|^2 dx ) + ν_0∫_Ωρ_10^2 |∇ y_tt|^2 dx = ∫_ωρ_10^2 v_tt· y_tt dx + ∫_Ωρ_10^2 f_tt· y_ttdx
+ 1/2∫_Ωd/dt( ρ_10^2 ) |y_tt|^2 dx.
We have
[ ∫_ωρ_10^2 v_tt· y_ttdx ⩽ C(∫_ωζ^2 |v_tt|^2 dx + ∫_Ωρ_9^2|y_tt|^2 dx ); ⩽ C[ ∫_ω( |(ζv_t)_t|^2 + |(ζ v)_t|^2 + ρ_4^2|v|^2)dx + ∫_Ωρ_9^2 |y_tt|^2 dx], ]
[ ∫_Ωρ_10^2 f_tt· y_tt dx ⩽ C(∫_Ωρ_10^2 |f_tt|^2 dx + ∫_Ωρ_9^2 |y_tt|^2 dx ), ]
∫_Ωd/dt( ρ_10^2)|y_tt|^2 dx ⩽ C ∫_Ωρ_9^2 |y_tt|^2,
and
[ y_tt(0)^2 ⩽ C(y_0_H^4(Ω)^N^2 + Δ v(0)_L^2(]0,T[×ω)^N^2 + Δ f(0)^2 .; . + v_t(0)_L^2(]0,T[×ω)^N^2 + f_t(0)^2 ); ⩽ Cκ_4(y_0,f). ]
Using (<ref>), (<ref>) and (<ref>) in (<ref>), then integrating in time and using (<ref>), we infer
sup_[0,T](∫_Ωρ_10^2|y_tt|^2 dx ) + ∫_Q ρ_10^2 |∇ y_tt|^2 d(t,x) ⩽ Cκ_4(y_0,f).
Estimates (<ref>), (<ref>), (<ref>) and (<ref>) are enough to conclude (<ref>).
Now, we use ρ_11^2(y_ttt - ν_0 A y_tt) as a test function in the equation of system (<ref>) twice differentiated in time, reaching
[ ∫_Ωρ_11^2|y_ttt|^2 dx + ν_0^2 ∫_Ωρ_11^2|Δ y_tt|^2 dx + ν_0 d/dt(∫_Ωρ_11^2 |∇ y_tt|^2 dx ); = ∫_ωρ_11^2 v_tt· (y_ttt-ν_0A y_tt) dx + ∫_Ωρ_11^2 f_tt· (y_ttt-ν_0 A y_tt)dx; + ν_0 ∫_Ωd/dt(ρ_11^2)|∇ y_tt|^2 dx. ]
For ϵ > 0,
[ ∫_ωρ_11^2 v_tt· (y_ttt-ν_0A y_tt) dx ⩽ C[1/ϵ∫_ω(|(ζv_t)_t|^2 + |(ζ v)_t|^2 + ρ_4^2|v|^2)dx; + ϵ∫_Ωρ_11^2( |y_ttt|^2 + |Δ y_tt|^2 )dx ], ]
∫_Ωρ_11^2 f_tt· (y_ttt-ν_0 A y_tt)dx ⩽ C[1/ϵ∫_Ωρ_10^2 |f_tt|^2 dx + ϵ∫_Ωρ_11^2( |y_ttt|^2 + |Δ y_tt|^2 )dx],
and we also notice that
∫_Ωd/dt(ρ_11^2)|∇ y_tt|^2 dx ⩽ C ∫_Ωρ_10^2 |∇ y_tt|^2 dx.
We easily check that
∇ y_tt(0)^2 ⩽ Cκ_5(y_0,f).
As in the proof of (<ref>), inequalities (<ref>)-(<ref>), we can infer from (<ref>), through an adequate choice of a small positive ϵ, and the aid of the previous estimates, the subsequent inequality
∫_Q ρ_11^2 ( |y_ttt|^2 +|Δ y_tt|^2) d(t,x) + sup_[0,T](∫_Ωρ_11^2 |∇ y_tt|^2 dx ) ⩽ Cκ_5(y_0,f).
Estimate (<ref>) is precisely (<ref>); hence, we have finished the proof of the present result.
§ NULL CONTROLLABILITY OF THE MODEL (<REF>)
§.§ Local right inversion theorem
It is possible to find a proof of the subsequent result in <cit.>. This is the inversion theorem that we will use to obtain our local null controllability result.
Let Y and Z be two Banach spaces, and H : Y → Z be a continuous function, with H(0) = 0. We assume that there are three constants δ,η^',M > 0 and a continuous linear mapping Λ from Y onto Z with the following properties:
(i) For all e ∈ Y, we have
e_Y ⩽ MΛ(e)_Z;
(ii) The constants δ and M satisfy δ < M^-1;
(iii) Whenever e_1,e_2 ∈ B_Y(0;η^'), the inequality
H(e_1) - H(e_2) - Λ(e_1-e_2)_Z ⩽δe_1-e_2_Y
holds.
Then, whenever k ∈ B_Z(0;η), the equation H(e) = k has a solution e ∈ B_Y(0;η^'), where η:= (M^-1 - δ)η^'.
A typical way of verifying condition (iii) is through the remark presented below.
Let Y and Z be two Banach spaces, and let us consider a continuous mapping H:Y→ Z, with H(0)=0, of class C^1(Y,Z). Then it has property (iii), for any positive δ subject to (ii), as long as we take η^' as the continuity constant of DH ∈ℒ(Y,ℒ(Y,Z)) at the origin of Y.
§.§ The setup
Let us set
X_1 := L^2(Q;ρ_3^2)^N, X_2 := L^2(]0,T[×ω; ρ_4^2)^N.
We define
[ Y := { (y,p,v) ∈ X_1 × L^2(Q) × X_2 : y_t ∈ L^2(Q)^N, ∇ y ∈ L^2(Q)^N× N,; (ζ v)_t, ζΔ v, (ζ v_t)_t, ζΔ v_t, ζD^4 v ∈ L^2(]0,T[×ω)^N,; for f:= Ly + ∇ p - χ_ω, ρ_0 f, ρ_8 f_t, ρ_9 Δ f, ρ_10f_tt∈ L^2(Q)^N,; ρ_10f_t ∈ L^2(0,T; H^1_0(Ω)^N), f(0) ∈[H^3(Ω)∩ H^1_0(Ω)]^N,; Af(0) ∈ H^1_0(Ω)^N, f_t(0) ∈ H^1_0(Ω)^N, y|_Σ≡ 0, ∇· y ≡ 0,; y(0) ∈[H^5(Ω)∩ V]^N, Ay(0), A^2y(0) ∈ H^1_0(Ω)^N, ∫_Ω p dx = 0 }. ]
We consider on Y the norm
[ (y,p,v)_Y^2 := ∫_Q(ρ_3^2|y|^2 + ρ_0^2 |f|^2+ ρ_8^2 |f_t|^2 + ρ_9^2 |Δ f|^2 + ρ_10^2|∇ f_t|^2 + ρ_10^2|f_tt|^2 )d(t,x); + ∫_0^T ∫_ω(ρ_4^2|v|^2 + |(ζ v)_t|^2 + |ζΔ v|^2 + |(ζ v_t)_t|^2 + |ζΔ v_t|^2 + |D^4( ζ v )|^2 ) dx dt; + f(0)_[H^3(Ω)∩ H^1_0(Ω)]^N^2 + f_t(0)_H^1_0(Ω)^N^2 + y(0)_H^5(Ω)^N^2, ]
where in (<ref>) we have written f:= Ly + ∇ p - χ_ω v. Then, endowing the space Y with ·_Y renders it a Banach space.
Now, we put
[ F := { f ∈ L^2(Q)^N : ρ_0 f, ρ_8 f_t, ρ_9 Δ f, ρ_10f_tt∈ L^2(Q)^N, ρ_10f_t ∈ L^2(0,T; H^1_0(Ω)^N),; f(0) ∈[H^3(Ω)∩ H^1_0(Ω)]^N, f_t(0) ∈ H^1_0(Ω)^N }, ]
f_F^2 := ∫_Q(ρ_0^2 |f|^2+ ρ_8^2 |f_t|^2 + ρ_9^2 |Δ f|^2 + ρ_10^2|∇ f_t|^2 + ρ_10^2|f_tt|^2 )d(t,x)
+ f(0)_[H^3(Ω)∩ H^1_0(Ω)]^N^2 + f_t(0)_H^1_0(Ω)^N^2,
and also consider the space of initial conditions
G := { y_0 ∈ H^5(Ω)^N ∩ V : Ay_0, A^2y_0 ∈ H^1_0(Ω)^N },
with the same topology as H^5(Ω)^N∩ V. Then, we define
Z := F × G,
The space Z with the natural product topology is also a Banach space.
Finally, we define the mapping H : Y → Z by
H(y,p,v) := (Dy/Dt - ∇·𝒯(y,p) - χ_ω v , y(0)).
§.§ Three lemmas and the conclusion
The mapping H : Y → Z is well-defined, and it is continuous.
We write H(y,p,v) = (H_1(y,p,v),H_2(y,p,v)), where
H_1(y,p,v) := Dy/Dt - ∇·𝒯(y,p) - χ_ω v;
H_2(y,p,v) := y(0).
There is nothing to prove about H_2, since it is cleary linear and continuous. We will consider only the mapping H_1 in what follows.
We decompose H_1(y,v) = h_1(y,p,v) + h_2(y,p,v) + h_3(y,p,v), where
h_1(y,p,v):= y_t -ν(0)Δ y + ∇ p - χ_ω v,
h_2(y,p,v):= - ∇·[(ν(∇ y)-ν(0))∇ y ],
h_3(y,p,v)=(y·∇) y.
By the definition of the norm of F, it follows promptly that
h_1(y,p,v)_F < ∞.
Next, we will prove that the quantity h_2(y,p,v)_F is finite.
CLAIM 1: Δ h_2(y,p,v)_9 < ∞.
We notice that
[ |Δ h_2(y,p,v)| ⩽ C[ (r+1)r|r-1||∇ y|^r-2|D^2 y|^3 + (r+1)r|∇ y|^r-1|D^2 y||D^3 y|+(r+1)|∇ y|^r|D^4 y|]; = C(D_1,1 + D_1,2 + D_1,3). ]
In the case r=1, the term D_1,1 vanishes and thus |Δ h_2| is bounded by C(D_1,2 + D_1,3). Otherwise, assuming r ⩾ 2, we have
[ ∫_Q ρ_9^2 D_1,1^2 d(t,x) ⩽ C(r)∫_0^T ρ_9^2 ∫_Ω |∇ y|^2(r-2) |D^2 y|^6 dx dt; ⩽ C(r)∫_0^T ρ_9^2D^3 y^2(r-2)D^2 y_L^6(Ω)^6 dt; ⩽ C(r) ∫_0^T ρ_9^2D^3 y^2(r-2)y_H^3(Ω)^6 dt; ⩽ C(r) ∫_0^T ρ_9^2D^3 y^2(r+2) dt; ⩽ C(r)( sup_[0,T]ρ_9D^3 y)^2(r+2)∫_Q ρ_9^-2(r+1) d(t,x) < ∞. ]
In the above equations, we used the continuous immersions: H^2(Ω) ↪ L^∞(Ω); H^1(Ω) ↪ L^6(Ω). These are valid for N ⩽ 3, see <cit.>, and we will use them tacitly henceforth.
Now, we obtain the estimate for D_1,2:
[ ∫_Q ρ_9^2 D_1,2^2 d(t,x) ⩽ C(r) ∫_0^T ρ_9^2 ∫_Ω |∇ y|^2(r-1)|D^2 y|^2|D^3 y|^2 dx dt; ⩽ C(r) ∫_0^T ρ_9^2D^3 y^2rD^4 y^2 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2r∫_Q ρ_9^2|D^4 y|^2 d(t,x) < ∞. ]
Likewise, we show D_1,3 to be finite, since
[ ∫_Q ρ_9^2 D_1,3^2 d(t,x) ⩽ C(r) ∫_0^T ρ_9^2 ∫_Ω |∇ y|^2r |D^4 y|^2dx dt; ⩽ C(r) ∫_0^T ρ_9^-2r(ρ_9D^3 y)^2r(ρ_9D^4 y)^2 dt; ⩽ C(r) sup_[0,T]ρ_9D^3 y^2r∫_Q ρ_9^2 |D^4 y|^2 d(t,x) < ∞. ]
CLAIM 2: ∂_t^2 h_2(y,p,v)_10 < ∞.
We begin with the pointwise estimate,
[ |∂_t^2 h_2(y,p,v)| ⩽ C[ (r+1)r|r-1||∇ y|^r-2|∇ y_t|^2|Δ y| + (r+1)r|∇ y|^r-1|∇ y_tt||Δ y|; + (r+1)r|∇ y|^r-1|∇ y_t||Δ y_t| +(r+1)|∇ y|^r|Δ y_tt|]; = C(D_2,1+D_2,2+D_2,3+D_2,4). ]
As in the previous claim, if r=1, then D_2,1≡ 0. For r ⩾ 2, the next estimate is valid:
[ ∫_Q ρ_10^2 D_2,1^2 d(t,x) ⩽ C(r) ∫_0^T ρ_10^2 ∫_Ω |∇ y|^2(r-2)|∇ y_t|^4|Δ y|^2 dx dt; ⩽ C(r) ∫_0^T ρ_9^-2rρ_9 D^3 y^2(r-2)ρ_9 D^4 y^2 ρ_10Δ y_t^4 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2(r-2)(sup_[0,T]ρ_10Δ y_t)^4 ∫_Q ρ_9^2|D^4 y|^2 d(t,x); < ∞. ]
Proceeding similarly, we prove the remaining inequalities:
[ ∫_Q ρ_10^2 D_2,2^2 d(t,x) ⩽ C(r) ∫_0^T ρ_10^2 ∫_Ω |∇ y|^2(r-1)|∇ y_tt|^2|Δ y|^2 dx dt; ⩽ C(r) ∫_0^T ρ_10^2ρ_9^-2rρ_11^-2ρ_9 D^3 y^2(r-1)ρ_9 D^4 y^2 ρ_11∇ y_tt^2 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2(r-1)(sup_[0,T]ρ_11∇ y_tt)^2 ∫_Q ρ_9^2|D^4 y|^2 d(t,x); < ∞; ]
[ ∫_Q ρ_10^2 D_2,3^2 d(t,x) ⩽ C(r) ∫_0^T ρ_10^2 ∫_Ω |∇ y|^2(r-1) |∇ y_t|^2|Δ y_t|^2 dx dt; ⩽ C(r) ∫_0^T ρ_9^-2(r-1)ρ_10^-2ρ_9 D^3 y^2(r-1)ρ_10 D^3 y_t^2 ρ_10Δ y_t^2 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2(r-1)(sup_[0,T]ρ_10Δ y_t)^2 ∫_Q ρ_10^2|D^3 y_t|^2 d(t,x); < ∞; ]
[ ∫_Q ρ_10^2 D_2,4^2 d(t,x) ⩽ C(r) ∫_0^T ρ_10^2 ∫_Ω |∇ y|^2r |Δ y_tt|^2 dx dt; ⩽ C(r) ∫_0^T ρ_10^2ρ_9^-2rρ_11^-2ρ_9 D^3 y^2rρ_11Δ y_tt^2 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2r∫_Q ρ_11^2|Δ y_tt|^2 d(t,x); < ∞. ]
This finishes the proof of the second claim.
CLAIM 3: |∂_t ∇ h_2(y,p,v)| _10 < ∞.
As before, we begin by considering the pointwise estimate:
[ |∂_t ∇ h_2(y,p,v)| ⩽ C[(r+1)r|r-1||∇ y|^r-2|∇ y_t||Δ y| + (r+1)r|∇ y|^r-1|Δ y_t| .; .+ (r+1)|∇ y|^r|D^3 y_t| ]; = C(D_3,1 + D_3,2 + D_3,3). ]
Again, if r=1, then we need not consider D_3,1, since it vanishes. For r⩾ 2,
[ ∫_Q ρ_10^2 D_3,1^2 d(t,x) ⩽ C(r)∫_0^T ρ_10^2 ∫_Ω |∇ y|^2(r-2)|∇ y_t|^2 |Δ y|^2 dx dt; ⩽ C(r) ∫_0^T ρ_10^2 ρ_9^-2rρ_9D^3 y^2(r-2)ρ_9D^4 y^2ρ_9∇ y_t^2 dt; ⩽ C(r) (sup_[0,T]ρ_9D^3 y)^2(r-2)(sup_[0,T]ρ_9D^4 y)^2∫_Q ρ_9^2 |∇ y_t|^2 d(t,x); < ∞, ]
[ ∫_Q ρ_10^2 D_3,2^2 d(t,x) ⩽ C(r)∫_0^T ρ_10^2∫_Ω |∇ y|^2(r-1)|Δ y_t|^2 dx dt; ⩽ C(r) ∫_0^T ρ_9^-2(r-1)ρ_9D^3 y^2(r-1)ρ_10Δ y_t^2 dt; ⩽ C(r)(sup_[0,T]ρ_9D^3 y^2 )^2(r-1)(sup_[0,T]ρ_10Δ y_t)^2 < ∞, ]
and
[ ∫_Q ρ_10^2 D_3,3^2 d(t,x) ⩽ C(r)∫_0^Tρ_10^2 ∫_Ω |∇ y|^2r|D^3 y_t|^2dx dt; ⩽ C(r)∫_0^T ρ_9^-2rρ_9D^3 y^2rρ_10D^3 y_t^2dt; ⩽ C(r)(sup_[0,T]ρ_9D^3 y)^2r∫_Q ρ_10^2|D^3 y_t|^2 d(t,x) <∞. ]
These inequalities confirm the third claim.
The remaining terms composing the F-norm of h_2(y,p,v), h_2(y,p,v)_F, are norms of lower order derivatives of it, compared to the ones considered above, in adequate weighted L^2 spaces. Therefore, these terms are even easier to handle. A similar remark is also true for h_3(y,p,v)_F. In addition, we can show the continuity of H via estimates which are very similar to the ones that we carried out in the claims above; hence, we omit these computations. This ends the proof of the Lemma.
The mapping H is strictly differentiable at the origin of Y, with derivative DH(0,0,0) = Λ∈ℒ(Y,Z) given by
Λ· (y,p,v) = ( y_t - ν_0 Δ y + ∇ p - χ_ω v, y(0)) = (Λ_1· (y,p,v),Λ_2· (y,p,v)).
In fact, H is of class C^1(Y,Z) and, for each (y,p,v) ∈ Y, its derivative DH(y,p,v) ∈ℒ(Y,Z) is given by
DH(y,p,v)· (y,p,v) = (Λ_1(y,p,v)· (y,p,v) , Λ_2 · (y,p,v) ),
where we have written
Λ_1(y,p,v)· (y,p,v) := Λ_1· (y,p,v)
- rν_1∇·[ χ_y |∇y|^r-2∇y : ∇ y ∇y + |∇y|^r∇ y ]
+ (y ·∇) y + (y·∇) y ,
∇y : ∇ y := ( ∇y^⊺∇ y ),
χ_y is the indicator function of the set {∇y≠ 0}.
We will only prove the first claim, i.e., that H is strictly differentiable at the origin (0,0,0) ∈ Y, with DH(0,0,0) being onto Z. There is no additional difficulty to prove the lemma in its full force.
We write H = (H_1,H_2) as in (<ref>) of Lemma <ref>. Again, it is only necessary to investigate H_1, since H_2 is linear and continuous, and therefore C^∞. Given (y,p,v),(y,p,v) ∈ Y, we note that
H_1(y,p,v) - H_2(y,p,v) - Λ_1 · (y-y, p - p,v-v) = -ν_1 D_1 + D_2,
where
D_1 := ∇·(|∇y|^r ∇y - |∇ y|^r∇ y ),
D_2 := (y·∇)y - (y ·∇) y.
Let us take two positive real numbers, ϵ and δ, and we suppose (y,p,v)_Y ⩽δ, (y,p,v)_Y ⩽δ. We must show that we can take δ = δ(ϵ) such that
H_1(y,p,v) - H_2(y,p,v) - Λ_1 · (y-y, p - p,v-v)_F ⩽ϵ(y-y, p - p, v-v)_Y.
We assume, without loss of generality, that δ < 1. It is enough to show that
ν_1D_1_F + D_2_F ⩽ϵ(y-y, p-p, v- v)_Y,
for a suitable δ = δ(ϵ). To begin with, we observe that
[ |Δ D_1| ⩽ C (r+1)r[|r-1|||∇y|^r-2 - |∇ y|^r-2||D^2y|^3; + |r-1||∇ y|^r-2(|D^2y|^2 + |D^2 y|^2 )|∇y -∇ y|; + |r-1|(|∇y|^r-2 + |∇ y|^r-2)|∇y -∇ y||D^2y||D^3y|; + |∇ y|^r-1|D^2y-D^2 y||D^3y| +|∇ y|^r-1|D^2 y||D^3(y-y)|; + |∇y|^r-1|∇(y-y)||D^4 y| + |∇ y|^r|D^4(y-y)| ]; = C(r+1)r(D_1,1 + ⋯ + D_1,7). ]
If r=1, then D_1,1≡ D_1,2≡ D_1,3≡ 0, whereas for r=2 we also have D_1,1≡ 0. If r ⩾ 3, we follow estimates similar to the ones we developed in Lemma <ref>, and make use of the immersions we described there, in such a way that
[ ∫_Q ρ_9^2 D_1,1^2 d(t,x); ⩽ C(r)∫_0^T ρ_9^2D^3 (y-y)^2(D^3 y^2(r-3) + D^3 y^2(r-3))D^2 y_L^6(Ω)^6 dt; ⩽ C(r) ∫_0^T ρ_9^2D^3 (y-y)^2(D^3 y^2(r-3) + D^3 y^2(r-3))D^3y^6 dt; = C(r) ∫_0^T ρ_9^-2rρ_9D^3 (y-y)^2(ρ_9D^3 y^2(r-3) + ρ_9D^3 y^2(r-3))ρ_9D^3y^6 dt; ⩽ C(r)δ^2r(y-y,p - p, v-v)_Y^2. ]
Next, for r⩾ 2,
[ ∫_Q ρ_9^2 D_1,2^2 d(t,x); ⩽ C(r)∫_0^T ρ_9^2D^3 y^2(r-2)D^3(y-y)^2(D^2y_L^4(Ω)^4 + D^2 y_L^4(Ω)^4 )dt; ⩽ C(r)∫_0^Tρ_9^2D^3 y^2(r-2)D^3(y-y)^2(D^3 y^4 + D^3 y^4 )dt; ⩽ C(r)δ^2r(y-y, p-p, v-v)_Y^2, ]
[ ∫_Q ρ_9^2D_1,3^2d(t,x); ⩽ C(r)∫_0^Tρ_9^2(D^3 y^2(r-2) + D^3 y^2(r-2))D^3(y-y)^2D^4 y^2D^3 y^2 dt; ⩽ C(r)δ^2r(y-y,p-p,v-v)_Y^2. ]
Now, for every r ⩾ 1,
[ ∫_Q ρ_9^2D_1,4^2d(t,x) ⩽ C(r)∫_0^T ρ_9^2D^3 y^2(r-1)D^4(y-y)^2D^3y^2 dt; ⩽ C(r)δ^2r(y-y,p-p,v-v)_Y^2, ]
[ ∫_Q ρ_9^2D_1,5^2d(t,x) ⩽ C(r)∫_0^T ρ_9^2 D^3 y^2(r-1)D^4 y^2D^3 (y-y)^2 dt; ⩽ C(r) δ^2r(y-y,p-p, v-v)_Y^2, ]
[ ∫_Q ρ_9^2D_1,6^2d(t,x) ⩽ C(r)∫_0^T ρ_9^2D^3 y^2(r-1)D^3(y-y)^2D^4y^2 dt; ⩽ C(r)δ^2r(y-y,p-p, v-v)_Y^2, ]
[ ∫_Q ρ_9^2D_1,7^2d(t,x) ⩽ C(r)∫_0^T ρ_9^2D^3 y^2rD^4(y-y)^2 dt; ⩽ C(r)δ^2r(y-y,p-p, v-v)_Y^2. ]
Summing up, the computations we carried out above yield
Δ D_1_9 ⩽ C(r)δ^r(y-y,p-p,v-v)_Y.
We can treat the remaining terms composing the F-norm of D_1 likewise, as we argued in Lemma <ref>. Dealing with D_2 is even simpler, since it involves lower order derivatives of y. In this way, we deduce that
ν_1D_1_F + D_2_F ⩽ C(r)δ(y-y,p-p,v-v)_Y.
Thus, it suffices to take any positive δ < min(1,ϵ/C(r)) in order to finish the proof.
The linear operator DH(0,0,0) : Y → Z is continuous and onto. Furthermore, there exists a constant M>0 such that
(y,p,v)_Y ⩽ MDH(0,0,0)· (y,p,v)_Z
The continuity of DH(0,0,0) follows promptly from the definition of the norms of Y and Z. As for the surjectiveness of this mapping, let us consider (f,y_0) ∈ Z. We take (y,p,v) as the state-pressure-control tuple given by Theorem <ref>. By the estimates we proved in subsection <ref>, namely (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), together with Lemma <ref>, the membership (y,p,v) ∈ Y is valid. Moreover,
DH(0,0,0)· (y,p,v) = (y_t - ν_0Δ y +∇ p - χ_ω v, y(0)) = (f,y_0),
where the last equality holds by the choice of (y,p,v); hence, DH(0,0,0) is onto Z. By the aforementioned estimates, (<ref>) follows easily. This establishes the lemma.
§.§ Proof of Theorem <ref>
According to Lemmas <ref>, <ref> and <ref>, it is licit to apply Theorem <ref>. This result allows us to deduce the existence of η > 0 such that, for each (f,y_0) ∈ Z subject to
(f,y_0)_Z < η,
the equation
H(y,p,v) = (f,y_0)
has a solution (y,p,v) ∈ Y which satisfies
(y,p,v)_Y < B η,
for a suitable constant B > 0 which is independent of η. Explicitly, we can take B := (M^-1 - δ)^-1, where M>0 is given by Lemma <ref> (cf. (<ref>)), and where we select the positive constant δ < M^-1 such that H satisfies condition (iii) of Theorem <ref>. Such a constant δ does in fact exist by Lemma <ref>.
In particular, taking f≡ 0, inequality (<ref>) reads
y_0_H^5(Ω)^N < η.
Since (y,p,v) ∈ Y, we have (<ref>), and alonside (<ref>), we see that (y,p,v) does solve (<ref>).
§ NUMERICAL ANALYSIS
§.§ Proof of the convergence of the algorithm
The proof of this result is straightforward once we have established Lemmas <ref> and <ref>. We present it here for completeness.
Firstly, we observe that Lemma <ref> ensures that (y^n+1,p^n+1,v^n+1) is well-defined in terms of (y^n,p^n,v^n), since in this lemma we showed that DH(0,0,0) is bijective. Furthermore, we have DH(0,0,0)^-1_ℒ(Z,Y)⩽ M, according to the notations of this lemma.
Next, we take y_0 ∈ G, with y_0_H^5(Ω)^N < η, and we let (y,p,v) ∈ Y be the solution of H(y,p,v) = (0,y_0). We also consider 0<ϵ < (2M)^-1. By Lemma <ref>, there exists δ >0 such that the relations
(y,p,v)∈ Y and (y,p,v) ∈ Y, (y-y, p-p, v-v)_Y ⩽δ
imply
DH(y,p,v) - DH(y,p,v)_ℒ(Y,Z)⩽ϵ.
Shrinking η, if necessary, we can assume η⩽δ. Employing Lemma <ref> once more, we find κ = κ(y,p,v) ∈]0,1[ such that (y,p,v) ∈ Y and (y-y,p-p,v-v)_Y ⩽κ together imply
H(y,p,v) - H(y,p,v) - DH(y,p,v)· (y-y,p-p,v-v)_Z ⩽ϵ(y-y,p-p,v-v)_Y.
We write e^n := (y^n, p^n, v^n) - (y,p,v), and let us assume e^0_Y ⩽κ. By the algorithm,
e^n+1 = - DH(0,0,0)^-1[H(y^n,p^n,v^n)-H(y,p,v) - DH(y,p,v) · e^n ]
- DH(0,0,0)^-1[DH(y,p,v) - DH(0,0,0) ]· e^n,
whence
e^n+1_Y ⩽ M {H(y^n,p^n,v^n)-H(y,p,v) - DH(y,p,v)e^n.
.+ [DH(y,p,v) - DH(0,0,0) ]· e^n}.
Assuming inductively that e^n_Y ⩽κ, which holds true for n=0, it follows that
e^n+1_Y ⩽ 2Mϵe^n_Y.
Thus, we also have e^n+1_Y ⩽κ. By induction, it follows that e^n_Y⩽κ, for every n; hence, it is always possible to pass from (<ref>) to (<ref>). Let us take θ := 2Mϵ. Applying inequality (<ref>) iteratively in n, we conclude that
e^n_Y ⩽θ^ne_0_Y.
This proves Theorem <ref>.
§.§ Implementation of the algorithm
To implement the fixed-point numerical algorithm, we proceed in two steps. Firstly, it is necessary to implement a solver for the control problem of the forced Stokes system. We begin with the variational problem (<ref>) and adequately reformulate it to achieve a mixed formulation, as in <cit.>. Below, we recall the main ideas for N=2. After treating the linear problem, we iterate it by updating the source term according to our algorithm.
Under the notations of the proof of Theorem <ref> (see (<ref>)), we define u := ρ_3^-1(L^*φ + ∇π), m := ρ_4^-1φ, and k := ρ_4^-1π. Let us introduce the spaces
Z := { (m^',k^') : m^'∈ L^2(0,T; H^1_0(Ω)^2, m^'_t ∈ L^2(Q)^2, k^'∈ L^2(0,T; H^1(Ω)), ∫_Ω k^' dx = 0 a.e. },
and
W:= L^2(Q)^2 × Z, M := L^2(0,T;H^1_0(Ω)^2)× L^2(Q),
as well as the bilinear forms b_1 : W × W →ℝ, B,B_1 : W × M →ℝ by
b_1((u,m,k),(u^', m^', k^')) := ∫_Q {u · u^' + χ m · m^'}d(t,x),
B((u,m,k),(λ,μ)) := ∫_Q {λ·[u+ρ_3^-1(ρ_4 m)_t + ∇(ρ_4 k ) ] - ∇( ρ_3^-1λ): ∇(ρ_4 m ) } d(t,x)
and
B_1((u,m,k),(λ,μ)) = B((u,m,k),(λ,μ)) - ∫_Q ρ_3^-1μ∇·(ρ_4 m ) d(t,x).
The last element we introduce is the linear form Λ : W →ℝ, which is given by
⟨Λ, (u,m,k) ⟩ := ∫_Q ρ_4 f m d(t,x) + ∫_Ω (ρ_4 m)(0)y_0 dx.
We reformulate problem (<ref>) as: find (u,m,k) ∈ W and multipliers (λ,μ) ∈ M such that
b_1((u,m,k),(u^',m^',k^')) + B_1((u^',m^',k^'),(λ,μ)) = ⟨Λ, (u^',m^',k^') ⟩, for all (u^',m^',k^') ∈ W,
B_1((u,m,k),(λ^',μ^')) = 0, for all (λ^',μ^') ∈ M.
After we solve it, we recover the control and corresponding velocity field of the linear control problem (<ref>) via
v = - χρ_4^-1 m and y = ρ_3^-1 u.
If we assume that Ω is polygonal, it is simple to find finite dimensional approximations W_h and M_h of the spaces W and M.
§.§ A numerical experiment
In the sequel, we will employ the FreeFem++ library of C++; see <http://www.freefem.org/ff++> for more informations. In Table <ref>, we describe the datum we used to apply the quasi-Newton method for (<ref>).
We illustrate in Figure <ref> the 2D mesh of Ω, and the 3D mesh of the cylinder Q. In Figure <ref>, we show both components of the initial state y(0) = y_0.
Our stopping criterion is
y^n+1-y^n_L^2(Q)/y^n_L^2(Q)⩽ϵ,
with ϵ = 10^-8. We took as the initial guess (y^0,p^0,v^0) = (0,0,0). We attained convergence after six iterates, with a rate of 4.68.
We begin to illustrate the overall behavior of the control and of the state we computed through the plots of some cross-sections in space of them. On the one hand, for the control, we plot the x_1 = 0.9 and x_1 = 2.1 cuts in Figures <ref> and <ref>, respectively. On the other hand, we provide the surfaces comprising the values of the state components, relative to these cuts, in Figures <ref> and <ref>.
The time evolution of the norms of the control and of the corresponding state is what we illustrate in Figure <ref>. It corroborates our theoretical findings as these norms decay exponentially. To further illustrate the control, we provide a surface of its values at initial time in Figure <ref>. Then, we give some insight into the dynamics of the problem by showcasing some heat maps of the control and of its corresponding state. Namely, in Figure <ref>, we illustrate the control at time t=0.15 — it is already considerably small, as we would expect from Figure <ref>. For several times, viz., for each t∈{0.15, 0.25, 0.35, 0.45}, we give a heat map of the first (respectively, second) component of the velocity field in Figure <ref> (respectively, Figure <ref>).
§ COMMENTS AND PERSPECTIVES
§.§ On the constitutive law for the shear stress
Upon scrutinizing the proof of Lemmas <ref> and <ref>, we conclude that they still hold for any function ν : ℝ^N× N→ℝ in (<ref>) having the following properties:
* ν⩾ν_0, for some constant ν_0>0;
* ν is of class C^3( ℝ^N× N\{ 0 });
* There exists r>0 such that
|D^k ν(A)|⩽ C(1 + |A|^(r-k)^+),
for k=0,1,2,3, and for every A ∈ℝ^N× N\{ 0}.
With Lemmas <ref> and <ref> at hand, we can follow the remaining steps towards the main result, i.e., Theorem <ref>, in the same manner as we proceeded in Section <ref>. This more general class of constitutive laws includes the one determining the reference model of this paper, namely, ν(A) := ν_0 + ν_1|A|^r, when r∈{ 1, 2 } or r⩾ 3. An example of another class of functions ν for which the properties we stated above hold are
ν(A) := ν_0 (1 + ν_1 |A|^2 )^r/2, r ∈{1,2}∪[3,∞[.
§.§ On the use of the gradient instead of the deformation tensor
We can replace the gradient of the velocity field in (<ref>) with the deformation tensor, Dy = ( ∇ y + ∇ y^T)/2, without losing any of the results we established. From a practical viewpoint, this form of the model is more realistic. Analyzing the estimates we carried out throughout the present work, it is easy to see that the techniques we employed work just as well under this substitution. In particular, we notice the new framework shares the linearization around the zero trajectory with the one we studied in Section <ref>. Using the estimates developed there, alongside Korn-type inequalities, we can prove all of the corresponding results in Sections <ref> and <ref> for this alternate version of the model (<ref>)-(<ref>).
§.§ On extensions of Theorem <ref> and some related open questions
Boundary controllability. We remark that a corresponding boundary local null controllability result follows from Theorem <ref>. In effect, let us assume that the initial data y_0 belongs to H^5_0(Ω)∩ V, being sufficiently small in the (strong) topology of this space, and that we act with a control on a smooth portion γ of the boundary ∂Ω (with γ≠∂Ω and γ≠∅). We can employ standard geometrical arguments to extend Ω to an open region Ω, with a smooth boundary ∂Ω, and in a way that ∂Ω\γ⊂∂Ω. Acting distributively over ω:= Ω\Ω, with y_0 extended to zero outside of Ω, we obtain a control v∈ L^2(]0,T[ ×ω) driving the corresponding state y to zero at time T. A boundary control for the original problem is y|_[0,T]×γ.
Local controllability to trajectories. Regarding the local exact controllability to trajectories, there are two key aspects to investigate. Firstly, we must prove a global Carleman inequality, analogous to Proposition <ref>, but for the adjoint system of the linearization around the given trajectory, cf. Lemma <ref>. Secondly, we have to extend the estimates of Section <ref> for this linearized problem. These endeavors are not straightforward, whence we leave this question open for future investigations.
On the restrictions on the exponent r. We notice that the estimates of Section <ref> are not immediately extensible for the values of r > 0 outside of {1,2}∪[3,∞[. However, we conjecture that our main result (viz., Theorem <ref>) is still in force for these values of r. A possible way to establish this is to parametrically regularize the function ν around zero, and attentively keep track of the regularization parameters throughout the estimates. We leave this question open here.
Requirements on the initial datum. Through another regularization argument, we possibly could require a less restrictive topology for the initial datum in the main theorem. Namely, if we assume y_0 ∈ H only, we ought to carry out estimates for the uncontrolled problem (corresponding to (<ref>) with v≡ 0) to show that there exists t_0 ∈]0,T[ for which y(t_0,·)_H^5(Ω)^N∩ V⩽η, as long as y_0_H is sufficiently small. We choose not to delve in the technicalities of these estimates here (see <cit.> for the application of such an argument in the case of the Navier-Stokes equations with the Navier boundary condition). However, we emphasize that this is a non-trivial task. Thus, assuming this is valid, Theorem <ref> asserts that there exists a control v ∈ L^2(]t_0,T[×ω) driving y(t_0,·) to zero at time T. From the exponential decay of solutions, see <cit.>, this argument immediately provides a large-time global null controllability result.
Remarks on other boundary conditions. We observe that, if instead of no-slip boundary conditions, we assume Navier boundary conditions, the method of <cit.>, used for the Navier-Stokes equations, may apply to the current model. If we manage to deal with the additional terms figuring in the expansions we must make after an appropriate time rescaling, especially the boundary layers, we should obtain a small-time global exact controllability to trajectories result (under Navier boundary conditions). Alternatively, if we consider the model (<ref>)-(<ref>) with Ω = 𝕋 (the N-dimensional torus) and periodic boundary conditions, then we can easily conduct the regularizing argument for the initial datum we outlined above, whence we can prove large-time global null controllability for this model — we omit the details here.
Stabilization results. It might be that, for ν_1 > 0, an appropriate use of the stabilizing effect of the power-law model makes it easier to establish stabilization results for this class of non-Newtonian fluids. In this way, we propose that our current contributions could bridge such results with global null controllability ones. We remark that, even for the Navier-Stokes equations (corresponding to ν_1 = 0) under no-slip boundary conditions, whether global null controllability holds is an open problem. We suggest that such results for (<ref>)-(<ref>) (with ν_1 > 0) could provide insight on this important open question.
apalike
|
http://arxiv.org/abs/2307.04554v1 | 20230710133817 | Non-unit quaternion parametrization of a Petrov-Galerkin Cosserat rod finite element | [
"Jonas Harsch",
"Simon R. Eugster"
] | math.NA | [
"math.NA",
"cs.NA",
"math-ph",
"math.MP"
] |
[EN]
The short title]Non-unit quaternion parametrization of a Petrov–Galerkin Cosserat rod finite element
[1][DE] Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany
[EN]
The application of the Petrov–Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a straight-forward and systematic exchange of the ansatz functions, specifically for centerline positions and cross-section orientations. In this concise communication, we present a total Lagrangian finite element formulation for Cosserat rods that attempts to come up with the least required concepts. The chosen discretization preserves objectivity and allows for large displacements/ rotations and for large strains. The orientation parametrization with non-unit quaternions results in a singularity-free formulation.
[
Simon R. Eugster1
August 12, 2023
=====================
§ INTRODUCTION
This article complements the two papers <cit.> on Petrov–Galerkin rod finite formulations for Cosserat rods. The cross-section orientations are parameterized using non-unit quaternions instead of total rotation vectors, which require additionally the concept of the complement rotation vector for a singularity-free parametrization. To keep the formulation as simple as possible, we opt for the ^12-interpolation for the ansatz functions, see <cit.>.
The paper is structured as follows. In Section <ref>, the Cosserat rod theory is recapitulated very briefly; mainly to introduce all quantities required for the further finite element formulation. For those interested in additional comments as well as a thorough introduction and explanation of the chosen notation, we recommend reading <cit.>. The Petrov–Galerkin finite element formulation in terms of nodal non-unit quaternions is presented in Section <ref>. The last section on numerical experiments, investigates the static analysis of a helical spring in line with <cit.>. Additionally, the Wilberforce example from <cit.> with a helical spring with three coils is discussed.
§ COSSERAT ROD THEORY
Let ξ∈𝒥 = [0, 1] ⊂ be the centerline parameter and let t denote time. The motion of a Cosserat rod is captured by a time-dependent centerline curve represented in an inertial I-basis _I_OP=_I_OP(ξ, t) ∈^3 augmented by the cross-section orientations _IK=_IK(ξ, t) ∈ SO(3)={∈^3 × 3| = 1_3 × 3∧()= 1}. The subscripts O and P in the centerline curve refer to the origin and the centerline point, respectively. The cross-section orientation _IK can also be interpreted as a transformation matrix that relates the representation of a vector in the cross-section-fixed K-basis to its representation in the inertial I-basis.
The derivatives with respect to time t and centerline parameter ξ are denoted by (̇∙̇)̇ and (∙)_,ξ, respectively. The variation of a function is indicated by δ(∙). With this, we can introduce the centerline velocity _I _P= (_I _OP)^· and the virtual displacement _I δ_P = δ(_I _OP).
The angular velocity of the cross-section-fixed K-basis relative to the inertial I-basis, in components with respect to the K-basis, is defined by _K _IK j^-1(_IK(_IK)^·),
where j ^3 →(3) = {∈^3×3 | = -} is the linear and bijective map such that = j() = × for all , ∈^3. Analogously, the virtual rotations and the scaled curvature are defined as
_K δ_IK j^-1( _IKδ(_IK) ) and _K _IK j^-1( _IK (_IK)_,ξ), respectively.
For the reference centerline curve _I _OP^0, the length of the rod's tangent vector is J = _I _OP, ξ^0. Thus, for a given centerline parameter ξ, the reference arc length increment is s = J ξ. The derivative with respect to the reference arc length s of a function = (ξ,t) ∈^3 can then be defined as _,s(ξ,t) _,ξ(ξ,t) /J(ξ).
The objective strain measures of a Cosserat rod are the curvature _K _IK = _K _IK / J, which measures torsion and bending, together with the measures for dilatation and shear strains contained in _K = _K / J determined by _K (_IK)_I _OP, ξ.
The internal virtual work of a Cosserat rod is defined as
δ W^int -∫_𝒥{
(_I δ_P,ξ)_IK_K + (_K δ_IK,ξ)_K
- (_K δ_IK)[ _K ×_K + _K _IK×_K ]
}[ξ] ,
where _K and _K denote the resultant contact forces and moments, respectively. For hyperelastic material models with a strain energy density with respect to the reference arc length W = W(_K , _K _IK; ξ), they can be determined by the constitutive relations _K = (∂ W / ∂_K ) and _K = (∂ W / ∂_K _IK).
Assume the line distributed external forces _I = _I (ξ,t) ∈^3 and moments _K =_K (ξ,t) ∈^3 to be given as densities with respect to the reference arc length. Moreover, for i∈{0,1}, point forces _I _i = _I _i(t) ∈^3 and point moments _K _i = _K _i(t) ∈^3 can be applied to the rod's boundaries at ξ_0=0 and ξ_1=1. The corresponding external virtual work functional is defined as
δ W^ext∫_𝒥{ (_Iδ_P)_I + (_K δ_IK)_K } J [ξ]
+ ∑_i = 0^1 [ (_Iδ_P)_I _i + (_K δ_IK)_K _i ]_ξ_i .
In case _I_OP is the line of centroids, the inertial virtual work functional of the Cosserat rod can be written as
δ W^dyn -∫_𝒥{(_I δ_P ) A_ρ_0 (_I_p)^· + (_K δ_IK)
(_K _ρ_0 (_K _IK)^· + _K _IK×_K _ρ_0_K _IK)} J [ξ] ,
where A_ρ_0 is the cross-section mass density and _K _ρ_0 the constant cross-section inertia tensor represented in the cross-section-fixed K-basis.
§ PETROV–GALERKIN FINITE ELEMENT FORMULATION
The rod's parameter space 𝒥 is divided into n_el linearly spaced element intervals 𝒥^e = [ξ^e, ξ^e+1) via 𝒥 = ⋃_e=0^n_el-1𝒥^e. For a p-th order finite element, the closure of each of the intervals 𝒥^e contains p + 1 evenly spaced points ξ^e_i ∈cl(𝒥^e) = [ξ^e, ξ^e+1] with i ∈{0, …, p} such that ξ^e_0 = ξ^e < ξ^e_1 < … < ξ^e_p = ξ^e+1. Note, for e ∈{0, …, n_el -2 }, the points ξ^e_p=ξ^e+1_0 denote the same point ξ^e+1, which is the boundary point of the adjacent element intervals. It is convenient to use both indexations in the following. For a given element interval 𝒥^e = [ξ^e, ξ^e+1), the p-th order Lagrange basis function and derivative of node i∈{0,…,p} are
N^p,e_i(ξ) = 0 ≤ j ≤ p
j≠ i∏ξ - ξ^e_j/ξ^e_i - ξ^e_j and
N^p,e_i,ξ(ξ) = N_i^p,e(ξ) k=0
k ≠ i∑^p1/ξ - ξ^e_k ,
where ξ^e_i, ξ^e_j, and ξ^e_k are the points contained in the set {ξ^e_0 = ξ^e, ξ^e_1, …, ξ^e_p = ξ^e+1}.
The centerline curve _I _OP and the cross-section orientations _IK are approximated by interpolating nodal centerline points _I_OP^e_i(t)∈^3 and nodal transformation matrices _IK^e_i(t)∈(3). For each node i ∈{0,…,p} within element e ∈{0,…, n_el-1}, it will hold that _I_OP^e_i(t) = _I _OP(ξ^e_i,t) and _IK^e_i(t) = _IK(ξ^e_i,t). In contrast to <cit.>, the nodal transformation matrices
_IK^e_i = (^e_i) = 1_3 × 3 + 2 ((^e_i)^2 + p^e_0,i ^e_i) / _i^e^2
are parametrized by nodal non-unit quaternions ^e_i(t) = (p^e_0,i(t), ^e_i(t)) ∈^4 with the scalar part p^e_0,i(t) ∈ and the vectorial part ^e_i(t) ∈^3, see <cit.>. Note that (<ref>) is formulated in such a way to return orthogonal matrices also for non-unit quaternions.
Accordingly, the N=(p n_el + 1) nodal generalized position coordinates ^e_i(t) = (_I _OP^e_i, ^e_i)(t) ∈^7 are given by the nodal centerline points _I _OP^e_i and the nodal non-unit quaternions ^e_i resulting in n_ = 7N positional degrees of freedom for the discretized rod. The nodal quantities can be assembled in the global tuple of generalized position coordinates (t) = (^0_0, …, ^0_p-1, …, ^e_0, …, ^e_p-1, …, ^n_el-1_0, …,^n_el-1_p-1, ^n_el-1_p)(t) ∈^n_. For e ∈{0, …, n_el -2 }, the coordinates ^e_p=^e+1_0 refer to the same nodal coordinates. Introducing an appropriate Boolean connectivity matrix _e ∈^7(p+1) × n_, the element generalized position coordinates ^e(t) = (^e_0, …, ^e_p)(t) ∈^7(p+1) can be extracted from via ^e = _e. Note that during a numerical implementation it is advisable to slice arrays instead of multiply them with Boolean matrices.
In the sense of <cit.>, both the nodal centerline points and the cross-section orientations are interpolated by p-th order Lagrangian polynomials. Using the characteristic function χ_𝒥^e𝒥→{0, 1}, which is one for ξ∈𝒥^e = [ξ^e, ξ^e+1) and zero elsewhere, together with the p-th order Lagrange basis functions (<ref>), the ansatz functions for centerline and cross-section orientations are
_I _OP(ξ, ) = ∑_e=0^n_el-1χ_𝒥^e(ξ) ∑_i=0^p
N^p,e_i(ξ) _I _OP^e_i and
_IK(ξ, ) = ∑_e=0^n_el-1χ_𝒥^e(ξ)
∑_i=0^p
N^p,e_i(ξ) (^e_i) .
The discretized version of the curvature strain is computed as
_K _IK = j^-1((_IK_IK,ξ)) / J ,
where the map () = 1/2( - ) ∈(3) extracts the skew-symmetric part of the matrix ∈^3×3. Hence, the curvature can efficiently be computed using j^-1(()) = 12(M_32 - M_23, M_13 - M_31, M_21 - M_12).
At the same N nodes as for the nodal generalized position coordinates, we introduce the nodal generalized virtual displacements δ^e_i(t) = (_I δ_P^e_i, _K^e_iδ_IK^e_i)(t) ∈^6 given by the nodal virtual centerline displacement _I δ_P^e_i(t) ∈^3 and the nodal virtual rotation _K^e_iδ_IK^e_i(t) ∈^3. In analogy to the generalized virtual displacements, we also introduce the nodal generalized velocities ^e_i(t) = (_I _P^e_i, _K^e_i_IK^e_i)(t) ∈^6 given by the nodal centerline velocity _I _P^e_i(t) ∈^3 and the nodal angular velocity _K^e_i_IK^e_i(t) ∈^3. Similar to the generalized position coordinates , the nodal generalized virtual displacements and velocities are assembled in the global tuple of generalized virtual displacements δ(t) ∈^n_ and velocities (t) ∈^n_. In contrast to the nodal position coordinates, there are only six nodal generalized virtual displacements or velocity coordinates resulting in n_ = 6N generalized virtual displacements or velocity degrees of freedom for the discretized rod.
Consequently, we require a new Boolean connectivity matrix _, e∈^6(p+1) × n_, which extracts the element generalized virtual displacements δ^e(t) = (δ^e_0, …, δ^e_p)(t) ∈^6(p+1) and velocities ^e(t) = (^e_0, …, ^e_p)(t) ∈^6(p+1) from the global quantities via δ^e = _,eδ and ^e = _,e. By further introducing the Boolean connectivity matrices _, i∈^3 × 6(p+1), the nodal virtual centerline displacements _I δ_P^e_i and centerline velocities _I _P^e_i can be extracted from the element generalized virtual displacements δ^e and velocities ^e via _I δ_P^e_i = _, iδ^e and _I _P^e_i = _, i^e, respectively. Identical extraction operations hold for the nodal virtual rotations _K^e_iδ_IK^e_i = _,iδ^e and angular velocities _K^e_i_IK^e_i= _,i^e, where _, i∈^3 × 6(p+1). The test functions are then given by interpolating the nodal generalized virtual displacements by p-th order Lagrangian basis functions (<ref>) in agreement with
_I δ_P(ξ, δ) = ∑_e=0^n_el - 1χ_𝒥^e(ξ) ∑_i=0^p N^p,e_i(ξ) _I δ_P^e_i and
_K δ_IK(ξ, δ) = ∑_e=0^n_el - 1χ_𝒥^e(ξ)∑_i=0^p N^p,e_i(ξ) _K^e_iδ_IK^e_i .
Note that the interpolation of the virtual rotations must be understood in the sense of a Petrov–Galerkin projection, where the virtual rotations are not obtained from a consistent variation of the ansatz functions (<ref>).
To obtain a constant and symmetric mass matrix in the discretized formulation, see (<ref>) below, the velocities are considered as independent fields and are interpolated with the same interpolation as the virtual displacements and rotations as
_I _P(ξ, ) = ∑_e=0^n_el - 1χ_𝒥^e(ξ)∑_i=0^p N^p,e_i(ξ) _I _P^e_i and
_K _IK(ξ, ) = ∑_e=0^n_el - 1χ_𝒥^e(ξ) ∑_i=0^p N^p,e_i(ξ) _K^e_i_IK^e_i .
The independent introduction of velocity fields (<ref>) demands an additional relation defining the coupling between position coordinates and velocity coordinates . This coupling is given by the nodal kinematic differential equations
^e_i =
[ _I_OP^e_i; ^e_i ] =
[ 1_3 × 3 0_3 × 3; 0_4 × 3 (^e_i) ][ _I _P^e_i; _K^e_i_IK^e_i ] =
(^e_i) ^e_i , where () = 1/2[ -; p_0 1_3 × 3 + ] ,
cf. <cit.>. The nodal kinematic equations (<ref>) can easily be assembled to a global kinematic differential equation of the form = (). Note that the kinematic differential equation is linear in too. This allows to write the relation also in the form = (), see <cit.> for more details.
Inserting the test functions (<ref>) together with the corresponding approximations for centerline, cross-section orientations (<ref>) and strain measures into (<ref>), the continuous internal virtual work is approximated by δ W^int(; δ) = δ^int(), where the internal generalized forces are computed element-wise by
^int() = ∑_e=0^n_el - 1_, e^int_e(_e ) ,
^int_e(^e) = -∫_𝒥^e∑_i=0^p{ N^p,e_i,ξ_, i_IK_K + N^p,e_i,ξ_, i_K
-N^p,e_i_, i(_K ×_K + _K_IK×_K ) }[ξ] .
Similarly, the external virtual work (<ref>) is discretized by δ W^ext(t, ; δ) = δ^ext(t, ) with
^ext(t, ) = ∑_e=0^n_el - 1_,e^ext_e(t, _e ) + _, 0[_, 0_I _0 +_, 0_K _0 ]_ξ=0 +
_, n_el - 1[_,p_I _1 +_, p_K _1 ]_ξ=1 ,
^ext_e(t, ^e) = ∫_𝒥^e∑_i=0^p{ N^p,e_i_, i_I + N^p,e_i_, i_K } J [ξ]
.
Finally, inserting (<ref>) and (<ref>) into the inertial virtual work functional (<ref>) yields the discrete counterpart δ W^dyn(;δ) = -δ( + ^gyr() ),
where we have introduced the symmetric and constant mass matrix
= ∑_e=0^n_el - 1_,e_e _,e , _e =
∫_𝒥^e∑_i=0^p∑_k=0^p N^p,e_i N^p,e_k{
A_ρ_0_, i_, k
+ _, i_K _ρ_0_, k} J [ξ] ,
and the gyroscopic forces
^gyr() = ∑_e=0^n_el-1_,e_e^gyr(_,e) , ^gyr_e(^e) = ∫_𝒥^e∑_i=0^p N^p,e_i {_, i (_K _IK×_K _ρ_0_K _IK) } J [ξ] .
Element integrals of the form ∫_𝒥^e f(ξ) [ξ] arising in the discretized external and gyroscopic forces, as well as in the mass matrix, are subsequently computed using a Gauss–Legendre quadrature rule with ceil[(p + 1)^2 / 2] quadrature points. To alleviate locking, the internal generalized forces (<ref>) are integrated by a reduced p-point quadrature rule.
Applying the principle of virtual work, which requires the total virtual work functional to vanish, we readily obtain the system dynamics in the form
= () ,
= ^-1(^gyr() + ^int() + ^ext(t, )) ,
where the two lines correspond to the global kinematic differential equation and the equations of motion, respectively. Even though deviations from unit length of ^e_i do not affect the kinematic differential equation, to avoid numerical issues due to quaternion magnitudes near zero or floating point overflow, the nodal quaternions are normalized after each time-step, i.e., ^e_i = ^e_i / _i. For static problems, the n_ = 6N nonlinear generalized force equilibrium equations
0 = ^int() + ^ext()
must be augmented by the N constraint equations
0 = () = (^0_0^2 - 1, …, ^n_el-1_p^2 - 1)
to ensure solvability.
§ NUMERICAL EXPERIMENTS
In the following, the quadratic strain energy density
W(_K , _K _IK; ξ) = 1/2(_K - _K ^0)_(_K - _K ^0) + 1/2(_K _IK - _K _IK^0)_(_K _IK - _K _IK^0)
is used. The superscript 0 refers to the evaluation in the rod's reference configuration. Moreover, _ = diag(EA, GA, GA) and _ = diag(G (I_y + I_z), E I_y, E I_z) denote the diagonal elasticity matrices with constant coefficients given by Saint-Venant’s relations from linear elasticity. Therein, E and G, respectively denote the Young's and shear modulus. The cross-sectional surface is denoted A and I_y, I_z are the respective second moments of area.
§.§ Helical spring
Following <cit.>, we investigate the elongation of an initially curved helical rod due to an applied external force at its tip, pointing in positive _z^I-direction. The rod has a Young's modulus E=10^11 N/m^2 and Poisson's ratio ν=0.2, i.e., a shear modulus G = E / 2 (1 + ν). It has an undeformed shape of a perfect helix with n_c=10 coils, coil radius R=10 mm, wire diameter d=1 mm and unloaded pitch k=5 mm, i.e., a total height of h=50 mm.
In the simulation, the spring was discretized using 75 elements of the presented finite element formulation with p=2. Reduced integration was performed with 2 quadrature points, while 5 points were used for all other integrals. The rod's curved initial configuration was obtained by solving the following minimization problem. Let ξ_j = jm - 1∈ [0, 1] for j ∈{0,1,…,m-1} denote the m linearly spaced evaluation points of the reference helix curve
_I (ξ) = R [ sinφ(ξ); -cosφ(ξ); c φ(ξ) ] , with c = k/2 π R and φ(ξ) = 2 π n_cξ .
Hence, the evaluation of the reference curve (<ref>) at all ξ_j's leads to m target centerline points _I _j = _I (ξ_j). Similarly, the corresponding cross-section orientations are given by evaluating the Serret–Frenet basis _IK_j = (_I _x^K_j _I _y^K_j _I _z^K_j) with _I _x^K_j = _I_,ξ(ξ_j) / _I_,ξ(ξ_j), _I _y^K_j = _I_,ξξ(ξ_j) / _I_,ξξ(ξ_j) and _z^K_j = _I _x^K_j×_I _y^K_j for the individual ξ_j's. Following <cit.>, the centerline positions and cross-section orientations can be assembled in the Euclidean transformations
_j = [ _IK_j _I _j; 0_1×3 1 ] and
(ξ_j) = [ _IK(ξ_j) _I _OP(ξ_j); 0_1×3 1 ] , with
_j^-1 = [ _IK_j^T -_IK_j^T _I _j; 0_1×3 1 ] .
Using the (3)-logarithm map Log_(3) introduced in <cit.>, the optimal initial generalized position coordinates _0 results from the nonlinear least squares problem
_0 = ∈ℝ^n_argmin K() , with K() = 1/2∑_j=0^m-1_j()^2 and _j() = Log_(3)(_j^-1(ξ_j)) ,
in terms of the metric of relative twists. The minimization problem (<ref>) can efficiently be solved using a Levenberg–Marquardt algorithm. The unity constraints of the nodal quaternions (<ref>) can be incorporated into the optimization problem as equality constraints, albeit at the expense of employing a complex constrained nonlinear least squares solver. To simplify the process, we initially solved the unconstrained minimization problem and subsequently applied a projection step to normalize all nodal quaternions.
Starting from _0, the maximal force of 100 N was applied within 500 linearly spaced force increments. During each iteration, the nonlinear equations (<ref>) and (<ref>) were solved up to an absolute error of 10^-8. As can be seen in Fig. <ref>, the helical spring initially elongates proportional to the applied load. This is in line with classical helical spring theory <cit.>, which assumes a linear force-displacement relation with linear equivalent stiffness G d^4 / (64 n_c R^3) ≈ 65.1 N/m. When the elongation exceeds a certain value (approx. 10 N), the linear theory does not longer agree with the numerically obtained nonlinear solution. This observation was also made by <cit.> and can be explained as follows. The helical spring unwinds gradually and approaches slowly a straight line with an altered linear stiffness EA. For comparison, we also solved the problem with the two-node (3)-interpolation strategy proposed in <cit.>, using the same number of unknowns. As depicted in Fig. <ref>, the results are in line with the proposed quaternion formulation.
§.§ Wilberforce pendulum
More than 100 years ago, Lionel Robert Wilberforce did investigations On the Vibrations of a Loaded Spiral Spring <cit.>. The experimental setup can be described as follows. While one end of a helical spring is clamped, at the other end a cylindrical bob is attached, see Fig. <ref>. When the cylinder in the gravitational field is displaced vertically, it starts oscillating up and down. Due to the coupling of bending and torsion of the deformed spring an additional torsional oscillation around the vertical axis of the cylinder is induced. When the cylinder's moment of inertia is properly adjusted, a beat phenomenon can be observed. In that case, the envelope of the vertical and torsional oscillations possess an almost perfect phase shift of π/2, i.e., the maximal amplitude of the vertical oscillations coincide with a zero torsional amplitude and vice versa.
To have a benchmark example that can be reproduced with reasonably computational effort, we introduce here a Wilberforce pendulum consisting of a spring with three coils modeled as a precurved rod. The rod has the properties of steel with mass density ρ_0=7850 kg/m^3, shear modulus G=81·10^9 N/m^2 and Poisson's ratio ν=0.23, i.e., a Young's modulus E = 2 G (1 + ν)=199·10^9 N/m^2. The undeformed shape is given by a perfect helix with n_c=3 coils, coil radius R=16 mm, wire diameter d=1 mm and an unloaded pitch of k=1 mm. The bob is modeled as a cylindrical rigid body with radius r=23 mm and height h=36 mm also having the mass density of steel.
In the simulations, the rod was discretized using 18 elements of the presented Cosserat rod finite element with p=2. Gravitational forces for the rod were neglected. Again, reduced integration was performed with 2 quadrature points, while for all other integrals 5 points were used. The bob was parameterized by the inertial position of the center of mass _I _OS together with a non-unit quaternion for the orientation. The bob was subjected to gravity with gravity constant g=9.81 m/s^2. For the governing equations describing such a parameterized rigid body under the influence of gravity, we refer to model 4 in <cit.>. Cylinder and rod were rigidly connected by perfect bilateral constraints <cit.>.
Again, the optimal helical initial configuration _0 was found by solving the minimization problem (<ref>). The system was initialized at rest with initial velocity _0 = 0. The resulting differential algebraic equations were solved using a first-order generalized-alpha method <cit.> for constrained mechanical systems of differential index 3, similar to the implementation found in <cit.>. A constant step-size Δ t = 5·10^-3 s was chosen and the governing equations were solved up to a final time of t_1 = 8 s. Since the example includes high-frequency oscillations, we chose a spectral radius at infinity of ρ_∞ = 0.8. The internal Newton–Raphson method satisfied a tolerance of 10^-8 with respect to the maximum absolute error. In Fig. <ref> the vertical position and the torsional angle of the rigid cylinder are plotted clearly showing the beat phenomenon of the Wilberforce pendulum.
[10]
Harsch2023a
J. Harsch, S. Sailer, and S. R. EugsterA total Lagrangian, objective and intrinsically locking-free
Petrov–Galerkin SE(3) Cosserat rod finite element formulation,
Int. J. Numer. Meth. Eng. 124(13) (2023).
Eugster2023a
S. R. Eugster and J. HarschA family of total
Lagrangian Petrov–Galerkin Cosserat rod finite element
formulations,
GAMM Mitt. 46(2) (2023).
Betsch2002
P. Betsch and P. SteinmannFrame-indifferent beam
finite elements based upon the geometrically exact beam theory,
Int. J. Numer. Meth. Eng. 54(12), 1775–1788 (2002).
Romero2002
I. Romero and F. ArmeroAn objective finite element
approximation of the kinematics of geometrically exact rods and its use in
the formulation of an energy-momentum conserving scheme in dynamics,
Int. J. Numer. Meth. Eng. 54, 1683–1716 (2002).
Marino2017
E. MarinoLocking-free isogeometric collocation formulation
for three-dimensional geometrically exact shear-deformable beams with
arbitrary initial curvature,
Comput. Method Appl. M. 324, 546–572 (2017).
Harsch2021a
J. Harsch, G. Capobianco, and S. R.
EugsterFinite element formulations for constrained spatial
nonlinear beam theories,
Math. Mech. Solids 26(12), 1838–1863 (2021).
Rucker2018
C. RuckerIntegrating rotations using nonunit quaternions,
IEEE Robot. Autom. Lett. 3(4), 2979–2986 (2018).
Berg1991
R. E. Berg and T. S. MarshallWilberforce
pendulum oscillations and normal modes,
Am. J. Phys. 59(1), 32–38 (1991).
Wilberforce1894
L. R. WilberforceOn the vibrations of a loaded spiral
spring,
Lond. Edinb. Dublin Philos. Mag. J. Sci. 38(233), 386–392
(1894).
Sailer2020
S. Sailer, S. R. Eugster, and R. I.
LeineThe tippedisk: a tippetop without rotational symmetry,
Regul. Chaotic Dyn. 25(6), 553–580 (2020).
Geradin2001
M. Géradin and A. Cardona,
Flexible Multibody Dynamics: A Finite Element Approach (Wiley, 2001).
Jansen2000
K. E. Jansen, C. H. Whiting, and G. M.
HulbertA generalized-α method for integrating the filtered
Navier–Stokes equations with a stabilized finite element method,
Comput. Method Appl. M. 190(3), 305–319 (2000).
Arnold2007
M. Arnold and O. BrülsConvergence of the
generalized-α scheme for constrained mechanical systems,
Multibody Syst. Dyn. 18(2), 185–202 (2007).
|
http://arxiv.org/abs/2307.06188v1 | 20230712142657 | On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions | [
"V. F. Babenko",
"V. V. Babenko",
"O. V. Kovalenko",
"N. V. Parfinovych"
] | math.FA | [
"math.FA",
"26D10, 41A17, 41A44"
] |
]On some sharp Landau – Kolmogorov – Nagy type inequalities in Sobolev spaces of multivariate functions
Department of Mathematical Analysis and Theory of Functions, Oles Honchar Dnipro National University, Dnipro, Ukraine
[email protected]
Department of Mathematics and Computer Science, Drake University, Des Moines, USA
[email protected]
Department of Mathematical Analysis and Theory of Functions, Oles Honchar Dnipro National University, Dnipro, Ukraine
[email protected]
Department of Mathematical Analysis and Theory of Functions, Oles Honchar Dnipro National University, Dnipro, Ukraine
[email protected]
[2020]26D10, 41A17, 41A44
For a function f from the Sobolev space W^1,p(C) (C⊂^d is an open convex cone), a sharp inequality that estimates f_L_∞ via the L_p-norm of its gradient and a seminorm of the function is obtained.
With the help of this inequality, a sharp inequality is proved, which estimates the L_∞-norm of the Radon–Nikodym derivative of a charge defined on Lebesgue measurable subsets of
C via the L_p-norm of the gradient of this derivative and a seminorm of the charge. In the case, when C=_+^m×^d-m, 0≤ m≤ d, we obtain inequalities that estimate the L_∞-norm of a mixed derivative of a function f C→ using its L_∞-norm and the L_p-norm of the gradient of the function's mixed derivative.
[
N. V. Parfinovych
July 2023
=====================
§ INTRODUCTION
Inequalities for norms of intermediate derivatives of univariate and multivariate functions have an important role in many branches of Analysis and its applications. The main attention is focused on sharp inequalities of this kind, and the results by Landau <cit.>, Kolmogorov <cit.> and Nagy <cit.> are among the brightest ones in the area. A survey of results and further references can be found in <cit.>.
Inequalities for Radon–Nikodym derivatives of charges defined on Lebesgue measurable subsets of an open cone C⊂^d, as well as for charges defined on measurable subsets of a metric space with a measure were obtained in <cit.>.
In this article for a function f from the Sobolev space W^1,p(C) (C⊂^d is an open cone) we obtain a sharp Nagy type inequality that estimates f_L_∞(C) via the L_p(C)-norm of its gradient and some seminorm of the function. Using this inequality, we prove a sharp Landau–Kolmogorov type inequality that estimates the L_∞(C)-norm of the Radon–Nikodym derivative via the L_p(C)-norm of the gradient of this derivative and the value of some seminorm of the charge. In the case C=_+^m×^d-m, 0≤ m≤ d, we obtain inequalities that estimate L_∞(C)-norm of a mixed derivative of a function f C→ via L_∞(C)-norm of the function and the L_p(C)-norm of the gradient of the mixed derivative. The obtained inequalities are sharp for m=0,1.
The results of the article together with the known general facts, allow to solve the problem of approximation of an unbounded operator by bounded ones, as well as several related problems (see e.g. <cit.>).
§ NOTATIONS AND SOME AUXILIARY RESULTS
For x,y∈, d≥ 1, we denote by (x,y) the dot products of the vectors x and y. Let K⊂ be a bounded open convex symmetric with respect to the origin θ set such that θ∈ int K. For x∈ we denote by
|x|_K the norm of the vector x that is generated by the set K i.e.,
|x|_K=inf{λ>0 x∈λ K}. Let also
|z|_ = sup{(x,z) |x|_K≤ 1} be the dual norm. Everywhere below C⊂^d is an open convex cone, μ is the Lebesgue measure in C, and for p≥ 1, we set p' = p/(p-1).
For a measurable set Q⊂ C by L_p(Q), 1≤ p≤∞, we denote the set of measurable functions f Q→ with corresponding norms ·_L_p(Q); by L_ loc(C) the space of locally integrable functions i.e., functions f C→ that are integrable on each compact Q⊂ C. In the space L_ loc(C) consider a family of seminorms
⌋ f⌈_h
=
sup_x∈ C| ∫_hK∩ Cf(x+u)du|, h > 0, and ⌋ f⌈ =sup_h>0⌋ f⌈_h.
By L_⌋·⌈_h(C) (L_⌋·⌈(C)) we denote the space of functions f∈ L_ loc(C) such that ⌋ f⌈_h<∞ (resp. ⌋ f⌈<∞). It is clear that L_1(C)⊂ L_⌋·⌈(C).
For a locally integrable function f by ∇ f we denote its gradient, where the derivatives are understood in the distributional sense. Let Q⊂ be an open set. By W^1,p(Q), 1≤ p≤∞, we denote the Sobolev space of functions f Q→ such that all their first order derivatives belong to L_p(Q).
For f∈ W^1,p(Q) one has |∇ f|_∈ L_p(Q).
For h>0 we consider the function g_h (0,h)→,
g_h(u)=1/d·μ(K∩ C)(1/u^d-1-u/h^d).
Using <cit.>, and transforming the obtained integral we obtain
g_h(|·|_K)_L_p'(hK∩ C) = (d·μ(K∩ C))^1/p'(∫_0^ht^d-1g_h^p'(t)dt)^1/p'=
A h^1-d/p/μ^-1/p(),
where (below B(·,·) denotes the Euler B-function)
A = A(d,p) = d^-1 B^1/p'(1-(d-1)p'/d, p'+1).
Define an operator S_h L_⌋·⌈_h(C)→ L_∞(C),
S_hf(x)=1/h^dμ()∫_hf(x+y)dy.
Now it becomes obvious that the following theorem is a partial case of <cit.>.
Let p∈ (d,∞], h>0 and f∈ W^1,p(h ). Then
| f(θ)-S_hf(θ)
|≤g_h( |·|_K) _L_p'(h)|∇ f|__L_p(h).
The inequality is sharp. It becomes equality for the function α· f +β, where α,β∈ and
f(y)= ∫_0^|y|_Kg_h^p'-1(u)du, y∈ h.
§ NAGY TYPE INEQUALITY
Using Theorem <ref>, for all x∈ C one obtains
|f(x) - S_hf(x)|
≤g_h(|·|_K) _L_p'(hK∩ C)| ∇ f(x+·)|__L_p( h)
≤g_h(|·|_K) _L_p'(hK∩ C)|∇ f|__L_p(C).
If h> 0, p∈ (d,∞] and f∈ W^1,p(C)∩ L_⌋·⌈_h(C), then f∈ L_∞(C) and the following inequality holds:
f_L_∞(C)≤g_h(|·|_K) _L_p'(hK∩ C)|∇ f|__L_p(C)+
μ^-1(K∩ C)h^-d⌋ f⌈_h.
Inequality (<ref>) is sharp. It becomes equality on the function
f_e,h(y)= ∫_|y|_K^hg_h^p'-1(u)du , y∈ h,
0, y∈ C∖ hK.
For each f∈ W^1,p(C)∩ L_⌋·⌈(C) the following multiplicative inequality holds:
f_L_∞(C)≤
a(d,p) μ^-α/d()⌋ f⌈ ^1-α|∇ f|__L_p(C) ^α,
where
α = pd/p+(p-1)d, a(d,p) = ((p-d)A(d,p)/pd)^α(pd/p-d+1),
and A(d,p) is defined in (<ref>).
Inequality (<ref>) is sharp. It becomes equality for each function f_e,h, h>0.
If f∈ W^1,p(C)∩ L_1(C), then in inequalities (<ref>) and (<ref>), ⌋ f⌈_h and ⌋ f⌈ respectively can be substituted by f_L_1(C), and the obtained inequalities remain sharp.
Using (<ref>), for each x∈ C we have
|f(x)| ≤|f(x) - S_hf(x)| + μ^-1(K∩ C)h^-d|∫_h f(x+y)dy|
≤g_h(|·|_K) _L_p'(hK∩ C)|∇ f|__L_p(C)+
μ^-1(K∩ C)h^-d⌋ f⌈_h,
which implies inequality (<ref>). Next we prove its sharpness. Since the function g_h is non-negative, one has
|x|_K≤ |y|_K f_e,h(x)≥ f_e,h(y).
The function f_e,h is continuous. Hence f_e,h_L_∞(C) = f(θ) and, moreover, the first inequality in (<ref>) becomes equality for x =θ and the function f_e,h. According to Theorem <ref>, the first inequality in (<ref>) for x = θ becomes equality for the function f_e,h. The function f_e,h vanishes outside the set h, and hence
|∇ f_e,h|__L_p(h) = |∇ f_e,h|__L_p(C) and ⌋ f_e,h⌈_h = ∫_h f_e,h(y)dy.
Thus the second inequalities in (<ref>) and (<ref>) also become equalities, and hence inequality (<ref>) becomes equality on the function f_e,h.
From (<ref>) and (<ref>) it follows that for all f∈ W^1,p(C)∩ L_⌋·⌈(C) and h>0
f_L_∞(C)≤ A(d,p) μ^-1/p() h^1 - d/p|∇ f|__L_p(C)
+
μ^-1(K∩ C)h^-d⌋ f⌈.
Moreover, for all h>0, ⌋ f_e,h⌈_h = ⌋ f_e,h⌈, hence inequality (<ref>) is sharp and becomes equality for the function f_e,h.
In order to prove (<ref>) it is sufficient to plug
h
=
(pdμ^-1/p'() ⌋ f⌈/(p-d) A(d,p)|∇ f|__L_p(C))^p'/p'+d
into the right-hand side of (<ref>).
Next we prove sharpness of inequality (<ref>). Indeed, from this inequality, using Young's inequality it is not hard to obtain that for each h>0 inequality (<ref>) holds. It turns into equality for the function f_e,h. Hence inequality (<ref>) also becomes equality on the function f_e,h.
The last statement of the theorem is obvious.
§ LANDAU–KOLMOGOROV TYPE INEQUALITIES FOR CHARGES
By 𝔑(C) we denote the family of charges ν defined on Lebesgue measurable subsets of the set C that are absolutely continuous with respect to the Lebesgue measure μ, see e.g. <cit.>.
The Radon–Nikodym derivative of a charge ν with respect to the measure μ will be denoted by D_μν.
The family 𝔑(C) is a linear space with respect to the standard addition and multiplication by a real number. For h>0 and ν∈𝔑(C) we consider seminorms
⌉ν⌊_h=sup_x∈ C|ν(x +hK)| and ⌉ν⌊=sup_h>0⌉ν⌊_h.
It is clear that
⌉ν⌊_h=⌋ D_μν⌈_h i ⌉ν⌊=⌋ D_μν⌈.
By 𝔑_⌉·⌊_h(C) (𝔑_⌉·⌊(C)) we denote the set of charges ν∈𝔑(C) that have finite seminorm ⌉·⌊_h (resp. ⌉·⌊).
Applying Theorem <ref> to the function f=D_μν, we obtain the following result.
If h>0 and the charge ν∈𝔑_⌉·⌊_h(C) is such that D_μν∈ W^1,p(C), then
D_μν_L_∞(C)≤
A μ^-1/p() h^1 - d/p|∇ D_μν|__L_p(C) +
μ^-1(K∩ C)h^-d⌉ν⌊_h,
where A(d,p) is defined in (<ref>).
Inequality (<ref>) is sharp. It becomes equality for the charge ν_e,h such that D_μν_e,h=f_e,h, where the function f_e,h is defined in (<ref>).
If the charge ν∈𝔑_⌉·⌊(C) is such that D_μν∈ W^1,p(C), then the following multiplicative inequality holds.
D_μν_L_∞(C)≤
a(d,p) μ^-α/d()⌋ν⌈ ^1-α|∇ D_μν|__L_p(C) ^α,
where α and a(d,p) are defined in (<ref>). Inequality (<ref>) is sharp. It becomes equality for each charge ν such that D_μν=f_e,h, h>0.
§ INEQUALITIES THAT CONTAIN THE L_P-NORM OF THE GRADIENT OF THE MIXED DERIVATIVE
Let 1≤ m≤ d, C=^d_m,+=^m_+×^d-m, and K=(-1,1)^d. Then hK∩ C=(0,h)^m × (-h,h)^d-m, h>0, and |x|_K=max{|x_1|,…,|x_d|}.
For I=(1,…,1)∈^d, a locally integrable function f C→ and the standard basis {e_i}_i=1^d in ^d set
∂_ If=∂ ^d f/∂ x_1…∂ x_d
(the derivatives are understood in the distributional sense),
Δ^+_i,hf(x)=f(x+he_i)-f(x)
and Δ_i,hf(x)=f(x+he_i)-f(x-he_i).
According to the Fubini theorem, for almost all x∈^d_m,+ we have
∫_x+hK∩ C∂_ If(u)du=(Δ^+_1,h∘…∘Δ^+_m,h∘Δ_m+1,h∘…∘Δ_d,h)f(x).
It is easy to see that for the operator 𝔖_h,m L_∞(C)→ L_∞(C)
𝔖_h,mf(x)=1/2^d-mh^d(Δ^+_1,h∘…∘Δ^+_m, h∘Δ_m+1,h∘…∘Δ_d,h)f(x),
𝔖_h,mf(x)=S_h∂_ If(x) ∀ f∈ L_∞(C), and
𝔖_h,m_L_∞(C)→ L_∞(C) = 2^mh^-d.
For h>0, K=(-1,1)^d, C = ^d_m,+ and the function f∈ L_∞(C) such that ∂_If∈ W^1,p(C), the following inequality holds:
∂_
If_L_∞(C)≤
A(d,p)h^1 - d/p2^m-d/p|∇∂_ If|__L_p(C)
+
2^mh^-d f_L_∞(C),
where A(d,p) is defined in (<ref>). It can be rewritten in the following multiplicative form:
∂_
If_L_∞(C)≤
a(d,p) 2^α(m/d - d/p) f_L_∞(C) ^1-α|∇∂_ If |__L_p(C) ^α,
where α and a(d,p) are defined in (<ref>).
For m=0 and m = 1 inequalities (<ref>) and (<ref>) are sharp.
Taking into account that μ()=2^d-m, and formulae (<ref>) and (<ref>), one has
∂_If_L_∞(C)≤∂_If-𝔖_h,mf_L_∞(C)
+ 𝔖_h,m_L_∞(C)→ L_∞(C)f_L_∞(C)
=
∂_If-S_h ∂_If_L_∞(C)
+ 2^mh^-df_L_∞(C)
≤ A(d,p) · 2^m-d/p h^1-d/p|∇∂_If|__L_p(C) + 2^mh^-df_L_∞(C),
and inequality (<ref>) is proved. From inequality (<ref>) for the function ∂_If and equality (<ref>), we obtain
∂_If_L_∞(C)≤
a(d,p) μ^-α/d()⌋∂_If⌈ ^1-α|∇∂_If|__L_p(C) ^α
≤
a(d,p) 2^(m-d)α/d(2^d f_L_∞(C))^1-α|∇∂_If|__L_p(C) ^α
=
a(d,p) 2^α(m/d - d/p) f_L_∞(C) ^1-α|∇∂_ If |__L_p(C) ^α,
and inequality (<ref>) is proved.
We prove sharpness of inequalities (<ref>) and (<ref>) for m = 0. For the function f_e,h defined in (<ref>), and the function
F_e,h(x) = ∫_0^x_1…∫_0^x_df_e,h(u)du,
we have ∂_ IF_e,h = f_e,h, |∇∂_ I F_e,h(·)|_=|∇ f_e,h(·)|_, and due to the symmetry considerations,
2^d F_e,h_L_∞(C) = 2^d∫_(0,h)^df_e,h(u)du = ∫_(-h,h)^df_e,h(u)du =
⌋ f_e,h⌈_h =
⌋ f_e,h⌈.
Since for the function f_e,h inequalities (<ref>) and (<ref>) become equalities, taking into account (<ref>), we obtain sharpness of inequalities (<ref>) and (<ref>) for m = 0.
Next we prove sharpness of inequalities (<ref>) and (<ref>) in the case m = 1. In this case h = (0,h)× (-h,h)^d-1 and μ () = 2^d-1. There exists a number a∈ (0,h) such that
∫_(0,a)× (-h,h)^d-1 f_e,h(u)du = ∫_(a,h)× (-h,h)^d-1 f_e,h(u)du
.
Consider the function
G_e,h(x) = ∫_a^x_1∫_0^x_2…∫_0^x_df_e,h(u)du.
We have ∂_ IG_e,h = f_e,h and hence |∇∂_ I G_e,h(·)|_=|∇ f_e,h(·)|_.
The hyperplanes x_1 = a and x_j=0, j=2,…, d, split the set (0,h)× (-h,h)^d-1 into 2^d parallelepipeds Π_1,…, Π_2^d; due to the symmetricity with respect to the coordinate hyperplanes of the plot of the function f_e,h and equality (<ref>), we have
∫_Π_if_e,h(u)du = 1/2^d∫_(0,h)× (-h,h)^d-1 f_e,h(u)du, i =1,…, 2^d,
and hence
G_e,h_L_∞(C) = 1/2^d∫_(0,h)× (-h,h)^d-1 f_e,h(u)du =
1/2^d⌋ f_e,h⌈_h =
1/2^d⌋ f_e,h⌈.
From these equalities and the fact that inequality (<ref>) and (<ref>) become equalities for the function f_e,h, we obtain sharpness of inequalities (<ref>) and (<ref>) for m = 1.
§ SOME APPLICATIONS
Let X, Y and Z be linear spaces with seminorms ·_X, ·_Y and ·_Z respectively. A linear operator S X→ Y is called bounded (and we write S∈ℒ(X,Y)), if
S_X→ Y=sup{ Sx_Y x_X≤ 1}<∞.
Let A X → Y, B X→ Z be two homogeneous operators with domains of definition D_A, D_B⊂ X, D_B⊂ D_A. Set 𝔐={ x∈ D_B Bx_Z≤1}.
For the operator A and an operator S∈ℒ(X,Y) set
U(A,S; 𝔐):=sup{Ax-Sx_Y x∈𝔐}.
For arbitrary S∈ℒ(X,Y) and each x∈ D_B the following Landau–Kolmogorov–Nagy type inequality holds:
Ax_Y≤ Ax-Sx_Y+ S_X→ Yx_X
≤ U(A,S;𝔐) Bx_Z+S_X→ Y·x_X.
From Theorem <ref> it follows that for the operators A f↦ f, B f↦ |∇ f|_K^∘ and S f↦ S_hf inequality (<ref>)
becomes equality on the functions f=f_e,h.
From Theorem <ref> it follows that for Aν↦ D_μν, Bν↦ |∇ D_μν|_K^∘ and Sν↦ν(x+hK∩ C)/μ(hK∩ C inequality (<ref>)
becomes equality for the charge ν=ν_e,h.
Finally, from Theorem <ref> it follows that for A f↦∂_If, B f↦ |∇∂_If|_K^∘ and S f↦𝔖_h,mf inequality (<ref>)
becomes equality for f=F_e,h in the case m=0, and for f=G_e,h in the cases m=1.
These observations together with known general facts (see e.g. <cit.> and <cit.>),
allow to solve the problems of approximation of the corresponding unbounded operators by bounded ones and related problems.
unsrt
|
http://arxiv.org/abs/2307.07444v1 | 20230714160648 | Complete characterization of robust perfect adaptation in biochemical reaction networks | [
"Yuji Hirono",
"Ankit Gupta",
"Mustafa Khammash"
] | q-bio.MN | [
"q-bio.MN",
"cond-mat.stat-mech",
"physics.bio-ph",
"physics.chem-ph"
] |
KUNS-2973
[email protected]
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
RIKEN iTHEMS, RIKEN, Wako 351-0198, Japan
[email protected]
Department of Biosystems Science and Engineering, ETH Zurich, 4058 Basel, Switzerland
[email protected]
Department of Biosystems Science and Engineering, ETH Zurich, 4058 Basel, Switzerland
Perfect adaptation is a phenomenon whereby the output variables of a system can reach and maintain certain values despite external disturbances.
Robust perfect adaptation (RPA) refers to an adaptation property that does not require fine-tuning of system parameters.
RPA plays a vital role for the survival of living systems in unpredictable environments, and there are numerous examples of biological implementations of this feature.
However, complex interaction patterns among components in biochemical systems pose a significant challenge in identifying RPA properties and the associated regulatory mechanisms.
The goal of this paper is to present a novel approach for identifying all the RPA properties that are realized for a generic choice of kinetics for general deterministic chemical reaction systems.
This is accomplished by proving that an RPA property with respect to a system parameter can be represented by a subnetwork with certain topological features.
This connection is then exploited to show that these special structures generate all kinetics-independent RPA properties, allowing us to systematically identify all such RPA properties by enumerating these subnetworks. An efficient method is developed to carry out this enumeration, and we provide a computational package for this purpose.
We pinpoint the integral feedback controllers that work in concert to realize each RPA property, casting our results into the familiar control-theoretic paradigm of the Internal Model Principle.
Furthermore, we generalize the regulation problem to the multi-output scenario where the target values belong to a robust manifold of nonzero dimension, and provide a sufficient topological condition for this to happen.
We call the emergence of this phenomenon as manifold RPA.
The present work significantly advances our understanding of regulatory mechanisms that lead to RPA in endogenous biochemical systems, and it also provides rational design principles for synthetic controllers.
We demonstrate these results through illustrative examples as well as biological ones.
The present results indicate that an RPA property is essentially equivalent to the existence of a “topological invariant", which is an instance of what we coin as the “Robust Adaptation is Topological” (RAT) principle.
Complete characterization of robust perfect adaptation
in biochemical reaction networks
Mustafa Khammash
August 12, 2023
============================================================================================
§ INTRODUCTION
§.§ Context and motivation
Maintaining stability in a variable environment is a crucial issue for biological systems <cit.>.
One strategy adopted by living cells to achieve this is perfect adaptation, which is a property of a system to maintain the level of certain quantities by countering the effects of disturbances within biochemical reaction networks <cit.>.
Perfect adaptation is said to be robust when no fine-tuning of system parameters is needed to achieve the adaptation.
Having this property of robust perfect adaptation (RPA)
contributes to the stability of a system,
and indeed there are numerous biological examples of RPA such as chemotaxis
of prokaryotes <cit.> and eukaryotes <cit.>,
calcium homeostasis <cit.>,
glucose uptake of cancer cells <cit.>,
yeast osmoregulation <cit.>,
cell signaling <cit.>,
scaling of morphogen gradient <cit.>, and so on.
Revealing how cells implement RPA in molecular networks is important not only for understanding the origin of robustness in living systems but also for designing synthetic biomolecular systems. Recent advances in synthetic biology techniques have afforded scientists an unprecedented ability to engineer biomolecular controllers, utilizing genetic components, and transport them into living cells, where they can accomplish novel functions, like RPA. This emerging field, known as Cybergenetics <cit.>, focuses on the analysis and design of genetic control systems and holds tremendous potential for various domains, including industrial biotechnology and medical therapy.
Most existing work on RPA in biological reaction systems borrows ideas from control theory and views RPA as the property of an input-output system to robustly reject constant-in-time disturbances to the input variable by ensuring that the steady-state level of the output remains unaffected <cit.>. Typically, the output is the concentration of some chemical species (e.g. a protein of interest) with a pre-defined steady-state value (called the set-point), and the input is the concentration of some other species (e.g. enzyme, chemical inducer etc.) or some exogenous variable. For such single-input single-output (SISO) reaction systems, it is known that only two types of RPA topologies emerge: the incoherent feedforward (IFF) loops and the negative feedback (NFB) networks. This was first discovered computationally through an exhaustive search over three-node networks <cit.> (see also Ref. <cit.>), but it has recently been proved mathematically that any arbitrarily-sized RPA network must essentially be composed of IFF and NFB modules <cit.>. The reason for such strict structural requirements for RPA networks is linked to the famous Internal Model Principle (IMP) <cit.> of control theory, which mandates that any RPA system must be internally organized (after possibly a change of coordinates) into two distinct components – an internal model (IM) that computes the time-integral of the deviation of the output and its set-point, and the rest of the network (RoN) which receives both input disturbances as well as counteracting signals from the IM to reject these disturbances (see Fig. <ref> in Section <ref>). This argument shows that an IM serves as an embedded controller that generates RPA by implementing the famous integral feedback mechanism. This mechanism not only resides at the forefront of modern control engineering but has also been identified in numerous endogenous biochemical RPA networks <cit.>. A specific type of SISO RPA is called Absolute Concentration Robustness (ACR) which refers to the robustness of the concentration of output species when the initial state of the system is perturbed. Traditionally, the study of ACR networks has relied on techniques from Chemical Reaction Network Theory (CRNT) <cit.>, starting from the seminal paper by Shinar and Feinberg in 2010 <cit.>. A recent development has established a connection between ACR and integral controllers <cit.>, resulting in a deeper understanding of the mechanisms underlying ACR.
Although robustness to a specific single input may be relevant in some scenarios, the notion of a single input source for disturbances appears to be generally limiting within biological contexts, where disturbances manifest as parameter perturbations that can arise due to multitude of reasons, such as temperature changes <cit.>, onset of stress conditions <cit.>, resource competition and burden effects <cit.>, etc.
As many of these disturbances are often simultaneously present, it is important to consider RPA networks with multiple inputs, and in such a setting, conditions have been identified that completely characterize biochemical networks that exhibit a maximal form of RPA (called maxRPA), whereby the set-point of one output species is robust to the most number of reaction rate parameters, i.e. it depends on the least number of parameters <cit.>.
The networks exhibiting maxRPA are examples of multiple-input single-output (MISO) RPA systems.
The characterization of more general multiple-input multiple-output (MIMO) RPA systems has been largely unexplored so far (see however Ref. <cit.> for the two-output case).
A major challenge for finding MIMO RPA structures in biological systems arises from the absence of a presumptive separation between a controlled system and a controller, which contrasts with control engineering, where these components are given from the outset.
Inspired by the connection between RPA and integral actions, recently an algebraic approach has been devised, that identifies all RPA properties, along with the hidden integral actions that generate this property <cit.>. This method assumes mass-action kinetics (see Eq. (<ref>)) to express the reaction rates as a function of reacting species' concentrations.
This ensures that the steady-states of the system satisfy a system of polynomial equations, and by investigating the ideal generated by these polynomials, through resource-intensive Gröbner basis computations, all RPA properties can in principle be found.
While the law of mass-action works remarkably well in in vitro (i.e. `test tube' ) conditions, where the reacting species are sufficiently dilute and well-mixed <cit.>, many experimental and computational studies have demonstrated that this law breaks down within living cells (see Ref. <cit.> and the references therein).
Several factors contribute to this breakdown, including macromolecular crowding in the cytoplasmic soup where reactions occur <cit.> and spatial compartmentalisation within the cells <cit.>.
These factors significantly impede molecular diffusion compared to in vitro conditions <cit.>, resulting in the loss of well-mixed characteristics.
Additionally, reaction networks serve as simplifications of the intricate reality found within cells. Kinetics-altering factors like conformational changes in protein structures <cit.>, allosteric perturbations <cit.>, resource competition <cit.>, and context dependence <cit.> are deliberately disregarded for the sake of model simplicity and manageability.
In such cases, the assumption of mass-action kinetics becomes tenuous. One could hypothesize that if RPA is crucial for a cell's survival, it would make evolutionary sense to select RPA-generating strategies that are independent of kinetics, meaning the RPA property remains invariant regardless of the chosen kinetics.
This notion holds significance not only in understanding living systems but also in the field of synthetic biology where one may aim to develop molecular controllers that can achieve RPA when interfaced with natural intracellular networks, irrespective of the specific kinetics employed <cit.>.
It is important to note that while the structure of a reaction network is determined by the chemical properties of the reacting species, which can be reliably determined through in vitro experiments, knowledge of reaction kinetics does not readily transfer from in vitro studies to the actual conditions within living cells <cit.>.
Distinctly from control-theoretic ideas, there exist alternative methods to characterize the response of reaction systems to parameter perturbations, based on the topological analysis of subnetworks containing the perturbed parameters <cit.>.
In such methods, a crucial role is played by the integers assigned to a given subnetwork based on its topological characteristics.
It has been shown that if a subnetwork is output-complete (meaning that all reactions that have a reactant species within the subnetwork are included in the subnetwork) and its influence index is zero, the steady-state values of the species-concentrations and reaction fluxes outside this subnetwork are insensitive to perturbations of parameters inside the subnetwork.
Namely, these special subnetworks confine the effect of perturbations inside them, and hence such subnetworks are called buffering structures.
This result can be interpreted as a sufficient condition for RPA: the concentrations and reaction rates outside the subnetwork exhibit RPA, if the subnetwork is output-complete and has a zero influence index.
More recently, a slightly different index called the flux influence index was introduced to identify parameters under the perturbation of which all the reaction rates (fluxes) exhibit RPA <cit.>.
A basic idea behind these topological approaches is that we can estimate the impact of parameter perturbations based on indices determined from network topology.
Although these indices provide us with sufficient conditions for RPA, whether all RPA properties can be explained in this way or not has been an open question.
The underlying regulatory mechanisms responsible for realizing these RPA properties and the relation of these results to control-theoretical approaches also remain unresolved.
§.§ Main results and structure of the paper
Motivated by these developments, the objective of this paper is to develop a systematic approach to identify all RPA properties that do not require a specific choice of kinetics for general deterministic chemical reaction systems (see Fig. <ref> for the sketch of our strategy).
Existing works on RPA define this notion as robustness of one or more quantities of interest (like steady-state species concentrations, reaction rates) with respect to certain system parameters (e.g. rate constants, values of conserved quantities, etc.).
To characterize regulatory patterns in biochemical system,
we shall regard a generic RPA property as a triplet (𝒱, ℰ, p) of subsets of species, reactions and system parameters respectively, which signifies that the steady-state concentrations of species in 𝒱 and rates of reactions in ℰ exhibit RPA to perturbations of any parameter in p.
It is easy to see that any generic RPA property can be viewed as a combination of elementary RPA properties that are each defined as subsets of species and reactions that exhibit RPA with respect to a single parameter p[See Sec. <ref> for definitions of the elementary and generic RPA properties].
The main result of the paper proves a one-to-one correspondence between each elementary RPA property in a deterministic reaction system and a buffering structure with additional information, which we call a labeled buffering structure (Theorem <ref>).
Therefore, to characterize and systematically identify all generic RPA properties, it suffices to enumerate all the labeled buffering structures. We have devised an efficient algorithm to accomplish this enumeration and implemented it in Mathematica as an open-source computational package named RPAFinder <cit.>.
This package can be used to identify the degrees of freedom with a desired RPA property, which is important for designing RPA-achieving synthetic controllers <cit.> and for finding yield-optimizing perturbations in metabolic pathways (see Sec. <ref>).
Furthermore, we uncover the hidden integral feedback control responsible for achieving a generic RPA property by explicitly constructing the Internal Model along with the set of integrators it generates.
Therefore, we discover a novel Internal Model Principle for kinetics-independent RPA networks, which is of independent interest in the control-theory community and has deep connections with Cybergenetic applications <cit.>.
In a number of examples, we find that the set of integrators forces the dynamics of multiple output species towards a manifold with nonzero dimension.
This phenomenon, which we call manifold RPA (see Fig. <ref> for a schematic illustration), implies that while the steady-state concentrations of the output species is sensitive to certain parameters, the relationship among the species is robustly maintained.
We show that any output-complete subnetwork gives rise to manifold RPA (Theorem <ref>), and the dimension of the target manifold is given by the influence index. This can be seen as a natural generalization the law of localization.
This form of multi-output control generalizes conventional approaches and has important implications for rational design in synthetic biology <cit.>.
At this point, we would like to compare our results with the characterization results obtained in Ref. <cit.> for RPA networks with the mass-action kinetics. Unlike the approach in Ref. <cit.>, our analysis does not rely on any computationally-expensive procedures. Instead, we employ elementary linear-algebraic computations to fully characterize and identify all RPA properties. This approach is particularly well-suited for application to large-scale networks, such as those encountered in metabolic engineering (refer to Section <ref>). Furthermore, our analysis yields a conventional IMP for RPA networks with a single IM that generates a global set of integral actions. This is in contrast to the non-standard IMP formulation presented in Ref. <cit.>, which considers each RPA network as a topologically organized collection of subnetworks, with each subnetwork possessing its own independent IM and localized integral action. Generally, these localized integral actions cannot be synthesized to form global integral actions, as required by the standard IMP. Nonetheless, the findings in Ref. <cit.> offer intriguing insights into the various ways in which RPA can arise due to intricate polynomial factorizations. Our results show that many of these ways `drop out' if deviations from mass-action kinetics are anticipated, revealing a linear structure that we meticulously unravel in this paper.
If we view the phenomenon of RPA in a broader context, the fact that a system with an RPA property is characterized by a “topological invariant” should not be restricted to deterministic chemical reaction systems and would also be true for more generic systems.
Since the “robustness” of adaptation does not allow the fine-tuning of reaction rates to achieve adaptation, the RPA conditions we obtain should be insensitive to deformations of the system, such as changes in parameters or modifications of rate functions.
Indeed, the condition of a zero index (or output-completeness) is topological, in the sense that it is insensitive to these deformations of reaction systems.
Generically, one can expect that the class of systems
with an RPA property, which does not necessarily have to be deterministic chemical reaction systems, should be of this feature.
Let us summarize this claim as the “Robust Adaptation is Topological” (RAT) principle,
by which we mean that the class of dynamical systems that exhibit robust adaptation is characterized by a topological invariant.
The RAT principle itself is not a concrete theorem,
but rather a template of theorems,
and different technical assumptions should be made
depending on the nature of the systems under consideration.
Namely, depending on technical details such as
the class of “dynamical systems” to be considered
or the choice of output variables,
the corresponding “topological invariant”
should be defined appropriately.
The law of localization <cit.>
and flux RPA <cit.>
are particular realizations of the RAT principle for deterministic chemical reaction systems.
The one-to-one correspondence of elementary RPA properties and labeled buffering structures indicates that topological characteristics can, in fact, exhaust all the RPA properties in these systems.
This leads us to hypothesize that topological characterization is crucial for other classes of dynamical systems that exhibit RPA properties beyond deterministic chemical reaction systems.
Let us describe the organization of the paper.
In Sec. <ref>, after introducing basic notions on the description of deterministic chemical reaction systems, we specify the technical setting, including the definitions of generic and elementary RPA properties.
In Sec. <ref>, we introduce a formalism to describe the response of steady states to parameter perturbations, and review topological approaches toward RPA.
In Sec. <ref>, we give a review of a control theoretical approach toward RPA, and we review the theory of maxRPA.
In Sec. <ref>, we show that all the elementary RPA properties in a network can be represented by labeled buffering structures, and generic RPA properties are generated by them. We also give a numerical algorithm for the identification of all the labeled buffering structures for a given chemical reaction network. We illustrate the method through several examples.
In Sec. <ref>, we discuss the topological characterization of kinetics-independent maxRPA, and interpret this result via the law of localization.
Hinted by this connection, in Sec. <ref>, we describe how to construct integral feedback control for a given buffering structure.
In Sec. <ref>, we discuss the phenomenon of manifold RPA, and provide a sufficient topological condition for this to be realized.
In Sec. <ref>, we discuss the naturality of a topological characterization of systems with RPA properties and propose the RAT principle.
Section <ref> is devoted to a conclusion and future outlook.
§ CHEMICAL REACTION SYSTEMS AND ROBUST PERFECT ADAPTATION
Various processes within biological cells can be viewed and modeled as chemical reaction networks.
In this section, we introduce the description of generic deterministic chemical reaction systems.
Then, we introduce notions for discussing the phenomenon of robust perfect adaptation along with basic technical assumptions.
§.§ Chemical reaction systems
We consider a chemical reaction network
Γ= (V, E), which consists of a set V of chemical species
and a set E of chemical reactions.
A reaction e_A ∈ E can be specified as
e_A :
∑_i s_iA v_i →∑_i t_iA v_i ,
where v_i ∈ V, and s_iA,t_iA∈ℤ_≥ 0 are stoichiometric constants of species v_i for reaction e_A.
We denote the stoichiometric matrix by
S, whose components are given by S_iA t_iA - s_iA.
The total numbers of chemical species and reactions will be denoted by
M |V| ,
N |E| ,
where |…| of a set indicates its cardinality.
When we consider a dynamical system based on a given reaction network Γ, there are different levels of description.
If the number of molecules is large, we can model the system
deterministically, and in that case, the system variables
are the concentrations x_i of chemical species v_i ∈ V.
On the other hand, if the number of molecules is small, stochastic nature of the system becomes important, and then the system has to be modeled as a stochastic reaction system <cit.> governed by a chemical master equation.
In the present paper, we focus on the former situation and consider deterministic dynamics.
The time evolution of the concentrations is governed by the rate equation[We use bold fonts to indicate vectors. Here, the components of a vector x are given by x_i.],
d/dt x (t) = S r ,
where r_A is the reaction rate of reaction e_A ∈ E.
To solve the rate equations, we need to express the reaction rates, r_A, as functions of reactant concentrations, x, and parameters, k_A, i.e., r_A = r_A ( x; k_A). Such a choice is called kinetics.
For example, in the case of mass-action kinetics, the rate function of a reaction e_A is proportional to the product of reactant concentrations,
r_A ( x; k_A) = k_A ∏_i x_i^s_iA.
In the present work, we do not assume any specific kinetics unless otherwise stated. Rather, we seek properties that do not depend on the choice of kinetics.
Since we are interested in RPA, we consider a situation where the system reaches an asymptotically-stable steady state in the long-time limit as we discuss in more detail in Sec. <ref>. The steady-state solution can be obtained by solving the following equations,[
Here we use different characters for
indices of
chemical species (i, j, …),
reactions (A,B,…),
the basis of the kernel of S (α, β, …), and that of cokernel of S (α̅, β̅, …).
We use the notation where |i| indicates the number of values the index i takes.
Thus, |i| indicates the number of chemical species,
|A| indicates the number of reactions,
|α| denotes the dimension of S,
and |α̅| denotes the dimension of S.
]
∑_A S_iAr̅_A ( x̅ ( k, ℓ); k_A ) = 0,
∑_i d^(α̅)_i x̅_i ( k, ℓ)
= ℓ^α̅,
where x̅_i and r̅_A indicate the steady-state values of
concentration x_i and reaction rate r_A, respectively.
Note that we here allow the stoichiometric matrix to have left null vectors, and the system can have conserved quantities.
The set of vectors { d^(α̅)}_α̅=1,…,|α̅| is a basis of S,
and we need to specify the values of conserved quantities
as in Eq. (<ref>) to obtain the steady-state solution (if S is nontrivial).
The set ( k, ℓ) ∈ K × L specifies the parameters of this chemical reaction system, where k is the vector of all reaction parameters, ℓ is the vector of the values of all the conserved quantities, and K and L are open sets of all admissible parameter vectors and values of conserved quantities, respectively.
§.§ Robust perfect adaptation
Let us now give the definition of robust perfect adaptation and present basic assumptions.
We say that a deterministic chemical reaction system exhibits perfect adaptation in species v_i ∈ V with respect to a certain parameter p ∈ ( k, ℓ)
when x_i(t) asymptotically reaches a steady-state value that is independent of p, i.e.
lim _t →∞ x_i(t) = x̅_i ,
where x̅_i is independent of p.
Similarly, we say that the system exhibits perfect adaptation in reaction e_A with respect to p if the corresponding reaction rate r_A reaches a value independent of p asymptotically.
A perfect adaptation property is said to be robust, if the adaptation occurs without fine-tuning of system parameters.
In this paper, we are concerned with such robust perfect adaptation (RPA) realized in deterministic chemical reaction systems.
If the system exhibits RPA in v_i ∈ V with respect to p, we will also say that the system has an RPA property in v_i with respect to p.
Observe that, if the system exhibits perfect adaptation in v_i ∈ V,
Eq. (<ref>) implies that the steady-state concentration
is unaffected by constant-in-time disturbances of parameter p,
p ↦ p + δ p .
Namely, the system has the ability to reject the effect of disturbances in p on x_i.[ We will discuss the control-theoretical aspects of this phenomenon from Sec. <ref>. ]
As a most general case of RPA, a subset 𝒱⊂ V of species and a subset ℰ⊂ E of reactions can exhibit RPA with respect to a subset of system parameters p ⊂ ( k, ℓ). Thus, a generic RPA property of a system is characterized by these three ingredients, (𝒱, ℰ, p).
How can we characterize all such generic RPA properties in a given chemical reaction system?
In a deterministic chemical reaction system,
the property of a steady state is specified by the values of concentrations and reaction rates at the steady state, x̅_i and r̅_A.
In general, the steady-state concentration x̅_i of species v_i ∈ V
is a function of a subset of system parameters ( k, ℓ).
Namely, concentration x_i exhibits RPA under the perturbation of the parameters on which x̅_i does not depend.
For example, if the steady-state concentration of v_1 depends only on k_1 and k_3, we can write it as x̅_1 (k_1, k_3), and hence it will exhibit RPA under the perturbation of k_2, k_4, or other system parameters except for k_1 and k_3.
If we identify the dependencies of all the concentrations and reaction rates on all the parameters, we will have a complete characterization of the RPA properties existing in the system.
Equivalently, we can perform the perturbation p ↦ p + δ p of a chosen parameter p ∈ ( k, ℓ), and identify the concentrations and reaction rates whose steady-state values are affected and unaffected by this perturbation.
The unaffected ones exhibit RPA with respect to p.
We call this separation of degrees of freedom whose steady-state values are affected and unaffected by p as an elementary RPA property with respect to parameter p.
Once we identify elementary RPA properties with respect to all the system parameters, we have a complete characterization of the RPA properties in the system, since a generic RPA property can be obtained by combinations of elementary ones.
In this work, we allow the reaction kinetics to be generic. While mass-action kinetics has often been assumed in prior studies, there exist multiple factors that can undermine this assumption. Moreover, experimental verification of its validity within cellular environments is highly challenging. Accordingly, the RPA properties that we discuss here are of the following nature:
An RPA property is said to be kinetics-independent if
the adaptation occurs regardless of the choice of kinetics
as long as the stability of the system is maintained.
For example, even if we observe an RPA property with a system with mass-action kinetics, it may disappear once we deform the kinetics
away from mass-action kinetics.
The RPA properties we discuss here are those that survive such deformations of a system.
Let us comment on the nature of perturbations under consideration.
In this paper, we consider the constant-in-time disturbances of
reaction parameters and the values of conserved quantities,
k_A
↦
k_A + δ k_A,
ℓ_α̅↦ℓ_α̅ + δℓ_α̅ .
When a reaction rate has multiple parameters,
the perturbation k_A ↦ k_A + δ k_A indicates that one of them is disturbed.
Since we allow the kinetics to be a generic one, the perturbation of a reaction parameter, k_A ↦ k_A + δ k_A, can be regarded as an arbitrary one-parameter deformation of the reaction rate function.
There can be situations where parameters of several reactions are simultaneously perturbed, for example, through the change of temperatures.
Once we know the response to the change of each parameter,
we can use this information to compute the response
under, for example, a temperature variation
by considering the temperature dependence of each parameter, k_A (T).
Let us summarize the assumptions we make throughout the paper.
As we are interested in the phenomenon of RPA, we are here concerned with the situation where a steady state is realized eventually, after adding a constant-in-time perturbation to the system.
Accordingly, we make the following assumption:
[Stability]
The reaction system has an asymptotically stable steady state.
In other words, the Jacobian matrix[
Here, we mean the Jacobian matrix of a reaction system where the values of conserved quantities are fixed. Thus, there is no zero eigenvalue associated with conserved quantities.
] at the steady state is non-singular and every eigenvalue has a strictly negative real part.
We also assume the following,
The steady-state concentration is nonzero, x̅_i ( k, ℓ) > 0,
for any v_i ∈ V, and
(
r_A ( x ; k_A)/ x_i)_ x = x̅( k, ℓ)≠ 0 if v_i ∈ V is a reactant of reaction e_A
= 0 otherwise.
§ RPA FROM A TOPOLOGICAL PERSPECTIVE
We here introduce a basic formalism for characterizing the responses
of steady states to parameter perturbations.
Then we introduce the law of localization <cit.>,
that gives us a sufficient condition for RPA based on the network topology.
§.§ Notations
In this subsection, let us summarize the notations we will use in the following.
Generically, we will use overlines to indicate quantities at steady state,
such as x̅ ( k, ℓ) and r̅ ( x̅ ( k, ℓ) ; k_A).
The arguments of a function may be omitted for simplicity.
The derivative of a steady-state concentration with respect to
rate parameter k_B
and the value ℓ^α̅ of a conserved quantity
will be denoted as
_B x̅_i
=
x̅_i, B/ k^Bx̅_i ( k, ℓ)
,
_α̅x̅_i
=
x̅_i, α̅/ℓ^α̅x̅_i ( k, ℓ)
As for reactions, note that each steady-state reaction rate,
r̅_A (x̅ ( k, ℓ) ; k_A),
has an implicit dependence on parameters ( k, ℓ) through x̅, as well as an explicit dependence on a particular parameter k_A.
In the following expressions, we mean the derivative of steady-state reaction rates with respect to parameter k_B
including the both contributions of implicit and explicit dependence,
_Br̅_A
=
r̅_A,B
/ k_Br̅_A (x̅ ( k, ℓ) ; k_A) ,
and similarly for the derivative with respect to ℓ^α̅.
On the other hand, the following expression means the derivative of
the steady-state reaction rate with respect to k_B
only through the explicit dependence,
_B r_A
=
r_A,B(
/ k_B r_A ( x ; k_A)
)_ x = x̅( k, ℓ) .
Note that this quantity is nonzero only when A=B,
i.e., r_A,B∝δ_AB, where δ_AB
denotes the Kronecker delta.
The following expressions indicate the derivative of
the rate function with respect to a reactant concentration,
that is evaluated at steady state,
r_A/ x_i
=
_i r_A
=
r_A,i
(
/ x_i
r_A( x ; k_A)
)_ x = x̅( k, ℓ).
In the following, we will often choose a subnetwork γ = (V_γ, E_γ)⊂Γ = (V, E).
The degrees of freedom inside γ will be denoted with indices with stars,
while those outside γ will be denoted with indices with primes.
For example, x_i^⋆
denotes the concentration of a species inside γ
(i.e., v_i^⋆∈ V_γ),
while
x_i' denotes the concentration of a species outside γ,
v_i'∈ V ∖ V_γ.
Similarly, r_A^⋆ denotes
the reaction flux of
a reaction inside γ,
e_A^⋆∈ E_γ,
while
r_A' denotes the reaction flux of
a reaction outside γ,
e_A'∈ E ∖ E_γ.
§.§ Response of steady states
Let us now discuss the response of steady states against parameter perturbations.
Equation (<ref>) indicates that the steady-state reaction rates are in the kernel of the stoichiometric matrix S. Hence, the rates can be written as
r̅_A (x̅ ( k, ℓ); k_A )
= ∑_αμ_α ( k, ℓ) c^(α)_A ,
where { c^(α)}_α=1,… ,|α|
is a basis of S.
Taking the derivative of
Eqs. (<ref>) and (<ref>)
with respect to k_B and ℓ^β̅,
we have
∑_i r_A/ x_ix̅_i/ k_B
+
r_A/ k_B = ∑_αμ_α/ k_B c^(α)_A ,
∑_i r_A / x_i x̅_i/ℓ^β̅ =
∑_αμ_α/ℓ^β̅ c^(α)_A ,
∑_i d_i^(α̅)x̅_i/ k_B = 0,
∑_i d_i^(α̅)x̅_i/ℓ^β̅ = δ^α̅β̅ .
As noted earlier, steady-state reaction rates r̅_A (x̅ ( k, ℓ); k_A) have an explicit dependence on k_A, as well as an implicit dependence on k and ℓ through steady-state concentrations,
x̅_i ( k, ℓ),
and r_A/ k_B indicates the derivative with respect to the explicit dependence.
Note also that r_A/ x_i is evaluated at steady state.
Equations (<ref>) – (<ref>)
can be simplified by introducing a matrix defined by[
The A-matrix is first introduced in Ref. <cit.> for
the case S = 0
and it was extended to the case S ≠ 0
in Ref <cit.>.
]
A[
[ _i r_A - c^(α)_A; d^(α̅)_i 0_|α̅| × |α| ]]
,
where _i / x_i.
The matrix A is (|A| + |α̅|) × (|i| + |α|) dimensional, and
because of the Fredholm's theorem, |A| + |α̅| = |i| + |α|, it is square.
Using this, Eqs. (<ref>)–(<ref>) are summarized compactly in the matrix form,
A _B
[ x̅; μ ]
= -
[ _B r; 0 ]
,
A _β̅[ x̅; μ ]
=
[ 0; _α̅ℓ ],
where, as mentioned earlier,
_B ∂ /∂ k^B
and
_β̅∂ /∂ℓ^β̅.
We note that the matrix A is invertible when the stability condition <ref> is satisfied.
For the case S = 0,
the invertibility of A results
from the one-to-one correspondence of
the eigenspectrum of the Jacobian matrix
and the generalized-eigenspectrum of A.
In Appendix <ref>,
we extend this correspondence to the case S ≠ 0,
from which the invertibility of A follows even in the presence of nontrivial S, as long as the stability condition <ref> is satisfied (see Corollary <ref>).
Thus, by multiplying A^-1 on Eq. (<ref>), we have
_B
[ x̅; μ ]
= -
A^-1[ _B r; 0 ]
,
_β̅[ x̅; μ ]
= A^-1[ 0; _α̅ℓ ].
Note that _B r_A is a diagonal matrix, i.e., _B r_A ∝δ_BA.
If we partition A^-1 as
A^-1 =
[ ( A^-1)_iA ( A^-1)_i α̅; ( A^-1)_α A ( A^-1)_αα̅ ],
the responses of steady-state concentrations and reaction rates to the perturbations of k_B and ℓ^β̅ are
proportional to the following components,
_B x̅_i ∝ ( A^-1)_iB ,
_β̅x̅_i ∝ ( A^-1)_i β̅ ,
_Bμ_α∝ ( A^-1)_α B ,
_β̅μ_α∝ ( A^-1)_αβ̅.
Let us introduce notations with which
the sensitivity can be expressed in a concise manner.
We organize the chemical concentrations and μ^α
as a single vector[We use ν,ρ,σ, … to denote the index of the vector y̅. ],
y̅^ν[ x̅^i; μ^α ].
We also denote the parameters k^A
and the values ℓ^α̅
of conserved quantities collectively,
q^ν[ k^A; ℓ^α̅ ].
Note that the vectors y̅^μ and q^μ have the same dimensions, |ν| = |A| + |α̅| = |i| + |α|.
Let us also introduce
L^ν[ r^A; -ℓ^α̅ ] .
The vector L^ν depends on both variables y
and parameters q,
L^ν = L^ν( y, q).
Its derivative with respect to the parameters
is represented as a matrix of the the following form,
L^ν/ q^ρ
=
[ r^A/ k^B 0; 0 - δ^α̅β̅ ] .
With these notations,
we can express Eq. (<ref>) as a single equation,
∑_ρ A_νρ y̅^ρ/ q^σ
=
- L^ν/ q^σ .
§.§ Law of localization
In this subsection, we introduce the law of localization <cit.>,
which gives us a sufficient topological condition
so that the effect of perturbations inside a subnetwork is localized inside it.
Let us choose a subnetwork γ,
which is specified by subsets of chemical species and reactions,
γ = (V_γ, E_γ).
A subnetwork γ is called output-complete
if E_γ includes all the reactions
whose reactants are in V_γ.
For a given output-complete subnetwork γ,
its influence index is defined by
λ(γ)
- |V_γ |
+
|E_γ |
-
|( S)_ supp γ|
+
| P^0_γ ( S) | .
The definitions of the spaces
that appear in the influence index
are given as follows:
( S)_ supp γ { c ∈ S
|
P^1_γ c = c
},
P^0_γ ( S)
{
P^0_γ d
| d ∈ S
},
where S is the stoichiometric matrix,
P^0_γ and P^1_γ
are the projection matrices to γ
in the space of chemical species and reactions, respectively.
Namely, ( S)_ supp γ
is the space of vectors of S
supported inside γ,
and P^0_γ ( S)
is the projection of S to γ.
We note that the influence index of an output-complete subnetwork is nonnegative under the assumption of stability (<ref>).
The statement of the law of localization is as follows:[
In Ref. <cit.>, S = 0 is assumed. The analysis is generalized to the case with S ≠ 0 in Ref. <cit.>.
]
Let γ⊂Γ be an output-complete subnetwork of
a deterministic chemical reaction system
satisfying the assumptions in Sec. <ref>.
When γ satisfies λ(γ)=0,
the steady-state values of
chemical concentrations and reaction rates
outside γ do not change
under the perturbation of rate parameters or
conserved quantities with nonzero support[
Let us clarify precise meaning of this expression.
For a given subnetwork, we first take basis { d^(α̅')}_α̅' = 1, …, |α̅'| of those vectors in S that have support only in V ∖ V_γ. We then extend this to a basis for the whole S by including vectors { d^(α̅^⋆)}_α̅^⋆ = 1, …, |α̅^⋆| that have nonzero support inside γ. Therefore if P_γ^0 denotes the projection matrix to γ,
we have
P_γ^0 d^(α̅^⋆)≠ 0
while
P_γ^0 d^(α̅') = 0. Note that the vectors {
P^0_γ d^α̅^⋆}_α̅^⋆ = 1, …, |α̅^⋆|
will be linearly independent by construction.
The perturbed parameter ℓ^α̅^⋆ is associated with any basis constructed this way.
]
inside γ.
Namely,
we have
_B^⋆x̅_i' = 0,
_B^⋆r̅_A' = 0,
_α̅^⋆x̅_i' = 0,
and
_α̅^⋆r̅_A' = 0
for any
e_B^⋆∈ E_γ,
v_i'∈ V ∖ V_γ,
e_A'∈ E ∖ E_γ,
and
_α̅^⋆
= /ℓ^α̅^⋆
is the derivative with respect to the value of a conserved quantity with nonzero support in γ.
For the proof of the theorem, see Refs. <cit.>.
This theorem tells us that the effect of perturbation is confined inside
a subnetwork with λ (γ)=0 for steady states,
and for this reason, a subnetwork with a zero influence index is called a buffering structure.
We note that the law of localization can in fact be interpreted as a form of RPA: the concentrations and reaction rates outside a buffering structure γ exhibit RPA with respect to the perturbation of parameters inside γ. In other words, the law of localization provides us with a sufficient condition for RPA.
In a similar spirit, the structural condition for the reaction fluxes to exhibit RPA has been identified.
It was shown that, by using a slightly different index, which is called the flux influence index, λ_ f(γ), we can find reaction parameters
under the perturbation of which all the reaction fluxes
exhibit RPA <cit.>.
A subnetwork with λ_ f(γ) = 0 is called
a strong buffering structure.
The flux influence is related to the influence index by
λ_ f(γ) =
λ(γ) + |( S)_ supp γ|.
A strong buffering structure is always a buffering structure,
i.e.
λ_ f(γ) =0 implies λ(γ)=0,
which is obvious from the relation between them.
Buffering structures are closed under union and intersection:
namely, if γ_1 and γ_2 are buffering structures,
so are γ_1 ∩γ_2 and γ_1 ∪γ_2.
This can be proven easily using the submodularity
of the influence index <cit.>.
The strong buffering structures also are closed
under union and intersection <cit.>.
§ RPA FROM A CONTROL-THEORETIC PERSPECTIVE
In this section, we discuss RPA from a control-theoretic perspective.
After introducing a general idea, we discuss maxRPA networks in which one distinguished species exhibit RPA with respect to maximally many parameters in a reaction system.
Then, we discuss how a buffering structure naturally arises for kinetics-independent maxRPA networks. In fact, the existence of such a buffering structure will be shown to be equivalent to the conditions for kinetics-independent maxRPA.
As we prove in Section <ref>, this equivalence goes beyond maxRPA networks, and it extends to all generic RPA properties.
§.§ RPA and integral feedback control
The biological notion of RPA is essentially equivalent to the notion of robust steady-state tracking, that is well known in control theory. This allows us to borrow control-theoretic concepts to understand the structural conditions for networks to exhibit RPA. One such concept is the famous Internal Model Principle (IMP) <cit.>, that sheds light into the organization of RPA networks.
In particular, it says that an RPA network Γ must contain a subsystem called the internal model (IM), that can generate the class of disturbances to which the RPA network adapts.
Furthermore, the IM generates the disturbance using only the regulated output variable (i.e. concentration of X) as input, and then passes restorative signals to the rest of the network (RoN) in order to eliminate the effect of the disturbance.
When the output species X belongs to RoN, this creates a natural “feedback” between IM and RoN (see Fig. <ref>).
Since we only consider constant-in-time perturbations of system parameters as disturbances, the function realized by the IM must be an “integrator” which computes the time-integral of the deviation of the output species concentration from its set-point (see Ref. <cit.> for more details).
There can be several instantiations of the IMP depending on the type of disturbances allowed and the nature of the robustness desired, and each version needs to be independently proven. So far the IMP has been shown in complete generality for linear systems <cit.> and for nonlinear systems that can be decomposed as affine functions of the disturbances <cit.> (see Ref. <cit.> for more details). It is important to note that typically RPA systems are not naturally structured into the IMP-mandated form (i.e. IM and RoN in feedback), but IMP asserts that they can be brought in this form after a nonlinear coordinate transformation (see the incoherent feedforward (IFF) example in Ref. <cit.>).
§.§ Characterization of maxRPA networks
Here, we discuss maxRPA networks <cit.> in detail and present their precise mathematical characterization, which generalizes the characterization result in Ref. <cit.>.
Consider a reaction network Γ =(V, E) with V= {v_1,…,v_M} consisting of M species and E = {e_1,…, e_N} consisting of N reactions. Suppose that this network regulates its last species v_M = X, which we shall call the output species. In particular, there is a set-point at which the concentration of X is maintained.
We say that a network satisfies the maxRPA property if the steady-state concentration of the output species v_M = X only depends on the parameters k_1̅ and k_2̅ of the last two reactions e_1̅ e_N-1
and
e_2̅ e_N. This implies that there is a function ϕ_ out such that
lim _t →∞ x(t) =
ϕ_ out(k_1̅, k_2̅)
for any ( k, ℓ) ∈ K × L,
where x(t) denotes the concentration of X at time t.
Here the function ϕ_ out provides an encoding between parameters k_1̅ and k_2̅, and the set-point for the maxRPA species. We assume that this function depends non-trivially on both k_1̅ and k_2̅, i.e. its partial derivative with respect to both parameters is nonzero.
When the last two reaction have mass-action kinetics (see Eq. (<ref>)), it can be shown that ϕ_ out can only depend on the ratio of k_1̅ and k_2̅ (see Ref. <cit.>), but we do not assume mass-action kinetics here and so ϕ_ out is arbitrary. Observe that Eq. (<ref>) implies that the steady-state output concentration is unaffected by constant-in-time disturbances that perturb parameters in ( k, ℓ), except k_1̅ and k_2̅.
The characterization result for maxRPA networks in Ref. <cit.> assumed that the cokernel of the stoichiometric matrix S is trivial, i.e. S = 0.
The extension of this characterization result, which we now state, will relax this assumption along with the assumption that reactions e_1̅ and e_2̅ have the mass-action kinetics.
Assuming that the chemical reaction system Γ =(V, E) is stable (see Assumption <ref>), it exhibits maxRPA for the output species v_M = X if and only if the following two conditions are satisfied:
* There exists a vector q ∈ℝ^M and a positive κ such that
q^⊤ S =
[ 0, ⋯, 0, κ, -1 ].
If such a pair ( q, κ) exists then the value of κ is unique and the value of vector q is unique up to addition of vectors in S.
* The ratio of the rate functions r_1̅( x; k_1̅) and r_2̅( x; k_2̅) for the last two reactions, depends on only the output species concentration x_M along with parameters k_1̅ and k_2̅, i.e. there exists a function Φ(x_M, k_1̅, k_2̅) such that
r_2̅( x; k_2̅)/r_1̅( x; k_1̅) = Φ(x_M, k_1̅, k_2̅).
Moreover if these conditions hold then the set-point x̅_M for the output species is uniquely determined by the implicit relation
Φ(x̅_M , k_1̅, k_2̅) = κ
where κ is the same constant as in Eq. (<ref>). In other words, there is a unique set-point encoding function ϕ_ out such that
Φ(ϕ_ out(k_1̅, k_2̅), k_1̅, k_2̅) = κ.
We present a proof in Appendix <ref>.
Note that the first condition (<ref>) is exactly the same as in the case with S = 0.
When S ≠ 0, the vector q (if it exists) will not be unique, as for any conservation relation
d∈ S, d^⊤ S = 0 and so ( q+ d)^⊤ S= q^⊤ S. However, as will be shown in the proof of Theorem <ref>, the value of κ is always unique, and the value of q in unique in the quotient subspace ℝ^M / S.
Therefore, a vector q ∈ ( S)^⊥ such that ( q, κ) satisfies Eq. (<ref>) is going to be unique[
Let us comment on how such a vector q can be identified, along with κ.
Let us pick a basis { d^(α̅)}_α̅=1,…,|α̅| for S, and define the M × |α̅| matrix D by horizontally stacking these basis vectors as columns. The condition q ∈ ( S)^⊥ is equivalent to q^⊤ D = 0, and hence ( q, κ) satisfies the augmented linear system
q^⊤[ S D ] =
[ 0, ⋯, 0, κ, -1, 0, ⋯, 0 ],
where [ S D ] is the M × (N+ |α̅|) augmented matrix formed by horizontally stacking S and D, and the vector on the right has (N-2) zeros at the start and |α̅| zeros at the end. Observe that the rows of this augmented matrix [ S D ] are independent because if d in any vector in [ S D ] then we must have d^⊤ S = 0 and d^⊤ D = 0, which means that d is present in both S and ( S)^⊥, and that is only possible when d =0. Let u_j be the N-dimensional vector whose j-th component is 1 and the rest are zeros. Then Eq. (<ref>) can be equivalently written as the linear system
[ S^⊤ - u_N-1; D^⊤ 0 ][ q; κ ] = -[ u_N; 0 ].
The pair ( q, κ) can be obtained by solving this linear equation.
], where ( S)^⊥ denotes the orthogonal complement of S.
Observe that if we assume the mass-action kinetics for the last two reactions, then they can be written as
r_1̅ ( x;k_1̅ )= k_1̅ m_1( x_M̅ )x^ν_1̅_M and r_2̅( x;k_2̅ ) = k_2̅ m_2( x_M̅ ) x^ν_2̅_M,
where x_M̅ is the concentration vector for all the species except X, m_1( x_M̅ ) and m_2( x_M̅ ) are monomials (see the definition of mass-action kinetics (<ref>)), and ν_1̅ and ν_2̅ are the numbers reactant molecules of X in reactions e_1̅ and e_2̅. Therefore Eq. (<ref>) implies that m_1 = m_2 and so reactions e_1̅ and e_2̅ have an equal number of all the species (except the output species X) as reactants. Setting νν_2̅ - ν_1̅, Eq. (<ref>) becomes equivalent to
x̅_M = ( κk_1̅/ k_2̅)^1/ν,
which is the set-point encoding function in Ref. <cit.>.
We end this section with a couple of important definitions.
A maxRPA network is called homothetic is its associated q vector has all its nonzero components of the same sign. Otherwise, the maxRPA network is called antithetic.
Let us define a subnetwork γ̅⊂Γ consisting of the output species v_M = X and the last two reactions e_1̅ and e_2̅
γ̅= ({X}, {e_1̅, e_2̅}).
The complement of this subnetwork,
γΓ∖γ̅,
consists of all the species except X and all the reactions except e_1̅ and e_2̅. Condition (<ref>) shows that for maxRPA to occur, the kinetics of the last two reactions must be fine-tuned to match the dependence on reactants other than the output species X. The only way to have maxRPA without fine-tuning is that the last two reactions do not have any species (other than X) as reactants, which is what we shall call kinetics-independent maxRPA.
A maxRPA network is said to be kinetics-independent if the reactions rates of the two set-point determining reactions (i.e. e_1̅ and e_2̅) do not depend on any species except the output species X. In other words, only X can be a reactant for these two reactants.
Note that a maxRPA network becomes kinetics-independent, precisely in the scenario where the subnetwork γ is output-complete, because all reactions involving species in γ are in γ. In fact, Theorem <ref> will show that γ is a buffering structure, i.e. its influence index is zero, λ(γ) = 0.
§.§ An Internal Model Principle for maxRPA networks
We now discuss how an Internal Model Principle can be formulated for maxRPA networks based on the unique vector q that characterizes them. Consider a maxRPA network with stoichiometric matrix S, and a pair ( q, κ) satisfying Eq. (<ref>) with q ∈ ( S)^⊥ and κ > 0. Recall that Φ(x_M, k_1̅, k_2̅) is the ratio of rate functions r_2̅( x; k_2̅)/r_1̅( x; k_1̅) which can only depend on the output species concentration x_M as per Condition 2 of Theorem <ref>. Defining
z q · x,
we see that this is an integrator for the maxRPA networks as its time-derivative is proportional to the “error" in Eq. (<ref>),
ż = q^⊤ S r( x, k) =
[ 0 ⋯ 0 κ -1 ] r( x, k) = κ r_1̅( x; k_1̅) - r_2̅( x; k_2̅) = r_1̅( x; k_1̅) (κ - Φ(x_M, k_1̅, k_2̅)).
Since the network dynamics is stable, the presence of this integrator ensures that the dynamics is driven to a steady-state where Eq. (<ref>) holds, and hence the set-point x̅_M for the output species is only a function of k_1̅ and k_2̅.
The Internal Model for the maxRPA network consists of species that form the support of the vector
q = [ q_1 ⋯ q_M ]^⊤,
namely,
V_ IM = { v_i ∈ V | q_i ≠ 0 }.
In order to satisfy Condition 2 of Theorem <ref>, at least one of the last two reactions e_1̅ and e_2̅ must have the output species X as a reactant. Without loss of generality, we may assume that e_2̅ has X as a reactant, and we call it the output sensing reaction, while we refer to the other reaction e_1̅ as the set-point encoding reaction.
In a number of situations, the output species v_M = X does not belong to V_ IM
[For example, this would happen when there is a combination of first (N-2) reactions and conservation relations in S that only modifies the output species v_M, i.e. there exists a vector y such that
[ S̅ D ] y = u_M,
where S̅ is the matrix formed by removing the last two columns from S, D is the matrix with basis vectors of S as columns and u_M is a M-dimensional vector whose last component is 1 and the rest are zeros. Note that since q satisfies Eq. (<ref>) we would have q_M = q^⊤ u_m = q^⊤[ S̅ D ] y= 0.
](i.e. q_M = 0) and so X belongs to the rest of the network, giving rise to the IMP decomposition shown in Fig. <ref>. The actuation reactions from IM to RoN are necessary to complete the feedback loop and ensure network stability, but the form of these reactions can be arbitrary.
The characterization result for maxRPA networks (Theorem <ref>) has two conditions, in which the first one is structural (i.e. Eq. (<ref>)) while the second condition (i.e. Eq. (<ref>)) depends on the kinetics of the last two reactions.
If we consider kinetics-independent maxRPA networks (see Definition <ref>) then the second condition also becomes structural, and in this situation the two conditions become equivalent to the subnetwork γ, defined by Eq. (<ref>), being a buffering structure.
The next theorem states this equivalence and one can find its schematic illustration in Fig. <ref>.
Consider a network Γ and define its subnetwork γ by Eq. (<ref>). Then, Γ is a kinetics-independent maxRPA network if and only if the subnetwork γ is a buffering structure, i.e. it is output-complete with a zero influence index λ(γ) = 0.
The proof of this theorem will be given in Sec. <ref>.
We now briefly discuss how this topological characterization of the kinetics-independent maxRPA property is connected to the more general results in the next section. As per the terminology introduced in Sec. <ref>, we can view maxRPA as a generic RPA property (𝒱, ℰ, p) where 𝒱 = {X}, ℰ = {e_1̅, e_2̅} and p is the set of all system parameters except k_1̅ and k_2̅ which appear in the kinetics of reactions in ℰ. Note that γ= Γ∖ (𝒱, ℰ) is the subnetwork consisting of all the species except X, and all the reactions except e_1̅ and e_2̅. It will be shown later that kinetics-independence and network stability, together imply that at the steady-state, the concentrations and rates of all the species and reactions in γ are sensitive to at least one of the parameters in p. This ensures that if for each parameter p_μ in p, if γ_μ is the subnetwork of all the species and reactions that are sensitive to p_μ, then γ is in fact the union of all such γ_μ-s. This is made mathematically precise in the next section, where each γ_μ is identified as a labelled buffering structure. As buffering structures are closed under the union operation, it follows that γ is itself a buffering structure, as asserted by Theorem <ref>, and it shows that the maxRPA property is generated by all the labelled buffering structures corresponding to parameters in p. The central goal of this paper, is to demonstrate that these arguments can be extended to any generic kinetics-independent RPA property (not just maxRPA) and hence the identification of all labelled buffering structures is sufficient to completely characterise and systematically identify all such generic RPA properties.
§ CHARACTERIZATION OF ALL RPA PROPERTIES
The law of localization discussed in Sec. <ref> gives us a sufficient condition for a generic RPA, and the condition is topological, meaning that it is determined by network topology.
A natural question arises as to whether any RPA property can be captured with topological criteria.
In this section, we show that
all the elementary RPA properties that are kinetics-independent
can be indeed characterized/found by labeled buffering structures,
which are buffering structures annotated with supplementary information.
Since generic RPA properties can be constructed from elementary ones, we thus have a complete characterization of all RPA properties.
We first introduce the transitivity of influence
in Sec. <ref>, that will be used later
when we show the one-to-one correspondence of the RPA properties
and labeled buffering structures in Sec. <ref>.
We then present a computational method to identify labeled buffering structures in Sec. <ref> and discuss examples in Sec. <ref>.
§.§ Transitivity of influence
Here we introduce the transitivity of influence.
The transitivity of influence for reaction perturbations is established first for the monomolecular case <cit.> and then for the multimolecular case <cit.> (for the case S = 0).
Let us prepare notations.
For reactions e_A, e_B ∈ E, if _A r̅_B ≠ 0 holds,
we say “e_A influences e_B” and
represent this in symbols as
e_A e_B .
On the other hand, if _A r̅_B = 0,
we say that “e_A does not influence e_B” and
we write this as e_A e_B.
Similarly, if _A x̅_i ≠ 0
for some v_i ∈ V and e_A ∈ E,
we express this as
e_A v_i.
When v_i ∈ V is a reactant of reaction e_A (i.e. s_iA≠ 0),
we write
v_i ⊢ e_A .
We may also express this with the corresponding indices
as i ⊢ A.
While the absence of conserved quantities is assumed in
Ref. <cit.>,
we here allow them and thus consider the perturbation of conserved quantities as well as reaction parameters.
When the perturbation of ℓ_α̅ affects
the flux of e_A (i.e., _α̅r̅_A≠ 0),
we express this as
ℓ_α̅ e_A.
The transitivity of influence
is stated as follows <cit.>:
If reaction e_A influences e_B and
e_B influences e_C, then e_A influences e_C.
In symbols,
e_A e_B e_C
e_A e_C.
The proof we describe here is slightly different from the one
given in Ref. <cit.>.
While S= 0 is assumed in Ref. <cit.>,
the following proof is valid for S ≠ 0.
We note that, to prove Eq. (<ref>),
it suffices to show that,
if reaction e_A influences v_i
and v_i is a reactant of e_B and
e_B influences e_C,
then e_A influences e_C.
In symbols,
e_A v_i ⊢ e_B e_C
e_A e_C.
Let us first show that Eq. (<ref>) implies
Eq. (<ref>).
When e_A = e_B or e_B = e_C is true,
Eq. (<ref>) is always satisfied,
so it is sufficient to consider the case where
e_A = e_C ≠ e_B or
e_A, e_B, e_C are all different.
Since e_A e_B,
0 ≠r̅_B, A
=
∑_i ⊢ B r_B,i x̅_i,A,
where the summation is over the species that are reactants of e_B.
For this to be nonzero, there must be a species v_i ⊢ e_B
such that e_A v_i.
By Eq. (<ref>),
this implies e_A e_C, and Eq. (<ref>) holds.
Now let us prove Eq. (<ref>).
We here use the formula for
the second-order response of steady-state fluxes[
Note that this formula is correct when
e_A, e_B, e_C are all different or
either of e_A ≠ e_B is equal to e_C.
],
which can be derived straightforwardly from Eq. (<ref>)
(see Appendix <ref> for derivation),
r̅_C,AB
=
r̅_C,A F_A,B
+
r̅_C,B F_B,A
+
∑_D∑_j ⊢ D∑_k ⊢ Dr̅_C,D r_D,jk/r_D,D x̅_j,A x̅_k,B ,
where
r̅_C,AB^2/ k^A k^Br̅_C (x̅( k, ℓ); k_C),
r_D,jk
(^2/ x^j x^k r_D ( x ; k_D) )_ x = x̅( k, ℓ),
r_A,A
(/ k_A r_A( x ; k_A) )_ x = x̅( k, ℓ),
F_Aln r_A,A
and
F_A,B/ k_B[
(ln r_A,A)_ x = x̅( k, ℓ)].
From the second term, we have the following
contribution,
r̅_C,AB
=
r̅_C,B∑_i ⊢ B
(ln r_B,B )_,i x̅_i,A
+
⋯ .
Thus, when we shift the value of parameter
as k_B ↦ k_B + δ k_B,
r̅_C,A as a function of k_B,
r̅_C,A (k_B), is modified as
(we here only denote k_B dependence)
r̅_C,A (k_B + δ k_B)
=
r̅_C,A (k_B)
+
r̅_C,B∑_j ⊢ B
(ln r_B,B)_,j x̅_j,Aδ k_B
+ ⋯ .
From the assumption, we have
r̅_C,B≠ 0
and
x̅_i,A≠ 0.
Thus, an infinitesimal shift of k_B induces nontrivial changes in the value of r̅_C,A.
This indicates that, without fine-tuning,
the value of r̅_C,A is nonzero.
This proves Eq. (<ref>).
§.§ RPA and labeled buffering structures
Suppose we would like to find all RPA properties for a given reaction system.
For each parameter of the system, we can hypothetically
perform its perturbation and observe the response.
In general, some of the steady-state concentrations and reaction rates
show nonzero response while others do not respond, and the latter exhibit perfect adaptation.
To test the robustness of adaptation,
we can change the system parameters (or the form of rate functions)
and repeat the hypothetical experiment.
Doing this procedure for all the parameters,
we can obtain all elementary RPA properties of the system (see Fig. <ref>)[Such a perturbation-based approach for systematically studying the steady-state response is similar to the one proposed in Ref. <cit.> for computing the influence matrix and also to the method described in Ref. <cit.> for deriving the influence graph.
A crucial difference is that we connect RPA properties to topological characteristics of subnetworks (via the one-to-one correspondence shown in this section), that facilitates the identification of integral feedback control for each RPA property.
].
Here, we show that an elementary RPA property with respect to a parameter
can be represented by a buffering structure with supplementary information,
by proving the following statement:
For a chosen reaction e_B ∈ E,
we collect the species and reactions influenced by e_B,
e_B ( V_B, E_B ) .
The subnetwork γ_0 ( V_B, E_B ) can be
made output-complete by adding reactions ℰ_B ⊂ E ∖ E_B that have reactants in V_B.
The resulting output-complete subnetwork
γ ( V_B, E_B ∪ℰ_B )
is a buffering structure, i.e., λ(γ)=0.
We note that a similar statement holds for the perturbation of conserved quantities.
Namely, if we find species and reactions that are affected by the perturbation of a conserved quantity as
ℓ_α̅ ( V_α̅, E_α̅),
then the corresponding output-complete subnetwork
γ ( V_α̅, E_α̅∪ℰ_α̅)
is a buffering structure.
In the following, we present the proof of the theorem for the case of perturbations of reaction parameters. The proof for conserved-quantity perturbations is completely analogous.
We first show that,
although γ_0 is identified as the subnetwork within which
the perturbation of the parameter of e_B is confined,
the effect of the perturbation of
any reaction e_C^⋆ in E_B
is in fact confined inside γ_0.
Namely, for any e_C^⋆∈ E_B, we have
e_C^⋆ { V ∖ V_B, E ∖ E_B }.
Suppose that this is not the case,
then there exists reaction e_D'∈ E ∖ E_B
which is influenced by a reaction inside E_B.
Then, because of the transitivity,
we should have e_B e_D'∉E_B.
This contradicts with the fact that
E_B includes all the reactions influenced by e_B.
Let us denote the set of reactions
in E ∖ E_B
that have a species of V_B as a reactant by ℰ_B.
Namely,
ℰ_B =
{
e_C ∈ E ∖ E_B
| ∃ v_i^⋆∈ V_B such that v_i^⋆⊢ e_C
} .
The subnetwork γ = (V_B, E_B ∪ℰ_B) is
obviously output-complete.
We show that the influence of the perturbations
of the reaction in ℰ_B is
also localized in γ.
If e_C ∈ℰ_B is equal to e_B,
this is trivially true, so we consider the case e_C ≠ e_B in the following.
Since x̅_i',B with v_i'∈ V ∖ V_B,
should vanish regardless of parameters,
the derivative of x̅_i',B with respect to the parameter
of e_C ∈ℰ_B should vanish as well,
0
= x̅_i',BC
=
x̅_i',C F_C,B + x̅_i',B F_B,C
+
∑_D∑_j ⊢ D∑_k ⊢ D
r_D,jk/r_D,D x̅_i',D x̅_j,B x̅_k,C .
The second term on the RHS is zero because x̅_i',B=0.
As for the third term,
we can divide the summation over all the reactions to
those inside/outside E_γ E_B ∪ℰ_B as
∑_D ⋯ = ∑_D^⋆⋯ + ∑_D'⋯,
and noting that x̅_i',D^⋆ = 0 and x̅_j',B=0,
the third term is now written as
(third term) =
∑_D'∑_j^⋆ ⊢ D'∑_k ⊢ D'r_D',j^⋆ k/r_D',D' x̅_i',D' x̅_j^⋆,B x̅_k,C .
Since there is no reaction outside γ
whose reactant is inside γ by construction,
the summation over j^⋆ is empty and
this term vanishes as well.
Thus, we have
0
=
x̅_i', C∑_i^⋆⊢ C
(ln r_C,C )_,i^⋆x̅_i^⋆, B.
There is at least one
species v_i^⋆
that is a reactant of e_C, by construction,
and (ln r_C,C)_,i^⋆≠ 0.
Since x̅_i^⋆, B≠ 0 by definition,
if the RHS of Eq. (<ref>) is to vanish
without fine-tuning, we should have x̅_i',C = 0.
Thus, there are no concentrations outside γ
that are influenced by e_C ∈ℰ_B.
As a result, no reactions outside γ are
influenced by e_C ∈ℰ_B.
Therefore, we have shown that the influence by any reaction e_C^⋆ inside γ is localized inside γ.
Let us here choose a basis of S in the following manner.
Given a subnetwork γ, we pick a basis
{ c^(α^⋆)}_α^⋆=1,…,|α^⋆|
of ( S)_ supp γ, and
we arrange the basis of S so that r̅_A is written as
r̅_A =
∑_α^⋆μ_α^⋆ c^(α^⋆)_A
+
∑_α'μ_α' c^(α')_A ,
where { c^(α')}_α'=1,…,|α'| are basis vectors
with nonzero support in E ∖ E_γ.
Note that c^(α^⋆)_A'=0, since it is supported in γ.
We employ the expansion the steady-state fluxes
in the form (<ref>).
From the response localization,
for e_C^⋆∈ E_γ
and
e_A'∈ E ∖ E_γ,
we have
0 = r̅_A', C^⋆
=
∑_α'μ_α',C^⋆ c^(α')_A' ,
where we used c^(α^⋆)_A'=0.
Let us denote a basis vector as
c^(α) =
[ c_1^(α); c_2^(α) ],
where
c_1^(α) and c_2^(α) are components
inside and outside of γ, respectively.
With the current choice of basis, the vectors
{ c^(α')_2 }_α'=1,…,|α'|
are linearly independent[
Suppose that the vectors { c^(α')_2 }_α'=1,…,|α'| are linearly dependent.
Then, by taking a linear combination of { c^α'}_α'=1,…,|α'|,
we can make a vector supported inside γ,
c' = [ c'_1; 0 ],
which is an element of ( S)_ supp γ.
Since the vector { c^(α)}_α=1,…,|α|
are linearly independent,
c'_1 should be linearly independent of
{ c_1^(α^⋆)}_α^⋆=1,…,|α^⋆|.
This contradicts the fact that
{ c^(α^⋆)}_α^⋆=1,…,|α^⋆|
is a basis vector of ( S)_ supp γ.
], and
Eq. (<ref>) implies that
μ_α', C^⋆ = 0.
The response of the steady-state concentrations and reaction fluxes to the perturbation of parameter k_C^⋆ with
e_C^⋆∈ E_γ
can be characterized by the following equation
using the A-matrix,
The response is determined by the following equation,
[
[ A_11 A_12; A_21 A_22 ]]
[
[ y_1,C^⋆; y_2,C^⋆ ]]
=
-
[
[ _C^⋆ r_1; 0 ]],
where we have defined
y_1,C^⋆_C^⋆[ x̅_i^⋆; μ_α^⋆ ]
,
y_2,C^⋆_C^⋆[ x̅_i'; μ_α' ].
We have y_2, C^⋆ = 0
for any e_C^⋆∈ E_γ
from the response localization.
Suppose that A_11 is an n × m matrix.
Since A_21 is a zero matrix,
the invertibility of A requires that
A_11 should be square or vertically long,
and we have n ≥ m.
On the other hand,
from the localization of responses,
its inverse A^-1 should have the structure
shown in Fig. <ref>,
with zero matrix in the lower-left sector,
and the upper-left part is an m × n matrix.
So that A^-1 be invertible,
( A^-1)_11 should be square or vertically long,
and we need m ≥ n.
Thus, the only possibility is n=m,
and A_11 is a square matrix.
Recall that
the set of vectors
{ c^(α^⋆)}_α^⋆=1,…,|α^⋆|
is a basis of ( S)_ supp γ.
The fact that A_11 is a square matrix
implies
|E_γ| + |P^0_γ ( S)|
- |V_γ| - |( S)_ supp γ|
=0,
meaning that λ(γ)=0.
Thus, we have the claim.
Motivated by Theorem <ref>,
we introduce labeled buffering structures
to characterize the elementary RPA properties in a reaction network.
Specifically, for each reaction e_A, we identify the
corresponding buffering structure
as in Theorem <ref>,
γ_A
=
(V_A, E_A ∪ℰ_A) .
The concentrations of V∖ V_A and
reaction fluxes of E ∖ E_A exhibit
RPA with respect to the perturbation of e_A.
We call Eq. (<ref>) as a labeled buffering structure.
The name comes from the fact that,
if we regard a labeled buffering structure
(V_A, E_A ∪ℰ_A) as a pair of sets and forget about the labels of parameters (and distinction of E_A and ℰ_A),
it reduces to an ordinary buffering structure.
Similarly, we also define a labeled buffering structure
associated with the perturbation of conserved quantity ℓ_α̅
γ_α̅
=
(V_α̅, E_α̅∪ℰ_α̅) .
By enumerating labeled buffering structures for all the parameters, we can identify all the RPA properties in a given reaction network,
via the one-to-one correspondence,
{ elementary RPA properties }⟷{ labeled buffering structures }.
There is a possibility that the RPA properties with respect to two or more reactions are exactly the same.
We use multiple indices in this case, such as γ_A,B,C
when the RPA properties with respect to reaction e_A,e_B, and e_B are the same.
As we stated in Sec. <ref>, we allow the kinetics to be generic,
and RPA properties that can be detected by labeled buffering structures are kinetics-independent ones.
For a specific choice of kinetics, such as mass-action kinetics, there can be further RPA properties that are robust only within the chosen kinetics.
The RPA properties represented by labeled buffering structures are also robust under the change of kinetics (as long as stability is not jeopardized).
We note that a generic RPA property can be generated by elementary RPA properties (see Fig. <ref>).
Let us consider, a subset of parameters p ⊂ ( k, ℓ).
Correspondingly to each element p_μ∈ p,
we have a labeled buffering structure
γ_μ = (V_μ, E_μ∪ℰ_μ).
Let us define
V_ p⋃_μ V_μ
and
E_ p⋃_μ E_μ.
Then, the subnetwork γ_ p defined by
γ_ p( V_ p, E_ p∪ℰ_ p)
is a buffering structure, where
ℰ_ p is the minimal set of reactions to make
( V_ p, E_ p∪ℰ_ p) output-complete.
Since γ_ p is the union of γ_μ with p_μ∈ p (as a pair of sets), the vanishing of the index λ(γ_ p) follows from the closure property of buffering structures under union <cit.>.
This buffering structure γ_ p is a minimal one with respect to p in the sense that for any species in V_ p and reactions in E_ p there exists a parameter p_μ∈ p to which it is sensitive to.
Then, if we define
𝒱 V ∖ V_ p ℰ E ∖ E_ p
then the triple (𝒱, ℰ, p) exhibits a generic RPA property, i.e. concentrations of the species in 𝒱
and reaction rates of the reactions in ℰ exhibit RPA with respect to any parameter in p.
The fact that all the generic RPA properties can be captured
by this construction
follows via the same logic of the proof for elementary RPA properties.
Let us summarize the argument of this paragraph in the following proposition:
For any generic RPA property (𝒱, ℰ, p),
there exists an associated buffering structure.
Once all the labeled buffering structures for a given reaction network are identified, we have the full characterization of dependencies of all the steady-state concentrations and reaction fluxes on system parameters, which can be used for various analyses.[
As we discuss in Sec. <ref>,
the enumeration of all the labeled buffering structures can be done in polynomial time.
]
Thus, labeled buffering structures can be used to find species and reactions with more complicated patterns of dependencies.
For example, if we are interested in
concentrations and reaction rates that are affected
by e_A
and not affected by e_B,
such species and reactions are identified as
(V_A ∩ (V∖ V_B) , E_A ∩ (E ∖ E_B)).
As another example, suppose that we have several target species
of interest V_ out⊂ V.
If we would like to identify the parameters which affect the concentrations of the species in V_ out,
this can be achieved by finding the largest buffering structure that does not include species in V_ out.
All the buffering structures in a reaction network can be
generated from the identified labeled buffering structures,
and for buffering structures, we can perform bifurcation analysis <cit.>,
or we can reduce the network to a smaller one without affecting
steady-state properties by eliminating buffering structures <cit.>.
Although we do not consider addition of new reactions or new species as a part of perturbations, we can study its effect on the dependencies of concentrations and reaction rates on system parameters, by comparing
the sets of labeled buffering structures before and after the addition of new reactions and species.
In this way, identified labeled buffering structures form a basis for various analyses. We shall refer to the identification of RPA properties through buffering structures as topological analysis, emphasizing the fact that these subnetworks are identified through topological characteristics. This name is also motivated from a more abstract viewpoint: robust adaptation is a topological phenomenon, as we discuss in Sec. <ref>.
§.§ Algorithm to enumerate labeled buffering structures
Let us present an algorithm to identify all
the labeled buffering structures for a given deterministic reaction system.
As we discussed so far, the qualitative responses (i.e., whether the response is zero or nonzero) of a reaction system against parameter perturbations
are captured by the inverse of the A-matrix.
The response of concentrations is given by
x̅_i,A∝ ( A^-1)_i A.
As for reaction rates,
noting that r̅_B = ∑_αμ_α c^(α)_B,
the response of a reaction rate can be written as
_A r̅_B =
∑_α,C c^(α)_B
( A^-1)_α C_A r_C .
Since _A r_C ∝δ_AC, we have
_A r̅_B
∝∑_α c^(α)_B
( A^-1)_α A .
We define the species-sensitivity matrix and
reaction-sensitivity matrix as
X_iA ( A^-1)_i A ,
R_BA∑_α c^(α)_B
( A^-1)_α A.
The numerical procedure goes as follows.
For a given chemical reaction network,
we first construct the A-matrix.
In doing to, we assign random numbers for
the entries with nonzero r_A,i.
Namely, we put a random number if v_i ⊢ e_A and
zero otherwise.
Then, numerically compute its inverse, A^-1,
with which we can obtain the species-sensitivity and reaction-sensitivity matrices (<ref>).
Based on these matrices,
for each reaction e_A (and conserved quantity ℓ_α̅),
we identify chemical species and reactions
that are affected by the perturbation of e_A (ℓ_α̅).
Namely, we identify (V_A, E_A) for each reaction e_A
that are defined by
V_A {
v_i ∈ V
| X_iA is nonzero},
E_A {
e_B ∈ E
| R_BA is nonzero}.
The subnetwork (V_A,E_A) is not necessarily output-complete.
It can be made output-complete by adding a set of reactions
ℰ_A
defined in Eq. (<ref>).
Then, we obtain the labeled buffering structure
associated with reaction e_A,
γ_A = (V_A, E_A ∪ℰ_A).
By repeating this for all the reactions,
we obtain all the labeled buffering structures within a given reaction system.
The bottleneck of this enumeration process is the inversion of matrix A, whose computational complexity is O(𝒩^3)
in the case of the Gaussian elimination, where 𝒩 is the dimension of A, which is roughly the same as the number of reactions.
According to the algorithm explained above, we have implement a method to enumerate all the labeled buffering structures for a given reaction network in RPAFinder, which is available on Github <cit.>.
§.§ Examples
Let us here discuss some examples of chemical reaction systems
to illustrate the use of topological analysis
based on the identification of labeled buffering structures.
§.§.§ A simple example
We consider a reaction network
Γ ({v_1,v_2,v_3 }, {e_1,e_2,e_3,e_4,e_5 })
whose reactions are given by
e_1 : ∅→ v_1,
e_2 : v_1 → v_2,
e_3 : v_2 → v_3,
e_4 : v_3 → v_1,
e_5 : v_2 →∅.
The network can be visualized as
[species] (v1) at (1.25,0) v_1;
[species] (v2) at (3.75,0) v_2;
[species] (v3) at (2.5,1.5) v_3;
(d1) at (-0.5,0) ;
(d2) at (5.5,0) ;
[-latex,draw,line width=0.5mm] (d1) edge node[below]e_1 (v1);
[-latex,draw,line width=0.5mm] (v1) edge node[below]e_2 (v2);
[-latex,draw,line width=0.5mm] (v2) edge node[above right]e_3 (v3);
[-latex,draw,line width=0.5mm] (v3) edge node[above left]e_4 (v1);
[-latex,draw,line width=0.5mm] (v2) edge node[below]e_5 (d2);
Corresponding to the perturbation of the parameter of each reaction,
we can identify the following labeled buffering structures,
γ_1 = ({v_1,v_2,v_3 }, { e_1,e_2, e_3,e_4,e_5}∪{}) = Γ,
γ^∗_2 = ({v_1}, {}∪{e_2 }),
γ_3 = ({v_1,v_3}, {e_2,e_3,e_4 }∪{} ),
γ^∗_4 = ({v_3 }, {}∪{e_4 }),
γ_5 = ({v_1,v_2,v_3 }, {e_2, e_3,e_4 }∪{ e_5 }) ,
where those with ∗ are strong buffering structures <cit.>.
Here, γ_1 and γ_3 are already output-complete without adding any additional reactions (i.e. ℰ_1 and ℰ_3 are empty).
For γ^∗_2, γ^∗_4, and γ_5,
we supplemented them with reactions so that they are output-complete.
The parameter of e_1 affects all the species and reactions,
and the corresponding labeled buffering structure is the entire network.
In the present example, all parameters
give rise to different labeled buffering structures.
From the identified labeled buffering structures, we can conclude the following:
* From γ_3, we learn that x̅_2, r̅_1, and r̅_5 exhibit RPA with respect to e_3.
* The strong buffering structure γ_2^∗∪γ_4^∗ tells us that all the reaction rates r̅ exhibit RPA with respect to e_2 and e_4.
* x̅_2, x̅_3 exhibit RPA with respect to e_2 (from γ_2^∗).
* x̅_1, x̅_2 exhibit RPA with respect to e_4 (from γ_4^∗).
* No species or reactions exhibit RPA with respect to e_1.
If we use mass-action kinetics,
the steady-state concentrations and rates are given by (k_A is the rate constant for e_A)
x̅_1 = k_1/k_3 k_5 (k_3+k_5),
x̅_2 = k_1/k_5,
x̅_3 = k_1 k_3/k_4 k_5,
r̅ = k_1 c^(1) + k_1 k_3/k_5 c^(2) ,
where
c^(1)[ 1 1 0 0 1 ]^⊤
and
c^(2)[ 0 1 1 1 0 ]^⊤.
These steady-state solutions are indeed compatible with
the predictions based on the labeled buffering structures.
§.§.§ Bacterial chemotaxis
We discuss an example describing bacterial chemotaxis <cit.>.
Bacterial chemotaxis is the ability of bacteria to move towards or away from a chemical stimulus in their environment.
E. coli are able to detect a variety of attractants and use a biased random walk to navigate towards them <cit.>.
Their motion is comprised of runs, during which the bacterium moves in a straight line, and tumbles, during which it stops and changes direction.
When a bacterium senses the increase of attractants
along its path, it reduces the frequency of tumbles, allowing it to move up gradients of attractants.
When an attractant is added uniformly in space,
the frequency of tumbles initially decreases
but eventually returns to its original level.
This adaptation is perfect, and the tumbling frequency at the steady state is independent of the attractant concentration.
We here use a simplified model <cit.> that captures this adaptation behavior.
Each receptor is bound to a protein kinase CheA, and we will denote the complex by X.
When a complex of a methylated receptor and a kinase is in an active state, X^⋆_m,
it phosphorylates a response regulator, which binds to the flagellar motor and changes the rotational direction, enhancing tumbling.
Thus, X^⋆_m determines the tumbling probability and it is the target output variable of the analysis here.
The set of reaction we consider is
e_1 : X_m^⋆→ X_m
e_2 : X_m → X_m^⋆,
e_3 : X → X_m ,
e_4 : X → X_m^⋆ ,
e_5 : X_m^⋆→ X .
Here, e_3 and e_4 are methylation reactions catalyzed by CheR. The demethylation is mediated by CheB.
When a ligand is bound to a receptor,
the probability of deactivation process e_1 increases,
which means that the parameter k_1 is an increasing function of the ligand concentration a, k_1 = k_1 (a).
Perfect adaptation is realized when
the following two conditions are met <cit.>:
* The demethylation by CheB only acts on active receptors, X_m^∗.
* The rate of methylation by CheR
does not depend on the substrate concentration, X.
Under these assumptions, the set of reactions above is
equivalently described by the following reaction network[
Here, we use ∅ to indicate the null species, whose abundance is assumed to be constant and its dynamics is not considered.
],
e_1 : X_m^⋆→ X_m
e_2 : X_m → X_m^⋆,
e'_3 :∅→ X_m ,
e'_4 :∅→ X_m^⋆ ,
e'_5 : X_m^⋆→∅ .
The time evolution of the concentrations of X_m
and X_m^⋆ is governed by[
Here, we use the same symbol to denote
the species and its concentration
to simplify notations.
]
Ẋ_m = - k_2 X_m + k_1 X^⋆_m + k_3 ,
Ẋ^⋆_m = k_2 X_m
- k_1 X^⋆_m - k_5 X_m^⋆ + k_4 .
Note that each rate constant
is proportional to the corresponding enzyme concentration,
k_5 ∝ B, and k_3, k_4 ∝ R,
which are treated as constants here.
At the steady state, the concentrations and reactions rates are given by
X̅_m^⋆ = k_3 + k_4/k_5,
X̅_m =
1/k_2(
k_3 + k_1 (k_3 + k_4)/k_5)
,
r̅ =
k_1 (k_3 + k_4)/k_5 c^(1)
+ k_3 c^(2)
+ k_4 c^(3),
where we have taken the basis of S as
c^(1)[ 1 1 0 0 0 ]^⊤,
c^(2)[ 0 1 1 0 1 ]^⊤,
and
c^(3)[ 0 0 0 1 1 ]^⊤.
Notably, X̅^⋆_m does not depend on k_1 (a), which means that
it is insensitive to the ligand concentration a.
This indicates that X^⋆_m exhibits RPA with respect to the ligand concentration.
As pointed out in Ref. <cit.>, RPA in this system is realized through an integral feedback control.
The corresponding integrator is identified as
d/dt( X^⋆_m + X_m )
= - k_5 X_m^⋆ + k_3 + k_4
= - k_5 δ X_m^⋆ ,
where δ X_m^⋆ X_m^⋆ - X̅_m^⋆ is the
deviation of X_m^⋆ from its steady-state value.
This equation drives the value of X_m^⋆ to X̅^⋆_m
regardless of the value of k_1 and k_2.
Let us try to understand this phenomenon using the topological analysis.
The labeled buffering structures in the reaction network (<ref>)
are identified as
γ_1 = ({ X_m }, { e_1,e_2 }∪{}),
γ^∗_2 =
({ X_m }, {}∪{ e_2 }),
γ_3 =
({ X_m, X^⋆_m }, { e_1,e_2, e_3,e_5 }∪{} ),
γ_4 =
({ X_m, X^⋆_m }, { e_1,e_2, e_4, e_5 }∪{} ),
γ_5 =
({ X_m, X^⋆_m }, { e_1,e_2 }∪{ e_5 }).
We can conclude the following from the identified buffering structures:
* The independence of X̅_m^⋆ on k_1 and k_2 is understood from
the buffering structure γ_1.
Since X^⋆_m is outside γ_1, the parameters k_1, k_2 do not affect the steady-state value of X^⋆_m.
* The independence of all the reaction fluxes on k_2
is understood from the strong buffering structure γ_2^∗.
The following two points are also consistent with the identified buffering structure,
although these are trivial (since the rates of e_3 and e_4 are given externally):
* The rate r̅_4 depends only on k_4, which can be explained by γ_3 (and partially by γ_5).
* The rate r̅_3 depends only on k_3, which can be explained by γ_4 (and partially by γ_5).
§.§.§ Sniffer system
We note that, the RPA properties found through the identification of buffering structures are kinetics-independent ones.
For example,
the perfect adaptation realized through
the incoherent feedforward (IFF) structure <cit.>
cannot be detected through buffering structures,
since the adaptation here depends on the choice of kinetics
and will disappear once the kinetics deviate from it.
In this sense, the perfect adaptation by IFF is not so robust.
As a example, let us consider the sniffer system <cit.>,
e_1 : v_3 → v_1 + v_3 ,
e_2 : v_1 + v_2 → v_2,
e_3 : v_3 → v_2 + v_3 ,
e_4 : v_2 →∅ ,
e_5 : ∅→ v_3 ,
e_6 : v_3 →∅ .
The rate equations in mass action kinetics are given by
ẋ_1 = k_1 x_3 - k_2 x_1 x_2 ,
ẋ_2 = k_3 x_3 - k_4 x_2 ,
ẋ_3 = k_5 - k_6 x_3 .
The effect of v_3 on v_1 enters in two ways:
directly via e_1 and indirectly via e_3 and e_2.
Adaptation occurs due to the cancellation of the two effects.
The steady-state concentrations and reaction rates are
x̅_1 = k_1 k_4/k_2 k_3,
x̅_2 = k_3 k_5/k_4 k_6,
x̅_3 = k_5/k_6,
r̅
=
k_1 k_5/k_6 c^(1)
+
k_3 k_5/k_6 c^(2)
+
k_5 c^(3) .
where we took a basis of S as
c^(1)[ 1 1 0 0 0 0 ]^⊤,
c^(2)[ 0 0 1 1 0 0 ]^⊤,
and
c^(3)[ 0 0 0 0 1 1 ]^⊤.
The steady-state concentration x̅_1 is independent of k_5 and k_6,
which determine the value of x̅_3.
However, this independence is due to the fact that
x_3 enters linearly in Eqs. (<ref>) and (<ref>).
Once the kinetics of r_1 or r_2 deviates from the linear form,
x̅_1 becomes dependent on k_5 and k_6.
In this sense, this adaptation is a kinetics-dependent feature
and hence is not quite robust.
The labeled buffering structures are identified as
γ_1 = ({v_1 },{e_1,e_2 }∪{}),
γ^∗_2 = ({v_1 }, {}∪{e_2 } ),
γ_3 = ({v_1,v_2 },{e_3,e_4 }∪{ e_2 } ),
γ^∗_4 = ({v_1,v_2 },{}∪{e_2,e_4 } ),
γ_5 = ({v_1,v_2,v_3 },{e_1,e_2,e_3,e_4,e_5, e_6}∪{} ) ,
γ_6 = ({v_1,v_2,v_3 },{e_1,e_2,e_3,e_4}∪{ e_6} ) .
Indeed, all the buffering structures include v_1,
and hence x̅_1 can depend on every parameter for a generic choice of kinetics.
We can infer the following RPA properties from the labeled buffering structures:
* x̅_3 ⊥ k_2, k_3, k_4 is explained by γ_3, where ⊥ means “is independent of.”
* x̅_2, x̅_3 ⊥ k_1, k_2 is explained by γ_1.
* r̅_1, r̅_5, r̅_6 ⊥ k_2, k_3, k_4 is explained by γ_3.
* r̅_5, r̅_6 ⊥ k_1,k_2,k_3,k_4 is explained by γ_1 ∪γ_3.
* r̅⊥ k_2, k_4, because of the strong buffering structure γ_4^∗.
The RPA properties above deduced from the buffering structures are kinetics-independent,
unlike the property x̅_1 ⊥ k_5, k_6, which is only true for mass-action kinetics and thus is not captured by buffering structures.
§.§.§ Ethanol production by yeasts
As a realistic example, let us discuss overexpression experiments
of metabolic enzymes of the yeast Saccharomyces cerevisiae,
which has been extensively used
in metabolic engineering <cit.>.
The quantity of interest here is the ethanol production.
Overexpression experiments
of glycolytic enzymes were performed
for the purpose of increasing the flux to ethanol, but they did not lead to a significant improvement.
We revisit the results of these experiments
from a topological perspective.
We employed the pathways shown in
Fig. <ref>,
which consists of glycolysis and the pentose phosphate pathway (PPP) (see Sec. <ref> for the list of reactions).
We identified all the labeled buffering structures in this network, which we list in Appendix <ref>.
Based on this,
we selected the reactions that are included
in buffering structures which do not include the ethanol production reaction (27 in the list
in Appendix <ref>).
Namely, the perturbations of the parameters
of the blue reactions do not affect ethanol production.
For example,
the overexpression of hexokinase (HK)
by 13.9 fold,
which catalyzes the phosphorylation
Glucose → G6P,
has led to the improvement of only 7 %
compared to the wild type (see Table <ref>).
Similarly, the overexpressions of
phosphoglycerate kinase (PGK),
phosphoglycerate mutase (PGM),
or
pyruvate kinase (PYK)
did not result in a significant increase in ethanol production.
Although the amount of enzymes are increased severalfold,
the ethanol production shows strong resistance to these changes.
These results are consistent with the expectation from buffering structures,
since the reactions catalyzed
by HK, PGK, PGM, PYK are
inside buffering structures which do not include the flux to ethanol.
Namely, the strong resilience of
S. cerevisiae against these overexpressions can be understood as a result of integral feedback control associated with buffering structures, meaning that it is a consequence of the network topology.
On the other hand,
the black reactions can in general affect ethanol production.
Thus, the overexpressions of
phosphofructokinase-1 (PFK-1),
pyruvate decarboxylase (PDC),
or
alcohol dehydrogenase (ADH)
can potentially increase ethanol production
(see the lower part of Table <ref>
for the corresponding experimental results).
Although we cannot make a definite judgement
as to whether the experimental results support this,
the identification of buffering structures allows us to narrow down the candidates for overexpression of enzymes.
§ TOPOLOGICAL CHARACTERIZATION OF KINETICS-INDEPENDENT MAXRPA NETWORKS
So far we have discussed two distinct approaches (topological and control-theoretical) toward the study of RPA properties in chemical reaction networks. A natural question is how these approaches are related. In this section, we provide this connection by proving the equivalence of the two approaches in the maxRPA setting, which is going to give us an insight into the construction of integrators for a generic RPA property.
In the following,
we first introduce the the decomposition of the influence index in Sec. <ref>,
which will be used in the proof.
In Sec. <ref>, we describe examples
to highlight the relation of the two approaches and gain an insight into the strategy to prove the equivalence.
The proof will be completed in Sec. <ref>.
§.§ Decomposition of the influence index
To prove the equivalence, a crucial role is played by
a decomposition of the influence index shown in Ref. <cit.>, which we introduce in this subsection.
The decomposition plays important roles in later analyses as well when we discuss integrators for generic RPA properties.
In the following, we often choose a subnetwork,
and let us first introduce the notations associated with such a choice.
A subnetwork is specified by subsets of species and reactions,
γ = (V_γ, E_γ),
with V_γ⊂ V and E_γ⊂ E.
We refer to the chemical species
and reactions inside γ
as internal, and
those in Γ∖γ as external.
Accordingly, the stoichiometric matrix S
can be partitioned into four blocks,
S =
[ S_11 S_12; S_21 S_22 ].
where 1 and 2 correspond to
internal and external degrees of freedom, respectively.
Now let us introduce the decomposition
of the influence index <cit.>,
λ (γ)
= c (γ) + d_l (γ) - d (γ) .
The definition and meaning of each term is as follows:
* The first term is defined as
c(γ) | C(γ)|,
which is the dimension of the following space,
C(γ)
S_11 /
( S)_ supp γ .
The term c(γ)
represents the number of emergent cycles in γ.
Intuitively, c(γ) is the number of cycles in subnetwork γ,
that are not cycles in the whole network Γ.
* The second term is
defined as d_l (γ) |D_l (γ)|
with
D_l (γ) ( S) / X(γ) ,
X (γ)
{[ d_1; d_2 ]∈ S
| d_1 ∈ S_11}.
This counts the number of conserved quantities in Γ
whose projections to γ are not conserved inside γ.
We call such conserved quantities as lost conserved quantities.
* The third term
d(γ) |D(γ)|
is the dimension of the space D(γ),
which is defined as
D(γ)
S_11 / D_11(γ) ,
D_11(γ)
{ d_1 ∈ S_11 | ∃ d_2 such that [ d_1; d_2 ]∈ S
} .
This counts the number of conserved quantities in γ
that cannot be extended to conserved quantities in Γ.
We call such them as
emergent conserved quantities in γ.
Note that
the integers c(γ),d_l(γ), and d(γ) are all nonnegative by definition.
We describe a linear-algebraic procedure to obtain
the bases of D(γ), C(γ)
in Appendix <ref>, that will be used later to construct integrator equations for a generic RPA property and also used in the discussion of manifold RPA.
RPAFinder <cit.> implements methods to obtain bases of these spaces.
For later purposes, let us also introduce the decomposition of S (see Ref. <cit.> for derivation),
S ≃ D_11(γ) ⊕ D_l (γ) ⊕D̅'(γ) ,
where D̅'(γ) denotes the space of conserved quantities of Γ supported in Γ∖γ,
D̅' (γ)
{[ 0; d_2 ]∈ S
}
.
We find that these quantities are in fact relevant in the description of maxRPA.
In the maxRPA setting, the subnetwork γ
contains all the species and reactions other than X and {e_1̅, e_2̅},
and the M × N dimensional stoichiometric matrix S is partitioned into four blocks where S_11, S_12, S_21 and S_22
are of dimensions
(M-1) × (N-2),
(M-1) × 2,
1 × (N-2) and
1 × 2 respectively.
When q_M = 0, we can write vector q as
q =
[ q_1 0 ],
and due to Eq. (<ref>) the (M-1) dimensional vector q_1 must satisfy
q_1^⊤ S_11 = 0, q_1^⊤ S_12
=
[ κ -1 ] and q_1^⊤D̅ = 0,
where D̅ is the (M-1) × |α̅| matrix formed by the first (M-1) rows of D. This shows that q_1 ∈ S_11 and q_1 cannot be extended to a vector in S by adding a component[This is because if there is a scalar q_2 such that [ q_1; q_2 ]∈ S, then since the columns of D span S, there exists a vector y such that D y = [ q_1; q_2 ]. This shows that
q_1^2 = [ q_1^⊤ 0 ][ q_1; q_2 ] =
[ q_1^⊤ 0 ]
D y = q_1^⊤D̅ y = 0 (due to the last relation in Eq. (<ref>)) and hence q_1 = 0 which is a contradiction as the second relation in Eq. (<ref>) must hold.]. Hence, the vector q_1 is an emergent conserved quantity for the subnetwork γ.
If the maxRPA network is kinetics-independent (see Definition <ref>),
then reactions e_1̅ and e_2̅ can only have the output species v_M = X as a reactant which implies that all nonzero entries of S_12 must be positive.
However, the second relation in Eq. (<ref>) can hold only when q has both positive and negative components.
Hence, a kinetics-independent maxRPA network can never be homothetic and it must be antithetic if q_M=0.
Interestingly, the same conclusion can be drawn for stochastic maxRPA networks (see Ref. <cit.>). Note that, as the example in Sec. <ref> shows, kinetics-independent maxRPA networks can be homothetic when q_M ≠ 0.
The following proposition summarises the discussion in the previous paragraph.
Suppose Γ is a kinetics-independent maxRPA network characterized by a pair ( q, κ) that satisfies Eq. (<ref>). Suppose that the last component of q is 0, and let q_1 be the vector obtained by removing this last component from q. Then this network must be antithetic and q_1 is an emergent conserved quantity for the subnetwork γ given by Eq. (<ref>).
§.§ Examples
In this subsection, we compare the
control-theoretical and topological approaches through simple examples that exhibit maxRPA to highlight the roles of the concepts introduced in the previous subsection.
The intuition from these examples is going to be straightforwardly generalized to give the proof of the equivalence in Sec. <ref>.
§.§.§ Example: Homothetic case
Let us discuss an example which corresponds a homothetic maxRPA network.
The network consists of two species {x,y} and
three reactions,
e_1 : ∅→ y ,
e_2 : y → x,
e_3 : x →∅ .
The stoichiometric matrix reads
S =
cccc
c[ccc]y -1 1 0
x 1 0 - 1
e_2 e_1 e_3
.
If we use mass-action kinetics, the rate equations are
ẏ = k_1 - k_2 y ,
ẋ = k_2 y - k_3 x .
The steady-state concentrations are given by
x̅ = k_1/k_3,
y̅ = k_1/k_2.
This is a maxRPA network for species x:
the q vector is identified as
q =
[ 1 1 ]^⊤ .
Indeed, it satisfies the condition (<ref>),
q^⊤ S =
[ 0 1 -1 ].
Since the components of q are all positive,
this is a homothetic maxRPA network.
The integrator is given by
d/dt q · x
=
d/dt (x + y)
= k_1 - k_3 x
= -k_3 δ x
,
where δ x x - x̅.
This equation drives x toward its steady-state value.
In this example,
e_3 is the sensing reaction, and e_1 is the set-point encoding reaction.
The maxRPA property of this network corresponds to the fact that
the target value of x, x̅, is independent of k_2.
Let us examine this example with the topological analysis.
Labeled buffering structures in this system are
γ_1 = ({x,y},{e_1, e_2,e_3 }∪{} ) (=Γ ) ,
γ^∗_2 = ({y},{}∪{ e_2 } ),
γ^∗_3 = ({x},{}∪{ e_3 } ).
The subnetworks γ_2^∗ and γ_3^∗
are strong buffering structures.
The fact that x̅ does not depend on k_2
is explained by γ_2^∗, so this subnetwork is responsible for the maxRPA behavior.
The decomposition of λ (γ_2^∗) reads
λ (γ_2^∗)
= -1 + 1 - 0 + 0
=
=c(γ_2^∗) 0
+
=d_l(γ_2^∗) 0
-
=d(γ_2^∗) 0
= 0.
In the stoichiometric matrix,
the subnetwork γ_2^∗ can be highlighted as
at (0, 0) S =
cccc
c[ccc]y -1 1 0
x 1 0 - 1
e_2 e_1 e_3
.;
[mydarkred, dashed,line width=1]
(-0.4,0.2) rectangle (0.4, 0.7);
at (0,1) S_11 ;
As we discuss later generically,
in this case, the q vector is identified as
q =
[ - S_21S^+_11 1 ]^⊤,
where S_11^+ denote the Moore-Penrose inverse of S_11.
In this example,
S_11=[ - 1 ]
and
S_21=[ 1 ],
which are 1 × 1 matrices.
Here, S_11 is invertible,
and S_11^+ = S_11^-1 = [ -1 ].
Thus, we can see that
Eq. (<ref>) reproduces
Eq. (<ref>).
§.§.§ Example: Antithetic case
We consider a reaction network consisting of the following set of reactions:
e_1 : z_1 → x + z_1,
e_2 : z_1 + z_2 →∅,
e_3 : ∅→ z_1 ,
e_4 : x → x + z_2 ,
e_5 : x →∅ .
The structure of the network is shown in Fig. <ref>.
The corresponding rate equation is
ẋ = k_1 z_1 - k_5 x,
ż_1 = - k_2 z_1 z_2 + k_3 ,
ż_2 = - k_2 z_1 z_2 + k_4 x.
In this network, e_3 is a set-point encoding reaction,
and e_4 is a sensing reaction.
Reaction e_1 is an actuation from the internal model to the
the rest of the network for countering the disturbances.
We here employed mass-action kinetics for definiteness,
but the conclusion does not depend on the choice of kinetics.
The stoichiometric matrix is given by
S =
cccccc
c[ccccc]z_2 -1 0 0 1 0
z_1 -1 0 0 0 1
x 0 1 -1 0 0
e_2 e_1 e_5 e_4 e_3
,
where rows and columns are arranged
for later convenience.
The steady-state concentrations are given by
x̅ = k_3/k_4,
z̅_1 = k_3 k_5/k_1 k_4,
z̅_2 = k_1 k_4/k_2 k_5,
and the steady-state fluxes are
r̅ = k_3k_5/k_4 c^(1) + k_3 c^(2),
where we have taken the basis vectors of S as
S = span {[ 0 1 1 0 0 ]^⊤,
[ 1 0 0 1 1 ]^⊤} span{ c^(1), c^(2)} .
Here, the components are arranged in the same order as the column of
S in Eq. (<ref>).
The value of x̅ is determined only by k_3 and k_4, and it does not depend on other parameters.
Namely, this is an example of a maxRPA network.
The vector q in this example is given by
q =
[ 1 -1 0 ]^⊤ .
Indeed, q satisfies the condition for maxRPA,
q^⊤ S
=
[ 0 0 0 1 -1 ].
With this q vector, one can identify the integrator as
d/dt
(z_1 - z_2)
= k_3 - k_4 x
= - k_4 δ x,
where δ x x - x̅.
Let us analyze this example based on the topological analysis.
We can identify the labeled buffering structures of this system as
γ^∗_1 = ( { z_1,z_2 }, {}∪{ e_1,e_2 }) ,
γ^∗_2 = ( { z_2 }, {}∪{ e_2 }) ,
γ_3 = ( {x,z_1, z_2 }, { e_1,e_2,e_3,e_4,e_5 }∪{}) (=Γ),
γ_4 = ( {x,z_1, z_2 }, {e_1,e_5 }∪{ e_2,e_4 }) ,
γ_5 = ( {z_1, z_2 }, {e_1,e_5 }∪{ e_2 }) .
From these buffering structures,
we can, for example, infer the following:
* x̅⊥ k_1,k_2,k_5 is understood from γ_5.
* z̅_1 ⊥ k_2 is explained by γ^∗_2.
* All the steady-state reaction fluxes satisfy
r̅⊥ k_1, k_2, can be understood from the strong buffering structure γ^∗_1.
Let us focus on γ_5, which is responsible for
the maxRPA property of x̅.
The subnetwork γ_5
corresponds to the following part of
S,
at (0, 0) S =
cccccc
c[ccccc]z_1 0 -1 0 1 0
z_2 0 -1 0 0 1
x 1 0 -1 0 0
e_1 e_2 e_5
e_3 e_4
,;
[mydarkred, dashed,line width=1]
(-0.75,0) rectangle (1.1, 0.9);
at (-0.1, 1.25) S_11 ;
where γ_5 is indicated by a red rectangle.
In this example, the internal part of q turns out to be an emergent conserved quantity, as we show below.
The spaces S_11 and S_11 are given by
S_11 =
span{[ 1 0 1 ]^⊤,
[ 1 0 -1 ]^⊤},
S_11 =
span{[ 1 -1 ]^⊤} ,
where colored vectors are emergent ones (note that [ 1 -1 ]∈ S_11, but it cannot be extended to an element in S, which is trivial in this example).
We can see that the internal part of the q vector (<ref>) is the same as the emergent conserved quantity in Eq. (<ref>).
The network γ_5
has one emergent conserved quantity
and one emergent cycle,
and the influence index of γ_5 is decomposed as
λ (γ_5)
= -2 + 3 - 1 + 0
=
=c(γ_5) 1
+
=d_l(γ_5) 0
-
=d(γ_5) 1
= 0.
This example suggests that, when the subnetwork has an emergent conserved quantity, d(γ)>0, we can use it to construct the integrator.
We will show that this observation is true for generic cases in
the proof of Theorem <ref>.
§.§ Proof of Theorem <ref>
From the examples in the previous subsection, we have learned that
the integrators (or equivalently the q vectors satisfying Eq. (<ref>)) are constructed
in different ways, depending on whether d(γ)=0 or not.
As we see below, this observation is true for a generic chemical reaction system, and we can now describe the proof of Theorem <ref>.
As we discussed at the end of Section <ref>, a network Γ is kinetics-independent maxRPA if the last two reactions do not have a species other than the output species X as a reactant.
This restriction of the kinetics of reactions in subnetwork γ̅ is equivalent to the subnetwork γ being output-complete.
Hence, to prove Theorem <ref>, it suffices to prove that the zero influence index condition (i.e λ(γ) = 0) is equivalent to the existence of a pair ( q, κ) satisfying Eq. (<ref>).
Let us first assume that γ is a buffering structure.
We note that when λ(γ)=0 is satisfied,
d_l (γ)=0 is always true
from the assumption of the existence of a steady state,
as we show later near Eq. (<ref>).
Thus, c(γ)=d(γ) holds under the current assumption.
We consider the following two cases,
c(γ)=0
and
c(γ)≠0,
separately.
* Suppose that c (γ)=0
and γ does not have an emergent cycle.
Since λ(γ)=0
by assumption,
we also have d(γ)=0.
As we show in Appendix <ref>,
when c(γ)=d(γ)=0,
we have the following isomorphisms,
S / S_11 ≃ S',
S / S_11 ≃ S',
where
S' S_22 - S_21 S^+_11 S_12 is the generalized Schur complement.
Note that S' is 1 × 2 matrix in the current setting.
We can decompose S as in Eq. (<ref>),
and since d_l(γ)=|D_l(γ)|=0, we have
S ≃ D_11(γ) ⊕D̅'(γ).
Since d(γ)=0, we have
S_11≃ D_11(γ) and
S / S_11≃D̅' (γ).
Recall that D̅'(γ) is the space of conserved quantities of Γ whose support is in γ̅= Γ∖γ, which contains only X as species.
Since X itself does not constitute a conserved quantity[ If that is the case, its concentration is solely determined by the value of the conserved quantity, which contradicts the assumption that the steady-state concentration of X depends nontrivially on k_1̅ and k_2̅ (see the text near Eq. (<ref>)).
],
we should have S' = 0.
Thus, S' is of rank 1 and is not
a zero matrix.
In this situation,
we can pick a vector q by
q =
c
[ - S_21S^+_11 1 ]^⊤,
where c is an overall constant.
Indeed,
q^⊤ S
=
c
[ - S_21S^+_11 1 ][ S_11 S_12; S_21 S_22 ]
=
c
[ S_21(1 - S_11^+ S_11) S' ].
Note that 1 - S_11^+ S_11 is the projection matrix to S_11.
Since here we have
c(γ)=0,
which is equivalent to S_11⊂ S_21,
we can write q^⊤ S as
q^⊤ S
=
c
[ 0^⊤ S' ] .
By choosing the constant c appropriately,
we can define q satisfying Eq. (<ref>).
* Suppose that c(γ) ≠ 0.
Since
c(γ) = d(γ),
there exists an emergent conserved quantity, q^⊤ =
[ q_1^⊤ 0 ],
which satisfies
q_1^⊤ S_11 = 0 ^⊤.
By multiplying q on S from the left,
q^⊤ S
=
[ 0^⊤ q_1^⊤ S_12 ],
where q_1^⊤ S_12 is not a zero matrix.
In both cases, we can construct a pair
( q, κ) satisfying Eq. (<ref>)
(note that
we assume the existence of a steady state solution, and
q^⊤ S r = 0 should be able to determine x̅,
which necessitates that the components of q^⊤ S should have opposite signs and both should be nonzero, since
reaction fluxes are positive).
Thus, we have shown that if γ is a buffering structure then kinetics-independent maxRPA also holds.
Conversely, let us assume that conditions for kinetics-independent maxRPA are satisfied and
prove that γ is a buffering structure. Since γ is output-complete (by definition of kinetics-independent maxRPA), we just need to show that its influence index vanishes, λ(γ) = 0. We first consider the case d_l (γ) ≠ 0. As γ̅ contains only one species, d_l (γ)=1, and there exists a conserved quantity [ d_1; d_2 ]∈ S
with d_2 ≠ 0. This implies that d_1^⊤ S_11 + d_2 S_21 = 0. So if we pick any c_1 ∈ S_11, then we have d_2 S_21 c_1 = 0 and as d_2 ≠ 0 we must have c_1 ∈ S_21. Since c_1 is an arbitrary element of S_11,
we have S_11⊂ S_21,
which is equivalent to
c(γ)=0.
This implies λ(γ)= 1 - d(γ). As λ(γ) must be nonnegative if the system is stable, in order to show that λ(γ) = 0 it suffices to prove that d(γ) ≥ 1. We denote the vector q satisfying Eq. (<ref>) as
q^⊤ =
[ q_1^⊤ q_2 ].
Letting q' = q - q_2/d_2[ d_1; d_2 ]
we see that q' also satisfies Eq. (<ref>) and its last component is 0. Then the first (M-1) components of q' form an emergent conserved quantity. This is because
q'^⊤ S =
[ q'^⊤_1 0 ][ S_11 S_12; S_21 S_22 ]
=
[ q'^⊤_1 S_11 q'^⊤_1 S_12 ]
=
[ 0^⊤ κ -1 ],
which implies q'_1 ∈ S_11 while q' ∉ S.
Thus, there must be at least one emergent conserved quantity d(γ) ≥ 1, which proves λ(γ) = 0 in the case d_l (γ)=1[
In fact, as we discuss in Sec. <ref>, λ(γ)=0 implies d_l (γ)=0. So the situation d_l(γ)=1 in fact does not happen when maxRPA is realized, although this does not affect the current proof.
].
Now we come to the case d_l (γ)=0 where the influence index can be written a λ(γ)= c(γ) - d(γ).
We denote the vector q satisfying Eq. (<ref>) as in Eq. (<ref>). From the assumption,
q^⊤ S
=
[ q_1^⊤ S_11 + q_2 S_21 q_1^⊤ S_12 + q_2 S_22 ]
=
[ 0^⊤ ⋯ ].
Thus, we have
q_1^⊤ S_11 + q_2 S_21 = 0^⊤ .
When S_11 is trivial, we have c(γ)=0
and
this implies the vanishing of the influence index (note that
λ(γ) and d(γ) are nonnegative).
When S_11 is nontrivial,
let us pick a nonzero element c_1 ∈ S_11.
By multiplying c_1 on Eq. (<ref>) from the right, we have
q_2 S_21 c_1 = 0 .
Suppose that
q_2 ≠ 0.
Since c_1 is an arbitrary element of S_11,
we have S_11⊂ S_21,
which is equivalent to
c(γ)=0.
This implies λ(γ)=0.
Thus, γ is a buffering structure.
Let us consider the case q_2=0.
Then, q_1 is an emergent conserved quantity,
and d (γ)=1 (note that 0 ≤d(γ) ≤ |V ∖ V_γ|).
Since λ(γ) = c(γ)
-
d(γ) is nonnegative for an asymptotically stable system,
c(γ) ≥ 1.
Thus, there exists at least one emergent cycle,
meaning that we have c_1 such that[
This equation means that susceptible reactions can independently affect the concentration x of the target species.
]
S
[ c_1; 0 ]
=
[ 0; v ],
with v ≠ 0.
In fact, there is only one emergent cycle, c(γ)=0.
To see this, suppose that there exists c_2 with the same property.
It can be normalized to satisfy
S
[ c_2; 0 ]
=
[ 0; v ].
By taking the difference
of Eqs. (<ref>) and (<ref>),
S
[ c_1 - c_2; 0 ]
=
0.
This means that
c_1 - c_2 ∈ ( S)_ supp γ.
Indeed, the choice of c_1 is unique
up to an element in ( S)_ supp γ,
which means that c(γ)=1.
Therefore, we have λ(γ)=c(γ) - d(γ)=0.
Thus, we have shown that, when the conditions for maxRPA are true,
γ is a buffering structure.
This concludes the proof of Theorem <ref>.
Let us make a comment.
In the proof, we have separated the cases depending on whether d(γ)=0 or 1.
In the case λ(γ) = d(γ)= c(γ) = 0, the vector q is given by Eq. (<ref>).
This vector is nothing but the matrix representation of
a reduction morphism <cit.>:
in Ref. <cit.>,
mappings between reaction networks are considered,
a reduction morphism is a map between reaction networks
under which all the complexes in V_γ
are mapped to a single complex in Γ∖γ.
Indeed, the expression (<ref>) corresponds
to a special case of Eq. (141) in Ref. <cit.>
(in the current situation, Γ∖γ
contains only one species and two reactions).
§ INTEGRAL FEEDBACK CONTROL FOR GENERIC RPA
In Sec. <ref>, we have shown that any generic RPA property in a deterministic reaction system can be represented by a buffering structure.
In this section, we construct the integral feedback controller corresponding to a given buffering structure.
This allows us to identify the integral control mechanism for any generic RPA property in a deterministic chemical reaction system.
§.§ Construction of integrators for a buffering structure
According to the IMP, we can expect that RPA with respect to constant-in-time disturbances is realized through integral control, and this expectation turns out to be correct.
To find the integrator equations, we reformulate the reaction system to an equivalent form that is better suited for this purpose.
For a given subnetwork, γ,
which we take to be a buffering structure,
meaning that γ is output-complete and λ(γ)=0[
In fact, the present formulation is applicable to the case λ(γ) >0. This leads to the phenomenon of manifold RPA, that we discuss in detail in Sec. <ref>.
],
we separate the chemical concentrations and reaction rates as
x =
[ x_1; x_2 ],
r =
[ r_1; r_2 ].
With the separation of internal and external degrees of freedom (to γ),
the rate equations of the whole reaction system is written as
d/dt[ x_1; x_2 ]
=
[ S_11 S_12; S_21 S_22 ][ r_1; r_2 ]
=
[ S_11 r_1 + S_12 r_2; S_21 r_1 + S_22 r_2 ].
While the internal reaction rates
r_1 = r_1 ( x_1, x_2)
in general depend on both the internal and external
chemical concentrations,
the external reaction rates are functions of only the concentrations of external species, r_2 = r_2 ( x_2)
because γ is chosen to be output-complete.
The first equation
of Eq. (<ref>) can be solved for
r_1 as
r_1 = S_11^+ ẋ_1- S_11^+ S_12 r_2 + c_11,
where S_11^+ is the Moore-Penrose inverse of S_11,
and c_11 is an arbitrary element in S_11.
Substituting this to the second equation of Eq. (<ref>),
d/dt(
x_2 - S_21 S_11^+ x_1
)
=
S' r_2 ( x_2) + S_21 c_11 ,
where S' is the generalized Schur complement,
S' S_22 - S_21 S_11^+ S_12.
The second term of the RHS of Eq. (<ref>)
vanishes if and only if the following condition is satisfied,
S_11⊂ S_21 ,
which is equivalent the absence of emergent cycles,
c(γ)=0[
When c(γ)=0 is satisfied,
the second term of Eq. (<ref>) vanishes, and
we have
d/dt(
x_2 - S_21 S^+_11 x_1
)
=
S' r_2 ( x_2),
This motivates us to consider the subnetwork
( x_2, r_2)
whose rate equation is given by
d/dt x_2
= S' r_2 ( x_2).
Namely, as long as steady states are concerned,
the subnetwork ( x_2, r_2)
satisfies the rate equation
whose stoichiometric matrix is S'.
Based on this observation,
Ref. <cit.> proposed
to use the generalized Schur complement as the stoichiometric matrix of the reduced system.
In particular, if γ is a buffering structure and d(γ)=0, the reduced system is guaranteed to have the same steady-state solution for x_2.
].
In general, c(γ) can be nonzero, and
the second term on the RHS of Eq. (<ref>)
cannot be dropped.
To account for this ambiguity,
there appear c(γ) undetermined variables.
Let us pick a basis for the space of emergent cycles,
C(γ) = S_11 / ( S )_ supp γ≃ S_11∩ ( S_21)^⊥ = span { c^(𝔠)}_𝔠 = 1, …, |𝔠|.
We introduce new variables
{w_𝔠}_c = 1, …, |𝔠| for C(γ)
and { w_α^⋆}_α^⋆=1,…,|α^⋆| for ( S)_ supp γ and parametrize c_11∈ S_11 by these variables as
c_11
=
∑_α^⋆ w_α^⋆ c_1^(α^⋆)
+
∑_𝔠w_𝔠 c^(𝔠).
Although we have solved the first equation of Eq. (<ref>) for r_1.
However, this does not work for certain combinations of ẋ_1 for which
r_1 does not appear on the RHS.
As we see below, this part can be captured by emergent conserved quantities.
We denote the set of linearly independent emergent conserved quantities by
{ d^(𝔞̅)_1 }_𝔞̅ = 1 … |𝔞̅|.
We extend each of them by
d^(𝔞̅)[ d^(𝔞̅)_1; 0 ].
The vector d^(𝔞̅) satisfies
d^(𝔞̅)⊤ S =
[ 0; c_2^(𝔞̅) ]^⊤ .
We note that the vectors
{ c_2^(𝔞̅)}_𝔞̅ = 1, …, |𝔞̅|
are linearly independent[
Suppose they are not independent.
Then, there is a certain linear combination such that
∑_𝔞̅ b_𝔞̅ c_2^(𝔞̅) = 0,
with b_𝔞̅∈ℝ
and not all of b_𝔞̅ are zero.
This implies that
∑_𝔞̅ b_𝔞̅ d^(𝔞̅)∈ S.
Namely, a certain linear combination of d^(𝔞̅) is in fact in S.
This contradicts the assumption that
d^(𝔞̅) are independent emergent conserved quantities.
].
Taking the time derivative of
the linear combinations
d^(𝔞̅)· x
= d^(𝔞̅)_1 · x_1,
d/dt d^(𝔞̅)· x
=
c_2^(𝔞̅)· r_2 ( x_2) .
Thus far, we have introduced a number equations as well as new variables ( w, w).
In fact, the set of these equations with additional variables is a equivalent description of the original system.
Namely, we have the following equivalence:
The reaction system with variables ( x_1, x_2)
under Eq. (<ref>)
and the system with variables ( x_1, w, x_2, w)
under Eqs. (<ref>), (<ref>), and (<ref>) (with parametrization (<ref>)) are equivalent as a dynamical system, meaning that they have the same solution for ( x_1 (t), x_2 (t)).
The equivalence can be checked by a straightforward computation.
Based on the reformulated description, we can identify the integrator equations realizing the RPA property represented by a buffering structure γ.
Equation (<ref>) gives us one set of integrator equations.
The other set can be obtained from Eq. (<ref>) as follows.
Equation (<ref>) can be written as
ẋ_2 - S_21 S_11^+ ẋ_1
=
S' r_2 ( x_2) +
∑_𝔠w_𝔠 u^(𝔠)
,
where u^(𝔠) S_21 c^(𝔠).
Note that { u^(𝔠)}_𝔠 = 1, …, |𝔠| are linearly independent, and we define
U span { u^(𝔠)}_𝔠 = 1, …, |𝔠|.
Since the new variables {w_𝔠}_c = 1, …, |𝔠| enter linearly, we can eliminate by projecting the dynamics on the space of original variables.
For this, we multiply Eq. (<ref>) by the projection matrix 𝒫 to the subspace U^⊥⊂ℝ^|V ∖ V_γ|, to obtain the second set of integrator equations as
𝒫
( ẋ_2 - S_21 S_11^+ ẋ_1 )
=
𝒫 S' r_2 ( x_2).
The combination of
Eqs. (<ref>) and (<ref>) and constitute the integrator equations associated with a buffering structure.
To see that these equations actually have the ability to realize the corresponding RPA property, let us show that the steady-state values of x_2 can be
determined from these equations and the obtained solution of x_2 is independent of the parameters inside γ.
To find the steady-state, we have to specify the values of conserved quantities. Recall that S is decomposed as
S ≃ D_11(γ) ⊕ D_l (γ) ⊕D̅'(γ) ,
where D_11(γ), D_l(γ),
and D̅'(γ) are defined
in Eq. (<ref>), Eq. (<ref>), and Eq. (<ref>) respectively.
The steady-state solution should satisfy the following equations,
S' r_2 ( x_2) + S_21 c_11 = 0 ,
c_2^(𝔞̅)· r_2 ( x_2) = 0 ,
d_2^(α̅')· x_2 = ℓ^α̅' ,
d_1^(α̅_γ)· x_1
+
d_2^(α̅_γ)· x_2
=
ℓ^α̅_γ,
d_1^(α̅^⋆)· x_1
=
ℓ^α̅^⋆,
where we further decomposed D_11 (γ) ≃D̅(γ) ⊕ D_11(γ) / D̅(γ) with
D̅(γ) ( S)_ supp γ
and
{[ d_1^(α̅_γ); d_2^(α̅_γ) ]}_α̅_γ = 1, …, |α̅_γ|
are a basis of D_11(γ) / D̅ (γ) ⊕ D_l (γ).
Namely, the vectors indexed by α̅_γ are conserved quantities of Γ with nonzero support
both in γ and Γ∖γ.
We took the basis of D̅'(γ) as
D̅' (γ)
=
span{[ 0; d_2^(α̅') ]}_α̅' = 1, …, |α̅'|
.
Here, we count the number of constraints on the variables ( x_2, w). The number of these variables is given by
(# of variables ( x_2, w))
= |V ∖ V_γ| + c(γ) .
In the presence of conserved quantities, not all the rows of Eq. (<ref>) are independent.
The number of independent equations in Eq. (<ref>)
is counted as
|V ∖ V_γ| - d̅' (γ) - d_l (γ),
where we defined d̅' (γ) |D̅'(γ)|.
We give the derivation of Eq. (<ref>) in Appendix <ref>.
Thus, the number of independent equations among
Eqs. (<ref>) – (<ref>)
that involve ( x_2, w) is given by
(# of independent eqs. that involve ( x_2, w)) =
|V ∖ V_γ| - d̅'(γ)
- d_l (γ)
_Eq. (<ref>)
+
d̅'(γ)
_Eq. (<ref>)
+
d (γ)
_Eq. (<ref>)
+
d(γ) - d^⋆(γ) + d_l(γ)
_Eq. (<ref>)
=
|V ∖ V_γ| + d (γ) + d(γ) - d^⋆(γ),
where we defined
d^⋆ (γ) | D̅(γ) |
and d(γ) |D_11(γ)|.
The difference between the number of variables ( x_2, w)
and the number of independent equations that involve ( x_2, w) is
(# of variables ( x_2, w))
-
(# of independent eqs. that involve ( x_2, w)
= λ(γ) - d_l (γ) -
(d(γ) - d^⋆(γ)) ,
where we used the decomposition (<ref>)
of the influence index.
Note that d(γ) - d^⋆(γ) ≥ 0, since
D̅(γ) ⊂ D_11(γ).
Thus, the relation (<ref>) implies that,
when λ(γ)=0,
we should have d_l(γ)=0 and d(γ) = d^⋆(γ).
This means that there is no equation of the form (<ref>) when λ(γ)=0.
This conclusion is consistent with the intuition that, if such a conserved quantity exists, the steady-state values of external concentrations seem affected by the change of internal parameters through this conserved quantity. When γ is a buffering structure, such a possibility is excluded.
Therefore, Eqs. (<ref>), (<ref>),
and (<ref>)
completely specify the steady-state values of x_2 when λ(γ)=0.
Since these equations do not involve any parameter in γ,
x_2 is independent of them.
On the assumption of the existence and stability of steady state,
x_2 is driven to values that are independent of the parameters in γ
by the action of integrator equations (<ref>) and (<ref>).
§.§ An Internal Model Principle for kinetics-independent RPA
We saw in Section <ref> that for maxRPA networks, a linear integrator for the dynamics can be constructed, based on the vector q satisfying Eq. (<ref>), and this provides us with an Internal Model Principle (IMP) for such networks in the case where the output species is not in the support of q.
As argued in the proof in Sec. <ref> of the equivalence of maxRPA and the law of localization, for such cases, the vector q can be viewed as an emergent conserved quantity associated with a buffering structure (see Theorem <ref>).
This argument extends to a general kinetics-independent RPA property characterized by a buffering structure γ, and it gives us the first set of integrators constructed in Sec. <ref> (see Eq. (<ref>)).
Observe that in this d(γ)-dimensional system of integrator equations, the LHS does not involve the species external to γ (i.e. variables x_2) as mandated by the IMP. Defining the Internal Model (IM) as the set of species in γ that form the support of these emergent conserved quantities establishes the IMP decomposition shown in Fig <ref>. Note that all the species external to γ are part of the Rest of the Network.
In many RPA examples (see Sections <ref> and <ref>), there exist integrators that do not belong to the first set, but rather they belong to the second set given by Eq. (<ref>).
While integrators belonging to this second set do not conform to the standard IMP (as their LHS involves the variables x_2), they do establish integral mechanisms that play a role in leading the concentrations of species external to γ to some manifold that is insensitive to parameters inside γ.
Note that while the integrators in the second set do not conform to the standard IMP in the natural coordinates of the system, it is possible that they become IMP-conformant under a suitably devised coordinate transformation.
§.§ Example
Let us look at the construction of integrators
for a simple example with
0 < d(γ) < |V ∖ V_γ |.
We consider a reaction network consisting
four species {x,y,z_1,z_2} and the following set of reactions:
e_1 : z_1 → z_1+x,
e_2 : z_1 + z_2 →∅ ,
e_3 : ∅→ z_1 ,
e_4 : 2x → 2x + z_2 ,
e_5 : x →∅ ,
e_6 : y → y+z_1 ,
e_7 : ∅→ y,
e_8 : y →∅ .
The rate equations under mass-action kinetics are
ż_̇1̇ = k_3 + k_6 y - k_2 z_1 z_2 ,
ż_̇2̇ = k_4 x^2 - k_2 z_1 z_2 ,
ẋ = k_1 z_1 - k_5 x ,
ẏ = k_7 - k_8 y .
The steady-state concentrations are given by
x̅ =
√(1/k_4(
k_3 + k_6 k_7/k_8)
)
,
y̅ = k_7/k_8,
z̅_1 = k_5/k_1 √(1/k_4(
k_3 + k_6 k_7/k_8)
)
,
z̅_2 =
k_1/k_2 k_5√(
k_4
(
k_3 + k_6 k_7/k_8)
)
.
The stoichiometric matrix is
S =
ccccccccc
c[cccccccc]z_2 -1 0 0 1 0 0 0 0
z_1 -1 0 0 0 1 1 0 0
x 0 1 -1 0 0 0 0 0
y 0 0 0 0 0 0 -1 1
e_2 e_1 e_5 e_4 e_3 e_6 e_8 e_7
.
The labeled buffering structures in this network are
γ_1^∗ = ({ z_1, z_2 },{}∪{e_1, e_2 }),
γ_2^∗ = ({ z_2 },{}∪{ e_2 }),
γ_3 = ({ x, z_1,z_2 }, {e_1,e_2,e_3,e_4,e_5}∪{}),
γ_4 = ({ x, z_1,z_2 }, {e_1,e_5}∪{e_2,e_4 }),
γ_5 = ({ z_1, z_2 },{e_1,e_5 }∪{ e_2 } ),
γ_6 = ({ x, z_1,z_2 }, {e_1,e_2,e_4,e_5,e_6 }∪{} ),
γ_7 = ({ x,y, z_1,z_2 }, {e_1,e_2,e_4,e_5,e_6,e_7,e_8}∪{}),
γ_8 = ({ x,y, z_1,z_2 }, {e_1,e_2,e_4,e_5,e_6}∪{e_8}).
We here consider RPA associated with
with γ_5.
The spaces S_11 and S_11 are spanned by
S_11 =
span{[ 0 1 1 ]^⊤,
[ 0 1 -1 ]^⊤},
S_11 =
span{[ 1 -1 ]^⊤},
where colored vectors are emergent.
The influence index of γ_5 is decomposed as
λ (γ_5)
= -2 + 3 - 1 + 0
=
=c(γ_5) 1
+
=d_l (γ_5) 0
-
=d(γ_5) 1
= 0.
Let us look at Eq. (<ref>) of this example.
The generalized Schur complement
of S with respect to this subnetwork is
S' =
cccccc
c[ccccc]x 0 0 0 0 0
y 0 0 0 -1 1
e_4 e_3 e_6 e_8 e_7
.
The combination of the LHS
of Eq. (<ref>) is
x_2 - S_21 S_11^+ x_1
=
[ x; y ].
We can pick an element of S_11 proportional to the emergent one,
c_11 =
w[ 0 1 -1 ]^⊤, where w = w(t) is an arbitrary function of time.
The RHS of Eq. (<ref>)
is given by
S' r_2 + S_21 c_11
=
[ 0 0 0 0 0; 0 0 0 -1 1 ][ k_4 x^2; k_3; k_6 y; k_8 y; k_7 ]
+
w[ 2; 0 ]
=
[ 2 w; - k_8 y + k_7 ].
Thus we have
d/dt[ x; y ]
=
[ 2 w; - k_8 y + k_7 ].
The second line of this equations gives us one integrator equation (the multiplication of the projection matrix in Eq. (<ref>) amounts to picking
the second line of Eq. (<ref>)).
We obtain another integrator equation
from the emergent conserved quantity
d_1 =
[ 1 -1 ]^⊤ in Eq. (<ref>).
By taking the time derivative of d_1 · x_1, we have
d/dt d_1 · x_1
=
d/dt (z_2 - z_1)
=
d^ ⊤ S r
=
r_4 (x) - r_3 - r_6 (y)
=
k_4 x^2 - k_3 - k_6 y .
Equation (<ref>)
and the second line of Eq. (<ref>)
constitute the integrators for the RPA associated with γ_5.
They drive the values of x and y
to their steady-state values that are independent of
the reaction parameters (k_2, k_1, k_5) inside γ_5.
§ REGULATION TO MANIFOLDS
In this section, we consider a generalization of a regulation problem
to situations where the target values of the output variables are in a manifold with nonzero dimension.
We shall call the emergence of this property as manifold RPA.
This problem can be treated naturally using the formulation developed in Sec. <ref>: we have reformulated the reaction system to an equivalent form to find the integrator equations for a given buffering structure, and this procedure is applicable even when λ(γ)≠ 0 as long as γ is output-complete.
This observation leads us to a natural generalization of the law of localization to manifold RPA, where the influence index turns out to give the dimension of the target manifold to which x_2 is regulated to.
§.§ Law of manifold localization
We have the following theorem:
Let γ⊂Γ be an output-complete subnetwork of
a deterministic chemical reaction system
satisfying the assumptions in Sec. <ref>.
Suppose that γ = (V_γ, E_γ)
is an output-complete subnetwork of Γ
whose influence index is given by λ(γ).
Then, the steady-state values of the concentrations of the species
outside γ are located in a λ(γ)-dimensional manifold,
which is invariant under the change of parameters in γ.
We shall call an output-complete subnetwork
as a generalized buffering structure of degree λ(γ),
which gives rise to manifold RPA to λ(γ)-dimensional submanifold
for the degrees of freedom outside γ.
Theorem <ref> is a generalization of the law of localization. Namely, the conventional law of localization corresponds to special cases of Theorem <ref> where the steady-state manifold for the external degrees of freedom is zero-dimensional.
Manifold RPA is very relevant for synthetic biology applications, where it is often important to design biomolecular controllers that robustly maintain some relationship between multiple output variables, without sacrificing the tunability of individual output variables <cit.>.
Basing on the formulation developed in the last section, a proof can be described succinctly.
For an output-complete subnetwork γ, we can perform the same procedure as in Sec. <ref> and obtain a set of steady-state equations (<ref>) – (<ref>),
which involve ( x_2, w).
The number of the external variables ( x_2, w) is given by Eq. (<ref>).
Among Eqs. (<ref>) – (<ref>),
if we count the number of independent equations that involve
x_2 or w but not x_1 and w (note that Eq. (<ref>) involve both x_2 and x_1),
(# of independent eqs. that involve ( x_2, w)
but not ( x_1, w) =
|V ∖ V_γ| - d̅'(γ)
- d_l (γ)
_Eq. (<ref>)
+
d̅'(γ)
_Eq. (<ref>)
+
d (γ)
_Eq. (<ref>)
= |V ∖ V_γ| + d (γ) - d_l (γ).
The difference between the number of variables ( x_2, w)
and the number of independent equations that involve ( x_2, w)
but not ( x_1, w) is
(# of variables)
-
(# of independent eqs. that involve ( x_2, w)
but not ( x_1, w)
= λ(γ) .
Thus, in general, the solutions of Eqs. (<ref>) – (<ref>) has λ(γ) undetermined variables,
and the external concentrations at steady-state are inside a manifold parameterized by them.
Since Eq. (<ref>) – (<ref>) do not
involve parameters in γ, this manifold is insensitive to them.
This concludes the proof.
Let us explain how manifold RPA emerges geometrically (for a fuller explanation, see the next subsection).
We denote the state space of internal and external variables by X_1 and X_2, i.e. x_1 ∈ X_1 and x_2 ∈ X_2.
As stated in the proof, there are |V ∖ V_γ|+c(γ) variables and |V ∖ V_γ|+d(γ) - d_l(γ) constraints for those variables,
and we have λ(γ)-dimensional steady-state manifold Σ(γ)|_X_2 of the external concentrations.
Namely, Σ(γ)|_X_2 is a submanifold
Σ(γ)|_X_2⊂ X_2 ⊂ℝ^|V ∖ V_γ|
determined by Eqs. (<ref>) – (<ref>).
The corresponding integral feedback control
for manifold RPA realized through a generalized buffering structure can be constructed in the same way as Sec. <ref> (namely, Eqs. (<ref>) and (<ref>) are the integrator equations),
and the dynamics of x_2 is steered toward the submanifold Σ(γ) |_X_2 through their action.
The system has a stable steady state by assumption, and this point in the external degrees of freedom can be seen as the intersection of Σ (γ)|_X_2
and another manifold 𝒮(γ)|_X_2⊂ℝ^|V ∖ V_γ| of (|V∖ V_γ| - λ(γ)) dimension.
While 𝒮(γ)|_X_2 is dependent on the parameters in γ, Σ(γ)|_X_2 is invariant under the change of parameters in γ.
Hence, by perturbing the parameters in γ, the location of the steady state may change but it will always lie in Σ(γ) |_X_2.
We illustrate this phenomenon in Fig. <ref>.
We discuss geometric interpretation of this phenomenon in more detail in the next subsection, and the manifolds Σ (γ) |_X_2 and 𝒮 (γ) |_X_2 turn out to be the shadows the RPA manifold and the susceptible manifold in the extended total state space.
For a concrete realization, see examples in Sec. <ref>.
§.§ Geometric interpretation of manifold RPA
Here, let us give a geometric interpretation of the manifold RPA phenomenon realized through the law of manifold localization.
In this subsection, we omit the dependence on γ for notational simplicity of quantities such as the number of emergent cycles.
Namely,
λλ (γ),
cc(γ),
d d (γ),
dd (γ),
d_l d_l (γ),
d^⋆ d^⋆ (γ),
and
d̅' d̅' (γ).
Additionally, we use the following abbreviations,
v |V_γ| ,
v' |V ∖ V_γ| = |V| - v',
e |E_γ|,
c | ( S)_ supp γ |,
With these notations, the definition of the influence index can be written as
λ = -v + e - c + d + d_l,
where we used the relation |P^0_γ ( S)| = d + d_l (see Eqs. (167) and (173) of Ref. <cit.>).
The decomposition of the influence index reads
λ = c + d_l - d.
By taking the difference of
Eqs. (<ref>) and (<ref>),
we have the following relation,
v + c + c
= e+ d + d,
which will be used later.
In the previous section, we have solved for r_1
using the Moore-Penrose inverse of S_11, and this process
introduces variables to account for the elements in S_11 (see Eq. (<ref>)).
We have parametrized c_11∈ S_11
as Eq. (<ref>).
By introducing the new variables ( w, w),
we are extending the state space
( x_1, x_2)
∈ X_1 × X_2
→
( ( x_1, w), ( x_2, w))
∈X̂_1 ×X̂_2.
Here,
we regard
( x_1 , w) to be in the extended internal state space,
( x_1 , w)∈X̂_1,
while ( x_2 , w) is in the extended external state space,
( x_2 , w)∈X̂_2.
The number of total variables is expressed as
(# of variables)
= |V| + c + c
=
v + c
+
v' + c
n_1 + n_2,
where we defined n_1 v+c and n_2 v'+c.
Together with the introduction of new variables, we have added
Eq. (<ref>) using emergent conserved quantities, which is necessary so that we have sufficient constraints to determine the steady state in the extended description[As we discussed in the previous section, the extended description is equivalent to the original system dynamically, although we here look at steady-state properties.].
See Fig. <ref> for the set of variables and steady-state equations.
Indeed, one can check that the number of variables is equal to number of independent equations.
The total number of independent equations is given by
(# of independent equations)
=
e + v' + d + d.
The number of variables minus the number of independent equations is
(# of variables)
-
(# of independent equations)
=
v+c+v'+c- e - v' - d - d
=0,
where we have used Eq. (<ref>).
Let us introduce key objects for describing manifold RPA.
In Fig. <ref>, we summarize the set of equations in the extended description together with the number of independent equations.
We define the RPA manifold Σ[
Note that this object depends on the choice of a subnetwork. We do not explicitly indicate this dependence for notational simplicity. The same comment is true for the susceptible manifold 𝒮.
]
associated with an output-complete subnetwork γ
to be the set of variables ( x_1, w, x_2, w) that satisfy
the conditions in the solid (light-blue) boxes.
We define the susceptible manifold 𝒮
to be the set of variables
( x_1, w, x_2, w) ∈X̂_1 ×X̂_2
satisfying
the conditions in dashed (green) boxes.
A key observation is that the conditions determining Σ is independent of parameters in γ and they only involve external variables,
( x_2, w) ∈X̂_2.
As a result, Σ has no dependence on parameters in γ.
On the other hand, the susceptible manifold 𝒮 is subject to parameters inside and outside γ.
The situation is illustrated in Fig <ref>.
Let us denote the numbers of constraints
determining Σ and 𝒮
by c_Σ and c_𝒮, respectively.
The dimension of these manifolds are given by
the total dimension minus the number of constraints,
Σ = n_1 + n_2 - c_Σ,
𝒮 = n_1 + n_2 - c_𝒮.
As we show in Fig. <ref>,
by counting the number of constraints,
we can express the dimensions of Σ and 𝒮
in terms of the influence index as
Σ =λ + n_1 , 𝒮 = n_2 - λ.
These manifolds intersect at a point,
corresponding to the existence of a steady state.
Indeed, the sum of the dimensions equal the dimension of the total space,
Σ+ 𝒮 = n_1 + n_2.
Now, let us consider the projection of Σ and 𝒮
to the space of external variables, ( x_2, w) ∈X̂_2[We will refer to the projected RPA and susceptible manifolds with the same names, indicating the total space they reside to avoid possible confusions.].
We denote these projections as Σ |_X̂_2 and
𝒮 |_X̂_2, respectively.
While the dimension of 𝒮 and 𝒮|_X̂_2 are the same (i.e. the dimension does not change under the projection),
the dimension of the RPA manifold is decreased by n_1,
since Σ has no dependence on the variables in X̂_1 (see Fig. <ref>).
Namely,
Σ |_X̂_2 = λ,
𝒮 |_X̂_2 = n_2 - λ .
The projected RPA and susceptible manifolds still intersect at a point in X̂_2,
Σ |_X̂_2 + 𝒮 |_X̂_2 = n_2.
We can further project Σ |_X̂_2
to X_2 by projecting out w, and obtain Σ |_X_2.
When the manifold Σ |_X_2 is in a general position, its projection to X_2 is also of dimension λ.
Since Σ |_X̂_2 does not depend on the parameters inside γ, its projection to X_2 does not either.
Thus, the steady state of x_2 is localized within a λ-dimensional manifold in X_2 under the change of parameters in γ.
§.§ Examples
Here, we demonstrate the manifold RPA phenomenon and the law of manifold localization using a few examples.
§.§.§ Simple example
Let us first describe a simple example that consists of two species { x,y } and three reactions,
e_1 : ∅→ y ,
e_2 : y → x,
e_3 : x →∅.
The stoichiometric matrix reads
S =
cccc
c[ccc]y -1 1 0
x 1 0 - 1
e_2 e_1 e_3
,
If we use mass-action kinetics, the rate equations are
ẏ = k_1 - k_2 y ,
ẋ = k_2 y - k_3 x .
At the steady state, the concentrations are given by
x̅ = k_1 / k_3 and y̅ = k_1 / k_2.
Let us choose the following output-complete subnetwork,
γ = ({}, {e_2 }) ,
which includes no species and one reaction.
The influence index is nonzero,
λ (γ)=1,
and the conventional law of localization does not apply.
Since the subnetwork is output-complete, the law of manifold localization is applicable.
The influence index of γ is decomposed as
λ (γ)
=
=c(γ) 1
+
=d_l(γ) 0
-
=d(γ) 0
= 1.
Thus, the subnetwork is a generalized buffering structure of degree 1.
In the current case, the submatrix S_11 is 0 × 1 dimensional, and the matrices S' and S_21 are given by
S' =
ccc
c[cc]y 1 0
x 0 - 1
e_1 e_3
,
S_21 =
[ -1; 1 ].
The subnetwork γ has an emergent cycle,
and S_11 is nontrivial,
S_11 = span {[ 1 ]}.
We can pick an element of S_11 as
c_11 = w(t)
[ 1 ], and we have
[ ẏ; ẋ ]
=
[ 1 0; 0 -1 ][ k_1; k_3 x ]
+ w[ -1; 1 ].
The second term on the RHS is orthogonal to [ 1 1 ],
and multiplying this vector, we get an integrator equation,
d/dt (y+x) = k_1 - k_3 x.
This integrator steers the variables (x,y) to the following
one-dimensional RPA manifold Σ (γ) |_X_2,
[ x̅; y̅ ]
=
[ k_1 / k_3; s ],
where s ∈ℝ_≥ 0 is a parameter.
This manifold is independent
of parameters in γ, which is k_2 in the current case.
On the other hand, the susceptible manifold 𝒮(γ) |_X_2 is given by the relation
k_1 - k_2 y̅ = 0 ,
which also defines a line in (x̅,y̅) plane.
The susceptible manifold 𝒮(γ) |_X_2 depends nontrivially on k_2.
§.§.§ Example with a lost conserved quantity
Here we discuss an example with a lost conserved quantity.
Let us take a reaction network with four species {v_1,v_2,v_3,v_4 } and four reactions,
e_1 : v_1 → v_2,
e_2 : v_2 → v_3,
e_3 : v_3 → v_4,
e_4 : v_4 → v_1.
Under mass-action kinetics,
the steady-state concentrations are given by
x̅_i = ℓ K / k_i for i=1,2,3,,4 with
1/K 1/k_1 + 1/k_2 + 1/k_3 + 1/k_4, where
ℓ = x̅_1 + x̅_2 + x̅_3 + x̅_4 is a conserved quantity,
and the steady-state rate is
r̅_A = ℓ K for A = 1,2,3,4.
We here take the following output-complete subnetwork,
γ = ( {v_1,v_2}, {e_1,e_2 } ) .
The subnetwork contains no emergent cycle and no emergent conserved quantity,
and it has one lost conserved quantity.
The decomposition of the influence index reads
λ (γ)
=
-2+2-0+1
=
=c(γ) 0
+
=d_l(γ) 1
-
=d(γ) 0
= 1.
Indeed, the projection of the conserved quantity d = [ 1 1 1 1 ]^⊤∈ S
to subnetwork γ is not conserved within γ.
Since there is no emergent conserved quantity,
the integrator equation is given by Eq. (<ref>),
d/dt[ x_1 + x_2 + x_3; x_4 ]
=
[ - k_3 x_3 + k_4 x_4; k_3 x_3 - k_4 x_4 ].
The first and second lines of these equations are equivalent via the conservation law, x_1 + x_2 + x_3 + x_4 = ℓ = const..
This integrator drives the variables to satisfy
k_4 x̅_4 - k_3 x̅_3 = 0,
This relation determines the RPA manifold Σ (γ)|_X_2, which is one-dimensional in (x̅_3, x̅_4).
Because of the existence of a lost conserved quantity,
(x̅_3,x̅_4) should satisfy an equation that involve x̅_1 and x̅_2 as well,
x̅_3 + x̅_4 = ℓ - x̅_1 - x̅_2.
The condition (<ref>) also defines a line and this corresponds to the susceptible manifold 𝒮 (γ) |_X_2.
While 𝒮 (γ) |_X_2 is subject to the parameters in γ, i.e. k_1 and k_2, the manifold Σ (γ)|_X_2 is insensitive to these parameters, as expected.
§.§.§ Example with an emergent conserved quantity
As a more nontrivial example, let us revisit the example discussed in Sec. <ref>.
We choose the following subnetwork
γ = ({z_1,z_2}, {e_1,e_2,e_5,e_7,e_8}) .
This subnetwork has nonzero influence index, λ (γ)=1, and cannot be discussed with the conventional law of localization.
It is output-complete, so we can apply the law of manifold localization.
The influence index is decomposed as
λ (γ)
=
-2 + 5 - 2 + 0
=
=c(γ) 2
+
=d_l(γ) 0
-
=d(γ) 1
= 1.
The spaces S_11 and S_11 are spanned by
S_11 =
span{[ 0 1 1 0 0 ]^⊤,
[ 0 0 0 1 1 ]^⊤,
[ 0 0 0 1 -1 ]^⊤,
[ 0 -1 1 0 0 ]^⊤},
S_11 =
span{[ 1 -1 ]^⊤},
where the components are ordered as
(e_2,e_1,e_5,e_8,e_7) for S_11
and (x,y) for S_11,
and colored vectors are emergent.
If we set
c^(1)_1 [ 0 0 0 1 -1 ]^⊤
and
c^(2)_1 [ 0 -1 1 0 0 ]^⊤,
we can parametrize an element of S_11∩ ( S_21)^⊥ as
c_11w_1 c^(1)_1 + w_2 c^(2)_2,
and we have
S' r_2 + S_21 c_11
=
[ -2 w_2; -2 w_1 ].
Hence, the rate equations do not give us an integrator and
they just fixes the values of w_1 and w_2 to zero at steady state.
The emergent conserved quantity gives the same integrator equation as Eq. (<ref>).
At steady state, this integrator enforces the relation,
k_4 x̅^2 = k_3 + k_6 y̅ .
Thus, the RPA manifold for the external concentrations
is given by (x,y) ∈ (ℝ_> 0)^2 subject to constraint (<ref>).
One can parametrize the one-dimensional manifold as
[ x̅; y̅ ]
=
[ √(1/k_4 (k_3 + k_6 s) ); s ],
where s ∈ℝ_> 0.
Figure <ref> illustrates manifold RPA of this example.
We plot the time evolution of states in (x,y) space.
The dashed line is the RPA manifold Σ (γ)|_X_2
associated with γ, that is specified by Eq. (<ref>),
and the solid line corresponds to 𝒮 (γ) |_X_2.
The states are absorbed to the intersection of Σ (γ) |_X_2
and 𝒮 (γ) |_X_2.
While he location of 𝒮 (γ) |_X_2 is subject to changes in the parameters in γ: it depends on k_7 and k_8,
the position of Σ (γ) |_X_2 is independent of
all the parameters (k_1,k_2,k_5,k_7,k_8) inside γ.
In other words, the relation (<ref>) is independent of the parameters (k_1,k_2,k_5,k_7,k_8).
§ ROBUST ADAPTATION IS TOPOLOGICAL
We have seen that the RPA properties and the corresponding integral feedback control can be identified based on indices determined by the topological characteristics of reaction networks.
Here, we argue that this feature is natural and generically true
because of the robustness of adaptation in broader classes of physical systems beyond deterministic chemical reaction systems.
Let us consider a class of dynamical systems (which do not necessarily have to be deterministic chemical reaction systems) and think of the set of all such systems.
Among these, we wish to find the systems that exhibit RPA.
Suppose also that we shall be able to detect such systems with
a real-valued quantity χ (S) associated with a system S[
In the case of deterministic chemical reaction systems,
the specification of a system S should
include the data of
a CRN Γ,
a subnetwork γ of Γ,
a set of parameters to be perturbed,
and a choice of output variables.
See Table <ref>.
],
and let us say that system S exhibits RPA if χ(S)=0.
When the perfect adaptation property is robust,
the adaptation does not require fine-tuning of system parameters.
If we have another system S', which only differs from S by the values of parameters, then we should also have χ (S') =χ (S) = 0
due to the robustness requirement.
Namely, the quantity χ (S) is invariant under
the change of parameters of the system S.
In this sense, the quantity χ should be a topological invariant.
Thus, we argue that the class of systems with an RPA property
should be generically identified by a topological invariant.
We summarize this feature as “Robust Adaptation is Topological”(RAT), and will refer to this motto as the RAT principle for short.
The RAT principle itself is not a single theorem,
but rather is a template for theorems, similarly to the Internal Model Principle of control theory.
Namely, depending on the technical details such as
the class of systems under consideration,
the choice of output variables,
and
the class of perturbations under consideration,
the corresponding topological invariant should be identified
appropriately (if it exists).
For example, the law of localization <cit.>
and the RPA of fluxes <cit.>
provide us with particular incarnations of the RAT principle (see Table <ref>).
The one-to-one correspondence of RPA properties and labeled buffering structures indicates that all the RPA properties can be identified topologically.
The law of manifold localization discussed in Sec. <ref>
generalizes the regulation problem to nonzero dimension of target manifolds,
and the condition of output-completeness is still topological, in line with the RAT principle.
The RAT principle gives us guidance for finding systems with RPA properties:
the question is to identify the appropriate topological invariant for the particular situation under consideration.
As we showed in Sec. <ref>,
buffering structures turned out to be also necessary for RPA to be realized.
The realizations of the RAT principle mentioned here are
for deterministic chemical reaction systems.
It would be highly intriguing to explore the possibility of formulating analogous theorems for various classes of dynamical systems, such as stochastic reaction systems, electrical circuits, and others.
It would also be interesting to examine the known results in control theory about robust control from this perspective: the condition for robust control
may allow for a topological interpretation.
As a further exploration, it would be interesting to identify the underlying mathematical object
behind the topological invariant, similarly to the case of the Euler characteristic, which can be written as the alternating sum of the ranks of homology groups of different degrees <cit.>.
§ CONCLUSIONS
The ability of cellular biomolecular networks to tightly regulate key quantities is of utmost importance, despite their complexity and various sources of disturbances, since the failure to achieve this precise control can have dire consequences for the cell population and, consequently, the entire organism.
The capacity of a reaction network to exhibit robustness against disturbances, known as Robust Perfect Adaptation (RPA), equips cells with the ability to effectively cope with such disruptions. Hence, unraveling the underlying mechanisms for RPA has been a crucial endeavor in understanding cellular resilience.
The goal of this paper is to develop a systematic method for finding all the kinetics-independent RPA properties and the embedded integral control mechanisms associated with them.
We have shown that every elementary RPA property in a deterministic chemical reaction system can be characterized by a labeled buffering structure,
which is a subnetwork with distinguished topological characteristics (output-completeness and vanishing of the influence index) (Theorem <ref>).
This finding gives us a systematic method for identifying all the RPA properties in any chemical reaction network through the identification of labeled buffering structures, with which any generic RPA property can be obtained.
We provided an efficient algorithm for the enumeration of all the labeled buffering structures for a given reaction network, and developed a computational package RPAFinder <cit.>.
We also gave a method for finding integral feedback controllers for a generic RPA property, which is represented by a buffering structure.
By doing so, we have shown that integral feedback control is necessary (and sufficient) for kinetics-independent RPA in deterministic chemical reaction systems, establishing the Internal Model Principle in these systems for constant-in-time disturbances. Unlike most existing works on this topic, our results apply to generic multi-output situations, where many different shades of the RPA property can naturally emerge.
These different shades correspond to the dimension of the RPA manifold to which the output-species dynamics is driven by the integral feedback mechanisms. This phenomenon, which we call manifold RPA, can be studied naturally using our approach.
Remarkably, every output-complete subnetwork gives rise to manifold RPA and the concentrations of the species external to the subnetwork is driven to a manifold whose dimension is given by the influence index (Theorem <ref>).
Importantly, this manifold remains unaffected by parameters that influence the species and reactions within the subnetwork, even though these parameters might impact the steady-state concentrations of species outside the subnetwork.
This observation highlights that perturbations in the subnetwork parameters can influence the external species, but they cannot disrupt the robust steady-state relationship among these species, which is determined by the manifold. Exploring this type of RPA holds crucial significance in systems biology and bears relevance to synthetic biology as well, particularly in the design of biomolecular controllers that can robustly maintain relationships between multiple output variables while retaining the adjustability of individual output variables <cit.>.
In biochemical systems, there is no pre-existing distinction between a “system” and a “controller” unlike the situations in control engineering, and we have to reverse engineer <cit.>
biological circuits to reveal embedded control mechanisms.
In the present method, a pair of a controlled subsystem and a controller emerges for each labeled buffering structure.
Typically, a reaction network is endowed with multiple labeled buffering structures and hence it possesses multiple distinct RPA properties, which we can systematically uncover with our method.
This is useful for understanding network resilience in systems biology, and it provides helpful insights into the design principles for synthetic biology <cit.>.
We believe it will also be useful for metabolic engineering <cit.>,
since we shall be able to narrow down the possible candidate reactions to modify for various objectives, like improving yields or reducing costs.
We have pointed out that the topological characterization of the condition for RPA
is not limited to the studies of deterministic chemical reaction systems
and should also be applicable to more generic dynamical systems.
This idea can be expressed as “Robust Adaptation is Topological,” and we call this motto as the RAT principle.
Results such as the one-to-one correspondence of the RPA property and labeled buffering structures,
the law of localization <cit.>,
and the RPA of reaction fluxes <cit.>,
can be seen as examples of theorems embodying the RAT principle.
We believe that the RAT principle provides us with a useful guideline
as to how to formulate theorems identifying dynamical systems with the RPA property,
depending on the technical details such as
the class of systems and the choice of target variables.
It will be interesting to reexamine existing results on robust control
from a topological perspective.
Finally, let us comment on possible further directions.
Labeled buffering structures provides us with a natural biological unit, in the sense that they can be controlled independently.
It would be interesting to investigate how and when labeled buffering structures change when we embed a network in a larger network.
Namely, the modularity <cit.> of labeled buffering structures is to be tested.
The present method of enumeration of buffering structures is specifically for subnetworks with vanishing influence index.
As shown in Theorem <ref>, any output-complete subnetwork can induce regulation to a λ(γ)-dimensional manifold. Developing an efficient method to find subnetworks with nonzero indices is important for uncovering manifold RPA in chemical reaction networks.
In the current work, we have considered deterministic reaction systems.
Whether a similar result can be established for stochastic reaction systems is a nontrivial and intriguing question.
As a generalization to a different direction, the structural conditions governing the regulation to time-dependent states, such as oscillatory ones <cit.>, are to be explored.
To have a more comprehensive understanding of the behavior of chemical reaction systems, it would be interesting to study the connection of the present results with other approaches based on, for example, information geometry <cit.>
or non-equilibrium thermodynamics <cit.>.
Y. H. is grateful to
Hyukpyo Hong,
Jae Kyoung Kim
for useful discussions.
The work of Y. H. was supported in part by JSPS KAKENHI Grant No. JP22H05111, and in part by an appointment of the JRG Program at the APCTP, which is funded by the Science and Technology Promotion Fund and Lottery Fund of the Korean Government and also by the Korean Local Governments of Gyeongsangbuk-do Province and Pohang City. A. G. and M. K. acknowledge funding from ETH Zurich and from the Swiss National Science Foundation (SNSF) under grant 182653.
§ SPECTRAL CORRESPONDENCE
In the current paper, we assume that the dynamics is asymptotically stable, i.e., the Jacobian matrix at steady state is non-singular and all its eigenvalues have strictly negative real parts, as discussed in Sec. <ref>.
In the case S = 0, this implies that the A-matrix is invertible due to Theorem 1 in Ref. <cit.>.
Let us here show that this continues to hold even when S ≠ 0.
Let us consider the time evolution of
fluctuations around a steady state, which is governed by
d/dtδ x_i =∑_j J_ijδ x_j ,
where
J_ij∑_A S_iA r_A,j is
the Jacobian matrix.
To be consistent with the conservation relations,
the fluctuations should satisfy the constraints
∑_i d^(α̅)_i δ x_i = 0.
The eigenvalue equations of the Jacobian matrix
satisfying the constraint from conserved quantities read
J δ x_λ = λδ x_λ,
D δ x_λ = 0 ,
where we have used the matrix notation
and the components of the matrix D is
given by D_α̅i d^(α̅)_i.
An eigenvector of the Jacobian
in the presence of conserved quantities
can be seen as an element of the kernel of the following matrix,
J_λ[
[ J - λ; D ]].
We will show that there is a one-to-one correspondence
between J_λ and
the generalized eigenvectors/values of the A-matrix.
We here use a matrix notation and express the A-matrix (<ref>) as
A
=
[
[ R -C; D 0 ]] .
We introduce the following matrix parametrized by λ∈ℂ,
A_λ[
[ R - λ S^+ -C; D 0 ]] .
We will show the following:
The following map F is a bijection,
F :
J_λ→ A_λ ,
where F is defined by
J_λ∋δ x_λ↦
F (δ x_λ)
[ δ x_λ; μ ],
Here, μ is determined uniquely by
C μ = (1 - S^+ S) R δ x_λ.
Let us show that
indeed F(δ x_λ) ∈ A_λ.
An element δ x_λ∈ J_λ satisfies
S R δ x_λ = λδ x_λ ,
D δ x_λ = 0 .
Equation (<ref>) indicates that
δ x_λ∈ ( S)^⊥.
Since S S^+ is a projection matrix to ( S)^⊥,
we can multiply this to the RHS of
Eq. (<ref>),
and we have
S
(
R - λ S^+
)
δ x_λ
= 0 .
This means that we can write
(
R
- λ S^+
)
δ x_λ
= C μ,
Since
S^+ δ x_λ∈ ( S)^⊥,
by multiplying the projection matrix to S, we obtain
Eq. (<ref>).
For a given set of basis vectors C of S,
μ is determined uniquely.
Equations (<ref>) and (<ref>)
can be rearranged as
[
[ R -C; D 0 ]]
[ δ x_λ; μ ]
=
λ[
[ S^+ 0; 0 0 ]]
[ δ x_λ; μ ],
which means that
[ δ x_λ; μ ]∈ A_λ.
The map F is obviously injective.
Conversely,
for a given
[ δ x_λ; μ ]∈ A_λ,
it can be easily checked that
δ x_λ∈ J_λ,
which means that F is surjective.
Thus, we have shown that F is a bijection.
This concludes the proof.
When a chemical reaction system has an asymptotically stable steady state, the corresponding A-matrix is invertible.
Because of the asymptotic stability,
J_λ is trivial when λ=0,
and consequently
A_λ is trivial when λ=0.
This implies that the square matrix A does not have a zero eigenvalue, which means that A is invertible.
§ FORMULA FOR SECOND-ORDER RESPONSES
Here we derive explicit formulas for second-order responses
of steady-state concentrations and reaction fluxes
with respect to parameters.
In the following, we derive the expression
for x̅_i,AB with A ≠ B,
and also for
y̅_ν,ρσ
with
ρ≠σ
in parallel.
We start with the formula for the first-order response,
x̅_i,A
=
- ( A^-1)_iA _A r_A.
y̅_ν,ρ
=
- ( A^-1)_νρ _ρ L_ρ.
By taking another derivative with respect to a parameter,
x̅_i,AB
=
-
( A^-1)_iA,B _A r_A
-
( A^-1)_iA (_A r_A)_,B .
y_ν,ρσ
=
- ( A^-1)_νρ,σ _ρ L_ρ
- ( A^-1)_νρ (_ρ L_ρ)_,σ .
The second term is written as
- ( A^-1)_iA (_A r_A)_,B
=
x̅_i,A (ln_A r_A)_,B
- ( A^-1)_νρ (_ρ L_ρ)_,σ
=
y̅_ν,ρ (ln_ρ L_ρ)_,σ .
Noting that
( A^-1)_iA,B
=
-
∑_C,D
( A^-1)_i C_B A_CD
( A^-1)_D A ,
( A^-1)_νρ,σ
=
-
∑_κ ,τ
( A^-1)_νκ_σ A_κτ
( A^-1)_τρ ,
and
_B A_Cj
=
δ_BC (_B r_C)_,j
+ ∑_k r_C,jk x̅_k,B ,
_σ A_C j
=
δ_σ C (_σ r_C)_,j
+ ∑_k r_C,jk y̅_k,σ ,
the first term is written as
- ( A^-1)_iA,B _A r_A
=
∑_C,D
( A^-1)_i C_B A_CD
( A^-1)_D A _A r_A
=
∑_C,j
( A^-1)_i C(
δ_BC (_B r_C)_,j + ∑_k r_C,jkx̅_k,B)
( A^-1)_j A _A r_A
=
∑_j
( A^-1)_i B
(_B r_B)_,j
( A^-1)_j A _A r_A
+
∑_C,j,k
( A^-1)_i C r_C,jk x̅_k,B
( A^-1)_j A _A r_A
=
∑_j
x̅_i,B
(ln_B r_B)_,j
x̅_j,A
+
∑_C,j,kr_C,jk/_C r_C x̅_i,C x̅_j,A x̅_k,B
=
x̅_i,B (ln_B r_B)_,A
+
∑_C,j,kr_C,jk/_C r_C x̅_i,C x̅_j,A x̅_k,B .
- ( A^-1)_νρ, σ _ρ L_ρ =
∑_κ, τ
( A^-1)_νκ_σ A_κτ
( A^-1)_τρ _ρ L_ρ
=
∑_C,j
( A^-1)_ν C(
δ_σ C (_σ r_C)_,j + ∑_k r_C,jk y_k,σ)
( A^-1)_j ρ _ρ L_ρ
=
∑_j
( A^-1)_νσ
(_σ r_σ)_,j
( A^-1)_j ρ _ρ L_ρ
+
∑_C,j,k
( A^-1)_ν C r_C,jk y_k,σ
( A^-1)_j ρ _ρ L_ρ
=
∑_j
y̅_ν,σ
(ln_σ r_σ)_,j
y̅_j,ρ
+
∑_C,j,kr_C,jk/_C r_C y̅_ν,C y̅_j,ρ y̅_k,σ
=
y̅_ν,σ (ln_σ L_σ)_,ρ
+
∑_C,j,kr_C,jk/_C r_C y̅_ν,C y̅_j,ρ y̅_k,σ .
Thus, we obtain the following formula,
x̅_i,AB
=
x̅_i,A F_A,B
+
x̅_i,B F_B,A
+
∑_Cx̅_i,C∑_j,k ⊢ Cr_C,jk/r_C,C x̅_j,A x̅_k,B ,
where F_A ln_A r_A.
y̅_ν,ρσ
=
y̅_ν,ρ F_ρ,σ
+
y̅_ν,σ F_σ, ρ
+
∑_Cy̅_ν,C∑_j,k ⊢ Cr_C,jk/r_C,C y̅_j,ρ y̅_k,σ ,
where F_σln_σ L_σ.
Note that this expression is manifestly symmetric
under the exchange A ↔ B
ρ↔σ
.
We can also derive a similar formula for reaction fluxes.
Noting that y̅_α = μ_α and r̅_C can be written as
r̅_C = ∑_α c^(α)_C
y̅_α ,
we find that second-order responses
of reaction fluxes are expressed as
r̅_C,AB
=
r̅_C,A F_A,B
+
r̅_C,B F_B,A
+
∑_Dr̅_C,D∑_j,k ⊢ Dr_D,jk/r_D,D x̅_j,A x̅_k,B .
Note that e_C can be taken to be the same as e_A or e_B, while we assume e_A ≠ e_B.
Including the conserved-quantity perturbations, we have
r̅_C,ρσ
=
r̅_C,ρ F_ρ,σ
+
r̅_C,σ F_σ,ρ
+
∑_Dr̅_C,D∑_j,k ⊢ Dr_D,jk/r_D,D y̅_j,ρ y̅_k,σ .
The formulas (<ref>) and (<ref>) (or (<ref>) and (<ref>))
can be seen as a decomposition of higher-order influence
in terms of lower-order influence.
We can diagrammatically represent Eq. (<ref>) as
[thick]
(0,0.7) node(eA) e_A
(0,-0.7) node(eB) e_B
(1,0) node(mdl)
(2.6,0) node(eC) v_i = ∑_j ⊢ A
(4,0) node(eB1) e_B
(5.6,0) node (ej1) v_j ⊢ e_A
(8.5,0) node(vi1) v_i + ∑_k ⊢ B e_A
(11.2,0) node(vk2) v_k ⊢ e_B
(13.7,0) node(vi2) v_i + ∑_C,j,k
(14.9,0.6) node(eA3) e_A
(14.9,-0.6) node(eB3) e_B
(16,0.6) node(vj3) v_j
(16,-0.6) node(vk3) v_k
(16.8,0) node(eC3) e_C
(18,0) node(vi3) v_i
(16.4,0.35) node[rotate=-45](vd1) ⊢
(16.4,-0.35) node[rotate=45](vd2) ⊢
;
[-, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (eA) – (1,0);
[-, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (eB) – (1,0);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (1,0) – (eC);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm]
(eB1) – (ej1);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (ej1) – (vi1);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (vi1) – (vk2);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (vk2) – (vi2);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (eA3) – (vj3);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (eB3) – (vk3);
[->, style=decorate, decoration=snake,segment length=2mm, amplitude=0.5mm] (eC3) – (vi3);
The LHS represents the influence of both e_A and e_B on v_i,
and there are different contributions.
The first term on the RHS represents a contribution
that e_B first influences a reactant of e_A and
then e_A influences v_i,
and
in the second term the roles of e_A and e_B are exchanged.
The third term on the RHS
takes into the contribution where
e_A and e_B affects
reactants of some reaction e_C,
and then e_C influences v_i.
Let us check the second-order formula with a simple example network,
e_1 : ∅→ v_1,
e_2 : v_1 →∅,
e_3 : v_1 + v_2 → v_1,
e_4 : ∅→ v_2 .
With mass-action kinetics,
the rate functions are written as
r_1 = k_1, r_2 = k_2 x_1, r_3 = k_3 x_1 x_2, r_4 = k_4.
The steady-state concentrations are given by
x̅_1 = k_1/k_2,
x̅_2 = k_2 k_4/k_1 k_3.
For example, the second-order derivative
x̅_2,21 = _k_2_k_1x̅_2
can be explicitly computed as
x̅_2,21 = - k_4/(k_1)^2 k_3.
Let us evaluate the RHS of Eq. (<ref>).
The first and second terms are
x̅_2,2 F_2,1
=
k_4/k_1 k_3·(∂_k_2 r_2)_,1/∂_k_2 r_2
=
k_4/(k_1)^2 k_3
,
x̅_2,1 F_1,2
=
x̅_2,1(∂_k_1 r_1)_,2/∂_k_1 r_1
= 0.
The third term is evaluated as
∑_Cx̅_2,C/r_C,C∑_j,k ⊢ C
r_C,jk x̅_j,2 x̅_k,1
=
x̅_2,3/r_3,3(
r_3,12 x̅_1,2 x̅_2,1
+
r_3,21 x̅_2,2 x̅_1,1)
=
-2 k_4/(k_1)^2 k_3.
The summation of
Eqs. (<ref>)
and (<ref>)
indeed reproduces Eq. (<ref>).
§ EXTENSION OF THE MAXRPA CHARACTERIZATION RESULT
Here, we prove Theorem <ref>, which is an extension of the characterization result for maxRPA networks <cit.>. In this extension, we no longer require that S = 0 and hence the full network Γ =(V, E) may have conserved quantities. The last two reactions, denoted by e_1̅ and e_2̅ no longer need to have mass-action kinetics, and we will denote their rate functions as r_1̅( x; k_1̅) and r_2̅( x; k_2̅) respectively, where k_1̅ and k_2̅ are parameters.
We start by proving the “if" part of the theorem, namely that if these two conditions are satisfied then the network exhibits maxRPA for the output species X. This is straightforward because by picking any ( q , κ) that satisfies Eq. (<ref>), we can construct a linear integrator z q · x whose dynamics is given by
ż = κ r_1̅( x;k_1̅ ) - r_2̅( x;k_2̅) = r_1̅( x;k_1̅) ( κ - Φ(x_M, k_1̅, k_2̅))
where the last relation follows from Eq. (<ref>). Setting the RHS to zero, and using the positivity of the fixed point we obtain that at steady-state the concentration of the output species X must satisfy Eq. (<ref>) and hence it is independent of all other parameters except k_1̅ and k_2̅. Note that if there exists another pair ( q' , κ') satisfying Eq. (<ref>), then we must necessarily have that κ = κ', or otherwise the condition of uniqueness of the steady-state will be violated. As both ( q' , κ) and ( q , κ) satisfy Eq. (<ref>), subtracting them yields that the vector ( q'- q) will be in S. This proves that the vector q is unique up to addition of vectors in S, or equivalently, we can say that q is unique in the quotient space ℝ^M / S.
We now prove the “only if" part of the theorem. We start from the steady-state equations Eqs. (<ref>) and (<ref>), where { d^(α̅)}_α̅=1,… ,|α̅| is a basis of S and for each α̅, ℓ^α̅ is the value of the conserved quantity corresponding to d^(α̅).
Differentiating Eq. (<ref>) with respect to parameter k_B and quantity ℓ^β̅ we arrive at:
∑_A S_iA r_A/ x_ix̅_i/ k_B = - ∑_A S_iA r_A/ k_B and ∑_A S_iA r_A/ x_ix̅_i/ℓ^β̅ = 0.
As we have seen before, differentiating Eq. (<ref>) with respect to k_B and ℓ^β̅ gives us relations,
∑_i d_i^(α̅) x_i/ k_B = 0 and ∑_i d_i^(α̅) x_i/ℓ^β̅= δ^α̅β̅.
We shall write these relations in matrix-form and for this we need to define certain quantities. Recall that r is the N-dimensional vector of reaction rates and S is the M × N stoichiometric matrix.
Let ∇_ x r = [_i r_A] be the N × M Jacobian matrix of r with respect to the explicit dependence on the state x, evaluated at the steady-state x = x̅ ( k, ℓ).
The M × M Jacobian for the dynamics is given by
J = S ∇_ x r.
Similarly, let ∇_ k r = [_B r_A] be the N × N
Jacobian matrix of r with respect to the explicit dependence on the parameters in k,
and let ∇_ k, ℓx̅ ( k, ℓ) be the M × (N +|α̅|) Jacobian matrix of the steady-state x̅ ( k, ℓ) with respect to both k and ℓ. Let D be the |α̅| × M matrix whose rows are the basis vectors { d^(α̅)}_α̅=1,… ,|α̅| for S. Relations (<ref>) and (<ref>) can be succinctly expressed as
[
[ J; D ]] ∇_ k, ℓx̅ ( k, ℓ) = -[[ S ∇_ k r 0_ M × |α̅|; 0_ |α̅| × M - 1_|α̅| × |α̅| ]],
where 0 and 1 denote zero and identity matrices of the dimensions indicated in the subscript. Note that the matrix (M + |α̅| )× M matrix
B [
[ J; D ]]
has independent columns, because otherwise B would be nontrivial, and this would imply that the steady state is not asymptotically stable.
Since the columns of B are independent, its Moore-Penrose inverse is explicitly given by
B^+ = (B^⊤ B)^-1 B^⊤,
and hence Eq. (<ref>) implies that
∇_ k, ℓx̅ ( k, ℓ) = - (B^⊤ B)^-1 B^⊤[[ S ∇_ k r 0_ M × |α̅|; 0_ |α̅| × M - 1_|α̅| × |α̅| ]].
Define q̃ to be the (M + |α̅| )-dimensional vector given by
q̃^⊤ = - u^⊤_M (B^⊤ B)^-1 B^⊤ ,
where u_M is a M-dimensional vector whose all components are zero, except the component at the M-th index (i.e. location of the output species X) which is 1. We can decompose q̃^⊤ as q̃^⊤ = [q̃_1^⊤ q̃_2^⊤], where q̃_1^⊤ and q̃_2^⊤ are vectors with dimensions 1 × M and 1 × |α̅| respectively. Multiplying Eq. (<ref>) by u^⊤_M on the left we see that the 1 × (M + |α̅|) vector of sensitivities of the steady-state x̅_M ( k, ℓ) of the output with respect to k, ℓ is given by
∇_ k, ℓx̅_M ( k, ℓ) u^⊤_M ∇_ k, ℓx̅ ( k, ℓ)
=
[ q̃_1^⊤ q̃_2^⊤ ][[ S ∇_ k r 0_ M × |α̅|; 0_ |α̅| × M - 1_|α̅| × |α̅| ]]
=
[ q̃_1^⊤ S ∇_ k r - q̃_2^⊤ ].
For maxRPA we require that
x̅_M ( k, ℓ) = ϕ_ out( k_1̅, k_2̅),
and hence the steady-state output has zero-sensitivity with respect to all components of ℓ and all components of k except k_1̅ and k_2̅.
Therefore Eq. (<ref>) implies that q̃^⊤_2 = - ∇_ℓx̅_M ( k, ℓ) = 0 and
q̃^⊤_1 S ∇_ k r =
[ 0 ∂_k_1̅ϕ_ out( k_1̅, k_2̅) ∂_k_2̅ϕ_ out( k_1̅, k_2̅) ].
Only the last two components of the vector on the RHS are non-zero, while the rest are zeros. Since for each reaction e_A, the reaction rate r_A (x̅ ( k, ℓ); k_A) has an explicit dependence on an independent k_A,
the N × N matrix ∇_ k r is a diagonal matrix.
Therefore, Eq. (<ref>) informs us that for any reaction A, which is not one of the last two reactions (i.e. A ≠1̅, 2̅) we must have
∑_i q̃_1 i S_i A = 0,
which says that vector q̃_1 is orthogonal to the stoichiometric vector for reaction e_A. Now for the last two reactions, Eq. (<ref>) implies that
∑_i q̃_1i S_i 1̅ = ∂_k_1̅ϕ_ out( k_1̅, k_2̅)/∂_k_1̅ r_1̅ (x̅ ( k, ℓ); k_1̅ ) and ∑_i q̃_1i S_i 2̅ = ∂_k_2̅ϕ_ out( k_1̅, k_2̅)/∂_k_2̅ r_2̅ (x̅ ( k, ℓ); k_2̅ ),
where for i=1,2, _k_i̅ r_i̅ (x̅ ( k, ℓ); k_i̅ ) denotes the partial derivative with respect to only the explicit dependence on parameter k_i̅ and not the implicit dependence through the steady-state x̅ ( k, ℓ).
Combining these relations with Eq. (<ref>), we see that the vector q̃_1 satisfies
q̃_1^⊤ S =
[ 0, ∂_k_1̅ϕ_ out( k_1̅, k_2̅)/∂_k_1̅ r_1̅ (x̅ ( k, ℓ); k_1̅ ), ∂_k_2̅ϕ_ out( k_1̅, k_2̅)/∂_k_2̅ r_2̅ (x̅ ( k, ℓ); k_2̅ ) ]
.
Now let c =
[ c_1 ⋯ c_N ]^⊤ be a vector in S whose last two components are not both zero. Choosing such a c is possible because the network has positive steady-states. Since S c= 0, we must have q̃_1^⊤ S c = 0 which implies that
c_N-1∂_k_1̅ϕ_ out( k_1̅, k_2̅)/∂_k_1̅ r_1̅ (x̅ ( k, ℓ); k_1̅ )
=
- c_N∂_k_2̅ϕ_ out( k_1̅, k_2̅)/∂_k_2̅ r_2̅ (x̅ ( k, ℓ); k_2̅ ).
Note that for any c fixed, this relation must hold for any ( k, ℓ).
We argued before that the matrix ∇_ k r has full row-rank. This implies that (S ∇_ k r) = S and hence the row-rank of the product matrix S ∇_ k r is just M - |α̅|, where |α̅| = | S|. This shows that the matrix
[[ S ∇_ k r 0_ M × |α̅|; 0_ |α̅| × M - 1_|α̅| × |α̅| ]]
has rank M. As matrix B^⊤ has full row-rank (because columns of B are independent) and matrix (B^⊤ B)^-1 has full row-rank (because it is an invertible matrix), from Eq. (<ref>) we can conclude that the Jacobian matrix ∇_ k, ℓx̅ ( k, ℓ), capturing the sensitivities of the steady-state x̅ ( k, ℓ) with respect to parameters ( k, ℓ) has rank M which is equal to the number of species. Hence, applying the constant rank theorem (see Theorem 11.1 in <cit.>) we can conclude that by perturbing the parameters in ( k, ℓ) we can independently perturb the components of the steady-state vector x̅ ( k, ℓ). This will be useful later in the proof.
Let Φ( x,k_1̅, k_2̅) be the ratio of the rate functions for the last two reactions, i.e.
Φ( x , k_1̅, k_2̅) = r_2̅( x; k_2̅)/r_1̅( x; k_1̅).
Then by differentiating with respect to both k_1̅ and k_2̅ we obtain
∂_k_1̅ r_1̅ ( x; k_1̅ ) = -r_1̅ ( x ; k_1̅ ) ∂_k_1̅Φ( x,k_1̅, k_2̅)/Φ( x,k_1̅, k_2̅) and ∂_k_2̅ r_2̅ ( x ; k_2̅ ) = r_1̅ ( x ; k_1̅ ) ∂_k_2̅Φ( x,k_1̅, k_2̅).
Setting x to be the steady-state vector x = x̅( k, ℓ), and substituting in Eq. (<ref>) we obtain
Φ( x̅( k, ℓ),k_1̅, k_2̅) _k_1̅ϕ_ out( k_1̅, k_2̅)/_k_1̅Φ(x̅( k, ℓ),k_1̅, k_2̅)
=
κ_k_2̅ϕ_ out( k_1̅, k_2̅)/_k_2̅Φ( x̅ ( k, ℓ),k_1̅, k_2̅),
where
κ c_N/ c_N-1. Let us now define the vector q as
q = - q̃_1r_1̅ (x̅( k, ℓ) ; k_1̅ ) _k_2̅Φ( x̅ ( k, ℓ),k_1̅, k_2̅) /_k_2̅ϕ_ out( k_1̅, k_2̅),
then Eqs. (<ref>), (<ref>) and (<ref>) together imply that pair ( q, κ) satisfies Eq. (<ref>). This proves the first condition of the theorem.
To prove the second condition, observe that if we pick a pair ( q, κ) satisfying Eq. (<ref>), then for the linear integrator z q · x, the dynamics is given by
ż = κ r_1̅( x;k_1̅ ) - r_2̅( x;k_2̅) = r_1̅( x;k_1̅) ( κ - Φ( x, k_1̅, k_2̅))
which proves that at the steady-state
Φ(x̅( k, ℓ) , k_1̅, k_2̅) = κ.
The only thing left to show is that the function Φ( x , k_1̅, k_2̅) does not depend on the concentrations x_1,…,x_M-1 of the first (M-1) species, i.e. this function can only depend on x_M. We shall prove this by showing that the gradient of this function with respect to the state vector x can only have one non-zero component, which is at the M-th location (corresponding to ∂_x_MΦ( x , k_1̅, k_2̅)). It suffices to prove this only at the steady-state x = x̅ ( k, ℓ ) because as we argued before, the components of x̅ ( k, ℓ) can be independently perturbed by perturbing the parameters in ( k, ℓ).
Recall that u_M is the M-dimensional vector whose all components are zero, except the component at the M-th index (i.e. location of the output species X) which is 1. In order to prove our claim we just need to show that u_M cannot be independent of the gradient vector
ϕ( k, ℓ) ∇_ xΦ(x̅( k, ℓ) , k_1̅, k_2̅).
We set q_Φ to be the (M + |α̅| )-dimensional vector
q_Φ^⊤ = - ϕ( k, ℓ)^⊤ (B^⊤ B)^-1 B^⊤.
and decompose it as q_Φ^⊤ = [ q_Φ,1^⊤ q_Φ,2^⊤], where q_Φ,1^⊤ and q_Φ,2^⊤ are vectors with dimensions 1 × M and 1 × |α̅| respectively. Similar to Eq. (<ref>) we can derive
ϕ( k, ℓ)^⊤∇_ k, ℓx̅ ( k, ℓ)
=
[ q_Φ,1^⊤ S ∇_ k r, - q_Φ,2^⊤ ]
.
Eq. (<ref>) implies that q_Φ,2^⊤ = - ∇_ℓΦ( x( k, ℓ) , k_1̅, k_2̅) = 0 and
q_Φ,1^⊤ S ∇_ k r=
[ 0, - _k_1̅Φ(x̅( k, ℓ),k_1̅, k_2̅), k_1̅, k_2̅), - ∂_k_2̅Φ(x̅( k, ℓ),k_1̅, k_2̅) ]
.
Due to Eq. (<ref>), the vector on the right is collinear with the vector on the right of Eq. (<ref>). Hence there exists constants ζ_1 and ζ_2 such that, the vector z_Φ = ζ_1 q̃_1 + ζ_2 q_Φ,1
satisfies
z_Φ^⊤ S ∇_ k r = 0,
which shows that the vector vanishes,
z_Φ^⊤ = ζ_1 q̃^⊤ + ζ_2 q_Φ^⊤ = -(ζ_1 u_M^⊤ + ζ_2 ϕ( k, ℓ)^⊤)(B^⊤ B)^-1 B^⊤ = 0.
However, since the rows of B^T are independent we must have that the vector ζ_1 u_M^⊤ + ζ_2 ϕ( k, ℓ)^⊤ = 0, which happen if and only if u_M and ϕ( k, ℓ) are dependent vectors.
Hence only the last component of ϕ( k, ℓ) can be nonzero and this proves that Φ( x, k_1̅, k_2̅) can only be a function of the output species concentration x_M. This completes the proof of this theorem.
Let us illustrate this result with an example. Consider the minimal reaction network that displays the phenomenon of Absolute Concentration Robustness (ACR), which refers to robustness with respect to a conserved quantity <cit.>. This network has just two species and reactions:
e_1 : v_1 → v_2,
e_2 : v_1 + v_2 → 2 v_1.
The rate equations in the mass action kinetics are given by
ẋ_1 = k_2 x_1 x_2 - k_1 x_1,
ẋ_2 = -k_2 x_1 x_2 + k_1 x_1.
It is immediate that ẋ_1 + ẋ_2 = 0 and so the total concentration ℓ = x_1(t) + x_2(t) is constant over time. Fixing the value of ℓ, the steady-state concentrations are given by
x̅_1 = ℓ - k_1/k_2 and x̅_2 = k_1/k_2.
Hence the output species X = v_2 is robust to ℓ, and this network exhibits maxRPA as the steady-state concentration of X is just a function of two mass-action reaction rate constants. The stoichiometric matrix for this network is
S =
ccc
c[cc]v_1 -1 1
v_2 1 -1
e_1 e_2
.
Hence S is one-dimensional and spanned by the vector d = [ 1 1 ]^⊤. Since S is non-empty the maxRPA characterisation result in Ref. <cit.> does not apply, but we can use the maxRPA result developed here to conclude that this network indeed exhibits maxRPA. Setting q =[ -1 0 ]^⊤ and κ =1, we see that the linear system (<ref>) holds and so the first condition of Theorem <ref> is satisfied. To check the second condition note that under the mass-action kinetics the ratio of the rate functions r_1(x_1,x_2) and r_2(x_1,x_2) for reactions e_1 and e_2 satisfy
r_2(x_1,x_2)/r_1(x_1,x_2) = k_2 x_1 x_2/k_1 x_1 = k_2/k_1x_2 Φ(x_2,k_1,k_2).
Hence the ratio is only a function of the output species concentration x_2 and the set-point is precisely determined by the relation
Φ(x_2,k_1,k_2) = κ = 1.
Observe that for any real number α, ( q +α d, κ) would also satisfy (<ref>) and these are in fact all the solutions to this linear system.
In this example, if we consider the subnetwork γ that is formed by excluding the output species and the two reactions from the full network, then γ has only one species v_1 and no reactions. Hence γ is not output-complete which happens because for this exampled maxRPA is not kinetics independent. Indeed, maxRPA does not hold for arbitrary kinetics for the two reactions.
§ DERIVATION OF ISOMORPHISMS (<REF>) AND (<REF>)
Let us give a proof of the isomorphisms (<ref>) and (<ref>), which are true when c(γ) = 0
and d(γ)=0 hold.
A proof of the isomorphisms is given in Ref. <cit.>.
We describe it here to make the paper self-contained
together with the explicit construction of the isomorphisms.
We first define the following vector spaces,
C_0 (Γ) {∑_i d_i v_i
|
v_i ∈ V, d_i ∈ℝ} ,
C_1 (Γ) {∑_A c_A e_A
|
e_A ∈ E, c_A ∈ℝ} .
We also define similar spaces for a subnetwork γ = (V_γ, E_γ) ⊂Γ, C_n (γ) for n=0,1,
to be those generated by V_γ and E_γ.
Similarly, we define the spaces for Γ' = Γ / γ,
to be the spaces spanned by (V ∖ V_γ, E ∖ E_γ).
We will denote the elements of
C_1(γ),
C_1(Γ)
and
C_1(Γ')
using a vector,
where each component represents the corresponding coefficient.
For example,
c_1 ∈ C_1(γ),
[ c_1; c_2 ]∈ C_1(Γ),
c_2 ∈ C_1(Γ').
In the above, the element of C_1(Γ) are partitioned
into those inside/outside γ.
Similar notation will be used for C_0(γ) and so on.
To prove the isomorphisms, we consider the following short exact sequence,
0
[d]
0 [d]
0 [d]
0 [r]
C_1 (γ)
[r]^ψ_1 [d]^_γ
C_1 (Γ)
[r]^φ_1[d]^
C_1(Γ' )
[r]
[d]^' 0
0
[r]
C_0(γ)
[r]^ψ_0 [d]
C_0 (Γ)
[r]^φ_0[d]
C_0 (Γ' )
[r]
[d]
0
0
0
0
where the columns are the chain complexes of γ,
Γ, and Γ', respectively.
The maps _γ, , and ' are defined by
_γ: c_1 ↦ S_11 c_1,
: c =
[ c_1; c_2 ]↦
S c,
': c_2 ↦ S' c_2.
The horizontal maps are given by
ψ_1:
c_1 ↦[ c_1; 0 ],
φ_1:
[ c_1; c_2 ]↦ c_2 ,
ψ_0:
d_1 ↦[ d_1; S_21 S^+_11 d_1 ],
φ_0:
[ d_1; d_2 ]↦ d_2 - S_21S^+_11 d_1 .
One can check that the diagram (<ref>) commutes
if and only if the following condition is satisfied:
S_21 (1 - S_11^+ S_11) c_1 = 0,
where c_1 ∈ C_1(γ).
The matrix (1 - S_11^+ S_11)
is the projection matrix to S_11,
and Eq. (<ref>) is equivalent to
S_11⊂ S_21 .
The condition (<ref>) is equivalent to
c (γ) = | S_11 / ( S)_ supp γ| = 0,
namely the absence of emergent cycles,
as shown around Eq. (182) of Ref. <cit.>.
Thus, the diagram (<ref>) commutes if and only if γ has no emergent cycle.
Applying the snake lemma to Eq. (<ref>), we obtain
an exact sequence,
0 [r] S_11[r]^ψ_1 S
[r]^φ_1 S'
[r]^δ_1
S_11[r]^ψ̅_0 S
[r]^φ̅_0 S'
[r]
0 ,
where ψ̅_0 and φ̅_0
are induced maps of ψ_0 and φ_0.
The map δ_1: S' → S_11
is called the connecting map.
For a given c_2 ∈ S',
the connecting map is given by[
The map is identified as follows.
Pick an element c_2 ∈ S', which
can be included in C_1(Γ').
Since φ_1 is surjective, there exists
c =
[ c_1; c_2 ]
such that φ_1 ( c) = c_2.
From the commutativity of the diagram (<ref>),
we have φ_0 (S c) = S' c_2 = 0.
Since the rows of Eq. (<ref>) are exact,
there exists d_1 ∈ C_0(γ)
such that ψ_0 ( d_1) = S c.
We obtain [ d_1]=[S_11 c_1 + S_12 c_2]= [S_12 c_2]∈ S_11 by identifying the differences in im S_11.
The mapping c_2 ↦ [S_12 c_2] is the connecting map.
]
δ_1:
c_2 ↦
[S_12 c_2] ∈ S_11,
where [...] means to identify the differences in im S_11.
When d(γ)=0 is satisfied,
the connecting map δ_1 is a zero map, as we show below.
Since d(γ)=0, for each d_1 ∈ S_11 there exists a d_2
such that [ d_1; d_2 ]∈ S. This implies that
d^⊤_2 S_21 = 0 ,
d^⊤_1 S_12 + d^⊤_2 S_22 = 0.
Multiplying the second equation on the right by c_2 ∈ S' and using Eq. (<ref>) and S' c_2= 0,
we obtain
d^⊤_1 S_12 c_2
= - d^⊤_2 S_22 c_2
= - d^⊤_2 S_21 S^+_11S_12 c_2 = 0,
where the last equality is because
d^⊤_2 S_21 = 0.
This shows that d_1 is orthogonal to S_12 c_2
and since d_1 is an arbitrary element of S_11 we can conclude that S_12 c_2 is in the image of S_11.
Therefore, δ_1 is a zero map.
The exact sequence (<ref>) and the fact that δ_1 = 0 implies that we have the following exact sequences,
0 [r] S_11[r]
S
[r]
S'
[r]
0
,
0 [r] S_11[r]
S
[r]
S'
[r]
0
.
These imply the isomorphisms (<ref>) and (<ref>).
Let us explicitly construct the isomorphism (<ref>).
We first define a map F: S → S'
by
F: S ∋ c =
[ c_1; c_2 ]↦ c_2 ∈ S'.
We can see that c_2 is indeed an element of S',
S' c_2 = - S_21 (1 - S^+_11 S_11) c_1 = 0,
where we have used S c= 0 and
S_11⊂ S_21 (since c(γ)=0).
The map F is a surjection as shown in the following.
Pick any c_2 ∈ S' and define c_1 - S^+_11S_12 c_2.
Then
S_11 c_1 + S_12 c_2 = (1 - S_11 S^+_11) S_12 c_2 ,
S_21 c_1 + S_22 c_2 = (- S_21 S^+_11S_12 + S_22 ) c_2
= -S' c_2 = 0.
The matrix (1 - S_11 S^+_11) is the projection matrix to S_11.
As we showed earlier, S_12 c_2 belongs to the image of S_11,
and the RHS of Eq. (<ref>) is zero,
proving that
[ c_1; c_2 ]∈ S and so the map F is surjective.
An element of the kernel of the map F is of the form
[ c_1; 0 ]∈
( S)_ supp γ⊂ S .
Thus, the kernel of F is identified as F = ( S)_ supp γ.
Therefore, the induced map
F̅ : S / ( S)_ supp γ→ S,
is an isomorphism.
Noting that ( S)_ supp γ = S_11
since c(γ)=0, we have obtained
the isomorphism (<ref>).
We now prove the isomorphism (<ref>). Recall the definitions of subspaces D_11(γ), X (γ) and D̅'(γ) from Eqs. (<ref>), (<ref>) and (<ref>). Since there is no emergent conserved quantities d(γ)=0, we must have S_11 = D_11(γ).
Consider the projection of S to X(γ). Clearly the kernel of this surjective map is S ∩ X (γ)^⊥, where X (γ)^⊥ denotes the orthogonal complement of X (γ). Hence S is isomorphic to the direct sum
S ≃ X(γ) ⊕ S/ X (γ).
Now if we consider the projection map from X(γ) to D_11(γ) by
[ d_1; d_2 ]↦ d_1,
then the image of this map is exactly D_11(γ) and the kernel of this map is exactly D̅'(γ). Hence X(γ) is isomorphic to the direct sum
X(γ) ≃ D_11(γ) ⊕D̅'(γ),
which together with Eq. (<ref>) shows that
S ≃ D_11(γ) ⊕D̅'(γ) ⊕ S / X (γ).
Since S_11 = D_11(γ) and the projection map φ̅_0 (given by Eq. (<ref>)) is an isomorphism between D̅'(γ) and S' ∩ S_21, in order to prove isomorphism (<ref>) we just need to establish the following isomorphism
S / X (γ) ≃ S' ∩ ( S_21)^⊥.
We claim that the projection map φ̅_0 also establishes this isomorphism. To see this consider any
[ d_1; d_2 ]∈ S/X (γ).
Without losing generality we can assume that d_1 ∈ ( S_11)^⊥ and [ d_1; d_2 ]∈ X(γ)^⊥. As
d_1^⊤ S_11 + d_2^⊤ S_21 = 0
and d_1 ∈ ( S_11)^⊥, we can express d_1^⊤ uniquely as
d_1^⊤ =- d_2^⊤ S_21S_11^+
which upon substitution in relation d_1^⊤ S_12 + d_2^⊤ S_22 = 0 yields
d_1^⊤ S_12 + d_2^⊤ S_22 = d_2^⊤ (S_22 - S_21 S_11^+ S_12) = d_2^⊤ S' = 0.
Hence d_2 ∈ S'. Let d'_2 be the projection of this vector on S' ∩ S_21. Then certainly [ 0; d'_2 ]∈ S ∩ X(γ) which implies that the dot product of this vector with [ d_1; d_2 ]∈ X(γ)^⊥
is zero. Therefore d'_2 = 0 and so d_2 ∈ S' ∩ ( S_21)^⊥. This shows that the image of S / X (γ) under the map φ̅_0 is in S' ∩ ( S_21)^⊥. Moreover this map is injective on S / X (γ) as its kernel is trivial, because if
[ d_1; 0 ]∈ S/X (γ)
then d_1 ∈ S_11 and so d_1 ∈ ( S_11)^⊥ can only happen when d_1 = 0. Next we establish the surjectivity of the map
φ̅_0 : S / X (γ) → S' ∩ ( S_21)^⊥.
Pick any d_2 ∈ S' ∩ ( S_21)^⊥. Then setting d_1^⊤ as in Eq. (<ref>) yields Eq. (<ref>). Moreover
d_1^⊤ S_11 + d_2^⊤ S_21 = d_2^⊤ S_21 (1 - S_11^+ S_11) = 0
because matrix (1 - S_11^+ S_11) is the projection matrix to S_11 and S_11⊂ S_21 as c(γ) =0. Eqs. (<ref>) and (<ref>) show that
[ d_1; d_2 ]∈ S. Noting that S_11^+ = S_11^⊤ = S_11, we from Eq. (<ref>) that d_1 ∈ ( S_11)^⊥. This also shows that
[ d_1; d_2 ]∈ X(γ)^⊥
because for any [ d_1; d_2 ]∈ X(γ)
we have d_1 ∈ S_11 (hence orthogonal to d_1) and since d_1^⊤ S_11 + d^⊤_2 S_21 = d^⊤_2 S_21 = 0, we must have d_2 ∈ S_21 while d_2 ∈ ( S_21)^⊥. This proves the surjectivity of the map φ̅_0 from S / X (γ) to S' ∩ ( S_21)^⊥, which establishes the isomorphism (<ref>), and therefore proves the isomorphism (<ref>).
§ DERIVATION OF EQ. (<REF>)
In this Appendix, we show that
the number of independent equations in Eq. (<ref>)
is given by
|V ∖ V_γ| - d̅' (γ) - d_l (γ).
We here count the dependent relations in Eq. (<ref>).
This amounts to finding independent vectors
d_2 ∈ℝ^|V ∖ V_γ| such that
d_2^⊤ S' r_2 = 0,
d_2^⊤ S_21 c_11 = 0,
for any c_11∈ S_11.
Since r_2 and c_11∈ S_11 is arbitrary,
this condition is rephrased as
finding d_2 ∈ℝ^|V ∖ V_γ| such that
d_2^⊤ S' = 0 ,
d_2^⊤ S_21 (1 - S^+_11 S_11) = 0 .
We show the following equivalence,
d_2 satisfies Eqs. (<ref>) and (<ref>) ⟺ d_2 ∈ ( S' ∩ S_21) ⊕φ̅_0 (D_l(γ)).
Let us first show the direction ⟹.
If d_2 ∈ S_21, then
d_2 ∈ S' ∩ S_21 and the relation holds.
Suppose d_2 ∉ S_21.
Then, if we define
d_1^⊤ - d_2^⊤ S_21 S^+_11,
Eqs. (<ref>) and (<ref>) implies that
[ d_1^⊤ d_2^⊤ ][ S_11 S_12; S_21 S_22 ]
= 0.
Namely, [ d_1; d_2 ]∈ S.
If we multiply the projection matrix to S_11
on d_1,
d_1^⊤ (1 - S_11 S^+_11)
=
- d_2^⊤ S_21 S_11^+ + d_2^⊤ S_21 S_11^+ S_11 S^+_11
= 0 ,
which means that d_1 ∈ ( S_11)^⊥.
This implies that
[ d_1; d_2 ]∈ S / X(γ) = D_l(γ) and d_2 ∈φ̅_0 (D_l (γ)).
We now show the opposite direction ⟸.
When d_2 ∈ S' ∩ S_21,
d_2 trivially satisfies Eqs. (<ref>) and (<ref>).
When d_2 ∈φ̅_0 (D_l (γ)),
there exists
[ d_1; d_2 ]∈ S such that d_1 ∈ ( S_11)^⊥.
Then, we have
d_2^⊤ S_21 (1 - S_11^+ S_11)
= - d_1^⊤ S_11 (1 - S_11^+ S_11)
= 0,
where we used
[ d_1; d_2 ]∈ S, and we also have
d_2^⊤ S'
=
d_2^⊤ (S_22 - S_21S_11^+ S_12)
=
-
d_1^⊤ S_12
+
d_1^⊤ S_11
S_11^+ S_12)
=
- d_1^⊤ (1-S_11 S_11^+) S_12
= 0,
because (1 - S_11 S_11^+) is a projection matrix to S_11.
This concludes the proof of the equivalence (<ref>).
Observe also that
φ̅_0 ( D̅'(γ) )
=
( S' ) ∩ ( S_21) .
Recall that the injective map φ̅_0 is defined on D̅'(γ) as
φ̅_0 :
D̅' (γ)
∋[ 0; d_2 ]↦ d_2
∈ S'.
Indeed, suppose that we are given
d_2 ∈φ̅_0 (D̅'(γ)).
Since it satisfies
[ 0 d_2^⊤ ][ S_11 S_12; S_21 S_22 ]
=
[ d_2^⊤ S_21 d_2^⊤ S_22 ]
= 0 ,
we have
d_2^⊤ S' = d_2^⊤ (S_22 - S_21S^+_11 S_12)
= 0
and
d_2 ∈ ( S') ∩ ( S_21).
Conversely, for a given d_2 ∈ ( S') ∩ ( S_21),
we have
[ 0 d_2^⊤ ][ S_11 S_12; S_21 S_22 ]
=
[ d_2^⊤ S_21 d_2^⊤ S_22 ]
=
[ d_2^⊤ S_21 d_2^⊤ S_21 S_11^+ S_12 ]
= 0 ,
where we have used d_2^⊤ S'= 0,
and thus d_2 ∈φ̅_0 (D̅'(γ)).
This proves Eq. (<ref>).
Combining Eq. (<ref>) and Eq. (<ref>),
the number of independent vectors satisfying
Eqs. (<ref>) and (<ref>)
is given by d̅'(γ) and d_l(γ).
Therefore, among |V ∖ V_γ| equations of Eq. (<ref>), the number of independent equations is given by Eq. (<ref>).
§ FINDING EMERGENT CYCLES, EMERGENT CONSERVED QUANTITIES, AND LOST CONSERVED QUANTITIES
We here describe how to find emergent cycles, emergent conserved quantities, and lost conserved quantities linear-algebraically.
As for emergent cycles, the space spanned by them can be written as
C(γ)
S_11 /
( S)_ supp γ
=
S_11 / { c_1 ∈ S_11 | [ c_1; 0 ]∈ S
}
≃ S_11 ∩ ( S_21)^⊥ ,
and the last expression can be used to find the basis of C(γ).
On the other hand, the space of emergent conserved quantities is written as
D(γ)
S_11 / D_11 (γ)
≃ S_11∩( D_11(γ) )^⊥ .
The space D_11(γ) can be written as
D_11(γ)
{ d_1 ∈ S_11 | ^∃ d_2 such that [ d_1; d_2 ]∈ S
}
=
{ d_1
| [ d_1^⊤ d_2^⊤ ]S
= 0
},
where we have defined an extended matrix
S[ S_11 S_12 S_11; S_21 S_22 0 ].
The last expression of Eq. (<ref>) can be used
to find the basis of D_11 (γ) and hence that of ( D_11 (γ) )^⊥, which can be combined with
Eq. (<ref>) to obtain emergent conserved quantities.
Similarly, we can also obtain the basis of lost conserved quantities
by noting that its space is written as
D_l (γ) S / X(γ)
≃ S ∩ (X(γ))^⊥
and
X (γ)
{[ d_1; d_2 ]∈ S
| d_1 ∈ S_11}≃{[ d_1; d_2 ] | [ d_1^⊤ d_2^⊤ ]S
= 0
} .
We have implemented methods to obtain bases of C(γ), D(γ), and D_l(γ) in RPAFinder <cit.>.
§ DETAILS OF THE EXAMPLE IN SEC. <REF>
§.§ List of reactions
1: Glucose → G6P,
2: G6P → F6P,
3: F6P → G6P,
4: F6P → F16P,
5: F16P → DHAP + G3P,
6: DHAP → G3P,
7: G3P → PGP,
8: PGP → 3PG,
9: 3PG → PGP,
10: 3PG → 2PG,
11: 2PG → 3PG,
12: 2PG → PEP,
13: PEP → 2PG,
14: PEP → PYR,
15: G6P → PG6,
16: PG6 → CO2 + Ru5P,
17: Ru5P → X5P,
18: Ru5P → R5P,
19: R5P + X5P → G3P + S7P,
20: G3P + S7P → R5P + X5P,
21: G3P + S7P → E4P + F6P,
22: E4P + F6P → G3P + S7P,
23: E4P + X5P → F6P + G3P,
24: F6P + G3P → E4P + X5P,
25: PG6 → G3P + PYR,
26: PYR → Acetal + CO2,
27: Acetal → Ethanol,
28: Ethanol → Acetal,
29: R5P → ∅,
30: CO2 → ∅,
31: ∅ → Glucose,
32: Ethanol → ∅,
33: Acetal → ∅,
34: PYR → Ala,
35: Ala → PYR,
36: Ala → ∅.
§.§ List of labeled buffering structures
γ^∗_1=({ Glucose}, {}∪{1}),
γ_2,3,4,15,16,17,18,25=({ Acetal, Ala, CO2, DHAP, E4P, Ethanol, F16P, F6P, G3P, G6P, PEP, 2PG, 3PG, PG6, PGP, PYR,
R5P, Ru5P, S7P, X5P}, {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34,
35, 36}∪{}),
γ^∗_5=({ F16P}, {}∪{5}),
γ^∗_6=({ DHAP}, {}∪{6}),
γ_7=({ E4P, G3P, S7P, X5P}, {19, 20, 21, 22, 23, 24}∪{7}),
γ^∗_8=({ PGP}, {}∪{8}),
γ_9=({ PGP}, {8, 9}∪{}),
γ_10=({ 3PG, PGP}, {8, 9}∪{10}),
γ_11=({ 3PG, PGP}, {8, 9, 10, 11}∪{}),
γ_12=({ 2PG, 3PG, PGP}, {8, 9, 10, 11}∪{12}),
γ_13=({ 2PG, 3PG, PGP}, {8, 9, 10, 11, 12, 13}∪{}),
γ_14=({ PEP, 2PG, 3PG, PGP}, {8, 9, 10, 11, 12, 13}∪{14}),
γ_19,20,21,22,23=({ E4P, S7P, X5P}, {19, 20, 21, 22}∪{23}),
γ_24=({ E4P, S7P, X5P}, {19, 20, 21, 22, 23, 24}∪{}),
γ_26,34,35,36=({ Acetal, Ala, CO2, Ethanol, PYR}, {26, 27, 28, 30, 32, 33, 34, 35, 36}∪{}),
γ_27,28,32,33=({ Acetal, Ethanol}, {27, 28, 32, 33}∪{}),
γ_29=({ E4P, R5P, S7P, X5P}, {19, 20, 21, 22}∪{23, 29}),
γ^∗_30=({ CO2}, {}∪{30}),
γ_31=({ Acetal, Ala, CO2, DHAP, E4P, Ethanol, F16P, F6P, G3P, G6P, Glucose, PEP, 2PG, 3PG, PG6, PGP, PYR,
R5P, Ru5P, S7P, X5P}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36}∪{})
Here, γ_31 coincides with the whole network.
|
http://arxiv.org/abs/2307.05623v1 | 20230711045845 | A DeepLearning Framework for Dynamic Estimation of Origin-Destination Sequence | [
"Zheli Xiong",
"Defu Lian",
"Enhong Chen",
"Gang Chen",
"Xiaomin Cheng"
] | cs.LG | [
"cs.LG",
"cs.AI",
"I.2.1"
] |
A DeepLearning Framework for Dynamic Estimation of Origin-Destination Sequence
Zheli Xiong2,Defu Lian2**Defu Lian is the corresponding author., Enhong Chen2, Gang Chen3 and Xiaomin Cheng23 2School of Data Science
University of Science and Technology of China, Hefei, China
Email: {liandefu,cheneh,wh5606}@ustc.edu.cn{zlxiong}@mail.ustc.edu.cn3Yangtze River Delta Information Intelligence Innovation Research Institute, China
Email: [email protected]
August 12, 2023
======================================================================================================================================================================================================================================================================================================================================================================================
OD matrix estimation is a critical problem in the transportation domain. The principle method uses the traffic sensor measured information such as traffic counts to estimate the traffic demand represented by the OD matrix. The problem is divided into two categories: static OD matrix estimation and dynamic OD matrices sequence(OD sequence for short) estimation. The above two face the underdetermination problem caused by abundant estimated parameters and insufficient constraint information. In addition, OD sequence estimation also faces the lag challenge: due to different traffic conditions such as congestion, identical vehicle will appear on different road sections during the same observation period, resulting in identical OD demands correspond to different trips. To this end, this paper proposes an integrated method, which uses deep learning methods to infer the structure of OD sequence and uses structural constraints to guide traditional numerical optimization. Our experiments show that the neural network(NN) can effectively infer the structure of the OD sequence and provide practical constraints for numerical optimization to obtain better results. Moreover, the experiments show that provided structural information contains not only constraints on the spatial structure of OD matrices but also provides constraints on the temporal structure of OD sequence, which solve the effect of the lagging problem well.
OD matrix estimation, deep learning, neural network
§ INTRODUCTION
With the development of big traffic data, a large amount of traffic data has been widely used in traffic applications such as route planning, flow prediction and traffic light control. Traffic demand describes the trips between the divided areas, commonly referred to as the OD (Origin-Destination) matrix. The OD matrix has significant value for various traffic tasks such as traffic flow prediction<cit.>, trajectory prediction<cit.> and location recommendation<cit.>. On the one hand, the change in traffic demand between regions will affect the flow of the road sections, leading to a change in the optimal vehicle path. On the other hand, OD estimation will lead to more effective traffic regulation since the already known OD demants can provide rationality for management strategies. However, traffic demand is the data that sensors cannot directly observe, and it must be obtained through other traffic data, such as the traffic flow, to estimate the matrix.
OD matrix estimation is mainly divided into two categories: static OD matrix estimation<cit.> and dynamic OD sequence estimation<cit.>. Static OD estimation uses integrated traffic counts to estimate the total traffic demand in the period. In comparison, dynamic OD sequence estimation uses time-varying traffic counts in a sequence of intervals to estimate the corresponding OD matrix sequence.
The bi-level framework is commonly used for OD estimation<cit.>, which is divided into upper and lower levels. The upper level adjusts the OD matrices by minimizing the numerical gap between real and estimated traffic counts by solving a least squares problem. The optimizing method is mainly gradient-based<cit.> and optimizes the OD matrices by the steepest descent method. The lower level assigns traffic demand to road sections by analysis<cit.> or simulation<cit.> method. The upper and lower layers updates iteratively and converge to an optimum point.
Due to the limited number of road sections and abundant parameters to be estimated in the OD matrices, the optimization problem is heavily underdetermined<cit.>. Some researchers have alleviated this problem by adding other observable information, such as travel speed<cit.>, cellular probe<cit.>, and bluetooth probe<cit.>to the estimator. In addition to the above challenge, dynamic OD sequence estimation faces another lag challenge: due to different traffic conditions such as congestion, identical vehicle will appear on different road sections during the same observation period, resulting in identical OD demands correspond to different trips<cit.>, so current OD matrix will refer to different time-varying traffic counts according to different traffic conditions. Furthermore, traffic conditions are caused by OD marices before and after the current OD matrix, which causes a temporal relationship between these OD matrices. To this end, some studies propose fixed maximum lags<cit.>, which assume that vehicles can complete their trip within a fixed maximum interval amount to eliminate the influence of observation intervals amount. And others like <cit.> propose using the temporal overlap in consecutive estimation intervals to alleviate the influence between OD matrices, as shown in Fig. 1.
To address these challenges above, we use deep learning to fit the mapping relationship between the time-varying traffic counts and the structure of the OD sequence to learn the impact of lagged traffic counts on the OD structure and the relationship between OD sequences.
Consider that numerical space makes the learning space too large for deep learning. However, distribution can constrain the learning space's size and reflect the structural information. Furthermore, simply using NN to infer numerical matrices will lose important information about assignments presented in the bi-level framework. Therefore, we propose a method that integrates deep learning and numerical optimization, which merges the inferred structural information into the upper-level estimator of the bi-level framework. Providing the spatial and temporal constraints for optimization can effectively avoid to fall into local optimum in advance and help to optimize to a better result.
In this paper, we further deliver the following contributions.
* We proposed a deep learning model to infer OD sequence structure which extracts the spatio-temporal constraints from time-varying traffic counts effectively.
* We novelly integrates deep learning method and bi-level framework to solve OD sequences estimation.
* Through our experiments, it is verified that the structural knowledge inferred by deep learning can provide great help for numerical optimization.
§ RELATED WORKS
OD matrix estimation can be mainly divided into static OD matrix estimation and dynamic OD matrices estimation.
§.§ static OD matrix estimation
For solving the static OD matrix estimation, gravity mode adopts a "gravitational behavior" for trip demand and builds a linear or nonlinear regression model<cit.>. The maximum likelihood technique<cit.> estimates the OD matrix over the sampled matrix on the presumption that OD pairs follow independent Poisson distributions. The entropy maximizing/information minimization method<cit.> seeks to select an estimated OD matrix that adds as little information as possible from traffic counts in order to match the underdetermined problem. The Bayesian method<cit.> additionally resolves OD matrices by maximizing the posterior probability, which utilizes a mix of the prior OD matrix and the observations. The maximizing/information minimization approach is a particular instance of Bayesian method when the prior is given only a minimal amount of confidence. A method that explicitly considers both observed flow biases and the target OD matrix is built using generalized least squares (GLS)<cit.>. All of the aforementioned techniques need to use the prior OD matrix, which may be out of date and lead to estimation bias. Additionally, travel time<cit.>, travel speed<cit.>, and turning proportions<cit.> are also employed directly in OD estimates due to the availability of massive traffic data and traffic simulation.
§.§ dynamic OD matrices estimation
Static OD matrix estimation can only estimate OD for a specific period. While a period is divided into multiple intervals, researchers propose dynamic OD matrix estimation, which aims to estimate the OD matrices in the corresponding time intervals and estimate the entire OD sequence further by using road traffic and other information. However, as mentioned above, the lagged flow may vary in successive intervals according to different traffic conditions, so the estimation by simply applying the static OD matrix is no longer applicable. Dynamic OD matrix estimation is divided into two types: off-line case and on-line case<cit.>.
§.§.§ Off-line case
The off-line method mainly discusses the direct estimation of the OD sequence when only given the corresponding observation sequence. <cit.> discusses off-line estimation methods, such as the simultaneous and sequential method, and further<cit.> proposes a correction method based on the average OD matrix. The simultaneous method focuses on establishing an estimator to optimize all OD matrix slices simultaneously, its idea is similar to the static OD matrix estimation. Compared to the simultaneous method, the sequential method only estimates one matrix slice at a time and infers the next OD matrix slice based on the historical OD matrices that have been estimated. The average-based correction method estimates an average OD over the observation period, then estimates coefficients, which are multiplied by the average OD to obtain the final OD sequence. Since each interval corresponds to an OD matrix slice, there are large amounts of parameters with non-negative constraints to be estimated. The main optimization methods proposed by researchers are the gradient projection method<cit.> and the Simultaneous Perturbation Stochastic Approximation(SPSA)<cit.>. Similar to static OD estimation, some researchers have integrated various traffic observations, such as vehicle transit time, travel speed, etc., into the estimators<cit.> to improve the accuracy.
§.§.§ On-line case
The on-line method has been widely studied. Unlike the off-line method, it requires the historical periodic OD sequence and the current observation. Traditional modeling methods include Kalman filter CF<cit.><cit.>, LSQR algorithm<cit.>, Canonical Polyadic (CP) decomposition<cit.>. <cit.> used principal component analysis (PCA) combine several machine learning models, and <cit.><cit.> utilize Dynamic Mode Decomposition (DMD) based method. At the same time, given the excellent performance of deep learning in prediction, some researchers use deep learning methods for the dynamic prediction of OD sequences. For example, <cit.> used a Graph Neural Network(GNN) to capture the spatial topology information of the graph structure and combined it with the traditional CF algorithm to improve the accuracy of the OD matrix prediction. <cit.><cit.> used a Recurrent Neural Network(RNN) to capture the temporal features of prior OD sequence evolution to predict the current OD matrix. The primary relationship is that the off-line methods can be used to provide a better initialization for on-line methods<cit.>.
§ PRELIMINARY
§.§ Definitions
As shown in Table 1, an OD node is a cluster created by grouping the intersections of road sections in the city network, and the roads connect the same OD pair are aggregated to one road section (see Fig. 3). Considering a city network consists of n_od OD nodes and n_sec road sections. We use a vector of length n_sec to represent OD node n_i, with n_ij=1 if road section j enters node n_i, -1 if it exits from n_i, and 0 if it is not connected to n_i.
During an estimation period divided into I equal intervals t = 1, 2, 3, . . ., I, and an obsevation period divided into o equal intervals τ = 1,2,3,...,o. ϵ_τ denotes the traffic counts of all road sections during observation interval τ. The traffic trips from n_i to n_j during estimation interval t is denoted by M_ijt. In addition, we transform the OD sequence into a tensor denoted by T∈ℝ^1× I × n_od^2× 1, and ε denotes a vector concatenated from ϵ_1 to ϵ_o. For ε_i ∈ε, we normalize its traffic count on each road section with respect to all observation intervals as D_E_ij = E_ij/∑_i^o∑_j^n_sec E_ij.
Production flows p̃_t is used to represent a vector records the number of trips leaving each node during estimation interval t as p̃_t=∑_j^n_odM_ijt. And attraction flows ã_t is to used to represent a vector records the number of trips that arrive at each node during estimation interval t as ã_t=∑_i^n_odM_ijt.
Additionally, the global production flows p is a vector concatenated from p̃_1 to p̃_I, and the global attraction flows a is a vector concatenated from ã_1 to ã_I, which are used to denotes the production flows and attraction flows of OD sequence, respectively. Moreover, we normalize their flow on each OD node with respect to all OD nodes and all estimation intervals as d_pi=p_i/∑_k^I × n_odp_k and d_ai=a_i/∑_k^I × n_oda_k, respectively.
In the inference phase, d̅_p or a denotes the inferred distribution of production flow or attraction flow from deep learning model when being trained. And the best inferred distribution is denoted by d_p or a^*.
In the optimization phase, T̂ denotes the tensor of optimized OD sequence from bi-level framewrok at each iteration, and correspondingly, d̂_p or a denotes the optimized distribution of production flows or attraction flows.
§.§ Bi-level framework
In the bi-level framework, the estimation will start from an initialized OD matrix T̃. In the lower level, for each observation interval, trips in every OD pair are allocated to the road sections in an analytical or simulative way, and then an allocation tensor P is obtained. P_τ t represents a matrix of proportion that OD matrix T_t allocated to road sections during the observation interval τ, and P_τ tT_t indicates the corresponding traffic counts. In the upper layer, the traffic counts assigned by T_1,T_1,...,T_t are summed to get the traffic counts ϵ_τ as shown in Fig 1. Optimizing the least squares estimator reduces the gap between optimized traffic counts ϵ̂_τ and real traffic counts ϵ_τ throughout the whole observation period τ=1,2,...,o to find a better estimated OD sequence. By iteration repeats the alternation of upper and lower levels, the final estimated OD matrix sequence is obtained when converges. The least squares estimator is formulated as follows:
c
Z(T̂)= min_T̂_1 ≥0,...,T̂_I ≥0 ∑_τ=1^o 1/2(ϵ_τ-ϵ̂_τ)^T(ϵ_τ-ϵ̂_τ)
where ϵ̂_τ= ∑_k=1^t P_τk T̂_k
The assignment tensor P∈ℝ^ o × I × n_sec× n_od^2 , can be derived by analysis, for example by taking into account stochastic user equilibrium on traffic counts or by simulation using a simulator like SUMO<cit.>. It displays the ratio of each OD trip to each road section during estimation intervals. Similar to<cit.>, our assignment tensor P is computed using a back-calculation technique based on traffic counts generated by the simulator during each iteration:
P=
( [ P_11 … P_1I; P_21 … P_2I; ⋮ ⋱; P_o1 … ( [ p_11 …p_1 n_od^2; p_21 …p_2 n_od^2; ⋮ ⋮; p_n_sec 1 …p_n_sec n_od^2; ] ) _oI ] )
§.§ Optimization
Considering Eq(1) is a problem with abundant non-negativity constraints on T, the gradient projection method is commonly used<cit.>. The idea is to evaluate the directional derivatives of the objective function at the current point, and to obtain a descent direction by making a projection on the non-negativity constraints.
It is worth noting that we adjust the update step T^k+1:=T^k⊕ (λ^k⊙d^k) to T^k+1:=T^k⊙ (e+λ^k⊙d^k). It has been proved that compared with the ordinary update step, this adjustment significantly improves the optimization speed of OD matrices<cit.>, since it proposed that update steps for larger variables should be greater. For the upper bound of step size λ^k_max at itertion k, we give the corresponding adjustment λ^k_max=min{-1/d^k_i|∀ i:d^k_i<0}. Since at the (k+1)^th iteration, 𝐀_2T^k > 0 and ensure 𝐀_2T^k+1 > 0, let 𝐀_2T^k⊙ (e+λ^k⊙d^k) > 0, which implies e+ λ^k⊙d^k>0. Therefore, λ^k<-1/d^k_i,∀ i:d^k_i<0, and then we have λ^k_max=min{-1/d^k_i},∀ i:d^k_i<0.
Finally, we search for the optimal step size λ^*k at iteration k based on Eq(2) and then determine the executable step size λ^k according to λ^*k and λ^k_max, that is, if λ^*k<λ^k_max, set λ^k=λ^*k; otherwies, set λ^k=λ^k_max.
min_λ^kZ(T̂^k⊙ (e+λ^k⊙d^k))
Where ⊙ denotes the element-wise product and e is a tensor of 1s with the same dimension as T̂^k.
§ METHOD
The pipline of our proposed method will be described in detail in this section, including sampling the probe flow to compose datasets, training and inference of NN models, and combining inferred spatial-temporal structural distributions into numerical optimization.
§.§ Probe Traffic Sampling
Firstly, as presented in our previous work on static OD estimation, most of the important trips are in a small part of OD pairs, leat to the values of other OD pairs are relatively small. So the production and attraction flows of an OD matrix will be uneven in the reality, and this property implies the structural information of the OD matrix. Moreover, as we mentioned in Part 1, we infer distributions rather than real numbers to reflect the structure, so the exact value is not a concern. Therefore, it is feasible to set a sparse matrix with limited values of non-zero elements to reconstruct the specific structure information of an real OD matrix. To this end, we set m to represent the maximum value of OD pairs and make it relatively small to speed up the sampling process.
Secondly, since the traffic congestion will cause lag problem, we need some probe vehicles to explore the traffic congestion. We form a original matrices sequence(OMS) by collecting m vehicles from these important OD pairs (with the number of trips >m) of the real OD sequence as shown in the left part of Fig. 2, these trips can be a combination of various data, such as car-hailing service data and GPS data since these vehicles can all be seen as probe vehicles. Then, we resample each OD paris on a scale of 0.0-1.0 from OMS to obtain a dataset composed of generated OD sequences, and calculate the corresponding global distributions d_p, d_a and D_E. The advantage of doing so is that, although we sample from a small number m of vehicles, it also can reconstruct the relationship between various traffic counts distribution and its corresponding golobal distributions under the real traffic conditions.
Finally, D_E are used as inputs of NN, d_p or d_a are used as labels, we form the dataset (D_E,d_p) and (D_E,d_a) and train the two models separately.
§.§ Dynamic Distribution Inference
The spatio-temporal evolution of traffic counts can effectively characterize the OD sequence. We utilize traffic counts of observation interval t to t × k+ δ (for t ∈ [0,I-1], k=o/I, δ>0) as input to characterize the OD matrix of estimation interval t. If δ>k, the observations overlap between two estimation intervals as shown in Fig 1, it indicates that the OD matrix of the current estimation interval t will affect the traffic counts of the following t × k+δ observation intervals due to the lag problem.
In order to obtain the global distributions, we need to consider the mutual influence relationship of each OD node in spatial and temporal. For example, if the trips of node n_i at observation interval t × k+δ_1 and node n_j at t × k+δ_2 both need to pass through road section e at t × k+δ when the traffic count has been given(δ >δ1; δ>δ_2), so there will be pairwise spatial-temporal dependencies between OD nodes.
§.§.§ DCGRU
In deep learning field, many studies have shown that the GNN+RNN based method can extract the spatio-temporal features well<cit.>. Therefore, we choose the DCGRU model as the feature extractor of each OD matrix. It combines Diffusion Convolutional Network(DCN, a spatial-based GNN model) and Gated Recurrent Unit(GRU,an improved RNN model) and has been studied to have an outstanding performance in capturing the long term spatio-temporal evolution characteristics of traffic flow<cit.> .
DCN can be adapted to deal with dependency between objects in non-Euclidean spaces according to node features χ and the adjacency matrix W. The K-step graph diffusion convolution is calculated to extract the upstream and downstream spatial dependencies between their surrounding K-order neighbor nodes and form an integrated graph signal, which is formulated as below:
c
χ_:,e⋆g=∑_k=0^K-1(θ_k,1(D_o^-1W)^k+θ_k,2(D_I^-1W^T)^k)χ_:,e
for e ∈{1,...,n_sec }
χ_⋆ g is the graph signal obtained after each OD node fuses the K order neighbors in every dimension e. θ∈ℝ^K × 2 are learnable parameters for the filter. D_o represents the diagonal matrix of the out-degree matrix of graph g, and D_I represents the diagonal matrix of the in-degree matrix, D_o^-1W, D_I^-1W^T represent the transition matrices of the diffusion process and the reverse one respectively.
GRU sends the integrated graph signal into the cell orderly to capture the temporal dependencies. The update process of feature in the GRU cell is as follows:
c
r^(τ)=σ(Θ_r ⋆g[χ^(τ),H^(τ-1)]+b_r)
u^(τ)=σ(Θ_u ⋆g[χ^(τ),H^(τ-1)]+b_u)
C^(τ)=tanh(Θ_C ⋆g[χ^(τ),(r^(τ) ⊙H^(τ-1)]+b_c)
H^(τ)=u^(τ) ⊙H^(τ-1)+(1-u^(τ)) ⊙C^(τ)
where χ^(τ), H^(τ) denote the input and output at observation interval τ, r^(τ), u^(τ) are reset gate and update gate at τ respectively. ⋆ g denotes the diffusion convolution defined in Eq(3) and Θ_r, Θ_u, Θ_C are learnable parameters for the corresponding filters.
§.§.§ Multihead Self-Attention(MSA)
Standard qkv self-attention(SA) compute the attention weight A over all value of elements v. A is based on the of query q and key k of elements , and calculate the pairwise dependency between two elements of input sequence ζ∈ℝ^(n_od^2+1) × d.
Self-attention (SA)<cit.> computes the attention weight A overall value of elements v to calculate the pairwise dependency between two elements of input sequence ζ∈ℝ^(I · n_od^2) × d where A is based on the query q and key k of elements.
[q,k,v]=ζU_qkv, U_qkv∈ℝ^d × 3d_h
A=softmax(qk^T / √(d_h))
SA(ζ)=Av
We projected concatenated outputs from MSA, which runs h SA procedures concurrently. The dimensions are kept constant by setting d_h to d/h, where h is the number of heads.
c
MSA(ζ)=[SA_1(ζ);SA_2(ζ);...;SA_h(ζ)]U_msa,
U_msa ∈ℝ^h ·d_h ×d
U_qkv and U_msa above are learnable parameters
§.§.§ Distribution Learner
We element-wise multiply the node vector n_i with the distribution of traffic counts ϵ_τ to obtain the feature of OD node i at observation τ. In order to get all OD nodes fetures χ during all the o observation intervals. We expand dimention of N to n_od× 1 × n_sec and D_E to 1 × o × n_sec, respectively, then do the broadcast operation ⊗ on these two tensor as shown in Eq(12) to get the shape of χ as (o, n_od, n_sec). Then, divide by the dimension o of the tensor, and take the t × k to t × k+δ (for t ∈ [0,I-1], k=o/I, δ>0) each time to obtain the input tensor (δ, n_od, n_sec). In our case, we estimate an OD sequence of 12 hours, with an estimation interval every hour and an observation interval every 10 minutes. So we have I=12, o=72, k=6.
χ=N⊗D_E
Subsequently, taking {χ^(t × k),...,χ^(t × k+δ) | t ∈ [0,I-1]} as the input of the DCGRU module orderly, a hidden tensor H={H^(t × k+δ) | t ∈ [0,I-1]} is as the output with its shape is (I, n_od, n_sec). Then expanded H by the dimension I to obtain (I × n_od, n_sec) as the input of the Transformer encoder. With a shared position embedding parameters added before and after Transformer encoder, we refer to the output as mutual vectors, there are I × n_od mutal vectors for each represents the spatio-temporal mutual information of corresponding OD node. Lastly, we operate element-wise addtion ⊕ on all these mutual vectors to one vector * containing the global infromation, and do inner production between each mutual vector and the global information vector * to give a scalar for each node, then perform the softmax operations to obtain the inferred global distribution d̅_p or a.
For model training, we choose Jensen-Shannon Divergence(JSD) as the loss function as Eq(13), which measures the distance between two distributions symmetrically.
c
Loss_p or a=JS(d̅_p or a||d_p or a)
§.§ Estimator
Like other studies in static OD estimation and the off-line case of dynamic OD sequence estimation, we adopt the bi-level framework. The difference is the least squares approach at the upper level merely seeks to reduce the gap between observed and simulated traffic counts, which only facilitates numerical similarity between estimated and real OD sequence. Therefore, we incorporate the optimization with the best inference global distributions d_p^* and d_a^* and choose KLD<cit.> as the objective function as following.
c
R(T̂)=min_T̂_1 ≥0,...,T̂_I ≥0 αN(T̂)+(1-α) S(T̂)
N(T̂)=∑_τ=1^o 1/2(ϵ_τ-ϵ̂_τ)^T(ϵ_τ-ϵ̂_τ)
S(T̂)=KL(d̂_p||d^*_p)+KL(d̂_a||D^*_a)
where ϵ̂_τ= ∑_k=1^t P_τk T̂_k
Our optimization process is shown in Algorithm 2. It is worth noting that since the optimization is alone the approximate distributions rather than the real distributions, the structure should not be optimal when S(T̂) converges. Therefore, we then slack the structure constraint (set α=1), which further does only numerical optimization and leads to a better point.
§ EXPERIMENTS AND RESULTS
We test our method on a large-scale real city network with a synthetic dataset. The effectiveness of NN for distributional inference is first validated. Then, a comparative experiment is used to demonstrate the advantage of our optimization method compared with traditional numerical optimization. The project uses Python programming and relies on the TensorFlow system<cit.> for gradient calculation and NN modeling. Sklearn library<cit.> and Scipy<cit.> are used for clustering and optimization program, respectively.
§.§ Study Network
We selected the 400km^2 area around Cologne, Germany, as our study network and used SUMO as the simulator program to test on a large-scale city network<cit.>. 71368 road sections and 31584 intersections make up the network (Fig. 3(a)). We employ the K-means<cit.> technique to gather OD nodes on the network according to Euclidean distance. In this case, we select n_od=15 (Fig. 3(b)), and we aggregate the directed road sections from 782 to 64.
§.§ Dataset
The synthetic data closely comparable to the actual conditions of urban traffic serves as the ground truth. Refer to <cit.> for more information.
As shown in Table 2, we took the ground truth OD matrix for 12 hours from 6:00 am to 18:00 pm from a 24-hours traffic simulation. Set the OD estimation interval to 1 hour and the traffic counts observation interval to 10 minutes. For the input of the DCGRU module, we set δ=4 to indicate that the traffic counts of the four flowing observation intervals from the current OD estimation interval are used as input to characterize the current OD matrix. Therefore, the whole length of observation intervals is 76 (12×6 since each OD estimation interval is with six observation intervals)
We set m=50 and sample the OMS from the ground truth OD matrix. Then resample from the original matrices sequence to get various OD sequences, and generate the corresponding observation traffic counts ϵ to form a data set (D_E,d_p) and (D_E,d_a), the size of each is 10k sample pairs.
In an 8:2 ratio, we split the dataset for training and validation. A model is trained using the training data, and its generalization performance is assessed using the validated data.
§.§ OD estimating evaluation
Referring to<cit.>, we use the numerical value indicators RMSN(T̂_t,T_t)<cit.> and structural indicators ρ(T̂_t,T_t)<cit.> to measure the gap between the estimated OD matrix T̂_t and the real OD matrix T_t.
RMSN(T̂_t,T_t)=√(n_od^2∑^n_od^2_i(T_ti-T̂_ti)^2)/∑^n_od^2_iT_ti
ρ(T̂_t,T_t)=(T_t-μ)^T(T̂_t-μ̂)/√((T_t-μ)^T(T_t-μ))√((T̂_t-μ̂)^T(T̂_t-μ̂))
where μ∈ℝ^n_od^2_≥0 is a vector with each element value equal to the mean of T_t, and μ̂, μ̃ corresponds to T̂_t and T̃_t, respectively.
§.§ Parameter settings
head number h 6
encoder layer N 2
learning rate r 1E-4
dimention d 128
diffusion convolution step K 2
maximum trips value m 50
OD estimation intervals I 12
observation intervals o 12 and 72
sequence length δ 4
§.§ Results and analyze
§.§.§ Training
Firstly, As shown in Fig. 4. We set the real OD sequence as our test set, the test set curve converges indicates our sampled training data is effective for model training. Moreover, the results of the test set is not as good as training and validation set since the distribution of real data and sampled data is not perfectly consistent. Which is a common problem in deep learning and implies it can be further alleviated through more appropriate sampling methods.
Secondly, we tested the impact of different observation intervals length δ on the inferred results. Our experiment in Fig. 5 shows that, when δ=4 the inferenced results by NN model is the best. It means our model does not completely utilize all the observation intervals information during one estimation interval(should be δ=6), and when there is an overlap (δ>6), the results get worse. It indicates that our current model has not effectively extract all the information introduced by longer observation intervals, more advanced NN model could further improve the inference results.
§.§.§ Distribution Inference
Firstly, Fig. 6(a) shows the best inference results of our NN model on the global production distribution d_p^*, it reflects the ability of our model to give the global spatio-temporal distribution of OD sequence. Secondly, we pick an OD matrix at 12-13 o’clock from the global distriution, as shown in Fig. 6(b), which shows that the model has a good performance on spatial distribution inference, it clearly reflects the proportional between different OD nodes in the same OD matrix. Finally, we picked the distributions of four OD nodes during the whole estimation period, as shown in Fig. 7, the model infers the temporal evolution trend of specific OD node also very well, moreover in Fig. 7 we demonstrate with different size of proportions(from le-2 to le-5) and it shows our model has a high sensitivity that, except 1e-5, since these proportions are too small to cause the model to infer the node has a proportions of 0.
§.§.§ Optimization
Firstly, we compare two traditional optimization methods. These methods only use numerical optimization with two different observation interval sets: one hour and 10 minutes, respectively. The one hour setting indicates there are 12 observation intervals since we have 12 hours during the whole estimation period, so we refer to this method as Traditional(o=12). And another is Traditional(o=72) since there are 72 observation intervals for 10 minutes setting. Although 10 minutes setting has the smallest scale and can provides more constraints to alleviate the underdetermined problem, as mentioned in<cit.>, it also introduces more noise in the optimization process, which may leading to a poor optimization result. As seen in Table 3, the final optimization result obtained by the 10 minutes setting is worse than the one hour setting, which indicates that the noise in the optimization process offset the benefits of excessive observation intervals. Secondly, our method namd Ours with d^* can mining significant information from these small sacle observation data to infer accurate global distributions d_p^* and d_a^* and then guide optimization process to find a better result. In addition, we also provide the results obtained from Ours with d, which utilizing the real global distribution d_p and d_a as guide, and provides the upper bound of our method, which is shown in Fig. 8. It can indicate that our method has a potent extendibility, such as through better data sampling methods or better models to infer more accurate distributions and get better optimization results.
In Fig. 9, we demonstrate the estimation results of four OD pairs along time series with four different orders of magnitude(1e1 to 1e4), respectively, to illustrate that our method has better performance on OD squence estimation tasks of different orders of magnitude. In there we only choose Traditional(o=12) and Ours with d^* since Traditional(o=12) has the better performance in numerical methods, and Ours with d^* is an only prctical way to use approximate distributions d^*_p and d^*_a from our distribution learner. And in Fig. 10 we show x-y plots of the estimation results for four different OD matrices.
As shown in Table 4, it is worth noting that the KLD of the best distribution d_p^* inferred by the our NN model is 0.0711. Our actual results from the final optimization is KL(d̂_p||d_p)=0.0560. And KLD of the best distribution d_p^* inferred by the our NN model is 0.0826, and our actual results from the final optimization is KL(d̂_a||d_a)=0.0591, which are much smaller than the approximate distributions be given since the approximate distributions are only used as a guide in the optimization process. Moreover, our convergence speed is also faster than traditional methods. It indicates that using the approximate distribution as a guide can help the optimization converge to a better point and faster.
§ CONCLUTION
In this paper, we propose a deep learning method to learn the relationship between traffic counts and OD structure information through mining information from a small amount of mixed observational traffic data. Then we use this structure information to constrain the traditional least squares numerical optimization method based on the bi-level framework. We validate that our method outperforms traditional numerical-only optimization methods on 12 hours of synthetic data from a large-scale city. Moreover, we present the space for future improvement of our method by improving the sampling method and deep learning models.
IEEEtran
|
http://arxiv.org/abs/2307.04313v1 | 20230710025609 | Unknotted Curves on Seifert Surfaces | [
"Subhankar Dey",
"Veronica King",
"Colby T. Shaw",
"Bülent Tosun",
"Bruce Trace"
] | math.GT | [
"math.GT",
"57K30, 57K10"
] |
Department of Mathematics
University of Alabama
Tuscaloosa
AL
[email protected]
Department of Mathematics
University of Texas Austin
Austin
TX
[email protected]
School of Mathematics
Georgia Institute of Technology
Atlanta
GA
[email protected]
Department of Mathematics
University of Alabama
Tuscaloosa
AL
[email protected]
Department of Mathematics
University of Alabama
Tuscaloosa
AL
[email protected]
[2010]57K33, 57K43, 32E20
We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in S^3, and in particular those that are unknotted or slice in S^3. We completely characterize all such curves for most twist knots: they are either positive or negative braid closures; moreover, we determine exactly which of those are unknotted. A surprising consequence of our work is that the figure eight knot admits infinitely many unknotted essential curves up to isotopy on its genus one Seifert surface, and those curves are enumerated by Fibonacci numbers. On the other hand, we prove that many twist knots admit homologically essential curves that cannot be positive or negative braid closures. Indeed, among those curves, we exhibit an example of a slice but not unknotted homologically essential simple closed curve. We further investigate our study of unknotted essential curves for arbitrary Whitehead doubles of non-trivial knots, and obtain that there is a precisely one unknotted essential simple closed curve in the interior of the doubles' standard genus one Seifert surface. As a consequence of all these we obtain many new examples of 3-manifolds that bound contractible 4-manifolds.
Unknotted Curves on Seifert Surfaces
Bruce Trace
====================================
§ INTRODUCTION
Suppose K ⊆ S^3 is a genus g knot with Seifert Surface Σ_K. Let b be a curve in Σ_K which is homologically essential, that is it is not separating Σ_K, and a simple closed curve, that is it has one component and does not intersect itself. Furthermore, we will focus on those that are unknotted or slice in S^3, that is each bounds a disk in S^3 or B^4. In this paper we seek to progress on the following problem:
Characterize and, if possible, list all such b's for the pair (K, Σ_K) where K is a genus one knot and Σ_K its Seifert surface.
Our original motivation for studying this problem comes from the intimate connection between unknotted or slice homologically essential curves on a Seifert surface of a genus one knot and 3-manifolds that bound contractible 4-manifolds. We defer the detailed discussion of this connection to Section <ref>, where we also provide some historical perspective. For now, however, we will focus on getting a hold on the stated problem above for a class of genus one knots, and as we will make clear in the next few results, this problem is already remarkably interesting and fertile on its own.
§.§ Main Results.
A well studied class of genus one knots is so called twist knot K=K_t which is described by the diagram on the left of Figure <ref>. We note that with this convention K_-1 is the right-handed trefoil T_2,3 and K_1 is the figure eight knot 4_1. We will consider the genus one Seifert surface Σ_K for K=K_t as depicted on the right of Figure <ref>.
The first main result in this paper is the following.
Let t≤ 2. Then the genus one Seifert surface Σ_K of K=K_t admits infinitely many homologically essential, unknotted curves, if and only if t=1, that is K is the figure eight knot 4_1.
Indeed, we can be more precise and characterize all homologically essential, simple closed curves on Σ_K, from which Theorem <ref> follows easily. To state this we recall an essential simple closed curve c on Σ_K can be represented (almost uniquely) by a pair of non-negative integers (m,n) where m is the number of times c=(m,n) runs around the left band and n is the number of times it runs around the right band in Σ_K. Moreover, since c is connected, we can assume gcd(m,n) = 1. Finally, to uniquely describe c, we adopt the notation of ∞ curve and loop curve for a curve c, if the curve has its orientation switches one band to the other and it has the same orientation on both bands, respectively (See Figure <ref>).
Let K=K_t be a twist knot and Σ_K its Seifert surface as in Figure <ref>. Then;
* For K =K_t, t≤ -1, we can characterize all homologically essential simple closed curves on Σ_K as the closures of negative braids in Figure <ref>. In case of the right-handed trefoil K_-1=T_2,3, exactly 6 of these, see Figure <ref>, are unknotted in S^3. For t<-1, exactly 5 of these, see Figure <ref>, are unknotted in S^3.
* For K=K_1=4_1, we can characterize all homologically essential simple closed curves on Σ_K as the closures of braids in Figure <ref>. A curve on this surface is unknotted in S^3 if and only if it is (1) a trivial curve (1,0) or (0,1), (2) an ∞ curve in the form of (F_i+1,F_i), or (3) a loop curve in the form of (F_i,F_i+1), where F_i represents the i^th Fibonacci number, see Figure <ref>.
For twist knot K=K_t with t>1 the situation is more complicated. Under further hypothesis on the parameters m,n we can obtain results similar to those in Theorem <ref>, and these will be enough to extend the theorem entirely to the case of K=K_2, so called Stevedore's knot 6_1 (here we use the Rolfsen's knot tabulation notation). More precisely we have;
Let K=K_t be a twist knot and Σ_K its Seifert surface as in Figure <ref>. Then;
* When t>1 and m<n, we can characterize all homologically essential simple closed curves on Σ_K as the closures of positive braids in Figure <ref>(a)(b). Exactly 5 of these, see Figure <ref>, are unknotted in S^3.
* When t>1 and m>n.
* If m-tn>0, then we can characterize all homologically essential simple closed curves on Σ_K as the closures of negative braids in Figure <ref> and <ref>. Exactly 5 of these, see Figure <ref>, are unknotted in S^3.
* If m-n<n and the curve is ∞ curve, then we can characterize all homologically essential simple closed curves on Σ_K as the closures of positive braids Figure <ref>. Exactly 5 of these, see Figure <ref>, are unknotted in S^3.
* For K =K_2=6_1, we can characterize all homologically essential simple closed curves on Σ_K as the closures of positive or negative braids. Exactly 5 of these, see Figure <ref>, are unknotted in S^3.
What Theorem <ref> cannot cover is the case t>2, m>n and m-tn<0 or when m-n<n and the curve is a loop curve. Indeed in this range not every homologically essential curve is a positive or negative braid closure. For example, when (m,n)=(5,2) and t=3 one obtains that the corresponding essential ∞ curve, as a smooth knot in S^3, is the knot 5_2, and for (m,n)=(7,3) and t=3, the corresponding knot is 10_132 both of which are known to be not positive braid closures–coincidentally, these knots are not unknotted or slice. Moreover we can explicitly demonstrate, see below, that if one removes the assumption of “∞” from part 2(b) in Theorem <ref>, then the conclusion claimed there fails for certain loop curves when t>2. A natural question is then whether for knot K =K_t with t>2, m>n and m-tn<0 or m-n<n loop curve, there exists unknotted or slice curves on Σ_K other than those listed in Figure <ref>? A follow up question will be whether there exists slice but not unknotted curves on Σ_K for some K=K_t? We can answer the latter question in affirmative as follows:
Let K=K_t be a twist knot with t>2 and Σ_K its Seifert surface as in Figure <ref> and consider the loop curve (m,n) with m=3, n=2 on Σ_K. Then this curve, as a smooth knot in S^3, is the pretzel knot P(2t-5, -3, 2). This knot is never unknotted but it is slice (exactly) when t=4, in which case this pretzel knot is also known as the curious knot 8_20.
We note that the choices of m,n values made in Theorem <ref> are somewhat special in that they yielded an infinite family of pretzel knots, and that it includes a slice but not unknotted curve. Indeed, by using Rudolph's work in <cit.>, we can show (see Proposition <ref>) that the loop curve (m,n) with m-n=1, n>2 and t>4 on Σ_K, as a smooth knot in S^3, is never slice. The calculation gets quickly complicated once m-n>1, and it stays an open problem if in this range one can find other slice but not unknotted curves.
We can further generalize our study of unknotted essential curves on minimal genus Seifert surface of genus one knots for the Whitehead doubles of non-trivial knots. We first introduce some notation. Let P be the twist knot K_t embedded (where t=0 is allowed) in a solid torus V⊂ S^3, and K denote an arbitrary knot in S^3, we identify a tubular neighborhood of K with V in such a way that the longitude of V is identified with the longitude of K coming from a Seifert surface. The image of P under this identification is a knot, D^±(K,t), called the positive/negative t–twisted Whitehead double of K. In this situation the knot P is called the pattern for D^±(K,t) and K is referred to as the companion. Figure <ref> depicts the positive -3–twisted Whitehead double of the left-handed trefoil, D^+(T_2,-3, -3). If one takes K to be the unknot, then D^+(K,t) is nothing but the twist knot K_t.
Let K denote a non-trivial knot in S^3. Suppose that Σ_K is a standard genus one Seifert surface for the Whitehead double of K. Then there is precisely one unknotted homologically essential, simple closed curves in the interior of Σ_K.
§.§ From unknotted curves to contractible 4-manifolds.
The problem of finding unknotted homologically essential curves on a Seifert surface of a genus one knot is interesting on its own, but it is also useful for studying some essential problems in low dimensional topology. We expand on one of these problems a little more. An important and still open question in low dimensional topology asks: which closed oriented homology 3-sphere [A homology 3-sphere/4-ball is a 3-/4- manifold having the integral homology groups of S^3/B^4.] bounds a homology 4-ball or contractible 4-manifold (see <cit.>). This problem can be traced back to the famous Whitney embedding theorem and other important subsequent results due to Hirsch, Wall and Rokhlin <cit.> in the 1950s. Since then the research towards understanding this problem has stayed active. It has been shown that many infinite families of homology spheres do bound contractible 4-manifolds <cit.> and at the same time many powerful techniques and invariants, mainly coming from Floer and gauge theories <cit.> have been used to obtain constraints.
In our case, using our main results, we will be able to list some more homology spheres that bound contractible 4-manifolds. This is because of the following theorem of Fickle <cit.>.
Let K be a knot in S^3 which has a genus one Seifert surface F with a primitive element [b]∈ H_1(F) such that the curve b is unknotted in S^3. If b has self-linking s, then the homology sphere obtained by 1/(s± 1) Dehn surgery on K bounds a contractible [Indeed, this contractible manifold is a Mazur-type manifold, namely it is a contractible 4-manifold that has a single handle of each index 0, 1 and 2 where the 2-handle is attached along a knot that links the 1-handle algebraically once. This condition yields a trivial fundamental group.] 4-manifold.
This result in <cit.> was generalized to genus one knots in the boundary of an acyclic 4–manifold W, and where the assumption on the curve b is relaxed so that b is slice in W. This will be useful for applying to the slice but not unknotted curve/knot found in Theorem <ref>.
The natural task is to determine self-linking number s, with respect to the framing induced by the Seifert surface, for the unknotted curves found in Theorem <ref> and <ref>. For this we use the Seifert matrix given by S = [ -1 -1; 0 t ] where we use two obvious cycles–both oriented counterclockwise–in Σ_K. Recall that, if c=(m,n) is a loop curve then m and n strands are endowed with the same orientation and hence the same signs. On the other hand for ∞ curve they will have opposite orientation and hence the opposite signs. Therefore, given t, the self-linking number of c=(m,n) loop curve is s=-m^2-mn+n^2t, and the self-linking number of (m,n) ∞ curve is s=-m^2+mn+n^2t. A quick calculation shows that the six unknotted curves in Figure <ref> for K_-1=T_2,3 share self-linking numbers s=-1, -3. As we will see during the proof of Theorem <ref> the infinitely many unknotted curves for the figure eight knot K_1=4_1 reduce (that are isotopic) to unknotted curves with s=-1 or s=1. The five unknotted curves in Figure <ref> for K_t, t<-1 or t>1, share self-linking numbers s=-1, t and t-2 (see <cit.> and references therein for some relevant work). Finally, Theorem <ref> finds a slice but not unknotted curve which is the curve (3,2) with t=4. One can calculate from the formula above that this curve has self-linking number s=1. Finally, the unique unknotted curve from Theorem <ref> has self linking s=-1. Thus, as an obvious consequence of these calculations and Theorem <ref> and its generalization in <cit.> we obtain:
Let K be any non-trivial knot. Then, the homology spheres obtained by
* -1/2 Dehn surgery on D^+(K,t)
* ±1/2 Dehn surgery on K_1=4_1
* -1/2 and -1/4 Dehn surgeries on K_-1=T_2,3
* -1/2 and 1/t±1 and 1/(t-2)±1 Dehn surgeries on K_t, t≠± 1
* 1/2 Dehn surgery on K_4
bound contractible 4-manifolds.
The 3-manifolds in part (3) are Brieskorn spheres Σ(2,3,13) and Σ(2,3,25); they were identified by Casson-Harer and Fickle that they bound contractible 4-manifolds. Also, it was known already that the result of 1/2 Dehn surgery on the figure eight knot bounds a contractible 4-manifold (see <cit.>) from this we obtain the result in part (2) as the figure eight knot is an amphichiral knot.
It is known that the result of 1/n Dehn surgery on a slice knot K⊂ S^3 bounds a contractible 4-manifold. To see this, note that at the 4-manifold level with this surgery operation what we are doing is to remove a neighborhood of the slice disk from B^4 (the boundary at this stage is zero surgery on K) and then attach a 2-handle to a meridian of K with framing -n. Now, simple algebraic topology arguments shows that this resulting 4-manifold is contractible.
It is a well known result that <cit.>; a nontrivial twist knot K=K_t is slice if and only if K=K_2 (Stevedore's knot 6_1). So, by arguments above we already know that result of 1/n surgery on K_2 bounds contractible 4-manifold for any integer n. But interestingly we do not recover this by using Theorem <ref>.
The paper is organized as follows. In Section <ref> we set some basic notations and conventions that will be used throughout the paper. Section <ref> contains the proofs of Theorem <ref>, <ref> and <ref>. Our main goal will be to organize, case by case, essential simple closed curves on genus one Seifert surface Σ_K, through sometimes lengthy isotopies, into explicit positive or negative braid closures. Once this is achieved we use a result due to Cromwell that says the Seifert algorithm applied to the closure of a positive/negative braid closure gives a minimal genus surface. This together with some straightforward calculations will help us to determine the unknotted curves exactly. But sometimes it will not be obvious or even possible to reduce an essential simple closed curve to a positive or negative closure (see Section <ref>, <ref> and <ref>). Further analyzing these cases will yield interesting phenomenon listed in Theorem <ref> and <ref>. Section <ref> contains the proof of Theorem <ref>.
§.§ Acknowledgments
We thank Audrick Pyronneau and Nicolas Fontova for helpful conversations. The first, second and third authors were supported in part by a grant from NSF (DMS-2105525). The fourth author was supported in part by grants from NSF (CAREER DMS-2144363 and DMS-2105525) and the Simons Foundation (636841, BT).
§ PRELIMINARIES
In this section, we set some notation and make preparations for the proofs in the next three sections. In Figure <ref> we record some basic isotopies/conventions that will be repeatedly used during proofs. Most of these are evident but for the reader's convenience we explain how the move in part (f) works in Figure <ref>. We remind the reader that letters on parts of our curve, as in part (e) of the figure, or in certain location is to denote the number of strands that particular curve has.
Recall also an essential, simple closed curve on Σ_K can be represented by a pair of non-negative integers (m,n) where m is the number of times it runs around the left band and n is the number of times it runs around the right band in Σ_K, and since we are dealing with connected curves we must have that m,n are relatively prime.
We have two cases: m>n or n>m. For an (m,n) curve with m>n, after the m strands pass under the n strands on the Seifert surface, it can be split into two sets of strands. For this case, assume that the top set is made of n strands. They must connect to the n strands going over the right band, leaving the other set to be made of m-n strands. Now, we can split the other side of the set of m strands into two sections. The m-n strands on the right can only go to the bottom of these two sections, because otherwise the curve would have to intersect itself on the surface. This curve is notated an (m,n) ∞ curve. See Figure <ref>(a). The other possibility for an (m,n) curve with m>n, has n strands in the bottom set instead, which loop around to connect with the n strands going over the right band. This leaves the other to have m-n strands. We can split the other side of the set of m strands into two sections. The m-n strands on the right can only go to the top of these two sections, because again otherwise the curve would have to intersect itself on the surface. The remaining subsection must be made of n strands and connect to the n strands going over the right band. This curve is notated as an (m,n) loop curve. See Figure <ref>(b). The case of (m,n) curve with n>m is similar. See Figure <ref>(c)&(d).
§ TWIST KNOTS
In this section we provide the proofs of Theorem <ref>, <ref> and <ref>. We do this in four parts. Section <ref> and <ref> contains all technical details of Theorem <ref>, Section <ref> contains details of Theorem <ref> and Section <ref> contains Theorem <ref> .
§.§ Twist knot with t<0
In this section we consider twist knot K=K_t, t≤ -1. This in particular includes the right-handed trefoil K_-1.
All essential, simple closed curves on Σ_K can be characterized as the closure of one of the negative braids in Figure <ref>.
It suffices to show all possible curves for an arbitrary m and n such that gcd(m, n) = 1 are
the closures of either braid in Figure <ref>. As mentioned earlier we will deal with cases where both m, n ≥1 since cases involving 0 are trivial. There are four cases to consider. The arguments for each of these will be quite similar, and so we will explain the first case in detail and refer to to the rather self-explanatory drawings/figures for the remaining cases.
Case 1: (m,n) ∞ curve with m>n>0. This case is explained in Figure <ref>. The picture on top left is the (m,n) curve we are interested. The next picture to its right is the (m,n) curve where we ignore the surface it sits on and use the convention from Figure <ref>(e). The next picture is an isotopy where we push the split between n strands and m-n strands along the dotted blue arc. The next three pictures are obtained by applying simple isotopies coming from Figure <ref>. For example, the passage from the bottom right picture to one to its left is via Figure <ref>(c). Finally, the picture on the bottom left, one can easily see that, is the closure of the negative braid depicted in Figure <ref>(a).
Case 2: (m,n) loop curve with m>n>0. By series isotopies, as indicated in Figure <ref>, the (m,n) curve in this case can be simplified to the knot depicted on the right of Figure <ref>, which is the closure of negative braid in Figure <ref>(b).
Case 3: (m,n) ∞ curve with n>m>0. By series isotopies, as indicated in Figure <ref>, the (m,n) curve in this case can be simplified to the knot depicted on the bottom left of Figure <ref>, which is the closure of negative braid in Figure <ref>(c).
Case 4: (m,n) loop curve with n>m>0. By series isotopies, as indicated in Figure <ref>, the (m,n) curve in this case can be simplified to the knot depicted on the right of Figure <ref>, which is the closure of negative braid in Figure <ref>(d).
Next, we determine which of those curves in Proposition <ref> are unknotted. It is a classic result due to Cromwell <cit.> (see also <cit.>) that the Seifert algorithm applied to the closure of a positive braid gives a minimal genus surface.
Let β be a braid as in Figure <ref> and K = β̂ be its closure. Let s(K) be the number of crossings and l(K) be the number of Seifert circles Seifert circles. Then;
(s(K), l(K))=
(m, |t|n(n-1) + (m - n)(m - n - 1) + n(m - n)) β as in Figure <ref>(a)
(m+n, (|t|+1)n(n-1) + (m - n)(m - n - 1) + nm+2n(m - n)) β as in Figure <ref>(b)
(n, (|t-1|)n(n-1) + (n - m)(n - m - 1) +m(m-1) + m(n - m)) β as in Figure <ref>(c)
(m+n, |t|n(n-1) + m(m - 1) + nm) β as in Figure <ref>(d)
Consider the braid β as in Figure <ref>(a). Clearly, it has m Seifert circles as β has m strands. Next, we will analyze the three locations in which crossings occur. First, the t negative full twists on n strands. Since each strand crosses over the other n-1 strands, we obtain |t|n(n-1) crossings. Second, the negative full twist on m-n strands produces additional (m-n)(m-n-1) crossings. Lastly, notice the part of β where m-n strands overpass the other n strands, and so for each strand in m-n strands we obtain an additional n crossings. Hence for K=β̂ we calculate:
l(β̂) = |t|n(n-1) + (m - n)(m - n - 1) + n(m - n).
The calculations for the other cases are similar.
We can now prove the first part of Theorem <ref>.
Proposition <ref> proves the first half of our theorem. To determine there are exactly six unknotted curves when t=-1 and five when t<-1, let B be the set containing the six and five unknotted curves as in Figure <ref> and <ref>, respectively. It suffices to show an essential, simple closed curve c on Σ_K where c ∉B, cannot be unknotted in S^3. We know by Proposition <ref>, c is the closure of one of the braids in Figure <ref> in S^3, where m,n ≥ 1, gcd(m,n) = 1. We show, case by case, that the Seifert surface obtained via the Seifert algorithm for curves c∉B in each case has positive genus, and hence it cannot be unknotted.
* Let c=(m,n) be the closure of the negative braid as in Figure <ref>(a) and Σ_c its Seifert surface obtained by the Seifert algorithm. There are m Seifert circles and by Proposition <ref>
l(c) = |t|n(n-1) + (m-n)(m-n-1) + n(m-n).
Hence,
g(Σ_c) = 1 + l - s/2= m(m-n-2) + n(|t|(n-1)+1) + 1/2.
If m=n+1, then we get g(Σ_c)=|t|n(n-1)/2 which is positive as long as n>1–note that when c=(2,1) we indeed get an unknotted curve. If m>n+1, then g(Σ_c)≥n(|t|(n-1)+1)+1/2>0 as long as n>0. So, c∉B is not an unknotted curve as long as m>n≥ 1.
* Let c=(m,n) be the closure of the negative braid as in Figure <ref>(b) and Σ_c its Seifert surface obtained by the Seifert algorithm. There are n+m Seifert circles and by Proposition <ref>
l(c) = (|t|+1)n(n-1) + (m - n)(m - n - 1) + nm+2n(m - n).
Hence,
g(Σ_c) = m(m+n-2)+n(|t|(n-1)-1)+1/2.
One can easily see that this quantity is always positive as long as n≥ 1. So, c∉B is not an unknotted curve when m>n≥ 1.
* Let c=(m,n) be the closure of the negative braid as in Figure <ref>(c) and Σ_c its Seifert surface obtained by the Seifert algorithm. There are n Seifert circles and by Proposition <ref>
l(c) = (|t|-1)n(n-1) + (n - m)(n - m - 1) + m(m-1)+m(n - m).
Hence,
g(Σ_c) = n(|t|(n-1)-m-1)+m^2+1/2.
This is always positive as long as m≥ 1 and |t|≠ 1–note that when c=(1,2) and |t|=1 we indeed get unknotted curve. So, c∉B is not an unknotted curve when n>m≥ 1.
* Let c=(m,n) be the closure of the negative braid as in Figure <ref>(d) and Σ_c its Seifert surface obtained by the Seifert algorithm. There are n+m Seifert circles and by Proposition <ref>
l(c) = |t|n(n-1) + m(m - 1) + nm.
Hence,
g(Σ_c) = |t|n(n-1)+m(m-2)+n(m-1)+1/2.
One can easily see that this quantity is always positive as long as m≥ 0. So, c∉B is not an unknotted curve when n>m≥ 1.
This completes the first part of Theorem <ref>.
§.§ Figure eight knot
The case of figure eight knot is certainly the most interesting one. It is rather surprising, even to the authors, that there exists a genus one knot with infinitely many unknotted curves on its genus one Seifert surface. As we will see understanding homologically essential curves for the figure eight knot will be similar to what we did in the previous section. The key difference develops in Case 2 and 4 below where we show how, under certain conditions, a homologically essential (m,n) ∞ (resp. (m,n) loop) curve can be reduced to the homologically essential (m-n, 2n-m) ∞ (resp. (2m-n, n-m) loop) curve, and how this recursively produces infinitely many distinct homology classes that are represented by the unknot, and we will show that certain Fibonacci numbers can be used to describe these unknotted curves. Finally we will show fort he figure eight knot this is the only way that an unknotted curve can arise. Adapting the notations developed thus far we start characterizing homologically essential simple closed curves on genus one Seifert surface Σ_K of the figure eight knot K.
All essential, simple closed curves on Σ_K can be characterized as the closure of one of the braids in Figure <ref> (note the first and third braids from the left are negative and positive braids, respectively).
The curves (1,0), (0,1) are clearly unknots. Moreover, because gcd(m,n)=1, the only curve with n=m is (1,1) curve, which is also unknot in S^3. For the rest of the arguments below, we will assume n>m or m>n. There are four cases to consider:
Case 1: (m,n) loop curve with m>n>0.
This curve can be turned into a negative braid following the process in Figure <ref>.
Case 2: (m,n) ∞ curve with m> n>0. As mentioned at the beginning, this case (and Case 4) are much more involved and interesting (in particular the subcases of Case 2c and 4c). Following the process as in Figure <ref>, the curve can be isotoped as in the bottom right of that figure, which is the closure of the braid on its left–that is the second braid from the left in Figure <ref>.
Case 3: (m,n) ∞ curve with n>m>0. This curve can be turned into a positive braid following the process in Figure <ref>.
Case 4: (m,n) loop curve with n> m>0. This curve can be turned into the closure of a braid following the process in Figure <ref>.
We next determine which of these curves are unknotted:
A homologically essential curve c characterized as in Proposition <ref> is unknotted if and only if it is (a) a trivial curve (1,0) or (0,1), (b) an ∞ curve in the form of (F_i+1,F_i), or (c) a loop curve in the form of (F_i,F_i+1).
Let c denote one of these homologically essential curve listed in Proposition <ref>. We will analyze the unknottedness of c in four separate cases.
Case 1. Suppose c=(m,n) is the closure of the negative braid in the bottom left of Figure <ref>. Note the minimal Seifert Surface of c, Σ_c, has (n)(m-n)+(m)(m-1) crossings and m Seifert circles. Hence;
g(Σ_c) = n(m-n)+(m-1)^2/2
This is a positive integer for all m,n with m>n. So c is never unknotted in S^3 as long m>n>0 .
Case 2. Suppose c is of the form in the bottom right of Figure <ref>. Since this curve is not a positive or negative braid closure, we cannot directly use Cromwell's result as in Case 1 or the previous section. There are three subcases to consider.
Case 2a: m-n=n. Because m and n are relatively prime integers, we must have that m=2, n=1, and we can easily see that this (2,1) curve unknotted.
Case 2b: m-n>n. This curve can be turned into a negative braid following the process in Figure <ref>. More precisely, we start, on the top left of that figure, with the curve appearing on the bottom right of Figure <ref>. We extend the split along the dotted blue arc and isotope m strands to reach the next figure. We note that this splitting can be done as by the assumption we have m-2n>0. Then using Figure <ref>(a) and further isotopy we reach the final curve on the bottom right of Figure <ref> which is obviously the closure of the negative braid depicted on the bottom left of that picture.
The minimal Seifert Surface coming from this negative braid closure contains m-n circles and (m-2n)n+(m-n)(m-n-1)
twists. Hence;
g(Σ_c)=(m-2n)n+(m-n)(m-n-2)+1/2.
This a positive integer for all integers m,n with m-n>n. So, c is not unknotted in S^3.
Case 2c: m-n<n. We organize this curve some more. We start, on the top left of Figure <ref>, with the curve that is appearing on the bottom left of Figure <ref>. We extend the split along the dotted blue arc and isotope m-n strands to reach the next figure, After some isotopies we reach the curve on the bottom left of Figure <ref>. In other words, this subcase of Case 2c leads to a reduced version of the original picture (top left curve in Figure <ref>), in the sense that the number of strands over either handle is less than the number of strands in the original picture.
This case can be further subdivided depending on the relationship between 2n-m and m-n, but this braid (or rather its closure) will turn into a (m-n, 2n-m) ∞ curve when m-n>2n-m:
Case 2c-i: 2n-m = m-n. This simplifies to 3n=2m. Because gcd(m,n)=1, this will only occur for m=3 and n=2, and the resulting curve is (1,1) ∞ curve. In other words here we observed that (3,2) curve has been reduced to (1,1) curve
Case 2c-ii: 2n-m > m-n. This means that we are dealing with a curve under Case 3, and we will see that all curves considered there are positive braid closures.
Case2c-iii: 2n-m < m-n. This means we are back to be under Case 2. So for m>n>m-n, the (m,n) ∞ curve is isotopic to the (m-n, 2n-m) ∞ curve. This isotopy series will be notated (m,n) ∼ (m-n, 2n-m). Equivalently, there is a series of isotopies such that (m-n, 2n - m) ∼ (m,n). If (k,l) denote a curve at one stage of this isotopy, then (k,l) ∼ ((k +l) + k, k+l). So, starting with k = l = 1, we recursively obtain:
(1,1) ∼ (3,2) ∼ (8,5) ∼ (21, 13) ∼ (55, 34) ∼⋯
In a similar fashion, if we start with k = 2, l = 1 we obtain:
(2,1) ∼ (5,3) ∼ (13,8) ∼ (34, 21) ∼ (89, 55) ∼⋯
Notice every curve c above is of the form c = (F_i + 1, F_i), i ∈ℤ_>0 where F_i denotes the i^th Fibonacci number. We will call these Fibonacci curves. We choose (1,1) and (2,1) because they are known unknots. As a result, this relation generates an infinite family of homologically distinct simple closed curves on Σ_K that are unknotted in S^3.
Case 3. Suppose a curve, c, is of the form (3), which is the closure of the positive braid depicted in the bottom left of Figure <ref>. An argument similar to that applied to Case 1 can be used to show c is never unknotted in S^3.
Case 4. Suppose c is of the form as in the bottom middle of Figure <ref>. Similar to Case 2, there are three subcases to consider.
Case 4a: m = n - m. Then 2m=n. Because gcd(m,n)=1, m=1 and n=2, resulting in unknot.
Case 4b: n-m>m. Then n-2m>0 and following the isotopies in Figure <ref>, the curve can be changed into the closure of positive braid depicted on the bottom right of that figure.
Identical to Case 2b, the curve c in this case is never unknotted in S^3.
Case 4c: m>n-m. Then 2m-n>0, and we can split the m strands into two: a n-m strands and a 2m-n strands.
This case can be further subdivided depending on the relationship between n-m and 2m-n, but this braid will turn into a (2m-n, n-m) loop curve when n-m>2m-n:
Case 4c-i: 2m-n = n-m.
This simplifies to 3m=2n. Because gcd(m,n)=1, this will only occur for m=2 and n=3, and the resulting curve is a (1,1) loop curve.
Case 4c-ii: n-m < 2m-n. This means that we are dealing with a curve under Case 1, and we saw that all curves considered there are negative braid closures.
Case 4c-iii: n-m > 2m-n. This means that we are back to be under Case 4. So for n>m>n-m, an (m,n) loop curve has the following isotopy series: (m,n) ∼ (2m-n,n-m). If (k,l) denote a curve at one stage of this isotopy, then the reverse also holds: (k,l) ∼ (k+l, (k+l)+l). As a result, much like Case 2c, we can generate two infinite families of unknotted curves in S^3:
(1,1) ∼ (2,3) ∼ (5,8) ∼ (13, 21) ∼ (34, 55) ∼⋯ and
(1,2) ∼ (3,5) ∼ (8,13) ∼ (21, 34) ∼ (55, 89) ∼⋯
Notice every curve c is of the form c = (F_i, F_i + 1), i ∈ℤ_>0. Finally, we show that this is the only way one can get unknotted curves. That is, we claim:
If a homologically essential curve c on Σ_K for K=4_1 is unknotted, then it must be a Fibonacci curve.
From above, it is clear that if our curve c is Fibonacci, then it is unknotted. So it suffices to show if a curve is not Fibonacci then it is not unknotted. We will demonstrate this for loop curves under Case 4. Let c be a loop curve that is not Fibonacci but is unknotted. Since it is unknotted, it fits into either Case 4a or 4c. But the only unknotted curve from Case 4a is (1,1) curve which is a Fibonacci curve, so c must be under Case 4c. By our isotopy relation, (m,n) ∼ (2m-n, n - m). So, the curve can be reduced to a minimal form, say (a,b) where (a,b) ≠ (1,1) and (a,b) ≠ (2,1). We will now analyze this reduced curve (a,b):
* If a = b, then (a,b) = (1,1); a contradiction.
* If a > b, then (a,b) is under Case 1; none of those are unknotted.
* If b - a < a < b, then (a,b) is still under Case 4c, and not in reduced form; a contradiction.
* If a < b - a < b, then (a,b) is under Case 4b; none of those are unknotted.
* If b-a = a<b, then (a,b) = (2,1); a contradiction.
So, it has to be that either (a,b) ∼ (1,1) or (a,b) ∼ (2,1). Hence, it must be that c = (F_i, F_i+1) for some i. The argument for the case where c is an ∞ curve under Case 2 is identical.
§.§ Twist knot with t>1–Part 1
In this section we consider twist knot K=K_t, t≥ 2, and give the proof of Theorem <ref>.
All essential, simple closed curves on Σ_K can be characterized as the closure of one of the braids in Figure <ref>.
It suffices to show all possible curves for an arbitrary m and n such that gcd(m, n) = 1 are
the closures of braids in Figure <ref>. Here too there are four cases to consider but we will analyze these in slightly different order than in the previous two sections.
Case 1: (m,n) ∞ curve with n>m>0. In this case the curve is the closure of a positive braid, and this is explained in Figure <ref> below. More precisely, we start with the curve which is drawn in the top left of the figure, and after a sequence of isotopies this becomes the curve in the bottom right of the figure which is obviously the closure of the braid in the bottom left of the figure. In particular, when n>m≥ 1, none of these curves will be unknotted.
Case 2: (m,n) loop curve with n>m>0. In this case too the the curve is the closure of a positive braid, and this is explained in Figure <ref> below. In particular, when n>m>1, none of these curves will be unknotted.
In the remaining two cases we will follow slightly different way of identifying our curves as braid closures. As we will see (which is evident in part (c) and (d) of Proposition <ref>) that the braids will not be positive or negative braids for general and m, n and t values. We will then verify how under the various hypothesis listed in Theorem <ref> these braids can be reduced to a positive or negative braids.
Case 3: (m,n) ∞ curve with m>n>0. We explain in Figure <ref> below how the (m,n) ∞ curve with m>n>0 is the closure of the braid in the bottom left of the figure. This braid is not obviously a positive or negative braid.
Case 3a (m,n) ∞ curve with m>n>0 and m-tn>0. We want to show the braid in the bottom left of Figure <ref> under the hypothesis that m-tn>0 can be made a negative braid. We achieve this in Figure <ref>. More precisely, in part (a) of the figure we see the braid that we are working on. We apply the move in Figure 7(f) and some obvious simplifications to reach the braid in part (d). In part (e) of the figure we re-organize the braid: more precisely, since m-tn>0 and m-n=m-tn+(t-1)n, we can split the piece of the braid in part (d) made of m-n strands as the stack of m-tn strands and set of t-1 n strands. We then apply the move in Figure <ref>(f) repeatedly (t-1 times) to obtain the braid in part (f). We note that the block labeled as “all negative crossings” is not important for our purpose to draw explicitly but we emphasize that each time we apply the move in Figure <ref>(f) it produces a full left handed twist between an n strands and the rest. Next, sliding -1 full twists one by one from n strands over the block of these negative crossings we reach part (g). After further obvious simplifications and organizations in parts (h)–(j) we reach the braid in part (k) which is a negative braid.
Case 3b (m,n) ∞ curve with m>n>0 and m-n<n. We want to show in this case the braid in the bottom left of Figure <ref> under the hypothesis that m-n<n can be made a positive braid (regardless of t value). This is achieved in Figures <ref>.
Case 4: (m,n) loop curve with m>n>0. The arguments for this case are identical Case 3 and 3a above. The (m,n) loop curve with m>n>0 is the closure of the braid that is drawn in the bottom left of Figure <ref>.
Case 4a (m,n) loop curve with m>n>0 and m-tn>0. We show the braid, which the (m,n) ∞ curve with m>n>0 is closure of, can be made a negative braid under the hypothesis m-tn>0. This follows very similar steps as in Case 3a which is explained through a series drawings in Figure <ref>.
Case 4b (m,n) loop curve with m>n>0 and m-n<n. Finally, we consider the (m,n) loop curve with m>n>0 and m-n<n. Interestingly, this curve for t>2 does not have to the closure of a positive or negative braid. This will be further explored in the next section but for now we observe, through Figure <ref>(a)-(c) that when t=2 the curve is the closure of a negative braid: The braid in (a) in the figure is the braid from Figure <ref>(d). After applying the move in Figure <ref>f, and simple isotopies we obtain the braid in (c) which is clearly a negative braid when t=2.
The proof of part (1) follows from Case 1 and 2 above. Part (2)a/b follows from Case 3a/b and Case 4a above. As for part (3), observe that when n>m by using Case 1 and 2 we obtain that all homologically essential curves are the closures of positive braids. When m>n, we have either m-2n>0 or m-2n<0. In the former case we use Case 3a and 4a to obtain that all homologically essential curves are the closures of negative braids. In the latter case, first note that m-2n<0 is equivalent to m-n<n, Now by Case 3b all homologically essential ∞ curves are the closures of positive braids, and by Case 4b all homologically essential loop curves are the closures of negative braids. Now by using Cromwell's result and some straightforward genus calculations we deduce that when m>n>1 or n>m≥ 1 there are no unknotted curves among (positive/negative) braid closures obtained in Case 1-4 above. Therefore, there are exactly 5 unknotted curves among homologically essential curves on Σ_K for K=K_t in Theorem <ref>.
§.§ Twist knot with t>1–Part 2
In this section we consider twist knot K=K_t, t≥ 3, and give the proof of Theorem <ref>.
We show that the loop curve (3,2) when t≥ 3 is the pretzel knot P(2t-5, -3,2). This is explained in Figure <ref>. The braid in (a) is from Figure <ref>(d) with m=3, n=2, where we moved (t-2) full right handed twists to the top right end. We take the closure of the braid and cancel the left handed half twist on the top left with one of the right handed half twists on the top right to reach the knot in (c). In (c)-(g) we implement simple isotopies, and finally reach, in (h), the pretzel knot P(2t-5, -3,2). This knot has genus t-1 (<cit.>[Corollary 2.7] , and so is never unknotted as long as t>1. This pretzel knot is slice exactly when 2t-5+(-3)=0. That is when t=4. The pretzel knot P(3,-3,3) is also known as 8_20. An interesting observation is that although P(2t-5, -3, 2) for t>2 is not a positive braid closure, it is a quasi-positive braid closure.
The (m,n) loop curve with m-n=1, n>3 and t> 4 is never slice.
By Rudoplh in <cit.>, we have that for a braid closure β̂ when k_+≠ k_-
g_4(β̂) ≥|k_+ - k_-| - n + 1/2
where β is a braid in n strands, and k_± is the number of positive and negative crossings in β. For quasi-positive knots, equality holds. In which case, the Seifert genus is also the same as the four ball (slice) genus.
Note that this formula can also be thought as a generalization to the Seifert genus calculation formula we used for positive/negative braid closures, since for those braids when, |k_+ - k_-| is the number of crossings and n, the braid number, is exactly the number of Seifert circles. Thus Rudoplh's inequality can also be used to state that the above calculations to rule out unknotted curves on various genus one Seifert surface can also be used to state that there are no slice knots other than the unknotted ones found.
Now for the loop curve c=(m,n) as in Figure <ref>(c), we have that
k_+ = (t-2)n(n-1), k_- = (m-n)(m-n-1) + 3(m-n)n
Hence, when m-n = 1, we get that k_- = 3n. Notice also that for n ≥ 3, t ≥ 4, we have k_+ > k_-. Thus, for n > 3, t > 4, m-n=1 we obtain c=β̂ is never slice as;
g_4(β̂=c) ≥(t-2)n(n-1) - 3n - m +1/2 = n((t-2)(n-1) - 4) > 0
It can be manually checked that the (4,3) loop curve when t = 3 is not slice either.
§ WHITEHEAD DOUBLES
In this section we provide the proof of Theorem <ref>
Let f:S^1× D^2→ S^3 denote a smooth embedding such that f(S^1×{0})=K. Set T=f(S^1× D^2). Up to isotopy, the collection of essential, simple closed, oriented curves in ∂ T is parameterized by
{mμ+nλ | m, n∈ℤ and gcd(m,n)=1}
where μ denotes a meridian in ∂ T and λ denotes a standard longitude in ∂ T coming from a Seifert surface. With this parameterization, the only curves that are null-homologous in T are ±μ and the only curves that are null-homologous in S^3∖int(T) are ±λ. Of course ±μ will bound embedded disks in T, but ±λ will not bound embedded disks in S^3∖int(T) as K is a non-trivial knot. In other words, the only compressing curves for ∂ T in S^3 are meridians.
Suppose now that C is a smooth, simple closed curve in the interior of T, and there is a smoothly embedded 2-disk, say Δ, in S^3 such that ∂Δ=C. Since C lies in the interior of T, we may assume that Δ meets ∂ T transversely in a finite number of circles. Initially observe that if Δ∩∂ T=∅, then we can use Δ to isotope C in the interior of T so that the result of this isotopy is a curve in the interior of T that misses a meridinal disk for T. Now suppose that Δ∩∂ T≠∅. We show, in this case too, C can be isotoped to a curve that misses a meridinal disk for T. To this end, let σ denote a simple closed curve in Δ∩∂ T such that σ is innermost in Δ. That is σ bounds a sub-disk, Δ' say, in Δ and the interior of Δ' misses ∂ T. There are two cases, depending on whether or not that σ is essential in ∂ T. If σ is essential in ∂ T, then, as has already been noted, σ must be a meridian. As such, Δ' will be a meridinal disk in T and C misses Δ'. If σ is not essential in ∂ T, then σ bounds an embedded 2-disk, say D, in ∂ T. It is possible that Δ meets the interior of D, but we can still cut and paste Δ along a sub-disk of D to reduce the number of components in Δ∩∂ T. Repeating this process yields that if C is smoothly embedded curve in the interior of T and C is unknotted in S^3, then C can be isotoped in the interior of T so as to miss a meridinal disk for T.
With all this in place, we return to discuss Whitehead double of K. Suppose that F is a standard, genus 1 Seifert surface for a double of K. See Figure <ref>. The surface F can be viewed as an annulus A with a a 1-handle attached to it. Here K is a core circle for A, and the 1-handle is attached to A as depicted in Figure <ref>
Observe that F can be constructed so that it lives in the interior of T. Now, the curve C that passes once over the 1-handle and zero times around A obviously misses a meridinal disk for T, and it obviously is unknotted in S^3. On the other hand, if C is any other essential simple closed curve in the interior of F, then C must go around A some positive number of times. It is not difficult, upon orienting, C can be isotoped so that the strands of C going around A are coherently oriented. As such, C is homologous to some non-zero multiple of K in T. This, in turn, implies that C cannot be isotoped in T so as to miss some meridinal disk for T. It follows that C cannot be an unknot in S^3.
10
CH
A. Casson and J. Harer, Some homology lens spaces which bound
rational homology balls, Pacific Journal of Mathematics 96 (1981),
no. 1, 23–36.
CG
A. Casson and C. McA. Gordon, On slice knot in dimension three, Proc. Smpos. Pure Math. XXXII Amer. Math. Soc. (1978), 39–53.
CD
T. D. Cochran C. W. Davis and , Counterexamples to Kauffman's conjectures on slice knots, Adv. Math. 274 (2015), 263–284.
Cr
P. R. Cromwell, Homogeneous links,
J. London Math. Soc. (series 2) 39 (1989), 535–552. 1002465
ET
J.B. Etnyre and B. Tosun, Homology spheres bounding acyclic smooth manifolds and symplectic fillings. , Michigan Math. Journal (2022).
Hirsch
M. W. Hirsch,
On imbedding differentiable manifolds in euclidean space, Ann. of Math. (2) 73 (1961), 566–571.
124915
Fickle
H. C. Fickle, Knots, Z-homology 3-spheres and
contractible 4-manifolds, Houston J. Math. 10 (1984), no. 4,
467–493. 774711
FintushelStern84
R. Fintushel and R. J. Stern, A μ-invariant one homology
3-sphere that bounds an orientable rational ball, Four-manifold theory
(Durham, N.H., 1982), Contemp. Math., vol. 35, Amer. Math. Soc.,
Providence, RI, 1984, pp. 265–268. 780582
Kirby:problemlist
R. Kirby, Problems in low dimensional manifold theory, Algebraic and
geometric topology (Proc. Sympos. Pure Math., Stanford Univ.,
Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Amer.
Math. Soc., Providence, R.I., 1978, pp. 273–312. 520548
KimLee
D. Kim and J. Lee, Some invariants of pretzel links, Bull. Austral. Math. Soc., 75 2007, 253–271
Manolescu:T
C. Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology
and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016),
no. 1, 147–176. 3402697
rudolph
L. Rudoplh
Quasipositivity as an obstruction to sliceness
Bulletin of the American Mathematical Society, 29, 1993
Rohlin
V. A. Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 160 (1965), 549–551. 0184246
Rohlin:3manembedding
V. A. Rohlin, The embedding of non-orientable three-manifolds into
five-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 160
(1965), 549–551. 0184246
Stern
R. Stern, Some Brieskorn spheres which bound contractible manifolds,
Notices Amer. Math. Soc (25) (1978).
St
A. Stoimenow, Positive knots, closed braids and the Jones polynomial,
Ann. Scuola Noem. Sup. Pisa Cl. Sci. (5) Vol. II, (2003) 237–285. 2004964
tosun:survey
B. Tosun, Stein domains in ℂ^2 with prescribed boundary, Adv. Geom. 22(1) (2022), 9–22. 4371941
Wall:embedding
C. T. C. Wall, All 3-manifolds imbed in 5-space, Bull. Amer.
Math. Soc. 71 (1965), 564–567. 175139
Zeeman
E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115
(1965), 471–495. 195085
|
http://arxiv.org/abs/2307.05662v1 | 20230711175118 | Charge Transfer and Zhang-Rice Singlet Bands in the Nickelate Superconductor $\mathrm{La_3Ni_2O_7}$ under Pressure | [
"Wéi Wú",
"Zhihui Luo",
"Dao-Xin Yao",
"Meng Wang"
] | cond-mat.str-el | [
"cond-mat.str-el",
"cond-mat.supr-con"
] |
APS/123-QED
[Corresponding author: ][email protected]
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
Recently, high-Tc superconductivity is reported in the bulk nickelate La_3Ni_2O_7 at pressures above 14 GPa. Here, we study an eleven-band Hubbard model with hopping parameters derived from ab initio calculations of La_3Ni_2O_7, by employing large scale determinant quantum Monte Carlo and cellular dynamical mean-field theory methods.
Our result suggests that the superexchange couplings in this system are comparable to that of cuprates.
The system is a charge transfer insulator as hole concentration becomes four per site at large Hubbard U. Upon hole doping, two low-energy spin-singlet bands emerge in the system exhibiting distinct correlation properties: while the one composed of the out-of-plane Ni-d_3z^2-r^2 and O- orbitals demonstrates strong antiferromagnetic correlations and narrow effective bandwidth, the in-plane singlet band consisting of the Ni-d_x^2-y^2 and O-/ orbitals is in general more itinerant. Over a broad range of hole doping, the doped holes occupy primarily the d_x^2-y^2 and / orbitals, as the d_3z^2-r^2 and orbitals retain underdoped. We propose an effective t-J model to capture the relevant physics and discuss the implications of our result for comprehending the La_3Ni_2O_7 superconductivity.
Charge Transfer and Zhang-Rice Singlet Bands in the Nickelate Superconductor La_3Ni_2O_7 under Pressure
Meng Wang
August 12, 2023
=========================================================================================================
Introduction -
Since the discovery of cuprate superconductors <cit.>, understanding and searching for novel high transition temperature (high-T_c) superconductors <cit.> has been one of the major topics in the condensed matter physics. The discovery of infinite layer nickelate superconductor <cit.> marks a notable recent advancement, although in which the superconductivity (SC) has been only observed in thin films on substrates <cit.>, but not yet in bulk samples <cit.>. The very recently discovered bulk superconductor La_3Ni_2O_7 under high pressures <cit.> , which exhibits a remarkable T_c of ∼ 80 Kelvins, then represents a significant breakthrough in this field.
As revealed by the density-functional theory (DFT) calculations <cit.>, a hallmark of the nickelate bi-layer La_3Ni_2O_7 <cit.> is the activating of both 3 and 3 orbitals in vicinity of Fermi level <cit.>. This distinct feature may lead to superconductivity that differs significantly from the cuprates and infinite layer nickelates. From a theoretical perspective, several crucial questions arise. First, to understand the driving force behind the SC, it is necessary to elucidate the magnetic exchange couplings <cit.> among the four active e_g orbitals in the NiO_2 bi-layer. Furthermore, the e_g orbitals of La_3Ni_2O_7 under pressure possess a large hole concentration, as
the nominal valence of Ni is Ni^2.5+ here, indicating an average of 1.25 holes per e_g orbital. This high hole filling level is on the verge of quenching SC by overdoping in the context of cuprates. Therefore, resolving the distributions of the holes in the Ni-3, Ni-3, and the correlated O-2p orbitals is crucial for understanding the correlation effects in La_3Ni_2O_7 under pressure.
To address above questions, here we study an 11-band Hubbard model that includes four 3 / 3 orbitals of nickel, and seven most relevant 2p orbitals of oxygen in the NiO_2 bi-layer per site. We carry out determinant quantum Monte Carlo simulations (DQMC) <cit.> and cellular dynamical mean-field theory (CDMFT) <cit.> calculations in the normal state of the system.
Our result suggests that the superexchange couplings in this system are in general comparable to that in cuprates <cit.>, supporting a magnetic correlation origin of the high T_c superconductivity in La_3Ni_2O_7 under pressure. We show that at large Hubbard U, the system is a charge-transfer insulator <cit.> in the Zaanen-Sawatzky-Allen (ZSA) scheme<cit.> at half-filling ( i.e., four holes per site). Upon hole doping, the spin-singlet band associating with the vertical Ni- - O- orbitals possesses strong antiferromagnetic correlations, as where a small hole doping level is retained. In contrast, the singlet band that consists of the in-plane Ni-d_x^2-y^2 and O- / orbitals, drawing an analogy to the Zhang-Rice singlet band (ZRSB) in cuprates, exhibits a higher propensity for hole doping and greater itinerancy. We discuss the interplay between the two spin-singlet bands and propose an effective t-J model that considers the leading-order exchange couplings in La_3Ni_2O_7 under pressure.
Model and Methods -
To fully take into account the superexchange couplings in La_3Ni_2O_7 at pressure, here we consider an 11-band
Hubbard model <cit.> that can be written as,
H = ∑_i,j,α,β,σt_i,j,α,βd_i ασ^†c_j βσ+∑_i,j,α,β,σt_i,j,α,βc_i ασ^†c_j βσ
+∑_i ασ(ϵ_α-μ)n_i ασ^d +∑_i ασ(ϵ_α-μ)n_i ασ^c
+∑_i αU_ddn_i α↑^d n_i α↓^d+∑_i, α≠β,σ , σ 'U_dd'n_i ασ^dn_i βσ '^d-∑_i ασE_dcn_i ασ^d
where t_i,j,α, β denote hoppings between electrons on sites (i,j) and orbital (α, β)( can be either Ni-d or O-p orbitals). d^†_α,i,σ (c^†_α,i,σ) is the creation operator for electrons on α∈ 3d (∈ 2p) orbital. ϵ_α is the site-energy of α - orbital. U_dd is the Hubbard interaction between two electrons on the same d-orbital ( or ) and U_dd' is for that on two different d-orbital. E_dc is the double counting (DC) term <cit.> to be subtracted in the DQMC or CDMFT. Here we use the Held's formula <cit.>: E_dc = 1/3(U_dd+2U_dd')(n^0_d -0.5), with n^0_d being the occupation number of d-orbitals in the non-interacting limit, n^0_d ≈ 2.16.
We have carefully checked that this DC term
does not shift
the non-interacting Fermi surface (FS) significantly <cit.>. We adopt the hopping parameters and site-energies proposed in Ref. <cit.>, which is obtained by downfolding the DFT result in the maximally localized Wannier orbitals. See Fig. <ref> for all the hopping parameters and site-energy values. In line with DFT result <cit.>, we assume that the chemical potential μ = 0 in above Hamiltonian corresponds to the single crystal La_3Ni_2O_7 material at pressures > 14GPa, without considering other potential doping effects (oxygen deficiency for example). We will also vary μ to explore regimes with different hole concentrations that defined as n_h = ( 22- ∑_α , σ n_α, σ^d)/ 4, namely, the average number of holes per d- orbitals per site. n_h = 1 corresponds to half-filling in our study. In this work, we neglect the Hund's coupling and keep U_dd' = 0.7 U_dd when U_dd is varied for the 11-band Hubbard model. Here we use two typical values of U_dd, U_dd=7 eV and U_dd=9 eV, where no qualitative difference is found between the two results. Below we use electron volt (eV) as the energy unit throughout the paper. For the calculations on cuprate, we employ a canonical set of parameters for the three-band Hubbard model <cit.> (one orbital and two / orbitals in the CuO_2 plane): t_pd = 1.39, t_pp = 0.64, t^'_pp = 0.103, Δ_dp≡ϵ_d - ϵ_p = 2.6, U_dd = 8.5, E_DC = 3.12. This set of parameters is assumed to be most relevant for the LSCO compound, which has been used in different studies of cuprates <cit.>.
For DQMC simulation, we use a two dimensional 6 × 6 × 11=396 orbitals lattice with periodic conditions for the 11-band Hubbard model, on which we have verified that the finite size effects are negligible in the parameter regime we study. For CDMFT study, we carry out computations in the normal state, where the 2 × 2× 11=44 orbitals cluster effective impurity model is used. The Hirsch-Fye quantum Monte Carlo (HFQMC) is used as impurity solver. The time discretions are Δτ = 0.0625 (DQMC) and Δτ = 0.078 (HFQMC).
Superexchanges- We first discuss the property of the magnetic exchange couplings in the system. As shown in Fig. <ref>, there are a few hopping processes can give rise to significant superexchanges. In the atomic limit of the charge-transfer picture, the spin singlet state of two Ni-d electrons acquires an energy gain of J = -4t^4_pd/(U_dd+Δ_pd)^2 × (1/U_dd+1/U_dd+Δ_pd) over the spin triplet states, where Δ_pd = ϵ_d - ϵ_p, and t_pd is the hopping between Ni-d and O-p orbitals. In principle, inclusion of the hopping integrals between O-p orbitals (t_pp) and interaction effects may modify this superexchange coupling <cit.>. The numbers shown in Fig. <ref>a are the magnitudes of spin correlation function ⟨ S_i,α· S_j, β⟩ for a few pairs of neighboring d- orbitals at T=0.25, where S_i,α is the spin operator at site i and orbital α. This result profiles the fact that there are three types of main exchange couplings in the system: the inter-layer (IT) on-site - antiferromagnetic exchange [ ⟨ S · S ⟩=-0.113(2)] dominates the exchange couplings in the system. Then it is the intra-layer (IR) nearest-neighboring -
exchange [⟨ S · S ⟩=-0.054(1)]. The intra-layer - superexchange, in contrast, is significantly weaker than the aforementioned two [⟨ S · S ⟩=-0.017(1)].
Finally, we note that the intra-layer - exchanges are less than 1/20 of inter-layer - coupling, hence can be neglected in further analysis. It is worthy noting that in CDMFT at lower temperatures (T ∼ 0.1), we observe similar relative strengths of the magnetic correlations at half-filling(not shown).
Fig. <ref>b and <ref>c show respectively ⟨ S_i,α· S_j, β⟩ as a function of temperature at half-filling, and spin structure factor S_α,β(Q) as a function of n_h at T=0.3, with comparison from the result of 3-band Hubbard model of cuprates. As one can see that the intra-layer - antiferromagnetic correlations are suggested to be in general comparable to its cuprate counterpart, while the inter-layer - coupling
seems to be essentially stronger than the former. This result implies that the antiferromagnetic correlations between inter-layer orbitals could be at the origin of the observed superconductivity in La_3Ni_2O_7 under pressure. Fig. <ref>a and <ref>b show ⟨ S_i,α· S_j, β⟩ respectively as a function of U_dd at n_h = 1.25, and as a function of n_h at U_dd = 7 in CDMFT, from which one sees that varying the value of U_dd in the range of (4 ∼ 9) eV does not change substantially the magnetic correlations between the inter-layer orbitals. More importantly, as shown in Fig. <ref>b, it does not vanish until a huge hole doping p = n_h-1 > 0.4 is approached.
Charge transfer insulator- We now focus on the metal- insulator transition in the 11-band Hubbard model. Fig. <ref> displays the hole filling per Ni-d orbital n_h as a function of hole chemical potential μ_h ≡ - μ. As one can see that the DQMC result (dots) of n_h exhibit an inflection point around n_h =1 ( μ_h ∼ -1.6) with small values of compressibility ∂ n_h / ∂μ_h, implying the forming of a charge gap. CDMFT result are obtained at lower temperatures (diamonds), where one indeed see a flat plateau in the n_h curve between μ_h ∈ (-1.6 ∼ -2.4), indicating the opening of a charge gap at half-filling. In Fig.<ref>, DQMC and CDMFT results at the same temperature T=0.3 are shown to be in good agreement. To understand the nature of this insulating behavior, we further study how the hole concentration changes in different orbitals as a function μ_h in the Inset of Fig. <ref>. In other words, as chemical potential μ_h is increased from the half-filling value μ_h ∼ -1.6, holes can be added into different Ni-d and O-p orbitals of the system, which is denoted by Δ n^α_h in the Inset of Fig. <ref>. As one can see that the doped holes go primarily to the oxygen orbitals, which unambiguously points to the charge-transfer nature of the insulating state at half-filling of this system. Similar to cuprates <cit.>, here orbital also has a sizable portion of the doped holes. It is remarkable that, however, the hole content of orbital almost do not change with μ_h for μ_h < 0 ( or for hole doping p ≲ 22% ). We note that in cuprates, a smaller portion of holes residing on cations in general indicates a larger superexchange and higher superconducting T_c <cit.>.
Zhang-Rice singlet band-
In Fig. <ref>a, and Fig. <ref>b, we plot the local density of states (DOS) for Ni- and Ni- orbitals respectively at U_dd=9. That of the in-plane O-/ orbitals, and of the out-of-plane O orbitals are also plotted along respectively. Here n_h ≈ 1.05 , T = 0.1. Near the Fermi level, the local DOS are shown to have mixed weights of Ni-d and O-p orbitals. To be more specific, as depicted in Fig. <ref>a, the low energy DOS of and O- / orbitals share similar structure (peak-dip-peak) near Fermi level, suggesting the presence of a Zhang-Rice singlet band<cit.> (ZRSB-I) with correlated -/ electrons. Likewise, in Fig. <ref>b, the Ni- orbital and O- orbitals also exhibit
similar structure (a single narrow peak) in the low-energy DOS, suggesting the formation of another singlet band (ZRSB-II) along the c-axis. It is remarkable that the vertical ZRSB-II singlet band are much narrower in bandwidth comparing to ZRSB-I, which suggests a more localized nature of the orbital.
There are a few points we would like to emphasize. First, the vertical singlet band is quite different from a conventional ZRSB in cuprates. In cuprates, a doped hole at oxygen sites hybridize with a Cu^2+ hole in terms of the superposition of four Op hole states adjacent to the Cu^2+ iron, forming a spin singlet. Then the singlet moves effectively in the antiferromagnetic background of Cu^2+ lattice with a bandwidth being (2 ∼ 3)eV <cit.>. Here the vertical singlet states of Ni--O- have a narrow bandwidth, and they barely interact with each other directly. Instead, the doped holes in the in-plane / orbitals can hybridize with the vertical singlets via - / hopping (t_2 = 0.75), or via the - hopping (t_5=0.49, t_7=0.43 , see also Fig. <ref>). As a result, the vertical singlet band may behave more like scattering centers with antiferromagntic characteristics in the system.
Finally, upon heavy hole doping, one can expect that the in-plane ZRSB will be destroyed, and, in contrast, the vertical singlet band can maintain intact, due to the imbalanced distributions of the doped holes in the two charge-transfer bands.
In Table-I we inspect specifically the hole filling level of the system at μ =0, which may arguably correspond to that of the real material of La_3Ni_2O_7 under pressure. As one can see that, the 11-band Hubbard model is about 21 ∼ 26% hole doped at μ =0, with small variations up to the value of Hubbard U and temperatures T. This result roughly coincides with nominal average doping level (=25%) of the e_g orbitals in La_3Ni_2O_7. We find that the hole doping in the out-of-plane and orbitals combined, p = (n^OP_h - 1) ≈ ( 5%∼ 8 %) , while that of the in-plane and / orbitals, p = (n^IP_h -1 ) is about ≈ 40% at μ = 0. Given this large value of hole doping of n^IP_h , one may expect that the in-plane and / orbitals can be seen as itinerant orbitals in La_3Ni_2O_7.
However, we find that in CDMFT at μ=0, the magnetic correlations between intra-layer - orbital is weak but not vanishing ( see Fig. <ref>b). It is notable that the strange metal (SM) phase can be very sensitive to weak magnetic correlations <cit.>. Hence, if the observed strange metal state in La_3Ni_2O_7 at P > 18 GPa is related to also the magnetic correlations in orbitals, then the latter may cannot be simplified as pure itinerant.
Effective t-J model- Based on our study above, we propose a four-band t-J model to describe the low-energy physics of the La_3Ni_2O_7 under pressure. The proposed Hamiltonian can be written as ℋ = ℋ_0+ℋ_J with ℋ_0=∑_ kσΨ_ kσ^†H( k)Ψ_ kσ, which reads,
H(k)_1,1 = H(k)_3,3 = -2t^x_1 [ cos(k_x) +cos(k_y)]
- 4t^x_2 cos(k_x)cos(k_y) + ϵ_x
H(k)_2,2 = H(k)_4,4 = -2t^z_1 [ cos(k_x) +cos(k_y)]
- 4t^z_2 cos(k_x)cos(k_y)+ ϵ_z
H(k)_1,2 = H(k)_3,4 = -2t^xz [ cos(k_x) -cos(k_y)]
H(k)_2,4 = -2V_⊥ [ cos(k_x) -cos(k_y)]
where Ψ_σ=(d_x_1 σ,d_z_1σ,d_x_1σ,d_z_2σ)^T, denoting the annihilation operators of and orbitals in the two NiO_2 layer. The ℋ_0 part is taken from the down-folded tight-binding model from Luo et al's work in Ref. <cit.>, namely,
t^x_1 ≈ 0.5 , t^x_2 ≈ 0.07, t^z_1 ≈ 0.11, t^x_2 ≈ 0.02, t^xz = -0.24 , V_⊥ = 0.64 . Note that the renormalization factors g_t <cit.> associating the corresponding hopping amplitudes in Eq. <ref> are not explicitly written down.
For the interacting part ℋ_J, we consider three main magnetic exchanges terms,
ℋ_J = J_1 ∑_ i ( S_i, z_1 S_i, z_2 -1/4 n_i,z_1n_i,z_2)
+ J_2 ∑_⟨ i,j⟩ , α = x_1,x_2 ( S_i, α S_j, α -1/4 n_i, αn_j, α )
+ J_3 ∑_⟨ i,j⟩
α,β = (x_1, z_1) / (x_2 ,z_2) ( S_i, α S_j, β -1/4 n_i, αn_j, β )
where J_1 captures the exchange couplings between the on-site inter-layer orbitals, J_2 the exchanges between the intra-layer orbitals on nearest neighboring (NN) sites, and finally J_3 for the intra-layer - exchanges on NN sites. Considering our magnetic correlation results compared with LSCO cuprate <cit.>, typical vales of the antiferromagnetic exchange couplings can be set around J_1 ∼ 0.18 eV, J_2 ∼ 0.09 eV , and J_3 ∼ 0.03 eV. Note that if the derived effects from Hund's coupling are considered <cit.>, then J_3 may be enhanced to be comparable to J_2.
Discussion and Conclusion - After the discovery of high-T_c superconductivity in La_3Ni_2O_7 under pressure <cit.>, a number of effective interacting models have been proposed to study the pairing symmetry <cit.>, as inspired by the ab initio calculations <cit.> or phenomenological insights. These effective models ignore the oxygen degrees of freedom, incorporating the magnetic correlations between Ni orbitals in terms of direct exchanges. Hence, the charge transfer property, which has shown to be one of the key ingredients in determining T_c of cuprate superconductors <cit.>, as well as the superexchange couplings, are beyond the scope of those effective models. In this work, we have studied an 11-band Hubbard model including both Ni-3d orbitals and relevant O-2p orbitals. We reveal the relative strengths of the superexchange couplings between different Ni-3d orbitals. We find two spin-singlet bands in the system with strikingly different hole concentrations and correlation strengths. These results suggest that the strong antiferromagnetic correlations within the inter-layer orbitals might be the driving force of the SC in La_3Ni_2O_7. The role of the more itinerant orbital in SM and SC states, owing to the remnant magnetic correlations within and the specific geometry of the α , β - sheets of the Fermi surface <cit.>, might be complicated though. Finally, we propose a four band t-J model with three different exchange coupling J_1, J_2, J_3 . Whether this t-J model can be further simplified, for example, by neglecting J_2 or J_3 terms <cit.>, can be scrutinized in further experimental or theoretical studies on the strange metal state <cit.> or the superconducting phase in La_3Ni_2O_7 under pressure.
Note added - During the preparation of this work, we become aware of an experimental work <cit.> probing the optical response in La_3Ni_2O_7, which suggests the proximity to Mottness of the electron correlations in the material. This finding is in accordance with our doped charge-transfer insulator description of the La_3Ni_2O_7.
Acknowledgment- We thank Hualei Sun, Mi Jiang, K. Le Hur, and Xunwu Hu for useful discussions. W.W is indebted to A. -M. Tremblay for useful discussions and insightful suggestions. W.W. acknowledge help from Dong Meng in preparing the illustrations in Figure 1. Work at Sun Yat-Sen University was supported by the National Natural Science Foundation of China (Grants No.12274472, No. 92165204, No.12174454, No.11974432), the National Key Research and Development Program of China (Grants No. 2022YFA1402802, 2018YFA0306001), the Guangdong Basic and Applied Basic Research Foundation (Grants No. 2022A1515011618, No. 2021B1515120015), Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices (Grant No. 2022B1212010008), Shenzhen International Quantum Academy (Grant No. SIQA202102), and Leading Talent Program of Guangdong Special Projects (201626003). We
acknowledge the support from GuangZhou National Supercomputing Center (Tianhe-II).
|
http://arxiv.org/abs/2307.04655v1 | 20230710155459 | Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters | [
"Francesco Benetti",
"Andrea Lapi",
"Giovanni Gandolfi",
"Balakrishna S. Haridasu",
"Luigi Danese"
] | astro-ph.CO | [
"astro-ph.CO"
] |
§ INTRODUCTION
Galaxy clusters constitute the largest bound structures in the Universe, with dark matter (DM) masses M∼ 10^14–15 M_⊙
and sizes extending out to R∼ a few Mpcs.
Most of the baryons are in the form of a hot diffuse gas, referred to as the intracluster medium (ICM), with a mass ratio over the DM very close to the cosmic
fraction Ω_b/Ω_M≈ 0.16 <cit.>.
The density n(r) and temperature T(r) distributions of the ICM throughout the cluster can be probed thanks to the copious X-ray powers
emitted by the ICM via thermal Bremsstrahlung and high-excitation lines <cit.>.
The inferred high average temperatures k_B T∼ several keVs and low average number densities make the ICM
the best plasma in the Universe ever, with thermal to electrostatic energy ratios k_B T/e^2 n^1/3∼ 10^12.
In addition, the pressure distribution p(r) can be probed thanks to the Sunyaev–Zel'dovich (SZ; <cit.>) effect, arising when the hot ICM electrons Compton upscatter the CMB photons crossing the cluster, tilting the latter's black-body spectrum toward high energies. In the microwave band, such a tilt mimics a diminution of the CMB temperature proportional to the Comptonization parameter y ∝∫ dℓ p(r), which encompasses the line-of-sight integral of the pressure profile. Combining X-ray and SZ data allows one to reconstruct the ICM thermodynamic profiles throughout most of the cluster volume, from the center to a few times R_500 or even beyond the virial boundaryHereafter, R_Δ indicates the radius where the average DM density is Δ times the critical density ρ_ c(z) at the redshift z of the cluster..
[-25]In massive and sufficiently relaxed clusters, the ICM is expected to settle in hydrostatic equilibrium within the overall gravitational potential well mainly provided by the DM component. Under this assumption, the gas density profile reconstructed from X-rays and the gas pressure profile from SZ data can be combined to probe the shape of the DM gravitational potential and check whether this is consistent with the DM density run extracted from N-body simulations in the ΛCDM cosmology. This is the rationale of many investigations aimed at exploiting galaxy clusters to probe modified gravity scenarios <cit.>, which have been developed to solve cosmological problems such as the origin of dark energy <cit.>, and/or to alleviate small-scales issues of the standard cold DM paradigm <cit.>.
In the latter vein, a prototypical example of such theories is the modified Newtonian dynamics (MOND) framework, which was originally designed to explain galactic dynamics through a modification of Newtonian gravity (or, more generally, Newton’s second law) that comes into action at accelerations well below a definite universal threshold; in its original formulation, DM was not included, and baryons were the only source of the gravitational field.
Although MOND can properly fit galactic RCs <cit.>, its performances at the scales of galaxy clusters are somewhat debated <cit.>.
More connected with the present work, in the last few years, various authors have put forward the idea that fractional calculus (i.e., the field of mathematics dealing with differentiation and integration of noninteger order) could be exploited to formulate modified gravity theories <cit.>. A relevant example is the theory of Newtonian fractional–dimensional gravity by <cit.>, which introduces a generalized law of Newtonian gravity in a spatial dimension smaller than three, representing the local effective Hausdorff dimension of the matter distribution. Another approach by <cit.> relies on multifractional spacetimes with variable Hausdorff and spectral dimensions directly inspired from
quantum gravity theories. The framework by <cit.> directly modifies the Laplacian operator in the Poisson equation to alter the dynamics followed by a test particle in a given gravitational well; a similar route is followed by <cit.>, using fractional Fourier derivatives. All these theories adopt a MONDian viewpoint where DM is not present, and the galaxy kinematics is interpreted as a pure geometrical effect.
Recently, in <cit.>, we suggested that the DM component itself may originate fractional gravity. In such a framework, the DM component exists, but the gravitational potential associated to its density distribution is determined by a modified Poisson equation including fractional derivatives (i.e., derivatives of noninteger type), which are meant to describe nonlocal effects; as such, this scenario is substantially different from the above theories where baryonic matter emulates DM-like effects via modifications of gravity. In <cit.>, we showed that DM in fractional gravity worked very well for reproducing the kinematics of disk-dominated galaxies, especially dwarfs. In addition, we found preliminary evidence that the strength of fractional effects tends to weaken toward more massive systems; however, the latter finding is still subject to large uncertainties since the rotation curves of massive spirals were not probed out to radii large enough for the DM contribution to clearly emerge.
In the present work, we aim to extend our previous investigation to much larger scales and test fractional gravity in galaxy clusters. Our aim is twofold: (i) perform an independent sanity check that it can accurately describe the distributions of the ICM in clusters; (ii) derive a clear-cut trend for the strength of its effects over an extended DM mass range, from dwarf galaxies to galaxy clusters. To this purpose, we forward model the density and pressure distributions of the ICM, working out the hydrostatic equilibrium equation in fractional gravity. Such theoretical framework is then compared with data from the XMM-Newton Cluster Outskirts Project (X-COPSee
<https://dominiqueeckert.wixsite.com/xcop/about-x-cop>.; <cit.>), which consists of 12 clusters with well-observed X-ray and SZ data, providing density and pressure profiles over an extended radial range of ∼0.2-2 Mpc. We then perform a Bayesian analysis of the thermodynamic profiles of the X-COP sample and infer constraints on the fractional gravity parameters, for individual clusters and also for clusters stacked together.
The structure of the paper is straightforward: in Section <ref>, we describe our methods and analysis; in Section <ref>, we present and discuss our results; in Section <ref>, we summarize our findings and highlight future perspectives. Throughout the work, we adopt the standard, flat ΛCDM cosmology <cit.> with rounded parameter values: a matter density Ω_M ≈ 0.3, a baryon density Ω_b ≈ 0.05, the Hubble constant H_0 = 100 h km s^-1 Mpc^-1, with h≈ 0.7.
§ THEORETICAL BACKGROUND AND DATA ANALYSIS
In this section, we recall the basics of the fractional gravity framework, illustrate how this can be exploited to derive the pressure profile of the ICM in hydrostatic equilibrium, and describe our Bayesian analysis to constrain the fractional gravity parameters.
§.§ DM in Fractional Gravity
The density distribution of virialized halos for collisionless DM as extracted from N-body simulations in the standard ΛCDM model is routinely described via the Navarro–Frenk–White profile <cit.>:
ρ(r) = ρ_s r_s^3/r (r+r_s)^2 ,
where r_s is a scale radius and ρ_s a characteristic density. The associated cumulative mass is given by
M(<r)=4π ∫_0^r dr' r'^2 ρ(r')=M_s [ln(1+r/r_s)-r/r_s/1+r/r_s] ,
with M_s≡ 4π ρ_s r_s^3.
In the standard (Newtonian) case, the potential Φ_ N(r) associated to a given density distribution ρ(r) is computed from the Poisson equation supplemented with appropriate boundary conditions (usually taken as a vanishing potential at infinity):
ΔΦ_ N(𝐫)=4π G ρ(𝐫)
where Δ is the Laplacian operator; this is an inherently local equation, in that the potential at a point depends only on the value of the density there. For the spherically symmetric NFW profile, one easily finds that
Φ_ N(r) = -G M_s/r log(1+r/r_s) ;
from the above expressions of the mass and potential, it is straightforward to verify that | dΦ_ N/ dr|=G M(<r)/r^2, as a direct consequence of Birkhoff's theorem.
In fractional gravity, the potential Φ_ F(r) is instead derived from the modified Poisson equation <cit.>
(-Δ)^s Φ_ F (𝐫) = -4π G ℓ^2-2s ρ(𝐫)
where (-Δ)^s is the fractional Laplacian operator (see <cit.> for details), s∈ [1,3/2] is the fractional index (this range of values for s is required to avoid
divergences; see Appendix A
in <cit.>), and ℓ is a fractional length scale that must be introduced for dimensional reasons. At variance with the standard case, the fractional Laplacian is inherently nonlocal; the index s measures the strength of this nonlocality, while the length scale ℓ can be interpreted as the typical size below which gravitational effects are somewhat reduced and above which they are instead amplified by nonlocality (around r≈ℓ, the dynamics is almost unaffected and indistinguishable from the standard case).
In <cit.>, we solved the fractional Poisson equation sourced by the NFW density distribution. For s∈ [1,3/2), the solution reads
Φ_ F(r) = -G M_s/r_s 1/2^2s √(π) (ℓ/r_s)^2-2s Γ(3/2-s)/Γ(s+1) r_s/r {2π s/sin(2π s) [(1+r/r_s)^2s-2..
-.. (1-r/r_s)^2s-2]+(r/r_s)^2s/1-(r/r_s)^2 [(1+r/r_s) _2F_1(1,1,2s+1,r/r_s) ..
+ ..(1-r/r_s) _2 F_1(1,1,2s+1,-r/r_s)-4s/2s-1] } , s∈[1,3/2)
with Γ(s) = ∫_0^∞ dx x^s-1 e^-x being the Euler Gamma function and _2F_1(a,b,c;x) =∑_k=0^∞ (a)_k (b)_k x^k/(c)_k k! being the ordinary hypergeometric function in terms of the Pochammer symbols (q)_k defined as (q)_0=1 and (q)_k=q (q+1) … (q+k-1); plainly, Φ_ F(r) for s=1 coincides with the usual expression Φ_ N(r) of Equation (<ref>). For the limiting case s=3/2, the computation requires some principal-value regularization and the solution reads
Φ_ F(r) = -G M_s/ℓ 1/π r_s/r {2 r/r_s [log(r/r_s)-1]-(1+r/r_s) log(r/r_s) log(1+r/r_s) .
+. (r/r_s-1) Li_2(1-r/r_s)-(1+r/r_s) Li_2(-r/r_s) + π^2/6} , s=3/2
with Li_2(x)=∑_k=1^∞ x^k/k^2 being the dilogarithm function.
Being a nonlocal framework, in fractional gravity, the Birkhoff theorem does not hold, but one can insist in writing | dΦ_ F/ dr| = G M_F(<r)/r^2 in terms of an effective mass M_F(<r), which plainly will be a function of the fractional gravity parameters s and ℓ. We illustrate the effective mass profile in Figure <ref>, suitably normalized so as to remove the dependence of dimensional quantities (including ℓ), for different values of the fractional index s. With s increasing from unity, the effective mass profile steepens: in the inner region, a uniform sphere behavior (corresponding to a cored density profile) tends to be enforced, while in the outskirts the effective profile resembles that of an isothermal sphere. Note that all the normalized mass profiles intersect at very close values of r/r_s; more in detail, the profile with a given s crosses the one with s=1 at r/r_s≈ 1.58 for s=1.1, at r/r_s≈ 1.49 for s=1.3, and at r/r_s≈ 1.36 for s=1.5; plainly, in log scale, all these points appear clustered around log r/r_s≈ 0.15 and are barely discernible by eye.
To have a quantitative grasp on the overall effect of fractional gravity, consider the s=3/2 case where the effective mass can be computed in terms of a relatively simple analytical expression; it reads
M_ F(<r) = M_s r_s/π ℓ {2 r/r_s [log(r/r_s)-1]-log(r/r_s) log(1+r/r_s).
- .Li_2(1-r/r_s)-Li_2(-r/r_s)
+π^2/6} , s=3/2 ,
and it is easily found to behave as M_ F(<r)∝ [1-3 log (r/r_s)] r^3 for r≪ r_s and as M_ F(<r)∝ r ln(r/r_s) for r≫ r_s; besides minor logarithmic corrections, the overall behavior is very similar to that of a cored isothermal sphere.
§.§ Forward Modeling of the ICM Thermodynamics
Assuming hydrostatic equilibrium and spherical symmetry, the distribution of the ICM in the overall gravitational potential well is ruled by the equation
1/ρ_ gas dP_ gas/ dr=-| dΦ dr| ,
where Φ=Φ_ DM+Φ_ gas is the total gravitational potential with main contributions from DM and gas, ρ_ gas is the gas mass density, and P_ gas is the gas pressure.
One can conveniently write ρ_ gas = μ m_p n_ gas in terms of the mean molecular weight μ≈ 0.6 and of the gas number density n_ gas, which is in turn easily related to the electron density by the expression n_ gas≈ 1.8 n_e, applying for a fully ionized plasma at high temperatures and a subsolar chemical composition typical of the ICM. The observed electron density profile n_e(r) of individual clusters inferred from X-ray observations is often empirically rendered by the (simplified version of the) Vikhlinin profile <cit.>
n_e(r) = n_0 (r/r_c)^-α/2 [1+(r/r_t)^-ϵ/6]/[1+(r/r_c)^2]^3 β/2-α/4 ;
where n_0 is the central density, r_c and r_t are a core and a transition radius (r_c<r_t), α, β, and ϵ<5 are three slopes characterizing the inner, intermediate, and outer radial behavior. The gas mass can then be computed as M_ gas(<r)=4π ∫_0^r dr' r'^2 ρ_ gas(r') and the gas contribution to the hydrostatic balance is fully specified by |dΦ_ gas/ dr| = G M_ gas(<r)/r^2.
As to the DM contribution, we can exploit the results of the previous section and write | dΦ_ DM/ dr| = G M_ F(<r)/r^2 in terms of the fractional gravity's effective mass M_ F(<r) illustrated in Figure <ref>, which depends on the parameters s and ℓ; in the standard case (corresponding to s=1), this is just the DM mass profile of Equation (<ref>). The mass profile is also a function of the NFW scale radius r_s and mass M_s; for the present analysis, it is convenient to trade off these parameters for the mass M_500 and the concentration c_500≡ R_500/r_s at the reference radius R_500.
The conversion between these variables can be performed easily using the relations M_500=4π 500 ρ_ c(z) R_500^3/3 and
M_500=M_s/[ln(1+c_500)-c_500/(1+c_500)] stemming from the definition of R_500 and from the adopted NFW mass distribution.
Then, the solution to the hydrostatic equilibrium equation is given by
P_ gas(r) = - 1.8 μ m_p ∫_r^∞ dr' n_e(r') G [M_ F(<r')+M_ gas(<r')]/r'^2 ,
where the zero pressure at infinity has been taken as a boundary condition.Note that in computing the overall gravitational potential Φ, we have neglected the stellar contribution Φ_⋆, mainly originated by the brightest central galaxy; this would add a term dΦ_⋆/ dr = G M_⋆(<r)/r^2 to the integrand on the right-hand side of Equation (<ref>). For the X-COP cluster sample exploited in this work (stellar profiles were available for 5 out of 12 clusters), the related contribution has been shown by <cit.> to become relevant only for r≲ 0.02 R_500∼ 20 kpc and as such can barely influence the innermost available data point of the pressure profile; as a consequence, our results were negligibly affected, as we also checked numerically.
Observationally, X-ray surface brightness and spectroscopic data can probe the electron density n_e(r) and the temperature T_ gas(r) profiles, whence the pressure profile P_ gas(r)∝ n_ gas(r) T_ gas can be derived, although sensitivity and background issues make such a determination robust only in the region out to R_500. In the outskirts, SZ observations can complement X-ray data in probing the pressure profile, though with some caveats about conversion from line-of-sight-integrated to spherically averaged quantities. The rationale of the above forward modeling of the hydrostatic equilibrium is to test the fractional gravity parameters entering in the effective mass profile M_ F(<r) by simultaneously fitting the observed electron density profile via Equation (<ref>) and the observed pressure profile via Equation (<ref>).
§.§ Bayesian Data Analysis
We tested the fractional gravity framework by exploiting the X-COP sample <cit.> of 12 massive galaxy clusters. The clusters are in the redshift range 0.04≲ z ≲ 0.1 and feature typical sizes R_500∼ 1–1.5 Mpc and masses M_500∼ 10^14–10^15 M_⊙. The X-COP clusters were selected to allow a robust reconstruction of the electron density and gas pressure profiles out to R_200 via a joint analysis of high-quality X-ray data from XMM-Newton and of high signal-to-noise SZ observations from Planck. Another important property of the sample is that the hydrostatic equilibrium holds to a high accuracy, with at most mild levels on nonthermal pressure support in the outermost regions, as demonstrated by the analysis of <cit.>; this is particularly important, since nonthermal effects can appreciably affect the mass estimation in the outer regions <cit.> and potentially induce spurious effects in constraining modified gravity parameters <cit.>. All in all, X-COP is currently the largest cluster sample available so far for robust mass-modeling studies over an extended radial range, and as such it has been extensively exploited to probe modified-gravity scenarios <cit.>.
To estimate the parameters θ_ F≡ (s,ℓ,c_500,M_500) describing the effective mass profile M_ F(<r), alongside with those θ_n_e≡(n_0,α,β,ϵ,r_c,r_t) describing the electron density profile n_e(r), we adopted a Bayesian framework and built the joint log-likelihood
logℒ(θ) = logℒ_P_ gas(θ_ F,θ_n_e) + logℒ_n_e(θ_n_e) .
Each term in the log-likelihood reads logℒ(θ)=-χ^2(θ)/2, where the chi-square χ^2(θ)=∑_i [ℳ(θ,r_i)-𝒟(r_i)]^2/σ_𝒟^2(r_i) was obtained by comparing our empirical model expectations ℳ(θ,r_i) to the data values 𝒟(r_i) with their uncertainties σ_𝒟(r_i), summing over the different radial coordinates r_i of the data (approximately 65 points for n_e and 20 points per P_ gas, with small variations around these numbers from cluster to cluster); note that for the pressure data from SZ observations, we took into account the full covariance matrix.
We adopted flat priors π(θ) on all the parameters; specifically, for those entering the effective mass profile in fractional gravity we took s∈ [1,3/2], logℓ (Mpc) ∈ [-3,3], log c_500∈ [-2,2], log M_500 (M_⊙)∈ [13,16]. We then sampled the parameter posterior distributions 𝒫(θ) ∝ℒ(θ) π(θ) via the MCMC Python package <cit.>, running it with 10^4 steps and 200 walkers for every individual cluster; each walker was initialized with a random position uniformly sampled from the (flat) priors. After checking the auto-correlation time, we removed the first 20% of the flattened chain to ensure burn-in; the typical acceptance fractions of the various runs were in the range 30–40%.
§ RESULTS
In Figures <ref> and <ref>, we illustrate the outcome of the fitting procedure on the 12 individual pressure and density profiles of the X-COP sample. In each panel, the best fit (solid lines) and the 2σ credible intervals sampled from the posterior (shaded areas) are shown. The reduced χ_r^2 value of the joint fit to the pressure and density profiles is also reported in Figure <ref>. Overall, the fits in the fractional gravity framework are very good. In a few cases (such as A3266 and A2319), the reduced χ_r^2 is somewhat large, but this should not raise any alarm, since the outcome is caused by some peculiar feature in the density profile reconstructed from X-ray data (oscillation in the data points at intermediate radii) or because of some outlier data in the pressure profile reconstructed from SZ (especially in the innermost or outermost radii); note that we retained all data points in our analysis, including them in the reduced χ_r^2 computation.
In Figure <ref>, we illustrate the MCMC posterior distributions for two representative clusters in the sample, namely A2255 and ZW1215; for clarity, we restricted the plot to the subspace of parameters entering the effective mass profile. Magenta/contour lines display the results in our fiducial setup, where no mass prior was imposed; the white cross marks the best-fit value of the parameters.
The corner plots illustrate a clear degeneracy between the fractional length-scale parameter ℓ and the DM mass M_500. This is somewhat expected since the effective mass profile entering the hydrostatic equilibrium equation scales like M_500 ℓ^2-2s. Therefore, it is possible to obtain the same normalization of the pressure profile, at a given density profile, by changing M_500 and ℓ in the same direction. Since s does not deviate much from unity, the ℓ dependence is weak, implying that to compensate a rather small change in mass requires a substantial variation in ℓ; on the other hand, this is also at the origin of the rather loose constraints that can be derived on the parameter ℓ with the present cluster sample.
The situation is expected to improve if a mass prior from other probes such as weak lensing (WL) is introduced in the analysis. However, one must be careful and use WL mass estimates that are independent from assumptions on the shape of the lensing potential; this is because in fractional gravity, the lensing potential corresponding to a given mass distribution would be different from the standard case, thus causing an inconsistency. For five X-COP clusters (A85, A1795, A2029, A2142, and ZW1215), such nonparametric WL mass determinations are available in the literature <cit.>.Actually, in principle, fractional gravity can also alter somewhat the total depth of the gravitational potential, thus biasing the overall WL mass estimates; however, given that the fractional gravity masses estimated without WL prior and the Newtonian ones are consistent with each other within 2σ (see fifth and last column in Table <ref>), we ignored such a small bias and used the Newtonian WL masses as prior, with their uncertainties, in the fractional gravity analysis. The outcome of exploiting the WL mass prior on the marginalized distributions of the parameters is illustrated by the cyan contours/lines in Figure <ref>. The DM mass posterior estimate of ZW1215 is made considerably more precise, and as a consequence of the above degeneracy, the estimate of the fractional length scale ℓ is also appreciably tightened.
In any case, the posterior distributions on all the parameters for the analysis without and with the WL mass prior are consistent within 1σ.
We also tested the performance of fractional gravity by stacking the X-COP data of all the clusters in the sample. Specifically, we built stacked electron density and pressure profiles by normalizing the individual profiles of the 12 clusters at a reference radius R_500, by co-adding them in radial bins of normalized radii r/R_500, and by computing the corresponding mean and standard deviation. The outcome of this procedure is illustrated in Figure <ref>: the crosses mark the stacked profiles, and for reference, the gray lines show the individual ones. All in all, the fractional gravity frameworks fit the stacked profiles to a remarkable accuracy.
Figure <ref> summarizes the posterior distributions of the fractional index s, fractional length scale ℓ, concentration c_500, and DM mass M_500. Table <ref> reports the marginalized posterior estimates (mean and 1σ credible intervals) of these parameters for all the individual X-COP clusters (including the WL mass prior when available), and for the stacked sample. On average, it is seen that the deviations of the fractional index s from unity are modest in clusters, and this originates rather loose constraints on the length scale ℓ. The inferred values of the DM mass M_500 and concentration c_500 are reasonable and consistent with that estimated by a variety of other methods in standard gravity <cit.>; we also checked that the same agreement applied for the gas fraction, as expected given the very good fits to the gas density profiles.
In Figure <ref>, we checked the concentration vs. the DM mass relation for the X-COP sample in fractional gravity. To fairly compare with the relation expected from N-body simulations in the ΛCDM framework, we converted our fitting variables c_500 and M_500 at a reference radius R_500 to the corresponding values c_200 and M_200 at R_200; this is a trivial rescaling given the adopted NFW density profile. In Figure <ref>, we show as filled magenta circles the outcome for individual X-COP clusters and with a magenta cross that for the stacked sample. It is seen that the estimates of c_200 and M_200 in fractional gravity are fairly consistent in shape and scatter with the concentration vs. mass relation extracted from N-body simulations <cit.>.
In passing, we note that the clusters A644, A1644, A2255, and A2319 have been shown not to favor an NFW mass profile, but rather a cored Burkert-like one (e.g., a Burkert, Hernquist, or pseudo-isothermal distribution) <cit.>. When forward modeling the pressure profiles in standard gravity with the NFW density distribution, this causes inconsistent results (especially in mass and concentration values) and/or a poor fit <cit.>. Contrariwise, such values and fits in fractional gravity stay reasonable and good, since the mass profile entering the hydrostatic equilibrium equation is not the true NFW mass, but the effective mass, which, as mentioned in Section <ref>, mirrors that of a cored profile. For these four clusters, we also checked that using a cored Burkert-like density distributions in place of the NFW one as an input in our fractional gravity framework did not substantially improve the fits to pressure profiles, and rather forced the fractional index to values s≈ 1 compatible with pure Newtonian gravity. In fact, fractional gravity actually reconciles the NFW density distribution from simulations with the observed galactic dynamics, which are empirically described via cored, Burkert-like profiles. Moreover, A2319 have been shown to be characterized by an appreciable nonthermal support in the outskirts <cit.>, which causes some difficulties in forward modeling and fitting the pressure profiles via the usual hydrostatic equilibrium equation in standard gravity. Instead, curiously, in fractional gravity, the fits stay good, suggesting that such a nonlocal framework may constitute an effective rendition for the effects of a nonthermal support on the pressure distribution.
In Figure <ref>, we explore the scaling of the fractional gravity parameters with the DM mass. For this purpose, we put together the analysis of the X-COP clusters from this work, and the constraints coming from the fitting of stacked galaxy rotation curves by <cit.>. These joint datasets covered six orders of magnitude in DM mass from M_200∼ 10^9 M_⊙ to 10^15 M_⊙. As to the fractional index s, we confirmed the decreasing trend with the DM mass, passing from values around s≈ 1.4 in dwarf galaxies, to s∼ 1.2-1.3 in intermediate mass galaxies, to s∼ 1.1 in massive galaxies and clusters. We described the s vs. M_200 relation by a linear fit (dashed line) with shape s = a + b (log M_200(M_⊙)-11) via an orthogonal distance regression (ODR) algorithm that took into account the error bars on both axis; we obtained the best-fit parameters a = 1.24± 0.02 and b = -0.057± 0.006 and a reduced χ_r^2≈ 1.87; a nonlinear fit (solid line) s = (5/4) + (1/4) tanh[c (log M_200 (M_⊙) - d)] interpolating between asymptotic values s = 1 and 1.5 at small and large masses yielded the best-fit parameters c = -0.39± 0.06, d = 10.76± 0.25 and a reduced χ_r^2≈ 1.34.
As to the fractional length scale, there was an increasing trend with the DM mass, extending the finding by <cit.> at the cluster scales. We fit the ℓ vs. M_200 relation with a linear shape ℓ (Mpc) = a + b (log M_200 (M_⊙) -11) via an ODR algorithm, to obtain the best-fit parameters a = -2.66 ± 0.09, b = 0.66 ± 0.06 and a reduced χ_r^2≈ 1.09. This relation was somewhat steeper than the scaling with the DM mass of the NFW scale radius r_s, in such a way that in dwarf galaxies ℓ/r_s≈ 0.25 but this ratio increased to around one at the cluster scales.
§ SUMMARY AND OUTLOOK
Extending the analysis carried out by <cit.> on galactic scales, in this paper, we tested fractional gravity in galaxy clusters. Our aim was twofold: (i) to perform an independent sanity check that fractional gravity can accurately describe such large and massive structures; (ii) to derive a clear-cut trend for the strength of fractional gravity effects in systems with different DM masses.
To fulfill this program, we forward modeled the density and pressure distributions of the intracluster medium (ICM), working out the hydrostatic equilibrium equation in fractional gravity. Then, we performed a Bayesian analysis of the X-COP galaxy cluster sample to infer constraints on the fractional gravity parameters for individual clusters and also by stacking them.
We found that fractional gravity performed remarkably well in modeling the ICM profiles for the X-COP sample. We also checked that the relationship between the concentration of the DM profile and the DM mass still remained consistent with the expectations of N-body simulations in the ΛCDM framework. Finally, we confirmed the weakening of the fractional gravity effects toward more massive systems and derived the overall scaling of the fractional gravity parameters from dwarf galaxies to massive clusters, over six orders of magnitude in DM mass. Such an overall trend implies that fractional gravity can substantially alleviate the small-scale issues of the standard DM paradigm, while remaining successful on large cosmological scales.
In future work, we plan to investigate a theoretical explanation for the empirical scaling of fractional gravity parameters with the DM mass. Hints may come from the connection of these parameters with different MONDian and fractional gravity theories, as partly explored by <cit.>. In fact, it has been pointed out that all these frameworks are characterized by an index (in our case s) interpolating between the Newtonian and a MOND-like regime, and by a length scale ℓ∼√(G M/a_0) that dimensionally can be written in terms of a MOND-like characteristic acceleration scale a_0 and of the system's mass M (baryons in the basic MOND theory, total mass dominated by DM in our case). However, the empirical scaling between ℓ∝ M^∼ 2/3 found here is barely consistent with this law within the uncertainties, and an ab initio explanation of the inverse dependence of s with mass M is difficult to be envisaged even in simple terms. This indicates that some crucial ingredient is missing to build a robust theoretical background behind DM in fractional gravity.
On the observational side, the present work clearly shows that whatever the ultimate origin of the fractional gravity behavior in the DM component is, most of its effects manifest in small DM masses; according to the canonical structure formation scenario, these objects must have formed at early cosmic times. Therefore, in the near future, we plan to look for signs of fractional gravity via kinematic (and possibly gravitational lensing) observations of low-mass galaxies at intermediate/high redshift, and of their relics in the local Universe; this could shed light on the mechanisms responsible for the origin and the emergence of fractional gravity across cosmic times.
Conceptualization: A.L. and L.D.; methodology: F.B. and G.G.; validation:
F.B., and B.S.H.; writing: A.L. All authors have read and agreed to the published
version of the manuscript.
This work was partially funded from the projects: “Data Science methods for MultiMessenger Astrophysics & Multi-Survey Cosmology” funded by the Italian Ministry of University and Research, Programmazione triennale 2021/2023 (DM n.2503 dd. 9 December
2019), Programma Congiunto Scuole; EU H2020-MSCA-ITN-2019 n. 860744 BiD4BESt: Big Data applications for black hole Evolution STudies; PRIN MIUR 2017 prot. 20173ML3WW, Opening the ALMA window on the cosmic evolution of gas, stars, and supermassive black holes; Fondazione ICSC, Spoke 3 Astrophysics and Cosmos Observations; National Recovery and Resilience Plan (Piano Nazionale di Ripresa e Resilienza, PNRR) Project ID CN-00000013 “Italian Research Center on High-Performance Computing, Big Data and Quantum Computing” funded by MUR Missione 4 Componente 2 Investimento 1.4: Potenziamento strutture di ricerca e creazione di “campioni nazionali di R&S (M4C2-19)”—Next Generation EU (NGEU).
N/A
We thank the anonymous referees for useful suggestions. We acknowledge C. Baccigalupi and P. Salucci for illuminating discussions.
The authors declare no conflict of interest.
-0cm
References
[custom]
999
Benetti23Benetti, F.; Lapi, A.; Gandolfi, G.; Salucci, P.; Danese, L. Dark Matter in Fractional Gravity I: Astrophysical Tests on Galactic Scales. Astrophys. J.
2023, 949, 65.
Planck20introAghanim, M. et al. [Planck Collaboration].
Planck 2018 results. I. Overview and the cosmological legacy of Planck. Astron. Astrophys. 2020, 641, 1.
Sarazin88Sarazin, C.L. X-ray Emission from Clusters of Galaxies; Cambridge University Press: Cambridge, UK, 1988.
Cavaliere13Cavaliere, A.; Lapi, A. The astrophysics of the Intracluster Plasma. Phys. Rep. 1988, 533, 69–94
Sunyaev80Sunyaev, R.A.; Zeldovich, I.B. Microwave background radiation as a probe of the contemporary structure and history of the universe. Annu. Rev. Astron. Astrophys. 1980, 18, 537–560.
Rephaeli95Rephaeli, Y. Comptonization of the Cosmic Microwave Background: The Sunyaev-Zeldovich Effect.
Annu. Rev. Astron. Astrophys. 1995, 33, 541–580.
Terukina14Terukina, A.; Lombriser, L.; Yamamoto, K.; Bacon, D.; Koyama, K.; Nichol, R.C. Testing chameleon gravity with the Coma cluster. J. Cosmol. Astropart. Phys. 2014, 4, 13.
Wilcox15Wilcox, H.; Bacon, D.; Nichol, R.C.; Rooney, P.J.; Terukina, A.; Romer, A.K.; Koyama, K.; Zhao, G.-B.; Hood, R.; Mann, R.G.; et al. The XMM Cluster Survey: Testing chameleon gravity using the profiles of clusters. Mon. Not. R. Astron. Soc. 2015, 452, 1171–1183.
Sakstein16Sakstein, J.; Wilcox, H.; Bacon, D.; Koyama, K.; Nichol, R.C.
Testing gravity using galaxy clusters: New constraints on beyond Horndeski theories. J. Cosmol. Astropart. Phys. 2016, 7, 19.
Haridasu21Haridasu, B.S.; Karmakar, P.; De Petris, M.; Cardone, V.F.; Maoli, R. Testing generalized scalar-tensor theories of gravity with clusters of galaxies. Phys. Rev. D 2021, submitted, arXiv:2111.01101.
Harikumar22Harikumar, S.; Biesiada, M. Moffat's modified gravity tested on X-COP galaxy clusters. Eur. Phys. J. C 2022, 82, 241.
Boumechta23Boumechta, Y.; Haridasu, B.S.; Pizzuti, L.; Butt, M.A.; Baccigalupi, C.; Lapi, A. Constraining Chameleon screening using galaxy cluster dynamics. Phys. Rev. D 2023, submitted, arXiv:2303.02074.
Gandolfi23Gandolfi, G.; Haridasu, B.S.; Liberati, S.; Lapi, A. Looking for Traces of Non-minimally Coupled Dark Matter in the X-COP Galaxy Clusters Sample. Astrophys. J. 2023, in press.
Clifton12Clifton, T.; Ferreira, P.G.; Padilla, A,; Skordis, C. Modified gravity and cosmology. Phys. Rep. 2012, 513, 1–189.
Nojiri17Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution. Phys. Rep. 2017, 692, 1–104.
Saridakis21Saridakis, E.N.; Lazkoz, R.; Salzano, V.; Moniz, P.V.; Capozziello, S; Beltran J.J.; De Laurentis, M.; Olmo, G.J. Modified Gravity and Cosmology: An Update by the CANTATA Network. arXiv 2021 arXiv:2105.12582.
Milgrom83Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J. 2017, 270, 365–370.
Famaey12Famaey, B.; McGaugh, S.S. Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions. Liv. Rev. Relat. 2012, 15, 10.
Verlinde17Verlinde, E.P. Emergent Gravity and the Dark Universe. Sci. Post. Phys. 2017, 2, 16.
Yoon23Yoon, Y.; Park, J.-C.; Hwang, H.S. Understanding galaxy rotation curves with Verlinde's emergent gravity. Class. Quant. Grav. 2023, 40, 02LT01.
Gandolfi21Gandolfi, G.; Lapi, A.; Liberati, S. Self-gravitating Equilibria of Non-minimally Coupled Dark Matter Halos. Astrophys. J. 2021, 910, 76.
Gandolfi22Gandolfi, G.; Lapi, A.; Liberati, S. Empirical Evidence of Nonminimally Coupled Dark Matter in the Dynamics of Local Spiral Galaxies? Astrophys. J. 2022, 929, 48.
deBlok98de Blok, W.J.G.; McGaugh, S.S. Testing Modified Newtonian Dynamics with Low Surface Brightness Galaxies: Rotation Curve FITS. Astrophys. J. 1998, 508, 132.
Sanders02Sanders, R.H.; McGaugh, S.S. Modified Newtonian Dynamics as an Alternative to Dark Matter. Annu. Rev. Astron. Astrophys. 2002, 40, 263–317.
Clowe96Clowe, D.; Bradac, M.; Gonzalez, A.H.; Markevitch, M.; Randall, S.W.; Jones, C.; Zaritsky, D. A Direct Empirical Proof of the Existence of Dark Matter. Astrophys. J. 1996, 648, L109.
Angus06Angus, G.W.; Famaey, B.; Zhao, H.S. Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry. Mon. Not. R. Astron. Soc. 2006, 371, 138–146.
Calcagni22Calcagni, G.; Varieschi, G.U. Gravitational potential and galaxy rotation curves in multi-fractional spacetimes. J. High Energy Phys. 2022, 8, 24.
Calcagni13Calcagni, G. Multi-scale gravity and cosmology. J. Cosmol. Astropart. Phys. 2013, 12, 41.
Varieschi20Varieschi, G.U. Newtonian Fractional-Dimension Gravity and MOND. Found. Phys. 2020, 50, 1608–1644.
Varieschi21Varieschi, G.U. Newtonian fractional-dimension gravity and rotationally supported galaxies. MNRAs 2021, 503, 1915–1931.
Giusti20aGiusti, A. MOND-like fractional Laplacian theory. Phys. Rev. D 2020, 101, 124029.
Giusti20bGiusti, A.; Garrappa, R.; Vachon, G. On the Kuzmin model in fractional Newtonian gravity. EPJP 2020, 135, 798.
Garcia22Garcia-Aspeitia, M.A.; Fernandez-Anaya, G.; Hernandez-Almada, A.; Leon, G.; Magana, J. Cosmology under the fractional calculus approach. Mon. Not. R. Astron. Soc. 2022, 517, 4813–4826.
Borjon22Borjon-Espejel, S.; Escalante-Martinez, J.E.; Padilla-Longoria, P. Newtonian gravity and MOND: A fractional Fourier approach. Indian Journ. Phys. 2022, 96, 3405–3411.
Eckert17Eckert, D.; Ettori, S.; Pointecouteau, E.; Molendi,
S.; Paltani, S.; Tchernin, C. The XMM cluster outskirts project (X-COP). Astron. Nachr. 2017, 338, 293–298.
Ettori19Ettori, S.; Ghirardini, V.; Eckert, D.; Pointecouteau, E.; Gastaldello, F.; Sereno, M.; Gaspari, M.; Ghizzardi, S.; Roncarelli, M.; Rossetti, M. Hydrostatic mass profiles in X-COP galaxy clusters. Astron. Astrophys. 2019, 621, 39.
Ghirardini19Ghirardini, V.; Eckert, D.; Ettori, S.; Pointecouteau, E.; Molendi, S.; Gaspari, M.; Rossetti, M.; De Grandi, S.; Roncarelli, M.; Bourdin, H.; et al. Universal thermodynamic properties of the intracluster medium over two decades in radius in the X-COP sample. Astron. Astrophys. 2022, 621, 41.
Eckert22Eckert, D.; Ettori, S.; Pointecouteau, E.; van der Burg, R.F.J.; Loubser, S.I. The gravitational field of X-COP galaxy clusters. Astron. Astrophys. 2022, 662, 123.
Aghanim20Aghanim, M. et al. [Planck Collaboration]. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. 2020, 641, A6.
Navarro97Navarro, J.F.; Frenk, C.S.; White, S.D.M. A Universal Density Profile from Hierarchical Clustering. Astrophys. J. 1997, 490, 493.
Vikhlinin06Vikhlinin, A.; Kravtsov, A.; Forman, W.; Jones, C.; Markevitch, M.; Murray, S.S.; Van Speybroeck, L. Chandra Sample of Nearby Relaxed Galaxy Clusters: Mass, Gas Fraction, and Mass-Temperature Relation. Astrophys. J. 2006, 640, 691.
Biffi16Biffi, V.; Borgani, S.; Murante, G.; Rasia, E.; Planelles, S.; Granato, G.L.; Ragone-Figueroa, C.; Beck, A.M.; Gaspari, M.; Dolag, K. On the Nature of Hydrostatic Equilibrium in Galaxy Clusters. Astrophys. J. 2016, 827, 112.
Ansarifard20Ansarifard, S.; Rasia, E.; Biffi, V.; Borgani, S.; Cui, W.; De Petris, M.; Dolag, K.; Ettori, S.; Movahed, S.M.S.; Murante, G.; et al. The Three Hundred Project: Correcting for the hydrostatic-equilibrium mass bias in X-ray and SZ surveys. Astron. Astrophys. 2020, 634, 113.
Pizzuti20Pizzuti, L.; Sartoris, B.; Borgani, S.; Biviano, A. Calibration of systematics in constraining modified gravity models with galaxy cluster mass profiles. J. Cosmol. Astropart. Phys. 2020, 4, 24.
Foreman-Mackey13Foreman-Mackey, D.; Hogg, D.W.; Lang, D.; Goodman, J. emcee: The MCMC Hammer. Publ. Astron. Soc. Pac. 2013, 125, 306.
Herbonnet20Herbonnet, R.; Sifon, C.; Hoekstra, H.; Bahe, Y.; van der Burg, R.F.J.; Melin, J.-B.; von der Linden, A.; Sand, D.; Kay, S.; Barnes, D. CCCP and MENeaCS: (updated) weak-lensing masses for 100 galaxy clusters. Mon. Not. R. Astron. Soc. 2020, 497, 4684–4703.
Dutton14Dutton, A.A.; Maccio, A.V. Cold dark matter haloes in the Planck era: Evolution of structural parameters for Einasto and NFW profiles. Mon. Not. R. Astron. Soc. 2014, 441, 3359–3374.
Eckert19Eckert, D.; Ghirardini, V.; Ettori, S.; Rasia, E.; Biffi, V.; Pointecouteau, E.; Rossetti, M.; Molendi, S.; Vazza, F.; Gastaldello, F.; et al. Non-thermal pressure support in X-COP galaxy clusters. Astron. Astrophys. 2019, 621, 40.
|
http://arxiv.org/abs/2307.03916v1 | 20230708064241 | Phased Geometric Controls of V-Shaped Three-Level System for Zero-field Quantum Sensing | [
"Zhijie Li",
"Xiangyu Ye",
"Xi Kong",
"Tianyu Xie",
"Zhiping Yang",
"Pengju Zhao",
"Ya Wang",
"Fazhan Shi",
"Jiangfeng Du"
] | quant-ph | [
"quant-ph"
] |
revtex4-2
These authors contributed equally to this work.
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
These authors contributed equally to this work.
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
[email protected]
The State Key Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093 Nanjing, China
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
[email protected]
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
School of Biomedical Engineering and Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China
[email protected]
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
School of Physics, Zhejiang University, Hangzhou 310027, China
Here we propose and demonstrate a phased geometric control protocol for zero-field double quantum gates in a V-shaped three-level spin system. This method utilizes linearly polarized microwave pulses and exploits the geometric qubit properties to prevent state leakage. By employing specific phased geometric controls, we realize a low-power multi-pulse zero-field sensing technique using single nitrogen-vacancy centers in diamond. Our method offers a novel approach to implement precise double quantum gate operations with an adaptable driving power, making it a valuable tool for zero-field spin-based quantum technology.
Phased Geometric Controls of V-Shaped Three-Level System for Zero-field Quantum Sensing
Jiangfeng Du
August 12, 2023
=======================================================================================
In recent years, quantum sensing techniques based on controllable quantum systems have seen significant development. One successful example is the nitrogen-vacancy (NV) center in diamond, which possesses numerous merits, including nanoscale size, biocompatibility, and long coherence time under ambient conditions <cit.>. Typically, solid-state quantum systems require a static external magnetic field to lift the degeneracy of their ground-state manifolds. However, the presence of an external magnetic field suppresses the anisotropic interactions within the target sample, resulting in the loss of anisotropic physical information and causing inhomogeneous spectral broadening. A well-known zero-field technology is the zero- to ultralow-field nuclear magnetic resonance (ZULF NMR) spectroscopy. This technique effectively mitigates the inhomogeneous broadening of the spectrum in heterogeneous environments by attenuating the broadening effects induced by magnetic susceptibility <cit.>. More zero-field scenarios can be found in the field of electromagnetic biology <cit.> and in the research of ferromagnetic film magnetization <cit.>. In order to extend the zero-field condition to solid-state quantum systems like NV centers, the implementation of high-fidelity quantum control for the three-level system (3LS) is imperative.
To address the near-degenerate quantum states in the absence of external fields, one approach is to employ circularly polarized microwave pulses <cit.>. While this method is effective when using a few pulses, it is limited in its ability to utilize double quantum (DQ) transitions with a multi-pulse method, which is crucial for sensing weak AC signals. Recent works have paved the way for realizing dynamical decoupling (DD) with linearly polarized microwave pulses at zero field by manipulating the 3LS via an effective Raman coupling <cit.>. This method enables the utilization of high-power multiple pulses, leveraging the advantage of DQ transitions at zero field to offer a significantly broader sensing bandwidth and expanded sensitivity range. However, the effectiveness of this method is compromised by the occurrence of state leakage due to the contradiction between the unavoidable hyperfine non-degeneracy and the limited driving field strength <cit.>. Subsequently, sequences that counteract the effects of the non-degeneracy detuning were proposed <cit.>. However, these methods, while relaxing the requirements for a strong driving field, lack versatility in their operations. In this study, we propose a method that prevents state leakage with a weak driving field by leveraging the geometric properties of the dressed states. Through this approach, a collection of effective DQ rotation operations can be achieved. Furthermore, we demonstrate a zero-field quantum sensing scheme utilizing single NV centers based on the proposed method.
A single NV center in diamond consists of a substitutional nitrogen and a neighboring vacancy, its electron ground states form a typical 3LS (Fig. <ref>(a)). The Hamiltonian of a single NV center driven by a linearly polarized microwave field can be given by (ħ=1) <cit.>
H= (D+d_∥Π_z) S_z^2+(Δ+δ/2)S_z+Ωcos(ω t+ϕ)S_x
+d_⊥[Π_x(S_y^2-S_x^2)+Π_y(S_xS_y+S_yS_x)],
where S=(S_x,S_y,S_z) is the spin-1 operator, D is the zero-field splitting, d_∥ and d_⊥ are the longitudinal and transverse electric dipole moment components, Δ refers to the Zeeman splitting induced by the external magnetic field along the NV center's principle axis, δ contains hyperfine couplings with the surrounding spin-1/2 nuclei, and Π=(Π_x,Π_y,Π_z) denotes the total effective electric field. Furthermore, Ω,ω and ϕ correspond to the amplitude, angular frequency, and phase of the linearly polarized microwave, respectively. Provided that the NV center's native nitrogen atom is a ^15N atom and there is no magnetic field along the NV center's symmetric axis, the splitting within each electronic state manifold is primarily attributed to hyperfine interactions and transverse electric dipole couplings. When a linearly polarized microwave pulse with angular frequency ω=D+d_∥Π_z is applied, it drives the oscillations |0⟩↔|+1⟩ and |0⟩↔|-1⟩ simultaneously. As a result, an effective Raman coupling emerges (Fig. <ref>(a)). By utilizing phase-fixed geometric controls <cit.> on the ground-state 3LS, it is possible to accumulate a geometric π phase on the state |+⟩ while keeping the state |-⟩ nearly unchanged, as long as the 2π cycle occurs rapidly compared to the detuning modulation (Fig. <ref>(b)). This approach enables the realization of a nearly π pulse within the {|+1⟩,|-1⟩} subspace. However, the presence of the hyperfine coupling δ and the transverse effective electric field (Π_x,Π_y) can induce state leakage to the |0⟩ state. Consequently, the imperfect controls in the dynamical decoupling sequence result in degraded spin coherence and distorted signal filtering, thereby diminishing the sensitivity.
In this Letter, we introduce a novel phased geometric control method that prevents state leakage and enables a diverse range of operations. With the resonance condition ω=D+d_∥Π_z and the microwave polarization perpendicular to the transverse projection of Π, the Hamiltonian of the system can be expressed as <cit.>
H̃'̃(Ω,ϕ)=(Ω e^iϕ/2 |0⟩ + δ'e^iψ/2 |-⟩)⟨+|+H.c.,
where δ'=√(δ^2+4d^2_⊥Π_y^2) and ψ=arctan(-2d_⊥Π_y/δ). Set Ω=δ', a complete transition between the states |0⟩ and |-⟩ is activated (Fig. <ref>(c)). The operation U_ϕ, which enables the complete transition |0⟩↔|-⟩, is defined by the incident microwave phase ϕ. Defining |ϕ'⟩=(e^iϕ|0⟩+e^iψ|-⟩)/√(2), the Hamiltonian Eq. (<ref>) can be written as
H̃'̃(δ',ϕ)=δ'/√(2)(|ϕ'⟩⟨+|+|+⟩⟨ϕ'|).
In the qubit spanned by {|+⟩,|ϕ'⟩}, Eq. (<ref>) is proportional to the Pauli-X operator, and U_ϕ acts as a 2π pulse defined by the duration T'=√(2)π/δ' (Fig. <ref>(c)). In this geometric spin qubit, any 2π cycle generates a microwave-phase independent factor of -1 before |+⟩ <cit.>. Moreover, the operation U_ϕ introduces conjugate phase factors in the {|0⟩,|-⟩} subspace (Fig. <ref>(d)), i.e.
⟨ -| U_ϕ |0⟩=-e^-i(ϕ-ψ),
⟨ 0| U_ϕ |-⟩=-e^i(ϕ-ψ).
Therefore, the 4π pulse defined as G_π=U_ϕ U_ϕ+π precisely leads to |+⟩→|+⟩ and |-⟩→-|-⟩ (Fig. <ref>(e)). Consequently, the π pulse in the {|+1⟩,|-1⟩} subspace can be achieved without any leakage to the state |0⟩, directly bringing about the zero-field dynamical decoupling (ZDD) sequence with equally spaced G_π operations. Generally, phased geometric gate G_θ=U_ϕ U_ϕ+θ is equivalent to the phase gate P(θ) in the {|+⟩,|-⟩} subspace <cit.>, thus the effect of G_θ can be depicted as a rotation on the Bloch sphere (Fig. <ref>(f)). Following the scheme outlined above, arbitrary effective rotations along z-axis in the {|+⟩,|-⟩} subspace can be implemented. In addition to the G_±π gates, the G_±π/2 gates are particularly relevant in quantum sensing protocols due to their ability to convert coherence into state population in the {|+1⟩,|-1⟩} basis, which can be used to perform correlation of phases accumulated in separate DD sequences.
We use a ^12C enriched diamond chip implanted with 40 keV ^15N^+ ions for our experiments. To counterbalance the geomagnetic field, a set of permanent magnets is employed, reducing the field strength to below 0.005 mT. In this regime, we ensure that Δ/δ<<1, where δ is dominated by the intrinsic ^15N hyperfine interaction A_∥. The transverse microwave polarization is aligned perpendicular to the transverse effective electric field vector, with the polarization direction along the x-axis. The resultant non-degenerate splitting is given by δ'=√(A_∥^2+4d^2_⊥Π_y^2)=2π×3.04(1) MHz. Therefore, the manipulating microwave can be determined by Ω=δ' and ω=D+d_∥Π_z=2π× 2870.79(1) MHz. Setting ϕ=0, the |+⟩↔|0'⟩ transition is driven with the angular frequency Ω=√(δ'^2+Ω^2)=√(2)δ', and the pulse length of the 2π operation U_ϕ is defined by T'=2π/Ω (Fig. <ref>(a)). Applying the Ramsey sequence with two separate 2π pulses, oscillation of the frequency δ'/2π emerges. The envelope of this oscillation directly reflects the dephasing occurring in the {|+ 1⟩,|- 1⟩} subspace. By inserting G_π in the middle of the Ramsey sequence, coherence revival is realized (Fig. <ref>(b)). With the specific 4π pulse available, we construct the ZDD-N sequence in the form of 2π (t'/2 4π t' 4π t'/2)^N/22π (Fig. <ref>(a)), where t'=t-2T' is the duration of each free evolution, t denotes the pulse interval, Nt is the total evolution time, and the superscript indicates the interchange of the phases of constituent 2π pulses. This interlaced sequence is designed to compensate fidelity errors caused by pulse imperfections up to the second order <cit.>. By applying the ZDD-N sequences, significant prolongation of the DQ coherence in the {|+ 1⟩,|0⟩,|- 1⟩} basis is observed as the pulse number N increases (Fig. <ref>(b)), indicating that there are sufficient manipulation fidelity and coherence resources available for quantum sensing purposes.
Measurements of an AC signal with a frequency of f=0.5 MHz are shown in Fig. <ref>(c, e). The ZDD-64 sensed frequency is f'=1/(2t_s)=0.499(1) MHz, corresponding to the coherence dip at t_s=1.002(1)µs (Fig. <ref>(c)). In nanoscale NMR applications, the correlation spectroscopy sequence <cit.> is utilized to achieve high-resolution spectroscopy or to mitigate the effects of unwanted harmonics <cit.>. However, conventionally performing this free precession technique at zero field is challenging due to the incomplete manipulation of the 3LS. Nevertheless, it can be implemented by inserting G_π/2 gates between separate DD sequences (Fig. <ref>(d)). The lowest order correlation reveals the signal frequency <cit.>, as expressed by
⟨sinψ_1sinψ_2⟩∼cos (2π f(2τ+t) ),
where τ is set to t_s according to the coherence dip in the ZDD spectrum, ψ_i is the phase accumulated during each individual ZDD sequence. The correlation signal of two ZDD-16 sequences for the AC field sensed in Fig. <ref>(c) is shown in Fig. <ref>(e).
In order to demonstrate the advantage of the ZDD sequence constructed with phased geometric gates, we conduct a comparison with other DD sequences. As shown in Fig. <ref>(a), state evolutions of different DD sequences with distinct driving powers are simulated in the absence of signal fields. The state evolution under normal DD sequence is significantly distorted by detuning, while the LDD and the OC sequences <cit.> which utilize detuning-resistant phase arrangements as well as optimal control techniques, effectively suppress the distortion. In comparison, the ZDD sequence ensures equivalent populations during the free evolution periods.
Measurements of the filter functions (FFs) F(t,ω) of different DD sequences at ω=0.5 MHz are presented in Fig. <ref>(b). With low driving fields, the signal filtering of the LDD and the OC sequences are distorted. However, the ZDD sequence operating with Ω=δ' exhibits a reasonable lineshape. The deviation between the ZDD-16 FF and the ideal FF is primarily caused by the finite duty cycle of the manipulating pulses. Nonetheless, this deviation is insignificant when the duty cycle is lower than 40% (Fig. <ref>(c)). In practice, the non-degenerate splitting δ' can be controlled by applying transverse strains, allowing for an adjustable duty cycle.
In this work, we introduce a phased geometric control protocol and demonstrate its application in a zero-field quantum sensing technique. The sequences employed for dynamical decoupling and correlation spectroscopy are specifically designed using phased geometric gates. Compared to previous approaches, our method provides a wider range of gate operations in sequence design and prevents the detrimental effects of state leakage by utilizing the properties of the geometric phase. In addition to the NV center, other solid spin systems such as divacancies in SiC <cit.> offer more alternatives for implementing the DQ manipulations with phased geometric gates. These systems possess a non-degenerate splitting that can be easily adjusted by strains or electric fields, enabling precise operations even with a short dephasing time. This allows for a broadened sensing bandwidth and the analysis of electric field noise. Furthermore, it is worth noting that our protocol can be extended to any other spin-based 3LS with similar energy configuration, thereby expanding its potential applications in various quantum technologies.
§ ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (Grant No. T2125011, 81788101), the National Key R&D Program of China (Grant No. 2018YFA0306600), the CAS (Grant No. XDC07000000, GJJSTD20200001, Y201984, YSBR-068), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302200, 2021ZD0303204), the Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000), Hefei Comprehensive National Science Center, and the Fundamental Research Funds for the Central Universities.
This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
unsrt
|
http://arxiv.org/abs/2307.04928v1 | 20230710223003 | Belle II status and prospects for studies of neutral currents | [
"Valerio Bertacchi"
] | hep-ex | [
"hep-ex"
] |
Turán number for bushes
Zoltán Füredi
Alfréd Rényi Institute of Mathematics, Budapest, Hungary.
E-mail: .
Research partially supported by National Research, Development and Innovation Office NKFIH grants 132696 and 133819.
Alexandr Kostochka
University of Illinois at Urbana–Champaign, Urbana, IL 61801
and Sobolev Institute of Mathematics, Novosibirsk 630090, Russia. E-mail: .
Research supported in part by NSF
grant DMS-2153507 and NSF RTG grant DMS-1937241.
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================
§ INTRODUCTION
The flavour chaning neutral current b→ s transitions are suppressed in the Standard Model (SM) and therefore sensitive to Beyond the Standard Model (BSM) amplitudes. The SM branching fractions are 𝒪(10^-5-10^-7), predicted with 10-30% uncertainties. Angular distributions and ratios can be used to improve the precision and eventually have access to new physics properties.
Belle II <cit.> and SuperKEKB <cit.> produce an optimal environment to study the neutral currents.
Belle II has similar and good performance in electron and muon channels, in term of efficiency, fake rate and particle identification capability. This is a key feature to perform lepton flavour universality (LFU) tests and lepton flavour violation (LFV) searches in the b→ sℓℓ^(') sector, where ℓ indicates a charged lepton. On the other hand, the b→ sγ and b→ sνν transitions represent a unique opportunity for Belle II, because of the almost complete hermeticity of the detector, the possibility to exploit the Υ(4S) initial state constraint and the relatively low combinatorial background of the SuperKEKB collisions.
One of the key tools of Belle II for the channels with missing energy in the final state is the B-tagging, a set of reconstruction techniques to identify the Υ(4S)→ BB events exploiting the initial state knowledge to constraint the missing information in the signal side. It consists in reconstructing the parter B meson, called B_tag, produced in association with the signal one, to infer the properties of the signal.
We refer to hadronic or semileptonic tagging according to the channels used for the B_tag reconstruction.
The B-tagging algorithm is called Full Event Interpretation (FEI) <cit.>, a boosted decision tree (BDT)-based tagging algorithm which exploits a hierarchical approach to reconstruct 𝒪(10^4) decay chains on the tag side. The efficiency for the hadronic (semileptonic) tag is 0.5% (2%) with a purity of 30% (10%).
§ FULLY INCLUSIVE B TO XS GAMMA
We present the measurement the B→ X_sγ branching ratio as a function of photon energy in the range 1.8 GeV<E_γ<2.7 GeV, where X_sγ is the inclusive final state involving a photon and a strange hadron. The measurement is performed on a 189 fb^-1 Belle II sample <cit.>.
The decays are reconstructed using the hadronic B-tagging, requiring a γ in the signal side with a threshold energy of 1.4 GeV. The main challenge of the analysis is to suppress the background without breaking the inclusivity of the measurement. The backgrounds are suppressed using a BDT and the residual X_d background is estimated using simulated events.
The signal is extracted by fitting the tag side M_ bc=√(E^*2_beam-p^*2_B) distribution (where E_beam^* is the beam energy and p_B* is the B meson momentum in the center-of-mass frame), as a function of E_γ. The result is competitive with previous measurements performed with hadronic B-tagging <cit.>. The result in term of the partial branching fraction as a function of the photon energy is shown inf Fig. <ref> (Left).
The prospects of this measurements with larger statistics depend on the chosen photon energy threshold <cit.>. With lower threshold the background will be higher, while with higher threshold the theoretical uncertainties will be higher. However, some improvements are expected both on background suppression side and by using additional tagging methods, which will allow to reach the percent level precision. Measurements of relative quantities, such as asymmetries, will allow for further reduction of systematic effect
§ MEASUREMENT OF B TO K* GAMMA BRANCHING FRACTIONS
We present the measurement of the branching fraction of B→ K^*γ, where K^* indicates both K^*+(892) and K^*0(892). The measurement is performed on a 63 fb^-1 Belle II sample <cit.>.
The decays are identified reconstructing only the signal B in the event. The misreconstructed γ background is suppressed with an energy selection, and with a veto on γ from π^0 and η decays. The e^+e^-→ q q background is suppressed with an MVA. The misreconstructed K^* background is suppressed using the K^* helicity angle distribution. A fit to Δ E=E_B^*-E_beam^* (where E_B is the energy of the B meson) is used to extract the signal, excluding higher-mass K^* resonances. The results are ℬ(B^0→ K^*0(K^+π^-)γ)=(4.5± 0.3± 0.2)× 10^-5, ℬ(B^0→ K^*0(K_S^0π^0)γ)=(4.4± 0.9± 0.6)× 10^-5, ℬ(B^+→ K^*+(K^+π^0)γ)=(5.0± 0.5± 0.4)× 10^-5, ℬ(B^+→ K^*+(K_S^0π^+)γ)=(5.4± 0.6± 0.4)× 10^-5, where the first uncertainty is statistical and the second systematic, compatible with the world averages <cit.>.
This measurement is performed as the cleanest exclusive channel in B→ X_sγ sector, and is a first step toward asymmetry measurements of radiative decays. In the latters several systematic uncertanties cancel out, and projections based on Belle result <cit.> shows that a precision below the percent level can be reached with few ab^-1 <cit.>.
§ SEARCH FOR B+ TO K+ NU NUBAR DECAYS
The search of B^+→ K^+νν decay is a unique opportunity for Belle II. This decays has never been observed before the amplitude <cit.> can receive sizeable contribution from BSM amplitudes. The measurement is performed on a sample with an integrated luminosity of 63 fb^-1 <cit.>. The reconstruction is performed with an inclusive tagging approach, reconstructing the B_sig using the highest p_T track compatible with a K^+, and assigning the rest of the event to the B_tag. The procedure is validated on B^+→ J/ψ(→μμ)K^+ decays. Two BDT in cascade are used to suppress the background exploiting the event shape, kinematics and vertex features.
No signal is observed and the result is shown in Fig. <ref> (Right) in term of upper limit. This correspond to ℬ( B^+→ K^+νν)=(1.9± 1.3 (stat)^+0.08_-0.07 (syst))× 10^-5, compatible with the SM prediction and the previous results <cit.>.
The projection with larger samples <cit.> shows that a 5σ observation can be achieved with an integrated luminosity of 5 ab^-1 with an expected 50% efficiency improvement coming from the use of exclusive tagging approaches in combination with the inclusive one. Moreover, additional channels (K^*, K_S^0) will be investigated.
§ MEASUREMENT OF RK(J/PSI)
We present the measurements of the branching fraction of B→ Jψ(→ℓℓ)K, ℓ=e,μ and K=K^+, K_S^0, performed on a 189 fb^-1 Belle II sample <cit.>. The ratios R_K(J/ψ)=ℬ(B→ Jψ(→μ^+μ^-)K)/ℬ(B→ Jψ(→ e^+e^-)K) are also measured. These channels have no sensitivity on BSM, so R_K(J/ψ)≈ 1 is expected. This analysis is used to validate the measurement of B→ K^*ℓℓ. The yields are extracted from a fit to (Δ E, M_bc) distribution. The results are R_K^+(J/ψ)=1.009±0.022±0.008, R_K_S^0(J/ψ)=1.042±0.042±.008, where the first uncertainty is statistical and the second systematic, in agreement with the expectations.
§ MEASUREMENT OF B TO K* L L BRANCHING FRACTIONS
We present the measurement of the branching fractions ℬ(B→ K^* ℓ^+ℓ^-) (where ℓ=e,μ and K^*=K^*+(892), K^*0(892)) performed on a sample with an integrated luminosity of 189 fb^-1 <cit.>. The backgrounds are suppressed using a BDT combined with a veto on the dilepton invariant mass for the J/ψ,ψ(2S)→ℓℓ background. An extended maximum likelihood fit is performed to the (Δ E, M_bc) distribution. The results are ℬ( B→ K^*μ^+μ^-)=(1.19± 0.31^+0.08_-0.07)× 10^-6, ℬ( B→ K^*e^+e^-)=(1.42± 0.48± 0.09)× 10^-6, ℬ( B→ K^*ℓ^+ℓ^-)=(1.25± 0.30^+0.08_-0.07)× 10^-6,
where the first uncertainty is statistical and the second systematic, compatible with the world average <cit.>.
These results prepare the ground for the measurement of R_K^(*)=ℬ(B→μ^+μ^-K^(*))/ℬ(B→ e^+e^-K^(*)), which will require a larger sample.
§ B TO K* TAU TAU PERSPECTIVES
The measurement of B→ K^*ττ is complementary to the previously discussed searches, investigating the new physics in the third generation. The SM branching ratio 𝒪(10^-7), but BSM amplitudes can enhance the signal of several order of magnitude <cit.>. Currently the decay has been never observed and an upper limit at 𝒪(10^-3) has been set <cit.>. Prospects extrapolating from the current upper limit with a larger samples shows that Belle II can investigate the branching ratios down to 10^-4 with 5 ab^-1, using hadronic and semileptonic B-tagging and reconstructing the τ leptons both in leptonic and hadronic decays <cit.>.
§ PERSPECTIVES OF LEPTON FLAVOR VIOLATION SEARCHES IN B TO K(*) L L' SECTOR
Several measurement have been performed in past years by BaBar, LHCb and Belle collaborations in the B→ K^(*)ℓℓ' sector, where ℓ=e,μ,τ, setting upper limits that span from 10^-5 to 10^-9 level. Belle II is planning to join the effort in the searches of new physics in this sector. Focusing on B→ K^(*)τℓ, the use of the hadronic or semileptonic tag allow to avoid the explicit reconstruction of the τ lepton. The signal is extracted from the τ recoil mass distribution, obtained from the B_tag and the signal K track information. The recent results performed on Belle sample using the FEI are very promising <cit.>.
This project has received funding from the European Union’s Horizon 2020 research and inno- vation programme under the ERC grant agreement No 819127.
JHEP
|
http://arxiv.org/abs/2307.04807v1 | 20230710180058 | The Dragon-II simulations -- III. Compact binary mergers in clusters with up to 1 million stars: mass, spin, eccentricity, merger rate and pair instability supernovae rate | [
"Manuel Arca Sedda",
"Albrecht W. H. Kamlah",
"Rainer Spurzem",
"Francesco Paolo Rizzuto",
"Mirek Giersz",
"Thorsten Naab",
"Peter Berczik"
] | astro-ph.HE | [
"astro-ph.HE",
"astro-ph.GA",
"gr-qc"
] |
firstpage–lastpage
Autonomous feedback stabilization of a cavity-coupled spin oscillator
Dan M. Stamper-Kurn
August 12, 2023
=====================================================================
Compact binary mergers forming in star clusters may exhibit distinctive features that can be used to identify them among observed gravitational-wave (GW) sources. Such features likely depend on the host cluster structure and the physics of massive star evolution. Here, we dissect the population of compact binary mergers in the Dragon-II simulation database, a suite of 19 direct N-body models representing dense star clusters with up to 10^6 stars and <33% of stars in primordial binaries. We find a substantial population of black hole binary (BBH) mergers, some of them involving an intermediate-mass BH (IMBH), and a handful mergers involving a stellar BH and either a neutron star (NS) or a white dwarf (WD). Primordial binary mergers, ∼ 30% of the whole population, dominate ejected mergers. Dynamical mergers, instead, dominate the population of in-cluster mergers and are systematically heavier than primordial ones. Around 20% of Dragon-II mergers are eccentric in the LISA band and 5% in the LIGO band. We infer a mean cosmic merger rate of ℛ∼ 12(4.4)(1.2) yr^-1 Gpc^3 for BBHs, NS-BH, and WD-BH binary mergers, respectively, and discuss the prospects for multimessenger detection of WD-BH binaries with LISA. We model the rate of pair-instability supernovae (PISNe) in star clusters and find that surveys with a limiting magnitude m_ bol=25 can detect ∼ 1-15 yr^-1 PISNe. Comparing these estimates with future observations could help to pin down the impact of massive star evolution on the mass spectrum of compact stellar objects in star clusters.
methods: numerical – galaxies: star clusters: general – stars: general, black holes
§ INTRODUCTION
In less than a decade, the LIGO-Virgo-Kagra (LVK) collaboration discovered 76 confident gravitational-wave (GW) sources associated to merging stellar black holes (BHs) and neutron stars (NSs) <cit.>. This number raises up to 90 if one considers the population of events with a probability to have an astrophysical origin > 0.5 <cit.>, and it is destined to further increase by the end of the fourth observation run. Measurable quantities like component masses, spins, or the orbital eccentricity, and the merger rate of different types of compact binary mergers can represent the keys to identify the signatures of different formation channels <cit.>. From the theoretical standpoint, there is a plethora of mechanisms proposed to explain the formation of compact binary mergers, like isolated binary evolution <cit.>, dynamical pairing in dense star clusters <cit.>, formation in AGN disks <cit.>, secular dynamics involving three compact objects <cit.> or a binary orbiting a supermassive black hole <cit.>, and primordial BH evolution <cit.>. The majority of the aforementioned mechanisms relies on the assumption that compact objects are the relic of massive stars, and therefore they suffer the uncertainties affecting stellar evolution.
For example, the insurgence of pair instability supernova (PISN) and pulsational pair instability supernova (PPISN) mechanisms can carve in the BH mass spectrum the so-called upper-mass gap, a region extending in the range m_ gap = 40-150 where no remnants are expected. The boundaries of the gap are highly uncertain and depend on many poorly constrained quantities, like stellar rotation, rate of nuclear reactions, stellar evolution model <cit.>. The presence of several upper mass-gap BH candidates in the LVK source catalogue poses the question about the origin of these BHs. Stellar mergers, star-BH interactions, and repeated BH mergers represent possible pathways to overcome (P)PISN <cit.> and produce merging compact objects in dense star clusters <cit.>.
Spins could carry crucial information on the BH formation scenario and help placing constraints on the evolution of massive stars, but little is known about the distribution of stellar BH natal spins. Observations of merging BHs indicate that the spin distribution follows a Maxwellian distribution, with a peak around χ_ BH∼ 0.2-0.5 <cit.>. However, stellar BHs detected in low-mass X-ray binaries (LMXBs) are characterised by spins broadly distributed in the whole allowed range <cit.>, whilst those in high-mass X-ray binaries (HMXBs) involve BHs almost maximally spinning <cit.>.
Despite these differences may suffer observation biases, they may represent peculiarities of different evolutionary pathways.
Efficient angular momentum transport driven by magnetic stars could trigger the formation of BHs with natal spins as small as χ_ BH≲ 0.01, a mechanism proposed for BHs forming from single stars and in binaries with a negligible mass transfer among the components <cit.>. Significant mass transfer, instead, has been proposed to produce BHs with spin in a broad range in LMXB, even for BHs spinless at birth <cit.>, and nearly extremal BHs in HMXBs <cit.>.
Common envelope evolution in massive stellar binaries can lead to merging BBHs consisting in a nearly non-rotating BH <cit.>, although this strongly depends on the stellar evolution adopted <cit.>, and a BH companion with a spin spanning the whole allowed range of values <cit.>.
Amplitude aside, also the alignment of the spin vectors among themselves and with the binary angular momentum can affect both the waveform, the final merger remnant mass and spin, and the recoil kick (e.g. see Equation <ref>). From an "observational" perspective, measuring the spin components is intrinsically hard and their directions generally vary owing to precession, thus the spin of observed mergers can be characterised through the so-called effective spin parameter <cit.>
χ_ eff = χ⃗_1 + qχ⃗_2/1+q·L⃗,
where q < 1 is the binary mass ratio, χ⃗_1,2 are the two component spin vectors and L⃗ is the binary orbital angular momentum.
Observations of BBH mergers suggest that χ_ eff may increase at increasing the binary merger mass ratio, although some merging binaries exhibit a negative value of χ_ eff <cit.>, a feature generally associated with dynamical sources.
The orbital eccentricity at merger could represent another distinguishing feature of compact binary mergers, as dynamical interactions could trigger the formation of fairly eccentric (>0.1) sources contrarily to mergers forming from isolated binaries <cit.>. It has been recently claimed that up to four LVK sources may be eccentric <cit.>, although the effects of eccentricity and precession can lead to degeneracies in GW data analysis, making the eccentricity a poorly constrained quantity <cit.>.
Alongside GWs, the detection of (P)PISNe can represent a key piece to understand the final stages of massive stars' life. So far, only a few, most of which controversial, PISN and PPISN candidates have been observed in the last two decades <cit.>. The rarity of PISNe observations sets an intrinsic limit on the frequency of PISNe in star clusters, a quantity poorly constrained in theoretical and numerical models.
Dynamical interactions among stars in dense and massive star clusters can trigger both the formation of merging binaries and the development of PISNe, either from single massive stars or from stellar merger products. Young and intermediate-age star clusters are particularly interesting environments where these sources can form, because they are still in their dynamical youth, when cluster mass-loss and expansion did not yet affected substantially the cluster structure and the interaction rate among stars is maximal. There is vast literature investigating the formation and evolution of merging BHs in star clusters via different techniques, e.g. direct N-body simulations <cit.>, Monte Carlo simulations <cit.>, and semi-analytic tools <cit.>. However, there is lack of direct N-body simulations of particularly dense (>10^5 pc^-3) and massive (>100,000) star clusters, owing to the computational cost required to simulate such systems. Exploring this range of mass and densities with N-body models can complement the already existing simulations and can offer a term of comparison to Monte Carlo simulations (see Figure 1 in Arca Sedda et al 2023a, hereafter AS-I).
In this work, which represents the third of a series, we present results from the star cluster database, a suite of 19 direct N-body simulations of young and intermediate-age star clusters comprised of up to 1 million stars and up to 33% of stars initially in binaries, characterised by typical densities ρ = (1.2×10^4 - 1.5× 10^7) pc^-3.
In our previous papers, we focused on the general properties of our cluster models and their compact object populations (paper AS-I) and the processes that regulate the formation and growth of IMBHs (Arca Sedda et al 2023b, hereafter AS-II).
Here, we dissect the properties of BH-BH, BH-NS, and BH-WD mergers developing in the star cluster database, a suite of 19 direct N-body simulations of star clusters comprised of up to 1 million stars and up to 33% of stars initially in binaries (details about these models are discussed in our companion paper AS-I), performed with the code[<https://github.com/nbody6ppgpu/Nbody6PPGPU-beijing>]. The paper is organised as follows: in Section <ref> we briefly summarise the main features of our models; Section <ref> discusses the main properties of compact binary mergers in our models, focusing on the component masses and mass ratios, the eccentricity at merger, and the possible signatures that can identify their formation history; in Section <ref> we explore the impact of BH natal spins onto the global properties of the population, and we adopt a cosmologically motivated framework to infer the compact binary merger rate, the detection perspectives for future low-frequency GW detections, and the frequency rate and detection perspectives in magnitude limited surveys of PISNe; Section <ref> summarises the main results of this work.
§ NUMERICAL METHODS
§.§ The clusters
The simulation database consists of 19 star cluster models characterised by an initial number of stars N = (1.2, 3, 6, 10)× 10^5, half-mass radius R_ = (0.48, 0.80, 1.76) pc, and an initial binary fraction f_b = 0.05-0.2. In the following, we briefly summarise the main properties of clusters, referring the interested readers to our companion paper AS-I for more details on the run properties.
To initialise the clusters we exploit the tool <cit.>.
Each cluster is modelled according to a <cit.> profile with adimensional potential well W_0 = 6. We adopt an initial metallicity Z = 0.0005, typical of several clusters possibly hosting a dense sub-system of compact objects or an IMBH, like NGC3201 or NGC6254 <cit.>.
Star masses are drawn according to a <cit.> initial mass function limited in the range m_ ZAMS = (0.08-150). Stars in primordial binaries are paired depending on their mass, with stars heavier than >5 paired according to a flat mass-ratio distribution, and lighter stars paired randomly. Binary eccentricities are distributed according to a thermal distribution, P(e) de = e^2 de, while initial semimajor axes are assigned according to a distribution flat in logarithmic values limited between the sum of stars' radii and a maximum value of 50 AU.
The host galaxy potential is modelled through a Keplerian potential assuming a total mass of M_ gal = 1.78×10^11. All clusters are placed on a circular orbit around this galaxy model at a distance of R_ clu = 13.3 kpc. The adopted galaxy mass and orbital radius lead to a value of the circular velocity compatible with what is observed in the Milky Way.
The resulting tidal radius is much larger than the cluster half-mass radius. Therefore, models are initially underfilling their Roche lobe, which implies that the initial impact of the host galaxy potential is negligible.
All simulations are terminated when either the mean BH mass falls below ⟨ m_ BH⟩≲ 15, there are no BHs with a mass above 30, or the simulated time exceeds at least one relaxation time. As a result, the simulation time in models spans a range T_ sim = 0.1-2.3 Gyr, corresponding to 0.8-80 times the initial half-mass relaxation time (see also Table <ref>).
Over the simulated time, we find a nice overlap (see also Figure 2 in paper AS-I) between the evolution of clusters' mass and half-mass radius and observed properties of young and intermediate-age massive clusters in the Milky Way <cit.>, the Magellanic clouds <cit.>, and other galaxies in the local Universe like Henize 2-10 <cit.> or M83 <cit.>. In this sense, models can represent one possible evolutionary pathways of (relatively) young massive clusters.
§.§ The code
simulations have been performed with the code <cit.>, a state-of-the-art direct N-body integrator that runs on high performance computing hardware equipped with graphic-processing-units <cit.>. The code is part of the famous NBODY code series that was pioneered almost sixty years ago by Sverre Aarseth <cit.>.
The code implements a 4th-order Hermite integrator scheme with adaptive time-step based on the Ahmad-Cohen scheme for neighbours <cit.>, and implements a treatment for close encounters and few-body dynamics via the Kustaanheimo-Stiefel regularisation <cit.> and chain regularisation <cit.>.
Stellar evolution in is based on an upgraded version of the population synthesis code <cit.>. The main features of this state-of-the-art version, named BSE++, are described in detail in <cit.> <cit.>. We adopt the so-called level B of stellar evolution <cit.>, whose main characteristics are: delayed supernova (SN) scheme <cit.>, pair- and pulsation pair-instability supernova (PISN and PPISN) treated following <cit.>, fallback prescription for NS/BH natal kicks, and metallicity-dependend winds for massive stars <cit.>. We refer the reader to <cit.> and paper AS-I for further details.
The common envelope phase in binaries is modelled through the widely known α_ CE-λ_ CE scheme, which enables us to regulate the fraction of orbital energy injected into the envelope (α_ CE) and to scale the binding energy of the envelope by a factor λ_ CE. In this work, we adopt α_ CE = 3 and λ = 0.5 <cit.>.
The adopted stellar evolution recipes imply that the stellar BH mass-spectrum in clusters is limited to m_ BH,max = 40.5 <cit.>, unless BHs form from stellar mergers or star-BH interactions. In the latter case, parametrises the amount of mass accreted in a strong star-BH interaction or collision via an accretion parameter f_c <cit.>, which we set to f_c=0.5. We refer the reader to <cit.> for a discussion about the impact of f_c on BH evolution.
§.§.§ Modelling the final stages of compact object binary mergers
The dynamics of relativistic binaries is followed via the orbit-average formalism <cit.>, which enables us to follow the evolution of compact binaries and their coalescence inside the cluster, similarly to previous works <cit.>.
In its current implementation, follows the dynamics of relativistic binaries also if they are part of triples <cit.> and multiple systems, as well as if they form via hyperbolic interactions.
However, the BBH evolution is not followed down to the merger, rather the binary is decoupled from dynamics and promtply merged when the BBH pericentre falls below a critical value, which we set to 10^2 Schwarzschild radii, i.e. a_ dec = 2kGm_ bin/c^2 = kR_ Sch with k=100.
Adopting such limiting separation ensures that the binary is unlikely to undergo any further interaction with surrounding stars before merging. Considering the range of binary masses (1-300), star cluster masses (<10^6) and half-mass radii (0.1-3 pc) explored in this work, it is easy to show that the binary—single interaction timescale t_2-1 = (n σΣ)^-1 – with n the cluster density, σ the cluster velocity dispersion, and Σ the binary cross section – is generally >10^8 larger than the binary inspiral timescale, t_ insp∝ a^4/(m_1m_2m_ bin) <cit.>.
Moreover, the typical merger time for a binary with mass m_ bin < 200 and separation a_ dec is generally t_ insp < 100 yr, i.e. much smaller than the cluster crossing time, t_ step∼ 10^5 yr.
Therefore, our procedure ensures reliabile results while reducing the computational effort required to simulate the evolution of a binary with an orbital period of minutes or hours.
The pre-merger stages of the merging binary orbits are reconstructed by retrieving the orbital parameters at decoupling and integrating the orbit via the <cit.> equations:
a/ t = -64/5β(m_1,m_2)F(e)/a^3,
e/ t = -304/15β(m_1,m_2)e G(e)/a^4,
with
F(e) = (1 - e^2)^-7/2(1 +73/24 e^2 + 37/96e^4);
β(m_1,m_2) = (G^3/c^5) m_1m_2(m_1+m_2);
G(e) = (1-e^2)^-5/2(1+121/304e^2).
Along with the orbital evolution we calculate the associated GW strain and frequency <cit.>.
Natal spins of stellar BHs can be assigned according to different distribution, three of which are based on physical stellar model, namely the “Geneva”, “MESA”, and “Fuller” models <cit.>, and four are rather generic, namely zero-spins, uniform spin distribution, Gaussian spin distribution with mean value χ = 0.5 and dispersion σ_χ = 0.2, and Maxwellian distribution with dispersion σ_χ = 0.2.
In this work, whenever spins are taken into account during the simulation we assume a Gaussian distribution with χ = 0.5 for stellar BHs, whilst for IMBHs we decide on a case by case basis, depending on the IMBH formation scenario (see paper AS-II).
Compact binary merger products are assigned a final mass and spin calculated via numerical relativity fitting formulae <cit.> and a relativistic recoil, generated by asymmetric GW emission <cit.>, expressed via the following relation:
v⃗_ = v_mê_,1 + v_(cosξê_,1 + sinξê_,2) + v_∥ê_∥,
v_m = Aη^2 √(1-4η) (1+Bη),
v_ = Hη^2/1+q_(S_2,∥ - q_ S_1,∥),
v_∥ = 16η^2/1+q_[ V_11 + V_A Ξ_∥ + V_B Ξ_∥^2 + V_C Ξ_∥^3 ] ×
×| S⃗_2, - q_S⃗_1,| cos(ϕ_Δ - ϕ_1).
Here, η≡ q_/(1+q_)^2 is the symmetric mass ratio, Ξ⃗≡ 2(S⃗_2 + q_^2 S⃗_1) / (1 + q_)^2, and the subscripts and ∥ mark the perpendicular and parallel directions of the BH spin vector (S⃗) with respect to the direction of the binary angular momentum. We assume A = 1.2 × 10^4 km s^-1, B = -0.93, H = 6.9× 10^3 km s^-1, and ξ = 145^∘ <cit.>, V_11 = 3677.76 km s^-1, and V_A,B,C = (2.481, 1.793, 1.507)× 10^3 km s^-1. The vector Δ⃗ is defined as Δ⃗≡ (M_a+M_b)^2 (S⃗_b - q_S⃗_a)/(1+q_). The angle between the direction of the infall at merger and the in-plane component of Δ⃗, i.e. ϕ_Δ, is drawn from a uniform distribution, while ϕ_1 = 0-2π, which represents the phase of the binary, is extracted between the two limiting values according to a uniform distribution.
In , the user can decide to set the GW recoil to zero or to a fixed value, or to calculate it self-consistently via Eqs. <ref> and <ref>, in which case the kick is assigned to the remnant and the resulting energy correction is included in a similar way as it is done for natal BH kicks.
As described in detail in paper AS-II, in this paper series we adopt a simplified approach to investigate the impact of GW recoil in the simulations, owing to the fact that the relatively small sample of mergers does not enable us to filter out the inevitable stochastic effect of the BH spin directions and amplitudes on the kick amplitude.
The approach consists of three steps. First, we run all simulations assuming no GW recoil. Second, for each merger event in each simulation we evaluate the GR recoil assuming different distribution for BHs natal spins and we determine whether the remnant is likely to be retained or not in the cluster. Third, if a BH undergoes n mergers in a simulation with zero GW kick, we restart the simulation shortly before the n-th merger event and enable GW kicks assuming a spin for the merging components that depends on the BH formation history. This enables us to verify whether the remnant can be retained in the cluster and eventually merge again in a n+1th merger generation. In paper AS-II, we have shown that this approach permits us to highlight the fact that even when GW kicks are not taken into account, Newtonian dynamics is sufficient to eject all BH remnants from the parent cluster via strong binary-single encounters.
We note that none of the mergers with component masses <100 undergo multiple mergers. This suggests that even adopting zero GW recoils may have a negligible impact on the formation of compact binary mergers with mass <100.
§ RESULTS
§.§ The population of black hole binary mergers in Dragon-II clusters
In this section we describe the main results of our simulations, focusing on the population of compact binary mergers. Table <ref> summarizes the main properties of clusters and their compact objects.
The population of BHs formed in models and, in general, in star clusters likely suffers both the effects of single and binary stellar evolution and stellar dynamics. To highlight this aspect we show in Figure <ref>, for the models with R_ = 0.8pc and N=300k, the so-called initial to final mass relation (IFMR) that links the masses of compact objects and their stellar progenitors. The plot is dissected into BHs with a progenitor initially single or in a primordial binary system.
The population of BHs forming from single stars generally follows the expectations of the adopted stellar evolution recipes <cit.>. Deviation from the general trend owes to initially single stars that got caught in a pair and underwent mass-transfer.
The IFMR of BHs formed from stars in primordial binaries is more complex, being characterised, for example, by BHs in the upper mass-gap with masses in the range 40.5 - 80. This highlights the crucial role of binary stellar evolution and dynamics in sculpting the population of BHs in star clusters <cit.>.
§.§.§ Component masses and formation channels
The population of compact binary mergers in consists in 75 BH-BH, 2 NS-BH, and 1 WD-BH. Among BH-BH mergers, 45 involve two BHs below the PPISN maximum mass (m_ BH < 40.5), 12 involve two mass-gap BHs, and 21 involve one BH below the gap and a mass-gap BH. Six BH-BH mergers involve a primary with a mass m_ BH,1=(5.4-7.1) and a companion with mass m_ BH,2=(2.55-3.6), i.e. just above the threshold separating NSs and BHs in our models. All these low-mass mergers are in primordial binaries.
As discussed in paper AS-II, the BHs in the upper-mass gap mostly form in a star-BH accretion event, either by purely dynamical interactions or stellar evolution. We stress that, throughout our models, we assume that a fraction f_c = 0.5 of the star mass is accreted onto the BH during an accretion event <cit.>.
When GW recoil are “switched off” 4 mergers involve a second or third generation BH, i.e. which underwent one or two previous mergers. The inclusion of GW recoil reduces the number of total mergers to 74. For a detailed discussion about the impact of GW recoil, see paper AS-II.
Figure <ref> shows the component masses and mass ratio of mergers and of mergers observed during the first three LVK observation campaign, collected in the so-called GWTC-3 catalogue <cit.>. The plot includes all mergers occurring in the cluster or outside the cluter after being ejected via dynamical interactions considering zero GW kicks.
This plot illustrates the wealth of information hid in the Dragon-II star clusters: we find mergers in the upper-mass gap, IMBHs[In this work we set a mass treshold of M_ IMBH,min = 100 to discern between BHs and IMBHs.], repeated mergers, and in a handful cases also BHs merging with either a NS or a WD.
Interestingly, we find that mergers occurring inside the cluster are characterised by a primary with mass m_ BH,1 > 30 and a companion with a mass in the range m_ BH,2 = (20-50). Conversely, mergers occurring outside the cluster — or ejected mergers — are characterised by a mass-ratio q>0.6 and a primary mass typically m_ BH,1 < 40.
The number of mergers occurring inside the cluster (31) is comparable to that of binaries that merge after being ejected from the cluster (47), thus suggesting that in-cluster mergers can made-up the 40% of the total merger population. Among all of them, 27 are from primordial binaries (3 inside, 24 ejected), whilst 51 (28 inside, 23 ejected) are from dynamical binaries.
Figure <ref> shows the primary and companion mass of mergers originated from primordial, dynamical, or mixed binaries, with the latter identifying binary mergers in which at least one component was originally in a primordial binary. The plot exhibits some interesting features: 1) mergers from primordial binaries tend to have nearly equal-mass components, 2) purely dynamical mergers have masses that occupy a tight region of the plane with m_ BH,1=(20-50) and m_ BH,2=(20-40), 3) mergers with one component previously in a primordial binary are characterised by a heavy primary, m_ BH, 1 > 40, and a heavy companion, m_ BH,2 > 20. A similar trend is observed in recent N-body simulations tailored to relatively light star clusters, i.e. with mass <8,000 <cit.>.
As deeply discussed in paper AS-II, the crucial role of primordial binary dynamics is highlighted by the fact that all the IMBHs in clusters but one have an ancestor that was member of a primordial binary, regardless of the IMBH formation scenario.
Dynamics and binary stellar evolution deeply impact also the properties of stellar-size mergers. For example, “dynamical” and “primordial” mergers occupy two well separated regions of the primary mass - mass ratio plane.
The vast majority of primordial binary mergers occupy a region delimited by q > 0.6 and m_1 = (5-40), with the mass-ratio weakly increasing at increasing the primary mass: note that for m_1≲ 15 mergers have mass-ratio q=0.6-1, whilst mergers with a heavier primary have mass ratio q>0.85.
Dynamical mergers, instead, form in the right hand-side of Figure <ref>, generally at m_1 > 40.5. In this case, the mass ratio decreases with the primary mass as expected from the mass function limit, with companion masses in the range m_2 = (30-50).
We can identify three relatively well separated regions: low BH masses (m_ BH,1 <15) and widely distributed mass ratio (q=0.6-1) dominated by primordial binary mergers, BH masses in the range m_ BH,1 = (15-40.5) and high mass ratios (q>0.9) dominated by primordial binary mergers, and heavy BH primaries (m_ BH,1>40.5) with relatively massive companions (m_2=30-50) dominated by dynamical mergers.
In clusters, most binaries merging outside the cluster originate from primordial binaries and their ejection is typically triggered by the BH natal kick. However, all ejected mergers with component masses m_1,2 > 30 have a dynamical origin, owing to the adopted stellar evolution recipes.
We note that, given the limited simulation time, the population of mergers in clusters may lack some element that could form later in the cluster life, beyond several relaxation times. These late mergers would unavoidably have a dynamical origin, or at most a "mixed" origin, because all BHs formed in primordial binaries undergo a binary exchange or have been ejected in clusters over the simulated time. Moreover, late mergers will likely have smaller masses compared to those shown in Figure <ref>. This is mostly due to the BH-burning process, by which the average BH mass decreases over time <cit.>.
As a consequence, some BH mergers forming at late time may have properties similar to the primordial binary mergers shown in Figure <ref>.
Figure <ref> shows the mass distribution of the primary BH in mergers, dissected into in-cluster/ejected mergers and primordial/dynamical ones. Ejected binaries dominate the m_ BH,1≲ 20 mass range, whilst at larger primary masses their number and distribution is similar to that of in-cluster mergers. Dynamical mergers completely dominate the population of mergers with m_ BH,1 > 20, while primordial mergers dominate the population of lighter mergers. Noteworthy, we see that the primary mass distribution for mergers nicely overlap with the sample of mergers in the GWTC-3 catalogue, i.e. the catalogue of BBH mergers detected by the LVK collaboration <cit.>. However, a thorough comparison between modelled and observed mergers would require to take into account observation biases <cit.>. For this reason, we also overlay to our data the cosmic BH mass distribution inferred from GW detections.
Comparing models and observations can be crucial to assess the impact of different formation channels on the population of BH-BH mergers <cit.>. Our models suggest, for example, that BH mergers developing in star clusters could produce a substantial amount of mergers from primordial binaries. The progenitor binary could, in some cases, suffer the impact of dynamical interactions which may alter their orbital parameters. Nonetheless, in most cases BH mergers from primordial binaries could represent "isolated binary merger impostors", because they have properties typical of merging binaries developing within the isolated formation scenario but form in a dynamical environment. Taking into account the impact of these sources with a sort of mixed formation channel is crucial to correctly quantify the role of different formation channels in determining the shape of the mass distribution of detected merging BHs <cit.>.
Moreover, models highlights the role of dynamics in determining the formation of BH mergers with masses inside, and beyond, the mass-gap, supporting and complementing previous works on the topic based either on smaller, or lower-density, N-body cluster models and Monte Carlo simulations <cit.>.
§.§.§ Delay times
The delay time of mergers (t_), defined as the time elapsed from the beginning of the simulation to the binary merger, is rather peculiar. As show in Figure <ref>, it exhibits three peaks at t_≃ (0.5 , 1.5 , 10) Gyr. However, when the delay time is normalised to the initial half-mass relaxation time (t_ rlx) of the cluster, the overall t_ nicely distribute around a main peak located at t_ GW/t_ rlx≃ 8-30. The exact location of the peak depends on the definition of t_ rlx. For the sake of clarity, in the plot we use three different expressions of t_ rlx taken from <cit.> (GR21), <cit.> (AR16), or <cit.> (RN21).
The three peaks that appear in the t_ distribution find a clear explanation looking at the t_/t_ rlx distribution. In fact, the first peak at t_ GW = 500 Myr corresponds to mergers happening in simulations with t_ rlx=50-100 Myr, whilst the second peak corresponds to mergers occurring in clusters with a longer relaxation time (see Table <ref>). This interesting features suggests, on the one hand, that the delay time depends intrinsically on the cluster initial properties, as they determine the relaxation time, and, on the other hand, that dynamical processes operate in a similar way over a relaxation time regardless of the cluster structure. The third peak, instead, corresponds to ejected binaries that merge outside the cluster, which are mostly products of primordial binaries ejected via SN explosion during the formation of one of the BHs in the pair.
§.§.§ Eccentricities
One intriguing question that arose since the first detection of GWs is whether it is possible to untangle different formation channels in the observed population of BH mergers. Among all the parameters at play, the orbital eccentricity could represent the key to answer this question. Broadly speaking, in fact, most BH mergers forming via binary stellar evolution are expected to feature a negligible eccentricity close to merger, either because the BBH progenitor undergoes common envelope, which shrinks and circularise the orbit, or because the BBH separation is initially so large that GW emission circularise the orbit before the merger. Binaries developing in star clusters, instead, can form with high eccentricity and sufficiently small separation that the merger occurs on a timescale shorter than GW circularisation.
At the lowest-order level, binaries merging in galactic fields, often called isolated binary mergers, are expected to be circular GW sources, whilst at least some of those developing in star clusters and galactic nuclei, named dynamical mergers, can preserve a significant eccentricity (i.e. e > 0.1) when entering the typical frequency bands of GW detectors.
This simplistic division between isolated and dynamical binaries does not take into account several layers of complication. For example, it is well known that star clusters and stellar nurseries may contain a large fraction of binaries, especially among the population of massive stars, where the percentage of paired stars attains values as large as 50-100% <cit.>. If primordial binaries evolve on a timescale shorter than the typical timescale of dynamical interactions, star cluster could harbor a sub-population of compact binary mergers with properties pretty similar to those forming in galactic fields, e.g. low eccentricities or peculiar component masses and mass-ratios.
With up to 33% of stars initially paired, simulations offer us the possibility to search for differences between mergers forming entirely via dynamics and those forming from the evolution of primordial binaries. Figure <ref> shows the semimajor axis and eccentricity of all BH-BH mergers in clusters calculated at the moment of decoupling, i.e. when the GW emission starts dominating over dynamical perturbations. The plot dissects the population of BH mergers into those coming from the evolution of primordial binaries, those assembled purely via dynamical interactions, and those involving at least one component that was former member of a primordial binary. The population of dynamical and mixed binaries seem to follow two different sequences, although the low statistics make hard to understand whether they actually exist. The population of nearly circular primordial binaries is evident. These mergers can be considered mimickers of the field merger population, and constitute the 33% of the whole population of mergers. Only two of the primordial binaries exhibit a significant eccentricity and a relatively small separation.
The first is a NS-BH binary, we postpone a discussion about this specific source to the next subsection.
The second one involves two low-mass BHs, with masses m_ BH1,2=(7.1+2.55) and eccentricity e=0.997. The progenitor of this merger was a binary that underwent a common envelope phase first, after which the first BH formed, and later undergo Roche lobe overflow, at the end of which also the second BH forms and receives a small kick (∼ 3 km/s) that triggers the eccentricity increase.
As the binary shrinks and circularises because of GW emission, its frequency will increase. Therefore, a first step to determine whether a binary merger can appear eccentric in the sensitivity band of a specific GW detector requires to compare the binary eccentricity and the corresponding GW frequency.
We show in Figure <ref> the characteristic strain - frequency evolution for all mergers in our sample, assuming that they are located at a redshift z = 0.05, i.e. at a luminosity distance of 230 Mpc. To calculate the GW strain of sources we follows the formalism laid out in <cit.> and the implementation described in <cit.> (see Eqs. 30-39). The GW signal from simulated mergers is overlaid to the sensitivity curves of current and future ground-based and space-based detectors like LIGO <cit.>, Einstein Telescope <cit.>, DECIGO <cit.>, and LISA <cit.>. The plot highlights how the eccentricity drops as the binary sweeps across different frequency bands.
The top panel in Figure <ref> shows the fraction of mergers with eccentricity above a given threshold calculated when mergers sweep through five frequency bands centered in f_ band = 10^-3-10^-2-10^-1-1-10 Hz, i.e. the typical sensitivity bands of space-borne detectors like LISA (<10^-2 Hz), mid-frequency detectors like DECIGO (10^-2-1 Hz), and ground-based detectors like LIGO-Virgo-Kagra or the Einstein Telescope (>1 Hz). The plot highlights the fact that around 20-40-5% of all mergers appear eccentric, i.e. e > 0.1, while moving through the f = 10^-3-10^-1-10^1 Hz frequency bands, respectively. Clearly, the detectability of these mergers depend on many parameters, among which the location of the merger and the detector properties. Nonetheless, the plot makes apparent the importance of future deci-Hz detectors in placing constraints on the population of eccentric BBH mergers. Moreover, comparing models with future observations will help to quantify the impact of star cluster dynamics on the cosmic population of merging BHs.
Noteworthy, the eccentricity carries information about the formation history of the merger. For example, we find that all mergers with an eccentricity e>0.1 in both the 0.05-1 Hz and 1-10 Hz frequency bands occur inside the cluster. The number of eccentric binaries doubles in the 10^-2-1 Hz frequency band, but these eccentric binaries appear almost circular while reaching the ground-based detector band, explaining why it is more likely to find a merging binary with significant eccentricity while sweeping through the deci-Hz band.
Any binary merger will spend some time in the detector band before merging. In order to characterise the evolution of the eccentricity as the binary inspirals, we calculate the average binary eccentricity weighted with the time to the inspiral, i.e. ⟨ e ⟩ = ∫_0^t_ merg e dt / ∫_0^t_ merg dt. Practically, we measure the binary eccentricity in subsequent time bins from the time of decoupling to the time of merger and weight it with the remaining time to the merger. This quantity is shown for all mergers in the bottom panel of Figure <ref>, along with the evolution of the eccentricity as a function of the peak frequency <cit.>
f_p = 0.29 Hz(m_1+m_2/30)^1/2(a/50 R_⊙)^-3/2×
× ceil[1.15(1+e)^1/2/(1-e)^3/2],
The step-like behaviour of the e-f_p is due to the ceil function in Equation <ref>, which returns the nearest integer larger than the function argument.
The majority of in-cluster mergers clearly show an average eccentricity ⟨ e ⟩ > 0.7 across the whole 0.01-100 Hz frequency spectrum, whilst ejected mergers preserve a moderate eccentricity ⟨ e ⟩ < 0.4 in the f<1 mHz band. This suggests that GW detectors operating in different bands can probe different sub-populations of mergers forming in dense star clusters, with high-frequency detectors being more suited to observe short-lived, highly eccentric mergers occurring inside star clusters, and low-frequency detectors more suited to observe GW sources merging outside their parent cluster.
§.§ Exotic mergers
Despite the relatively small simulation grid, we also find some exotic mergers: a dynamical WD-BH and 2 NS-BH mergers, one dynamical and one from a primordial binary. The three mergers occur in the most dense simulations in our sample: the WD-BH merger occurs in a simulation with N=120k, R_=0.47pc, the dynamical BH mergers develop in a simulation with N=300k, R_=0.47pc, and the one forming from a primordial binary in a simulation with N=120k, R_=0.8pc.
This type of mergers are particularly rare in star clusters, because dynamical exchanges favor the replacement of the light component with another BH. Given their rarity, we discuss in the following the details of the formation and evolution of these interesting sources.
§.§.§ White dwarf - black hole mergers: implications for low-mass X-ray binaries
The WD-BH binary consists in a BH with mass m_ BH = 23.1 and a carbon-oxygen white dwarf (COWD) with mass m_ WD = 1.18. Initially, dynamical interactions pair the BH with the WD progenitor, a MS star with mass m_ MS,pro = 4.89. The two objects undergo common envelope during the late AGB phase of the companion, at the end of which the star turns into a WD, after ∼ 105 Myr. The resulting WD-BH binary has an "initial" eccentricity of e = 0.2 and period of 900 days. The binary undergoes a series of strong scatterings that cause a rather chaotic variation of the binary semimajor axis and a systematic increase of the eccentricity from e=0.6 up to e=0.99994930 after 135 Myr, corresponding to ∼ 4 relaxation times. At this stage, GW emission becomes sufficiently effective to drive binary coalescence. Figure <ref> shows the time variation of the WD-BH binary semimajor axis and eccentricity before coalescence.
The WD Roche lobe is larger than the BH innermost stable circular orbit, hence the WD will likely undergo disruption and start feeding the BH, possibly evolving into a low-mass X-ray binary. In these regards, it is interesting noting the observation of a ultracompact X-ray binary in the galactic cluster 47 Tuc <cit.>, likely comprised of a COWD and a BH <cit.>, with the BH being probably heavier than m_ BH > 9 <cit.>.
Our model confirms the possibility to form such type of low-mass X-ray binary via interactions of stars and BHs in a dense cluster, even in a relatively short time (t < 200 Myr).
Ultimately, the binary shrinkage driven by GW emission will disrupt the WD and the mass falling onto the BH could possibly power jets that can give rise to transients with peak energy 10^47-10^50 erg^-1 and duration of a few minutes <cit.>, giving rise to a tidal disruption event (TDE).
Despite this source is the only one undergoing coalescence, we find a total of 50 WD-BH binaries by the end of the simulation in all clusters. None of them have orbits such to trigger a TDE within a Hubble time, unless a strong interaction with some cluster member pushes the WD on an extremely eccentric orbit. Pushing the orbit to at least e > 0.9999(0.99999) would lead to 1(26) further WD-BH mergers.
Note that the eccentricity value required to trigger a WD TDE may seem extreme, but it is comparable to the eccentricity achieved by the WD-BH merger, hence testifying that it is possible to reach such extreme eccentricity values in clusters.
§.§.§ Neutron star - black hole mergers: implications for multimessenger astronomy
Concerning NS-BH binaries, we find two mergers, one of dynamical origin and the other forming from the evolution of a primordial binary.
The dynamical NS-BH has a NS with mass m_ NS=1.28 and a BH with mass m_ BH = 14.96.
The BH, whose progenitor had a mass m_ MS = 26.7, undergoes a series of chaotic interactions with a primordial binary containing the NS and its companion, which eventually leads to the merger. When the binary decouples from the cluster dynamics, it has a semimajor axis of a = 0.33 AU and an eccentricity e = 0.99778817, corresponding to a GW peak frequency f_ GW = 0.01 Hz.
After decoupling, the binary evolution is completely dominated by GW emission and the variation of its orbital parameters can be described, at first order, via the <cit.> formalism. We find that as the binary sweeps through the 0.01-0.5-1-10 Hz GW frequency band the NS-BH merger has a residual eccentricity of e_ NSBH = 0.99779-0.9974-0.21-0.02, thus in principle measurable with future GW detectors, especially with those operating in the deci-Hz frequency band.
The chirp mass of this merger, ℳ_ chirp = 3.4, is typical of dynamically assembled NS-BH mergers <cit.>, but hard to produce with isolated binary evolution <cit.>, although this strongly depends on the adopted stellar evolution scheme <cit.>.
The primordial NS-BH binary merger, instead, forms from a primordial binary with initial mass components m_1,2 = (26.3 + 18.7) and evolves through a common envelope phase initiated by the primary, which eventually forms the BH. Shortly after, the binary undergoes a second common envelope phase and eventually the companion evolves into a NS. Eventually, the merging binary consists of a BH with mass m_ BH = 5.6 and a NS mass m_ NS=1.88. Note that these properties, are intriguingly similar to GW200115, a GW source detected by the LVK during the O3 observation campaign, which was characterised by a BH with m_ BH= 5.7^+1.8_-2.1 and a NS with m_ NS = 1.5^+0.7_-0.3.
When the NS forms, the common envelope has shrunk the binary from 2.5 R_⊙ to a = 0.6 R_⊙, whilst the natal kick imparted at formation onto the NS causes an enhancement of the eccentricity from nearly zero to e= 0.57. The new orbital parameter as such that GW emission dominates over dynamics and the binary coalesces in ∼ 7× 10^4 yr. At decoupling, the binary peak frequency is f_∼ 2 mHz, right in the middle of LISA sensitivity band.
The development of a NS-BH binary merger from a primordial binary in a dense star cluster highlights the impact of primordial binaries in contributing to the population of mergers with properties similar to those forming in isolation, making quite hard untangling their actual origin.
Merging NS-BH binaries are thought to be possible progenitors of several electromagnetic (EM) transients, like short Gamma Ray Bursts (sGRBs) <cit.> and kilonovae <cit.>. A basic condition for the possible development of an EM transient is that part of the NS material remains bound to the BH, forming a disk. The fraction of NS mass in the disk depends on several quantities, among which the BH-to-NS mass ratio m_ BH/m_ NS, the BH spin χ, and the NS compactness C≡ Gm_ NS/c^2 R_ NS <cit.>. As numerical simulations have shown, in general the larger the m_ BH/m_ NS the larger the minimum spin required for the NS material to form a disk around the BH, and the larger the spin the larger the amount of matter bound to the BH <cit.>. Depending on the orbital parameters, the BH tidal field can tear apart the NS before it enters the BH event horizon, provided that the NS tidal radius
R_ tid = R_ NS(3m_ BH/m_ NS)^1/3,
exceeds the BH innermost stable circular orbit (ISCO), which for a spinning BH can be expressed as <cit.>
R_ ISCO = Gm_ BH/c^2[3 + Z_2 - sign(χ) [(3 - Z_1)(3 + Z_1 + 2Z_2 )]^1/2],
where Z_1,2 are functions of the BH adimensionless spin parameter χ. Whilst the condition R_ tid / R_ ISCO < 1 implies that the merger has no EM emission, the opposite does not ensure the EM counterpart detectability, as it depends on the geometry of the merger with respect to the observer and other possible observation biases.
In clusters, the dynamical NS-BH merger is characterised by m_ BH/m_ NS = 11.7 and compactness C = 0.19 (assuming a NS radius of 10 km). As shown in Figures 6-8 of <cit.>, the minimum BH spin required for an accretion disk to form with a mass 10% of the NS mass around such type of binary is χ_ BH > 0.98. The BH formed in this binary did not undergo any major interaction with stellar companions that could spin-up it <cit.>. Hence, it is possible that the BH formed with low-spin, according to the <cit.> model, hampering the formation of a massive accretion disk around the BH and minimizing the probability for a EM counterpart to develop.
The isolated NS-BH merger, instead, is characterised by m_ BH/m_ NS = 2.98 and C = 0.27. Even in this case, the spin required for an accretion disk to form is χ_ BH > 0.9. The BH in this binary undergoes a RLO phase, which could, in principle, spin-up the BH up to extremal values <cit.>, although this strongly depends on the stellar evolution recipes and the binary properties <cit.>.
The development of just 2 NS-BH mergers highlights how rare are these type of objects on the one hand, and make any statistical analysis poor, on the other hand. Nonetheless, the fact that the NS-BH mergers developed in clusters seem to be unlikely to feature an EM counterpart supports the idea that most NS-BH mergers proceed unseen in star clusters <cit.>. For comparison, note that for isolated binaries typically m_ BH∼ 12 and m_ NS = 1.6 <cit.>, which implies a minimum BH spin of χ_ BH≳ 0.8 to permit the formation of a fairly massive (mass > 0.1m_ NS) disk around the BH <cit.>.
§ DISCUSSION
§.§ The impact of natal spins on the properties of stellar black hole mergers
Spin amplitude and mutual orientation at merger represent two possible quantities that can help discerning whether a BBH merger results from isolated stellar binary evolution or stellar dynamics <cit.>.
In order to explore the impact of different spin prescriptions on mergers, we devise two models.
The first model (hereafter STEV) assumes that the spin is intrinsically related to the BH evolutionary pathways. For BHs formed from single stellar evolution, we assume a negligible spin (χ_ BH = 0.01) owing to efficient angular momentum transport triggered by the Tayler-Spruit dynamo <cit.>. For upper-mass gap BHs formed from massive binary evolution we assume that final spins spans the χ_ BH = 0.8-1 range <cit.>. For BHs in primordial binaries, instead, we assign to one BH a spin value of χ_ BH=0.01 and to the other χ_ BH = 0.1-1 <cit.>.
The second model (GAUS model) assumes, instead, that the spin distribution follows a Gaussian distribution with mean χ̅_ BH = 0.5 and dispersion σ_χ = 0.2, regardless the BH past evolution, a case possibly supported by the population of observed BH-BH mergers <cit.>.
In our analysis, we assume that the spin vectors in dynamical mergers are isotropically distributed, whilst for primordial mergers we proceeds as follows. We define an ad-hoc distribution function for the cosine of the angle between the spin of the i-th binary component and the binary angular momentum, θ_i, such that <cit.>
P(cosθ) = [(cosθ + 1)/2]^n_θ+1.
We set n_θ = 8, which implies that binaries have a 20(55)% probability to have θ_1,2 that differ by less than 5(20)%. Note that n_θ = 0 implies the isotropic distribution whilst n_θ≫ 1 implies fully aligned spins, i.e. θ_1 = θ_2.
For each BBH merger in our sample we select 1,000 values of the spin and spin directions depending on the aforementioned assumptions, in order to assess statistically the properties of mergers. The top panels in Figure <ref> show the median value and 95th percentile of the effective spin parameter and remnant BH mass for all BBH mergers in models. As, expected, we can clearly see a difference between primordial binaries, which have mildly aligned spins and thus χ_ eff>0, and dynamical binaries, for which χ_ eff∼ 0. The plots suggest that the STEV model, based on stellar evolution models, leads primordial binaries to have a χ_ eff smaller, on average, compared to the GAUSS model. The bottom panels of Figure <ref> overlay to a single realisation of the simulated data the observed mergers from GWTC-3, for comparison's sake.
Noteworthy, the assumption that BHs form with a negligible spin unless matter accretion processes are at play (STEV model) leads to a sub-population of mergers with χ_ eff∼ 0 and m_ bin = (40-100), a feature that disappears when a global Gaussian spin distribution is adopted (GAUS model) as shown in Figure <ref>.
If BH spins do not strongly depend on stellar evolution processes, but rather are well described by a general distribution, like a Gaussian, we can identify two populations in the plot, one with clearly positive χ_ eff values and m_ BH < 40, and one widely distributed around zero χ_ eff involving massive BHs, m_ BH > 40.
In order to improve the poor statistics, we proceed as follows: from the list of mergers we create an oversampled catalogue by repeating the spin assignment 100 times and, at each time, selecting a new "mock" BH mass in the range 2.5-40.5 if the BH merger mass is below the upper-mass gap, and in the range 40.5-100 otherwise. This way, each real merger will have 100 counterparts with BHs of the same class (upper mass-gap or not, merger in primordial or dynamical binary), but enabling to build-up a catalogue sufficiently rich to analyse the overall χ_ eff distribution. Figure <ref> shows the distribution of χ_ eff for the augmented sample in STEV and GAUS models.
We see that the STEV model follows a narrower distribution compared to the GAUS model, and exhibits a clear peak around zero owing to the population of BHs formed from single stars <cit.>.
§.§ Compact binary merger rates
§.§.§ Merger efficiency
As described in the previous section, we have simulated a total mass of M_ sim = 3.65× 10^6 and find in total 78 mergers when GW recoil is not accounted for, and 74 otherwise. Therefore, the resulting BH merger efficiency, defined as the ratio between the number of mergers and the total simulated mass <cit.>, is given by
η_ GW = N_ GW/M_ sim≃ (2.0-2.1)×10^-5^-1,
similar to what inferred for young and open clusters with a similar metallicity <cit.>. Note that given the limited simulation time our estimate could represent a lower limit to the total merger efficiency in massive young and intermediate-age clusters. Nonetheless, we note that as the cluster loses mass and expands, the binary formation rate and binary–single interaction rate will sharply decrease until the point in which it will be unlikely for tight binaries to form and merge within a Hubble time.
Interestingly, at fixed value of the half-mass radius, the merger efficiency changes sensibly with the initial binary fraction, being
η_ GW, fb =
2.3 × 10^-5^-1 f_b = 0.20,
1.2 × 10^-5^-1 f_b = 0.05.
This highlights the role of primordial binaries in determining the formation of merging compact objects. For comparison, note that the merger efficiency derived in <cit.> is based on star cluster models containing ∼ 40% of stars in primordial binaries.
To further explore the impact of cluster properties on the merger efficiency, we show in Figure <ref> the average merger efficiency per cluster, ϵ_(R_), as a function of the average cluster density ⟨ρ_ sim⟩, using the following definitions
ϵ_(R_) = N_/(M_ sim/N_ sim),
⟨ρ_ sim⟩ = M_ sim/N_ simR_^3
where M_ sim is the total simulated mass and N_ sim is the number of simulations performed for a given value of the half-mass radius, R_.
At fixed value of the binary fraction, this relation is well described by a power-law in the form ϵ_ = a (⟨ρ_⟩ / 1 M_⊙ pc^-3) ^ b, with a = (0.15±0.07)× 10^-5 and b = 0.25 ± 0.03.
The plot makes clear that increasing the cluster density by two orders of magnitude leads to ∼ 2.5× more mergers. Moreover, it further highlights the role of primordial binaries, showing that clusters with a lower binary fraction have a probability ∼ 50% smaller to develop a merger, at least in the case of R_ = 1.75 pc.
§.§.§ Merger rate for black hole binaries
We define the cosmic merger rate following <cit.>
ℛ(z) = / t_ lb(z)∫_0^z_ maxψ_ clus(z') t_ lb(z)/ z' z'
×∫_Z_ min^Z_ maxη_ GW(Z)ℱ(z',z,Z) Z,
where t_ lb(z) is the lookback time at merger, ψ_ clus(z') is the star cluster formation rate when the merging binary formed, η_ GW(Z) is the merger efficiency at the metallicity Z, ℱ(z',z,Z) is the number of mergers forming at redshift z' and merging at redshift z in environments with metallicity Z.
The adoption of Equation <ref> enables us to compare simulation results with those obtained for low-mass star clusters <cit.>. Note that this procedure does not take into account possible effects related to the initial cluster mass function, which could indeed have an impact on the overall merger rate <cit.>. Nonetheless, the similarity between the merger efficiency derived from simulations and that obtained by <cit.> for low-mass clusters suggests that it is possible to safely utilise the merger efficiency as a proxy of the overall number of mergers per unit mass in the whole range of possible cluster masses. This choice, although representing an approximation, permits us to avoid the inclusion of a cluster mass function in Equation <ref> and all the related uncertainties, like the cluster mass function boundaries and functional form.
We adopt a cosmic star cluster formation rate in the form
ψ_ clus(z) = 0.01 (1+z)^2.6f_ CFE/ 1 + [(1+z)/3.2]^6.2 yr^-1 Mpc^-3,
i.e. we rescale the stellar star formation rate derived by <cit.> by a factor f_ CFE, which represents the cluster formation efficiency, i.e. the fraction of star formation that goes into bound clusters. Although uncertain, observations and models suggest that the cluster formation efficiency (CFE) can be as large as f_ CFE,YC = 0.3 for young clusters <cit.> and f_ CFE,GC = 0.08±0.03 <cit.> for globular clusters, regardless of the star formation history. In the following, we adopt both young and globular cluster CFE values to constrain the BBH merger rate in our simulations.
For dynamical mergers, it has been shown that the merger efficiency η_ GW(Z) remains almost constant in the range Z<10^-3, and decreases roughly by an order of magnitude at solar values <cit.>. Since our models have all the same metallicity, Z = 0.005, to infer the merger rate we assume that the merger efficiency is constant at Z<0.005 and reduces by 10 times at larger metallicities <cit.>. Moreover, we factorise the function F(z,z',Z) = p(Z,z') N(z,z'), thus assuming that the number of mergers at redshift z that formed at z' is independent on the metallicity distribution. The p(Z,z') term represents the cosmic fraction of clusters with metallicity in the (Z,Z+dZ) bin at redshift z'.
We assume that the metallicity follows a log-normal distribution peaked at <cit.>
Log⟨Z(z)/ Z_⊙⟩ = 0.153 - 0.074z^1.34,
with dispersion either σ_Z = 0.2-0.5-0.8 <cit.>.
Since all models have the same metallicity, to infer the simulated merger rate we integrate Equation <ref> under two assumptions, one conservative and one optimistic. In the conservative case, we consider only clusters with a metallicity Z<0.005 and assume that they have a similar merger rate efficiency <cit.>. In the optimistic case, instead, we include in the integration also clusters with metallicity larger than the simulated one, reducing for metal-rich clusters the simulated merger efficiency by a factor 10, as expected from low-N simulations of young clusters <cit.>.
To compare with similar estimates in the literature, we first set f_ CFE = 1, i.e. that all stars form in star clusters, and calculate a merger rate of ℛ = 27, in broad agreement with the rate inferred for low-mass star clusters (N=10^2-5× 10^4) <cit.> and semi-analytic models of young and globular clusters <cit.>.
A more reliable estimate of the merger rate is shown in Figure <ref> for both the conservative and optimistic cases, and assuming different values of the cluster formation efficiency, f_ CFE=0.08-0.3.
As shown in the plot, we find a simulated merger rate of ℛ_ GW = (12±7) at redshift z=0.2. At the same redshift, the BBH merger rate inferred by the LVK is ℛ_ LVK=17.9-44 <cit.>.
§.§.§ Merger rate for exotic mergers
In simulations we find 3 elusive mergers: one WD-BH and two NS-BH mergers. Despite they are evidently too scarce to allow a statistical treatment, we can exploit them to attempt a rough, order of magnitude, estimate of the merger rates for these two classes of GW sources assembled in star clusters as:
R_ xBH(<D) = N_ x/M_ sim f_xδ M_g* N(<D) t_ rel^-1,
where M_g* is the galaxy stellar mass, δ=0.001-0.01 is the fraction of galaxy mass made up by star clusters <cit.>, f_x is the fraction of clusters with a given property (e.g. age within a certain range),
t_ rel is the cluster relaxation time, and N(<D) is the number of MW equivalent galaxies within a given cosmological distance D <cit.>
N(<D) = 4π/3 (2.26)^-3(D/ Mpc)^3 ρ_g/ Mpc^-3,
where ρ_g = 0.0116 Mpc^-3 is the galaxy number density in the local Universe <cit.>.
Moreover, we consider typical relaxation times of either globular clusters, t_ rel = 10^9 yr <cit.>, or massive and relatively young clusters in the Small Magellanic Cloud (SMC), t_ rel = 3.2× 10^7 yr <cit.>.
Note that the relaxation time of Galactic clusters is inferred from their present time properties. Depending on the amount of mass lost and the level of cluster expansion, it could be possible that the initial relaxation time was relatively shorter and therefore the number of dynamically old globular clusters is larger than what we see at present. In these regards, note that the relaxation time of SMC clusters, which are generally younger than a few Gyr, is sensibly smaller compared to Milky Way globulars, possibly because relaxation processes did not have time to sufficiently influence the cluster dynamics.
In the following calculations, we consider Milky Way-like galaxies only, M_g* = 6× 10^10 <cit.> located within D = 1 Gpc. In the Milky Way, there are only ∼ 4 out of 155 globular clusters with an age larger than 1 relaxation time, whilst around half of clusters in the SMC satisfy this requirement, thus f_x∼ 0.025 - 0.5.
This implies a frequency rate for WD-BH mergers in the local Universe of R_ WDBH = (1.8×10^-3 - 10.8) yr^-1, corresponding to a volumetric merger rate ℛ_ WDBH = RV_ com^-1(1 Gpc) = (3.8×10^-4-2.3).
In the case of NS-BH mergers, instead, the event occurs over a timescale of t_ = (0.04-0.5)t_ rel. The fraction of cluster with an age longer than t_ is f_x∼ 0.94 for clusters in both the Milky Way and the SMC, the resulting frequency rate for NS-BH mergers is R_ NSBH = (0.13-40.7) yr^-1, which implies a volumetric merger rate of ℛ_ NSBH = (0.027-8.7).
§.§ Multimessenger sources: prospects for LISA detection
Over the next decade, the network of ground-based detectors will be complemented by LISA, possibly the first space-borne low-frequency detector. LISA will be able to possibly catch the GW echoes of merging stellar BHs, IMBHs, and nearby WD and NS binaries. While we postpone a detailed discussion about BBHs forming in young massive clusters detectable with LISA to a forthcoming paper, we focus in the following on the handful exotic mergers that develop in our models.
Let us consider the case of a WD-BH merger. We have shown in Section <ref> that such a source could appear as an X-ray binary and give rise to a TDE once the WD approaches too closely the BH. Assuming that the binary evolves solely via GW emission, and adopting the <cit.> formalism to evolve the binary until the merger, we find that around 6 months prior to the merger, the WD will overfill the Roche lobe and start the X-ray binary phase.
At disruption, the frequency of the associated GW emission is given by <cit.>
f_ GW≃ 0.09 Hz(1+M_ WD/M_ BH) ×
×(M_ WD/0.6)^1/2(R_ WD/10^4 km)^-3/2 = 0.13 Hz,
where we have assumed R_ WD = 10^4 km. Note that an eccentricity between 0 and 1 would affect f_ GW by less than 20% <cit.>. The amplitude of the emitted signal at disruption will be <cit.>
h_c≃ 2×10^-20(T_ obs/4 yr)^1/2(D_L/10 Mpc)^-1(M_ BH/10)^0.66×
×(M_ WD/0.6)^1.58(R_ WD/10^4 km)^-1.75≃ 10^-19.
Since the WD will disrupt completely as crossing its Roche limit, the associated GW emission will appear as a burst <cit.>. For such source, the corresponding signal-to-noise ratio (S/N) for LISA can be written as <cit.>
( S/ N) = f^2/3h_c/S_c = 1.2(D_L/10 Mpc)^-1,
where S_c is the detector sensitivity curve in terms of characteristic strain <cit.> and we have exploited the intrinsic dependence on the measurable GW strain and the source luminosity distance D.
If the merger occurs inside the Milky Way, i.e. at D < 0.1 Mpc, it would appear as a loud source in LISA, with (S/N)> 120. More in general, the maximum distance at which LISA could detect such merger with a minimum signal-to-noise ratio of (S/N)>8(15) is D < 1.5 Mpc(0.7 Mpc).
Note that the Andromeda galaxy is ∼ 0.7-0.8 Mpc away from us, therefore to roughly estimate the probability for a closeby WD-BH merger we can replace in Equation <ref> N(<D) = 2 and find an upper limit to the local merger rate of closeby WD-BH mergers of R_ WDBH,close < (8.4× 10^-10 - 5.1× 10^-6) yr^-1.
§.§ The pair-instability supernova rate for massive star clusters: perspectives for detection via magnitude limited surveys
The onset of IMBH formation and the development of BBH mergers depend intrinsically on the cluster radius and initial density, the amount of stars initially in a binary, and the stellar evolution recipes adopted – e.g. BH matter accretion efficiency, the physics of PISNe and PPISNe.
In these regards, the fact that PISNe are rare events for which a smoking gun has not been observed yet <cit.>, offers us the possibility to use this physical process as a diagnostic quantity in models.
In practice, we can infer the PISN rate in simulations and compare such rate with current observation limits to explore whether our simulations produce unrealistically large PISN frequency rates.
As described in paper AS-II, in models PISNe develop either in single stars or in stellar merger products, provided that their core Helium reaches a mass in the range (64-130). This offer us a unique possibility to explore the impact of PISNe in star clusters, taking simultaneously into account the impact of stellar mergers in the overall population of PISN progenitors. According to the adopted stellar evolution, in a simple stellar population only stars heavier than m_ ZAMS≥ m_ PISN = 150 could undergo a PISN event, i.e. larger than the maximum stellar mass adopted for the initial mass function. Instead, in models we find 23 stars that undergo supernova. All these stars are either in a primordial binary or are captured in a binary before the explosion and undergo one or more stellar merger and accretion events that bring the star mass above m_ PISN. Typical masses for PISN progenitors are in the range (150-282).
The simulated PISN efficiency can be defined similarly to the compact object merger rate, i.e. η_ PISN = N_ PISN/M_ sim = 6.2× 10^-6^-1.
To calculate the PISN rate, we follow the approach adopted by <cit.>. Firstly, we assume that the Ni mass of the massive star that goes off as a PISN can be calculated via the following equation:
Log(M_ Ni/) = r (M_ He, f/)^s + t,
where r = -5.02× 10^4, s = -2.159, and t = 2.887 <cit.>, and M_ He,f is the final mass of the star He core. The Ni mass is used to infer the peak bolometric magnitude exploiting an Arnett-like relation <cit.>
Υ_ bol, Ni^ = -19.2-2.5 Log(M_ Ni/0.6),
which can be converted into an apparent bolometric magnitude via the Pogson's relation
μ_ bol^ = Υ_ bol, Ni^ + 5 Log(D_L / 10 pc),
being D_L the luminosity distance. To simplify the calculations, we adopt for the He mass, which is the main ingredient to calculate the Ni mass, an average value of M_ He,f = 90.4 as extracted from our models.
The value of μ_ bol^ is used to calculate whether a PISN can be detected in a magnitude limited survey. Assuming to have a population of PISNe with apparent magnitude distributed according to a Gaussian around μ_ bol^ and a magnitude detection threshold μ_ lim, we define the fraction of detectable sources as
f_ GSS = 0.5[1+ erf(μ_ lim-μ_ bol^/√(2)σ_μ)] ,
where we adopted σ_μ = 0.2[We verified that varying σ_μ in the range 0.1-0.3 has little effect on our results.].
The PISN rate as a function of the redshift can thus be evaluated as:
ℛ_ PISN(z) = ∫_z_1^z_2dV/dzψ(z) η_ PISN f_ GSS(z) f_ Z(z) dz,
where dV/dz is the comoving volume element and ψ(z) is the cosmic star formation rate, for which we assume the cosmic star formation history in Equation <ref> <cit.> and the same limits for f_ CFE described in Section <ref>, and that only stars with a metallicity Z ≤ 0.008 undergo PISNe <cit.>. Figure <ref> shows the PISNe rate for the intrinsic cosmic population and assuming different detection threshold in magnitude limited surveys, namely μ_ bol = 17, 20, 25. Note that these threshold roughly corresponds to the typical maximum detectable magnitude of already completed, like the Sloan Digital Sky Survey (SDSS[SDSS home: <http://www.sdss.org>]) or the Palomar Transient Factory (PTF[PTF home: <http://www.ptf.caltech.edu>]), ongoing, e.g. the Dark Energy Survey (DES[DES home: <http://www.darkenergysurvey.org>]), and future surveys, like the Large Synoptic Survey Telescope (LSST[LSST home: <http://www.lsst.org>]), the Zwicky Transient Facility (ZTF[ZTF home: <https://www.ztf.caltech.edu>]), or the EUCLID mission[EUCLID home: <https://sci.esa.int/web/euclid>] <cit.>.
From Figure <ref> we see that only future surveys (μ_ bol≥ 25) will be able to probe the cosmological properties of PISNe, whilst current surveys could in principle place constraints on PISNe within a redshift z<0.3.
Integrating Equation <ref> over the redshift returns the number of detected sources per year. The possible number of PISN detections per year for different values of the limiting bolometric magnitude, μ_ bol, and the cluster formation efficiency, f_ CFE, is summarized in Table <ref>. From the table is clear that the detection of PISNe from star clusters is still highly unlikely in completed and ongoing surveys, but it could lead to ∼ 8 detections per year with the next generation of detectors. Comparing future PISNe detections with numerical models could have a twofold aim. On the one hand, it will permit us to shed light on the actual contribution of massive stars in dense clusters to the overall population of PISNe. On the other hand, it will provide us with an useful term of comparison to determine the reliability of cluster simulations.
§ CONCLUSIONS
In this paper we have presented and discussed the properties of compact binary mergers and PISNe in the simulations, a suite of direct N-body models representing star clusters with up to 1 million stars and a relatively large (10%-33%) binary fraction. Our main results can be summarised as follows:
* We find a population of 75 BBH, 2 NS-BH, and 1 WD-BH mergers. Among them, 4 BBHs avoid merger when GW recoils are enabled. Mergers occurring inside the cluster make-up the ≳ 40% of the whole population and are mostly due to mergers formed via dynamical interactions (dynamical mergers). The population of ejected mergers, which merge outside the parent cluster, are equally contributed by mergers formed dynamically and from primordial binaries (primordial mergers). Typically, in-cluster mergers have primaries with masses m_ BH,1 > 30 and companion in the m_ BH,2 = 30-50 mass range, whilst ejected mergers involve lighter primaries, m_ BH,1 < 40, and are characterised by fairly large mass ratios, q > 0.6;
* Mergers forming from primordial binaries are characterised by large mass ratios and component masses clearly smaller than those formed dynamically. Among dynamical mergers, the most massive ones are those in which at least one component had an ancestor in a primordial binary;
* BBH mergers are characterised by a delay time that nicely distribute around a value of 10-30 cluster relaxation time. This highlights the fact that the processes that trigger BBH formation and merger are intrinsically related to the cluster dynamical evolution;
* The population of mergers forming from dynamical interactions or primordial binaries is clearly distinguishable from the residual eccentricity of the binary as it enters in the typical frequency bands of GW detectors, i.e. f = 0.001-100 Hz. We find that practically all primordial binaries are circular at merger, this implying that primordial binaries merge before dynamics can have an impact on their evolution, whilst around 20-40-5% of mergers preserve an eccentricity e > 0.1 when entering the LISA-DECIGO-LIGO bands. All mergers with e > 0.1 in the 0.05-1 Hz and 1-10 Hz bands occur inside the cluster, whilst half of eccentric mergers in the mHz band are ejected. This hints at the possibility to distinguish the formation history of a BBH merger from the frequency band in which it is observed;
* We identify three exotic mergers in our sample: a WD-BH binary formed dynamically and two NS-BH mergers, one formed dynamically and the other from a primoardial binary. A WD-BH merger that forms after 4 cluster relaxation time and it is triggered by chaotic interactions that increase the eccentricity up to an extremal value of e = 0.99994930. Once the WD approaches sufficiently close the BH, this type of sources could appear as an ultraluminous X-ray sources and, ultimately, be a source detectable by LISA if it occurs within 700 kpc from us, i.e. within the distance between the Milky Way and Andromeda. The dynamical NS-BH binary is characterised by a chirp mass ℳ = 3.4, larger than what predicted by the isolated stellar evolution scenario, and preserve an eccentricity of e= 0.9974(0.21) when crossing a frequency of f = 0.5(1) Hz, thus future observations with ET could help probing the population of closeby, dynamically formed, NS-BH mergers. The primordial NS-BH binary is not affected by dynamics at all, thus they can be mistaken for a merger occurring in isolation. This highlights the importance of star clusters with a large binary fraction as contributors of the isolated scenario of compact binary mergers. None of the NS-BH mergers are expected to release EM emission, unless the BHs have a spin χ > 0.9;
* We find that comparing the remnant mass and spin of BBH mergers could help untangling their origin. Using a model based on stellar evolution theories, we show that primordial binary mergers are characterised by remnant masses systematically smaller and effective spin parameters systematically larger than dynamical mergers;
* We derive a BBH merger efficiency of ∼ 2× 10^-5^-1, comparable with the value estimated for low-mass star clusters. Interestingly, we find that the merger efficiency depends on the star cluster properties. Decreasing the binary fraction by a factor 4, for example, leads to a decrease of the merger efficiency by a factor ∼ 2. Moreover, the merger efficiency increases with the cluster density following a power-law with slope ∼ 0.25. We adopt a series of cosmologically motivated assumptions for the cosmic star formation history, and use them to infer a merger rate density at redshift z < 0.2 of ℛ = 5-19 (0.027-8.7) (3.8×10^-4 - 2.3) for BBHs(WD-BH)(NS-BH) mergers, respectively. We predict that, in a 4 yr-long mission, LISA could detect N_ BBH = 12±7(5±3) BBH mergers (IMRIs) and can identify the WD-BH merger with a signal-to-noise ratio SNR> 8(15) if it occurs within D_L < 1.5(0.7) Mpc from us.
* We retrieve the cosmic frequency rate of PISNe, in order to explore the reliability of our simulations on the one hand, and to make predictions for PISNe detection from star clusters on the other hand. We find that future surveys with a limiting magnitude of m_ bol = 25 could detect N_ PISN = 0.7-8.8 PISNe per year. Comparing these estimates with future surveys could help placing constraints on the population of massive stars in dense star clusters.
The clusters represent a further step forward in the modelling of young and intermediate-age star clusters, providing the first suite of simulations that models clusters with both N>120,000 stars (up to 10^6), a high binary fraction (up to 33%), and an initial density of ρ = (1.2× 10^4-1.6×10^6) pc^-3. These simulations complement the vast literature of N-body simulations of lower-mass and lower density star clusters <cit.>, and provide the largest catalogue of BH mergers obtained in direct N-body simulations of metal-poor, dense and massive young clusters.
§ ACKNOWLEDGEMENTS
The authors thank the referee for their constructive and helful report. The authors warmly thank Agostino Leveque for their help and assistance in using their implementation of the code, and Giuliano Iorio, Sara Rastello, and Michela Mapelli for useful comments and discussion.
This work benefited of the support from the Volkswagen Foundation Trilateral Partnership through project No. 97778 “Dynamical Mechanisms of Accretion in Galactic Nuclei” and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 “The Milky Way System”), and by the COST Action CA16104 “GWverse”. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS Booster at Jülich Supercomputing Centre (JSC). Data analysis and part of the runs were conducted on the GRACE-BH HPC workstation, funded by the European Union's under the research project GRACE-BH.
MAS acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101025436 (project GRACE-BH, PI: Manuel Arca Sedda).
AWHK is a fellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD).
The work of PB was supported by the Volkswagen Foundation under the special stipend No. 9B870.
PB acknowledge the support within the grant No. AP14869395 of the Science Committee of the Ministry of Science and Higher Education of Kazakhstan ("Triune model of Galactic center dynamical evolution on cosmological time scale").
The work of PB was supported under the special program of the NRF of Ukraine Leading and Young Scientists Research Support - "Astrophysical Relativistic Galactic Objects (ARGO): life cycle of active nucleus", No. 2020.02/0346.
RS thanks Max Planck Institute for Astrophysics (Thorsten Naab) for hospitality during many visits
MG was partially supported by the Polish National Science Center (NCN) through the grant No. 2021/41/B/ST9/01191.
FPR acknowledges the support by the European Research Council via ERC Consolidator Grant KETJU (no. 818930).
TN acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311 of the DFG Cluster of Excellence "ORIGINS”.
§ DATA AVAILABILITY
The data from the runs of these simulations and their initial models
will be made available upon reasonable request by the corresponding author.
The Nbody6++GPU code is publicly available[<https://github.com/nbody6ppgpu/Nbody6PPGPU-beijing>]. The McLuster version used in this work will soon be available. A similar version is described in <cit.>.
mnras
|
http://arxiv.org/abs/2307.04615v1 | 20230710145809 | Numerical quantification of the wind properties of cool main sequence stars | [
"Judy Chebly",
"Julián D. Alvarado-Gómez",
"Katja Poppenhäger",
"Cecilia Garraffo"
] | astro-ph.SR | [
"astro-ph.SR",
"astro-ph.EP"
] |
firstpage–lastpage
: Multi-input Transformer for Age and Gender Estimation
t]c@8emc
Maksim Kuprashevich Irina Tolstykh
[email protected] [email protected]
Layer Team, SaluteDevices
August 12, 2023
============================================================================================================================================
As a cool star evolves, it loses mass and angular momentum due to magnetized stellar winds which affect its rotational evolution. This change has consequences that range from the alteration of its activity to influences over the atmosphere of any orbiting planet.
Despite their importance, observations constraining the properties of stellar winds in cool stars are extremely limited. Therefore, numerical simulations provide a valuable way to understand the structure and properties of these winds. In this work, we simulate the magnetized winds of 21 cool main-sequence stars (F-type to M-dwarfs), using a state-of-the-art 3D MHD code driven by observed large-scale magnetic field distributions. We perform a qualitative and quantitative characterization of our solutions, analyzing the dependencies between the driving conditions (e.g., spectral type, rotation, magnetic field strength) and the resulting stellar wind parameters (e.g., Alfvén surface size, mass loss rate, angular momentum loss rate, stellar wind speeds). We compare our models with the current observational knowledge on stellar winds in cool stars and explore the behaviour of the mass loss rate as a function of the Rossby number. Furthermore, our 3D models encompass the entire classical Habitable Zones (HZ) of all the stars in our sample. This allows us to provide the stellar wind dynamic pressure at both edges of the HZ and analyze the variations of this parameter across spectral type and orbital inclination. The results here presented could serve to inform future studies of stellar wind-magnetosphere interactions and stellar wind erosion of planetary atmospheres via ion escape processes.
exoplanets – stars: atmospheres – stars: magnetic fields – stars: mass-loss – stars: winds, outflows
§ INTRODUCTION
For many decades, scientists have known that the Sun has a mass outflow, which is most visible in the behavior of comet tails (e.g., ). It has also been established that solar wind is a natural byproduct of the heating processes that produce the hot solar corona (T ∼ 10^6 K). As a result, all cool main-sequence stars (M_ ⩽ 1.3 M_⊙) with analogous hot coronae, evidenced from their measured X-ray properties (), should have similar winds ().
Magnetic fields are thought to play a key role as an energy source for the corona and the expanding solar atmosphere (e.g., ). Recent theories have shown that in addition to magnetic fields, wave dissipation (via turbulence) and magnetic reconnection could also play a role in energizing and shaping the spatial properties of the solar wind (see, ).
Winds, even if relatively weak, play an important role in stellar evolution for stars of different spectral types causing the star to lose angular momentum and slow its rotation over time (). As a result, the magnetic activities that constitute the space weather (i.e., stellar winds, flares, coronal mass ejections)
will decrease with age in low-mass stars
().
These changes in the host star will also affect the evolution of planetary atmospheres and habitability ().
Direct measurements of the solar wind by spacecraft such as the Advanced Composition Explorer (ACE, ), Ulysses <cit.>, and Parker solar probe <cit.> have improved our knowledge and understanding of its properties.
On the other hand, detecting a solar-like wind emitted by another star has proven extremely challenging. This is not surprising, given how difficult it is to observe the solar wind remotely. The latter carries a very low mass loss rate (Ṁ_⊙ = 2 × 10^-14 Ṁ_̇⊙̇ yr^-1, see ), which implies relatively low densities (near the heliopause: ∼ 0.002 cm^-3, ). Similarly, its high temperature and elevated ionization state, make it difficult to detect with simple imaging or spectroscopic techniques.
As a result, properties such as the associated mass loss rates, angular momentum loss rates, and terminal velocities, crucial to understand stellar winds in low-mass stars, remain poorly constrained.
Attempts to directly detect thermal radio emission from the plasma stream in cool stars have not yet led to any discovery ().
Current radio telescopes are not optimized for this method; they can only detect winds much stronger than those from the Sun. Moreover, the coronae of these active stars are also radio sources, making it difficult to determine the exact source of the emission.
Nevertheless, this method has been able to establish upper limits for solar analogs of 1.3 × 10^ -10 Ṁ_⊙ yr^ -1 ().
Another proposed method for direct detection is to look for X-ray emission from nearby stars. As the star's winds propagate, they collide with the Local Interstellar Medium (ISM), forming "astrospheres" similar to the Sun's heliosphere <cit.>. The charge exchange between the highly ionized stellar wind and the ISM produces X-ray photons with energies ranging from 453 to 701 eV. However, this method was unable to detect circumstellar charge exchange X-ray emission even from the nearest star, Proxima Centauri <cit.>.
Similar to the charge exchange X-ray emission method, the Ly-α absorption technique assumes the presence of the charge exchange phenomenon. In this case, however, we are interested in the neutral hydrogen wall formed at the astrospherical outer boundary by the interaction between the stellar wind and the ISM. This exchange has been detected as excess HI Ly-α absorption in Hubble Space Telescope UV stellar spectra <cit.>. With nearly 30 measurements to date, spectroscopic analyses of the stellar HI lines have proven to be the best method to unambiguously detect and measure weak solar-like winds as well as some evolved cool stars <cit.>.
Using this method,<cit.> found evidence for some increase in Ṁ with magnetic activity, corresponding to a power-law relation in the form Ṁ∝ F_ X^ 1.34 ± 0.18 with F_ X < 10^ 6 erg cm^ -2 s^ -1. However, this relation does not seem to hold anymore for more active stars (F_ X > 10^ 6 erg cm^-2 s^-1), mainly M-dwarfs <cit.>. Recently, <cit.> established a power law (Ṁ∝ F_ X^ 0.77 ± 0.04) between the Ṁ per unit surface area and the X-ray surface flux for coronal winds for a broader selection of stars, including G, K, and new Ṁ estimates for M-dwarfs. They found that the relation breaks even for stars with F_ X < 10^ 6 erg cm^-2 s^-1 (e.g., GJ 436, which has F_ X = 4.9 × 10^ 4 erg cm^-2 s^ -1, where the Ṁ was estimated by using the planet as a probe for the stellar wind ) with the magnetic topology being a possible factor for the scatter.
While extremely useful, the search for astrospherical absorption is influenced by a number of critical factors. For instance, this method is strongly dependent on the relative velocity of the stellar rest frame and the ISM flow velocity (V_ ISM). As well as on the angle, θ, between the upwind direction of the ISM flow and the line-of-sight to the star <cit.>.
It also requires prior knowledge of the properties of the ISM such as the density and its ionization state (; ). Finally, its applicability is limited to relatively nearby stars (≲ 15 pc) due to the absorption of the ISM.
Due to the scarcity of observational data and associated limitations, numerical simulations can be used to improve our understanding of stellar winds. Models based on Alfvén waves are more commonly used to simulate the stellar wind from stars other than the Sun <cit.>. This is because these waves are considered to be key mechanism for heating and accelerating the solar wind (; ).
In this study, we present a detailed numerical characterization of the stellar wind properties of cool main-sequence stars (early F to M-dwarfs) covering a range of rotation rates and magnetic field strengths. We compute steady-state stellar wind solutions using a state-of-the-art 3D MHD model and provide consistent qualitative and quantitative comparisons. Our goal is to better understand the different stellar wind properties as a function of the driving parameters, allowing us to explore the expected stellar wind conditions in the circumstellar region around planet-hosting stars.
This paper is organized as follows: Section <ref> describes the numerical model and properties of the selected stellar sample. In Sect. <ref>, we present our numerical results, discuss the derived trends in the stellar wind properties, and compare our results with observations. This information is then used to quantify the stellar wind conditions and explore their implications in the context of the classical habitable zone (HZ) around cool main-sequence stars. Conclusions and summary are provided in Sect. <ref>.
§ MODEL DESCRIPTION
We simulate stellar winds in cool main-sequence stars using the state-of-the-art Space Weather Modeling Framework (SWMF; ). The SWMF is a set of physics-based models (from the solar corona to the outer edge of the heliosphere) that can be run independently or in conjunction with each other <cit.>.
This model uses the numerical schemes of the Block Adaptive Tree Solar Roe-Type Upwind Scheme (BATS-R-US; ) MHD solver.
For a detailed description of the model, see <cit.>.
The multi-domain solution starts with a calculation using the Solar/Stellar Corona (SC) module which incorporates the Alfvén Wave Solar Model (AWSoM; ). This module provides a description of the coronal structure and the stellar wind acceleration region. The simulation is then coupled to a second module known as the Inner Heliosphere/Astrosphere[This module is formally labeled IH within the SWMF, but since we are working with low-mass main sequence stars, we will refer to it as the Inner Astrosphere (IA) domain.] (IA). In this way, it is possible to propagate the stellar wind solution up to Earth's orbit and beyond. The model has been extensively validated and updated employing remote sensing as well as in-situ solar data (e.g., ).
AWSoM is driven by photospheric magnetic field data, which is normally available for the Sun in the form of synoptic magnetograms <cit.>.
A potential field source surface method is used to calculate the initial magnetic field (more details in the following section). This information is used by AWSoM to account for heating and radiative cooling effects, as well as the Poynting flux entering the corona, and empirical turbulent dissipation length scales. With the interplay between the magnetic field distribution, the extrapolation of the potential field, and the thermodynamic properties, the model solves the non-ideal magnetohydrodynamic (MHD) equations for the mass conservation, magnetic field induction, energy (coronal heating), and momentum (acceleration of the stellar wind). These last two aspects are controlled by Alfvén waves propagating along and against the magnetic field lines (depending on the polarity of the field). In the momentum equation, the heat and acceleration contributions are coupled by an additional term for the total pressure and a source term in the energy equation. The numerical implementation is described in detail in <cit.>.
Once these conditions are provided, the simulation evolves all equations locally until a global steady-state solution is reached.
§.§ Simulation parameters and setup
In our work, we apply the SWMF/AWSoM model to main-sequence F, G, K, and M-type stars by assuming that their stellar winds are driven by the same process as the solar wind.
We analyze the properties of the stellar wind by a coupled simulation covering the region of the stellar corona (SC, spherical) and the resulting structure within the inner astrosphere (IA, cartesian). Figure <ref> illustrates the coupling procedure in one of our models. This coupling was necessary only in the case of F, G, and K stars, in order to completely cover the habitable zones (HZ)[The range of orbits around a star in which an Earth-like planet can sustain liquid water on its surface.], which are larger and farther away from the star. Parameters such as stellar radius (R_), mass (M_), and rotation period (P_ rot), are also taken into account in the simulations. We followed the approach in <cit.> in order to determine the optimistic HZs boundaries of each star in our sample.
§.§.§ Simulation domain
The star is positioned in the center of the SC spherical domain. The radial coordinate in SC ranges from 1.05 R_ to 67 R_, except for M-dwarfs, where it extends to 250 R_. The choice of the outer edge value of the SC domain was chosen in a way to obtain both edges of the HZ in one domain. The habitable zones limits were calculated using <cit.> approach and the reported measured L_ and T_ eff for each star in our sample (see Table <ref>). As will be discussed in Sect. <ref>, in the case of M-dwarfs, the extension had to be performed in order to cover the entire Alfvén surface (AS)[This structure sets the boundary between the escaping wind and the magnetically coupled outflows that do not carry angular momentum away from the star.], while keeping the default parameters for AWSoM fixed (see Sect. <ref>). The domain uses a radially stretched grid with the cartesian z-axis aligned with the rotation axis.
The cell sizes in the meridional (ϕ) and azimuthal (θ) directions are fixed at ∼ 2.8 ^∘. The total number of cells in the SC domain is ∼ 8 × 10^5.
The steady-state solutions obtained within the SC module are then used as inner boundary conditions for the IA component. An overlap of 5 R_ (from 62 R_ to 67 R_) is used in the coupling procedure between the two domains for F, G, and K stars (more details on the necessity of the overlap when coupling between domains can be found in ). The IA is a cube that extends from 62 R_ to 600 R_ in each cartesian component. Adaptive Mesh refinement (AMR) is performed within IA, with the smallest grid cell size of ∼ 1.17 R_ increasing up to 9.37 R_ with a total of 3.9 million cells. As the simulation evolves, the stellar wind solution is advected from SC into the larger IA domain where the local conditions are calculated in the ideal MHD regime.
§.§.§ Magnetic boundary conditions
In the initial condition of the simulation, observations are used to set the radial component of the magnetic field B_ r [G] anchored at the base of the wind (at the inner boundary). As mentioned earlier, a finite potential field extrapolation procedure is carried out to obtain the initial configuration of the magnetic field throughout SC <cit.>. This procedure requires setting an outer boundary (source surface, r_ s), beyond which the magnetic field can be considered to be purely radial and force-free. The magnetic field can therefore be described as a gradient of a scalar potential and determined by solving Laplace's equation in the domain. For the simulations discussed here, we set r_ s at 45% of the SC domain size for F, G, and K stars, and 70 % for M-dwarfs. While the choice of this parameter does not alter significantly the converged solutions, it can modify the required run time of each model to achieve convergence. Therefore, our selection was done to guarantee convergence to the steady-state in a comparable number of iterations between all spectral types.
The stellar magnetic field as reconstructed from Zeeman Doppler Imaging (ZDI)[A tomographic imaging technique that allows the reconstruction of the large-scale magnetic field (strength and polarity at the star’s surface from a series of polarized spectra (see e.g., ).], is used as the inner boundary condition of SC (Fig. <ref>). Therefore, the resulting wind solutions are more realistic than models based on simplified/idealized field geometries <cit.>. Although the reconstructed maps provide the distribution of vector magnetic fields, we use only the radial component of the observed surface field.
The magnetogram is then converted into a series of spherical harmonic coefficients with a resolution similar to that of the original map.
The order of the spherical harmonics should be chosen so that artifacts such as the "ringing" effect do not appear in the solution <cit.>. In our models, we performed the spherical harmonics expansion up to l_ max = 5.
§.§.§ Input parameters
After we set the initial conditions, we define several parameters for the inner boundary.
In order to reduce the degree of freedom of the parameter set, we only modify the parameters related to the properties of the stars, such as mass, rotation period, and radius.
As for the other parameters, we implement the same values that are commonly used in the solar case ().
The Poynting flux (S/B_ = 1.1× 10^6 J m^-2 s^-1 T) is a parameter that determines the amount of wave energy provided at the base of a coronal magnetic field line.
The other parameter is the proportionality constant that controls the dissipation of Alfvén wave energy into the coronal plasma and is also known as the correlation length of Alfvén waves (L_⊥ = 1.5× 10^5 m √(T)). We use the values given in <cit.> to define the base temperature (T_ o = 2 × 10^6 K) and the base density (n_ o = 2× 10^11 cm^-3).
We note that the choice of these parameters will affect the simulation results, as reported in several studies that followed different approaches (e.g., ). Recently, <cit.> performed a global sensitivity analysis to quantify the contributions of model parameter uncertainty to the variance of solar wind speed and density at 1 au. They found that the most important parameters were the phostospheric magnetic field strength, S/B_, and L_⊥. Furthermore, in <cit.>, an increase in the mass loss rate (Ṁ_), and angular momentum loss rate (J̇_) was reported when S/B_ is increased from the solar value to 2.0× 10^6 J m^-2 s^-1 T), which is expected because S/B_ drives the energy of the Alfvén wave, resulting in higher Ṁ_ and J̇_.
In this work, however, we are interested in isolating the expected dependencies with the relevant stellar properties (e.g., mass, radius, rotation period, photospheric magnetic field) which can only be analyzed consistently if the AWSoM related parameters are kept fixed between spectral types. Moreover, as will be discussed in detail in Sect. <ref>, the results obtained using the standard AWSoM settings are either consistent with current stellar wind observational constraints for different types of stars or the apparent differences can be understood in terms of other physical factors or assumptions made in the observations. For these reasons, we have chosen not to alter these parameters in this study, which also reduces the degrees of freedom in our models.
§.§ The sample of stars
Our investigation is focused on main sequence stars, with effective temperatures ranging from 6500 K down to 3030 K, and masses M_ < 1.34 M_⊙ (spectral types F to M). All of these stars are either fully or partially convective. We use a sample of 21 stars whose large-scale photospheric magnetic fields were reconstructed with ZDI ( and references therein). Some of these stars were observed at different epochs. In this case, the ZDI map with the best phase coverage, signal-to-noise ratio, and most spectra used in the reconstruction was chosen.
The sample includes radial magnetic field strengths in the ZDI reconstruction between 5 G and 1.5 kG corresponding to HD 130322 (K0V) and EV Lac (M3.5V), respectively. Spectral types range from F7 (τ Boo, M_ = 1.34 M_⊙, R_ = 1.46 R_⊙) to M6 (GJ 1245 B, M_ = 0.12 M_⊙, R_ = 0.14 R_⊙).
The rotation periods vary between fractions of a day to tens of days, with GJ 1245 B (M6V) having the shortest rotation period (P_ rot = 0.71 d) and HD 219134 (K3V) the longest one (P_ rot = 42.2 d). Table <ref> contains the complete list of the sample stars and a summary of the stellar properties incorporated in our models.
§ RESULTS & DISCUSSION
§.§ The effect of star properties on the wind structure
The Alfvén surface (AS) is defined by the collection of points in the 3D space that fulfils the Alfvén radius criterion[The Alfvén radius (R_ A) is defined as the distance around a star at which the kinetic energy density of the stellar wind equals the energy density of the astrospheric magnetic field.]. Numerically, it is determined by finding the surface for which the wind velocity reaches the local Alfvén velocity, v_A = B/√(4 πρ), where B and ρ are the local magnetic field and plasma density, respectively. The Alfvén surface can be interpreted as the lever arm of the wind torque –the "position" at which the torque acts to change the angular rotation of the star[In other words, the angular momentum per unit mass within the stellar wind can be computed as if there were solid body rotation, at an angular velocity Ω_ = 2π/P_ rot, out as far as the Alfvén surface.]. The Alfvén Surface is used in numerical models to characterize (Ṁ_) and (J̇_) (e.g., ).
We compute J̇_ by performing a scalar flow rate integration over the AS and another one over a closed spherical surface (S) beyond the AS to determine Ṁ_:
Ṁ_ = ∫_Sρ (u·dA)
J̇_ = ∫_ASΩρ R^2sin^2θ (u·dA)
Here J̇_ is the component of the change in angular momentum in the direction of the axis of rotation. The distance to the Alfvén surface is represented by R. The angle between the lever arm and the rotation axis is denoted by θ, which depends on the shape/orientation of the AS with respect to the rotation axis (and accounted for in the surface integral). The stellar angular velocity is represented by Ω = 2 π/ P_ rot. The surface element is denoted by dA.
Figure <ref> shows the AS of the stellar wind, with plasma streamers along with the equatorial section flooded with the wind velocity (U_ r) for three K stars in our sample (HIP 12545, HD 6569, 61 Cyg A). If we compare two stars with similar P_ rot but different B_ R^ max, we can clearly see that the size of AS increases with increasing magnetic field strength. This is a direct consequence of the dependence of the Alfvén velocity on these quantities (Eq. <ref>) and the distance from the star at which the Alfvén velocity is exceeded by the wind. For instance, for very active stars with stronger magnetic fields, the expected coronal Alfvén velocity is greater than for less active stars, increasing the radial distance that the wind velocity must travel to reach the Alfvén velocity. The associated Alfvén surface has a characteristic two-lobe configuration (Fig. <ref>, gray translucent area), with average sizes of 27 R_, 18 R_ and 13 R_ for HIP 12545, HD 6569, and 61 Cyg A, respectively (see Table <ref>).
When we compare two stars with similar magnetic field strengths but different P_ rot (see Fig. <ref>, panels B and C), the change in AS size is not as dramatic. The rotation period has primarily a geometric effect on the resulting AS. The Alfvén surface assumes a different tilt angle in all three cases. This tilt is mainly connected to the open magnetic field flux distribution on the star's surface <cit.>. We also notice in Fig. <ref> that the stellar wind distribution is mainly bipolar with a relatively fast component reaching up to ∼ 891 km s^−1 for HIP 12545 ∼, 702 km s^−1 for HD 6569, and ∼ 593 km s^−1 for 61 Cyg A. In section <ref> we will discuss further the relation between the wind velocity with regard to P_ rot and B_ R.
Figure <ref> shows the Ṁ_, J̇_, AS as estimated by the previously described method, against the sub-spectral type of our star sample (left column) and the average radial magnetic field strength (B_ R^ avg, right column). Similar relations have been obtained for the maximum radial magnetic field strength and are presented in Appendix <ref>. The average Alfvén surface size was calculated by performing a mean integral over the radius at each point of the 3D AS. The extracted quantities are represented by different colors and symbols for each spectral type (F, G, K, and M).
As expected, the AS increases as we move toward more magnetically active stars (Fig. <ref>, top-right panel). From our simulations, we were able to establish a relation between AS and B_ R^ avg using the bootstrap technique (1000 realizations) to find the mean of the slope and the intercept along with their uncertainties. We use this approach to determine all relations from our simulations.
The relation is as follows:
logAS_ R = (0.42 ± 0.06) log B_ R^ avg + (0.71 ± 0.07)
Our simulated steady-state Ṁ_ show a scatter within the range [0.5 Ṁ_⊙/R_⊙^2, 30 Ṁ_⊙/R_⊙^2], which is comparable to that estimated from the observed Lyα absorption method of G, K, and M-dwarfs in <cit.>.
The variations in Ṁ_ are related to differences in the strength and topology of the magnetic field driving the simulations (see ), as well as to the Alfvén wave energy transfer to the corona and wind implemented in the model (). For this reason, we tried to isolate the effects introduced by the star (e.g., M_, R_, P_ rot, magnetic field strength) over the ones from the Alfvén wave heating (i.e., n_ o, T_ o, S/B_, L_⊥).
In terms of mass loss rate, stronger winds are expected to be generated by stronger magnetic fields (see Fig. <ref>) implying that the winds are either faster or denser. This interplay determines Ṁ_ (Eq. <ref>), which increases with increasing magnetic field strength regardless of spectral type. We see a common increase for F, G, K, and M-dwarfs (excluding EV Lac) in the saturated and unsaturated regime that can be defined from the simulations as follows:
logṀ_ / R^ 2_ = (0.48 ± 0.09) log B_ R^ avg + (0.11 ± 0.10)
On the other hand, we observe a slightly different behavior for M-dwarfs, whose Ṁ_ and J̇_ values tend to be lower. As discussed by <cit.>, the magnetic field complexity could also affect Ṁ_ for a given field strength. We consider this possibility in the following section. Note that, as has been shown in previous stellar wind studies of M-dwarfs (e.g., ), modifications to the base AWSoM parameters (either in terms of the Poynting flux or the Alfvén wave correlation length) would lead to strong variations in Ṁ_. This would permit placing the M-dwarfs along the general trend of the other spectral types in particular, the Ṁ_ value obtained for the star with the strongest B_ R in our sample (EV Lac). While these modifications have physical motivations behind them (i.e. increased chromospheric activity, stronger surface magnetic fields), in most regards, they remain unconstrained observationally. Furthermore, the values we obtain in our fiducial AWSoM models are still within the range of observational estimates available for this spectral type (see Sect. <ref>), with the added benefit of minimizing the degrees of freedom and isolating the effects of the stellar parameters on the results.
Similarly, we see a large scatter of J̇_ with respect to the spectral type (Fig. <ref>, bottom left column), ranging from 10^26 g cm^2 s^-2 to 10^31 g cm^2 s^-2. This range is within the expected J̇_ values estimated for cool stars with the lowest value corresponding to M-dwarfs ( and references therein). The maximum J̇_ values reached in our simulations are comparable to J̇_⊙ reached at solar minimum and maximum (7× 10^ 30 and 10 × 10^ 30 g.cm^2s^-2, ; ).
We note that this is the only parameter for which we have retained units in absolute values (as is commonly done in solar/stellar wind studies; see ; ; ). Using absolute units, we expect a decrease in J̇_ as we move from F to M-dwarfs, since J̇_ is a function of R^2 (Eq. <ref>). The scatter around this trend is dominated by the relatively small Ṁ_ values, the distribution of Ω_ in our sample (variations up to a factor of 5), and the equatorial AS size where the maximum torque is applied (sinθ in Eq. <ref>).
We also note that the sample is biased toward weaker magnetic field strengths. To better estimate how the magnetic field affects the properties of the stellar winds, we need a larger sample, not only in terms of stellar properties but also with stellar wind constraints such as Ṁ_. The latter is so far the only stellar wind observable parameter for which comparisons can be made. For this reason, we will focus on the behavior of the Ṁ_ as a function of different stellar properties in the following sections of the analysis.
§.§ Stellar mass-loss rate and complexity
Coronal X-ray luminosity is a good indicator of the level of magnetic activity of a star and the amount of material heated to 10^6 K temperatures. The dependence of magnetic activity on dynamo action (i.e., dynamo number D = R_ o^-2, ) has led a number of authors to use the Rossby number to characterize stellar activity, for a wide range of stellar types <cit.>.
The Rossby number is defined as R_ o = P_ rot/τ_ c, where P_ rot is the stellar rotation period and τ_ c is the convective turnover time (). We adopted the approach of <cit.> to calculate τ_ c. In this case, the latter is only a function of the stellar mass (M_):
logτ_ c = 2.33 - 1.50 (M_/M_⊙) + 0.31 (M_/M_⊙)^2
As it was mentioned in Sect. <ref>, the study of <cit.> suggests that coronal activity increases with Ṁ_. The overall increase in Ṁ_ with X-ray flux F_ X (Ṁ_∝ F_ X^0.77±0.04), is most likely due to their dependence on magnetic field strength (see Sect. <ref>). However, they report a scatter of about two orders of magnitude of Ṁ_ around the trend line. This suggests that coronal activity and spectral type alone do not determine wind properties. The geometry of the magnetic field may also play a role.
The correlation between Ṁ_ and magnetic complexity has already been suggested by <cit.>, which could in principle contribute to the scatter in (, Fig. 10).
The large-scale distribution of the magnetic field on the stellar surface is mainly determined by the rotation period and the mass of the star, namely R_ o
().
The Rossby number was used to determine the complexity function in <cit.>, which was able to reproduce the bimodal rotational morphology observed in young open clusters (OCs). The complexity function of <cit.> is defined as
n = a/R_ o+ 1 + bR_ o
The constant 1 reflects a pure dipole. The coefficients a = 0.02 and b = 2 are determined from observations of OCs. The first term is derived from the ZDI map observation of stars with different spectral types and rotation periods. The third term is motivated by Kepler's observations of old stars ().
We emphasize that the complexity number (n), estimated from Eq. <ref>, differs from the complexity derived from the ZDI maps themselves (e.g., ). The complexity number from R_ o is expected to be higher. This is due to the fact that many of the small-scale details of the magnetic field are not captured by ZDI.
We expect to lose even more information about the complexity of the field given that the ZDI maps are not really available to the community (apart from the published images). Image-to-data transformation techniques (which we applied to extract the relevant magnetic field information from the published maps) can lead to some losses of information, both spatially and in magnetic field resolution. These vary depending on the grid and the projection used to present the ZDI reconstructions (i.e., Mercator, flattened-polar, Mollweide). Using the star's raw ZDI map would prevent these issues and would aid with the reproducibility of the simulation results.
Finally, note that the expected complexity is also independent of the spherical harmonic expansion order used to parse the ZDI information to the simulations. The obtained R_ o and n values for each star in our sample are listed in Table <ref>.
Figure <ref> shows the behaviour of coronal activity and Ṁ_ with respect to the expected magnetic field complexity (n). The coronal activity is denoted by full and empty symbols corresponding to saturated and unsaturated stars, respectively. We consider stars with R_ o≤ 0.1 in the saturated regime and stars with R_ o > 0.1 in the unsaturated regime based on X-ray observations (). The colors correspond to the different spectral types, whereas the numbers indicate the ID of each star in our sample. The symbol size represents the maximum radial magnetic field strength of each star extracted from the ZDI observations.
We anticipate seeing a trend in which the Ṁ_ decreases as the magnetic field complexity increases (leading to an increment of closed loops on the stellar corona), for stars in saturated and unsaturated regimes. For instance, ϵ Eri (#10, B_ R^ max= 25 G, n = 2.21724) has an Ṁ_ = 4.53 Ṁ_⊙/R_⊙^ 2 lower than HD 6569 (#9, B_ R^ max= 29 G, n = 1.80346 ) with Ṁ_ = 6.70 Ṁ_⊙/R_⊙^ 2. This is also true for τ Boo and HD 179949 where τ Boo (#1, B_ R^ max= 14 G, n = 1.84728, Ṁ_ = 2.30 Ṁ_⊙/R_⊙^ 2) has a higher Ṁ_ compared to HD 179949 (#2, B_ R^ max= 12 G, n = 2.65746, Ṁ_ = 1.90 Ṁ_⊙/R_⊙^ 2).
We also noticed that as we go to more active stars, like in the case of M-dwarfs, the field strength starts to dominate over the complexity in terms of contribution to the Ṁ_. For example, GJ 1245 B (#21, B_ R^ max= 404 G , n = 5.02602 , Ṁ_ = 9.27 Ṁ_⊙/R_⊙^ 2) has an Ṁ_ higher than DT Vir even though the complexity of the former is almost 5 times higher (DT Vir, #17, B_ R^ max= 327 G, n = 1.41024, Ṁ_ = 3.81 Ṁ_⊙/R_⊙^ 2). However, in order to better understand the contribution of the complexity in Ṁ_, we will need to run simulations for a wider range of stars with sufficiently high resolution of the driving magnetic field to capture directly the complexity of the field (and not estimate it from a scaling relation as it was performed here).
Moreover, our results show that whenever we have a case in which the star properties (M_, R_, and P_ rot), magnetic field strength and complexity are comparable, we end up with similar Ṁ_. This will be the case of TYC 6878-0195-1 (#13, B_ R^ max= 162 G, n = 1.48069, Ṁ_= 17.42 Ṁ_⊙/R_⊙^ 2) and HIP 12545 (# 15, B_ R^ max= 184 G, n = 1.41505, Ṁ_ = 20.11 Ṁ_⊙/R_⊙^ 2).
Furthermore, two stars with similar coronal activity with respect to X-ray flux, i.e., EV Lac and YZ CMi (F_ X≈ 10^7 ergs cm^-2 s^-1), but with slightly different magnetic field complexity, result in different wind properties: respectively Ṁ_ = 0.62 Ṁ_⊙/R_⊙^ 2, and Ṁ_ = 20.57 Ṁ_⊙/R_⊙^ 2.
A similar situation occurs when two stars have a comparable field complexity but different coronal activity i.e., YZ CMi and GJ 205 (#18, Ṁ_ = 2.32 Ṁ_⊙/R_⊙^ 2, F_ X≈ 10^5 ergs cm^-2 s^-1).
The lowest Ṁ_ corresponds to the saturated M-dwarf EV Lac (#19), which has the strongest B_ R (1517 G) and one of the simplest complexities in our sample (n = 1.46331). The low complexity of the field means that the wind is dominated by open field lines, leading to very high wind velocities in the standard AWSoM model, but with a very low density, which in turn leads to small Ṁ_ values. We remind the reader that the base density of the stellar wind is fixed at the stellar surface and is the same for all the stars in the sample (Sect. <ref>).
§.§ Stellar wind mass-loss rate and Rossby number
Using the results of our stellar winds models, we can study how the Ṁ_ changes as a function of the Rossby number (R_ o). The Rossby number is a useful quantity because it not only removes the dependence on spectral type, but also relates the rotation period to magnetic field strength, complexity, and even stellar coronal activity.
The latter is also important because cool stars exhibit a well-defined behavior between L_ X (or F_ X) and R_ o (saturated and unsaturated regimes). Thus, if we analyze Ṁ_ using this parameter, we can see (to some extent) all dependencies simultaneously.
Figure <ref> shows the stellar mass-loss rate per unit surface area (Ṁ_/R^ 2_) as a function of the Rossby number (R_ o). The circles show our 3D MHD numerical results, while the empty, filled, and the plus sign within a square corresponds to observational estimates of astrospheres <cit.>, slingshot prominences <cit.>, and absorption during an exoplanetary transit <cit.>, respectively. We use the same method as for the simulated stars (Eq. <ref>) to calculate the R_ o of stars with constraints on their mass loss rate.
Spectral types are indicated by different colors: cyan (F), yellow (G), orange (K), and red (M). The Sun is represented by a yellow star symbol. Dashed lines connect the common stars in our models and the observations. In this section, we will focus only on the resulting Ṁ_ from the numerical results.
As was mentioned earlier, our 3D MHD simulated Ṁ_ values are in the same range as the Ṁ_ estimates from the Ly-α astrospheric absorption method. Note that since we are only simulating steady-state stellar winds, our comparison is mostly focused on the steady mass loss Ṁ_ (filled squares and squares with a plus sign). As such, it is not surprising that our Ṁ_ values appear 1 - 2 orders of magnitude below the estimates associated with sporadic mass loss events such as slingshot prominences in very active stars in the saturated regime (filled squares, ).
Based on the relation between F_ X and R_ o <cit.>, and the broad correlation observed between Ṁ_ and F_ X <cit.>, we expect to see traces of a two-part trend (albeit with significant scatter) between Ṁ_ and R_ o: a flat or saturated part that is independent of stellar rotation (R_ o≲ 0.1, rapidly rotating stars), and a power law showing that the stellar wind mass loss rate decreases with increasing R_ o (R_ o > 0.1, slowly rotating stars).
For stars in the unsaturated regime, we do see a trend in which Ṁ_ increases with decreasing R_ o. The relationship between Ṁ_ and R_ o retrieved from our simulations is
logṀ_/R_^2 = (-1.13± 0.23) log R_ o + (0.50± 0.07) .
The majority of the Ṁ_ derived from observation appears to follow the established relationship Ṁ_–R_ o, with some scatter within the error range. We do, however, notice four outliers, including three K stars and one G star. The K stars with the high Ṁ_ correspond to the binary 70 Oph A (K0V) and 70 Oph B (K5V). As for the 3^ rd K star and the G star, they correspond to evolved stars: δ Eri (K0IV, Ṁ_ = 0.6 Ṁ_⊙/R_⊙^ 2, R o ∼ 21) and DK UMA (G4III-I, Ṁ_ = 0.0077 Ṁ_⊙/R_⊙^ 2, R_ o ∼ 2.51).
We do not expect evolved stars to follow the same trend as unsaturated main sequence stars because their winds might be generated from a different mechanism (such as pulsations, see ).
As for 70 Oph A and B, we do not have much insight into their eruptive activity levels in order to rule out whether or not the Ṁ inferred from the astropsheric technique was influenced by slingshot prominences or CME activity.
As can be seen in Fig. <ref>, our numerical results in this region are essentially bracketed by the observations for which the R_ o reaches larger values. The largest Rossby number from our star sample corresponds to HD 219134 (K3V, R_ o = 2.02732), which is comparable to the accepted solar value. Since our models use ZDI maps as inner boundary conditions to simulate stellar winds, this implies that extending our numerical models to even larger R_ o would be very challenging as those ZDI reconstructions would require prohibitively long observing campaigns.
While we have limited data points, we see that for objects with R_ o≲ 0.15, we do not obtain larger numerical values Ṁ_ even when the magnetic field strengths increase dramatically. For example, in the case of YZ CMi (B^ max_ R = 822 G, Ṁ_/R_^ 2 = 20.57 Ṁ_⊙/R_⊙^ 2) and GJ 1245 B (B^ max_ R = 404 G, Ṁ_ = 9.27 Ṁ_⊙/R_⊙^ 2).
All stars on the left-hand side of Fig. <ref> lie beneath the maximum Ṁ_ value obtained for YZ CMi (B^ max_ R = 822 G, Ṁ_/R_^ 2 = 20.57 Ṁ_⊙/R_⊙^ 2).
This is true even when R_ o varies by more than one dex, magnetic field strength by factors of 100, and the expected complexity number by ∼ 4.
These results indicate that the contribution from the steady wind will only account for a small fraction of the Ṁ_ budget in the case of very active stars. Furthermore, the obtained behaviour hints of a possible saturation of the steady-state stellar wind contribution to Ṁ_, while the star could still lose significant mass through other mechanisms such as slingshot prominences or CME activity due to flares among others.
According to <cit.> and references therein, cool stars can support prominences if their magnetospheres are within the centrifugal regime (i.e. R_ K < R_ A, where R_ K = √(GM_/Ω_^2) is the co-rotation radius). They provide estimates for the prominence masses (m_ p) and the ejection time-scales (t_ p) for a sample of cool stars. According to their analysis, DT Vir would have m_ p = 1.5 × 10^ 15 g and t_ p = 0.1 d, while the values for GJ 1245 B would be m_ p = 4.4 × 10^ 14 g, t_ p = 0.3 d. Using these values, they also reported the expected mass loss rate from prominences for these two stars in absolute units. In order to compare with the steady state wind, we convert their results to units of Ṁ_⊙/R_⊙^ 2. For DT Vir we have Ṁ_^ p/R_^ 2 = 0.49 Ṁ_⊙/R_⊙^ 2 and for GJ 1245 B the resulting value is Ṁ_^ p/R_^ 2 = 0.68 Ṁ_⊙/R_⊙^ 2.
For the CMEs contribution, we can obtain an order of magnitude estimate by following the approach in <cit.>. They estimate the mass-loss rate from the CME (Ṁ_^ CME) as a function of L_ X and the power law index (α) of the flare frequency distribution. For the X-ray luminosity, we used the <cit.> database, and for the flare frequency distribution exponent we took α = 2 <cit.>.
For DT Vir, with log(L_ X) = 29.75, we obtain Ṁ_^ CME/R_^ 2 ∼ 160 Ṁ_⊙/R_⊙^ 2. For GJ 1245 B, with log(L_ X) = 27.47, the estimated CME-mass loss rate is Ṁ_^ CME/R_^ 2 ∼ 12.8Ṁ_⊙/R_⊙^ 2.
We emphasize here that this approach assumes that the solar flare-CME association rate holds for very active stars (see the discussion in ). As such, it does not consider the expected influence due to CME magnetic confinement (e.g. ) which currently provides the most suitable framework to understand the observed properties of stellar CME events and candidates <cit.>.
Still, we can clearly see that the input from CMEs to the total Ṁ_ could be higher than the steady wind and prominences for these two stars (with the latter contributing less in these cases). For instance, the estimated contribution of CMEs to the total Ṁ_ of DT Vir is almost 40 times higher than the value obtained for the steady stellar wind.
We will discuss the cases of EV Lac and YZ CMi in Section <ref>
§.§.§ Comparison between simulations and observations
In addition to analyzing the general trends, we can compare the models for common stars between our sample and the observations in <cit.> and references therein.
The stars in <cit.> contain a total number of 37 stars with a mix of main-sequence and evolved stars. The sample includes 15 single K-G stars among them 4 evolved stars, and 4 binaries. <cit.> reports individual Ṁ values for the G-K binary pairs (this means that it was possible to model their individual contribution to the astrosphere of the system or they were separated enough not to share a common astrosphere). This is important as, in principle, one could treat the binary pairs as individual stars. The rest of the star sample includes 22 M-dwarfs with 18 single M-dwarfs, 3 binaries, and 1 triple system. Unlike the G-K stars, Ṁ_ values for the M-dwarf binaries/triple system are listed as a single value (therefore, it means that it has to be taken as the aggregate of all the stars in the system). For the binary system GJ 338 AB we were unable to include it in the plot of Fig. <ref> due to a lack of needed information to estimate its R_ o.
Following on the results from Sect. <ref>, our simulated mass loss rates for stars in the unsaturated regime agree well with those estimated from astrospheric detections (see Fig. <ref>). Specifically, for GJ 205 (M1.5V), 61 Cyg A (K5V), and HD 219134 (K3V) we obtain Ṁ_/R_^ 2 of 2.32, 3.98, and 1.50, respectively. These values are all consistent with their respective observational estimates, taking into account the typical uncertainties of the astrospheric absorption method[Astrospheric estimates on Ṁ_ should have an accuracy of about a factor of 2 with substantial systematic uncertainties <cit.>.]. While further observations could help to confirm this, the agreement between our asynchronous models and the observations indicates that, within this R_ o range, the temporal variability of Ṁ_ is minimal. This is certainly the case for the Sun (R_ o∼ 2.0) in which long-term monitoring has revealed only minor variability of the solar wind mass loss rate over the course of the magnetic cycle (, ).
On the other hand, Ṁ_ from the 3D MHD simulations appear to fall short by an order of magnitude or more from the available estimates for ϵ Eri (K2V), EV Lac (M3.5V), and YZ CMi (M4.5V) with Ṁ_/R_^ 2 of 4.53, 0.62 and 20.57, respectively. We will discuss different possibilities for these discrepancies on each star in Sect. <ref>. However, it is important to remember that the Ṁ_ estimates from the Ly-α absorption technique contain systematic errors that are not easily quantified. One example is that they depend on the assumed properties and topology of the ISM <cit.>, which have not been fully agreed upon in the literature (e.g., ). While studies have provided a detailed characterization of the local ISM (see ), intrinsic uncertainties and additional observational limitations can greatly alter the estimated mass-loss rate values. These include column densities, kinematics, and metal depletion rates (), as well as local temperatures and turbulent velocities <cit.>.
Furthermore, we would also like to emphasize the variation of the Ṁ_ in the astrospheric estimates with the assumed stellar wind velocity, as we believe that this factor is one of the largest potential source of uncertainty and discrepancy with our models. As discussed by <cit.>, this parameter is used as input in 2.5D hydrodynamic models to quantify the stellar wind mass loss rate. The Ly-α absorption signature, leading to Ṁ_, is determined to first order by the size of the astrosphere. The latter depends on the stellar wind dynamic pressure (P_ dyn∝Ṁ_ U_ sw), which implies an inverse relation between Ṁ_ and U_ sw <cit.>.
The astrospheric analysis of <cit.> assumed a stellar wind velocity of 450 km s^-1 at 1 au (matching models of the heliosphere) for all main-sequence stars. However, we find that stellar wind velocities can vary significantly between different types of stars and even among the same spectral type for different magnetic field strengths and rotation periods. To quantify this, we compute the average terminal velocity of the wind, (U_ R^ T), by averaging U_ R over a sphere extracted at 99% of the maximum extent of each simulation domain (594 R_ for F, G, and K stars and 248 R_ for M-dwarfs; see Sect. <ref>). In the cases in which the spatial extension of our numerical domain allowed, we also computed the average wind velocity at 1 au. The resulting values, listed in Table <ref>, indicate variations in the wind velocity by factors of 5 or more when moving from F-type stars (U_ R^ T∼ 325 km s^-1) to M-dwarf (U_ R^ T∼ 1500 km s^-1). This is also illustrated in Fig. <ref>, which portrays the simulated stellar wind environment for HD 179949 (F8V), HD 73256 (G8V), HD 189733 (K2V), and DT Vir (M0V). We include a green iso-surface that corresponds to the wind velocity at 1 au for F, G, and K stars as for M-dwarfs it represents the average terminal wind velocity in the domain. The visualizations also include the equatorial projection of the wind dynamic pressure (P_ dyn = ρ U^ 2), normalized to the nominal Sun-Earth value, as well as on a sphere highlighting the wind 3D structure at 0.5 au.
What is clear from this analysis is that is not ideal to use the same wind velocity for all spectral types. Even within the same spectral type, we can observe a wide range of terminal velocities (e.g., the velocity in K stars ranges from 400 km s^-1 to 700 km s^-1). As such, for models that require wind velocity as an input parameter, we recommend using the average radial wind velocity among a given spectral type.
For G-K stars, we obtain wind velocities at 1 au in the range of 400 to 700 km s^-1 which is not too different from the wind velocity assumption of . This is also consistent with the fact that for these spectral types, we have a better agreement between Ṁ estimated from our simulations and those from the astropsheric technique <cit.>.
For lower mass stars with relatively small R_ o we obtain velocities higher than 450 km s^-1 up to 3675 km s^-1.
Note that due to computational limitations, the extent of our M-dwarf simulations does not reach up to 1 au (varying from 0.6 au for DT Vir to 0.16 au for GJ 1245 B). Nevertheless, as indicated by the calculated terminal velocities, even at closer distances the wind velocity is already >450 km s^-1, a situation that should still hold when propagated out to 1 au. Wind velocities on the order of 1000-1500 km s^-1 at distances of 1 au and beyond had been reported in high-resolution AWSoM simulations of the environment around the M5.5V star Proxima Centauri <cit.>. This helps to explain why our simulated mass-loss rates for EV Lac, YZ CMi, and ϵ Eri were lower than the observed ones (differences larger than a factor of 2). We discuss these cases in more detail in the following section.
§.§.§ Exploring the cases of EV Lac, YZ CMi, ϵ Eri
* YZ CMi & EV Lac
Frequent stellar flares have been observed at YZ CMi in several wavelength ranges (). The flaring energy distribution of this star ranges from 10^30.6 to 10^34.09 erg <cit.> with a total flaring time that varies from 21 to 306 minutes. Likewise, there is also significant flare activity on EV Lac (). From spectroscopic and photometric studies of EV Lac, <cit.> reports to have found 27 flares (∼ 5.0 flares per day) in H α with energies between 1.61 × 10^31 erg −1.37 × 10^32 erg and 49 flares (∼ 2.6 flares per day) from the TESS lightcurve with energies of 6.32 × 10^31 erg −1.11 × 10^33 erg.
With such high flare activity, it is possible that a large fraction of the Ṁ_ estimated in <cit.> for these stars could arise from transient phenomena (e.g., prominences, CMEs).
Following the same approach described at the end of Section <ref>, we can obtain a rough estimate of Ṁ from CMEs for EV Lac and YZ CMi.
For EV Lac we find
Ṁ_^ CME/R_^ 2 = 55.5 Ṁ_⊙/R_⊙^ 2 assuming log(L_ X) = 28.69. In the case of YZ CMi, an log(L_ X) = 28.53 yields Ṁ_^ CME/R_^ 2 = 47.6 Ṁ_⊙/R_⊙^ 2. However, given the magnetic field strength observed in EV Lac and YZ CMi (a few kG, ), we expect that the magnetic confinement of CMEs would play an important role in these objects (see , , ). Therefore, it is not straightforward to estimate exactly how large the contribution of CMEs to Ṁ_ is for these stars.
In addition, as discussed by <cit.>, EV Lac and YZ CMi are considered in the slingshot prominence regime. For EV Lac they estimate m_ p = 2.0 × 10^ 16 g and t_ p = 0.6 d, while for YZ CMi values of m_ p = 4.5 × 10^ 16 g and t_ p = 0.6 d are given. Using the associated mass loss rate values reported in <cit.>, we obtain Ṁ_^ p/R_^ 2 = 3.16 Ṁ_⊙/R_⊙^ 2 for EV Lac and Ṁ_^ p/R_^ 2 = 8.32 Ṁ_⊙/R_⊙^ 2 for YZ CMi.
This suggests another possible explanation for the discrepancies between our models and the astrospheric estimates is that some of the stellar wind detected for EV Lac and YZ CMi contains material from the slingshot prominences. Indeed, the location of the latter in the Ṁ_ – R_ o diagram (Fig. <ref>) appears more consistent with the mass loss rate estimates from slingshot prominences by <cit.>.
Moreover, <cit.> noted that the YZ CMi astrospheric absorption comes primarily from neutrals near and inside the astropause, rather than from the hydrogen wall where neutral H density is highest. Therefore, using Ly alpha absorption to calculate Ṁ_ from YZ CMi will result in substantial uncertainty.
Finally, as mentioned in Sect. <ref>, there is a significant difference between the wind velocity assumed by <cit.> and our results. Our average terminal wind velocity for YZ CMi (1709 km s^-1) and EV Lac (3675 km s^-1) is significantly higher than the wind velocity of 450 km s^-1 assumed in <cit.> at 1 au. While the wind velocity in EV Lac might be overestimated in our models (due to the usage of fiducial AWSoM parameters), we still expect relatively large wind velocities for this star (∼ 1000-1500 km s^-1) given its magnetic field strength and Rossby number (see e.g., ). As was discussed in Sect. <ref>, while our terminal wind velocity for M-dwarfs is calculated closer to the star (0.33 au for YZ CMi and 0.16 au for EV Lac), we do not expect a large reduction in the average velocity between these distances and 1 au. As such, the fast wind velocity resulting in our simulations of YZ CMi and EV Lac would imply lower Ṁ_ values when analyzed following the astrospheric technique of <cit.>.
* ϵ Eri
With a relatively slow rotation period (11 d), and weak large-scale magnetic field (< 50 G), ϵ Eri cannot be considered within the slingshot prominence regime (like in the cases of YZ CMi and EV Lac). Because of this, we do not expect a significant presence of slingshot prominences in the Ṁ value of this star. On the other hand, the analysis of <cit.>, estimated the contribution of flare-associated CMEs to the mass loss rate. They reported an upper limit of 1.09 Ṁ_⊙ / R _⊙^ 2, which is insignificant when compared to the star's overall estimated Ṁ_ value by <cit.> and the astrospheric technique (56 Ṁ_⊙ / R _⊙^ 2).
Therefore, the contribution from CMEs is also most likely not responsible for the elevated astrospheric Ṁ_ value on this star and its discrepancy with our steady-state models.
On the other hand, multiple observations of the large-scale magnetic field geometry of ϵ Eri reveal that it evolves over a time-scale of months <cit.>. According to <cit.>, the maximum field strength can reach up to 42 G. As shown in Fig. <ref>, a global increase in the magnetic field strength causes an increase in Ṁ_.
The Zeeman Doppler Imaging map of ϵ Eri used to drive the 3D MHD model has a B_ R^ max = 25 G leading to Ṁ_/R_^2 = 4.53 Ṁ_⊙/R_⊙^2. This value is comparable to the numerical result obtained by <cit.> for this star (Ṁ_/R_^2∼ 5.3 Ṁ_⊙/ R_⊙^ 2). Increasing the surface magnetic field strength of ϵ Eri to the maximum value reported in observations will raise the mass loss rate to ∼ 10 Ṁ_⊙/ R_⊙^ 2. As such, the variability of the stellar magnetic field and its expected modulation of the stellar wind properties could account for some of the differences between the simulated and the observed mass loss rates. However, corroborating this would require contemporaneous ZDI and astrospheric measurements which, to our knowledge, have not been performed on any star so far.
As ϵ Eri goes through a magnetic/activity cycle (), we can expect relatively large variations in Ṁ_ values in our Alfvén-wave driven stellar wind models.
Finally and following the discussion for YZ CMi and EV Lac, the average wind velocity for ϵ Eri at 1 au (554 km s^-1) resulting from our models exceeds the one assumed in <cit.>. This will result in a smaller estimated Ṁ_ value from the pressure-balance astrospheric technique.
In this way, the deviation between our models and the astrospheric detection of ϵ Eri could be due to the combined contribution of all the preceding elements (i.e., CMEs, cycle-related variability of the magnetic field, higher stellar wind velocity), and therefore we do not consider this discrepancy critical to our analysis.
§.§ Stellar wind and Circumstellar region
This section focuses on using the stellar wind results obtained from the 3D MHD simulations to assess the conditions an exoplanet would experience. This includes the characterization of the Alfvén surface for the various stellar wind solutions, the properties of the stellar wind in the habitable zone of these stars (in terms of the dynamical pressure of the wind), and the resulting magnetosphere size for these stellar wind conditions (assuming that a planet with the same properties/magnetization as Earth is in the HZ of these stars). The obtained quantities are listed in Table <ref> and <ref>.
§.§.§ Stellar wind properties and orbital distances
* Alfvén surface size
Figure <ref> summarizes our results showing the stellar wind environment around cool main sequence stars. We include the average size of the AS, resulting from our 21 3D MHD models, indicated in filled diamonds. To complement this information, empty diamonds correspond to the expected average AS size employing the scaling relation provided in Sect. <ref>, and using the ZDI information from 29 additional stars ( and reference therein). The green region corresponds to the optimistic HZ, calculated using the approach provided by <cit.> and the expected behaviour of the luminosity, temperature as a function of stellar mass on the main sequence (). Each square indicates the limits of the optimistic HZ for each star in our sample. These have been color-coded by the stellar wind dynamic pressure, normalized to the average Sun-Earth value. The position of the Earth is indicated by the ⊕ symbol. In the background, a sample of the semi-major axis of some exoplanets is included.
There are a few noteworthy aspects of Fig. <ref>. First of all, the 3D MHD simulated AS values (filled diamonds) do not show a clear trend with stellar mass. Instead, we see more or less similar AS regardless of the spectral type of the star (Table <ref>). We see a similar behavior for stars whose AS were extracted from the scaling relationship presented in Eq. <ref> (empty diamonds). There is a significant scatter in the obtained distribution of AS against M_, indicating that the intrinsic dependency with the surface magnetic field properties can in principle be replicated among multiple spectral types. However, we remind the reader that this result is also partly a consequence of our fixed choice for the base parameters of the corona and stellar wind solution (Sect. <ref>), which could in principle vary among different spectral types and activity stages (i.e. ages). As such, the generalization of the results presented here requires further investigation from both, observational constraints and numerical simulations.
We can also see that for late K and M-dwarfs, AS reaches orbital distances comparable to their HZ limits. Examples of this from our sample are GJ 1245 B (AS = 0.028 au, HZ_ inner = 0.033 au) and YZ CMi (AS= 0.178 au, HZ_ inner= 0.09 au). This situation has been also identified in previous case studies of stellar winds and exoplanets (e.g, ).
The location of the HZ relative to the stellar Alfvén surface must be considered when studying the interactions between a star and a planet. A planet orbiting periodically or continuously within the AS region could be directly magnetically connected to the stellar corona, which could have catastrophic effects on atmospheric conservation <cit.>.
On the other hand, a planet with an orbit far outside this limit will be decoupled from the coronal magnetic field and interact with the stellar wind in a manner similar to the Earth (e.g. ). In the case of a planet orbiting in and out of the AS, the planet will experience strongly varying wind conditions, whose magnetospheric/atmospheric influence will be greatly mediated by the typical time-scale of the transition <cit.>.
* Dynamic pressure
We also see a general trend in Fig. <ref> in which the dynamic pressure at the HZ boundaries increases as we move from earlier to later spectral types. For example, P_ dyn^ Inn, HZ for the lowest-mass star GJ 1245 B is 447.39 P_⊕ nearly 200 times stronger than for the highest-mass star τ Boo with P_ dyn^ Inn, HZ = 2.76 P_⊕.
Our results also show a large variability in P_ dyn as we move from the inner to the outer edge of the HZ of G, K, and M-dwarfs (Table <ref>). For these stars, the P dyn at the inner HZ is almost 6 times stronger than that at the outer edge of the HZ (i.e., EV Lac P_ dyn^ Inn, HZ = 33.18 P_⊕, P_ dyn^ Out, HZ = 6.27 P_⊕). For F stars, the difference is smaller, around a factor of 2 like in the case of HD 179949, where P_ dyn^ Inn, HZ = 2.51 P_⊕ and P_ dyn^ Out, HZ = 0.48 P_⊕. The reason is that the HZs of these stars are farther from the star, where the wind density starts to become less variable.
Moreover, in some cases, we have P_ dyn at the inner and outer edge of the star HZ comparable to the typical range experienced by the Earth (0.75 and 7 nPa, ). For example, HD 73256 (G8V, 6.45 - 1.18 P_⊕ ∼ 9.675 - 1.77 nPa), HD 130322 (K0V, 2.56 - 0.46 P_⊕ ∼ 3.84 - 0.69 nPa), τBoo (F7V, 2.76 - 0.40 P_⊕ ∼ 4.14 - 0.6 nPa). For the case of M-dwarfs, we have dynamic pressures higher than those experienced by Earth, as in the case of DT Vir (M0V, 88.97 - 16.20 P_⊕ ∼ 133.455 - 24.3 nPa). This is because the HZ is located near the star where the density is highest. This indicates that planets orbiting at very close distance to the star (∼ 0.03 - 0.05 au) would experience extreme space weather conditions with P_ dyn up to 10^3 and 10^4 P_⊕. These values are comparable to the ones estimated in <cit.> for Proxima d and for Proxima b in <cit.>. However, the reader is reminded here that any point from our simulations should be interpreted as an indication of the average conditions, but should not be treated as a specific absolute value (since it will change depending on the instantaneous local density and velocity of the wind (both a function of the evolving stellar magnetic field).
In addition, we notice a scatter in P_ dyn estimates at the HZ when comparing stars of the same spectral type. This is not surprising since the P_ dyn depends on the wind velocity and density at a given place. This also translates into having a range of dynamic pressure that a planet will experience within the HZ. This will defer from one orbital distance to the other as we can see in Fig. <ref> where we show the equatorial plane color-coded by the dynamic pressure.
We can use our 3D models to investigate also the influence due to the orbital inclination. To illustrate this, Fig. <ref> shows a 2D projection of the normalized dynamic pressure P_ dyn extracted from spherical surfaces matching the midpoint of the HZ of HD 179949 (F8V), TYC 198-509-1 (G7V), 61 Cyg A (K6V), and GJ 205 (M1.5V). We notice that in the case of F and G stars (i.e., HD 179949, and TYC-198-509-1) we have a large P_ dyn variation with inclination around a factor 7. However, P_ dyn values, are still relatively small in terms of absolute units (i.e., 0.01 - 10 P_⊕ ∼ 0.015 - 15 nPa). For K and M-dwarfs, we see less variability in the P_ dyn for the different inclinations, a more homogeneous P_ dyn, especially in the case of the K star. However, in these cases, the P_ dyn can reach values > 100 P_⊕ (> 150 nPa). Our results also show that even with an extreme orbit around the G-type star (TYC 198-509-1) with an inclination matching the current sheet, we would most likely not reach the very high P_ dyn values as in the case of the K and M-dwarfs as we move closer to the star. As such, the inclination of the orbit plays a secondary role compared to the distance. This is clearly seen in the color gradient that gets redder and redder as we move toward lower masses (so the HZ is closer).
On the other hand, the variability of P_ dyn, which we can see in Fig. <ref> while represented in the same 'spatial scale', it does not coincide in terms of 'temporal scales'. In other words, the x-axis in Fig. <ref> do not correspond to the same timescale units for each star, where the 360 degrees of longitude correspond to “1 orbital period”. However, the orbital period is very different for a planet in the HZ of an F-type star (within a few au) compared to a planet orbiting an M-dwarf (within a fraction of an au). A planet orbiting an M-dwarf star experiences the variations in P_ dyn on a much faster timescale (∼ 1 day for each current sheet crossing), while these variations are much longer for more massive stars. This means that even if the P_ dyn values were the same, the faster variability over the orbital period for low-mass stars would result in planets and their magnetospheres/atmospheres having less time to recover from passing through regions of high P_ dyn than planets around more massive stars.
Finally, following the results compiled by <cit.>, if we consider the presence of a rocky exoplanet with an atmosphere similar to those of Venus and Mars at those mid-HZ locations, we would expect atmospheric ion losses between 2 × 10^24 ions s^-1 and 5 × 10^24 ions s^-1. This of course assumes that all processes occur in the same way as in the solar system (which might not be necessarily true for some regions of the vast parameter space of this problem). The ion losses will depend heavily on the type of stars that the exoplanet orbits, both in terms of the high-energy spectra and the properties of the stellar wind (see e.g. ). If the rocky exoplanet is found around the HZ of an M-dwarf, the planet might suffer from unstable stellar wind conditions as previously stated that might increase the ion losses in the exoplanetary atmosphere. We will consider the case of Earth with its magnetosphere in the following section.
§.§.§ Magnetopause Standoff Distances
Using the dynamic pressure, we can define a first-order approximation to determine the magnetosphere standoff distance (R_ M) of a hypothetical Earth-like planet orbiting at the HZ around each star in our sample. This is done by considering the balance between the stellar wind dynamic pressure and the planetary magnetic pressure (Eq. <ref>, ):
R_ M = R_ E [B_ p^2/ 8 π P_ dyn]^1/6
The Earth's equatorial dipole field and radius are represented by B_ p and R_ E respectively. Normally the total wind pressure should be considered (i.e., thermal, dynamic, and magnetic), but in all the cases here considered, we can neglect the contributions of the magnetic and thermal pressures. For this calculation, we assume an equatorial dipole magnetic field of 0.3 G, similar to that of the Earth <cit.>.
The magnetospheric standoff distance is expressed in Earth's radii (Eq. <ref>). The different R_ M, HZ values for the different stars in our sample are listed in table <ref>. Note that we only estimate the R_ M in the cases where the HZ is in the super-Alfvénic regime ().
Our estimated R_ M, HZ for F, G, and early K stars have values closer to the standard size of Earth's dayside magnetosphere (∼ 10 R_⊕, see ). This is comparable to the value obtained by <cit.> for Proxima c (∼ 6 - 8 R_⊕ in both activity levels), assuming an Earth-like dipole field on the planet surface.
For the late K and M-dwarfs in our star sample, R_ M starts to reach lower values < 50% from that of Earth.
This suggests that a planet orbiting these stars must have a stronger dipole magnetic field than that of the Earth to withstand the wind conditions since R_ M∝ B_ p ^1/3.
However, in <cit.> they show that contrary to what we have seen so far, the magnetosphere might actually not act as a shield for the stellar wind-driven escape of planetary atmospheres. In fact, they reported an ion loss for Earth that ranges from 6 × 10^24 ions s^-1 - 6 × 10^26 ions s^-1 which is higher than what Venus and Mars lose. Further modeling studies are needed in order to characterize the stellar wind influence on the atmospheric loss of rocky exoplanets (e.g., ), whose input stellar wind parameters can be extracted from this investigation.
§ SUMMARY & CONCLUSIONS
In this study we employed a state-of-the-art 3D MHD model (SWMF/AWSoM) to investigate the dependencies between different star properties (R_, M_, B_ R, and P_ rot) and a number of stellar wind parameters (AS, Ṁ_, J̇_, P_ dyn) of cool main sequence stars. We present numerical results of 21 stars going from F to M stars with magnetic field strengths between 5 and 1.5 kG and rotation periods between 0.71 d and 42.2 d.
The large-scale magnetic field distribution of these stars, obtained by previous ZDI studies, were used to drive the solutions in the Stellar Corona domain, which are then self-consistently coupled for a combined solution in the Inner Astrosphere domain in the case of F, G, and K stars.
Our results showed a correlation between the average AS size and B_ R^ avg, regardless of the spectral type of the star (Eq. <ref>). We also obtained a strong correlation between Ṁ_ and B_ R^ avg for the different spectral types (excluding EV Lac, Eq. <ref>). The correlation between J̇_ and B_ R, on the other hand, was dominated by the absolute dependence on the stellar size, with significant scatter resulting mainly from the variability in Ṁ_, the distribution of Ω_ in our sample and the equatorial AS size where the maximum torque is applied.
Having established these star-wind relations, we looked in detail at Ṁ_, since it is the only observable parameter of the stellar wind for which comparisons can be made. Using the complexity number as a function of the Rossby number R_ o–defined previously in the literature– we were able to investigate the dependence of magnetic complexity on Ṁ_. Our results showed that for more active stars, as in the case of M-dwarfs, the field strength starts to dominate over the complexity in the contribution on shaping Ṁ_. Also, for cases in which the magnetic field strength and complexity were comparable, we obtained similar Ṁ_. This indicates that in these cases the stellar properties (R_, M_, and P_ rot) play a secondary role in changing Ṁ_.
We then used our stellar wind results to investigate its behaviour with respect to the well-known stellar activity relationship (F_ X vs R_ o with the saturated and unsaturated regimes). For stars in the unsaturated regime, we see a trend where Ṁ_ increases with decreasing R_ o (Eq. <ref>). For stars in the saturated regime, we find that the contribution of the steady wind is only a small part of the Ṁ_ budget. This suggests that there could be saturation in Ṁ_ due to the steady stellar wind, while the star could lose even more mass through other mechanisms, such as transient events (i.e. prominences, coronal mass ejections).
In addition to analyzing the general trends, we compared the model results of stars in our sample and objects with astrospheric Ṁ_ constraints. Our simulated Ṁ_ for stars in the unsaturated regime agree well with those estimated from astrospheric detections (namely for GJ 205, 61 Cyg A, and HD 219134). On the other hand, Ṁ_ from the 3D MHD simulations appear to differ by an order of magnitude or more from available estimates for ϵ Eri, EV Lac, and YZ CMi. We discussed how these results might be connected with the underlying assumption made by the observational analysis with respect to the stellar wind speed. Indeed, for all the stars in which our models differed largely from the literature estimates, we obtained much larger stellar wind speeds than the ones used in the astrospheric method. As such, we emphasized the importance of using the appropriate wind velocity when estimating Ṁ_ from observations.
We further discussed various possibilities for the discrepancies in EV Lac, YZ Cmi, ϵ Eri. For the two flaring stars, EV Lac and YZ CMi, we suspect that the high Ṁ_ estimates from the Ly-α absorption technique could be dominated by material from slingshot prominences and possibly CMEs (uncertain due to the expected magnetic confinement of CMEs in these stars). Note that this possibility was also considered by <cit.> in the original astrospheric analysis. In the case of ϵ Eri, we do not expect a large contribution from prominences or CMEs to the observed Ṁ_. However, as ϵ Eri undergoes a magnetic cycle, the stellar magnetic field and its expected modulation of stellar wind properties could explain some of the differences between the simulated and observed Ṁ_.
Moreover, we used the stellar wind results from the 3D MHD simulations to assess the conditions that an exoplanet would experience, and provide the stellar wind conditions in the entire classical Habitable Zones of our target stars. Our results show a scatter in the obtained distribution of AS versus M_, suggesting that the intrinsic dependence with the surface magnetic field properties can be reproduced for several spectral types. With respect to the stellar wind dynamic pressure, our results show that the orbital inclination plays a secondary role compared to the orbital distance. We have also found that a planet orbiting K and M stars must have a stronger dipole magnetic field than that of Earth to withstand the wind conditions, if the planetary magnetic field is indeed acting as a shield (this paradigm, however, is starting to be challenged by solar system observations).
Finally, the properties of the stellar wind in the HZ of different spectral types obtained here can be used in future studies to, for instance, estimate the expected radio emission due to wind-magnetosphere interactions or the planetary atmospheric mass loss due to erosion of the stellar wind from ion escape processes.
§ ACKNOWLEDGEMENTS
The authors would like to thank the referee for valuable comments that improved the quality of the paper.
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (<www.gauss-centre.eu>) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (<www.lrz.de>) under application ID 21761 (PI: Alvarado-Gómez). JJC and KP acknowledge funding from the German Leibniz Community under project number P67/2018. CG was supported by NASA
contract NAS8-03060 to the Chandra X-ray Center. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
§ DATA AVAILABILITY
The data would be made available to the community on reasonable request due to the volume of the 3D simulations. Extractions of specific quantities discussed in the paper could be requested from the corresponding author.
mnras
§ TRENDS WITH MAXIMUM RADIAL MAGNETIC FIELD
We have also quantified AS_ R and Ṁ_ / R^ 2_ as a function of the absolute maximum radial magnetic field strength (|B_ R|^ max|). It is important to also investigate |B_ R|^ max, since the average radial magnetic field strength may suffer from cancellations, especially if the star has a symmetric surface magnetic field distribution. Figure <ref> shows the simulated average Alfvén surface area (AS, top) and the mass-loss rate per unit surface area (Ṁ_/R^ 2_, bottom) as a function of the maximum absolute radial magnetic field on the stellar surface (|B_ R|^ max). We see a trend where AS and Ṁ_/R^ 2_ increase with increasing magnetic field strength.
We fit this trend to a power law by applying the bootstrap method used to derive this parameter as a function of the average radial magnetic field (similar to the procedure used in Sect. <ref>, Eqs. <ref> and <ref>).
logAS_ R = (0.44 ± 0.05) log |B_ R|^ max + (0.54 ± 0.08)
logṀ_ / R^ 2_ = (0.83 ± 0.07) log |B_ R|^ max - (0.48 ± 0.10)
|
http://arxiv.org/abs/2307.04121v1 | 20230709082717 | A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I | [
"Rajat Arora"
] | cs.LG | [
"cs.LG",
"cond-mat.mtrl-sci",
"cs.NA",
"math.AP",
"math.NA"
] | |
http://arxiv.org/abs/2307.04953v2 | 20230711005025 | Measuring Cause-Effect with the Variability of the Largest Eigenvalue | [
"Alejandro Rodriguez Dominguez",
"Irving Ramirez Carrillo",
"David Parraga Riquelme"
] | q-fin.PM | [
"q-fin.PM",
"stat.AP",
"58C40, 37M10, 60B12",
"G.3"
] |
Joint Radio Frequency Fingerprints Identification via Multi-antenna Receiver
Xiaofang Chen, Student Member, IEEE,
Wenbo Xu, Member, IEEE,
and Yue Wang, Senior Member, IEEE
XXX.
XXX.
XXX.
XXX.
XXX.
October 2023
==================================================================================================================================================
We present a method to test and monitor structural relationships between time variables. The distribution of the first eigenvalue for lagged correlation matrices (Tracy-Widom distribution) is used to test structural time relationships between variables against the alternative hypothesis (Independence). This distribution studies the asymptotic dynamics of the largest eigenvalue as a function of the lag in lagged correlation matrices. By analyzing the time series of the standard deviation of the greatest eigenvalue for 2× 2 correlation matrices with different lags we can analyze deviations from the Tracy-Widom distribution to test structural relationships between these two time variables. These relationships can be related to causality. We use the standard deviation of the explanatory power of the first eigenvalue at different lags as a proxy for testing and monitoring structural causal relationships. The method is applied to analyse causal dependencies between daily monetary flows in a retail brokerage business allowing to control for liquidity risks.
§ OVERVIEW
The Marcenko-Pastur paper <cit.> on the spectrum of empirical correlation matrices turned out to be useful in many, very different contexts (neural networks, image processing, wireless communications, etc.). It became relevant in the last two decades, as a new statistical tool to analyse large dimensional data sets and can be used to try to identify common causes (or factors) that explain the dynamics of N quantities.
The realization of the i^th quantity (i = 1,…, N) at “time” t (t = 1,…, T ) will be denoted r_t^i, which are demeaned and standardized. The normalized T × N matrix of returns will be denoted as X: X_ti = r^t_i/√(T). The Pearson estimator of the correlation matrix is given by:
E_ij = 1/T∑_i=1^Tr_i^tr_j^t≡(X^TX)_ij
where E will denote the empirical correlation matrix on a given realization, in contrast ti the true correlation matrix C of the underlying statistical process. The difference can be analysed by the Marcenko-Pastur result <cit.>. The empirical density of eigenvalues (the spectrum) is strongly distorted when compared to the ‘true’ density in the special asymptotic limit. When T→∞, N→∞, the spectrum has some degree of universality with respect to the distribution of the r_ti’s.
The lagged correlation matrix between past and future returns 𝒞_ij(τ) can be defined as:
𝒞_ij(τ)=⟨ r_i^t,r_j^t+τ⟩
such that 𝒞_ij(τ=0)=𝒞_ij is the standard correlation coefficient. Whereas 𝒞_ij is clearly a symmetric matrix, 𝒞_ij(τ>0) is in general non symmetric, and only obeys 𝒞_ij(τ)).
§ THE TRACY-WIDOM REGION
The Tracy-Widom result is that for a large class of N× N matrices (e.g. symmetric random matrices with i.i.d elements with a finite fourth moment, or empirical correlation matrices of i.i.d random variables with a finite fourth moment), the re-scaled distribution of λ_max - λ^∗ converges towards the Tracy-Widom distribution, usually noted F_1:
Prob(λ_max≤λ_+ + γ N^-2/3u)=F_1(u)
where γ is a constant that depends on the problem. Everything is known about the Tracy-Widom density f_1(u) = F^'_1(u), in particular its left and right far tails:
lnf_1(u)∝-u^3/2, (u→∞); lnf_1(u)∝-|u|^3, (u→-∞)
The left tail is much thinner: pushing the largest eigenvalue inside the allowed band implies compressing the whole Coulomb-Dyson gas of charges, which is difficult. Using this analogy, the large deviation regime of the Tracy-Widom problem (i.e. for λ_max - λ^+ = O(1)) can be obtained <cit.>.For square symmetric random matrices, the celebrated semicircle law of Wigner <cit.> describes the limiting density of eigenvalues. There is an analog for covariance matrices <cit.>, and independently, Stein <cit.>. The Marcenko-Pastur result is stated here for Wishart matrices with identity covariance E = I, but is true more generally, including non-null cases. Suppose that both n and p tend to ∞, in some ratio n/p →γ≥ 1. Then the empirical distribution of the eigenvalues converges almost surely,
G_p(t)=1/p#{l_i:l_i≤ nt}→ G(t)
and the limiting distribution has a density g(t) = G^'(t):
g(t)=γ/2π t√((b-t)(a-t)), a≤ t ≤ b
where a = (1 - γ^1/2)^2 and b = (1 + γ^1/2)^2. Consider now the right-hand edge, and particularly the largest eigenvalue. Why the interest in extremes? In the estimation of a sparse mean vector, the maximum of n i.i.d. Gaussian noise variables plays a key role. Similarly, in distinguishing a "signal subspace" of higher variance from many noise variables, one expects the largest eigenvalue of a null (or white) sample covariance matrix to play a basic role.
The bulk limit (<ref>) points to a strong law for the largest eigenvalue. Indeed, <cit.> shows that:
n^-1l_1→(1 + γ^1/2)^2, a.s
that is l_1 ∼ (√(()n)+√(()p))^2. Later Bai, Krishnaiah, Silverstein and Yin established that strong convergence occurred iff the parent distribution had zero mean and finite fourth moment. For more details, full citations and results on the smallest eigenvalue <cit.>. However, these results say nothing about the variability of the largest eigenvalue, let alone about its distribution. For a survey of existing results <cit.>. For example, [<cit.>, page 1284] gives an exact expression in terms of a zonal polynomial series for a confluent hypergeometric function of matrix argument:
P(l_1 ≤ nt) = d_p,nt^pn/2_1F_1(1/2n;1/2(n+p+1);-1/2btℐ_p))
where d_p,n is a constant depending only on p and n [cf. also <cit.>, page 421]. There are explicit evaluations for p = 2, 3, but in general the alternating series converges very slowly, even for small n and p, and so is difficult to use in practice. For fixed p and large n, the classic paper by <cit.> gives the limiting joint distribution of the roots, but the marginal distribution of l_1 is hard to extract even in the null case X = I. <cit.> gives a series approximation again for p = 2, 3. In general, there are upper bounds on the d.f. using p independent χ^2(n). Overall, there is little that helps numerically with approximations for large p.
We now turn to what can be derived from random matrix theory (RMT) methods. Suppose that X = (X_jk)_n× p has entries which are i.i.d. X_jk∼ N(0, 1). Denote the sample eigenvalues of the Wishart matrix X^' X by
l_1 >…> l_p. Define center and scaling constants:
μ_np=(√(n-1)+√(p))^2
σ_np=(√(n-1)+√(p))(1/√(n-1)+1/√(p))^1/3
The Thacy-Widom law of order 1 has distribution function defined by:
F_1(s)=exp{-1/2∫_s^∞q(x)+(x-s)q^2(x)dx},s∈ℝ
where q solves the (nonlinear) Painleve II differential equation:
q^''(x)=xq(x)+2q^3(x)
q(x) ∼ A_i(x) as x→ +∞
and A_i(x) denotes the Airy function. This distribution was found by Tracy
and Widom (1996) as the limiting law of the largest eigenvalue of an n by n
Gaussian symmetric matrix.
Let 𝒲 be a white Wishart matrixand l_1 be its largets eigenvalue. Then:
l_1-μ_np/σ_np𝒟⟶W_1∼ F_1
where the center and scaling constants are:
μ_np=(√(n-1)+√(p))^2 , σ_np=μ_np((n-1)^-1/2+p^-1/2)^1/3
and F_1 stands for the distribution function of the Tracy-Widom law of order 1.
The theorem is stated for situations in which n > p. However, it applies equally well if n < p are both large, simply by reversing the roles of n and p in (<ref>) and (<ref>). The limiting distribution function F_1 is a particular distribution from a family of distributions F_β. For β = 1, 2, 4 functions F_β appear as the limiting distributions
for the largest eigenvalues in the ensembles GOE, GUE and GSE, correspondingly. For the largest eigenvalue l_max(𝒜) of the random matrix 𝒜 (GOE (β = 1), GUE (β = 2) or GSE (β = 4)) its distribution function F_N,β(s) = P(l_max(𝒜) < s), β = 1, 2, 4 satisfies the limit law in (<ref>) with l_1=l_max(𝒜), and F_1=F_β given by (<ref>).
From (<ref>), the Airy special function A_i(s) is one of the pairs of linearly independent solutions to the differential equation: w^''-zw=0 such that:
lim_s→∞q(s)/A_i(s)=1
The Painleve II is a second-order ordinary differential equation of the form d^2w/dz^2=F(z,w,dw/dz). In Figure <ref> <cit.>, we can see some simulations for square cases n - p = 5, 10 and 100, using R = 10,000 replications (s is the lag value).
The main conclusion is that the Tracy-Widom distribution F_1 provides a usable numerical approximation to the null distribution of the largest principal component from Gaussian data even for quite moderate values of n and p. In particular, we have the following simple approximate rules of thumb:
* About 83% of the distribution is less than μ_np = (√(n-1) +√(p))^2
* About 95% and 99% lie below μ_np + σ_np and μ_np + 2σ_np respectively
§ METHOD
We can monitor the empirical value of σ_np in time from data. However, our method focuses on σ_λ, the standard deviation of the explanatory power for the first eigenvalue in 2× 2 matrices, λ_1/λ_1+λ_2, instead of just λ_1-λ_2 as with σ_np. This allows us to deal with a measure with empirical sense for the method. With extreme values of σ_λ indicating deviations from the Tracy-Widom distribution F_1 which implies asymptotically independence versus structural time (causal) relationships. In Algorithm <ref> we present the method for any dataset.
In Figure <ref>, we show the average of the σ_λ for 10 i.i.d time series generated from a Normal distribution. For this, Algorithm <ref> is applied to sets of 2 different time-series. In Figure <ref>, we compare the average values of σ_λ for the i.i.d series from the previous case with the average values for Algorithm <ref> applied to time series variables from a financial dataset. The hypothesis can be simply verified from data as for independent time series the values of the indicator σ_λ are much smaller than the values for time series with structural relationships.
§ EXPERIMENTS
For the experiments, we analyse daily brokerage activities with time series data of 15000 clients approximately. Our goal is to analyse the power of the structural temporal (causal) relationship between cash balance and other variables in time. Experiments are performed in cause-effect pairs with effect being cash balance, but for illustration we plot the results in thematic buckets. The buckets are:
* The account status includes the daily amount in account currency of open positions in different products including Shares, Bonds, Mutual Funds, CFDs, Derivatives, FX Spot and ETO.
* Bookkeeping cash includes all the daily cash movements in the account currency, both internal and external cash transactions, including the financial product that is the subject of the transaction.
* Trades Executes is focused in the daily buying and selling transaction activities in the account currency for all clients in all broker products.
* From experiments we can see that account status bucket has more mixed results, with Mutual Funds from November 2022 until March 2023 holding majority. After that, the majority of cash movements could be explained by Stocks (Shares) (Figure <ref>).
* In the Bookkeeping cash bucket we see a defined pattern of the majority of cash transactions coming from shares activities. Until March 2023 this majority was shared with Mutual Funds however, this stopped from march on-wards and taking some importance the derivatives and FX products since March. The relationship is studied with respect to daily variations of cash balance gross amount (net outflows plus inflows) and not the total amount of flows (Figure <ref>).
* Finally, the trades executed bucket shows very similar behaviour than the Bookkeeping cash serving as validation point for our results (Figure <ref>). The reason why Mutual Funds are not in this picture is because the execution process is different for Mutual Funds, much of operations not having to be converted to cash. This could mean that, although Mutual Funds has been a product that at least historically until March 2023 has been responsible for the movement of cash balance of our clients (seen Bookkeeping), this has not been due to daily executions in funds from cash but in the form of internal and external transfer funds. With regards to trades executed, we can see that from June 2022 until December 2022 the relationships where mixed with shares not holding the majority. From December 2022 until now shares have hold the majority with some derivatives gaining importance since march.
In Figure <ref>, we show the average σ_λ for lags 2 and 5 of the top candidates with greater values in the sample dataset (causal candidates). We perform the Granger causality test <cit.> with the sample dataset to compare results with our method. In Figures <ref> and <ref>, we show the values for the logarithm of the inverse of the p-value for the Granger causality test for lags 2 and 5 respectively. We can see discrepancies between the Granger Causality test and our method, however our method is closer to the structural causal relationship based on empirical evidence from the dataset. In the case of the Granger test, the CFD product seems to be the one most causally related with the cash balance, in contrast to the product shares or stocks for our method. The issue with the Granger causality test is that is window based, which means that it depends on the sample time interval you choose without a dynamic sense of the causal relationships. On the other hand, a big structural relationship in one data point out of the full sample can bias the results. In contrast, our method is rolling window based, which means that is adaptive and able to capture changes in causal or structural relationships. It is less biased to causal point relationships focusing more on persistent dynamics. The CFDs business is highly correlated with the equity business, reason why the Granger causal test chooses CFDs against our method choosing Shares or stocks. However, our method is able to infer causality better in that, the Shares business is the biggest in size and flows and the CFDs business is much smaller. To measure true causal relationships between these flows, the time series and correlations are not necessary as the flow size matters, and our method is able to capture causality better, as it avoids window biases coming from extreme point values which are more related to correlation than to causal dynamics.
We can conclude that with greater lags, 5 or more, values only get smoother but patterns look the same (In Figure <ref> we can see results for lag 10). This verifies the theoretical setup presented in this document in that, for greater lags the standard deviation of the explanatory power of their largest eigenvalue (σ_λ) is higher, and this implies a higher probability that the two variables deviate from independence based on the Tracy-Widom distribution towards a structured (causal) relationship.
§ CONCLUSION
A systematic approach to measure and monitor structural relationships in time, which are related to causal relationships, is presented. The method is based on monitoring the time series of the standard deviation of the explanatory power of the first eigenvalue for multiple lags in lagged correlation matrices, which is related to the Tracy-Widom distribution from RMT. These matrices consist on 2× 2 correlation matrices between an hypothetical causal variable and the respective effect variable. The different time series for different causal variables given the same effect variable are compared. The method is simple and fast, allowing to avoid biases produced by other statistical tests such as Granger Causality test. The method is applied to analyse the structural or causal dependencies between daily monetary flows in a retail brokerage business. This allows practitioners to understand the causal dynamics between these flows being able to control for liquidity risk in banks or other financial institutions. The method can be applied to monitor causal or structural dependencies in time in any particular dataset. Extreme values of the indicator can serve for risk management or alpha signal purposes.
unsrt
|
http://arxiv.org/abs/2307.05045v3 | 20230711065139 | Stationary striations in plasma, created by a short microwave pulse in a waveguide filled with a neutral gas | [
"Y. Bliokh",
"Y. Cao",
"V. Maksimov",
"A. Haim",
"J. G. Leopold",
"A. Kostinsky",
"Ya. E. Krasik"
] | physics.plasm-ph | [
"physics.plasm-ph"
] |
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Corr.author
[email protected]
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
Physics Department, Technion, Israel Institute of Technology, Haifa 320003, Israel
It was observed experimentally that after crossing a waveguide filled with a neutral gas, a short powerful microwave pulse leaves a periodic glow of plasma along the waveguide, persisting several tens of nanoseconds. A theoretical model is presented which in combination with numerical simulations proposes a possible explanation of this phenomenon.
Stationary striations in plasma, created by a short microwave pulse in a waveguide filled with a neutral gas
Ya.E. Krasik
August 12, 2023
=============================================================================================================
§ INTRODUCTION
Interaction of strong electromagnetic waves with plasma has always attracted considerable attention. Charged particles acceleration in the wake wave exited by a powerful laser pulse propagating in plasma <cit.> is at present of particular interest because of significant experimental and theoretical achievements and practical importance (see, e.g., <cit.>).
The non-linear interaction of electromagnetic pulses with plasma when the microwave and plasma frequencies are of the same order of magnitude, remains permanently in researchers' field of view. Recent progress in generating extremely intense, hundreds of MW, sub-nanosecond microwave pulses <cit.> allows one to observe experimentally and study such phenomena as ionization-induced self-channeling of a microwave beam,<cit.>, frequency shift in the wake excitated in a plasma-filled waveguide<cit.>, complete absorption of a microwave pulse in plasma,<cit.> frequency up-shift and pulse compression in a propagating self-generated ionization front <cit.>.
In this paper a new phenomenon is described which accompanies a neutral gas ionization by intense sub-nanosecond microwave pulse propagating in a waveguide. It is suggested that a periodic stratification of the plasma forms near the waveguide wall and its glow is observed (see Fig. <ref>), for helium within a pressure range of 7-20 torr and for air within 1-3 torr. The plasma glow appears approximately 5 ns after the microwave pulse enters the gas tube and persists for 70 ns.
There is nothing unexpected in that perturbations remain in the plasma after the electromagnetic pulse's passage. The wake excited by a short laser pulse is an example of such perturbation which has received most of the attention. This perturbation propagates together with the pulse and behind it, that is, in the pulse's frame of reference, it does not move. The periodic stratification demonstrated in Fig. <ref>, is static in the laboratory frame of reference and appears in the same positions along the waveguide from shot to shot. The purpose of the present article is to propose a physical model which explains this phenomenon.
§ EXPERIMENTAL SETUP
The experimental setup (see Fig. <ref>) used in this research is similar to that described in <cit.>.
The high-power microwave (HPM) pulse (∼ 250 - 300 MW, ∼ 0.35 ns, ∼ 9.6 GHz, TM_01 mode) is generated by a super-radiant backward wave oscillator (SR-BWO) (see <cit.> for details). The output of the SR-BWO is connected to a 2.8 cm diameter circular waveguide with a calibrated coupler (#1) at its center. This waveguide is followed by the gas tube, filled with helium or air at the desired pressure. Inside the gas tube, the HPM pulse propagates along a circular waveguide consisting of 12 parallel stainless steel cylindrical wires (1 mm diameter) distributed uniformly (5-mm apart) along its 2.8 cm perimeter. Electromagnetic simulations confirm that the wave propagating along this waveguide is practically indistinguishable from the wave propagating along a solid-wall cylindrical waveguide. A second calibrated coupler (#2) is connected at the exit from the gas tube followed by an impedance-matched horn antenna. Couplers #1 and #2 measure the incident, transmitted, and reflected wave forms acquired by an Agilent DSO81204B oscilloscope (12 GHz, 40 Gs/s). At the entrance and exit of the gas tube, 0.3 mm thick Mylar interface windows are installed to separate the vacuum/gas/vacuum media. A fast framing intensified 4QuikE ICCD camera (Stanford Computer Optics) operating with a frame duration of 1.5 ns was used to capture the light emission from the plasma formed by the HPM pulse.
The plasma light emission patterns are shown in Fig. <ref>. One can see periodic plasma glow (striations) near wires which appears approximately 5 ns after the HPM pulse has left the wire waveguide and this glow is present for 70 ns. Such striations were obtained in helium within the pressure range of 7-20 torr, and in air at pressures of 1 - 3 Torr, which corresponds to the transparency window of the HPM pulse in air. By capturing frames at different time delays between the HPM pulse registered by coupler #1 and the 4QuilE camera frame, it can be stated that the striations are stationary and do not move along the path of the HPM pulse propagation. A typical longitudinal period of these striations is 1.9 cm.
§ IONIZATION
The electric field of a symmetric TM microwave mode propagating in a cylindrical waveguide and the energy of electrons, oscillating in the wave fields, are maximal near the axis. When the microwave power is moderate (a few MW) the plasma is created mostly near and its density is maximal on the axis. When the microwave power is sufficiently high, the region of maximal ionization rate shifts from the axis toward the waveguide wall. The reason is a non-monotonic dependence of the electron impact ionization cross section σ(w_e) on the electron energy w_e. The cross section is maximal when the electron energy is of the order of 100 eV and decreases with increasing electron energy. For microwave power of the order of tens of MW or higher, the electron oscillating energy near the axis is so large that the ionization cross sections is smaller than its maximal value. In contrast, near the waveguide wall, where the wave electric field is smaller than near the axis, the ionization cross section is close to its maximal value.
The time evolution of the electron density, n_e(r,t), is described by the equation
d/dtln n_e(t)=n_gσ[w(r,t)]|v(r,t)|,
where n_g is the neutral gas density, v is the electron velocity, and σ[w(r,t)] is the ionization cross section, which depends on the electron kinetic energy w=mv^2/2. If the duration of the electromagnetic pulse, t_ pulse, is large compared to the field oscillation period T=2π/ω, ω t_ pulse≫ 1 (where ω is the wave frequency), it is convenient to use the ionization rate I(r,t)=σ[w(r,t)]|v(r,t)| averaged over a period of the oscillations:
I(r;P)=ω/2π∫_π/ω^π/ωdt σ[w(r,t)]|v(r,t)|
The dependence of the ionization rate I(r;P) on the radius for various values of the microwave power P are shown in Fig. <ref>.
The radial profile of the plasma density, which remains in the waveguide behind the pulse, can be calculated as
n_e(r,t)=n_0exp[n_g∫_-∞^tI̅(r;P(t^'))dt^'],
where n_0 is the initial electron density, and P(t) is the pulse power's temporal profile. An example of the electron density evolution is presented in Fig. <ref>. One can see that the electron density in a narrow layer near the waveguide wall is almost an order of magnitude larger than near the axis. This difference increases rapidly with the neutral gas density growth.
Note that the plasma density grows rapidly at the rear tail (late time) of the pulse, and the characteristic rise time of the plasma density is much smaller than the pulse duration. The importance of this will be discussed below.
§ EIGENMODES OF A WAVEGUIDE FILLED PARTIALLY WITH PLASMA
The neutral gas ionization process induced by a powerful microwave pulse described in Sect. III assumes that the structure of the microwave fields remains unchanged by the presence of plasma, which is correct for plasma density smaller than critical n_ crit=ω^2 m/4π e^2. When the plasma density is close to critical, the structure of the wave's fields vary considerably. In order to trace this variation, let us consider the simple model of assuming a step-wise three-layer density distribution:
n(r)={[ n_1, r<r_1 ,; n_2, r_1<r<r_2 ,; n_3, r_2<r<R ].
where R is the waveguide radius. This plasma density profile is a rough description of the expected density distribution described in Sect. III.
Matching tangential components of the electromagnetic fields at the interfaces between the different regions, one can find the spectrum ω(k) of the layered waveguide eigenmodes as the solution of a boundary value problem.
Even such a simple form of the plasma density distribution contains many parameters. Below, only such values of the parameters will be considered which correspond to the experimental conditions. In particular, because the plasma density in region II can exceed the ones in regions I and III, the model can be further simplified, by assuming n_1=n_3=0.
An example of the eigenmodes spectrum is shown in Fig. <ref>.
The presence of the plasma with a ring-shaped transversal profile enriches the spectrum by two surface waves: a backward- and a forward-propagating mode. The spectrum of the electromagnetic wave changes only little if the plasma layer's thickness is small compared to the waveguide radius. On the contrary, the change in the structure of the wave's fields are very strong in Fig. <ref>.
Compared to an empty waveguide, in the plasma-filled waveguide the amplitudes of the longitudinal component of the electric field, E_z(r), and the azimuthal component of the magnetic field, H_ϕ(r), decrease over all cross sections, but not that much. In contrast, the variation in the radial electric field, E_r(r), profile is considerably stronger. This component of the field increases strongly in region II, occupied by the plasma which means that the oscillatory energy of the plasma electrons w∝ E_z^2+E_r^2 exceeds the value calculated ignoring the change in the eigenmode fields. Because the ionization cross section is a decreasing function of the electron energy (in the energy range of interest), changes in the eigenmode fields structure suppresses ionization and the region with the most effective ionization is shifted closer to the waveguide axis, widening the plasma layer.
In the numerical example relevant to Fig. 5, the plasma density was chosen so that the Langmuir frequency ω_p is equal to the empty waveguide cut-off frequency ω_c. This is an arbitrary choice, but the analysis allows one to assume that the plasma density in the expanded layer is close to this value. Indeed, if the plasma density is small, so that ω_p≪ω_c, the deformation of the eigenmode fields is small, and also the ionization in this region continues. When the density is large enough, so that ω_p≃ω_c, the ionization slows down. The density cannot be so large that ω_p≫ω_c, because the waveguide becomes opaque for the incoming microwaves if the frequency ω_ EMW is close to the cut-off frequency, ω_c, as is the case in the experiments.
Thus, a high-power microwave pulse, propagating in the waveguide filled with a neutral gas, can create a hollow, tubular plasma. Note that a similar hollow plasma column can be created in an unbounded system by a microwave pulse with a bell-shaped radial profile when the required power density can be achieved by focusing the microwave beam <cit.>, so that the hollow plasma is created in the vicinity of the focal plane only. A hollow plasma can also be created by a tubular electron beam, propagating in a neutral gas along a strong guiding magnetic field <cit.>. Ionization of a neutral gas by a powerful tubular laser beam can also create a plasma with specified density distribution <cit.>.
§ BACKWARD SURFACE WAVE EXCITATION AND PLASMA GLOW STRIATIONS
Propagation of a powerful electromagnetic pulse in a preliminarily prepared plasma is accompanied by the excitation of a wake, which is a slow eigenmode of either an unbounded plasma or a plasma-filled waveguide. Phase velocity of this eigenmode coincides with the group velocity of the pulse. It seems reasonable to assume that a similar wake is excited and remains in the plasma behind the pulse even though the plasma was created by the pulse itself.
Indeed, for the case considered, the perturbation, produced by an electromagnetic pulse, remains in a tubular plasma in the form of a backward surface wave. The wave number and the frequency of the exited surface wave are defined by Čherenkov's resonance condition ω_ BSW(k)=k_ BSWv_g, as shown in Fig. <ref>. Here ω_ BSW and k_ BSW are assigned to the backward surface wave and v_g is the group velocity of the electromagnetic pulse. This wave can persist in the plasma for a long time after the pulse passage. Radial distribution of the electric and magnetic fields of this backward propagating surface wave is shown in Fig <ref>.
Axially-symmetric surface waves with negative group velocity in waveguides containing a plasma column in a dielectric tube, separated from the conducting wall, were considered in <cit.>. Backward surface waves in a waveguide with a step-wise annular plasma column, similar to that considered in this article, were studied in <cit.>. It was shown here that the backward and forward propagating modes are associated with the inner and outer plasma surfaces, respectively. Surface waves in waveguides filled with smoothly distributed inhomogeneous plasma were studied in <cit.>. It was shown that the absence of sharp boundaries between different layers leads to collision-less dissipation of the surface waves. The reason is the presence of resonant electrons in the smooth transition plasma layer.
Negative group velocity of the surface wave means that the wave, excited along the plasma-filled section of the waveguide, propagates in the opposite direction (upstream) towards the input window. Here the wave is reflected and propagates towards the downstream window. Because the phase velocity of the reflected wave is negative, these two incident and reflected waves form a standing wave near the input window. Electrons, oscillating in the electric field of the standing wave antinodes, can excite and/or ionize molecules of the neutral gas, producing a periodically distributed plasma glow.
It is worth noting that, under certain conditions, stationary striations in a glow discharge are the result of interference between two ionization waves <cit.>. Unlike backward-propagating surface waves, ionization waves are excited only in the discharge current and disappear immediately after current interruption.
As shown in the Sect. IV, the process of the plasma creation by the microwave pulse can be roughly divided into two phases. Initially, plasma is created in a narrow layer near the waveguide boundary. When the electron Langmuir frequency ω_p(r) in this layer approaches the cut-off frequency ω_c, the plasma density growth slows down, and the second phase of the process begins. Now, ionization leads to widening of this layer with approximately constant plasma density. In the step-wise model, Eq. (<ref>), it means that n_2(t)≃ const and r_1(t) decrease with time. Simultaneously, the cut-off frequency of the plasma-filled waveguide grows and, if the pulse duration is long enough, it reaches the incoming wave frequency ω_ EMW (which, in the experiment, is close to the cut-off frequency of the empty waveguide, ω_ EMW≃ 1.17 ω_c) while the retained part of the pulse is reflected from the waveguide input window.
When the layer thickness d=r_2-r_1 varies with time, the pulse group velocity v_g and the backward surface wave dispersion characteristics ω_ BSW(k_ BSW) vary too. In this case, effective excitation of the surface wave is possible if the resonant frequency ω_ BSW and wave number k_ BSW remain essentially unchanged during the pulse duration. In Fig. <ref> the dependence of the resonant surface wave frequency and wavelength on the plasma layer thickness demonstrate relatively small dispersion. The characteristic wavelength is about 2 cm, so the distance between nodes of the standing wave is about 1 cm. Taking into account the roughness of the model, one can consider that this value agrees well with the experimentally observed period of the glow ( 1.8 cm).
§ DISCUSSION
The resonant excitation of a wave of any nature by a moving perturbation is effective when the characteristic spatial scale of the perturbation is small compared with its wavelength, or otherwise, the amplitude of the wake behind the perturbation is small. As seen in Fig. <ref>, the expected wavelength of the backward surface wave is smaller than the electromagnetic pulse wavelength and much smaller than the longitudinal dimension of the pulse. Thus, the electromagnetic pulse with parameters corresponding to the experimental conditions, moving through a tubular plasma, can excite a surface wave of a small amplitude. Results of numerical simulation, presented in Fig. <ref>, confirm this conclusion.
The main peak of the wake spectrum, depicted in Fig. <ref>, is located in the region of 8-9 GHz, that is in good agreement with Fig. <ref>. Note that the spectral width of the wake is smaller than that of the pulse because of the following reason. A short pulse (such as that considered) is characterized by a wide spread of group velocity values rather than a single velocity. Consequently, the resonant condition ω_ BSW(k_ BSW)=k_ BSWv_g is satisfied for a set of surface waves, but, due to the weak dependence ω_ BSW(k_ BSW) (see Fig. <ref>), the spectrum of the excited surface waves remains narrow. This property of the wake spectrum and its position strongly supports the backward flowing surface wave description.
Despite that the excitation of the backward flowing surface wave by the electromagnetic pulse is confirmed in numerical simulations, the assumption that this wave is responsible for the observed spatially-periodic plasma glow casts doubt, because of the small amplitude of the surface wave. This discrepancy can be resolved in the following way.
In the performed numerical simulation it was assumed that the plasma density is stationary and homogeneously distributed along the waveguide. However, in the experiment the plasma is created by the propagating electromagnetic pulse itself. Then, the temporal (and longitudinal) profile of plasma density, which is shown in Fig. <ref>, looks like shock wave and propagates along the waveguide together with the electromagnetic pulse. In other words, the surface wave propagates in the media with time-dependent, traveling parameters. The characteristic spatial scale of the plasma density variation is much smaller than the pulse length and can be of the order of the surface wave wavelength. The amplitude of the resonant wave, excited by such abrupt traveling variation of the medium parameters, is much larger compared to the case, when the medium parameters are varied smoothly <cit.>. Due to this reason, one can assume that the stationary periodic plasma glow is indeed associated with the excitation of the backward surface wave.
§ CONCLUSIONS
To summarize, it was shown that a sub-nanosecond microwave pulse of hundreds MW power propagating in a neutral gas filled cylindrical waveguide produces a tubular plasma, with the density mainly concentrated in a thin layer near the waveguide wall. A surface wave with negative group velocity can be excited in such plasma-filled waveguide. The wake in the form of this backward propagating surface wave remains in the waveguide long after the pulse's passage. A backward propagating surface wave is reflected from the input window and in combination with the incident wave forms a standing wave. Plasma electrons, oscillating in the electric field of this standing wave's antinodes, excite and/or ionize molecules of the neutral gas, producing the experimentally observed periodically distributed plasma glow along the waveguide.
This scenario is based on a theoretical model supported by numerical simulations and analysis. We cannot present indisputable experimental facts that the observed spatially-periodic plasma glow appeared exactly in that way as it is described in the paper. Should this be the case then this phenomenon would be the first non-direct observation of the backward surface wave, propagating along a tubular plasma in the absence of magnetic field. In future experiments we will attempt to confirm the mechanism behind this interesting phenomenon.
§ ACKNOWLEDGEMENTS
The authors are grateful to E. Flyat for his technical support. This study was supported by Pazy Foundation Grant No. 2032056.
99
Tajima-1979 T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979).
Esarey-2009 E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev.Mod. Phys. 81, 1229 (2009).
Hooker-2013 S. M. Hooker, Nature Photonics 7, 775 (2013).
Tajima-2020 T. Tajima and V. Malka, Plasma Phys. Control. Fusion 62, 034004 (2020).
Eltchaninov-2004 A. A. Eltchaninov, S. D. Korovin, G. A. Mesyats, I. V. Pegel, V. V. Rostov, V. G. Shpak, and M. I. Yalandin, IEEE Trans. Plasma Sci. 32, 1093 (2004).
Rostov-2016 V. V. Rostov, I. V. Romanchenko, M. S. Pedos, S. N. Rukin, K. A. Sharypov, V. G. Shpak, S. A. Shunailov, M. R. Ul’Masculov, and M. I. Yalandin, Phys. Plasmas 23, 093103 (2016).
Shafir-2018 G. Shafir, Y. E. Krasik, Y. P. Bliokh, D. Levko, Y. Cao, J. G. Leopold, R. Gad, V. Bernshtam, and A. Fisher, Phys. Rev. Lett. 120, 135003 (2018).
Cao-2020 Y. Cao, Y. P. Bliokh, J. G. Leopold, A. Li, G. Leibovitch, and Ya. E. Krasik, Phys. Plasmas 27, 053103 (2020).
Cao-2023 Y. Cao, Y. P. Bliokh, V. Maksimov, J. G. Leopold, and Ya. E. Krasik, Phys. Rev. E 107, 045203 (2023).
Cao-2021
Cao-2018 Y. Cao, J. G. Leopold, Y. P. Bliokh, and Ya. E. Krasik, Phys. Plasmas 25, 103101 (2018).
Kuzelev-1982 M. V. Kuzelev, F. Kh. Mukhametzyanov, M. S. Rabinovich, A. A. Rukhadze, P. S. Strelkov, and A. G. Shkvarunets, Sov. Phys. JETP 56, 780 (1982).
Laser M. Châteauneuf, S. Payeur, J. Dubios, and J.-C. Kieffer, Appl. Phys. Lett. 92, 091104 (2008).
Trivelpiece-1958 A. W. Trivelpiece, “Slow wave propagation in plasma waveguides,” Ph.D. Dissertation, California Institute of Technology,
Pasadena-California, 1958; A. W . Trivelpiece, Slow Wave Propagation in Plasma Waveguides, San Francisco Press Inc. 1967.
Paik-1962 S. F. Paik, Journal of Electronics and Control 13, 515 (1962).
Kuzelev-2005 M. V. Kuzelev, R. V. Romanov, and A. A. Rukhadze, Fiz.
Plazmy 31, 172 (2005) [Plasma Phys. Rep. 31, 147
(2005)].
Kuzelev-2014 M. V. Kuzelev and N. G. Orlikovskaya, Fiz.
Plazmy 40, 345 (2014) [Plasma Phys. Rep. 40, 276
(2014)].
Cooper-1958 A.W.M. Cooper, J.R.M. Coulter, and K.G. Emeléus, Nature 4619, 1327 (1958).
Stirand-1967 O. Štirand and Láska, Czech. J. Phys. B 17, 928 (1967).
Maruyama-1990 T. Maruyama, S. Nishina, H. Kitamura, K. Itagaki, and H. Mizuochi, Contrib. Plasma Phys. 30, 497 (1990).
Kravtsov-1990 Yu.A. Kravtsov, Yu.I. Orlov, Geometric Optics of Inhomogeneous Media, Springer-Verlag, 1990.
Stepanov-1993 N.S. Stepanov, Izv.VUZov Radiofizika, 36, 623 (1993) (in Russian).
|
http://arxiv.org/abs/2307.07260v1 | 20230714102126 | A Dynamic Points Removal Benchmark in Point Cloud Maps | [
"Qingwen Zhang",
"Daniel Duberg",
"Ruoyu Geng",
"Mingkai Jia",
"Lujia Wang",
"Patric Jensfelt"
] | cs.RO | [
"cs.RO",
"cs.AI"
] |
A Dynamic Points Removal Benchmark in Point Cloud Maps
Qingwen Zhang^1, Daniel Duberg^1, Ruoyu Geng^2, Mingkai Jia^2, Lujia Wang^2, Patric Jensfelt^1
^1Authors are with the Division of Robotics, Perception, and Learning (RPL), KTH Royal Institute of Technology, Stockholm 114 28, Sweden. (email: [email protected])
^2Authors are with Robotics Institute, The Hong Kong University of Science and Technology, Hong Kong SAR, China.
August 12, 2023
======================================================================================================================================================================================================================================================================================================================================================================================
In the field of robotics, the point cloud has become an essential map representation.
From the perspective of downstream tasks like localization and global path planning, points corresponding to dynamic objects will adversely affect their performance.
Existing methods for removing dynamic points in point clouds often lack clarity in comparative evaluations and comprehensive analysis.
Therefore, we propose an easy-to-extend unified benchmarking framework for evaluating techniques for removing dynamic points in maps.
It includes refactored state-of-art methods and novel metrics to analyze the limitations of these approaches.
This enables researchers to dive deep into the underlying reasons behind these limitations.
The benchmark makes use of several datasets with different sensor types.
All the code and datasets related to our study are publicly available for further development and utilization.
§ INTRODUCTION
Point clouds are widely used in the domains of robotics, given their effectiveness in facilitating key components such as localization and path planning. Current SLAM (Simultaneous Localization and Mapping) packages <cit.> fuse data from multiple sensors to obtain corresponding poses. These poses can be used to integrate point cloud frames into a global map shown in Fig. <ref>.
Removing dynamic points from maps is crucial for accurate representations of the environment. Failing to detect dynamic points while integrating point cloud data can result in the inclusion of ghost points, as illustrated in Fig. <ref> yellow part.
In the localization task, ghost points may reduce robustness as they introduce ambiguous features or mislead the matching process between the current observation and the global map. For global path planning, the presence of ghost points can lead to suboptimal path selection.
If the planning algorithm interprets points corresponding to dynamics as part of the static environment's structure, it will mistake these points as obstacles and classify the region as untraversable, resulting in unnecessarily long path allocation or even failure in path planning.
Various methods are proposed to tackle the issue of removing dynamic points, where different metrics are tailored to showcase the benefits of their own approaches. For example, Lim et al. <cit.> utilize the voxel-wise preservation rate to evaluate their results. However, the existing evaluation metrics neglect the classification accuracy in the sub-voxel scale.
Our benchmark adopts a set of new metrics for point-wise evaluation with a unified standard.
Existing methods including <cit.> are mainly evaluated on SemanticKITTI <cit.>, which solely includes small town scenarios by a single type of LiDAR.
Our benchmark performs evaluation on various datasets to analyze robustness towards different scenarios and sensor setups.
We also prepared a dataset in a semi-indoor scenario where dynamic objects are moving close to the static structure, and the ego agent is equipped with a sparse LiDAR.
For qualitative results, we additionally choose the latest Argoverse 2.0 dataset <cit.> that contains various streetscapes in big cities and has more dynamic objects compared with SemanticKITTI.
These diverse datasets enable a comprehensive assessment of the existing techniques to compare their adaptability to a range of scenarios and sensor configurations.
Based on our benchmarking result, we summarise the strengths and weaknesses of each technique revealed from our proposed metrics, facilitating further development and innovation in the field.
For instance, the occupancy mapping approach, Octomap <cit.>, is also frequently used as a dynamic point removal baseline.
Guided by the benchmarking result, we demonstrate how we improve its dynamic point removal performance by incorporating ground fitting into the pipeline.
We contribute the benchmark implementation and extended datasets to the research community at <https://github.com/KTH-RPL/DynamicMap_Benchmark>.
The main additional contributions include the following:
* Refactoring existing methods to establish a unified benchmark for removing dynamic points in the map.
* Introducing new metrics and evaluating the performance of all methods, detailing the challenges associated with this task.
* Introducing an extension of Octomap better adapted to the map clean task.
§ RELATED WORK
In the field of point cloud processing on dynamic points removal, methods can be broadly categorized into two main approaches: learning-based and traditional algorithms.
Learning-based methods have been increasingly popular in detecting dynamic points or objects. However, they require training data and a network to learn latent space representations, often lacking explainability. Therefore, this paper focuses on traditional approaches to removing dynamic points. In the below sections, we will review both learning-based and traditional methods in detail.
§.§ Learning-based
Learning-based methods typically involve deep neural networks and supervised training with labeled datasets.
Mersch et al. <cit.> employ sparse 4D convolutions to segment receding moving objects in 3D LiDAR data, efficiently processing spatiotemporal information using sparse convolutions.
Sun et al. <cit.> develop a novel framework for fusing spatial and temporal data from LiDAR sensors, leveraging range and residual images as input to the network. Toyungyernsub et al. <cit.> predict urban environment occupancy by considering both spatial and temporal information, incorporating environmental dynamics to improve moving object segmentation performance.
Huang et al. <cit.> propose a novel method for unsupervised point cloud segmentation by jointly learning the latent space representation and a clustering algorithm using a variational autoencoder.
Lastly, Khurana et al. <cit.> use differentiable raycasting to render future occupancy predictions into future LiDAR sweep predictions for learning, allowing geometric occupancy maps to clear the motion of the environment from the motion of the ego-vehicle.
However, they share common drawbacks, such as the need for extensive labeled datasets, unbalanced data during training <cit.>, and potential limitations when applied to different sensor types they were not trained on.
§.§ Traditional Algorithm
In light of these challenges, our focus shifts towards traditional methods, which typically exhibit greater robustness and flexibility in handling diverse sensor types and data distributions.
Various approaches have been proposed, often categorized into ray-casting, visibility-based, and visibility-free.
Occupancy grids, often in the form of Octomap <cit.>, are popular techniques that employ ray casting to update the occupancy value of the grid map space by counting the hits and misses of scans. Additionally, other data structures have been proposed, e.g., by representing the truncated signed distance field (TSDF) <cit.> instead of occupancy. They rely on the concept of occupancy values or truncated signed distances to detect dynamic points in point clouds. These methods update the values for each voxel, frame by frame, based on the measurements obtained from the sensor.
If the values within a voxel deviate significantly from a specified threshold, the points inside that voxel are considered dynamic.
Despite their effectiveness, these methods can be computationally expensive when performing ray-casting steps, leading to the development of visibility-based methods to reduce computational costs.
Visibility-based methods assume that if a query point is observed behind a previously acquired point in the map, then the previously acquired point is dynamic. Kim et al. <cit.> constructs a static point cloud map using multi-resolution range images based on visibility.
Both ray casting and visibility-based methods suffer from the problems illustrated in Fig. <ref>.
(a) shows that rays are far from the ground, and the angle between the rays and the ground line becomes very small. In such scenarios, ray-based methods update the free value when the rays pass through the area, which may cause some ground points to be incorrectly seen as dynamic.
(b) means after accumulating multiple scan frames, noise below the ground in some frames can cause previous regions to be updated as free, mislabeling ground points.
(c) illustrates how these methods fail to remove dynamic points when no object is behind them. In this example, only some hits on the big truck will later be cleared by hits on the wall, while others will not. The purple hits will erroneously remain, as no new hits pass through them.
Lim et al. observed these limitations in <cit.> and proposed a novel approach based on the height difference between the raw map and the query.
They compare the ratio between the difference in the minimum and maximum z-values in regions between a query scan and the map.
If the ratio is larger than a predefined threshold, the region is considered to contain dynamic objects. This approach improved the handling of dynamic objects from unlabeled classes.
We have discussed several traditional methods for removing dynamic points from point clouds.
They often involve numerous parameters that need to be tuned.
For instance,
Lim's method <cit.> requires knowledge of the sensor height, making it highly sensitive to height values. This approach also necessitates tuning the maximum and minimum height ranges, as it cannot handle scenarios such as pedestrians walking under trees in Fig. <ref>.
This paper remains focused on traditional methods because they do not require the creation of a large labeled dataset or training on various datasets to ensure generalization, as compared to learning-based approaches.
Nevertheless, it is still possible to include learning-based methods with the effort dedicated to creating various labeled datasets, training networks, and performing inference under unified setups.
§ METHODS
In this section, we provide a summary of the methods <cit.> included in our benchmarks, discussing their algorithm design and frameworks. We prepare the processing dataset and scripts to extract data from several open-dataset and refactored methods without ROS (Robot Operating System) for easier benchmarking and faster running speeds.
Guided by our benchmark analysis and addressing the angle problem and sparse points problems in Fig. <ref>, we adapt Octomap <cit.> to estimate the ground, followed by the same ray casting process for hit-and-miss detection in non-ground points.
§.§ Removert
Kim et al. <cit.> proposes an offline method that requires a prior raw map to compare the difference between query and raw as shown in Fig. <ref>. Firstly, they convert the query and prior raw map point cloud to depth range images using OpenCV <cit.>. Subsequently, they compute the difference between these two image matrices I_k^Q, I_k^M as follows:
I_k^Diff = I_k^Q - I_k^M.
Finally, the dynamic map points are defined in <cit.>,
𝒫_k^DM = 𝐩_k,ij^M ∈𝒫_k^M | I_k,ij^Diff > τ _D,
where 𝒫_k^M is the raw global point cloud map, 𝒫_k^DM is the set of dynamic points, and 𝐩_k,ij^M is the set of points in the pixel (i,j), τ _D is a threshold.
Fig. <ref> illustrates their framework.
§.§ ERASOR
Lim et al. <cit.> propose an approach based on the observation that most dynamic objects in urban environments are in contact with the ground. They introduce the novel concept of pseudo-occupancy to represent the occupancy of unit space and discriminate spaces with varying occupancy levels. Subsequently, they determine potential dynamic candidate bins 𝒫_Dynamic based on the height difference between the raw map and query frame, as briefly described in <cit.>:
Δ h_(i,j), t = sup{ Z_(i,j), t} - inf{ Z_(i,j), t}
where Z_(i,j), t={z_k∈𝐩_k|𝐩_k∈𝒮_(i,j), t}, and z represents the point's z-value concerning the sensor origin. sup, inf separately means the highest and lowest point height value in the Z.
The condition for determining potential dynamic candidate bins is:
if Δ h_(i,j), t^Query/Δ h_(i,j), t^Map < 0.2,
then 𝒮(i,j), t∈𝒫_Dynamic
Finally, they employ Region-wise Ground Plane Fitting (R-GPF) to distinguish static points from dynamic points within the candidate bins that potentially contain dynamic points. Fig. <ref> illustrates their framework.
§.§ Octomap and Improvement
Hornung et al. <cit.> offer a popular mapping framework to generate volumetric 3D environment models in the robotics field. It is based on octrees and uses probabilistic occupancy estimation. Although it is not initially designed for dynamic point removal, it has frequently been used as a baseline.
First, Octomap rasterizes all points to 3D voxels, where each voxel is a leaf node n. Then, the probability of n being occupied is updated given the sensor measurements z_1:t according to:
P (n | z_1: t)
= [1+1-P(n | z_t)/P(n | z_t)1-P(n | z_1: t-1)/P(n | z_1: t-1)P(n)/1-P(n)]^-1
After updating the whole map with all scan frames, each node in the map will have a final occupancy value. If it exceeds a threshold, we consider the node as a static point. During this process, the occupancy probability of nodes containing dynamic points will decrease as rays pass through these nodes in some frames, reducing their occupancy values.
However, as mentioned earlier, it is not designed for dynamic point removal tasks, and in Section <ref>, we can observe the challenges mentioned in Section <ref>. Guided by our benchmarking analysis, we will enhance the original Octomap by incorporating noise filtering and ground estimation techniques. The performance differences between our improved Octomap and the original version are examined through ablation studies in Section <ref>, demonstrating the benefits of our modifications.
To minimize the impact of noise and abnormal points or reduce the computational burden of our ray casting, we employ the Statistical Outlier Removal (SOR) technique for filtering. Then we perform ground estimation using Sample Consensus (SAC) segmentation <cit.> on the output point clouds. In the last, we optimize the process by setting the grid cells occupied by the estimated ground points as free, ensuring no ray casting occurs in these regions. This approach prevents the mislabeling of ground points as dynamic and their subsequent removal from the static map, preserving the integrity of the final representation. We then only integrate and update the octree based on the non-ground points throughout all frames.
When exporting the final map, we use a threshold to query the occupancy grid points and integrate the ground points.
This approach ensures that the occupancy values of grid cells containing dynamic points are updated when the dynamic objects move away, and rays pass through the area once more, providing an accurate and efficient representation of the environment.
§ BENCHMARK SETUP
§.§ Metric
Although nowadays datasets provide point-wise labels, most methods downsample the ground truth to voxel-wise level for evaluation. To provide a more accurate evaluation, we propose a new benchmark based on point-wise assessments.
The map clean task aims for two goals: remove true dynamic points and keep true static points. This process involves maintaining high recall in the classification of both dynamic and static points, often referred to as Dynamic Accuracy (DA%) and Static Accuracy (SA%) respectively.
Additionally, we utilize the Associated Accuracy (AA %) calculated using the geometric mean as a comprehensive metric that combines both accuracies, offering an overall assessment of the algorithm's performance. AA is more sensitive to smaller values compared to the harmonic mean used in the F1 score.
AA = √(SA × DA).
There are also distance distribution plots that show the distance from mislabeled points to their nearest correct dynamic points. It serves as a metric to illustrate where errors typically occur, helping researchers identify and address the shortcomings of methods for further improvements.
§.§ Implementation details
We conduct experiments on three primary datasets: KITTI, Argoverse 2.0, and a semi-indoor dataset. The first two have their own ground truth pose files, while semi-indoor dataset poses are obtained using the SLAM package <cit.>. All datasets are integrated into a unified PCD format with point cloud data and pose in it.
KITTI has ground truth labels for dynamic objects from SemanticKITTI <cit.>. Part of Argoverse 2.0 and the semi-indoor dataset collected by us have manually labeled dynamic points ground truth. In the point-wise evaluation, if an algorithm rasterizes the grid, we query all the points in ground truth to search for the corresponding grid and label the point as static or dynamic according to the algorithm's output.
§ BENCHMARK RESULTS
In this section, our benchmark contains both quantitative and qualitative evaluations. We use it to conduct a detailed analysis of the performance of the methods in Section <ref> on various datasets, identify specific failure scenarios for each method, and explain the reasons for these failures in relation to their theoretical foundations. Additionally, we provide a table outlining the time cost and the number of parameters required for tuning to achieve a better static map.
All experiments are conducted on a desktop computer equipped with a 12th Gen Intel® Core™ i9-12900KF processor featuring 24 cores.
The benchmark link includes all the parameters used for the experiments presented in this paper.
Methods marked with an asterisk (*) in tables and figures indicate offline methods, which require a raw global point cloud map as a prior for comparison. More detail can be found in Section <ref> and Fig. <ref>.
§.§ Quantitative
In Table <ref>, we present a quantitative comparison of dynamic object removal methods in various scenarios datasets. Removert mostly retains the complete static map but labels only a few correct dynamic points. In contrast, ERASOR performs better in removing dynamic points and balancing static and dynamic points. The original Octomap suffers from angle problems in the ground plane and noise points shown in Section <ref> which cause their score on SA to be lower than others. Octomap w G denotes the method with ground estimation, resulting in a higher score across most of the sequences.
Also adding the noise filter (Octomap w GF), we achieve a 20%-30% speed-up, as shown in Table <ref>, since the noise points do not undergo the ray casting process.
Considering Table <ref> and Table <ref>, there is a trade-off between speed and performance, as the method with the lowest score on AA achieves the fastest processing time for a single frame. It is important to note that Removert requires multiple resolutions to produce better results, which may increase the processing time depending on the number of resolutions. The speed of ray-based methods like Octomap has the potential for further optimization and improvement.
To better future analyze, Fig. <ref> illustrates where errors typically occur.
We observe that most of the false negative points (true dynamic points labeled as static) are close to the true positive points (correctly labeled dynamic points), with all methods' false negative points ranging from 10cm to 30cm away from the true positive points.
In such cases, other techniques, such as clustering around the true positive points, can be employed to address this issue.
The largest scale difference occurs in Removert, corresponding to the challenges we mentioned earlier in visibility-based methods that involve occlusion behind the true positive points. Techniques that apply understanding object relationships and establishing connections between them may help address this issue more effectively.
§.§ Qualitative
To complement the quantitative results discussed earlier, We present the cleaned map in the Argoverse 2.0 and semi-indoor dataset, where the ground truth map is marked with yellow points to represent dynamic objects.
In one sequence of the Argoverse 2.0 LiDAR dataset, Fig. <ref> presents the cleaned maps produced by different methods in the Argoverse 2.0 dataset.
As this dataset contains more recent and challenging scenarios from various US cities, it features many poles and trees that effectively illustrate the disadvantages of each method.
As seen in the raw map, there are dynamic cars, cyclists, and pedestrians near the building. Removert retains the most complete static points but fails to remove the points near the object center. ERASOR keeps the cleanest map among all methods but, removes tree trunks and the ground near the pedestrian due to its sensitivity to height and slightly different pavement heights compared to driving roads.
A comparison with the improved Octomap version in Fig. <ref> (d) and Fig. <ref> (e) demonstrates significant improvements in error reduction for ground points.
By incorporating ground estimation, most ground points are preserved, ensuring a more accurate representation of the static environment. As these points are considered definitively static, ray casting is not performed to remove ground points, further enhancing the accuracy of the map.
There is room for further improvement by using better ground estimation and clustering techniques to label the missing dynamic points in the map, as discussed in Fig. <ref>.
Fig. <ref> displays the cleaned map for a custom dataset with one VLP-16 LiDAR. As clearly illustrated, the issues discussed in Section <ref> are apparent. Removert has difficulty removing points that are behind dynamic obstacles moving around. Without fine-tuning the parameters and using the same settings as for the KITTI dataset, ERASOR fails to remove points higher than the threshold, and the original Octomap exhibits the sparse LiDAR ground problem, leading to many ground points being removed regularly by LiDAR rings.
The improved Octomap still requires some fine-tuning of the occupancy probability values, as people are standing in the same place for an extended period, making it challenging to remove them using the default parameters.
§ CONCLUSION
In this paper, we conducted a comprehensive review and benchmark of methods for removing dynamic points from point clouds. We refactored three existing methods and contributed them to a unified benchmarking framework. The proposed metric on error distribution offers a novel perspective for analyzing where errors occur and gaining insights for researchers.
In benchmarking evaluation, we provide detailed analyses of each method's strengths and weaknesses for future researchers.
Guiding by our evaluation, we also propose a modified Octomap version tailored for this task by filtering and estimating ground points first. Through analysis, there is potential to generalize methods to similar scenarios that minimize reliance on parameter tuning and prior knowledge, as well as accelerate the algorithms for efficient execution.
The future direction of this benchmark extends beyond merely removing dynamic points to encompass generating labels in perception datasets, as demonstrated in studies such as <cit.>, or performing real-time detection in point clouds, as illustrated by <cit.>.
In conclusion, we hope this benchmark, open-source code, and dataset will serve as valuable resources for researchers and practitioners in this field, fostering further advancements and innovations in point cloud processing.
§ ACKNOWLEDGEMENT
Thanks to RPL's members: Yi Yang, and HKUST Ramlab's members: Bowen Yang, Jin Wu, and Yingbing Chen, who gave constructive comments on this work.
Thanks to Shenzhen Unity-Drive Inc. for providing essential experimental devices and services for this work.
We also thank the anonymous reviewers for their constructive comments.
This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
IEEEtran
|
http://arxiv.org/abs/2307.05908v1 | 20230712042841 | Predictive Pipelined Decoding: A Compute-Latency Trade-off for Exact LLM Decoding | [
"Seongjun Yang",
"Gibbeum Lee",
"Jaewoong Cho",
"Dimitris Papailiopoulos",
"Kangwook Lee"
] | cs.CL | [
"cs.CL",
"cs.LG"
] |
[
Predictive Pipelined Decoding:
A Compute-Latency Trade-off for Exact LLM Decoding
equal*
Seongjun Yangequal,yyy
Gibbeum Leeequal,yyy
Jaewoong Choyyy
Dimitris Papailiopoulosyyy,sch
Kangwook Leeyyy,sch
yyyKRAFTON
schDepartment of Electrical and Computer Engineering, University of Wisconsin-Madison
Kangwook [email protected]
Machine Learning, ICML
0.3in
]
This paper presents “Predictive Pipelined Decoding (PPD),” an approach that speeds up greedy decoding in Large Language Models (LLMs) while maintaining the exact same output as the original decoding. Unlike conventional strategies, PPD employs additional compute resources to parallelize the initiation of subsequent token decoding during the current token decoding. This innovative method reduces decoding latency and reshapes the understanding of trade-offs in LLM decoding strategies. We have developed a theoretical framework that allows us to analyze the trade-off between computation and latency. Using this framework, we can analytically estimate the potential reduction in latency associated with our proposed method, achieved through the assessment of the match rate, represented as p_correct. The results demonstrate that the use of extra computational resources has the potential to accelerate LLM greedy decoding.
§ INTRODUCTION
The recent advances in LLMs, especially transformers <cit.>, have brought a breakthrough to the domain of natural language processing.
The notable generative language models include GPT-3 <cit.>, GPT-4 <cit.>, PaLM <cit.>, LaMDA <cit.>, OPT <cit.>, and LLaMA <cit.>.
The power of LLMs is primarily driven by their enormous scale, often involving hundreds of billions of parameters <cit.>.
However, the considerable size of these models can present challenges in practical applications where immediate responses are crucial <cit.>.
Generative transformers usually create text sequences using auto-regressive decoding.
After passing through all layers of the transformer, each token is generated using the hidden representation from the final layer of the transformer <cit.>.
Some studies <cit.>, however, suggest that it is not necessary to pass through all layers, but
similar results can be obtained using a sub-network of the transformer's layers.
Still, these methods do not always ensure the same output as when all transformer layers are utilized.
In this paper, we introduce Predictive Pipelined Decoding (PPD), a new approach that lowers latency by utilizing additional compute resources, while keeping the exact same decoding results, as illustrated in <ref>.
Our methodology is motivated by early-exiting, specifically as described by <cit.>.
Early-exiting allows the generation process to exit before reaching the final layer, enabling predictions to be made earlier in the process.
PPD shares similarities with early exit strategies as it also utilizes intermediate representations to make early predictions on the next token.
However, PPD distinguishes itself by continuing the current token decoding without exiting.
In other words, the main process remains focused on the current token while other subprocesses early start the generation process with predicted next token(s).
PPD accelerates decoding by parallelizing processes, each of which begins decoding from the top-k token predictions of the specific transformer layer.
Simultaneously, the main process continues to compute the output of the final layer and predicts the next token.
By aligning the results with the next token prediction from the final layer, we can maintain the original decoding result.
To assess the potential benefits of our method, we conduct an analysis to determine the extent of latency reduction and the associated compute resource costs.
Also, we measure the match rate, the probability that the early top-k predictions match the top-1 prediction from the final layer, with the commonly utilized dataset in NLP such as SQUAD 1.1 <cit.>, WMT EN-FR <cit.>, and CNN/DM <cit.>.
We could estimate the potential savings in latency and the corresponding compute resources based on the match rate. However, it is essential to mention that we have not implemented the algorithm, and this work is purely providing a performance modeling and analysis. We believe that once properly implemented, the predicted latency reduction can be realized, but we leave it to the future work.
In summary, our contributions are: (1) a framework, which we call PPD, that boosts the speed of the decoding, (2) a theoretical analysis of latency savings versus computing resource costs, and (3) an empirical measurement of match rate to estimate how effective PPD would be in an actual situation.
§ RELATED WORK
Various strategies have been proposed to improve the inference speed of large-scale transformer models.
These include employing model pruning techniques <cit.>; implementing knowledge distillation methods to downsize the models <cit.>; and adopting quantization procedures <cit.>.
However, these approaches do not necessarily guarantee the original inference quality since they do not have a mechanism that verifies the validity of the generated token.
Our research is inspired by early-exiting approaches <cit.> that utilize only the initial segments of transformer layers for inference, rather than the entire network.
Especially, <cit.> implements an early-exiting approach for decoder-only models in which one can select the layer to exit and check the confidence measure of each token using a threshold function.
However, the approach could not be as exact as conventional decoding due to its dependency on a threshold-based confidence measure.
Similarly, with the goal of reducing the inference time of transformers, numerous studies <cit.> have utilized two language models which are one smaller and one larger.
The smaller model rapidly generates output, while the larger model verifies its validity.
Despite the potential speed advantage, this method might not consistently match the exact output of larger models, resulting in discrepancies since the larger model relies on the smaller model's confidence score.
§ PREDICTIVE PIPELINED DECODING
We introduce Predictive Pipelined Decoding (PPD), a low-latency decoding method that leverages multiple compute resources.
PPD utilizes an intermediate output of a transformer to predict the next token, which is typically produced by the final layer's output.
This allows PPD to start the forward propagation of the next sequence earlier than the conventional decoding. Despite this early start, the original forward propagation continues uninterrupted up to the final layer. This parallel approach accelerates the conventional greedy decoding process while ensuring the same decoding result.
In the following, we elaborate on the process of PPD. This method predicts the next token early at an intermediate transformer layer. PPD employ an intermediate hidden representation h, e.g., d/2-th layer's output, to estimate the probability p(x|h) of the next token. This is done by applying a language modeling classifier and a softmax activation to the hidden representation. Subsequently, PPD identifies the top-k candidate tokens with regard to p(x|h) and initiates k parallel sub-processes. Each sub-process then inputs the selected token into the transformer. In parallel, the main process continues to forward the intermediate output up to the final layer.
Once the main process completes the forward propagation to the final layer, PPD checks if the decoded next token from the final output matches any of the top-k next token candidates previously predicted from the intermediate output. If a match is found, PPD only continues the forward propagation associated with the matching token, disregarding the results from other processes. In cases where no matches are found, all results from sub-processes are disregarded, and PPD proceeds with the output from the final layer. This approach enables us to achieve decoding results identical to those of the original method while improving latency efficiency. Figure <ref> provides an overview of the proposed method. For subsequent rounds, the main process repeatedly employs the output from the intermediate layer of sub-processes. The algorithm description is provided in Algorithm <ref>.
§ THEORETICAL ANALYSIS
§.§ Model
For fixed k, PPD makes an early next-token prediction at the d̅-th intermediate layer out of the total d layers in a transformer, and we model that one of the top-k early predictions will match the actual top-1 at the final layer with probability 0 < p_correct < 1.
Furthermore, we model that these events are independent of all the others.
We define a sequence of consecutively generated tokens as a run.
PPD begins a run and continues until all early predictions no longer match the final prediction, at which point all sub-processes are disregarded.
Counting from the beginning of the generated text, we denote the i-th run's length by X_i, where X_i ≥ 1. Note that X_i ∼Geom(1-p_correct) except for the last run, where Geom denotes a geometric distribution, and [X_i] = 1/(1-p_correct).
Assume that the length of the text to be generated is ℓ tokens, where ℓ≥ 1.
Then, we have ∑_i=1^N X_i= ℓ, where N is a random variable that denotes the number of runs required to completely generate ℓ tokens. We assume an equal amount of computational time is required for each layer of the transformer, which is mainly due to the majority of layers being composed of the same self-attention and feed-forward network. We refer to this consistent time requirement for one layer as one `time unit'. Consequently, forwarding through d layers of the transformer would demand d time units.
§.§ Latency Analysis
Before we show our exact analysis, we first present an approximate analysis for ℓ≫ 1.
Also, for simplicity, let us assume d̅ = d/2, i.e., we make an early prediction after processing through the middle layer.
Let us first find the expression for N.
Since ℓ≫ 1, we also have N ≫ 1.
Thus, we have ℓ= N ·X_1 + X_2 + ⋯ + X_N/N≈ N [X_1],
where the last approximation is derived from the law of large numbers, with the assumption
that X_is are i.i.d.
Now, we compute the expected latency to generate ℓ tokens with PPD.
Recall that for a run of length X, it takes d+(X-1)d/2=d(X+1)/2 time units to generate the run.
Thus, the total time to generate the ℓ tokens is
∑_i=1^Nd(X_i+1)/2 = d(∑_i=1^NX_i + N)/2 = d(ℓ + N)/2.
By dividing the total latency by ℓ, we get the per-token latency:
d(ℓ + N)/2ℓ = d(1 + N/ℓ)/2 ≈d(1 + 1/[X_1])/2
= d(1-p_correct/2).
This reveals an intuitive relationship between the per-token latency and the probability of successful early token prediction.
If p_correct is close to 1, then the per-token latency becomes 0.5d, while if p_correct is close to 0, then the average per-token latency remains as d.
To run PPD with a fixed choice of k, one needs k+1 compute resources.
However, at the beginning of each run, only one compute resource is used.
Thus, to compute the average compute resources required for running PPD, we need the following calculation.
For a run of length X, the first d/2 time units requires one compute resource, while the remaining Xd/2 time units use k+1 compute resources.
Therefore, the total compute resources spent for the run of length X is (k+1)dX+d/2,
and the total compute resources spent for the entire text generation is
∑_i=1^N(k+1)dX_i+d/2 = (k+1)dℓ+dN/2.
By dividing the total compute resources by the total generation time, we get the average compute resources per time unit:
(k+1)dℓ+dN/2/d(ℓ + N)/2≈(k+1)+1/[X_1]/1 + 1/[X_1]
= k+2-p_correct/2 - p_correct.
If p_correct is close to 1, then the average compute resources per time unit becomes k+1.
Note that this makes sense since when p_correct is 1, one can generate the whole text in one run, and hence all k+1 compute units will be used almost all the time.
If p_correct is close to 0, then the average compute units per time unit becomes k+2/2.
This also makes sense as if the early prediction is always wrong, the run length is always 1.
For the first half unit time, we use one compute unit.
For the second half unit time, we use k+1 compute units.
Thus, on average, we use k+2/2 compute units throughout the generation.
Recall that the above analysis was approximate, assuming ℓ≫ 1 and d̅=d/2.
The following theorem gives the exact analysis without the assumption, when d̅ is greater than or equal to 0.5d.
Given p_correct, k, and for fixed ℓ, if PPD makes an early prediction at the d̅-th intermediate layer out of the total d layers (d̅≥d/2), then the expected latency to generate a sequence of ℓ tokens is
dℓ - (d-d̅)(ℓ-1)p_correct,
and the expected total compute units is
dℓ-(d-d̅)(ℓ-1)p_correct
+k(d-d̅)ℓ.
The proof is in Appendix <ref>.
Conventional decoding requires dℓ time units to generate ℓ tokens. However, in PPD, there is an expectation that a proportion of (ℓ-1)p_correct tokens accurately match the predictions made at the intermediate layer. For these instances, parallel pre-computations up to the (d-d̅)-th layer result in time savings. Consequently, it allows PPD to reduce the expected latency by (d-d̅)(ℓ-1)p_correct time units.
To achieve these savings, PPD employs one computational unit dedicated to the main process for dℓ - (d-d̅)(ℓ-1)p_correct time units. In addition, PPD allocates k computational units for each of the ℓ tokens to pre-compute the output of the (d-d̅)-th layer along with the predictions.
§.§ Simulations
Experimental Setup In order to theoretically estimate the potential improvements in decoding latency in real-world NLP tasks, we examine the match rate, denoted by p̂_correct.
This match rate is empirically estimated across multiple token geration processes by verifying if any of the top-k predicted tokens from the intermediate layer match the top-1 token from the final layer.
We test the NLP tasks on three benchmark datasets: SQUAD 1.1 <cit.>, WMT EN-FR <cit.>, and CNN/DM <cit.>. We use their respective test datasets for evaluations.
The model for the test is Vicuna-13B <cit.>, a transformer with a total of 40 layers.
We specifically probe the early prediction at the 15th, 20th, 30th, 35th, and 37th layers to derive the match rate (see <ref>).
Furthermore, our analysis includes two different utilization of the language modeling classifier for estimating the distribution of tokens over vocabulary.
The first employs the common classifier across all layers, trained specifically for the final layer, and the second uses the classifier trained uniquely for each layer.
An in-depth explanation of the experimental setup can be found in <ref>.
Result <ref> reveals that the accuracy of token prediction enhances with an increase in both the top-k and layer values.
Indeed, the p̂_correct can be represented with the term k as you can see in Section 4.2.
Additionally, the overall performance shows improvement when the language modeling classifier is individually trained for each layer.
However, it is critical to note that as the number of layers increases, the method starts to resemble the vanilla approach, implying a potential loss in latency benefits.
Nevertheless, it becomes clear that training the classifier with an increase in top-k values, or computational resources, can effectively contribute to reducing latency.
Further results based on the token positions can be found in <ref>.
<ref> shows the theoretical results of the trade-off between latency and computational resources from Equation (<ref>) and (<ref>). The figure, which represents theoretical estimations rather than empirical data, sets the intermediate layer at 20, using p̂_correct values from <ref> for scatter plots. With normalized latency and computational resources of the original decoding, we observe latency per token varying from 0.629 (SQUAD+“tr”, k=5) to 0.892 (WMT, k=1). These findings suggest a speed improvement between 10.8% and 37.1%, whilst maintaining output quality equivalent to the original decoding. It is crucial to note, however, that this speed increase requires more computational resources, escalating by 1.561 to nearly 4.973 times the original usage. Additionally, we present the trade-off graph between average compute resource per token and latency per token in <ref>.
§ LIMITATIONS
While our method has the potential for latency improvements, this comes at the cost of increased computational requirements. To reduce the computation costs, future research should focus on better utilization of GPU resources. It is also crucial to consider other factors that impact latency, such as GPU synchronization, data transfer overhead, and communication and memory management overhead, as highlighted in <cit.>. The scope of our current work specifically targets greedy decoding, yet it is worth acknowledging that other decoding algorithms <cit.> have demonstrated superior performance. Thus, future endeavors intend to extend our methodology to other decoding methods.
§ CONCLUSION
We introduced PPD, a method aimed at reducing the decoding latency while maintaining the original decoding result of LLM. Based on our theoretical analysis and empirical measurements, we identified the potential of PPD to reduce latency. Furthermore, we demonstrated that training the language modeling classifier for an intermediate transformer layer can effectively enhance early prediction accuracy, potentially leading to further reductions in latency.
icml2023
§ DATASETS
SQUAD 1.1 <cit.> is a Question Answering dataset which has 10,570 test pairs. WMT15 FR-EN <cit.> is a machine translation dataset that includes 1,500 test pairs of English to French translations. CNN/DM <cit.> is a dataset for text summarization which has 11,487 test pairs. For these datasets, we set the token length for text generation at 16 for SQUAD 1.1, and 128 for both the WMT EN-FR and CNN/DM.
§ EXPERIMENT SETTING
We measured match rate, p̂_correct, based on Vicuna-13B <cit.> which was fine-tuned using the LLaMA <cit.> on user-shared dialogues collected from the website ShareGPT. To the best of our knowledge, <cit.> is claimed to be the best performance among the open-source models available. Also, we conduct our experiments using the Huggingface Transformers library <cit.>.
For the language modeling classifier training of each layer in the Vicuna-13B model, we utilize the ShareGPT dataset used in fine-tuning. We freeze all model parameters except for the language modeling classifier. Training is conducted using 8 A100 GPUs, and the hyperparameters for the training can be found in <ref>. Additionally, please refer to <ref> for the prompts we use in our evaluation.
§ ADDITIONAL RESULTS
In this section, we present the results of token prediction for the generated tokens in three text generation tasks, based on their respective positions. The corresponding findings can be observed in <ref>. For the sake of clarity, token prediction was represented by grouping 2 tokens together for SQUAD, and by grouping 16 tokens together for WMT EN-FR and CNN/DM. Within the three tables, the “N” notation signifies evaluation using the pretrained lanage modeling classifier of Vicuna-13B, while “Y” indicates evaluation with the trained language modeling classifier for each layer. Overall, the analysis of the results reveals a consistent pattern of token prediction across various token positions when utilizing the same layer, k, and language modeling classifier.
§ PROOF OF THEOREM 4.1
For a run of length X, the time required to generate the run T_X is given by
T_X = d + (X-1)d̅.
Therefore, the total per-token latency is
∑_i=1^N T_X_i = ∑_i=1^N d+(X_i-1)d̅
= d̅∑_i=1^N X_i + (d-d̅)N
= d̅ℓ + (d-d̅)N.
To compute the expected value of this quantity without assuming ℓ, N ≫ 1, we first need to identify the distribution of N.
Note that the expectation here is over different instances of sequence generations.
Since N is the number of runs, N-1 is the number of times early predictions fail.
Thus, N-1 = Bin(ℓ-1, 1-p_correct).
Hence, N = 1 + Bin(ℓ-1, 1-p_correct).
Thus, [N] = 1 + (ℓ-1)(1-p_correct)=ℓ-(ℓ-1)p_correct.
With this, we have
[d̅ℓ + (d-d̅)N] = d̅ℓ + (d-d̅)[N]
= d̅ℓ+(d-d̅)ℓ - (d-d̅)(ℓ-1)p_correct
= dℓ - (d-d̅)(ℓ-1)p_correct.
For a run of length X, the (d-d̅)X time units require k+1 compute resources while the remaining T_X-(d-d̅)X time unit requires one compute resource.
Therefore, the total compute resources spent for the run of length X are
T_X-(d-d̅)X+(k+1)(d-d̅)X
=d+(X-1)d̅-(d-d̅)X+(k+1)(d-d̅)X
=d̅X+d-d̅+k(d-d̅)X
=(d̅+k(d-d̅))X+(d-d̅),
and the total compute resources spent for the entire text generation is
∑_i=1^N(d̅+k(d-d̅))X_i+(d-d̅)
= (d̅+k(d-d̅))ℓ+(d-d̅)N.
By computing the expected value of it, we have
[(d̅+k(d-d̅))ℓ+(d-d̅)N]
=(d̅+k(d-d̅))ℓ+(d-d̅)[N]
=(d̅+k(d-d̅))ℓ+(d-d̅)(ℓ-(ℓ-1)p_correct)
=(d̅+(k+1)(d-d̅))ℓ-(d-d̅)(ℓ-1)p_correct
=(d+k(d-d̅))ℓ-(d-d̅)(ℓ-1)p_correct
=dℓ-(d-d̅)(ℓ-1)p_correct
+k(d-d̅)ℓ.
§ ADDITIONAL TRADE-OFF GRAPH
We illustrate the trade-off between latency and
compute resources per token. The “compute resources per token” is calculated by multiplication of equation (<ref>) and (<ref>).
We set d̅=d/2 for the following equation.
Average compute resources per token≈2+k-p_correct/2.
|
http://arxiv.org/abs/2307.04409v1 | 20230710081749 | Violation of a Leggett-Garg inequality using ideal negative measurements in neutron interferometry | [
"Elisabeth Kreuzgruber",
"Richard Wagner",
"Niels Geerits",
"Hartmut Lemmel",
"Stephan Sponar"
] | quant-ph | [
"quant-ph"
] |
[email protected]
[email protected]
^1Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria
^2Institut Laue-Langevin, 38000, Grenoble, France
=800=800
We report on an experiment that demonstrates the violation of a Leggett–Garg inequality (LGI) with neutrons. LGIs have been proposed in order to assess how far the predictions of quantum mechanics defy `macroscopic realism'.
With LGIs, correlations of measurements performed on a single system at different times are described.
The measured value of K =1.120±0.007, obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macro-realistic theories.
Violation of a Leggett–Garg inequality using ideal negative measurements
in neutron interferometry
Stephan Sponar^1
August 12, 2023
===================================================================================================
Introduction.—The question whether measurable quantities of a quantum object have definite values prior to the actual measurement is a fundamental issue ever since quantum theory has been introduced more than a century ago. Examples include Bell's inequality <cit.>, which sets bounds on correlations between measurement results of space-like separated components of a composite (entangled) system. A violation of Bell's inequality thus demonstrates that certain predictions of quantum mechanics cannot be reproduced by realistic theories, more precisely, by local hidden variable theories (LHVT). Another prime example is found in the Kochen-Specker theorem <cit.>, which stresses the incompatibility of quantum mechanics with a larger class of hidden-variable theories, known as noncontextual hidden-variable theories (NCHVTs). Here it is assumed that the result of a measurement of an observable is predetermined and independent of a suitable (previous or simultaneous) measurement of any other compatible (co-measurable or commuting) observable, i.e., the measurement context. While both, Bell's inequality and tests of the Kochen-Specker theorem, require composite or multiple spatially-separated systems Leggett-Garg inequalities (LGIs) <cit.> study temporal correlations of a single system, therefore they are often referred to as Bell inequalities in time.
Violation of a Bell inequality is a direct witness of entanglement - a very specific feature of quantum mechanics. Contrary, in the case of LGIs the violation occurs due to the coherent superposition of system states, which is essentially the most fundamental property of quantum mechanics. In other words LGIs quantify coherence in quantum systems and can consequently be seen as a measure or test of quantumness.
Leggett-Garg inequalities were proposed in 1985 <cit.> in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying macro-realistic theory <cit.>. Within the framework of a macro-realistic theory a single macroscopic system fulfills the following two assumptions of macrorealism measured at successive times: (A1) at any given time the system is always in only one of its macroscopically distinguishable states, and (A2) the state of the system can be determined in a non-invasive way, meaning, without disturbing the subsequent dynamics of the system. Quantum mechanics predicts the violation of the inequalities since it contradicts with both assumptions (A1) and (A2). The (quantum) system under observation has to be measured at different times. Correlations that can be derived from sequences of this measurements
let us formulate the LGI. The result of these correlation measurements either confirm the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it <cit.>. This will also refuse a well-defined pre-existing value of a measurement.
Recent violations of LGI have been observed in various systems, including photonic qubits <cit.>, nuclear spins in a diamond defect center<cit.>, superconducting qubits in terms of transoms <cit.> and flux qubits <cit.>, nuclear magnetic resonance <cit.>, and spin-bearing phosphorus impurities in silicon <cit.>. Proposed schemes for increasing violations of Leggett-Garg inequalities range from action of an environment on a single qubit in terms of generic quantum channels <cit.> to open many-body systems in the presence of a nonequilibrium <cit.>. In a recent paper <cit.> the authors propose to test a violation of the Leggett-Garg inequality due to the gravitational interaction in a hybrid system consisting of a harmonic oscillator and a spatially localized superposed particle <cit.>, aiming to probe the quantumness of gravity <cit.>.
The violation of an LGI in an interferometric setup has been proposed in literature theoretically for electrons in <cit.>. The requirement of non-invasive measurements from (A2) is realized in most experiments by utilizing the concept of weak measurements, or by introducing an ancilla system, as implemented in <cit.>. Note that even a weak measurement in practice can never be completely non-invasive (due to a non-vanishing measurement strength) and the preparation of the ancilla system will also always be imperfect. However, the experimental procedure from <cit.> realizes ideal negative measurements in an interferometer experiment in order to fulfill the requirement of non-invasive measurements from (A2) without the need for an ancilla.
In this Letter, we present a neutron interferometric experiment, demonstrating a violation of the LGI. In our measurement scheme the single system is represented by the neutron's path in an interferometer. A respective observable is defined and measured non-invasively according to the LGI protocol.
Leggett–Garg inequality.—For dichotomous variables Q_i, accounting for two macroscopically distinguishable states, having outcomes q_i=±1, the correlation function for measurements at times t_i, t_j is given by
C_ij=⟨ Q_i Q_j⟩=∑_q_i q_j=± q_i q_j P(q_i(t_i),q_j(t_j)),
where P(q_i(t_i),q_j(t_j)) denotes the joint probability of obtaining the measurement results q_i at time t_i and q_j at time t_j.
Considering Eq.(<ref>) for three experimental sets with i,j∈{1,2,3} yields the LGI
K ≡ C_21 + C_32-C_31,
where K denotes the Leggett-Garg correlator, with limits -3≤ K ≤ 1. Since the three correlators are derived from probabilities with |C_ij|≤ 1, the lower limit cannot be violated. However, quantum mechanics allows for a violation of the upper bound. In a two-level system, the maximum obtainable violation is K=1.5 <cit.>.
The basic idea behind the experimental procedure as proposed by Emary et al. in <cit.>, is to map the temporal structure (or measurement time t_i) of LGI onto real-space coordinates, more precisely onto three distinct regions of the interferometer, indicated by the index α∈{1,2,3}, cf. Fig. <ref>. Within each region the two paths of the interferometer constitute a qubit. The measurement of the qubit's state, denoted as q_i=±1, therefore results in a “which-way” measurement <cit.> in the particular region of interest. While a click of a detector in e.g. the + arm of region 2 (q_2=+1) is a strongly invasive measurement, on the other hand the absence of a detector response implies q_2=-1 and does not disturb the system at all. It accounts for the required non-invasive measurement (A2) in terms of an ideal negative measurement.
In our neutron interferometric realization of <cit.> neutrons enter the IFM via the + port of region 1. Hence, it is not necessary to measure in region 1 and the noninvasive measurability is granted. The first plate of the IFM consists of a tunable beamsplitter characterized by parameter ϑ_A, which is schematically illustrated in Fig. <ref>. The theoretical maximum of K=1.5 is obtained for ϑ_A=ϑ_B=π/3 and phase shift χ=0. However, in our setup with fixed ϑ_B=π/2 (usual 50:50 beamsplitter), the maximal possible violation is K=√(2) (for ϑ_A=π/4).
We define P_α±,β±(n_α,n_β) as the joint probability that two detectors placed at position α± and β± respectively detect (n=1) or don't detect a neutron (n=0), where α and β specify the region and ± the path. Then the correlator, as defined in Eq.(<ref>), between regions α and β is given by
C_αβ=∑_q_α,q_β=±q_α q_β P_α q_α,β q_β(1,1).
Hence the correlation function for regions 1 and 3, denoted as C_31, can simply be expressed as
C_31=P_3+,1+(1,1)-P_,3-,1+(1,1),
since the neutrons always enter from 1+.
Therefore, the correlation function C_31 can also be expressed in terms of mariginal probabilities as C_31=P_3+(1)-P_3-(1).
Although not particularly necessary here,
it is instructive to express C_31 in terms of ideal negative measurements as
C_31= ∑_q_1,q_3=±q_1 q_3 P_3 q_α(1)(1-P_1q_β(0))
=-∑_q_1,q_3=±q_1 q_3 P_1q_2,3q_3(1,0),
since P_1q_1(0)=1-P_1q_1(1). A similar expression gives the correlator C_21=P_1+,2+(1)-P_1+,2-(1) which is measured with detectors directly placed in region 2, shown in Fig. <ref> (a).
For C_32 all four terms of the sum from Eq.(<ref>) contribute, taking both paths of section 2 into account.
C_32=∑_q_2,q_3=±q_2 q_3 P_3q_3,2q_2(1,1)
Using again P_2q_2(0)=1-P_2q_2(1) we write the sum as
C_32=-∑_q_2,q_3=±q_2 q_3 P_3q_3,2q_2(1,0)
in order to account for the non-invasive or ideal negative measurement in section 2. The two pobabilities P_3±,2-(1,0) are determined by counting the neutrons in path 3+ and 3- respectively under the condition that they have not been counted in pah 2-. The latter is ensured by placing a beam blocker in path 2-, cf. Fig. <ref>(b). The other two pobabilities are measured similarly as shown in Fig. <ref>(c).
The correlators according to <cit.> for the regions in our setup are calculated as follows
C_21= cosϑ_A
C_32= cosϑ_B
C_31= cosϑ_A cosϑ_B - cosχsinϑ_A sinϑ_B
K= cosϑ_A+cosϑ_B-cosϑ_A cosϑ_B
+ cosχsinϑ_A sinϑ_B,
which in our setup, with fixed sinϑ_B=π/2, K becomes
K=cosϑ_A + cosχsinϑ_A.
Figure <ref> shows the regions in the parameter space (ϑ_A,χ) of our experimental LGI test (with fixed value ϑ_B=π/2), where it is in theory possible to violate the LGI with a value K=√(2). ϑ_A represents the mixing angle of the first interferometer plate, and χ the phase shifter angle. The resulting K values are shown in green for areas where no violation is possible, and in orange for a possible violation of the LGI. The dashed red line indicates our measurement result in an ideal interferometer.
Neutron interferometer setup.—Neutron interferometry <cit.> provides a powerful tool for investigation of fundamental quantum mechanical phenomena. Entanglement between different degrees of freedom (DOF), e.g., the neutron’s spin, path, and energy DOF has been confirmed, and the contextual nature of quantum mechanics has been demonstrated successfully <cit.>. In more recent experiments the concept of weak measurements and weak values has been utilized for direct state reconstruction <cit.>, demonstration of the canonical commutator relation <cit.> and studies of which way information <cit.>.
The experiment was carried out at the neutron interferometer instrument S18 at the high-flux reactor of the Institute Laue-Langevin (ILL) in Grenoble, France (the experimental data can be found on the ILL data server under <cit.>. A monochromatic unpolarized neutron beam with mean wavelength λ=1.91Å (δλ/λ∼0.02) and 3 × 3 mm^2 beam cross section was used to illuminate the interferometer. In order to observe a violation of an LGI in an interferometric experiment, it is necessary to implement a non-50:50 beam splitter at the first plate of the interferometer. This is achieved by placing a partial absorber behind the first interferometer plate in one of the neutron paths.
The absorber is an Indium slab, about 3 thick, placed in path I, resulting in an intensity ratio between paths I and II of about 10:90. The interferometer itself is a symmetric three-plate silicon perfect crystal (triple Laue type), with a plate thickness of 3 and a length of 140. A schematic illustration of the interferometric setup is given in Fig. <ref>. To obtain interference fringes, a 5 Aluminium phase shifter was used. Additional beam blockers for the detection of single path intensities were made of Cadmium. Both the `O' and `H' detectors outside the interferometer and the additional detector for C_21 measurements were ^3He proportional counting tubes.
Determination of correlators C_31 and C_21 is straightforward. In both cases it is not necessary to measure non-invasively, since no subsequent measurement on the same state is performed. For C_31, the measurement is that of a standard interferogram Fig. <ref>, with measurement time 180 seconds per phase shifter position. The correlator C_31 is calculated via
C_31=N_3+1+(χ)-N_3-1+(χ)/N_3+1+(χ)+N_3-1+(χ),
where N_3+1+(χ) denotes the counts in the H detector and N_3-1+(χ) the counts in the O detector. Due to the cosine behaviour of the recorded interferogram, this correlator is dependent on the position χ of the phase shifter. For the largest possible violation, the maximum counts in O and minimum in H are used, which corresponds to the position χ=2 n π (where n∈ℕ_0) in Fig. <ref>.
Similarly, the correlator C_21 is calculated as
C_21=N_2+1+-N_2-1+/N_2+1++N_2-1+
and is performed as a transversal scan with a pencil-size He-3 detector mounted on a translation stage in region 2 of the interferometer, with measurement time 300 seconds per detector position. Moving first through path I and then through path II, the resulting neutron counts are shown in Fig. <ref>, where the separation between both paths is also clearly visible. The N_2i1+ are the neutron counts in the peak of the respective Gaussian fit to the intensity profiles.
For correlator C_32, however, it is crucial to measure non-invasively. This is done by measuring the absence of a neutron in a given path due to the Cd blocker, meaning that the neutron has to take the path without the Cd blocker. This is represented by the minus sign in Eq. (<ref>). Four measurements are performed, with each of the paths blocked in turn and the resulting intensity in detectors O and H recorded for a measurement time of 600 seconds. These results are shown in Fig. <ref>. C_32 becomes
C_32=N_3+2-+N_3-2+-N_3+2+-N_3-2-/N_3+2-+N_3-2++N_3+2++N_3-2-,
with N_3+2- and N_3+2+ the neutron counts in the H detector with blocked path II and path I, respectively, and likewise for the O detector in N_3-2±.
Results.—In order to demonstrate the experimental violation of the Leggett–Garg inequality, we calculate the correlator K, Eq. (<ref>). The resulting curve is shown in Fig. <ref>, with the maximum at a phase shift of χ=0. With the Indium absorber in path I of the interferometer, a violation of the limit K=1 is clearly visible (Fig. <ref>(a)). Our results show a significant violation of the LGI by 18 standard deviations σ (denoted as n_σ=18) at the maximum, K =1.120±0.007. The violation is visible over a wide range of phase shifter values χ. Numeric values of the individual correlators C_ij and the final value of K in case of the maximal violation of the LGI are presented in Tab. <ref>.
For comparison, Fig. <ref>(b) shows the same measurement procedure for a symmetric beam splitter (ϑ_A=π/2), i.e. without absorber,
where no violation is possible, resulting in K=0.540±0.023.
Concluding remarks and discussion.—Our measurement results demonstrate a violation of an LGI by n_σ=18.0, while the absorberless measurements show no violation. Hence we conclude that neutrons in an interferometer must be understood quantum mechanically.
An even higher violation can be achieved when the signs in region 3 are switched, and detector O becomes 3+, detector H 3-. The correlators C_31 and C_32 have to be recalculated accordingly, resulting in K=1.162±0.006 with n_σ=28.
This `additional' violation is due to the asymmetric nature of the perfect crystal interferometer. Since successive reflections on the crystal lamellas enhance the reflectivity <cit.> the H detector always receives some phase-independent intensity offset.
The detection loophole is closed due to the high efficiency of our neutron detectors, close to unity. The fair sampling assumption is needed, especially for the correlator C_21, which is the case for a wide range of experiments of this kind, since simultaneous detection of everything is impossible.
Finally, we want to emphasize that the interferometric scheme applied in the present work is not limited neutrons, but is in fact completely general and can be used for any quantum particle with nonzero or even zero mass.
This work was supported by the Austrian science fund (FWF) Projects No. P 30677 and No. P 34239.
30
fxundefined [1]
ifx#1
fnum [1]
#1firstoftwo
secondoftwo
fx [1]
#1firstoftwo
secondoftwo
noop [0]secondoftwo
ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]
rl [1]href #1
@bib@innerbibempty
[Bell(1964)]Bell64
author author J. S. Bell, title title On the
Einstein-Podolsky-Rosen paradox, @noop journal
journal Physics (Long Island City, N.Y.) volume 1, pages 195 (year 1964)NoStop
[Bell(1966)]Bell66
author author J. S. Bell, title title On the problem of hidden
variables in quantum mechanics, https://doi.org/10.1103/RevModPhys.38.447 journal journal Rev. Mod. Phys. volume 38, pages 447 (year 1966)NoStop
[Kochen and Specker(1967)]Kochen67
author author S. Kochen and author E. P. Specker, title title The problem of hidden
variables in quantum mechanics, @noop journal
journal J. Math. Mech. volume 17, pages 59 (year 1967)NoStop
[Leggett and Garg(1985)]leggett_quantum_1985
author author A. J. Leggett and author A. Garg, title title Quantum mechanics
versus macroscopic realism: Is the flux there when nobody looks?, https://doi.org/10.1103/PhysRevLett.54.857 journal journal Phys. Rev. Lett. volume 54, pages 857 (year 1985)NoStop
[Emary et al.(2014)Emary,
Lambert, and Nori]emary_leggettgarg_2014
author author C. Emary, author N. Lambert, and author F. Nori, title title Leggett–Garg inequalities, https://doi.org/10.1088/0034-4885/77/1/016001 journal
journal Rep. Prog. Phys. volume 77, pages 016001 (year
2014)NoStop
[Ruskov et al.(2006)Ruskov,
Korotkov, and Mizel]Ruskov06
author author R. Ruskov, author A. N. Korotkov, and author A. Mizel, title title Signatures of quantum
behavior in single-qubit weak measurements, https://doi.org/10.1103/PhysRevLett.96.200404 journal
journal Phys. Rev. Lett. volume 96, pages 200404 (year 2006)NoStop
[Jordan et al.(2006)Jordan,
Korotkov, and Büttiker]Jordan06
author author A. N. Jordan, author A. N. Korotkov, and author M. Büttiker, title title Leggett-garg
inequality with a kicked quantum pump, https://doi.org/10.1103/PhysRevLett.97.026805 journal
journal Phys. Rev. Lett. volume 97, pages 026805 (year 2006)NoStop
[Dressel et al.(2011)Dressel, Broadbent, Howell, and Jordan]Dressel11
author author J. Dressel, author C. J. Broadbent, author J. C. Howell, and author A. N. Jordan, title title Experimental violation of
two-party Leggett-Garg inequalities with semiweak measurements, https://doi.org/10.1103/PhysRevLett.106.040402 journal
journal Phys. Rev. Lett. volume 106, pages 040402 (year 2011)NoStop
[Goggin et al.(2011)Goggin,
Almeida, Barbieri, Lanyon,
O’Brien, White, and Pryde]Goggin11
author author M. E. Goggin, author M. P. Almeida,
author M. Barbieri, author B. P. Lanyon, author
J. L. O’Brien, author
A. G. White, and author
G. J. Pryde, title title Violation of the Leggett–Garg inequality with weak measurements of
photons, https://doi.org/10.1073/pnas.1005774108 journal journal Proc. Natl. Acad. Sci. USA volume 108, pages 1256
(year 2011)NoStop
[Waldherr et al.(2011)Waldherr, Neumann, Huelga, Jelezko, and Wrachtrup]Waldherr11
author author G. Waldherr, author P. Neumann,
author S. F. Huelga, author F. Jelezko, and author
J. Wrachtrup, title title Violation of a temporal Bell inequality for single spins in a
diamond defect center, https://doi.org/10.1103/PhysRevLett.107.090401 journal
journal Phys. Rev. Lett. volume 107, pages 090401 (year 2011)NoStop
[Palacios-Laloy et al.(2010)Palacios-Laloy, Mallet, Nguyen,
Bertet, Vion, Esteve, and Korotkov]Palacios10
author author A. Palacios-Laloy, author F. Mallet, author F. Nguyen,
author P. Bertet, author D. Vion, author
D. Esteve, and author
A. N. Korotkov, title
title Experimental violation of a bell's inequality in time with
weak measurement, https://doi.org/10.1038/nphys1641 journal journal Nat. Phys. volume
6, pages 442 (year 2010)NoStop
[Knee et al.(2016)Knee,
Kakuyanagi, Yeh, Matsuzaki,
Toida, Yamaguchi, Saito,
Leggett, and Munro]Knee16
author author G. C. Knee, author K. Kakuyanagi,
author M.-C. Yeh, author Y. Matsuzaki, author
H. Toida, author H. Yamaguchi, author S. Saito, author A. J. Leggett, and author W. J. Munro, title title A strict
experimental test of macroscopic realism in a superconducting flux qubit, https://doi.org/10.1038/ncomms13253 journal journal Nat. Commun. volume 7, pages 13253 (year 2016)NoStop
[Athalye et al.(2011)Athalye, Roy, and Mahesh]Athalye11
author author V. Athalye, author S. S. Roy, and author T. S. Mahesh, title title Investigation of the Leggett-Garg
inequality for precessing nuclear spins, https://doi.org/10.1103/PhysRevLett.107.130402 journal
journal Phys. Rev. Lett. volume 107, pages 130402 (year 2011)NoStop
[Souza et al.(2011)Souza,
Oliveira, and Sarthour]Souza11
author author A. M. Souza, author I. S. Oliveira, and author R. S. Sarthour, title title A scattering quantum
circuit for measuring Bell's time inequality: a nuclear magnetic resonance
demonstration using maximally mixed states, https://doi.org/10.1088/1367-2630/13/5/053023 journal
journal New J. Phys. volume
13, pages 053023 (year 2011)NoStop
[Knee et al.(2012)Knee,
Simmons, Gauger, Morton,
Riemann, Abrosimov, Becker,
Pohl, Itoh, Thewalt,
Briggs, and Benjamin]Knee2012
author author G. C. Knee, author S. Simmons,
author E. M. Gauger, author J. J. Morton, author
H. Riemann, author N. V. Abrosimov, author P. Becker, author H.-J. Pohl, author K. M. Itoh, author M. L. Thewalt,
author G. A. D. Briggs, and author S. C. Benjamin, title title Violation of a Leggett–Garg
inequality with ideal non-invasive measurements, https://doi.org/10.1038/ncomms1614 journal journal Nat. Commun. volume 3, pages 606 (year 2012)NoStop
[Emary(2013)]Emary13
author author C. Emary, title title Decoherence and maximal
violations of the Leggett-Garg inequality, https://doi.org/10.1103/PhysRevA.87.032106 journal journal Phys. Rev. A volume 87, pages 032106 (year 2013)NoStop
[Mendoza-Arenas et al.(2019)Mendoza-Arenas, Gómez-Ruiz, Rodríguez, and Quiroga]Arenas19
author author J. J. Mendoza-Arenas, author F. J. Gómez-Ruiz, author F. J. Rodríguez, and author L. Quiroga, title title Enhancing violations of
Leggett-Garg inequalities in nonequilibrium correlated many-body systems by
interactions and decoherence, https://doi.org/10.1038/s41598-019-54121-1 journal journal Sci. Rep. volume 9, pages 17772 (year 2019)NoStop
[Matsumura et al.(2022)Matsumura, Nambu, and Yamamoto]Matsumura22
author author A. Matsumura, author Y. Nambu, and author K. Yamamoto, title title Leggett-Garg inequalities for testing
quantumness of gravity, https://doi.org/10.1103/PhysRevA.106.012214 journal journal Phys. Rev. A volume 106, pages 012214 (year 2022)NoStop
[Bose et al.(2018)Bose,
Home, and Mal]Bose18
author author S. Bose, author D. Home, and author S. Mal, title title Nonclassicality of the harmonic-oscillator
coherent state persisting up to the macroscopic domain, https://doi.org/10.1103/PhysRevLett.120.210402 journal
journal Phys. Rev. Lett. volume 120, pages 210402 (year 2018)NoStop
[Bose et al.(2017)Bose,
Mazumdar, Morley, Ulbricht,
Toro šš, Paternostro, Geraci, Barker, Kim, and Milburn]Bose17
author author S. Bose, author A. Mazumdar,
author G. W. Morley, author H. Ulbricht, author
M. Toro šš,
author M. Paternostro, author A. A. Geraci, author
P. F. Barker, author
M. S. Kim, and author
G. Milburn, title title Spin entanglement witness for quantum gravity, https://doi.org/10.1103/PhysRevLett.119.240401 journal
journal Phys. Rev. Lett. volume 119, pages 240401 (year 2017)NoStop
[Marletto and Vedral(2017)]Marletto17
author author C. Marletto and author V. Vedral, title title Gravitationally induced
entanglement between two massive particles is sufficient evidence of quantum
effects in gravity, https://doi.org/10.1103/PhysRevLett.119.240402
journal journal Phys. Rev. Lett. volume 119, pages 240402 (year
2017)NoStop
[Emary et al.(2012)Emary,
Lambert, and Nori]emary_leggett-garg_2012
author author C. Emary, author N. Lambert, and author F. Nori,
title title Leggett-Garg inequality in electron interferometers, journal volume 86, https://doi.org/10.1103/PhysRevB.86.235447 Phys. Rev. B volume 86 (year 2012)
NoStop
[Englert(1996)]Englert96
author author B.-G. Englert, title title Fringe visibility and
which-way information: An inequality, https://doi.org/10.1103/PhysRevLett.77.2154 journal journal Phys. Rev. Lett. volume 77, pages 2154 (year 1996)NoStop
[Rauch and Werner(2000)]RauchBook
author author H. Rauch and author S. A. Werner, @noop title Neutron
Interferometry (publisher Clarendon Press, Oxford, year 2000)NoStop
[Klepp et al.(2014)Klepp,
Sponar, and Hasegawa]klepp2014fundamental
author author J. Klepp, author S. Sponar, and author Y. Hasegawa, title title Fundamental phenomena of quantum mechanics
explored with neutron interferometers, https://doi.org/10.1093/ptep/ptu085
journal journal Prog. Theor. Exp. Phys volume 2014, (year 2014)NoStop
[Sponar et al.(2021)Sponar,
Sedmik, Pitschmann, Abele, and Hasegawa]Sponar21
author author S. Sponar, author R. I. P. Sedmik, author M. Pitschmann,
author H. Abele, and author Y. Hasegawa, title
title Tests of fundamental quantum mechanics and dark
interactions with low-energy neutrons, https://doi.org/10.1038/s42254-021-00298-2 journal journal Nat. Rev. Phys volume 3, pages 309 (year 2021)NoStop
[Denkmayr et al.(2017)Denkmayr, Geppert, Lemmel, Waegell, Dressel, Hasegawa, and Sponar]Denkmayr17
author author T. Denkmayr, author H. Geppert,
author H. Lemmel, author M. Waegell, author
J. Dressel, author Y. Hasegawa, and author S. Sponar, title title
Experimental demonstration of direct path state characterization by strongly
measuring weak values in a matter-wave interferometer, https://doi.org/10.1103/PhysRevLett.118.010402 journal
journal Phys. Rev. Lett. volume 118, pages 010402 (year 2017)NoStop
[Wagner et al.(2021)Wagner,
Kersten, Danner, Lemmel,
Pan, and Sponar]Wagner21
author author R. Wagner, author W. Kersten,
author A. Danner, author H. Lemmel, author
A. K. Pan, and author
S. Sponar, title title Direct experimental test of commutation relation via imaginary weak
value, https://doi.org/10.1103/PhysRevResearch.3.023243 journal journal Phys. Rev. Research volume 3, pages 023243 (year
2021)NoStop
[Geppert-Kleinrath et al.(2018)Geppert-Kleinrath, Denkmayr, Sponar,
Lemmel, Jenke, and Hasegawa]Geppert18
author author H. Geppert-Kleinrath, author T. Denkmayr, author S. Sponar,
author H. Lemmel, author T. Jenke, and author
Y. Hasegawa, title title Multifold paths of neutrons in the three-beam interferometer
detected by a tiny energy kick, https://doi.org/10.1103/PhysRevA.97.052111 journal journal Phys. Rev. A volume 97, pages 052111 (year 2018)NoStop
[Lemmel et al.(2022)Lemmel,
Geerits, Danner, Hofmann, and Sponar]Lemmel2022
author author H. Lemmel, author N. Geerits,
author A. Danner, author H. F. Hofmann, and author S. Sponar, title
title Quantifying the presence of a neutron in the paths of an
interferometer, https://doi.org/10.1103/PhysRevResearch.4.023075
journal journal Phys. Rev. Research volume 4, pages 023075 (year
2022)NoStop
[ILL et al.(2021)Sponar,
Kreuzgruber, and Lemmel]S18data
author author Stephan Sponar, author Elisabeth Kreuzgruber,
and author Hartmut Lemmel, @noop title Leggett-Garg Inequality, (year 2019), note
https://doi.ill.fr/10.5291/ILL-DATA.CRG-2643 https://doi.ill.fr/10.5291/ILL-DATA.CRG-2643NoStop
[Petrascheck and Rauch(1984)]petrascheck1984
author author D. Petrascheck and author H. Rauch, title title Multiple Laue rocking
curves, @noop https://doi.org/10.1107/S0108767384000878journal journal Acta Crystallogr. A volume 40, pages 445 (year
1984)NoStop
|
http://arxiv.org/abs/2307.04265v1 | 20230709210635 | Enhancement and anisotropy of electron Lande factor due to spin-orbit interaction in semiconductor nanowires | [
"J. Czarnecki",
"A. Bertoni",
"G. Goldoni",
"P. Wójcik"
] | cond-mat.mes-hall | [
"cond-mat.mes-hall"
] |
[email protected]
AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland
[email protected]
CNR-NANO S3, Istituto Nanoscienze, Via Campi 213/a, 41125 Modena, Italy
[email protected]
Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, via Campi 213/a, 41125 Modena, Italy
CNR-NANO S3, Istituto Nanoscienze, Via Campi 213/a, 41125 Modena, Italy
[email protected]
AGH University of Science and Technology, Faculty of Physics and Applied Computer cience, Al. Mickiewicza 30, 30-059 Krakow, Poland
We investigate the effective factor in semiconductor nanowires with strong Rashba spin-orbit coupling. Using the 𝐤·𝐩 theory and the envelope function approach we derive a conduction band Hamiltonian where the tensor is explicitly related to the spin-orbit coupling contants . Our model includes orbital effects from the Rashba spin-orbit term, leading to a significant enhancement of the effective factor which is naturally anisotropic. For nanowires based on the low-gap, high spin-orbit coupled material InSb, we investigate the anisotropy of the effective factor with respect to the magnetic field direction, exposing a twofold symmetry for the bottom gate architecture. The anisotropy results from the competition between the localization of the envelope function and the spin polarization of the electronic state, both determined by the magnetic field direction.
Enhancement and anisotropy of electron factor due to spin-orbit interaction in semiconductor nanowires
Paweł Wójcik
February 2023
=======================================================================================================
§ INTRODUCTION
Semiconductor nanowires (NWs) continue to attract significant interest due to the abundance of physical phenomena observed in such nanostructures, as well as the wealth of potential applications, including optoelectronics,<cit.> quantum computing,<cit.> or spintronics.<cit.> Applications in spintronics are largely driven by the spin-orbit (SO) interaction, which – in low energy gap semiconductors, such as InAs or InSb – is sufficiently strong to enable electrical control of the electron spin. In general, the SO interaction originates from the lack of the inversion symmetry, which could be an intrinsic feature of the crystallographic structure (Dresselhaus SO coupling <cit.>) or induced by the asymmetry of the confinement potential (Rashba SO coupling<cit.>). The latter has the essential advantage of being tunable by external fields, e.g., using gates attached to the nanostructures, as predicted theoretically<cit.> and demonstrated in recent experiments.<cit.>
The significant progress in heteroepitaxy, which has been made over the last decade, enables the growth of a thin superconducting layer on the surface of semiconductor.<cit.> In this respect hybrid NWs with a large SO interaction are recently intensively studied as the basic building blocks for topological quantum computing based on Majorana zero modes.<cit.> These exotic states are formed at the ends of NWs when the system becomes spinless, which is achieved in experiments by applying a magnetic field and the corresponding spin Zeeman effect.<cit.> The induced topological gap strongly depends on the strength of the SO coupling and the energy of the Zeeman splitting,<cit.> usually expressed in terms of a linear response to the magnetic field with a proportionality constant – the effective factor. In other words, the electron determines the strength of the magnetic field required to trigger the system into the topological phase. For this reason, it is desirable to make it as large as possible, as the magnetic field needed for the topological transition is required to be lower than the critical magnetic field of the superconducting shell.<cit.>
In semiconducting materials is significantly different from the free-electron factor g_0, due to coupling between the valence and the conduction band. In the second-order perturbation 𝐤·𝐩 theory it leads to the Roth-Lax-Zwerdling () formula,<cit.> which for low gap semiconductors gives ≫ g_0, e.g. ≈ -49 for InSb. In particular, for semiconductor nanostructures the formula predicts a reduction of the effective factor,<cit.> as the subband confinement increases the energy gap, which is inversely proportional to .<cit.> However, unexpectedly, recent experiments in NWs based on InAs and InSb exhibit opposite behaviour - the extracted is up to three times larger than the bulk value.<cit.> Furthermore, in Ref. Marcus2018 a step like evolution of has been reported as a function of the gate voltage.
It has been recently proposed that this surprising behaviour arises from the 𝐋·𝐒 coupling, which for higher subbands (characterized by the large orbital momentum) leads to an enhancement of by about one order of magnitude.<cit.>
In this paper we develop a full 8 × 8 k·p theory of the effective factor in semiconductor NWs which takes into account the orbital effects in the SO coupling terms induced by an external magnetic field of arbitrary direction. We show explicitly that the response to the magnetic field can be described in terms of a tensor , whose elements originate from the vector of Rashba coupling constants . For a nanowire based on the low-gap, strongly SO coupled material InSb, we performed fully self-consistent calculations taking into account on equal footing orbital and Zeeman effect of the applied magnetic field, SO coupling and the electrostatic environment. We demonstrate that the orbital component of ensuing from the SO interaction may be greater than the one determined by the bulk and may represents a major component of , leading to the enhancement of the effective factor by an order of magnitude, even for the lowest subband, the one usually considered in Majorana experiments. Finally, we also evaluate the anisotropy of the SO induced tensor with respect to the magnetic field rotated in different planes. Our results qualitatively agree with recent experiments<cit.> reproducing the enhancement of g^* and its anisotropy.
The paper is organized as follows. In Sec. II A the tensor is derived from the 8 × 8 k·p model within the envelope function approximation. Details on the numerical method are given in Sec. II B. Sec. III contains results of our calculations for homogeneous InSb NWs and their discussion with respect to recent experiments. Sec. IV summarizes our results.
§ THEORETICAL MODEL
Below we shall derive a k·p formulation of the factor in semicondutor NWs. We shall specifically consider a homogeneous InSb, with hexagonal cross section, grown in the zincblede crystallographic structure along the [111] direction. This particular orientation preserves the crystal inversion symmetry, resulting in the reduction of the Dresselhaus SO coupling term.<cit.>
The system is subjected to a uniform external magnetic field with intensity B.
The direction of the applied magnetic field with respect to the NW axis is determined by two angles: θ, the angle between the field and the NW axis (z), and φ, the angle between the x axis (oriented along the corner-corner direction) and the projection of the field on the xy plane – see Fig. <ref>. Hence,
B = [B_x,B_y,B_z]^T
= B [sin(θ)cos(φ), sin(θ)sin(φ),cos(θ) ]^T .
We adopt the symmetric vector potential
A(r) = [ -yB_z/2, xB_z/2, yB_x - xB_y ]^T .
If not stated otherwise, we assume that the back gate is attached directly to the bottom facet of NW, generating an electric field parallel to the nanowire section in the xy plane. Although in real experiments a dielectric layer separating the NW from the gate is usually used, it plays a role of screening for the electric field. Hence, the value of the factor obtained for a particular gate voltage V_g can be considered as the maximum achievable value at that specific V_g.
§.§ 𝐤·𝐩 theory of the tensor
Our model is based on the 8 × 8 k·p approximation described by
_8 × 8 =
[ _c _cv; _cv^† _v ],
where _c is Hamiltonian of the conduction band electrons corresponding to Γ_6c band. In the presence of the magnetic field _c can be written as
_c = H_Γ_6cI_2 × 2+1/2μ_Bg_0 σ·𝐁,
where the second term corresponds to the Zeeman spin effect, where μ_B is the Bohr magneton, g_0 is the factor of the free electron and σ=(σ_x,σ_y, σ_z) is the vector of Pauli matrices, while
Ĥ_Γ_6c = P̂^2/2m_0 + E_c + V(𝐫),
where P̂ = p̂ - eA, e is the electron charge, m_0 is the free electron mass and E_c is the conduction band minima. The potential V(𝐫) in (<ref>) contains interaction of electrons with the electric field generated by the external gates and the electron-electron interaction included in our model at the mean field level (Hartree potential),
V(𝐫)=V_g(𝐫)+V_H(𝐫).
Below we shall use a folding procedure of _8 × 8 to the conduction band sector, where in the Hamiltonian _v, related to valance bands Γ_8v and Γ_7v, all off-diagonal elements are neglected. Then, _v can be written as
_v = H_Γ_8vI_4 × 4⊕ H_Γ_7vI_2 × 2,
with
H_Γ_7v = E_v' = E_c + V(𝐫) - E_0 - Δ_0 ,
H_Γ_8v = E_v = E_c + V(𝐫) - E_0 ,
where E_0 is the energy gap and Δ_0 is the energy of spin-orbit splitting in the valence band.
The coupling between the conduction band and the valence band is described by the off-diagonal matrix _̋cv,
_̋cv = P_0/ħ[ P̂_+/√(6) 0 P̂_-/√(2) -√(2)P̂_z/√(3) -P̂_z/√(3) P̂_+/√(3); ; -√(2)P̂_z /√(3) -P̂_+/√(2) 0 -P̂_-/√(6) P̂_-/√(3) P̂_z /√(3) ] ,
where P̂_± = P̂_x ±P̂_y and the parameter P_0 = -iħ/m_0⟨ S|p̂_x|X⟩ accounts for the coupling between conduction and valence bands at the Γ point of the Brillouin zone.
Using the standard folding-down transformation, we can reduce the 8× 8 𝐤·𝐩 model (<ref>) into the effective 2× 2 Hamiltonian for conduction electrons
_̋𝑒𝑓𝑓 = _̋c + _̋cv(_̋v - E)^-1_̋cv^† = _̋c + _c.
In the above formula, _c can be written in terms of Pauli matrices
_c = λ_0I_2 × 2 + λ·σ,
where
λ_0 = P_0^2/3ħ^2 [P̂_x (2/E_v + 1/E_v' )P̂_x + P̂_y (2/E_v + 1/E_v' )P̂_y ],
λ_x = iP_0^2/3ħ^2 [P̂_z (1/E_v - 1/E_v')P̂_y - P̂_y(1/E_v - 1/E_v' )P̂_z ],
λ_y = iP_0^2/3ħ^2[P̂_x(1/E_v - 1/E_v')P̂_z - P̂_z(1/E_v - 1/E_v')P̂_x ],
λ_z = iP_0^2/3ħ^2[P̂_y (1/E_v - 1/E_v' )P̂_x - P̂_x(1/E_v - 1/E_v' )P̂_y ].
The first term in Eq. (<ref>) leads to the standard formula for the effective mass
1/m^* = 1/m_0+2P_0^2/3ħ^2 ( 2/E_v + 1/E_v' ) ,
while the second term corresponds to the Rashba SO coupling. If we assume that E_0 and Δ_0 are the largest energies in the system
we can expand E_v(v') in Eqs. (<ref>-<ref>) to the second order in energy. Then, Eqs. (<ref>)-(<ref>) can be rewritten as
λ_x = - α_R^y ( k_z - e/ħA_z ) - eP_0^2/3ħ ( 1/E_0 - 1/E_0 + Δ_0 )B_x,
λ_y = α_R^x ( k_z - e/ħA_z ) - eP_0^2/3ħ ( 1/E_0 - 1/E_0 + Δ_0 )B_y,
λ_z = α_R^y ( k̂_x - e/ħA_x ) - α_R^x ( k̂_y - e/ħA_y )
- eP_0^2/3ħ ( 1/E_0 - 1/E_0 + Δ_0 )B_z,
where
α_R = (α_R^x, α_R^y, α_R^z)
= P_0^2/3 ( 1/E_0^2 - 1/ ( E_0 + Δ_0 )^2 )∇ V(x,y)
is the Rashba SO coupling constant. Note that in Eqs. (<ref>, <ref>) we have already omitted α_R^z terms since the magnetic field does not break translational invariance along the wire axis, i.e.,
Ψ_n,k_z(x,y,z) =ψ_n,k_z(x,y)e^ik_zz
=[ψ_n,k_z^↑(x,y),ψ_n,k_z^↓(x,y)]^Te^ik_zz .
Finally, the effective Hamiltonian for conduction electrons can be written as
_̋𝑒𝑓𝑓 = ( P^2/2m^* + E_c + V(𝐫) ) I_2 × 2+(α_R^xσ_y-α_R^yσ_x)k_z
+ (α_R^y k̂_x-α_R^x k̂_y)σ_z + 1/2μ_B 𝐁σ
where is a tensor given by
= g_RLZ𝐈_3×3+𝐠_SO ,
where
g_RLZ = g_0 - 2E_p/3(1/E_0 -1/E_0 + Δ_0) .
corresponds to the well-know formula<cit.>, with E_p = 2m_0P_0^2/ħ^2 and the tensor 𝐠_SO results from the orbital effects of the magnetic field in the SO Hamiltonian
𝐠_SO =
[ g^xx_SO g^xy_SO 0; g^yx_SO g^yy_SO 0; 0 0 g^zz_SO ].
For the assumed vector potential, the individual elements of this tensor can be expressed as
g^xx_SO = 2e/μ_Bħα_R^y y,
g^yy_SO = 2e/μ_Bħα_R^x x,
g^zz_SO = e/μ_Bħ( α_R^y y - α_R^x x ),
g^xy_SO = -2e/μ_B ħα_R^x y,
g^yx_SO = -2e/μ_B ħα_R^y x.
which shows that depends linearly on the vector of Rashba SO coupling constants α_R.
Since the Rashba coefficients and the factor are functions of space [see Eqs. (<ref>, <ref>)], they may not be easily compared to experiments. Therefore, in the following part of the paper we discuss the matrix elements of the Rashba SO coupling constants
⟨α_R^x(y)(k_z) ⟩ _n = ⟨ψ_n,k_z |α_R^x(y)σ_y(x)|ψ_n,k_z⟩
and the individual diagonal and off-diagonal matrix elements of 𝐠_SO, respectively defined as
⟨ g_SO^xx(yy,zz)(k_z) ⟩ _n = ⟨ψ_n,k_z |g^xx(yy,zz)_SOσ_x(y,z)|ψ_n,k_z⟩,
⟨ g_SO^xy(yx)(k_z) ⟩ _n = ⟨ψ_n,k_z |g^xy(yx)_SOσ_y(x)|ψ_n,k_z⟩,
where |ψ_n,k_z⟩ is the in-plane part of the n-th envelope functions of NW, to be calculated as described in the following section.
§.§ Numerical calculations
To understand the physics behind the behaviour of the factor in NWs with strong SO coupling, we use a numerical approach taking into account important key ingredients, namely the orbital and Zeeman effect, SO coupling and electrostatic environment. For this purpose, we employ a standard Shrödinger-Poisson approach. Assuming the translational invariance along the growth axis z, the envelope functions ψ_n,k_z(x,y)=[ψ_n,k_z^↑(x,y),ψ_n,k_z^↓(x,y)] can be determined from the Schrödinger equation
[ ( P̂_2D^2/2m^* + 1/2m^*ω_c^2[(ycosθ-xsinθ)sinϕ-k_xl_B^2]^2 + E_c
+ V(𝐫) ) I_2 × 2+(α_R^xσ_y-α_R^yσ_x)k_z
+ (α_R^y k̂_x-α_R^x k̂_y)σ_z
+ 1/2μ_B 𝐁σ ] ψ_n,k_z(x,y)=E_n,k_zψ_n(x,y),
where α_R^x(y) and are functions of the position (x,y), ω_c=eB/m^* is the cyclotron frequence, l_B=√(ħ/eB) is the magnetic length and
P̂_2D^2= ( p̂_x+eBycosϕ/2 )^2 + ( p̂_y-eBxcosϕ/2 )^2.
Note that in the presence of magnetic field and spin-orbit coupling the Hamiltonian (<ref>) depends on the k_z vector. The calculations are carried out on a uniform grid in the range [-k_z^max,k_z^max] where k_z^max is chosen to be much larger than the Fermi wave vector.
The self-consistent potential V(𝐫) in Eq. (<ref>) is determined at the mean field level by solving of the Poisson equation
∇ _2D^2V(x,y)=-n_e(x,y)/ϵ_0ϵ
where ϵ is a dielectric constant and the electron density n_e can be calculated based on the formula
n_e(x,y)=∑_n ∫_-k_z^max^k_z^max1/2π |ψ_n,k_z(x,y)|^2 f(E_n,k_z-μ,T) dk_z
where μ is the chemical potential, T is the temperature and f(E,T) is the Fermi-Dirac distribution.
In the applied Shrödinger-Poisson approach, equations (<ref>) and (<ref>) are solved alternatively until the self-consistency is reached, which we consider to occur when the relative variation of the charge density between two consecutive iterations is lower than 0.001. In each iteration a spatial distribution of α_R^x(y) and g_SO^ab, where a,b={x,y,z}, are determined based on Eqs. (<ref>) and (<ref>). The numerical calculations are carried on the triangular grid corresponding to the hexagonal symmetry of the nanowire cross-section, to avoid artifacts at the boundaries which may appear when using smaller grid densities.<cit.> We assume Dirichlet boundary condition corresponding to the assumed gate architecture. Finally, the self-consistent potential V(x,y) and the corresponding wave functions ψ_n,k_z(x,y) are used to determine ⟨α_R^x,(y)⟩ _n, ⟨ g^xx(yy,zz)_SO⟩ _n and ⟨ g^xy(yx)_SO⟩ _n tensor elements according to Eqs. (<ref>, <ref>).
Calculations have been carried out for the material parameters corresponding to InSb: E_0=0.235 eV, Δ_0=0.81 eV, m^*=0.014, E_P= 2m_0P/2 = 23.3 eV, T=4.2 K, and for the nanowire width W=100 nm (corner-to-corner). We keep the constant linear electron density at the low level n_e=8× 10^7 cm^-1 which guarantees that only the lowest subband is occupied in the range of the considered magnetic field B=[0,4] T.
§ RESULTS
We shall now discuss the tensor as a function of the magnetic field intensity and direction. As g_RLZ evaluated from the formula (g_RLZ=-49 for the present material) does not depend on the magnetic field, we put particular emphasis on the role of the SO-induced component 𝐠_SO in terms of the tensor elements, Eqs. (<ref>). As shown in the previous section, corrections to the factor coming from the SO interaction are in general wave-vector dependent, which results from the orbital effects of the magnetic field. For this reason, we shall investigate 𝐠_SO as a function of both the wave vector and the magnetic field. We limit our study to the lowest subband assuming an the electrical potential is applied to the bottom gate to induce SO coupling. For simplicity, in the rest of the paper we omit the subband index in Eqs. (<ref>), (<ref>), i.e. ⟨…⟩ _n=1=⟨…⟩.
§.§ Enhancement of due to SO coupling
First, we show that a magnetic field oriented along the x axis, i.e., perpendicular to the NW axis and to the direction of ⟨α_R⟩, results in a substantial enhancement of the effective factor. For this purpose, we assume that V_g=0.2 V is applied to the bottom gate, generating an electric field that mantains reflection symmetry with respect to the y axis, hence ⟨α_R⟩ is directed along y by symmetry.
In Fig. <ref>(a), we show the diagonal element ⟨ g_SO^xx⟩ as a function of the wavevector and the magnetic field intensity. Note that with this field configuration the off-diagonal elements vanish by symmetry. Indeed, the reflection symmetry of the electric field with respect to the y axis leads to ⟨α_R^x ⟩ =0, hence ⟨ g_SO^xy⟩ =0 [see Eq. (<ref>)]. Moreover, the even symmetry of the envelope function is unaffected by the magnetic field directed along the x, hence ⟨ g_SO^yx⟩=0 [see Eqs. (<ref>)]. Fig. <ref>(a) clearly demonstrates that the correction to the effective factor arising from the orbital effects in the SO coupling term reaches a value similar to that obtained from the RLZ formula. Under certain conditions, this enhancement can lead to a significant increase of g^*, almost doubling it, as observed in recent experiments. <cit.>
In Fig. <ref>(a) we distinguish three regions, with positive (yellow), negative (purple) and vanishing (black) ⟨ g_SO^xx⟩. The abrupt change of sign between positive and negative regions is simply understood as the crossing of subbands of opposite spin, since only the value for the lowest subband is shown here. Indeed, as shown in Fig. <ref>(b), the subband of opposite spin cross at k_z=0 at vanishing field. When the field is switched on, both subband shift to negative k_z and shift in energy due to Zeeman term. Hence, the crossing shifts linearly with the field to more negative wavevectors, as shown in Fig. <ref>(a).
For sufficiently large k_z>0 and field intensity, ⟨ g_SO^xx⟩ almost vanishes, as shown in Fig. <ref>(a) - black region. This is due to the localization and symmetry of the envelope functions, which are determined by the orbital coupling to the magnetic field. Figure <ref>(a) illustrates maps of the position-dependent SO coupling constants α_R^x(y) [see Eq. (<ref>)] calculated in a self-consistent cycle at B=0. The spatial distribution is primarily influenced by the electric field generated by the bottom gate and do not undergo significant changes as the magnetic field increases. Since the value of 𝐠_SO elements depends on the Rashba SOC constant, the SO-induced modification of the factor for a specific subband is most significant when its envelope function is localized in the regions of strong Rashba SOC, which, in turn, is determined by both the electric and the magnetic field, as we discuss below.
In Fig. <ref>(b), we report the squared envelope functions of the lowest subbands at k_z=0 and k_z=0.4 nm^-1 at increasing magnetic fields. At k_z=0 there is no kinetic coupling to the magnetic field and the localization of the envelope function is only determined by the electric field; hence, it concentrates near the bottom gate, where the SOC is strong. For a positive wave vectors k_z, instead, the orbital effects shift the wave function towards the opposite facet of the NW, where the SO coupling is weak, leading to vanishing ⟨ g^xx_SO⟩, which explains the black region in Fig. <ref>(a). As shown in Fig. <ref>(a), the stronger the magnetic field, the lower k_z is required to push the wave function away from the region with large SO coupling, near the bottom facet. Naively, one might expect that the state k_z=0 would not be affected by this phenomenon as there is not orbital coupling to the magnetic field for this state. However, it should be noted that for high magnetic fields, diamagnetic effects become dominant, causing the wave functions to localize in the middle of NW along the field direction, resembling dispersionless Landau levels, as shown in Fig. <ref>(b). As the position of this wave function is associated with low SO coupling regions, ⟨ g^xx_SO⟩ gradually decreases towards zero, even for k_z=0, as illustrated in Fig. <ref>(a).
We next discuss the behavior of the SO-induced factor with the magnetic field directed either parallel to α_R (along the y axis) or to the NW axis (along the z axis). When the magnetic field is applied parallel to α_R, ⟨ g^yy_SO⟩ 0. However, its magnitude, shown in Fig. <ref>(a), is not as large as ⟨ g^xx_SO⟩ in the perpendicular orientation – compare with Fig. <ref>(a). In this configuration the off-diagonal element ⟨ g_SO^yx⟩ is non-negligible, in contrast to ⟨ g_SO^xy⟩ which is nearly zero as the avarage value of α_R^x is vanishing due to the gate symmetry. Again, the evolution of both ⟨ g_SO^yy⟩ and ⟨ g_SO^yx⟩ as a function of the magnetic field, shown in Fig. <ref>(a) and Fig. <ref>(b), respectively, is determined by the localization and symmetry of the wave function.
In Fig. <ref>(b), one can observe that at zero magnetic field, the wave function sets itself at the center-bottom of the NW. In this region, α_R^x is antisymmetric with respect to the x axis, resulting in the significant suppression of ⟨ g_SO^yy⟩ and ⟨ g_SO^xy⟩, which vanish at k_z=0. The symmetry of the wave function is broken by the magnetic field, as depicted in Fig. <ref>(b). For k_z = 0.4 nm^-1, for increasing magnetic fields, the wave function is first localized at the bottom-left corner, where the contribution from negative α_R^x leads to non-zero values of ⟨ g_SO^yy(yx)⟩, and eventually in the left corner, where α_R^x is significantly lower, resulting in a decrease in ⟨ g_SO^yy(yx)⟩. This field-induced evolution leads to the maximum of ⟨ g_SO^yy(yx)⟩ at a certain k_z value, as illustrated in Fig. <ref>(a,b).
We next consider a magnetic field applied in z-direction, i.e., along the NW axis. The finite value of ⟨ g_SO^zz⟩, shown in Fig. <ref> has a different nature, since the orbital effects of magnetic field are highly reduced by the confinement. In this case the localization of the wavefunction is not measurably changed with the magnetic field, regardless of k_z, and thus it does not determine the evolution of ⟨ g_SO^zz⟩ with k_z and B_z. Rather, it is governed by the interplay between the Zeeman effect and the SO interaction, which mixes the spin states. Note that all tensor elements ⟨ g_SO^xx(yy,zz)⟩ [see Eqs. (<ref>)], are in fact defined by the energy splitting caused by the orbital effects from the SO interaction what makes ⟨ g_SO^zz⟩ sensitive to the relative distribution of spin up and down component in the spinor. Since the SO coupling depends on the wave vector, for a small k_z the ordinary Zeeman effect is dominant, aligning the electron spin along the magnetic field direction and - in the limit of k_z=0 - makes the system spin polarized along the z axis. The expectation value of σ_z in this case is the largest, resulting in the large value of ⟨ g_SO^zz⟩. In other words, the value of ⟨ g_SO^zz⟩ for small k_z results from the finite Rashba couplings near the bottom gate, where the wave function is localized and the almost complete z-spin polarization of electrons induced by the magnetic field. As a consequence, ⟨ g_SO^zz⟩ is independent of the magnetic field magnitude at k_z=0,.
On the other hand, for a large value of k_z and low magnetic field, the SO coupling plays a major role, forcing the electron spin to align along the effective Rashba field directed in the x axis. In this scenario, the spin-up and spin-down components of the spinor become almost equal, resulting in a decrease in ⟨ g_SO^zz⟩. It is worth noting that even for a large k_z and strong SO coupling, an increasing magnetic field can deviate the electron spin direction from the x towards the z axis, leading to an overall increase in ⟨ g_SO^zz⟩ with the magnetic field, as depicted in Fig. <ref>.
Finally, note that results presented in Fig. <ref> for the magnetic field directed along the z-axis at k_z=0 corresponds to the physical situation considered theoretically in Ref. Winkler2017, where the enhancement of the effective factor has been recently predicted in semiconductor NWs. The predicted effect was however restricted to the higher subbands characterized by the large orbital momentum. Here, we show that the enhancement of g^* can be also observed for the lowest state for which it can be induced by the orbital effects form the spin-orbital term.
To summarize this section, we conclude that the significant enhancement of the effective factor observed in recent experiments can be explained as resulting from the orbital effects from the SO coupling. Our results demonstrate that this enhancement is most pronounced when the magnetic field is directed perpendicular α_R. In Fig. <ref> we show the gate voltage dependence of ⟨ g_SO^xx⟩, calculated for a magnetic field directed along the x axis with B_x = 1 T. It can be observed that the inclusion of the SO effects may lead to a substantial increase of the effective factor 𝐠_SO, reaching up to three times the value obtained from the RLZ formula.
§.§ Spin-orbital induced factor anisotropy
We next analyze the anisotropy of 𝐠_SO with respect to the field direction. For this purpose we consider a magnetic field with intensity B=1 T rotated in (i) the xz plane (φ = 0), (ii) the xy plane (θ = π/2) and (iii) the yz plane (φ = π /2). To induce Rashba SO coupling we apply a gate voltage V_g = 0.2 V.
Figures <ref>(a),(b) show maps of ⟨ g_SO^xx⟩ and ⟨ g_SO^zz⟩ as a function of the wave vector k_z and θ when the magnetic field is rotated in the xz plane. The black region on the right sides of both panels originates from the localization of the wave function far away from the bottom gate, in the region where the SO coupling is weak. This is apparent in Fig. <ref>, which shows the squared wave function for k_z = 0.4 nm^-1 under different magnetic field orientations.
Interestingly, we observe unusual behavior in the region where ⟨ g_SO^xx⟩ changes sign. As discussed earlier, when the magnetic field is directed along the x-axis, this sign change is due to subband crossing. However, here the finite z-component of the magnetic field, perpendicular to the effective Rashba field, causes anticrossing of the subbands. The magnitude and position of the anticrossing in wave vector space depend on the orientation of 𝐁. The behavior of ⟨ g_SO^xx⟩ damping to zero at the sign change region, accompanied by a maximum in |⟨ g_SO^zz⟩ |, can be explained by considering the evolution of electron spin at the anticrossing. Figure <ref> presents the z-spin polarization of the lowest subbands, defined as P = ∫ (|ψ^↑_k_z(x,y)|^2 - |ψ^↓_k_z(x,y)|^2) dxdy, as a function of k_z for different angles, θ. We observe that at the anticrossing, the states become completely z-spin polarized, which maximizes |⟨ g_SO^zz⟩|. Simultaneously, the average value of σ_x, which determines |⟨ g_SO^xx⟩| [see Eq. (<ref>)], becomes zero, which explains its vanishing for a specific k_z vector.
The evolution of the SO-induced factor in the other rotation planes, as depicted in Figs. <ref>(c-i), is in general a result of the interplay between the wavefunction localization, which is determined by orbital effects, and the electron spin direction, which is defined by both the SO interaction and the external magnetic field. It is worth noting that when the magnetic field has a component along the y or the z axis, the off-diagonal elements of the 𝐠_SO tensor may also contribute significantly to the effective factor - the magnitudes of ⟨ g_SO^xy(yx)⟩ in Figs. <ref>(d,e,g) are comparable to those of the diagonal elements.
Although the maps of the 𝐠_SO tensor elements presented so far provide valuable information and offer a precise representation of the physical phenomena underlying their evolution, it becomes challenging to directly compare them with results of recent experimental evidence. In experiments, the k_z vector is often not well-defined, and what is typically obtained is an average value of over all electronic states involved in the transport. For this reason we define the mean value of 𝐠_SO tensor elements averaged over all occupied states
g^ab_SO = ∑_k_z |⟨ g^ab_SO(k_z) ⟩ | f(E_n=1,k_z - μ, T)/∑_k_z f(E_n=1,k_z - μ, T),
where a,b={x,y,z}.
Such an approach has been recently used for analyzing the SO coupling in NWs and good agreement with experiments has been obtained.<cit.>
In Fig. <ref> we show the mean value of the tensor elements g^ab_SO and the Rashba SO constant α^y_R (defined in the same manner) for three different rotation planes. We observe that irrespective of the rotation plane, all elements g^ab_SO exhibit strong anisotropy with a two-fold symmetry, closely corresponding to the evolution of the SO coupling, shown in Fig. <ref>(d-f) (with a bottom gate α^x_R=0 due to the symmetry along the y axis and it is not shown). A similar two-fold symmetry with respect to the magnetic field direction has been recently observed in the Rashba SO coupling measured for suspended InAs NW.<cit.> In both cases, the symmetry arises from the bottom gate architecture, which induces a large SO coupling near the bottom facet, while the rotating magnetic field alters localization of the wave function, due to the orbital effects. Since g_SO^zz is rather sensitive to the spin polarization of electronic states than the orbital effects, we do not observe a direct correspondence between g_SO^zz and α^y_R - compare Figs. <ref>(c,f).
It is noteworthy that g_SO^xx remains the most robust against the rotation in the xy plane [see Fig. <ref>(b)], and it dominates over other terms for the considered gate setup. This can be attributed to the large coupling constant α^y_R induced by the bottom gate voltage and the broken symmetry with respect to the x-axis – see Eq. (<ref>). Finally, it should be emphasized that the off-diagonal tensor components are one order of magnitude smaller than the diagonal ones. This observation holds true for the considered bottom gate configuration, which preserves symmetry around the y axis, but it may differ for more sophisticated gate configurations as presented in the next subsection.
§.§ Different gate configuration
In order to analyze in detail the magnitude of the off-diagonal elements of the 𝐠_SO tensor let us now consider an asymmetric gate configuration with two gates attached to the top and left-top facet. In this case the voltage applied to the gates generate both the x and y component of the Rashba SO coupling - see Fig. <ref>(d-f). In particular, the negative voltage generates the effective band bending near the gates similar to that observed in the Majorana NWs at the superconductor/semiconductor interface. Thus, in some sense the presented architecture can be treated as a first approximation of typical Majorana NWs with a superconducting shell covered the top and left-top facet, but with one important difference, i.e. that the factor in the Majorana NWs are additionally modulated by the presence of metallic shell.<cit.>
As shown in Fig. <ref>(a), in this configuration, the off-diagonal elements of 𝐠_SO are of the same order of magnitude as the diagonal elements. This additional contribution plays a role in enhancing the overall effective factor. While the general principle that the largest SO-induced factor occurs when the magnetic field is perpendicular to α _R is observed also for the this gate configuration, it is remarkable that even for the magnetic field aligned along the NW axis, the configuration relevante to Majorana experiments, there is a significant enhancement in g_SO^zz. Consequently, we believe that our model, when applied to higher gate voltages, can account for the observed twofold enhancement of the effective factor, as recently observed in Majorana NWs.<cit.>
§ SUMMARY
Based on the 𝐤·𝐩 theory within the envelope function approximation, we have analyzed the effective factor induced by the SO coupling in homogeneous semiconductor NWs under different magnetic field and gate configurations. By considering the orbital effects in the kinetic and SO terms, we have obtained the 𝐠_SO tensor and studied its elements with respect to the magnetic field magnitude and orientation.
Our findings demonstrate that the effective factor induced by SO interaction is proportional to the Rashba coupling constant, which arises from the electric field generated by the adjacent gates. We have found that 𝐠_SO is determined by two factors: 1) position and symmetry of the electron’s wave function, which can be tuned by the orbital effects, 2) the z spin polarization of the electronic state. Specifically, when we apply the magnetic field perpendicular to NW, the inversion symmetry of the envelope functions is broken and the wave function is squeezed to the NW surface by a k_z-dependent effective potential. This effect results in an enhancement of 𝐠_SO in a situation when the envelope function is squeezed to the facet near the gate where the electric field and consequently the Rashba SO coupling is larger. The opposite magnetic field (or k_z) results in the squeezing of wave function to the opposite facet where electric field from the gate and the corresponding SO coupling is weak, which results in nearly zero 𝐠_SO.
On the other hand, for 𝐁 directed along the NW axis the orbital effects are strongly reduced by the confinement and 𝐠_SO depends on the z component of spin polarization, which is a resultant of the magnetic and effective Rashba field. Our results explains the recently demonstrated enhancement of the effective factor observed in semiconductor NWs as well as its anisotropy.<cit.>
Note that although our simulations have been limited to the regime where only the lowest subband is occupied, from our previous papers we expect that the electron-electron interaction, here introduced at the mean-field level, could be essential in estimating factor, via charge localization. At the high concentration regime total energy is minimized by reducing repulsive Coulomb energy, moving electrons outwards, and charge localizes at the six quasi-1D channels at the edges. As we discussed in Ref. Wojcik_anizotropy, this strong localization is almost insensitive to the gate potential and the magnetic field direction.
Finally, we would like to underline that our model does not include the hole bands coupling expressed in the 𝐤·𝐩 model by the Lüttinger parameters.<cit.> Note however, that as recently shown in Ref. Escribano the applied conduction band approximation underestimates the SO coupling constant for the considered zinc-blende crystal structure. As the considered SO induced factor depends on the Rashba SO constants, we expect that the renormalization of the effective observed in the experiments should be even greater than predicted by our results.
§ ACKNOWLEDGEMENT
The work was supported in part by PL-Grid Infrastructure, grant no. PLG/2022/015712.
*
§ SIZE DEPENDENCE
Calculations presented in the paper have been carried out for the NW width W=100 nm for two reasons. First, it is a typical diameter of NWs fabricated by the commonly used fabrication methods and second, for this range of NW width, orbital effects considered here become significant. For completeness, in Fig. <ref> we present g_SO^xx and g_SO^zz calculated with a magnetic field along the x and z directions, respectively. As expected, for a small diameter, when the orbital effect are highly reduced, the spin-orbital induced factor approaches zero, which shows that the predicted enhancement of is observable only for NWs of moderate or large width.
49
fxundefined [1]
ifx#1
fnum [1]
#1firstoftwo
secondoftwo
fx [1]
#1firstoftwo
secondoftwo
noop [0]secondoftwo
ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]
rl [1]href #1
@bib@innerbibempty
[Reimer et al.(2011)Reimer,
van Kouwen, Barkelind, Hocevar, van Weert, Algra, Bakkers, Björk, Schmid, Riel, Kouwenhoven, and Zwiller]Reimer
author author M. E. Reimer, author M. P. van
Kouwen, author M. Barkelind,
author M. Hocevar, author M. H. M. van Weert, author R. E. Algra, author
E. P. A. M. Bakkers, author
M. T. Björk, author
H. Schmid, author H. Riel, author L. P. Kouwenhoven, and author V. Zwiller, @noop journal journal J. Nanophotonics volume 5, pages 053502 (year 2011)NoStop
[Stettner et al.(2016)Stettner, Zimmermann, Loitsch,
Döblinger, Regler, Mayer,
Winnerl, Matich, Riedl,
Kaniber, Abstreiter, Koblmüller, , and Finley]Stettner
author author T. Stettner, author P. Zimmermann, author B. Loitsch,
author M. Döblinger, author A. Regler, author
B. Mayer, author J. Winnerl, author S. Matich, author H. Riedl, author M. Kaniber, author G. Abstreiter, author G. Koblmüller, , and author J. J. Finley, @noop journal
journal Appl. Phys. Lett. volume
108, pages 011108 (year 2016)NoStop
[Li et al.(2006)Li,
Qian, Xiang, and Lieber]Li
author author Y. Li, author F. Qian, author J. Xiang, and author
C. M. Lieber, @noop journal journal Materials Today volume
9, pages 18 (year 2006)NoStop
[Czaban et al.(2009)Czaban,
Thompson, and LaPierre]Czaban2009
author author J. A. Czaban, author D. A. Thompson, and author R. R. LaPierre, @noop journal journal Nano
Lett. volume 9, pages 148 (year 2009)NoStop
[Nadj-Perge et al.(2010)Nadj-Perge, Frolov, Bakkers, and Kouwenhoven]NadjPerge
author author S. Nadj-Perge, author S. Frolov,
author E. P. A. M. Bakkers, and author L. P. Kouwenhoven, @noop journal journal Nature volume 468, pages 1084 (year
2010)NoStop
[Frolov et al.(2013)Frolov,
Plissard, Nadj-Perge, Kouwenhoven, and Bakkers]Frolov
author author S. M. Frolov, author S. R. Plissard, author S. Nadj-Perge, author L. P. Kouwenhoven, and author E. P. A. M. Bakkers, @noop journal journal MRS Bulletin volume 38, pages 809 (year 2013)NoStop
[Schroer et al.(2011)Schroer, Petersson, Jung, and Petta]Schroer
author author M. D. Schroer, author K. D. Petersson, author M. Jung, and author J. R. Petta, 10.1103/PhysRevLett.107.176811 journal journal Phys. Rev. Lett. volume 107, pages 176811 (year 2011)NoStop
[Pribiag et al.(2013)Pribiag, Nadj-Perge, Frolov, van den Berg, van Weperen, Plissard,
Bakkers, and Kouwenhoven]Pribiag
author author V. S. Pribiag, author S. Nadj-Perge,
author S. M. Frolov, author I. van den Berg, J W G., author I. van Weperen,
author S. R. Plissard, author E. P. A. M. Bakkers, and author L. P. Kouwenhoven, @noop journal journal Nature
Nanotechnology volume 8, pages 170
(year 2013)NoStop
[Miladi ćć et al.(2020)Miladi ćć,
Stipsi ćć, Dobardžži ćć, and Milivojevi ćć]Miladi
author author S. Miladi ćć, author P. Stipsi ćć, author E. Dobardžži ćć, and author
M. Milivojevi ćć, 10.1103/PhysRevB.101.155307 journal journal Phys. Rev. B volume
101, pages 155307 (year 2020)NoStop
[Nadj-Perge et al.(2012)Nadj-Perge, Pribiag, van den Berg,
Zuo, Plissard, Bakkers,
Frolov, and Kouwenhoven]NadjPergePRL
author author S. Nadj-Perge, author V. S. Pribiag, author J. W. G. van den Berg, author K. Zuo,
author S. R. Plissard, author E. P. A. M. Bakkers, author S. M. Frolov, and author L. P. Kouwenhoven, 10.1103/PhysRevLett.108.166801 journal journal Phys. Rev. Lett. volume 108, pages 166801 (year 2012)NoStop
[Wójcik et al.(2014)Wójcik, Adamowski, Spisak, and Wołoszyn]Wojcik2014
author author P. Wójcik, author J. Adamowski,
author B. J. Spisak, and author M. Wołoszyn, @noop
journal journal Journal of Applied Physics volume 115, pages 104310 (year 2014)NoStop
[Dresselhaus(1955)]Dresselhaus
author author G. Dresselhaus, 10.1103/PhysRev.100.580 journal journal Phys. Rev. volume
100, pages 580 (year 1955)NoStop
[Rashba(1960)]Rashba
author author E. I. Rashba, @noop journal journal Phys.
Solid State volume 2, pages 1109
(year 1960)NoStop
[Campos et al.(2018)Campos,
Faria Junior, Gmitra, Sipahi, and Fabian]Campos
author author T. Campos, author P. E. Faria Junior, author M. Gmitra, author G. M. Sipahi,
and author J. Fabian, 10.1103/PhysRevB.97.245402 journal journal Phys. Rev. B volume 97, pages 245402 (year 2018)NoStop
[Kokurin(2015)]Kokurin2015
author author I. A. Kokurin, @noop journal journal
Physica E volume 74, pages 264
(year 2015)NoStop
[Kokurin(2014)]Kokurin2014
author author I. A. Kokurin, @noop journal journal Solid
State. Commun. volume 195, pages 49
(year 2014)NoStop
[Wójcik et al.(2018)Wójcik, Bertoni, and Goldoni]Wojcik2018
author author P. Wójcik, author A. Bertoni, and author G. Goldoni, 10.1103/PhysRevB.97.165401 journal journal Phys. Rev. B volume 97, pages 165401 (year 2018)NoStop
[Wójcik et al.(2021)Wójcik, Bertoni, and Goldoni]Wojcik2021
author author P. Wójcik, author A. Bertoni, and author G. Goldoni, https://link.aps.org/doi/10.1103/PhysRevB.103.085434 journal
journal Phys. Rev. B volume 103, pages 085434 (year 2021)NoStop
[Escribano et al.(2020)Escribano, Yeyati, and Prada]Escribano
author author S. D. Escribano, author A. L. Yeyati, and author E. Prada, 10.1103/PhysRevResearch.2.033264 journal journal Phys. Rev. Res. volume
2, pages 033264 (year 2020)NoStop
[Wójcik et al.(2019)Wójcik, Bertoni, and Goldoni]Wojcik2019
author author P. Wójcik, author A. Bertoni, and author G. Goldoni, @noop journal journal Applied Physics
Letters volume 114, pages 073102
(year 2019)NoStop
[Furthmeier et al.(2016)Furthmeier, Dirnberger, Gmitra,
Bayer, Forsch, Hubmann,
Schüller, Reiger, Fabian,
Korn, and Bougeard]Furthmeier
author author S. Furthmeier, author F. Dirnberger, author M. Gmitra,
author A. Bayer, author M. Forsch, author
J. Hubmann, author C. Schüller, author E. Reiger, author J. Fabian, author T. Korn, and author D. Bougeard, @noop journal
journal Nat Commun. volume 7, pages 12413 (year 2016)NoStop
[van Weperen et al.(2015)van
Weperen, Tarasinski, Eeltink, Pribiag, Plissard, Bakkers, Kouwenhoven, and Wimmer]vanWeperen2015
author author I. van
Weperen, author B. Tarasinski,
author D. Eeltink, author V. S. Pribiag, author
S. R. Plissard, author
E. P. A. M. Bakkers, author
L. P. Kouwenhoven, and author
M. Wimmer, @noop journal journal Phys. Rev. B volume
91, pages 201413(R) (year 2015)NoStop
[Kammhuber et al.(2017)Kammhuber, Cassidy, Pei, Nowak, Vuik, Car, Plissard,
Bakkers, Wimmer, and Kouwenhoven]Kammhuber2017
author author J. Kammhuber, author M. C. Cassidy, author F. Pei,
author M. P. Nowak, author A. Vuik, author
D. Car, author S. R. Plissard, author E. P.
A. M. Bakkers, author M. Wimmer, and author L. P. Kouwenhoven, @noop journal journal Nat Commun. volume 8, pages
478 (year 2017)NoStop
[Dhara et al.(2009)Dhara,
Solanki, Singh, Narayanan,
Chaudhari, Gokhale, Bhattacharya, and Deshmukh]Dhara2009
author author S. Dhara, author H. S. Solanki,
author V. Singh, author A. Narayanan, author
P. Chaudhari, author
M. Gokhale, author A. Bhattacharya, and author M. M. Deshmukh, 10.1103/PhysRevB.79.121311 journal journal Phys.
Rev. B volume 79, pages 121311
(year 2009)NoStop
[Scherübl et al.(2016)Scherübl, Fülöp, Madsen,
Nygård, and Csonka]Scherubl2016
author author Z. Scherübl, author G. m. H. Fülöp, author M. H. Madsen, author J. Nygård,
and author S. Csonka, 10.1103/PhysRevB.94.035444 journal journal Phys. Rev. B volume 94, pages 035444 (year 2016)NoStop
[Liang and Gao(2012)]Liang2012
author author D. Liang and author X. P. A. Gao, @noop journal journal Nano
Lett. volume 12, pages 3263–3267
(year 2012)NoStop
[Gazibegovic et al.(2017)Gazibegovic, Car, Zhang, Balk, Logan, de Moor, Cassidy, Schmits, Xu, Wang,
Krogstrup, Op het Veld, Zuo,
Vos, Shen, Bouman,
Shojaei, Pennachio, Lee,
van Veldhoven, Koelling, Verheijen, Kouwenhoven, Palmstrøm, and Bakkers]gazibegovic_epitaxy_2017
author author S. Gazibegovic, author D. Car,
author H. Zhang, author S. C. Balk, author
J. A. Logan, author
M. W. A. de Moor, author
M. C. Cassidy, author
R. Schmits, author D. Xu, author G. Wang, author P. Krogstrup,
author R. L. M. Op het Veld,
author K. Zuo, author
Y. Vos, author J. Shen, author D. Bouman, author B. Shojaei,
author D. Pennachio, author J. S. Lee, author
P. J. van Veldhoven, author
S. Koelling, author
M. A. Verheijen, author
L. P. Kouwenhoven, author
C. J. Palmstrøm, and author E. P. A. M. Bakkers, 10.1038/nature23468 journal journal
Nature volume 548, pages 434
(year 2017)NoStop
[Krogstrup et al.(2015)Krogstrup, Ziino, Chang, Albrecht, Madsen, Johnson, Nygård, Marcus, and Jespersen]krogstrup_epitaxy_2015
author author P. Krogstrup, author N. L. B. Ziino, author W. Chang,
author S. M. Albrecht, author M. H. Madsen, author
E. Johnson, author J. Nygård, author C. M. Marcus, and author T. S. Jespersen, 10.1038/nmat4176 journal journal Nat. Mater. volume
14, pages 400 (year 2015)NoStop
[Chang et al.(2015)Chang,
Albrecht, Jespersen, Kuemmeth, Krogstrup, Nygård, and Marcus]chang_hard_2015
author author W. Chang, author S. M. Albrecht,
author T. S. Jespersen, author F. Kuemmeth, author
P. Krogstrup, author
J. Nygård, and author
C. M. Marcus, 10.1038/nnano.2014.306 journal journal Nat.
Nano. volume 10, pages 232 (year 2015)NoStop
[Kjaergaard et al.(2016)Kjaergaard, Nichele, Suominen,
Nowak, Wimmer, Akhmerov,
Folk, Flensberg, Shabani,
Palmstrøm, and Marcus]kjaergaard_quantized_2016
author author M. Kjaergaard, author F. Nichele,
author H. J. Suominen, author M. P. Nowak, author
M. Wimmer, author A. R. Akhmerov, author J. A. Folk, author K. Flensberg, author J. Shabani,
author C. J. Palmstrøm, and author C. M. Marcus, 10.1038/ncomms12841 journal journal Nat. Commun. volume 7, pages
12841 (year 2016)NoStop
[Mourik et al.(2012)Mourik,
Zuo, Frolov, Plissard,
Bakkers, and Kouwenhoven]mourik_signatures_2012
author author V. Mourik, author K. Zuo,
author S. M. Frolov, author S. R. Plissard, author
E. P. a. M. Bakkers, and author L. P. Kouwenhoven, 10.1126/science.1222360 journal journal
Science volume 336, pages 1003
(year 2012)NoStop
[Deng et al.(2012)Deng,
Yu, Huang, Larsson,
Caroff, and Xu]deng_anomalous_2012
author author M. T. Deng, author C. L. Yu,
author G. Y. Huang, author M. Larsson, author
P. Caroff, and author
H. Q. Xu, 10.1021/nl303758w journal journal Nano Lett. volume 12, pages 6414 (year
2012)NoStop
[Albrecht et al.(2016)Albrecht, Higginbotham, Madsen,
Kuemmeth, Jespersen, Nygård, Krogstrup, and Marcus]albrecht_exponential_2016
author author S. M. Albrecht, author A. P. Higginbotham, author M. Madsen, author F. Kuemmeth,
author T. S. Jespersen, author J. Nygård, author
P. Krogstrup, and author
C. M. Marcus, 10.1038/nature17162 journal journal Nature volume 531, pages 206 (year
2016)NoStop
[Zhang et al.(2018)Zhang,
Liu, Gazibegovic, Xu,
Logan, Wang, van Loo,
Bommer, de Moor, Car,
Veld, van Veldhoven, Koelling, Verheijen, Pendharkar,
Pennachio, Shojaei, Lee,
Palmstrom, Bakkers, Sarma, and Kouwenhoven]zhang_quantized_2017
author author H. Zhang, author C.-X. Liu,
author S. Gazibegovic, author D. Xu, author
J. A. Logan, author
G. Wang, author N. van Loo, author J. D. S. Bommer, author M. W. A. de Moor, author D. Car, author R. L. M. O. h. Veld, author P. J. van
Veldhoven, author S. Koelling,
author M. A. Verheijen, author M. Pendharkar, author
D. J. Pennachio, author
B. Shojaei, author J. S. Lee, author C. J. Palmstrom, author E. P. A. M. Bakkers, author S. D. Sarma, and author L. P. Kouwenhoven, @noop journal journal Nature volume 556, pages
74 (year 2018)NoStop
[Finck et al.(2013)Finck,
Van Harlingen, Mohseni, Jung, and Li]finck_anomalous_2013
author author A. D. K. Finck, author D. J. Van Harlingen, author P. K. Mohseni, author K. Jung, and author X. Li, 10.1103/PhysRevLett.110.126406 journal journal Phys. Rev. Lett. volume 110, pages 126406 (year 2013)NoStop
[Oreg et al.(2010)Oreg,
Refael, and von Oppen]oreg_helical_2010
author author Y. Oreg, author G. Refael, and author F. von Oppen, 10.1103/PhysRevLett.105.177002 journal journal Phys. Rev. Lett. volume 105, pages 177002 (year 2010)NoStop
[Sau et al.(2010)Sau,
Lutchyn, Tewari, and Das Sarma]sau_generic_2010
author author J. D. Sau, author R. M. Lutchyn,
author S. Tewari, and author S. Das Sarma, 10.1103/PhysRevLett.104.040502 journal journal
Phys. Rev. Lett. volume 104, pages
040502 (year 2010)NoStop
[Lutchyn et al.(2010)Lutchyn, Sau, and Das Sarma]Lutchyn
author author R. M. Lutchyn, author J. D. Sau, and author S. Das Sarma, 10.1103/PhysRevLett.105.077001 journal journal Phys. Rev. Lett. volume 105, pages 077001 (year 2010)NoStop
[Roth et al.(1959)Roth,
Lax, and Zwerdling]Roth-Lax
author author L. Roth, author B. Lax, and author S. Zwerdling, @noop
journal journal Phys. Rev. volume 114, pages 90 (year
1959)NoStop
[Lommer et al.(1985)Lommer,
Malcher, and Rössler]Lommer
author author G. Lommer, author F. Malcher, and author U. Rössler, 10.1103/PhysRevB.32.6965 journal journal Phys. Rev. B volume 32, pages 6965 (year 1985)NoStop
[Kiselev et al.(1998)Kiselev, Ivchenko, and Rössler]Kiselev
author author A. A. Kiselev, author E. L. Ivchenko, and author U. Rössler, 10.1103/PhysRevB.58.16353 journal journal Phys. Rev. B volume
58, pages 16353 (year 1998)NoStop
[Gawarecki and Zieliński(2020)]Gawarecki2020
author author K. Gawarecki and author M. Zieliński, @noop journal journal
Scientific Reports volume 10, pages
22001 (year 2020)NoStop
[vanWeperen et al.(2013)vanWeperen, Plissard, Bakkers,
Frolov, and Kouwenhoven]vanWeperen2013
author author I. vanWeperen, author S. R. Plissard, author E. P. A. M. Bakkers, author S. M. Frolov,
and author L. P. Kouwenhoven, @noop journal journal
Nano Lett. volume 13, pages 387
(year 2013)NoStop
[Vaitiek ėėnas et al.(2018)Vaitiek ėėnas,
Deng, Nygård, Krogstrup, and Marcus]Marcus2018
author author S. Vaitiek ėėnas, author
M.-T. Deng, author
J. Nygård, author
P. Krogstrup, and author
C. M. Marcus, 10.1103/PhysRevLett.121.037703 journal journal
Phys. Rev. Lett. volume 121, pages
037703 (year 2018)NoStop
[Winkler et al.(2017)Winkler, Varjas, Skolasinski, Soluyanov, Troyer, and Wimmer]Winkler2017
author author G. W. Winkler, author D. Varjas,
author R. Skolasinski, author A. A. Soluyanov, author
M. Troyer, and author
M. Wimmer, 10.1103/PhysRevLett.119.037701 journal journal
Phys. Rev. Lett. volume 119, pages
037701 (year 2017)NoStop
[Bertoni et al.(2011)Bertoni, Royo, Mahawish, and Goldoni]Bertoni2011
author author A. Bertoni, author M. Royo,
author F. Mahawish, and author G. Goldoni, 10.1103/PhysRevB.84.205323 journal journal Phys. Rev. B volume 84, pages 205323 (year 2011)NoStop
[Iorio et al.(2019)Iorio,
Rocci, Bours, Carrega,
Zannier, Sorba, Roddaro,
Giazotto, and Strambini]Ioro
author author A. Iorio, author M. Rocci,
author L. Bours, author M. Carrega, author
V. Zannier, author L. Sorba, author S. Roddaro, author F. Giazotto, and author E. Strambini, https://doi.org/10.1021/acs.nanolett.8b02828 journal
journal Nano Letters volume 19, pages 652 (year 2019), http://arxiv.org/abs/https://doi.org/10.1021/acs.nanolett.8b02828
https://doi.org/10.1021/acs.nanolett.8b02828 NoStop
[Wójcik et al.(2021)Wójcik, Bertoni, and Goldoni]Wojcik_anizotropy
author author P. Wójcik, author A. Bertoni, and author G. Goldoni, 10.1103/physrevb.103.085434 journal journal Physical Review B volume 103 (year 2021), 10.1103/physrevb.103.085434NoStop
[Vezzosi et al.(2022)Vezzosi, Bertoni, and Goldoni]Vezzosi2022
author author A. Vezzosi, author A. Bertoni, and author G. Goldoni, 10.1103/PhysRevB.105.245303 journal journal Phys. Rev. B volume 105, pages 245303 (year 2022)NoStop
|
http://arxiv.org/abs/2307.04415v1 | 20230710084328 | Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors | [
"Armin Lederer",
"Jonas Umlauft",
"Sandra Hirche"
] | eess.SY | [
"eess.SY",
"cs.LG",
"cs.SY",
"stat.ML"
] |
Journal of Class Files, Vol. 14, No. 8, August 2021
Shell et al.: A Sample Article Using IEEEtran.cls for IEEE Journals
Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors
Armin Lederer, Graduate Student Member, IEEE,
Jonas Umlauft,
Sandra Hirche, Fellow, IEEE,
Armin Lederer, Jonas Umlauft and Sandra Hirche are with the Chair of Information-oriented Control (ITR), School of Computation, Information and Technology, Technical University of Munich, 80333 Munich, Germany (email: armin.lederer, jonas.umlauft, [email protected]).
Received / Accepted
================================================================================================================================================================================================================================================================================================================================================================================
Due to the increasing complexity of technical systems, accurate first principle
models can often not be obtained. Supervised machine learning can mitigate this
issue by inferring models from measurement data. Gaussian process regression
is particularly well suited for this purpose due to its high data-efficiency
and its explicit uncertainty representation, which allows the derivation of
prediction error bounds. These error bounds have been exploited to show tracking accuracy
guarantees for a variety of control approaches, but their direct dependency on
the training data is generally unclear. We address this issue by deriving a
Bayesian prediction error bound for GP regression, which we show to
decay with the growth of a novel, kernel-based measure of data density. Based
on the prediction error bound, we prove time-varying tracking accuracy guarantees
for learned GP models used as feedback compensation of unknown nonlinearities,
and show to achieve vanishing tracking error with increasing
data density. This enables us to develop an episodic approach for learning
Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed.
The effectiveness of the derived theory is demonstrated in several simulations.=-1
Gaussian processes, machine learning, uncertain systems, data-driven control.
§ INTRODUCTION
For many technical systems, no or only partial first principle models are available due to their complexity or a priori unknown operating conditions. Since measurement data of such systems can typically be obtained, inferring models using supervised machine learning techniques has become increasingly popular in recent years <cit.>. In particular, Gaussian process (GP) regression <cit.> is a popular method since it is very data-efficient <cit.> and exhibits closed-form expressions for model updates allowing on-line learning <cit.>. Moreover, GP models provide an explicit measure for prediction uncertainty, which enables the confidence-based distributed aggregation of GP models <cit.>, and allows to tune the behavior of control towards curiosity <cit.> or cautiousness <cit.>.
In addition to these beneficial properties, GP regression is particularly appreciated in safety-critical control due to the existence of prediction error bounds <cit.>. These bounds are typically based on the close relationship between kernel methods
and GPs <cit.>, such that the reproducing kernel Hilbert space norm induced by the GP can be used as a measure of function complexity. By combining bounds on this norm and assumptions about observation noise distributions, statistical prediction error bounds
can be derived <cit.>. They can be efficiently computed on-line in an optimization-based fashion <cit.>, but data-dependent closed-form expressions also exist <cit.>. Moreover, they reduce to deterministic bounds when the observation noise is bounded <cit.>.
Based on the prediction error bounds for learned GP models, tracking accuracy guarantees for a large variety of control laws have been derived. This can be achieved using Lyapunov theory, e.g., for feedback linearization <cit.>, computed torque control <cit.> and sliding mode control <cit.>, by extending stability properties of nominal model predictive control, e.g., using continuity arguments <cit.>, or robust linear control, e.g., through integral quadratic constraints <cit.>. However, these approaches suffer from the crucial drawback that accuracy guarantees are global, even though the prediction error bounds from GP models are state-dependent. Therefore accuracy guarantees can be very loose
in cases with inhomogeneously distributed training data over the state space. In such a case, the guarantees would be dominated globally by the most conservative bound derived from the region with the fewest training data.
In general, the data dependency of such accuracy guarantees for model-based control methods has barely been analyzed in detail. While it can
be shown for feedback linearization with event-triggered on-line learning that the tracking error vanishes
with growing noise-free data set <cit.>, similar results for noisy data do not exist. Moreover, this result is limited to feedback linearizing controllers to the best of our knowledge and does not extend to other approaches. Finally, on-line learning with GPs can be realized using suitable approximations in principle <cit.>, but it remains computationally expensive, such that it is not applicable to systems with limited computational resources. The computationally less demanding approach of episodic, off-line learning has been investigated in the context of optimization-based controller tuning approaches <cit.>, which can be shown to provide data-dependent performance guarantees due to the close relationship to Bayesian optimization <cit.>.
While these guarantees can be extended to model-based
reinforcement learning <cit.>, they strongly rely on the solved optimization problems, such that they do not generalize to a wider class of control techniques. Therefore, no guarantees and conditions for the convergence of accuracy guarantees for model-based control laws employing GP models exist to the best of our knowledge. Consequently, it is an open question how we can learn a GP model in order to ensure a desired tracking error bound with such learning-based controllers.
§.§ Contribution and Structure
The main contribution of this article is a novel episodic learning approach
for GP models in order to ensure arbitrary tracking accuracy when the GP is used
to compensate unknown nonlinearities in control. Such nonlinearities can be found in a wide range of applications ranging
from underwater vehicles, where unmodeled hydrodynamic forces due to currents can appear <cit.>, to physical human-robot interaction, where humans introduce generally unknown torques <cit.>.
For the development of this approach, we first derive an easily interpretable prediction error bound
for GPs by exploiting their Bayesian foundations.
In order to allow its straightforward computation, we
provide probabilistic Lipschitz bounds for unknown
functions based on the GP prior.
Based on these results, we propose a kernel-based measure to evaluate the training data
density, whose flexibility we demonstrate by exemplarily illustrating it for squared
exponential (SE), Matérn class and linear kernels. Moreover, we show that prediction error
bounds directly depend on this data density measure, which allows us to prove vanishing prediction
errors with growing data density. Based on this analysis of the GP prediction error, we derive
a novel, data density-dependent tracking error bound for
control laws in linear systems which employ
the GP model for compensation of an unknown nonlinearity.
Finally, we extend these accuracy guarantees to establish a direct relationship with
the proposed data density measure, which allows us to develop an episodic approach
for learning a GP model ensuring a specified tracking error bound.
This article is based on our prior work <cit.>, which purely focuses on the derivation
of probabilistic prediction error bounds depending on the posterior variance of Gaussian processes. It
significantly extends these preliminary results by establishing a direct relationship between the training data
density and prediction error bounds. Due to this relationship, we can bound the tracking error of linear systems with an unknown
nonlinearity compensated by a learned model directly in terms of the data density. This allows
us to actively generate training data for achieving arbitrary tracking accuracy in an episodic approach, while <cit.>
only bounds the tracking error of feedback linearizing controllers with models learned from a given data set. Therefore, we extend
the analysis framework from our prior work <cit.> to a design method.
The remainder of this article is structured as follows: We briefly introduce
Gaussian process regression and formalize the considered problem setting in
<ref>. In <ref>, we derive a novel Bayesian
prediction error bound for GP regression and provide methods to determine all relevant
parameters based on the prior distribution. We develop a kernel-dependent
measure of data density and establish a straightforward relationship to the GP variance, which
allows us to investigate the asymptotic
behavior of the error bound with increasing data set size in <ref>.
In <ref>, we exploit these results to derive time-varying and time-independent tracking
error guarantees, which we exploit to develop a novel episodic learning algorithm for
ensuring arbitrary tracking accuracy. Finally, in <ref>, we evaluate the developed theoretical
framework in different simulations to demonstrate its effectiveness, before we conclude the
paper in <ref>.
§.§ Notation
Vectors/matrices
are denoted by lower/upper case bold symbols, the
n× n identity matrix by I_n, the Euclidean norm
by ·, and λ_min(A) and λ_max(A) the
minimum and maximum real parts of the eigenvalues of a matrix A, respectively. Sets are
denoted by upper case black board bold letters, and sets restricted to positive/non-negative
numbers have an indexed +/+,0, e.g., ℝ_+ for
all positive real valued numbers. The cardinality of
sets is denoted by |·| and subsets/strict subsets are indicated by .
Class 𝒪 notation is used to provide
asymptotic upper bounds on functions.
The ceil and floor
operator are denoted by ⌈·⌉ and ⌊·⌋,
respectively. The Gaussian distribution with mean μ∈ℝ and
variance σ^2∈ℝ_+ is denoted by 𝒩(μ,σ^2).
A chi-squared distribution with N degrees of freedom is denoted by χ^2_N.
The expectation operator E[·] can have an additional index to specify
the considered random variable. Finally, a function α:ℝ_0,+→ℝ_0,+
is in class 𝒦_∞ if it is monotonically increasing and α(0)=0, lim_x→∞α(x)=∞.
=-1
§ PRELIMINARIES AND PROBLEM SETTING
In this paper, we consider the problem of controlling linear systems perturbed by an unknown nonlinearity such that they track reference trajectories with a prescribed accuracy. In order to achieve this, we employ models learned via Gaussian process regression as compensation. Therefore, we first introduce the fundamentals of Gaussian process regression in <ref>, before we formalize the problem setting in <ref>.
§.§ Gaussian Process Regression
A Gaussian process is a stochastic process such
that any finite number of outputs, N∈ℕ,
is assigned a joint Gaussian distribution with
prior mean function m:ℝ^d→ℝ and covariance
defined through the kernel
k:ℝ^d×ℝ^d→ℝ
<cit.>. Without loss of generality, we assume m(·) to equal 0 in the following.
In order to perform regression with Gaussian processes,
they are considered as a a prior distribution. This allows to employ Bayes' theorem to
calculate the posterior distribution given a training data set 𝔻={(x^(n),y^(n)}_n=1^N
consisting of N inputs x^(n)∈ℝ^d and targets y^(n)∈ℝ,
which are Gaussian perturbed measurements of an unknown function f:ℝ^d→ℝ,
i.e., y^(n)=f(x^(n))+ϵ^(n), ϵ^(n)∼𝒩(0,σ_on^2), σ_on^2∈ℝ_+.
Due to the properties of Gaussian distributions, the posterior is again a Gaussian process,
which yields the posterior mean μ(·) and variance σ^2(·) functions
μ(x) =k^T(x)( K+σ_on^2I_N)^-1y,
σ^2(x) =k(x,x)-k^T(x)(K+σ_on^2I_N)^-1k(x),
where we define the kernel matrix K and the
kernel vector k(x) through K_ij=k(x^(i),x^(j)) and k_i(x)=k(x,x^(i)), respectively, with i,j=1,…,N, and y = [y^(1)⋯ y^(N)]^T.
§.§ Problem Formulation
We consider single-input linear dynamical systems with nonlinear input perturbation of the form
ẋ=Ax+b(u+f(x))
with initial condition x(0)=x_0∈𝕏⊆ℝ^d and scalar control input u:ℝ_0,+→𝕌⊆ℝ.
The matrix A∈ℝ^d× d and vector b∈ℝ^d are assumed to be known, while we consider f:𝕏→ℝ to be an unknown nonlinearity. This system structure covers a wide range of practical systems and can represent, e.g., systems controlled via approximate feedback linearization <cit.> or backstepping controllers for certain classes of dynamics <cit.>.
Note that we merely consider the restriction to single-input systems for notational convenience, but our derived results can be easily generalized to multi-input dynamics.
The considered task is to track a bounded reference trajectory x_ref:ℝ_0,+→ℝ^d with the state x(t).
In order to enable the accurate tracking of the reference trajectory x_ref(·), we restrict ourselves to references of the form
ẋ_ref=Ax_ref+br_ref,
where r_ref:ℝ_0,+→ℝ is a reference signal. For tracking the reference trajectory, we can employ a control law
u = θ^T(x-x_ref)+r_ref-f̂(x),
where θ∈ℝ^d is a control gain vector and f̂:𝕏→ℝ is a model of the unknown nonlinear perturbation f(·). This control law leads to closed-loop dynamics of the tracking error e(t)=x(t)-x_ref(t) given by
ė=A_θe + b(f(x)-f̂(x)),
where A_θ=A-bθ^T. In order to ensure the stability of these dynamics in the case of exact model knowledge f(x)=f̂(x), we employ the following assumption on A_θ.
The matrix A_θ has distinct and non-positive eigenvalues, which decrease monotonically with the parameters θ, i.e., there exists a class 𝒦_∞ function α:ℝ_0.+→ℝ_0,+ such that λ_max(A_θ)≤-α(θ).
This assumption essentially requires the controllability of the pair (A,b) <cit.>, which allows the eigenvalues of the matrix A_θ to be considered as design parameters, e.g., using methods such as pole placement. Since controllability is a common requirement in linear systems theory, <ref> is not restrictive. Note that the requirement of distinct eigenvalues is only required to simplify the presentation in the following sections by ensuring diagonalizability of A_θ, but can be avoided by generalizing the derivations using Jordan blocks <cit.>.
While <ref> ensures that the error dynamics (<ref>) do not diverge, the tracking precision crucially relies on the accuracy of the model f̂(·). Therefore,
we assume to learn it from measurements (x^(n),y^(n)) using Gaussian process regression, such
that we can use f̂(x)=μ(x) in the control law (<ref>). Since this merely leads
to an approximate compensation of the nonlinearity, exact tracking cannot be ensured in general. Therefore, we
consider the problem of learning a Gaussian process model of f(·),
such that the tracking error is guaranteed to be probabilistically bounded by a prescribed constant e̅∈ℝ_+, i.e.,
ℙ(x(t)-x_ref(t)≤e̅, ∀ t≥ 0)≥ 1-δ
for δ∈(0,1).
Due to the complexity of this problem, we decompose it into the subproblems of deriving a probabilistic error bound for Gaussian process regression, analyzing the dependency of the error bounds on the training data density, and developing an approach for generating training data with sufficiently high density, such that the prescribed tracking error bound e̅ is satisfied. These subproblems are described in more detail in the following.
§.§.§ Probabilistic Regression Error Bounds
In order to be able to ensure any bound for the tracking error x-x_ref, it is necessary
to find an upper bound for the learning error f(x(t))-μ(x(t)) along the system trajectory x(t).
Since we do not know the exact system trajectory x(t) in advance, we consider the problem of bounding the regression error in a compact domain 𝕏⊂ℝ^d. Since the bound must hold jointly for all states x in the domain 𝕏, we refer to it as probabilistic uniform error bound, which is formally defined as follows.
Gaussian process regression exhibits a uniformly bounded prediction error on a compact set 𝕏⊂ℝ^d with probability 1-δ if there exists a function η:𝕏→ℝ_0,+ such that
P( |f(x)-μ(x)|≤η(x), ∀x∈𝕏)≥ 1-δ.
In general, we cannot expect to guarantee a uniformly bounded regression error without any regularity assumptions about the unknown function f(·). Due to the Bayesian foundation of
Gaussian processes, we employ their prior distribution for this purpose, which we formalize in the following assumption.
The unknown function f(·) is a sample from the
Gaussian process 𝒢𝒫(0,k(x,x')).
This assumption, which has similarly been used in, e.g., <cit.>, has a twofold implication. On the one hand, it specifies the admissible functions for regression via the space of sample functions, which depends on the employed kernel k(·,·). For example, it is straightforward to see that polynomial kernels can be used to learn polynomial functions of the same degree. Moreover, it is well known that the sample space of GPs with squared exponential kernel contains all continuous functions <cit.>. Therefore, choosing a suitable kernel for ensuring that the unknown function lies in the space of sample functions is usually not a challenging problem in practice. On the other hand, <ref> induces a weighting between possible sample functions due to the Gaussian process probability density. Since we base the derivation of the uniform error bound on this weighting, an unknown function f(·) with low prior probability density would lead to sets {f'(·): |f'(x)-μ(x)|≤η(x) } with a high probability under the GP prior, even though they do not contain the unknown function f(·).
Hence, the true function f(·) should have a high probability density under the GP prior. This can be efficiently achieved in practice using suitable kernel tuning methods, e.g., <cit.>, or via a re-calibration of the probability distribution after training <cit.>. Therefore, ensuring
a suitable prior distribution is not a severe limitation, such that <ref> is not restrictive in
practice.
§.§.§ Dependency of Error Bounds on Data Density
After a probabilistic uniform error bound η(·) has been derived, we consider the problem of deriving conditions for the training data 𝔻 which ensure that the error bound η(·) stays below a desired value η̅∈ℝ_+. This requires the design of a suitable measure of data density ρ:𝕏→ℝ_+, which reflects the dependency of the error bound η(·) on the data distribution. Therefore, the measure ρ(·) must consider the information structure of the GP induced by the employed kernel k(·,·).
Based on the derived density measure ρ(·), the problem of ensuring a learning error bound η̅ reduces to showing that the existence of a lower bound ρ∈ℝ_+ for the data density ρ(·) leads to the implication
ρ(x)≥ρ ⇒ η(x)≤η̅(ρ).
As we want to be able to ensure arbitrary small learning error bounds η̅(ρ), it must additionally hold that
lim_ρ→∞η̅(ρ)=0.
§.§.§ Data Generation for Guaranteed Tracking Accuracy
Finally, we consider the problem of developing an episodic approach for training data generation, which
achieves the necessary data density ρ(·) to ensure the satisfaction of the tracking error bound
(<ref>). Firstly, this requires the derivation of a tracking error bound, such that for
a given learning error bound η̅, we have
η(x_ref(t))≤η̅ ⇒ ℙ(x(t)-x_ref(t)≤υ̅(η̅))≥ 1-δ
for some function υ̅:ℝ_0,+→ℝ_0,+.
Similarly as in (<ref>), this bound must also vanish asymptotically, i.e.,
lim_η̅→ 0υ̅(η̅) = 0,
in order to admit arbitrarily small tracking error guarantees.
Using this tracking error bound and the derived dependency of the learning error bound η(·) on the data density ρ(·), the problem of developing a data generation approach simplifies to finding an episodic roll-out strategy satisfying
ρ_i+1>ρ_i, lim_i→∞ρ_i = ∞,
where the index i is used to denote the roll-out episode. This ensures that there exists a finite number of episodes N_E∈ℕ such that υ̅(η̅(ρ_N_E))≤e̅. Therefore, finding a roll-out strategy ensuring (<ref>) solves the overall problem of learning a Gaussian process model of f(·) such that a prescribed error bound e̅ is satisfied.
§ PROBABILISTIC UNIFORM ERROR BOUND
In this section, we derive an easily computable uniform
error bound for Gaussian process regression based on the prior distribution addressing
the problem described in <ref>.
We first present the uniform error bound and approaches to compute its
parameters in <ref>. Since the bound also relies on the Lipschitz
constant of the unknown function, which is not always known a priori, we show
how a probabilistic Lipschitz constant can be derived from the prior Gaussian process
distribution in <ref>.
§.§ Uniform Error Bound based on Lipschitz Continuity
Since the prior Gaussian process induces a probability distribution for each point in a compact
set 𝕏, we can discretize this set and exploit standard tail bounds for Gaussian
distributions to obtain point-wise error bounds <cit.>. If all involved functions are
continuous, we can straightforwardly extend these point-wise guarantees yielding the uniform
error bound presented in the following.
Consider a zero mean prior Gaussian process defined
on a compact set 𝕏 and let f:𝕏→ℝ be a continuous unknown function with Lipschitz constant L_f which satisfies <ref>. Assume the GP posterior mean μ(·) and standard deviation σ(·)
are continuous with Lipschitz constant L_μ and modulus of continuity ω_σ(·).
Moreover, pick δ∈ (0,1), τ∈ℝ_+ and set
β_𝕏(τ) =2log(M(τ,𝕏)/δ),
γ(τ) =( L_μ+L_f)τ+√(β_𝕏(τ))ω_σ(τ),
where M(τ,𝕏) denotes the τ-covering number of 𝕏[The τ-covering number of a set 𝕏 is the smallest number, such there exists a set 𝕏_τ
satisfying |𝕏_τ|=M(τ,𝕏) and ∀x∈𝕏 there exists
x'∈𝕏_τ with x-x'≤τ.].
Then, the prediction error is uniformly bounded with probability of at least 1-δ on 𝕏
with bound
η(x)=√(β_𝕏(τ))σ(x)+γ(τ).
We exploit the continuity properties of the posterior mean, variance and the
unknown function to prove
the probabilistic uniform error bound by exploiting the fact that for every
grid 𝕏_τ with |𝕏_τ| grid points and
max_x∈𝕏min_x'∈𝕏_τx-x'≤τ
it holds with probability of at least
1-|𝕏_τ|e^-β_𝕏(τ)/2 that <cit.>
|f(x)-μ(x)|≤√(β_𝕏(τ))σ(x)
∀x∈𝕏_τ.
Choose ,
then
|f(x)-μ(x)|≤√(β_𝕏(τ))σ(x)
∀x∈𝕏_τ
holds with probability of at least 1-δ. Due to continuity of
f(x), μ(x) and σ(x) we obtain
min_x'∈𝕏_τ|f(x)-f(x')| ≤τ L_f ∀x∈𝕏
min_x'∈𝕏_τ|μ(x)-μ(x')| ≤τ L_μ ∀x∈𝕏
min_x'∈𝕏_τ|σ(x)-σ(x')| ≤ω_σ(τ) ∀x∈𝕏.
Moreover, the minimum number of grid points satisfying (<ref>) is
given by the covering number M(τ,𝕏). Hence, we obtain
P(|f(x)-μ(x)|≤√(β_𝕏(τ))σ(x)+γ(τ),
∀x∈𝕏)≥ 1-δ,
for β_𝕏(τ) and γ(τ) defined in (<ref>) and (<ref>), respectively.
The virtual grid constant τ used in (<ref>) and (<ref>)
balances the effect of the state space discretization and the inherent uncertainty
measured by the posterior standard deviation σ(·). Therefore,
γ(τ) can be made arbitrarily small by choosing a sufficiently fine
virtual grid. This in turn increases β_𝕏(τ) and thus the
effect of the posterior standard deviation σ(·) on the bound.
However, β_𝕏(τ) depends merely logarithmically on τ such that
even poor Lipschitz constants L_μ, L_f and moduli of continuity ω_σ(·) can
be easily compensated by small virtual grid constants τ.
Since the standard deviation σ(·) varies within the state space 𝕏,
an optimal virtual grid constant τ, which minimizes the expression
√(β_𝕏(τ))σ(x)+γ(τ) for all x∈𝕏, does
not exist in general. While simple approaches such as choosing τ such that
γ(τ) is negligible for all x∈𝕏 provide satisfying
results in our simulations, more complex approaches remain open research questions.
It is important to note that most of the parameters in <ref>
do not require a difficult analysis such that the bound (<ref>)
can be directly evaluated. While the computation of the exact covering number
M(τ,𝕏) is a difficult problem for general sets 𝕏,
it can be easily upper bounded as illustrated in <ref>. For this reason,
we overapproximate the set
𝕏 through a d-dimensional hypercube 𝕏̃ with edge length r.
Then, the covering number of 𝕏̃ is bounded by <cit.>
M(τ,𝕏̃)≤(r√(d)/2τ)^d,
which is by construction also a bound for the covering number of 𝕏,
i.e.,
M(τ,𝕏)≤(r√(d)/2τ)^d.
The Lipschitz constant L_μ of the posterior mean in (<ref>) can
be straightforwardly bounded when the prior Gaussian process has a Lipschitz continuous
kernel, as shown in the following lemma.
Consider a zero mean prior Gaussian process defined through the L_k-Lipschitz
kernel k(·,·). Then, its
posterior mean μ(·)
is continuous with Lipschitz constant=-1
L_μ ≤ L_k√(N) (K+σ_on^2I_N)^-1y.
The norm of the difference between
the posterior mean μ(x) evaluated at two different points is given by
μ(x)-μ(x') =
(k(x)-k(x'))
α,
with
α=(K+σ_on^2I_N)^-1y.
Due to the Cauchy-Schwarz inequality and the Lipschitz continuity of the
kernel we obtain
μ(x)-μ(x') ≤ L_k√(N)αx-x',
which proves Lipschitz continuity of the mean μ(x).
Moreover, the assumption of a Lipschitz continuous kernel also suffices to compute the modulus of
continuity ω_σ(·) for the posterior standard deviation in (<ref>),
as shown in the following lemma.=-1
Consider a zero mean prior Gaussian process defined through the L_k-Lipschitz
kernel k(·,·). Then, its
posterior standard deviation σ^2(·)
is continuous with modulus of continuity=-1
ω_σ(τ) ≤√(2L_kτ).
The difference between two different evaluations of the posterior standard deviation is bounded by
|σ(x)-σ(x')|≤ d_k(x,x')
as shown in <cit.>, where the kernel metric is defined as
d_k(x,x')=√(k(x,x)+k(x',x')-2k(x,x')).
Due to Lipschitz continuity of the kernel, we have
d_k(x,x')≤√(2L_kx-x'),
which concludes the proof.
For the special case of stationary kernels , the convergence
rate of the modulus of continuity ω_σ(·) can even be improved, as shown in the
following.
Consider a zero mean prior Gaussian process defined through the stationary, L_k-Lipschitz
kernel k(·,·). Then, its
posterior standard deviation σ(·)
is continuous with modulus of continuity ω_σ(τ)=L_στ, where =-1
L_σ
= sup_x-x'∈𝕏√(1/2k(0)-2k(x-x'))∇ k(x-x').
For stationary kernels, we can express the kernel metric as
d_k(x,x')=d_k(x-x')=√(2k(0)-2k(x-x')).
The simplified kernel
metric is only a function of x-x', such that the supremum of the norm of the
derivative of d_k(·,·) with respect to x-x' is the Lipschitz constant of σ(·).
This derivative directly follows from the chain rule of differentation as
∇ d_k(x-x') = √(1/2k(0)-2k(x-x'))∇ k(x-x'),
which concludes the proof.
While computing the Lipschitz constant L_σ requires the computation of a supremum in general,
this optimization problem can be straightforwardly solved analytically for specific kernel choices, e.g.,
squared exponential kernels <cit.>. Thereby, it allows the efficient computation of a tight
modulus of continuity.
The remaining open parameter in (<ref>) is the Lipschitz constant L_f of the unknown
function f(·). In many applications, in particular in control, rough knowledge of the
unknown function is known in advance, which can allow to specify L_f. Even if this constant
is a rather poor estimate of the true Lipschitz constant, conservative estimates are not a crucial
issue as discussed after <ref>. If no such knowledge of the unknown function f(·)
is available, the prior Gaussian process distribution can be employed to derive a probabilistic
Lipschitz constant as shown in the following section.
§.§ Probabilistic Lipschitz Constants for Gaussian Processes
In order to derive a probabilistic Lipschitz constant L_f of the unknown function f(·) from
the prior Gaussian process distribution, we exploit the fact that the derivative of a Gaussian process
is again a Gaussian process. Therefore, Lipschitz constants can be obtained by adapting results from
the well-studied theory of suprema of Gaussian processes. This yields the following
lemma, which is based on the metric entropy criterion <cit.>.
Consider a Gaussian process with a continuously differentiable covariance
function k(·,·) and let L_k denote its Lipschitz constant on
the compact set 𝕏 which is included in a cube with edge length r.
Then, the expected supremum of a sample function f(·) of this
Gaussian process satisfies
E[sup_x∈𝕏f(x)]≤
12√(6d)max{max_x∈𝕏√(k(x,x)),√(rL_k)}.
We prove this lemma by making use of the metric entropy criterion for the
sample continuity of Gaussian processes <cit.>.
This criterion allows to bound the expected supremum of a sample function f(·)
by
E[ sup_x∈𝕏f(x) ]≤∫_0^max_x∈𝕏√(k(x,x))√(log(N_k(ϱ,𝕏)))dϱ,
where N_k(ϱ,𝕏) is the ϱ-packing number of 𝕏
with respect to the kernel metric (<ref>).
Instead of bounding the ϱ-packing number, we bound the ϱ/2-covering
number, which is known to be an upper bound of the packing number. The covering number can be easily bounded
by transforming the problem of covering 𝕏 with respect to the metric
d_k(·,·) into a coverage problem in the original metric of 𝕏. For
this reason, define
ψ(ϱ')=sup_x,x' ∈𝕏
x-x' _∞≤ϱ' d_k(x,x'),
which is continuous due to the continuity of the covariance kernel k(·,·).
Consider the inverse function
ψ^-1(ϱ)=inf{ϱ'>0: ψ(ϱ')>ϱ}.
Continuity of ψ(·) implies ϱ=ψ(ψ^-1(ϱ)). In particular,
this means that we can guarantee d_k(x,x')≤ϱ/2 if
. Due to this relationship
it is sufficient to construct a uniform grid with grid constant 2ψ^-1(ϱ/2)
in order to obtain a ϱ/2-covering net of 𝕏. Furthermore, the cardinality
of this grid is an upper bound for the ϱ/2-covering number, such that we obtain
N_k(ϱ,𝕏)≤⌈r/2ψ^-1(ϱ/2)⌉^d.
Due to the Lipschitz continuity of the covariance function, we can bound
ψ(·) by ψ(ϱ')≤√(2L_kϱ').
Hence, the inverse function satisfies
ψ^-1(ϱ/2)≥(ϱ/2√(2L_k))^2
and consequently
N_k(ϱ,𝕏)≤(1+4rL_k/ϱ^2)^d
holds, where the ceil operator is resolved through the addition of 1.
Substituting this expression in the metric entropy bound
(<ref>) yields
E[sup_x∈𝕏f(x)]≤ 12√(d)∫_0^max_x∈𝕏√(k(x,x))√(log(1+4rL_k/ϱ^2))dϱ.
As shown in <cit.> this integral can be bounded by √(6)max{max_x∈𝕏√(k(x,x)), √(rL_k)}, which concludes the proof.
While <ref> provides a bound merely for the expected supremum of
a sample function, a high probability bound for the supremum can be obtained using
the Borell-TIS inequality <cit.>. This is shown in the following
result.
Consider a Gaussian process with a continuously differentiable covariance
function k(·,·).
Then, with probability of at least 1-δ_L the supremum of a sample
function f(·) of this Gaussian process is bounded by
f_sup(δ_L,k(·,·),r)= √(2log( 1/δ_L))max_x∈𝕏√(k(x,x))
+12√(6d)max{max_x∈𝕏√(k(x,x)),
√(rL_k)}.
We prove this lemma by exploiting the wide theory of concentration inequalities
to derive a bound for the supremum of the sample function f(x). We apply the
Borell-TIS inequality <cit.>, which ensures for arbitrary c∈ℝ_0,+ that
P( sup_x∈𝕏f(x)-
E[ sup_x∈𝕏f(x) ] ≥
c )≤exp( -c^2/2max_x∈𝕏
k(x,x)).
Due to <ref> we can directly bound E[sup_x∈𝕏f(x)]. Therefore,
the lemma follows from
substituting (<ref>) in (<ref>) and choosing
c=√(2log( 1/δ_L))max_x∈𝕏√(k(x,x)).
Since the derivatives of sample functions from Gaussian processes with sufficiently smooth kernels are the
sample functions of the derivative Gaussian processes <cit.>, <ref> directly
allows to compute a high probability Lipschitz constant for the unknown function f(·) from the prior
Gaussian process distribution. This is summarized in the following Theorem.
Consider a zero mean Gaussian process defined through the
covariance kernel k(·,·)
with continuous partial derivatives up to the fourth order
and partial derivative kernels
k^∂ i(x,x') =∂^2/∂ x_i∂ x_i'
k(x,x') ∀ i=1,…, d.
Then, a sample function f(·) of the Gaussian process is almost
surely continuous on 𝕏 and with probability of at least 1-δ_L,
L_f≤L̂_f=[ f_sup(δ_L/2d,k^∂ 1(·,·),r); ⋮; f_sup(δ_L/2d,k^∂ d(·,·),r) ]
for f_sup(·,·,·) defined in (<ref>).
Continuity of the sample function f(x) follows directly from
<cit.>. Furthermore, this theorem guarantees that the derivative
functions ∂/∂ x_if(x) are samples from derivative Gaussian
processes with covariance functions k^∂ i(x,x').
Therefore, we can apply <ref> to each of the derivative processes and
obtain with probability of at least 1-δ_L/d
sup_x∈𝕏|∂/∂ x_if(x)|
≤ f_sup(δ_L/2d,k^∂ i(·,·),r).
Applying the union bound over all partial derivative
processes i=1,…,d finally yields the result.
Since many practically employed kernels such as, e.g., the squared exponential, the Matern 5/2,
satisfy the required smoothness assumption of <ref>, this assumption does not pose
a severe restriction. Therefore, this theorem allows to straightforwardly determine high probability Lipschitz constants
for the unknown function f(·), which can be directly used in <ref>, while barely requiring additional
assumptions.
§ DATA DEPENDENCY OF LEARNING ERROR BOUNDS
In order to derive conditions for ensuring that the learning error bound in <ref> is
below a given threshold as described <ref>, we need to analyze its dependency on the training data density. For this purpose, we investigate the decay behavior of the probabilistic uniform error bound (<ref>) depending on the decrease rate of the GP standard deviation in <ref>. A kernel-dependent measure of data density is proposed in <ref> in order to bound the decrease rate of the GP standard deviation. Finally, it is shown in <ref> how the kernel-dependent density measure can be bounded using straightforwardly computable Euclidean distances.
§.§ Asymptotic Bounds for the Learning Error
Since the probabilistic uniform error bound (<ref>) consists of two summands, a vanishing posterior standard deviation σ(x) is not by itself sufficient to guarantee a decreasing value of η(x). Therefore, it is necessary to additionally vary the parameter τ, such that γ(τ) decreases with growing number of training samples N. Even though this leads to a growing value of β_𝕏(τ), it ensures an asymptotically vanishing learning error bound in the limits N→∞ and σ(x)→ 0 as shown in the following theorem.
Consider a zero mean Gaussian process defined by the continuously differentiable
kernel k(·,·). Let f:𝕏→ℝ be a continuous unknown function with Lipschitz constant L_f on the compact domain 𝕏 which satisfies <ref>.
Then, for τ∈𝒪(1/N), the learning error asymptotically behaves as
η(x)∈𝒪(√(log(N/δ))σ(x)+1/N).
Due to Theorem <ref> with suitable value of β_𝕏(τ) it holds that
sup_x∈𝕏|f(x)-μ(x)|≤√(β_𝕏(τ))σ(x)+γ(τ)
with probability of at least 1-δ/2 for δ∈(0,1). A trivial
bound for the covering number can be obtained by considering a uniform grid over the
cube containing 𝕏. This approach leads to
M(τ,𝕏)≤(r√(d)/2τ)^d.
Therefore,
we have
β_𝕏(τ)≤ 2dlog(r√(d)/2τ)-2log(δ).
In order to derive a bound for γ(τ), we employ the bounds for the Lipschitz constants and
modulus of continuity.
The Lipschitz constant L_μ in (<ref>) is bounded by
L_μ ≤ L_k√(N) (K+σ_on^2I_N)^-1y
due to <ref>. Since the Gram matrix K
is positive semidefinite and f(·) is bounded by some f̅ due to Lipschitz continuity and a compact domain 𝕏, we can
bound (K+σ_on^2I_N)^-1y by
(K+σ_on^2I_N)^-1y ≤y/λ_min(K+σ_on^2I_N)
≤√(N)f̅
+ϵ/σ_on^2,
where ϵ is a vector of N i.i.d. zero mean Gaussian random variables with
variance σ_on^2. Therefore, it follows that
ϵ^2/σ_on^2∼χ_N^2. Due to
<cit.>, with probability of at least 1-exp(-log(2/δ)) we have
ϵ^2≤(2√(Nlog(2/δ))+2log(2/δ)+N)σ_on^2.
Hence, the Lipschitz constant of the
posterior mean function μ(·) satisfies with probability of at least
1-δ/2
L_μ≤ L_kNf̅+√(N(2√(Nlog(2/δ))+2log(2/δ)+N))σ_on/σ_on^2.
It can clearly be seen that the fastest growing term is increasing linearly, such that it holds that L_μ∈𝒪(N) with
probability of at least 1-δ/2.
The modulus of continuity in (<ref>) can be bounded by
ω_σ(τ)≤√(2L_kτ)
due to <ref>.
Since the unknown function f(·) is assumed
to admit a Lipschitz
constant L_f, we obtain
γ(τ)≤ L_kτNf̅+√(N(2√(Nlog(2/δ))+2log(2/δ)+N))σ_on/σ_on^2
+√(2β_𝕏(τ)L_kτ)
+L_fτ.
with probability of at least 1-δ/2 by substituting (<ref>) and (<ref>) into (<ref>). In order to admit asymptotically vanishing
error bounds, (<ref>) must converge to 0 for N→∞, which is only ensured if τ decreases faster
than 𝒪(1/N). Therefore, set τ∈𝒪(1/N) in order
to guarantee
γ_N(τ)∈𝒪( 1/N).
However, this choice of τ implies that β_𝕏(τ)∈𝒪(log(N/δ))
due to (<ref>). Therefore, it directly follows that
√(β_𝕏(τ))σ(x)+γ(τ)∈𝒪(√(log(N/δ))σ(x)+1/N),
which concludes the proof.
Due to the linear dependency of the bound for the Lipschitz constant L_μ on the number
of training samples, the virtual grid constant must decay faster than 𝒪(1/N).
This in turn leads to a logarithmic growth of β_𝕏(τ), which causes the √(log(N))
increase of the scaling factor of the posterior standard deviation σ(x). Note that this is a
common phenomenon in uniform error bounds for GP regression and can also be found in RKHS based approaches, where
similar bounds as (<ref>) are used to bound the effect of the noise <cit.>.
§.§ Asymptotic Bounds for the Posterior Variance
In order to compensate the growth of the scaling factor in <ref>, a sufficiently
fast decay of the standard deviation σ(x) must be ensured. Therefore, we investigate
the behavior of the posterior variance σ^2(x) depending on the training data density of an input data set
𝔻^x={x^(i)}_i=1^N. The starting point of this analysis
is the following lemma, which provides a straightforward upper bound for the
posterior variance σ^2(x).
Consider a GP trained using a
data set with input training samples 𝔻^x. Then,
the posterior variance is bounded by=-1
σ^2(x) ≤σ_on^2k(x,x)+NΔ k(x)/N
max_x'∈𝔻^x
k(x',x')+σ_on^2,
where
Δ k(x)= k(x,x)max_x'∈𝔻^x
k(x',x') -min_x'∈𝔻^x
k^2(x',x).
Since K+σ_on^2I_N is a
positive definite, quadratic matrix, it follows
that
σ^2(x) ≤ k(x,x)-
k(x)^2/λ_max(K)+σ_on^2.
Applying the Gershgorin theorem <cit.>
the maximal eigenvalue is bounded by
λ_max(K)≤ N
max_x'∈𝔻^x
k(x',x').
Furthermore, due to the definition of
k(x) we have
k(x)^2≥ N
min_x'∈𝔻^x
k^2(x',x).
Therefore, σ^2(x) can be bounded by
σ^2(x) ≤ k(x,x)-
Nmin_x'∈𝔻^x
k^2(x',x)/N
max_x'∈𝔻^x
k(x',x')+σ_on^2.
Finally, the proof follows from the definition of Δ k(x).
This theorem does not pose any restriction on the employed kernel, but strongly depends on
the particular choice of kernel. Therefore, it can be difficult to interpret. However, it can be significantly
simplified for specific kernels, as shown in the following corollary for stationary
covariance functions.
Consider a GP with stationary kernel and input training samples 𝔻^x. Then,
the posterior variance is bounded by=-1
σ^2(x)≤ k(0)-min_x'∈𝔻^xk^2(x-x')/k(0)
+σ_on^2/N.
The proof follows directly from
<ref> and the fact that max_x'∈𝔻^xk(x',x')= k(0)
since the kernel is stationary.
In this special case of <ref>, which has been previously stated,
e.g., in <cit.>, the kernel induces a notion of proximity, where the absence
of training inputs x' with k(x-x')≈ 0 leads to a large bound for the
posterior variance σ^2(x). Therefore, this corollary shows that it is
desirable to have data close to the test point x as measured by k(·) for
stationary kernels.
Since <ref> and <ref> still consider the full input data
set 𝔻^x, a single sample with k(x',x)≈ 0 can practically lead to the
trivial bound σ^2(x)≲ k(x,x). This is clearly an undesired behavior for
a bound since it would imply that additional data can potentially increase the posterior variance bound.
In order to avoid this effect, we make use of an important property of
Gaussian process posterior variances, which is the fact that σ^2(x) is
non-increasing with the number of training samples N <cit.>. Therefore,
we can consider subsets of 𝔻^x to compute the posterior variance
bounds in <ref> and <ref>, which exclude these training samples with
a negative effect on the bound.
Due to the importance of Δ k(x) for these bounds, we make use of the following subset
𝕂_ρ'(x) ={x'∈𝔻^x: k^2(x,x)≤ k^2(x',x')≤1/ρ'+k^2(x',x) }
for this purpose. It can be easily seen that considering only the subset 𝕂_ρ'(x)⊂𝔻^x in (<ref>) ensures
k(x,x)max_x'∈𝕂_ρ'(x)
k(x',x') -min_x'𝕂_ρ'(x)
k^2(x',x)≤1/ρ'.
Since the consideration of a subset of 𝔻^x also reduces the number of considered training
samples in (<ref>), we trade-off the size of 𝕂_ρ'(x) and the
ensured value for Δ k(x) by defining ρ' using the following optimization problem
ρ(x)= max_ρ'∈ℝ_+ρ'
such that |𝕂_ρ'(x)|≥ρ'σ_on^2k(x,x).
It can easily be seen that ρ(x) is well-defined since the optimization problem is always
feasible for ρ'→ 0. Moreover, it can be directly used as a measure of data density
as shown in the following proposition.
Consider a zero mean Gaussian process defined by the
kernel k(·,·). If k(x,x)≠ 0, the posterior standard deviation at x
satisfies
σ(x)≤√(2/ρ(x)k(x,x))
such that it behaves as σ(x)∈𝒪( 1/√(ρ(x)) ).
By exploiting the fact that the posterior variance σ^2(x) is non-increasing with the number of training samples N <cit.>
and considering only samples inside the set
𝕂_ρ(x)(x) for the computation of the
posterior standard deviation, we obtain=-1
σ^2(x) ≤σ_on^2k(x,x)+|𝕂_ρ(x)(x)| Δ k(x)/|𝕂_ρ(x)(x)|
max_x'∈𝕂_ρ(x)(x)
k(x',x')+σ_on^2
due to <ref>. Since x'∈𝕂_ρ(x)(x) implies k(x',x')≥ k(x,x), we can simplify this expression to
σ^2(x) ≤σ_on^2/|𝕂_ρ(x)(x)|
+Δ k(x)/k(x,x).
Moreover, it can be straightforwardly checked that the restriction to 𝕂_ρ(x)(x)
implies Δ k(x)≤1/ρ(x), which yields
σ^2(x) ≤σ_on^2k(x,x)/|𝕂_ρ(x)(x)|
k(x,x)+1/ρ(x)k(x,x)
Since |𝕂_ρ(x)(x)| is lower bounded by ρ(x)σ_on^2k(x,x) by definition, we obtain
σ^2(x) ≤2/ρ(x)k(x,x),
which directly implies σ(x)∈𝒪(1/√(ρ(x))).
concluding the proof.
It can be clearly seen that ρ(x) is a measure of data density which is highly specific for each particular GP and therefore is capable of reflecting the requirements on good data distributions posed by the employed kernel k(·,·). Moreover, it immediately follows from <ref> that a sufficiently fast growth of ρ(x), i.e., ρ(x)∉𝒪(log(N)), guarantees a vanishing error bound |μ(x)-f(x)|→ 0. Therefore, ρ(·) satisfies the requirements posed on a suitable measure of data density in <ref>.
§.§ Conditions for Specific Kernels
The high flexibility of <ref> allows its application to GPs with arbitrary kernels, but comes at the price of a difficult interpretability. However, when we fix a specific kernel, it is often possible to derive more accessible and intuitive subsets contained in 𝕂_ρ'(x), as shown in the following lemma for linear, squared exponential and Matérn class kernels.
Geometrically interpretable subsets of 𝕂_ρ'(x) defined in (<ref>) are given by
* the set
ℍ_ρ'^c(x)={ x'∈𝔻^x: x'^2(x'^2-cx^2) ≤1/ρ',
x≤x', |x^Tx'|≥ cxx'}⊂𝕂_ρ'(x)
for every c∈(0,1);=-1
* the Euclidean ball
𝔹_√(1/2L_∂ kσ_f^2ρ')(x)=
{x'∈𝔻^x: x-x'≤√(1/2L_∂ kσ_f^2ρ')}⊂𝕂_ρ'(x)
for isotropic SE or Matérn kernels with ν≥3/2 and σ_f^2=k(x,x).
Due to the definition of the linear kernel, we have the identity
k^2(x',x')-k^2(x',x)= x'^4-(x^Tx')^2.
For |x^Tx'|/(xx')≥ c, we therefore obtain
k^2(x',x')-k^2(x',x)≤x'^2(x'^2-cx^2).
Finally, the first inequality in (<ref>) yields the requirement
k^2(x,x)=x^4≤x'^4= k^2(x',x'),
which concludes the first part of the proof.
For the second part of the proof, we exploit the continuous differentiability of Matérn kernels with ν≥3/2 and squared exponential kernels together with the fact that their derivative at r=x-x'=0 is 0. Therefore, we have
k(x-x')≥σ_f^2-L_∂ kx-x'^2.
where L_∂ k∈ℝ_+ is the Lipschitz constant of the kernel derivative.
Using this lower bound, we obtain
k^2(0)-k^2(x-x') ≤ 2L_∂ kσ_f^2x-x'^2-L_∂ k^2x-x'^4,
which we can simplify to
k^2(0)-k^2(x-x') ≤ 2L_∂ kσ_f^2x-x'^2
due to non-negativity of the norm. Therefore, x-x'^2≤ρ'/2L_∂ kσ_f^2 implies |k^2(x,x)-k^2(x,x')|≤ρ'. Since k(x,x)=k(x',x') for isotropic kernels, the first inequality is always satisfied, concluding the proof.
This lemma illustrates the flexibility of quantifying the data density using 𝕂_ρ'(x). While this set can
be innerapproximated by a ball for Matérn and SE kernels as illustrated in <ref>, it looks more like segments of a sphere for linear kernels. Since we can easily determine the volume of such simple geometrical structures, <ref> enables the derivation of a straightforward relationship between the sampling distributions and data density ρ(x). For example, when training samples in 𝔻^x are generated by drawing from a uniform distribution, the number of points in a Euclidean ball is proportional to the volume of the ball, i.e., 𝔹_ρ'(x)∝N/ρ'^d. Therefore, it follows from (<ref>) that ρ(x)∈𝒪(N^1/d+1) for SE or Matérn kernels with uniformly drawn input training samples. This in turn implies that σ(x)∈𝒪(1/N^1/2d+2) due to <ref> and consequently
|μ(x)-f(x)|∈𝒪(log(N)/N^1/2d+2)
due to <ref>. This demonstrates the flexibility and effectiveness of the derived formalism for bounding the asymptotic decay of the prediction error |μ(x)-f(x)| presented in this section.=-1
§ SAFETY GUARANTEES FOR CONTROL OF UNKNOWN DYNAMICAL SYSTEMS
We employ the theoretical results for GP error bounds introduced in the previous sections
to develop an iterative approach for ensuring arbitrary tracking accuracy with the considered
control law (<ref>). For this purpose, we derive a time-varying tracking
error bound in <ref> which depends explicitly on the uniform GP error bound along
the reference trajectory. This result allows us to analyze the asymptotic decay of the tracking
error bound depending on the training data density measured by ρ(x) in <ref>.
Finally, we employ the obtained insight to develop an episodic approach for ensuring arbitrary
tracking accuracy in <ref>.
§.§ Probabilistic Tracking Error Bound
Since <ref> ensures distinct eigenvalues of the matrix A_θ defining the closed-loop behavior of the dynamics (<ref>) of the tracking error e=x-x_ref, we can compute the eigendecomposition A_θ=UΛU^-1, where Λ is a diagonal matrix consisting of the eigenvalues of A_θ. This allows the derivation of a dynamic bound for
the tracking error e inspired by the comparison principle <cit.>, as shown in the following theorem.=-1
Consider a linear system (<ref>) satisfying <ref>, which is perturbed by
a L_f-Lipschitz nonlinearity f(·) satisfying <ref>. Assume that a zero
mean Gaussian process with L_k-Lipschitz stationary kernel is used to learn a model f̂(·)=μ(·) of f(·),
such that a controller (<ref>) is used to track the bounded
reference x_ref. Then,
the tracking error is bounded by
x(t)-x_ref(t)≤υ(t)
with probability of at least 1-δ, where υ(t) is the solution of the linear dynamical system
υ̇=(λ_max(A_θ)+L_σζ√(β_𝕏(τ)))υ + ζη(x_ref)
with initial condition υ(0)=UUe(0) and constant
ζ=UU^-1b.
Due to the error dynamics in (<ref>), its solution is given by
e(t) = e^A_θte(0)+∫_0^t e^A_θ (t-t') b f_e(t')dt',
where f_e(t)=f(x(t))-μ(x(t)). Therefore, we directly obtain
e(t)≤e^A_θte(0)+∫_0^t e^A_θ (t-t') b |f̅_e(t')|dt',
where f̅_e(t) can be any function such that |f_e(t)|≤f̅_e(t).
Using the eigendecomposition of A_θ=UΛU^-1, it can be directly seen that
e^A_θtb≤UU^-1be^λ_max(A_θ)t.
Hence, we obtain
e(t)≤ UU^-1e(0)e^λ_max(A_θ)t
+UU^-1b∫_0^t e^λ_max(A_θ) (t-t') |f_e(t')|dt'.
The right handside of this inequality is again the solution of a differential equation such hat e(t)≤υ̃ for
υ̇̃̇=λ_max(A_θ)υ̃+UU^-1bf̅_e(t)
with υ̃(0)=UU^-1e(0).
It remains to derive a bound f̅_e(t) for |f_e(t)| in (<ref>). Due to <ref>, it holds that |f_e(t)|≤η_N(x(t)) for all x∈𝕏 with probability of at least 1-δ. Moreover, we have η_N(x(t))≤η_N(x_ref(t))+L_σ√(β_𝕏(τ))e(t) due to Lipschitz continuity of σ(·) guaranteed by <ref>. Therefore, it follows that
υ̇̃̇≤(λ_max(A_θ)+L_σζ√(β_𝕏(τ)))υ̃ + ζη(x_ref),
which concludes the proof.
Since η(x_ref) can be directly computed at any time instant, determining the tracking error bound using <ref> simply requires simulating the linear dynamical system (<ref>). This can be straightforwardly done for a given time horizon in contrast to similar prior approaches <cit.>, where the uniform error bound needs to be determined at the actual system state x. In order to achieve this improved practical applicability, additional requirements on the stability of the linear dynamics described by A_θ are necessary. It is obvious that (<ref>) only remains bounded if the linear dynamics (<ref>) are stable, which can be straightforwardly shown to require λ_max(A_θ)<-L_σζ√(β_𝕏(τ)).
Due to the dependency of the eigenvalue λ_max(A_θ) on the parameters θ, this condition can be satisfied if
θ≥α^-1(-L_σζ√(β_𝕏(τ))).
Therefore, this condition effectively poses a lower bound on the admissible control gains.
§.§ Dependency of Accuracy Guarantees on Data Density
While <ref> provides an accurate bound for the tracking error depending on the local data density,
it is challenging to apply this result to the asymptotic analysis of the tracking error. Therefore, we bound the maximum tracking
error along the reference trajectory as shown in the following proposition.
Consider a linear system (<ref>) satisfying <ref>, which is perturbed by
a L_f-Lipschitz nonlinearity f(·) satisfying <ref>. Assume that a zero
mean Gaussian process with L_k-Lipschitz stationary kernel is used to learn a model f̂(·)=μ(·) of f(·),
such that a controller (<ref>) is used to track the bounded
reference x_ref. If (<ref>) is satisfied, then, for e(0)=0, the maximum tracking error is bounded by sup_t≥ 0e(t)≤υ̅ with probability of at least 1-δ, where
υ̅ = -ζ/λ_max(A_θ)+L_σζ√(β_𝕏(τ))sup_t≥ 0η(x_ref(t)).
It immediately follows from (<ref>) that
e(t) ≤
ζ∫_0^t e^(λ_max(A_θ)+L_σζ√(β_𝕏(τ))) (t-t') dt'sup_0≤ t'≤ tη(x_ref(t')).
Since the integral can be straightforwardly calculated, we obtain
sup_t≥ 0e(t)≤ -ζsup_t≥ 0η(x_ref(t))/λ_max(A_θ)+L_σζ√(β_𝕏(τ)),
which concludes the proof.
Note that the restriction to a zero initial condition is only considered to simplify the derivation, but the extension to non-zero initial conditions is straightforward. Therefore, the assumptions of <ref> are not more restrictive than those of <ref>.
In order to analyze the asymptotic behavior of the tracking error, we combine <ref> with <ref>. Using the shorthand notation ρ=inf_t≥ 0ρ(x_ref(t)), this results in the following theorem.
Consider a linear system (<ref>) satisfying <ref>, which is perturbed by
a L_f-Lipschitz nonlinearity f(·) satisfying <ref>. Assume that a zero
mean Gaussian process with L_k-Lipschitz stationary kernel is used to learn a model f̂(·)=μ(·) of f(·),
such that a controller (<ref>) is used to track the bounded
reference x_ref.
Choose τ such that β_𝕏(τ)≥γ^2(τ)ρk(0)/2 and θ such that
κ=-2ζ√(β_𝕏(τ))/λ_max(A_θ)+L_σζ√(β_𝕏(τ))
is constant and (<ref>) is satisfied.
Then, for e(0)=0, the maximum tracking error bound asymptotically behaves as
υ̅∈𝒪(1/√(ρ)).
We first focus on the asymptotic behavior of the maximum learning error bound along the reference sup_t≥ 0η(x_ref(t)), which can be expressed as
sup_t≥ 0η(x_ref(t)) = √(β_𝕏(τ))sup_t≥ 0σ(x_ref(t))+γ(τ).
Due to <ref>, the considered parameter β_𝕏(τ) implies
sup_t≥ 0σ(x_ref(t))≥γ(τ)/√(β_𝕏(τ)),
such that we can simplify the learning error bound to
sup_t≥ 0η(x_ref(t)) ≤ 2√(β_𝕏(τ))sup_t≥ 0σ(x_ref(t)).
Therefore, it follows from proposition <ref> that
υ̅ = κsup_t≥ 0σ(x_ref(t)),
whose asymptotic behavior only depends on σ(x_ref(t)) due to the assumed constant value of κ̃, i.e., υ̅∈𝒪(sup_t≥ 0σ(x_ref(t))).
Due to <ref>, we have sup_t≥ 0σ(x_ref(t))∈𝒪(1/√(ρ)),
which concludes the proof.
This theorem establishes a direct relationship between the minimum data density ρ along the reference trajectory x_ref(t) and the maximum of the tracking error e, showing that an arbitrarily small tracking error can be guaranteed when suitable data is available. Since this requires a vanishing γ(τ), β_𝕏(τ) must grow. The chosen β_𝕏(τ) in <ref> satisfies this property. In order to see this note that √(β_𝕏(τ)) is growing with decreasing τ and γ(τ)∈𝒪(Nτ) holds for stationary kernels. Therefore, we can set τ∝1/(N√(ρ)), which directly yields β_𝕏(τ)∝log(N√(ρ)). Due to condition (<ref>), this increase rate of β_𝕏(τ) finally requires reducing eigenvalues -λ_max(A_θ)∝√(log(N√(ρ))).
While this increase requirement might seem like a restrictive assumption, it is important to note that without learning, it follows from the proof of <ref> that -λ_max(A_θ)∝1/υ̅. In contrast, we immediately obtain ρ∝1/υ̅^2 from (<ref>), such that -λ_max(A_θ)∝√(log(N/υ̅)) holds. Assuming the number of training samples N grows at most polynomially with ρ as ensured, e.g., for the case of SE or Matérn kernels with uniformly distributed training data discussed in <ref>, this finally implies -λ_max(A_θ)∈𝒪(√(log(1/υ̅))). Therefore, the requirement on the growth rate for ensuring arbitrarily small tracking errors reduces from hyperbolic to log-hyperbolic with suitable training data.=-1
§.§ Episodic Data Generation for Prescribed Performance
Although <ref> provides conditions for training data to ensure an arbitrarily small tracking error e,
it does not provide direct insights how suitable training data sets can be obtained. Therefore, we develop an episodic approach for generating training data sets in this section. For simplicity, we consider a constant sampling time T_s∈ℝ_+ during each episode with execution time T_p∈ℝ_+, which yields data sets of the form
𝔻_N^T_s={(x(iT_s),f(x(iT_s))+ϵ^(i)) }_i=0^N_p,
where N_p = ⌊ 1+T_p/T_s⌋ denotes the number of training samples gathered during one episode. Therefore, the tracking error bound υ̅ from one episode immediately provides guarantees for the training data of the next episode. We exploit this by adjusting the sampling time T_s and the maximum eigenvalue λ_max(A_θ) as demonstrated in <ref> in order to ensure a sufficiently small error bound for the next episode. This dependency on the sampling time is emphasized by an index T_s in the posterior standard deviation σ_T_s(·). As shown in the following theorem, this approach guarantees the termination of <ref> after a finite number of iterations.
Consider a linear system (<ref>) satisfying <ref>, which is perturbed by
a L_f-Lipschitz nonlinearity f(·) satisfying <ref>. Assume that a zero
mean Gaussian process with L_k-Lipschitz stationary kernel is used to learn a model f̂(·)=μ(·) of f(·),
such that a controller (<ref>) is used to track the bounded
reference x_ref. If θ and T_s are chosen such that
-λ_max(A_θ) ≥8√(L_∂ k) )+ξ L_σ/ξζ√(β_𝕏(τ))
max_0≤ t ≤ T_pσ^2_T_s(x_ref(t)) ≤ 16L_∂ kυ̅_i-1^2
holds in every episode for ξ<1,
<ref> terminates after at most
N_E=⌈log(4e̅√(L_∂ k))-log(√(k(0)))/log(ξ)⌉
episodes with probability of at least 1-N_Eδ.
It is straightforward to see that (<ref>) together with <ref> implies
υ̅_0 =κ√(k(0)),
υ̅_i+1 =4√(L_∂ k)κυ̅_i
for τ such that (<ref>) is satisfied,
where the index i is used to denote the episode.
Since 4√(L_∂ k)κ≤ξ<1 holds due to (<ref>), it immediately follows that υ̅_i decays exponentially, i.e., υ̅_i=ξ^iυ̅_0 with probability of at least 1-δ for each episode. Therefore, <ref> is guaranteed to terminate after N_E episodes with probability of at least 1-N_Eδ due to the union bound. =-1
Due to the exponential decay of the tracking error bound υ̅ ensured by <ref>, <ref> quickly terminates. This comes at the price of higher requirements (<ref>) on the eigenvalues of A_θ compared to <ref>. However, the difference is merely a constant factor, and it is indeed straightforward to see that -λ_max(A_θ)∝1/√(log(e̅)) is sufficient to compensate the effect of an increasing β_𝕏(τ) for all polynomially growing data sets. Therefore, this requirement is still significantly lower compared to ensuring the tracking error bound e̅ without learning as discussed in <ref>.
While the results in previous sections posed requirements on the data distribution in terms of the data density ρ(x), <ref> explicitly considers the data generation process by providing an upper bound for the sampling time T_s in (<ref>). Due to the form of this condition, it cannot be computed before the controller is applied to the system, but it can easily be verified a posteriori. Therefore, we can ensure it via a sufficiently high sampling rate during the application of the controller, such that we simply can downsample the obtained data to the necessary sampling time T_s. The required maximum sampling rate can be bounded using the following proposition.
Consider a linear system (<ref>) satisfying <ref>, which is perturbed by
a L_f-Lipschitz nonlinearity f(·) satisfying <ref>. Assume that a zero
mean Gaussian process with L_k-Lipschitz stationary kernel is used to learn a model f̂(·)=μ(·) of f(·),
such that a controller (<ref>) is used to track the continuous, bounded
reference x_ref.
Then, the sampling time T_s required by condition (<ref>) in <ref> is bounded by
T_s≥T_s=16L_∂ ke̅^3/σ_on^2max_0≤ t ≤ T_pẋ(t).
We prove this proposition by deriving a value of T_s which satisfies (<ref>)
Due to <ref>, (<ref>) is guaranteed to hold if ρ≥1/(8L_∂ kσ_f^2 υ̅_i-1^2). Set ρ'=1/(8L_∂ kσ_f^2 υ̅_i-1^2). Then, it follows from <ref> that
𝔹_2υ_i-1(x_ref(t))⊂𝕂_ρ'(x_ref(t)).
The Euclidean ball around x_ref(t) on the left handside can be inner bounded by a Euclidean ball with half the radius around the actual trajectory, i.e.,
𝔹_υ̅_i-1(x(t))⊂𝔹_2υ̅_i-1(x_ref(t)).
The smaller Euclidean ball has a diameter of υ̅_i-1 and the actual trajectory passes through its center. Moreover, the distance between two samples can be bounded by T_s max_0≤ t ≤ T_pẋ(t). Note that the maximum temporal derivative of the state is bounded. In order to see this, note that we can express the dynamics of the system as
ẋ=ẋ_ref+A_θe+b(f(x)-μ(x).
Due to the bounded prediction error, the bounded tracking error and the continuous reference trajectory, we can therefore bound the state derivative by
max_0≤ t≤ T_pẋ(t) ≤(A_θ+√(β_𝕏(τ))L_σ)υ̅_i+max_0≤ t≤ T_pη(x_ref(t))
+max_0≤ t≤ T_pẋ_ref(t).
This allows us to bound the number of points in 𝕂_ρ'(x_ref(t)) by
|𝕂_ρ'(x_ref(t))|≥ |𝔹_υ̅_i-1(x(t))|≥2υ̅_i-1/
T_smax_0≤ t ≤ T_pẋ(t).
For ρ≥ρ', it must hold that
2υ_i-1/
T_smax_0≤ t ≤ T_pẋ(t)≥ρ'σ_on^2k(0)=σ_on^2/8L_∂ kυ_i-1^2
due to (<ref>). This inequality can be ensured to hold by setting
T_s=16L_∂ kυ̅^3_i-1/σ_on^2max_0≤ t ≤ T_pẋ(t),
which concludes the proof.
§ NUMERICAL EVALUATION
In order to demonstrate the flexibility and effectiveness of the derived theoretical results, we compare the tracking error bounds with empirically observed tracking errors in different simulations. In <ref>, we evaluate the time-varying
tracking error bound for training data unevenly distributed over the relevant part of the state space 𝕏. The behavior of the asymptotic error bound is investigated in <ref>. Finally, we demonstrate the effectiveness of the proposed episodic data generation approach for ensuring a desired tracking accuracy in <ref>.
§.§ Data-dependency of Safety Regions
For evaluating the time-varying tracking error bound, we consider a
nonlinear dynamical system
ẋ_1=x_2, ẋ_2=f(x)+g(x)u,
where and g(x)= 1+1/2sin(x_2/2), which is a marginal variation of the system considered in <cit.>.
Assuming exact knowledge of g(·), we can approximately feedback linearize this system and apply a linear tracking controller u_lin=-θ_1θ_2 x_2-θ_2 x_2, where θ_1,θ_2∈ℝ_+ are
design parameters. This yields a
two-dimensional system of the form (<ref>)
with
A_θ=[ 0 1; -θ_1θ_2 -θ_2 ] b=[ 0; 1 ].
In order to demonstrate the effect of the distribution, we use a uniform
grid over [0 3]×[-4 4] with 25 points and σ_on^2 = 0.01 as
training data set, such that half of the considered state
space 𝕏 =[-5 5]^2 is not covered by training data.
A SE kernel with automatic
relevance determination is employed for Gaussian process regression
and the hyperparameters are optimized using likelihood maximization.
For computing the uniform prediction error bound in
<ref>, we
set τ=0.01, δ=0.01 and L_f=2.
The task is to track the circular reference trajectory x_d(t) = 2sin(t) with state
x_1, which leads to the reference trajectory x_ref(t)=[2sin(t) 2cos(t)]^T.
We aim to achieve this using θ_1=10 and θ_2=20, which can be shown to satisfy condition
(<ref>).
Snapshots of the resulting trajectory together with visualizations of the tracking error bounds
obtained using <ref> are illustrated in <ref>. When
the GP standard deviation σ(x_ref) is large, the tracking error
bound υ(t) starts to increase, such that it reaches its maximum just before the system
enters the region with low standard deviation. Afterwards, the feedback
controller reduces the tracking error until the standard deviation starts
to increase again. This leads to the minimum tracking error bound illustrated on the left of
<ref>.
This effect can also be seen at the observed tracking error as illustrated in <ref>, which has its peaks at times when the tracking error bound υ is large. Therefore, the tracking error bound υ reflects the behavior of the observed error e well, even though it is rather conservative. The sources of this conservatism can be easily investigated by determining the bound obtained when using the true model error |f(x_ref)-μ(x_ref)|
as input in (<ref>).
It is clearly visible that even with the knowledge of the true prediction error, the tracking error bound
exhibits some conservatism due the linearization around the reference trajectory x_ref.
The remaining conservatism is a consequence of the prediction error bound η(x_ref) as visualized at the bottom of <ref>.
Even though this bound reflects the availability of data well, it needs to capture the probabilistic worst case and is therefore considerably larger than the actual prediction error |f(x_ref)-μ(x_ref)|. This leads to the fact that the tracking error bound υ conservatively reflects the behavior of the observed tracking error e.
Note that the usage of a probabilistic Lipschitz constant L̂_f obtained via <ref>
does not significantly change this behavior. The corresponding tracking error bound merely
becomes slightly larger since we can compensate the conservative value of L̂_f using a smaller value τ=10^-3. Therefore, <ref> enables the effective computation
of prediction error bounds without knowledge of a Lipschitz constant of the unknown function f(·).
§.§ Dependency of the Tracking Accuracy on the Data Density
In order to investigate the dependency of the tracking error bound υ on the data density ρ in more detail, we consider the same setting as in <ref>, but use grids with different grid constants defined on [-4,4]^2 as training data sets, such that they cover the whole relevant domain. Due to the varying size of the training data set, we determine τ by finding the maximum value satisfying (<ref>) using a line search. We set θ_1=θ_2=θ, such that we can compute a gain θ ensuring κ=10 in (<ref>) for the obtained value of τ.
The resulting tracking errors e and bounds sup_t≥0υ(t) obtained with <ref> for different data densities ρ are illustrated in <ref>. Moreover, the asymptotic decay rate of υ̅ guaranteed by <ref> is depicted. It can be clearly seen that the asymptotic decay rate closely reflects the actual decay rate of the error bound sup_t≥0υ(t). Analogously to <ref>, the tracking error bound is rather conservative, but the
observed error e exhibits a decay rate with high similarity to its bound sup_t≥0υ(t). Despite this conservatism, the necessary maximum eigenvalues λ_max(A_θ) for ensuring a low desired tracking error bound sup_t≥0υ(t) with such training data are significantly
larger than without a controller compensating the nonlinearity as depicted in <ref>. This baseline comparison can be straightforwardly obtained as λ_max(A_θ)≥ζf̅/e̅
by slightly adapting the proof of <ref> using |f(x)|≤f̅ and μ(x)=0. Due to the linear growth of this
condition with 1/e̅, it quickly exceeds the maximum eigenvalue λ_max(A_θ) ensuring the same tracking error bound through the learned controller, even though we use
the non-conservative bound f̅=3. This clearly demonstrates the benefits of the derived theoretical results.
§.§ Episodic Data Generation
For evaluating the episodic data generation using <ref>, we consider the same setting as in <ref>. Moreover, we set θ_1=θ_2=θ analogously to the previous section and choose θ such that ξ=0.95 holds in every iteration.
A high frequency data set with sampling time 3· 10^-4 is generated in every episode, such that a line search can be used to determine the maximum value of T_s satisfying (<ref>).
The tracking error bounds obtained form <ref> with these parameters are exemplarily illustrated for several
different episodes in <ref>. Due to the constant sampling time, the training data density along the
reference is very similar within an episode, which directly leads to the rather minor variations in the tracking error bound
over time. Moreover, it can be seen that decrease of the tracking error bound υ is significantly larger during the first few episodes, before it slows down. This becomes even clearer when plotting the behavior of the error bound over the number of episodes as depicted in <ref>. During the first 10 episodes the error bound sup_t≥0υ(t) decays faster than the guaranteed rate of ξ^N_Eυ̅_0, which is guaranteed by <ref>. This can be attributed to the fact that even a single additional data point reduces the posterior variance more than required for (<ref>) at the beginning. Once a sufficiently large number of additional training samples is necessary to ensure (<ref>), this inaccuracy is overcome and the error bound sup_t≥0υ(t) closely follows the guaranteed decrease rate. In fact, the tracking error bound sup_t≥0υ(t), while being rather conservative similar to the
previous simulations, even reflects the behavior of the actually observed tracking error e accurately after 10 episodes.
Note that this unexpected fast decay at the beginning has no influence on the
required maximum eigenvalues λ_max(A_θ) as depicted in <ref>. While smaller eigenvalues are required for the episodic approach
compared to the asymptotic analysis in <ref>, the maximum eigenvalue λ_max(A_θ) used in <ref> closely follow
the expected 𝒪( log(1/sup_t≥0υ(t))) behavior. Moreover, it can be directly seen that <ref> offers a significant advantage over a direct reduction of the tracking error bound using the maximum eigenvalue λ_max(A_θ) without a compensation of the nonlinearity. Note that the sampling time T_s necessary to achieve this behavior quickly decays as illustrated in <ref>. However, since it remains significantly larger than its theoretical bound T_s, it remains
in magnitudes which can be realized in practice. Therefore, <ref> provides an effective method for generating data, such that an arbitrary tracking error can be ensured when using a GP model for compensating unknown nonlinearities in systems of the form of (<ref>).=-1
§ CONCLUSION
This paper presents a novel, episodic approach for learning GP models in order
to ensure an arbitrarily high desired tracking accuracy using the GP to
compensate unknown nonlinearities in linear systems. We first derive
a novel Bayesian prediction error bound for GP regression and demonstrate the
straightforward computability of all required parameters. In order to establish
a straightforwardly interpretable connection between training data and prediction accuracy,
we propose a kernel-dependent measure of data density and show that the prediction error
bound vanishes with increasing data density. We exploit the Bayesian error bounds to
derive a time-varying tracking error bound when using
the GP model to compensate unknown nonlinearities, and show that the tracking accuracy grows
with increasing data density. These theoretical results allow us to develop an
episodic approach for learning a GP model, such that a desired tracking error bound
can be guaranteed. The effectiveness of our theoretical results is demonstrated in
several simulations.=-1
IEEEtran
[
< g r a p h i c s >
]Armin Lederer
(S'20)
received the B.Sc. and M.Sc. degree in electrical engineering and information technology from the Technical University of Munich, Germany, in 2015 and 2018, respectively.
Since June 2018, he has been a PhD student at the Chair of Information-oriented Control, Department of Electrical and Computer Engineering at the Technical University of
Munich, Germany. His current research interests
include the stability of data-driven control systems and machine learning in closed-loop systems.
[
< g r a p h i c s >
]Jonas Umlauft
(S’14) received the B.Sc. and M.Sc. degree in electrical engineering and information technology from the Technical University of Mu- nich, Germany, in 2013 and 2015, respectively. His Master’s thesis was completed at the Computational and Biological Learning Group at the University of Cambridge, UK. Since May 2015, he has been a PhD student at the Chair of Information-oriented Control, Department of Electrical and Computer Engineering at the Technical University of Munich, Germany. His current research interests
include the stability of data-driven control systems and system identification based on Gaussian processes.
[
< g r a p h i c s >
]Sandra Hirche
(M'03–SM'11–F'20)
received the Dipl.-Ing degree in aeronautical engineering from the Technical University of Berlin, Berlin, Germany, in 2002, and the Dr. Ing. degree in electrical engineering from the Technical University of Munich, Munich, Germany, in 2005. From 2005 to 2007, she was awarded a Post-doctoral scholarship from the Japanese Society for the Promotion of Science at the Fujita Laboratory, Tokyo Institute of Technology, Tokyo, Japan. From 2008 to 2012, she was an Associate Professor with the Technical University of Munich. Since 2013, she has served as Technical University of Munich Liesel Beckmann Distinguished Professor and has been with the Chair of Information-Oriented Control, Department of Electrical and Computer Engineering, Technical University of Munich. She has authored or coauthored more than 150 papers in international journals, books, and refereed conferences. Her main research interests include cooperative, distributed, and networked control with applications in human–machine interaction, multirobot systems, and general robotics.
Dr. Hirche has served on the editorial boards of the IEEE Transactions on Control of Network Systems, the IEEE Transactions on Control Systems Technology, and the IEEE Transactions on Haptics. She has received multiple awards such as the Rohde & Schwarz Award for her Ph.D. thesis, the IFAC World Congress Best Poster Award in 2005, and – together with students – the 2018 Outstanding Student Paper Award of the IEEE Conference on Decision and Control as well as Best Paper Awards from IEEE Worldhaptics and the IFAC Conference of Manoeuvring and Control of Marine Craft in 2009.
|
http://arxiv.org/abs/2307.04493v1 | 20230710113115 | Geometric Constraints in Probabilistic Manifolds: A Bridge from Molecular Dynamics to Structured Diffusion Processes | [
"Justin Diamond",
"Markus Lill"
] | cs.LG | [
"cs.LG",
"q-bio.QM"
] |
[
Geometric Constraints in Probabilistic Manifolds: A Bridge from Molecular Dynamics to Structured Diffusion Processes
equal*
Justin Diamondyyy
Markus Lillyyy
yyyDepartment of Pharmaceutical Sciences, University of Basel, Basel, Switzerland
Justin [email protected]
Machine Learning, ICML
0.3in
]
Understanding the macroscopic characteristics of biological complexes demands precision and specificity in statistical ensemble modeling. One of the primary challenges in this domain lies in sampling from particular subsets of the state-space, driven either by existing structural knowledge or specific areas of interest within the state-space.
We propose a method that enables sampling from distributions that rigorously adhere to arbitrary sets of geometric constraints in Euclidean spaces. This is achieved by integrating a constraint projection operator within the well-regarded architecture of Denoising Diffusion Probabilistic Models, a framework founded in generative modeling and probabilistic inference.
The significance of this work becomes apparent, for instance, in the context of deep learning-based drug design, where it is imperative to maintain specific molecular profile interactions to realize the desired therapeutic outcomes and guarantee safety.
§ INTRODUCTION
Infinitesimal Dynamics in classical mechanics is commonly formalized by Lagrangians.
By solving for functionals that extremize the Lagrangian one obtains equations of motion. In molecular systems, e.g. Molecular Dynamics, the EOM are:
Md^2x/dt^2=-∇U - ∑_aλ_a∇σ_a,
where M is the diagonal mass matrix, x the cartesian coordinates, t is time,
and U is the potential energy. The σ_a are a set of holonomic constraints and λ_a are the Lagrange multiplier coefficients. To generalize from holonomic to nonholonomic constraints, one can use slack variables to transform the latter into the first.
Starting with z_x, z_h = f(x,h) = [x(0), h(0)] + ∫_0^1ϕ(x(t), h(t))dt with z being a latent vector sampled from Gaussians and the indexes x and h indicate the latent variables associated
to the coordinates of each particle and the vector embedding of each particle, ϕ is the parameterized transformation defined by a equivariant graph neural network. This defines a Neural ODE <cit.> which generalizes to Denoising Diffusion Probabilistic Models <cit.>.
This form of transformation has the same infinitesimal nature as our previous EOM which makes it acceptable to apply sets of constraints via Langrange's Multipliers, analogous to solving our EOM and thus one can insure the continual satisfaction of a set of constraints
using the Shake algorithm from Molecular Dynamics.
The study of constrained dynamics in Molecular Dynamics and Machine Learning, has traditionally focused on mostly linear constraints: e.g. removing high-frequency oscillations by constraining bond distances in the first and in-painting in the latter by thresholding certain pixel values to predetermined values. From a high level these can be seen as linear constraint problems as the constrained subset affects the unconstrained subset to minimal degrees. In addition, our task is more challenging as different constraints induce different geometric topological structures, such that some sets of distance constraints can determine uniquely the solution, and small modifications in the constraints may lead to vast changes in the solution set.
The problem we hope to model are non-linear constraints where constrained subsets of atoms determine the unconstrained subset to a high degree. We argue these types of non-linear constraints are important in the field of generative drug development where generated molecules must satisfy certain structural or analytic properties a priori. Take for instance, the optimization of lead molecules which is crucial at the final stages of drug development pipeline where off target interactions are attempted to be minimized. Since these off-target reactions can often be described by structural or analytic properties, then we can generate precisely molecules that satisfy a constraint profile of the target of interest, while specifying the subspace of generated molecules to not lie within the subspace of off-target interaction profiles.
In the following, we will give a summary of the Shake algorithm and segments of the equivariant normalaizing flow necessary to elaborate on how to combine them. Next, it will be elaborated that the spaces of latent embeddings and output samples are generally of very different nature, and constraints defined in one space will not necessarily be useful in the other. We suggest a continuous transformation of
the constraints such that they are always satisfied in the latent space, and become more restrictive throughout the integration. Lastly, we show simple examples where complex constraints are satisfied within small molecules. We leave to future work the study of this methodology to larger systems, and more application based studies. Our approach builds a fruitful junction where probabilistic inference, structured data representation, and generative modeling meet, while emphasizing the necessity to encode domain knowledge effectively in these settings, offering a way to formally verify the distributions from which samples are drawn.
§ PREVIOUS RESEARCH
Generative models of graphs have been a subject of interest in recent years. A number of different approaches have been proposed in the literature. <cit.> generates valid Euclidean distance matrices ensuring the resulting molecular structures are physically realistic which are then reconstructed in 3D space. In <cit.>, Boltzmann Generators sample equilibrium states of many-body systems with deep learning, useful for generating molecular configurations that obey thermodynamics distributions.
<cit.> proposed Equivariant Graph Neural Networks, which can be applied to model molecules and proteins while ensuring that their predictions are consistent under different orientations and permutations of the molecule.<cit.> further extended the concept to the diffusion process for 3D molecule generation. <cit.> applied similar methodologies to diffusion models on protein ligand complexes, and <cit.> devise a method of protein generation models that diffuse over harmonic potentials.
The Shake algorithm, described in a parallelized fashion by <cit.>, enforces linear constraints on molecular dynamics simulations of chemicals and biomolecules. This algorithm is conventionally used in simulations to get rid of high frequency motions, i.e. those seen in bonds between atoms.
§ CONSTRAINED GENERATIVE PROCESSES
§.§ Geometric Constraints in Shake
First, we define the constraint functions for the pairwise distance (not necessarily between bonded atoms), bond angle, and dihedral angle.
σ_d_ij = (d_ij - d_ij,0)^2 = 0
σ_θ_ijk = (θ_ijk - θ_ijk,0)^2 = 0
σ_ψ_ijkl = (ψ_ijkl - ψ_ijkl,0)^2 = 0
These constraint functions compare the current pairwise distance, bond angle, and dihedral angle with their target values, and the goal is to minimize the difference. We can additionally create nonholonomic constraints via slack variables. For example, we can add a slack variable y ≥ 0 and define d_j as the boundary of a nonholonomic constraint. Then, we can express the constraint as:
σ_a := ||x_aj - x_ak||^2_2 - d_j ≤ 0 → ||x_aj - x_ak||^2_2 - d_j + y= 0.
Next, modify the constraint matrix in the Shake algorithm to include pairwise distance, bond angle, and dihedral angle constraints seen in equation 4, where ij, ijk, and ijkl sum over the pairwise, bond angles, and torsion constraints indicating the number of atoms in each type of constraint type.
The constraint matrix now accounts for the pairwise distance, bond angle, and dihedral angle constraints by including their second-order derivatives with respect to the Cartesian coordinates by including their contributions to the Lagrange multipliers. After solving for the Lagrange multipliers, update the coordinates using the adjusted coordinate set equation like before.
It is also possible to try to optimize the coordinates via other optimization algorithms like ADAM or SGD.
In this section, we discuss the methods needed to understand how constraints can be represented, and define a novel diffusion process which projects the dynamics onto the submanifold defined by arbitrary sets of geometric constraints.
§.§ Shake Algorithm
The Shake algorithm takes as input a set of coordinates x of a molecular system and a set of constraints σ. At each time step the coordinates
are updated according to the equations of motion (EOM) at hand (without constraint terms) and subsequently are corrected. In general, the EOM will lead to dynamics that do not
satisfy the constraints, and thus this correction is mandatory.
Assuming masses of all the particles and delta time are unit we have the following equation for updating x_i iteratively until the constraints are satisfied.
x_i^(n)= x_i^(n-1) - ∑_bλ_b^(n-1)∇σ_b(x_i)
where x_i^(n) is the updated coordinate after n iterations of
satisfying constraints at each time step, x_i is the initial coordinates at each time step, and λ_b^(n-1) is the lagrange multiplier for each
constraint σ_a. The equation to solve at each iteration of each time step is
∑_βλ_β^(n-1)A_αβ^(n-1)= σ_α(x_i^(n-1))
with
A_αβ^(n-1)= ∇σ_α(x_i^(n-1)) ∇σ_β(x_i).
The matrix A^(n-1)_αβ is a symmetric matrix that describes how changes in particle positions affect both potential energy and constraint violations. The elements of the matrix are given by:
A^(n-1)_αβ = ∂^2 U/∂ x_α∂ x_β + ∑_k=1^N_cλ^(n-1)_k∂^2 σ_k/∂ x_α∂ x_β
where N_c is the number of constraints. The matrix A^(n-1)_αβ is used to solve for the Lagrange multipliers λ^(n)_β , which are then used to adjust particle positions.
§.§ Constraint-Induced Diffusion Process
Suppose we want to incorporate a constraint, such as a distance constraint between two atoms. Let's denote this constraint by f(x) = 0 for simplicity. We can modify the diffusion process to satisfy this constraint by projecting the noise term onto the nullspace of the gradient of the constraint function, analagous to the A matrix in Shake. This gives us:
dx = √(2D) (I - ∇ f(x) (∇ f(x))^T) dB - D ∇log p_t(x) dt
where D is the diffusion constant, B is a standard Brownian motion, and ∇log p_t(x) is the gradient of the log-probability density, which is equivalent to the negative of the potential energy function of the system.
Here, I is the identity matrix, and ∇ f(x) (∇ f(x))^T is the outer product of the gradient of the constraint function, which represents the direction in which the constraint is changing. This projection ensures that the noise term does not push the system out of the constraint-satisfying space.
The covariance matrix of the perturbed Gaussian distribution of the denoising process can be understood formally using the Schur complement method, available in the Appendix. The key takeaway is the relation between constraints and correlations via projecting out the constraints in the Covariance matrix of a Multivariate Gaussian. This modified covariance matrix then defines the perturbed Gaussian distribution from which we can sample at each time step of the diffusion process. This is a good approximation when the constraints are nearly linear or when the changes in the variables are small. One note is that in if the projection operator is non-linear than the the process is no longer Gaussian, but since we deal with linearized constraints, or small changes at each time step, this is negligible as seen in the original Shake formalism. However, the Schur Complement method gives a more general formalism to ensure Gaussian-ness.
§.§ Constraints as Correlations
Consider, for instance, a scenario involving pairwise distance constraints between a set of variables denoted as d = d_ij, where d_ij signifies the distance separating variables i and j. These constraints can be mathematically expressed through the set of functions C_ij(ϵ) = ||ϵ_i - ϵ_j|| - dij = 0, which is applicable to all corresponding variable pairs (i, j) ∈d, influencing the samples drawn from a Multivariate Normal distribution.
The introduction of these geometric constraints essentially interrelates variables that were initially independent in the Gaussian distribution. In order to comprehend the implications of these constraints, the covariance matrix Σ' of the perturbed distribution p'(ϵ') is worth examining:
Σ' = 𝔼_ϵ' ∼ p' [ϵ' (ϵ')^T] - 𝔼_ϵ' ∼ p' [ϵ'] 𝔼_ϵ' ∼ p' [ϵ']^T,
Here, the expectations are calculated over the perturbed distribution. The covariance matrix Σ' elucidates the correlations among variables that emerge as a result of the geometric constraints.
Importantly, these correlations, which are encoded within the covariance matrix of a multivariate Gaussian distribution, represent the constraints in the distribution. This provides a way to naturally incorporate constraint-based information into the model.
§.§ Training and Sampling Algorithms
§.§.§ Training Process
During training, in Algorithm 2, we first sample a time step t and noise vector ϵ from uniform and Gaussian distributions respectively. Then subtract the center of gravity from the noise vector to ensure that it lies on a zero center of gravity subspace. Then compute the latent variable z_t by scaling and adding the input coordinates [x,h] with the noise vector. Finally, minimize the difference between the estimated noise vector and output of the neural network to optimize EDM. For each molecule between 5 and 15 constraints are sampled from x for each batch element. The constraints are uniformly sampled from the pairs, triples, and quadruplets of the atom set of each molecule. This adds an extra layer of complexity due to the constraint distribution which we need to sample from the true data distribution.
§.§.§ Generative Process
In this generative process, we first sample a latent variable z_T from a Gaussian distribution. Then iterate backwards through time and sample noise vectors ϵ at each step. Subtract the center of gravity of the coordinates from the noise vector to ensure that it lies on a zero center of gravity subspace. Then compute the latent variable z_s by scaling and adding the input coordinates with the noise vector and previous latent variable. Finally, sample the input coordinates [x,h] from a conditional distribution given the initial latent variable z_0. The Shake algorithm enforces the constraints, as in training, at each sampling step during generation.
§ EXPERIMENTS
In the experimental section of our study, we evaluate our proposed method by generating molecules with cyclic constraints in Figure 1. The cyclic constraints impose specific geometric relationships among atoms in a molecule, such as the bond distances, bond angles, and torsional angles, which are essential for maintaining the chemical stability and physical plausibility of the generated molecules.
During the training phase, constraints are sampled from the dataset. This approach encourages the model to learn the distribution of constraints inherent in the training data, which reduces the Kullback-Leibler (KL) divergence between the data distribution and the model distribution. Consequently, the KL divergence during training is always minimized, promoting the model to generate molecules that closely resemble those in the training set.
For the practical implementation of this training procedure, we began with a pre-trained model provided by Welling et al.Our methodology then fine-tuned this pre-existing model using our constraint projection method. Due to time considerations and simplicity, our training and experiments focused on molecules consisting of 21 atoms.
§ DISCUSSION
Our method serves as a potent tool for incorporating complex constraints in denoising diffusion processes, specifically when dealing with multi-constraint specifications. Its iterative nature allows it to address nonlinear constraint problems and extends the power of denoising diffusion probabilistic models to work with constraints. Thus allowing these models to leverage the structure inherent in many physical systems. Indeed, many of these systems come with prior structural knowledge, including geometric information like distances, torsions, bond angles, and generalizeable to other piece-wise polynomial terms. Such information can significantly enhance the training process and enable explicit sampling of subsets of the state space.
Although constraints can guide generation towards more physically plausible structures, there can be potential instability in the generation process. This instability may originate from discrepancies between constraints used during training and those applied during generation. It underlines the need for further work to establish robust training procedures that align more closely with the generation constraints. Especially, with application focused studies like generating peptides or ligands with specific interaction profiles.
Though the language of our work is steeped in the semantics of Molecular Generation, the way we use geometric constraints to guide sampling mirrors a more general need of generative models in ML, which must navigate complex, structured probability spaces.
Further exploration could include adapting our methodology to discern constraints intrinsically or applying it to optimization processes like gradient-based learning and potentially lead to more efficient or robust learning algorithms.
§ APPENDIX A: GENERALIZED SCHUR COMPLEMENT FOR MULTIPLE CONSTRAINTS
To obtain a generalized approach of Schur Complement for multiple distance constraints, let's consider a set of M pairwise constraints between atoms. We can express each constraint as a function of the positions of the corresponding atoms:
f_m(𝐱_i, 𝐱_j) = ||𝐱_i - 𝐱_j||^2 - d_ij^2 = 0, m = 1, 2, …, M,
where d_ij is the distance constraint between atoms i and j.
To incorporate all the constraints, we can form the combined gradient and Hessian matrices by stacking the corresponding matrices for each constraint:
∇𝐟 = [ ∇ f_1 ∇ f_2 ⋮∇ f_M ],
∇^2 𝐟 = [ ∇^2 f_1 ∇^2 f_2 ⋮∇^2 f_M ].
To project the Gaussian distribution with the original covariance matrix Σ onto the space of distance constraints, we can use the following generalized Schur complement:
Σ' = Σ - Σ∇^2 𝐟^T (∇^2 𝐟Σ∇^2 𝐟^T)^-1∇^2 𝐟Σ.
While the Schur complement method can be implemented iteratively for non-linear systems, it is computationally intensive due to the inversion of the Hessian matrix. However, it serves as an excellent theoretical tool, providing a precise representation of how constraints can be formally incorporated into the diffusion process.
On the other hand, the Schur complement method provides a direct way to project the covariance matrix of the atomic positions onto the space that satisfies the distance constraints. It essentially modifies the covariance matrix in a way that embeds the constraints, without needing to adjust the atomic positions. This approach formally modifies the probability distribution of interest, and may be more useful for theoretic insight.
§ APPENDIX C: NONHOLONOMIC CONSTRAINTS
We are more interested in nonholonomic constraints where each constraint has possibly a lower and upper bound. As we mentioned earlier,
by adding a slack variable one can translate the nonholonomic constraints to holonomic ones. To formalize this, one sees that a constraint having
a lower and upper bound will either be completely satisfied or fail to satisfy a single boundary. Thus, we only have to consider
at most one holonomic constraint at each call to Shake meaning each constraint with a lower and upper bound may be replaced by a lower, upper,
or no bound for each call.
To calculate the slack variable y from σ_jk:=‖ x^l_i-x^l_j ‖ - d_jk which is ≤ or ≥ 0, one has
y={[ max(0,||x^l_i-x^l_j||-d^u_jk), if ≤; max(0,d^l_jk-||x^l_i-x^l_j||), if ≥ ].
where d_jk is the lower or upper bound in case of nonholonomic constriants and the defined constraint
value for holonomic constraints.
In the generative process, we define the initial values of d_jk such that the constraints have little effects. The constraints are then linearly interpolated throughout the ODE until the predetermined boundary values of d_jk are reached.
§ APPENDIX B: INCORPORATION OF LOGICAL OPERATORS IN GEOMETRIC CONSTRAINTS
The application of logical operators such as'AND', 'OR' and 'NOT' within geometric constraints enables a more flexible and representative modeling of physical and chemical systems. Real-world scenarios frequently require the satisfaction of multiple constraints following complex logical rules. Below, we detail the basic implementation of 'OR' and 'NOT' logical operators within the geometric constraints of our diffusion process while noting that the 'AND' operator is the basis of the formalism:
§.§ 'OR' Logic
The 'OR' condition necessitates that at least one of two (or more) constraints be met. Let's denote two constraint functions as f_1(x) and f_2(x). The 'OR' logic can be integrated by constructing a composite constraint function that is satisfied when any of its constituent constraints is met. We can express this as:
g(x) = min(f_1(x), f_2(x))
In this case, if either f_1(x) = 0 or f_2(x) = 0 (or both), g(x) = 0, thereby meeting the 'OR' condition. Alternatively, we can employ a product of the constraints:
g(x) = f_1(x) · f_2(x)
If either f_1(x) = 0 or f_2(x) = 0 (or both), g(x) = 0, again adhering to the 'OR' logic. This method requires that both f_1(x) and f_2(x) are always non-negative.
§.§ 'NOT' Logic
The "NOT" operator in the context of geometric constraints could be defined using the following equations. Let's say we have a constraint f(x) = 0. We want to define a NOT operator for this constraint. We can then define "NOT f(x)" as regions where f(x) does not equal zero, which can be represented with two inequality constraints which can be combined via the 'OR' operator to designate the 'NOT' operator.
We denote ϵ as a small positive number, then "NOT f(x)" can be represented as:
g_1(x) = f(x) + ϵ < 0
g_2(x) = f(x) - ϵ > 0
In the equations above, we have defined two regions (when f(x) is smaller than -ϵ and larger than ϵ) where "NOT f(x)" is true, thus defining a NOT operator for our constraints. Note that these regions depend on the choice of ϵ.
|
http://arxiv.org/abs/2307.04853v1 | 20230710185012 | Spatially-Resolved Recent Star Formation History in NGC 6946 | [
"Debby Tran",
"Benjamin Williams",
"Emily Levesque",
"Margaret Lazzarini",
"Julianne Dalcanton",
"Andrew Dolphin",
"Brad Koplitz",
"Adam Smercina",
"O. Grace Telford"
] | astro-ph.GA | [
"astro-ph.GA"
] |
0000-0002-6440-1087]Debby Tran
Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195, USA
0000-0002-7502-0597]Benjamin Williams
Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195, USA
0000-0003-2184-1581]Emily Levesque
Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195, USA
0000-0002-0786-7307]Margaret Lazzarini
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 1200 E California Boulevard, Pasadena, CA 91125, USA
0000-0002-1264-2006]Julianne Dalcanton
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
0000-0001-8416-4093]Andrew Dolphin
Raytheon Technologies, 1151 E. Hermans Road, Tucson, AZ 85756, USA
University of Arizona, Steward Observatory, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
0000-0001-5530-2872]Brad Koplitz
School of Earth & Space Exploration, Arizona State University, 781 Terrace Mall, Tempe, AZ 85287, USA
0000-0003-2599-7524]Adam Smercina
Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195, USA
0000-0003-4122-7749]O. Grace Telford
Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA
The nearby face-on star forming spiral galaxy NGC 6946 is known as the Fireworks Galaxy due to its hosting an unusually large number of supernova. We analyze its resolved near-ultraviolet (NUV) stellar photometry measured from images taken with the Hubble Space Telescope's (HST) Wide Field Camera 3 (WFC3) with F275W and F336W filters. We model the color-magnitude diagrams (CMD) of the UV photometry to derive the spatially-resolved star formation history (SFH) of NGC 6946 over the last 25 Myr. From this analysis, we produce maps of the spatial distribution of young stellar populations and measure the total recent star formation rate (SFR) of nearly the entire young stellar disk. We find the global SFR(age≤25 Myr)=13.17 +0.91
-0.79 M_⊙/ yr. Over this period, the SFR is initially very high (23.39+2.43
-2.11 M_⊙/ yr between 16-25 Myr ago), then monotonically decreases to a recent SFR of 5.31+0.19
-0.17 M_⊙/ yr in the last 10 Myr. This decrease in global star formation rate over the last 25 Myr is consistent with measurements made with other SFR indicators. We discuss in detail two of the most active regions of the galaxy, which we find are responsible for 3% and 5% of the total star formation over the past 6.3 Myr.
§ INTRODUCTION
Star formation rate (SFR) is one of the defining characteristics in determining the current evolutionary state of a galaxy. The SFR strongly affects evolution through metal production <cit.>, gas consumption <cit.>, cold gas content <cit.>, and feedback in the galaxy <cit.>. Thus, SFR is a key property in tests of galaxy evolution models <cit.>. Because of its significance, many methods of measuring star formation rate with observational data have been developed, such as measuring UV emission from young (≲10 Myr) massive stars <cit.>, Hα emission from the youngest (≲5 Myr) massive stars <cit.>, and estimating SFR from the rate of core-collapse supernova (ccSN) <cit.>, which probes SFR at timescales at which stars supernova (30-100 Myr ago). These methods probe a range of timescales, making direct comparisons between SFR measured with different indicators challenging. Star formation histories (SFHs) avoid this problem by providing the SFR over time, allowing us to compare and calibrate SFR measurements obtained with different methods, while revealing how galaxies change over time.
Even more powerful are spatially-resolved star formation histories, which provide both temporal and spatial information. With these, we can trace local mechanisms that could be triggering star formation. For nearby galaxies with resolved stellar photometry, we can construct and fit observed color-magnitude diagrams (CMDs) to infer a star formation history for a specific region, assuming a specific initial mass function (IMF), stellar evolutionary model, binary fraction, and distribution of dust. By tiling together SFHs from multiple regions, we can construct a spatially-resolved star formation history for a galaxy. This kind of work has been done in the Small and Large Magellenic Clouds <cit.>, M31 <cit.>, M33 <cit.>, and M81 <cit.>.
In this paper, we apply this technique to NGC 6946, which has been widely studied due to its active star formation (<cit.> classify it as a circumnuclear starburst) and the high frequency of supernovae in the past century <cit.>. Among these studies, there have been inconsistent measurements of the galaxy's global star formation rate, ranging widely from 3-12 M_⊙/ yr, due to the diverse methods of measuring star formation rate and wide range of different distances used. Throughout this paper, we use a distance of 7.83 ± 0.29 Mpc <cit.> and an inclination of 32.8 <cit.>. <cit.> has explored the accuracy of these various diagnostics for a sample of regions in NGC 6946, finding discrepancies of up to factors of 5. To better constrain the recent star formation history across the entire galaxy, we have carried out a NUV HST survey to obtain photometry of the young massive stellar population of NGC 6946. This dataset provides the most detailed and complete probe to date of the global, localized, and episodic star formation in NGC 6946.
In Section <ref>, we present the HST observations, alignment of the data, photometry, artificial star tests for measuring photometric uncertainties, gridding schema, and method for measuring star formation rates. In Section <ref>, we present the recent star formation rates, the reliability of the SFRs at the youngest and oldest time bins, total stellar mass formed over the past 25 Myr, and foreground and differential extinction of each cell. In Section <ref>, we discuss two highly star-forming regions of interest, the decline in global star formation rate, and the correlation between stellar density and age. In Section <ref>, we summarize our methods and findings.
§ OBSERVATIONS AND DATA ANALYSIS
Observations for this program (GO-15877; PI <cit.>) were obtained between May 11 2020 and November 21 2021 using HST's WFC3 Ultraviolet- (UVIS) channel in filters F275W and F336W. Details of the observations are found in Table <ref>. NGC 6946 was imaged in a 4x4 grid excluding the northernmost and southernmost regions (Figure <ref>). This covers all of the UV-bright regions and the locations of observed core collapse supernovae. Each neighboring pointing overlaps at the edges to ensure there are no gaps in the catalog due to poorer photometric quality at the edges. Each pointing in both filters was dithered with small offsets to control for hot pixels and cosmic rays. Unfortunately, even with the careful selection of observing strategy, there are two small gaps of 10×1 and 30×1 approximately centered at 20:34:39.40 +60:06:80 and 20:34:25.00 +60:08:80, respectively. These gaps are due to adjusting the rotation of two pointings to obtain a sufficient number of guide stars. Upon comparison with existing optical data, there do not appear to be dense star clusters in these two gaps.
cccccccc
1
Details of Fields Observed
0pt
Field Name R.A. Dec Filters
Exposure Time Number of Date Roll Angle
(hh:mm:ss.sss) (::.)
(s) Exposures (YYYYMMDD) (PA_V3)
NGC6946-2 20:35:06.007 +60:12:48.93 F275W 1432 2 20201106 257.0
NGC6946-2 20:35:06.007 +60:12:48.93 F275W 1414 2 20201106 257.0
NGC6946-3 20:34:38.218 +60:12:59.17 F275W 1432 2 20201109 256.5
NGC6946-3 20:34:38.218 +60:12:59.17 F275W 1414 2 20201109 256.5
NGC6946-4 20:35:19.114 +60:10:49.51 F275W 1432 2 20201110 255.7
NGC6946-4 20:35:19.114 +60:10:49.51 F275W 1414 2 20201110 255.7
NGC6946-5 20:34:51.355 +60:10:59.92 F275W 1432 2 20201103 257.0
NGC6946-5 20:34:51.355 +60:10:59.92 F275W 1414 2 20201103 257.0
NGC6946-6 20:34:23.591 +60:11:09.97 F275W 1432 2 20201103 257.0
NGC6946-6 20:34:23.591 +60:11:09.97 F275W 1414 2 20201103 257.0
NGC6946-7 20:35:32.194 +60:08:50.01 F275W 1432 2 20201112 257.0
NGC6946-7 20:35:32.194 +60:08:50.01 F275W 1414 2 20201111 257.0
NGC6946-8 20:35:04.464 +60:09:00.59 F275W 1432 2 20201112 253.5
NGC6946-8 20:35:04.464 +60:09:00.59 F275W 1414 2 20201112 253.5
NGC6946-9 20:34:36.730 +60:09:10.81 F275W 1371 2 20210504 75.5
NGC6946-9 20:34:36.730 +60:09:10.81 F275W 1390 2 20210504 75.5
NGC6946-10 20:34:09.061 +60:09:21.20 F275W 1432 2 20200515 73.0
NGC6946-10 20:34:09.061 +60:09:21.20 F275W 1414 2 20200515 73.0
NGC6946-11 20:35:17.548 +60:07:01.19 F275W 1432 2 20201112 253.0
NGC6946-11 20:35:17.548 +60:07:01.19 F275W 1414 2 20201112 253.0
NGC6946-12 20:34:49.842 +60:07:11.57 F275W 1432 2 20201113 253.0
NGC6946-12 20:34:49.842 +60:07:11.57 F275W 1414 2 20201113 253.0
NGC6946-13 20:34:22.203 +60:07:22.13 F275W 1432 2 20200511 73.0
NGC6946-13 20:34:22.203 +60:07:22.13 F275W 1414 2 20200511 73.0
NGC6946-14 20:35:02.928 +60:05:12.26 F275W 1432 2 20201107 257.0
NGC6946-14 20:35:02.928 +60:05:12.26 F275W 1414 2 20201107 257.0
NGC6946-15 20:34:35.318 +60:05:22.98 F275W 1362 2 20211114 257.0
NGC6946-15 20:34:35.318 +60:05:22.98 F275W 1361 2 20211115 257.0
NGC6946-2 20:35:06.007 +60:12:48.93 F336W 880 3 20201106 257.0
NGC6946-3 20:34:38.218 +60:12:59.17 F336W 880 3 20201109 256.5
NGC6946-4 20:35:19.114 +60:10:49.51 F336W 880 3 20201110 255.7
NGC6946-5 20:34:51.355 +60:10:59.92 F336W 880 3 20201103 257.0
NGC6946-6 20:34:23.591 +60:11:09.97 F336W 880 3 20201103 257.0
NGC6946-7 20:35:32.194 +60:08:50.01 F336W 880 3 20201111 257.0
NGC6946-8 20:35:04.464 +60:09:00.59 F336W 880 3 20201112 253.5
NGC6946-9 20:34:36.730 +60:09:10.81 F336W 865 3 20210504 75.5
NGC6946-10 20:34:09.061 +60:09:21.20 F336W 880 3 20200515 73.0
NGC6946-11 20:35:17.548 +60:07:01.19 F336W 880 3 20201112 253.0
NGC6946-12 20:34:49.842 +60:07:11.57 F336W 880 3 20201113 253.0
NGC6946-13 20:34:22.203 +60:07:22.13 F336W 880 3 20200511 73.0
NGC6946-14 20:35:02.928 +60:05:12.26 F336W 880 3 20201106 257.0
NGC6946-15 20:34:35.318 +60:05:22.98 F336W 870 3 20211114 257.0
Config Mode - WFC3/UVIS Imaging
§.§ Source Detection and Photometry
HST WFC3 NUV photometry were measured using DOLPHOT <cit.>, a stellar photometry package using point spread function (PSF) fitting, described in detail in <cit.>. We generated separate catalogs for each of the 14 overlapping pointings using the same DOLPHOT parameters as in <cit.>. We then combined all of the measurements into a single catalog, described in Section <ref>. In this catalog, we identified sources as reliable, high quality photometry using the metrics of sharpness^2 < 0.2; crowding < 0.7; signal-to-noise ratio (SNR) > 4 in both F275W and F336W; and F275W-F336W color < -1.3, as anything blueward of this color is unphysical for young massive stars, see Figure <ref> for comparison with Padova isochrones. For the analysis in the paper, we used ∼ 81,000 sources that passed the aforementioned quality cuts. The brightest single stars in the Padova log(age)=6.6 isochrone <cit.>, the youngest age we could fit, had a F336W magnitude of 20, so sources with magnitudes brighter would be likely blends. These likely blends, which are noted in the catalog, were included in our analysis as we were interested in the high crowding regions. The impact of including the blends is further discussed in Section <ref>.
§.§ Astrometry and Foreground Stars
Using the high quality photometry, we cross-match our stellar catalog to Gaia Data Release 2 (Gaia DR2; <cit.>; <cit.>). We shifted each frame by the median of the residuals of the sources matched between our catalog and Gaia DR2. The residuals have mean magnitudes on milliarcsecond scales in both right ascension and declination, which is roughly a hundred times smaller than a WFC3 UVIS pixel. The values of the overlapping sources were then averaged.
After finding Gaia matches, we removed likely foreground stars from our analysis. First, we utilized the matched Gaia sources to remove anything with a measured proper motion, as it is a likely foreground star. Then, we applied the following F275W-F336W color cuts to remove brighter, redder sources that are likely foreground stars: F336W < 21, color > 0.7; F336W < 22, color > 1; F336W < 22.5, color > 2; F336W < 23, color > 2.5. The photometric catalog used in this paper can be found as a High Level Science Product in MAST (the Mikulski Archive for Space Telescopes) via doi:[10.17909/gveq-8820]http://dx.doi.org/10.17909/gveq-8820.
§.§ Spatial Mapping
To recover the spatially-resolved SFH, we first divide our full photometric catalog into a custom grid pattern (Figure <ref>), allowing us to recover the SFH in each grid cell independently. We choose a grid pattern such that the size of the cell is based on the stellar density of the cell, which helps equalize the number of stars per cell. This gridding schema ensures the denser regions are divided into finer spatial bins, or cells, taking advantage of the large number of stars available for age constraints. Conversely, the less dense regions are divided into coarser cells to ensure each cell has enough stars to measure reliable ages (see Section <ref> for details).
To generate the cell vertices, we implemented a quadtree algorithm, which operates as follows. First, it counts the number of stars in the specified region. If the number of stars in the region is higher than a certain threshold (100 stars in this study), then it will subdivide into four equal parts. This iterates until it hits a minimum cell size, which is roughly 3×3 (∼ 100 pc×100 pc), chosen because that is the approximate size of clusters in NGC 6946.
§.§ Artificial Star Tests and Completeness
We use artificial star tests (ASTs) to measure the effects of noise, crowding, and bias on the photometry. We injected artificial stars into regions of the galaxy of different stellar densities to assess how well artificial stars of different colors and magnitudes are recovered as a function of stellar density. The input and recovered colors and magnitudes are included as a parameter in the derivation of the SFHs (Section <ref>) to account for these biases.
Because the impacts of crowding are largely density-dependent, we must ensure that our ASTs are fully sampling the wide range in the environment. For each cell, we calculate a stellar density by taking the number of stars that pass the quality cut described in Section <ref> above 25 mag in the F336W filter in the cell and dividing it by size of cell in arcsec^2. We attempted several ways of binning the cells by density, but ultimately separated them into a low density regime (cells with densities less than 11.5 stars/arcsec^2) and a high density regime (cells with densities greater than 11.5 stars/arcsec^2). We illustrate the differences in the depth of the observed data in the low and high density regimes in Figure <ref>.
We further bin these cells by density to generate at least 20,000 artificial stars per density bin to ensure that we have a sufficiently fine grid of artificial stars of different colors and magnitudes. These artificial stars were then run through DOLPHOT and flagged as recovered or unrecovered. Artificial stars are defined as recovered if they pass the same quality cuts we apply to our dataset, described in Section <ref>. For each density bin, we divided recovered and unrecovered stars into bins of width of 0.2 magnitude. We then convolved the ratio of recovered to unrecovered stars with a boxcar function to smooth this ratio and interpolated the magnitudes at which 50%, of the stars are recovered, or simply the 50% completeness limits. The F275W and F336W 50% completeness limits define the magnitude ranges that we fit with models to obtain star formation history measurements, as described in detail in Section <ref>.
We took the mean of the 50% completeness limits in the low density regime to smooth over some of the stochasticity (± 0.1 mag variation in both filters) and determined a mean 50% completeness of 26.02 and 25.88 in the F275W and F336W filters, respectively (Figure <ref>). With so few of the cells in the high density regime, it was computationally feasible to run these artificial star tests for those individual high density cells to determine the cell's individual completeness limit.
§.§ Derivation of the SFHs
We used the CMD-fitting code, MATCH <cit.>, to derive the star formation history of each cell. For each cell, MATCH creates Hess-diagrams or binned CMDs of stars in the cell. MATCH then takes user-defined ranges in age, metallicity, distance, extinction, IMF, and binary fraction to create individual synthetic CMDs for each possible combination of parameters. The individual CMDs generated from given parameters are linearly combined to form composite CMDs, which are compared to the observed CMDs. The best-fit composite synthetic CMDs are then used to infer what ages and metallicities make up the observed cell and its resulting star formation history.
We choose a Kroupa IMF <cit.>, binary fraction of 0.35, and the Padova stellar evolutionary models <cit.>. We use the distance of 7.83 ± 0.29 Mpc to be consistent with <cit.>, who used the same CMD-fitting technique. Because of the short timescale, we fix the metalliciities to be between log(Z)= -0.5 to 0.1 and fix the most recent time bins to have near solar metallicities. The youngest age we could fit was log(age)=6.6. As seen in Figure <ref>, the NUV data barely graze the log(age)=7.5 isochrone for the low density bin and the log(age)=7.4 isochrone for the high density bin. However, for completeness, we fit up to ages of log(age)=7.5. For a more detailed discussion on the reliability of the age range fit, see Sections <ref> and <ref>. A summary of these parameters is provided in Table <ref>.
cc[h!]
2
MATCH Fitting Parameters
0pt
Parameter Values
IMF Model Kroupa
Evolutionary Models Padua2006
Distance 7.83 ± 0.29 Mpc
Distance Modulus 29.47 ± 0.079
A_V 0.8-2.2, steps of 0.1
log(Z) -0.5 - 0.1, steps of 0.1
Binary fraction 0.35
F336W step size 0.1
F275W-F336W step size 0.05
CMD smoothing param 3
F275-F336W -1.3-3.3
Ages (log(yr)) 6.6-7.4 for ρ<11.5
6.6-7.5 for ρ≥ 11.5
Age step size 0.1
ρ = stellar density in stars/arcsec^2
First, we determine the best fit values of foreground extinction (A_V) and differential, or circumstellar, extinction (dA_V) of each cell by running SFH calculations over a coarse grid of A_V and dA_V with A_V between 0.8 to 2.2 with the parameters in Table <ref>. Finding the highest likelihood value of A_V and dA_V, we then redo the same SFH calculations over a finer grid of values in 0.05 increments to find the best fit A_V and dA_V, described in detail in <cit.>. Second, we adopt the best fit A_V and dA_V and rerun the SFH calculations to determine the best fit star formation rate and metallicity per time bin in each cell. Third, we measure the uncertainties of the star formation rates and metallicities by running a hybrid Monte Carlo algorithm, described in detail in Section <ref>. We then combine the SFH of each cell to create a spatially-resolved map of NGC 6946's star formation history, presented in Section <ref>.
§.§.§ Uncertainties
There are a few systematic uncertainties associated with this analysis. First, the choice of binary fraction could have a systematic impact our results. For consistency with other work deriving the recent SFH of galaxies <cit.>, we adopt a binary fraction of 0.35, knowing that massive stars have a binary fraction greater than 0.7 <cit.>. <cit.> showed that uncertainties introduced by choice of binary fraction are small compared to the uncertainties due to dust. Second, choice of stellar evolutionary model has a systematic impact on our results. <cit.> showed that the SFH measured using the Padova versus MIST models differed at ages less than 20 Myr. However, results from individual cells fit with both models agreed within less than 1%. Third, choice of IMF could impact our results. For consistency, we used the Kroupa IMF, which has been widely used for measuring star formation rate in NGC 6946.
To characterize the random uncertainties, we used a hybrid Monte Carlo (MC; <cit.>) implemented within MATCH. These uncertainties scale with the number of stars in each cell, where more stars in a cell result in lower random uncertainties. From each cell's CMD, we generate 10,000 possible SFHs. We then calculate the 1-sigma error by calculating the 68th-percentile of the samples for the cell.
§ RESULTS
In Table <ref>, we present the best fit star formation rates per time bin, number of stars in the cell (N), area in arcsec^2, mean age of the population, A_V, and dA_V, along with their cell indices and vertices. The mean age of the population in each cell is only included for a convenient reference. Some cells can be have a bimodal or trimodal age distribution, so please use this mean age with caution. Some numbers in this table have been rounded to save space, but the full machine readable table for all 2658 cells contain the measurements with full precision.
§.§ Star Formation Rate and Mass Maps
In Figure <ref>, we present maps of the spatially-resolved star formation rate for NGC 6946 in linear time bins, and include a color image from the unWISE catalog <cit.>) in W1 and W2 filters to illustrate that the star formation in the youngest ages is mostly recovered despite the dust and we are not missing much embedded star formation. For every time bin, labeled in the upper left corner of each panel, we create maps with the best fit star formation rate setting the value of intensity of each pixel. These rates are then converted to star formation rate intensity by dividing by the area of the cell in corrected for the inclination of 32.8 degrees <cit.>. In Figure <ref>, we present the spatially-resolved star formation history for NGC 6946 with log time bins for higher time resolution at younger ages.
cccccccccccccccccccccc[h]
3
Sample of SFRs over Time
0pt
i RA-NE Dec-NE RA-NW Dec-NW RA-SE Dec-SE RA-SW Dec-SW N Area SFR 0-6.7 SFR 6.7-6.8 SFR 6.8-6.9 SFR 6.9-7.0 SFR 7.0-7.1 SFR 7.1-7.2 SFR 7.2-7.3 SFR 7.3-7.4 A_V dA_V Age
^2 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr 1e-3M_⊙/yr Myr
8 308.51220 60.150921 308.48380 60.150921 308.48380 60.136787 308.51220 60.136787 17 2588.9 0.00+0.14
-0.00 0.00+0.57
-0.00 0.00+0.64
-0.00 0.06+0.62
-0.06 1.19+0.12
-1.04 0.00+0.64
0.00 0.00+0.58
0.00 0.00+0.90
0.00 0.80 0.00 11.1
9 308.51220 60.165054 308.48380 60.165054 308.48380 60.150921 308.51220 60.150921 6 2588.9 0.00+0.10
-0.00 0.00+0.41
-0.00 0.00+0.36
-0.00 0.00+0.38
-0.00 0.00+0.43
-0.00 0.00+0.42
-0.00 1.04+0.26
-0.83 0.00+0.86
0.00 0.95 0.05 17.8
12 308.59740 60.263989 308.48380 60.263989 308.48380 60.207455 308.59740 60.207455 18 41422.6 0.00+0.22
-0.00 0.00+1.11
-0.00 0.00+1.22
-0.00 2.01+0.00
-1.98 0.00+1.22
-0.00 0.00+1.20
-0.00 0.00+2.76
0.00 5.80+0.21
5.80 0.95 0.30 18.9
14 308.54060 60.136787 308.51220 60.136787 308.51220 60.122654 308.54060 60.122654 7 2588.9 0.23+0.10
-0.20 0.00+0.78
-0.00 0.00+0.64
-0.00 0.00+0.78
-0.00 0.00+0.82
-0.00 0.00+0.09
-0.00 0.00+1.88
0.00 0.00+4.20
0.00 1.35 0.00 4.4
15 308.54060 60.150921 308.51220 60.150921 308.51220 60.136787 308.54060 60.136787 60 2588.9 4.29+0.81
-1.76 0.00+7.40
-0.00 0.00+5.96
-0.00 10.0+0.07
-9.62 0.00+8.68
-0.00 0.00+13.9
-0.00 17.5+3.98
17.5 0.00+31.6
-0.00 1.20 1.05 13.2
16 308.54060 60.165054 308.51220 60.165054 308.51220 60.150921 308.54060 60.150921 54 2588.9 1.95+0.27
-1.05 0.00+3.11
-0.00 0.00+3.32
-0.00 2.42+1.80
-2.42 0.00+2.81
-0.00 3.37+2.82
-2.83 0.00+5.23
0.00 11.9+10.1
8.37 1.40 0.00 17.5
17 308.54060 60.179188 308.51220 60.179188 308.51220 60.165054 308.54060 60.165054 40 2588.9 1.15+0.69
-0.67 0.00+4.72
-0.00 7.30+2.64
-4.74 0.00+5.01
-0.00 0.00+6.28
-0.00 13.4+0.00
-11.7 0.00+15.6
0.00 0.00+46.4
0.00 1.50 0.05 11.3
18 308.56900 60.108520 308.54060 60.108520 308.54060 60.094387 308.56900 60.094387 41 2588.9 0.00+0.91
-0.00 0.00+14.1
-0.00 57.3+0.21
-27.7 0.00+16.6
-0.00 0.00+13.3
-0.00 0.00+11.0
-0.00 0.00+16.7
0.00 0.00+37.0
0.00 1.50 1.50 7.1
19 308.55480 60.115587 308.54060 60.115587 308.54060 60.108520 308.55480 60.108520 3 647.2 0.12+0.11
-0.12 0.00+0.76
-0.00 0.00+0.67
-0.00 0.00+0.62
-0.00 0.00+0.53
-0.00 0.00+0.82
-0.00 1.24+2.02
0.92 0.00+2.81
0.00 1.40 0.00 16.6
20 308.55480 60.122654 308.54060 60.122654 308.54060 60.115587 308.55480 60.115587 6 647.2 0.00+0.24
-0.00 0.00+1.21
-0.00 2.25+0.52
-1.83 0.00+1.29
-0.00 0.00+1.17
-0.00 0.00+1.23
-0.00 0.00+1.96
0.00 0.00+4.79
0.00 1.45 0.00 7.1
Note: Some of the values have been rounded to save space. The machine readable table provided will have the full precision. Area listed is not corrected for inclination. i= index; N = number of stars
We integrate the time bins to calculate the total mass formed over the last 25 Myr. In the left panel of Figure <ref>, we show the resulting mass surface density map. A majority of the recently formed mass is in the spiral arms, though there is a significant population of young stars outside of the spiral arms. The spatial distribution of the mass formed traces the resolved UV photometry of the galaxy fairly well. Our data and mass map look far more extended, particularly in the northwest and southeast arms, than the GALEX color image (right panel of Figure <ref>) due to the increased sensitivity of our data (Figure <ref>). Our methods seem to be more sensitive to older star formation than that of GALEX (which would probe <10Myr), as these features appear most prominently in the 16-20 Myr time bins of Figures <ref> and <ref>.
§.§ Extinction Maps
We recovered foreground extinction fairly uniform with mean of 1.4 and standard deviation of 0.3 (Figure <ref>, left panel). This is slightly higher than the extinction of A_V=0.938 from <cit.>. We present the differential extinction map in the right panel of Figure <ref>. The areas of high differential extinction are in the spiral arms and appear to be very clumpy. Approximately 17% of the cells have high measured differential extinction (dA_V>1), which could mean that we cannot detect some of the older stars in those grids. For more detail, see Section <ref>. The measured differential extinction show no correlation with ages measured in Section <ref>.
§.§ Global SFH
To derive a global star formation history for the galaxy, at each time bin, we integrate the SFR over all cells. We calculate the uncertainties due to the number of stars in the cells by adding the uncertainties of each spatial bin in quadrature. Then we bootstrap the uncertainties across spatial bins by sampling the number of cells 10,000 times with replacement to account for uncertainties due to binning the stars into cells. We then add the uncertainties obtained via bootstrapping to the random uncertainties in quadrature. We present these global star formation rates in Table <ref> and plot them in Figure <ref>.
To obtain the star formation rates in the past 10 Myr, we integrated the star formation rates over time to obtain total mass formed in the past 10 Myr, then divided that total mass by 10 Myr. We calculated the uncertainties by adding the uncertainties of each time bin in quadrature. We find the global star formation rate over the past 10 Myr to be roughly constant at 5.31+0.88
-0.78 M_⊙/yr, shown in Figure <ref>. We did the same for the global star formation rate 16-25 Myr ago, obtaining an SFR of 23.38+4.65
-4.25 M_⊙/yr. The SFR 16-25 Myr ago was roughly five times larger than the current (≤ 10 Myr) star formation rate, with a monotonically decreasing SFR in the 6 Myr in between the two epochs.
§.§ Reliability of the Younger Time Bins
As with SFR measurement techniques that rely on UV data, dust is a big challenge to measuring young star formation. Stars are born from giant molecular clouds and are obscured by dust until a massive star forms, ionizes its birth cloud, clears out the material <cit.>. This makes young stars incredibly challenging to observe. Our technique requires that the young stars are observable to measure star formation. The uncertainties due to dust for SFRs older than 8 Myr decrease significantly. Young stellar clusters emerge from the giant molecular clouds on timescales of 8 Myr, with very few stars remaining embedded after that <cit.>. Additionally, upon visual inspection of WISE and GALEX images (Figures <ref> and <ref>, respectively), the star formation in the 0-5 Myr time bin seems well recovered. This suggests that it is unlikely that a significant fraction of the star formation is being missed, though additional measurements of SFR using infrared are necessary better constrain the impacts of dust.
We check the impact of the blends on the reliability of the SFR in the young time bins. Our choice to include the 16 sources flagged as blends in 14 cells had been made to include as many young stars as possible in our MATCH fits. We ran MATCH again on two of these cells removing the blends from the observed CMD. We find that there is no impact on the measured SFH of these two cells. However, without the blends, the uncertainty of the SFR in the youngest time bin decreased by two orders of magnitude. The inclusion of the blends in our CMD-fitting gives us more conservative uncertainties on the SFRs of the younger time bins.
Additionally, we compare the SFR measurements to that from the literature. The current SFR is consistent with SFRs measured via methods probing the youngest stars. Previous measurements of the SFR within the last 5 Myr from Hα measurements find an SFR of ∼ 4 M_⊙/yr (no reported uncertainties, <cit.>) and 5.7 ± 1.7 M_⊙/yr <cit.>, which both agree with our measurement of 4.93 +0.22
-0.23 (Table 4, row 1) within uncertainties. <cit.> measured a ≲ 5 Myr SFR of ≃ 7.1 M_⊙/yr with Hα and 24 μm observations, with no uncertainties reported. This higher star formation rate is more consistent with our SFR in the 5-6.3 Myr time bin of 7.21 +0.58
-0.52 ((Table 4, row 2). Measurements of the SFR obtained with far-ultraviolet (FUV) tend to probe timescales over the past 10 Myr. <cit.> measures a FUV SFR 9.1 ± 2.7 M_⊙/yr, which is more consistent with our measurements of the SFRs 10-16 Myr ago, 7.33 +0.70
-0.66 and 12.81 +1.04
-0.95 (Table 4, rows 5-6).
§.§ Reliability of Older Time Bins
Another challenge of utilizing UV data to measure star formation rates is that UV primarily probes young star formation, as seen in Figure <ref> where the log(age)=7.4 isochrone lies very close to the completeness limit of our data. This limitation is reflected in the high uncertainties (at least an order of magnitude higher than the rest of the time bins) in the star formation rate in the oldest time bin, in Table <ref>. Thus we exclude the log(age)=7.4-7.5 time bin from the our analysis in the paper. In addition, we perform several tests to check the reliability of the SFR in the two oldest time bins (log(age)=7.2-7.3 and log(age)=7.3-7.4) by simulating model CMDs from the resulting SFRs.
We perform tests to check the impact of our chosen completeness limits on our measured star formation rates in the two oldest time bins. First, we check that the model CMDs created accurately model the observed CMDs for a selection of cells at varying stellar densities. We create these model CMDs by using the same parameters (i.e. completeness limits, binary fraction) we used to fit the star formation histories, as well as the output best fit metallicities and star formation rates from our results. We then check that the model has enough stars in the oldest two time bins log(age)=7.2-7.3 and log(age)=7.3-7.4 to allow for good measurements.
We also test for density-dependent effects which might arise due to the brighter completeness limit in the highest density regimes. We want to ensure that the measured SFRs of the oldest two time bins is consistent with the number of stars in their observed CMDs and reflect an accurate measurement. We check this by performing two tests. First, to measure the minimum expected percentage of older stars, we simulate constant star formation histories at different constant SFRs to measure the percentage of older (log(age)=7.2-7.4) stars out of the total (log(age)=6.6-7.4) at varying stellar densities. To check the impact of the input SFR on the percentage of older stars, we choose an SFR of 13.17 M_⊙/yr (the average global SFR), 0.1 M_⊙/yr (the highest measured SFR of a cell), and 0.005 M_⊙/yr (an average SFR value in a cell). We present the resulting systematics from these tests below (Table <ref>). There is some stochasticity in the percentage of modeled older stars for a constant SFH at SFR=0.005 M_⊙/yr likely due to the small numbers of stars in the model. Second, we measure the percentage of observed stars that fell in the two oldest time bins for varying stellar densities in a selection of spatial cells that have a spike in SFR in the older time bins. For the highest stellar densities (≥ 12 stars/arcsec^2), the log(age)=7.2-7.3 and 7.3-7.4 time bins contain 20 ± 10% of the total number of stars, whereas the lower stellar density bins have 26 ± 10% of the total number of stars. Fortunately, there are only 5 cells that have these high densities, which is not significant enough to impact the high measured global SFR in the older time bins. The lower density bins have a sufficient percentage of stars above the minimum expected percentage to have the accurate SFRs measured in the oldest time bins, appearing to be less impacted by the completeness limit concerns than the 5 cells in the high density regime.
Finally, we compare the SFR measured in the oldest time bins to that obtained via supernova rates, which probes older star formation, from 30 Myr (assuming single star evolution) to 100 Myr (assuming binary star evolution; <cit.>) and requires an SFR of at least 12.1 ± 3.7 M_⊙/yr <cit.>. We convert the measured SFR, ψ(t), of the two oldest time bins to an estimated core-collapse supernova rate, R(t)_CC. We utilize the formalism from <cit.>, using the same assumptions (all stars in the suitable mass range m_u^CC-m_l^CC supernova, the number fraction of stars that supernova in the time range and number of stars per unit mass of the stellar generation do not vary with time, since the time range we are looking at is very short) and resulting equation (Equations <ref> and <ref>) from <cit.>.
R(t)_CC = K_CC×ψ(t)
where
K_CC = ∫_m_l^CC^m_u^CCϕ(m) dm/∫_m_l^m_u mϕ(m) dm
For consistency, we choose the IMF, ϕ(m) ∝ m^-α_i, to be the Kroupa IMF (α_0 = +0.3, 0.01 ≤ m/M_⊙ < 0.08; α_1 = +1.3, 0.08 ≤ m/M_⊙ < 0.50; α_2 = +2.3, 0.50 ≤ m/M_⊙), <cit.>). We utilize the minimum progenitor masses to supernova for log(age)=7.2-7.3 to be 11.73 M_⊙ and log(age)=7.3-7.4 for 10.28 M_⊙ from the Padova stellar evolutionary models <cit.> used in the calculation of the SFH.
For log(age)=7.2-7.3, we derive a core-collapse supernova rate of 0.15 +0.02
-0.02 SN/yr. For log(age)=7.3-7.4, a ccSN rate = 0.19+0.03
-0.02 SN/yr, with uncertainties only propagated from SFR in the two time bins. This doesn't account for the systematic uncertainties due to the IMF and estimated progenitor mass. We estimate the uncertainty due to choice of IMF by comparing the ccSN rates obtained with the high-mass IMF from <cit.>. The ccSN rate obtained for log(age)=7.2-7.3 is 0.09+0.01
-0.01 SN/yr and 0.12+0.02
-0.02 SN/yr for log(age)=7.3-7.4, which is more aligned with the observed supernova rate of 0.1 SN/yr <cit.>. A 6% change in the slope of the IMF changes the ccSN rate by 60%, dominating the uncertainty of this calculation. We estimate the uncertainty due to progenitor mass by selecting progenitor masses from two additional stellar evolutionary models, PARSEC <cit.> and Geneva <cit.>, and taking the standard deviation the ccSN rate obtained with the three different models. The uncertainty due to choice of progenitor mass is 0.7% and 4% for log(age)=7.2-7.3 and log(age)=7.3-7.4, respectively.
§ DISCUSSION
In Section <ref>, we analyze two regions with high recent star formation and present their star formation histories. Finally, in Section <ref>, we examine the relationship between stellar density and age.
cccccccc[h]
5
Percentage of Modeled Older Stars for Simulated Constant SFH at SFR of 13.17 M_⊙/yr (the average global SFR), 0.1 M_⊙/yr (the highest measured SFR of a cell), and 0.005 M_⊙/yr (an average SFR value in a cell)
45pt
Density Bin SFR = 13.17 SFR = 0.1 SFR = 0.005
[stars/arcsec^2] M_⊙/yr M_⊙/yr M_⊙/yr
0-2 18% 18% 25%
2-4 18% 19% 15%
4-6 18% 18% 14%
6-8 17% 18% 15%
8-10 17% 17% 20%
10-11 17% 16% 10%
11-12 18% 17% 22%
12-13 15% 11% 3%
13-18 14% 11% 19%
ccc[h]
4
Global SFR over Time
45pt
Time Bin
Time Bin SFR
[Myr]
[log(yr)] [M_⊙/yr]
4-5 6.6-6.7 4.93 +0.22
-0.23
5-6.3 6.7-6.8 7.21 +0.58
-0.52
6.3-8 6.8-6.9 4.37 +0.42
-0.36
8-10 6.9-7.0 5.80 +0.45
-0.40
10-12.5 7.0-7.1 7.33 +0.70
-0.66
12.5-16 7.1-7.2 12.81 +1.04
-0.95
16-20 7.2-7.3 22.84 +2.67
-3.18
20-25 7.3-7.4 23.82 +3.81
-2.81
§.§ Local Star Formation History
We analyze two regions with high star-forming activity, the Hodge Complex <cit.> roughly centered at 20:34:34.80 +60:08:18.60 and the HII region at the tip of the northeast spiral arm roughly centered at 20:35:22.57 +60:10:14.70. The cells of the regions of interest used in this analysis are flagged in the machine readable table.
To obtain the star formation histories of these regions, we sum the star formation rates of each cell in the region per time bin. We then add the random uncertainties in quadrature. The locations of the regions and their star formation histories are presented in Figure <ref>.
The Hodge Complex is a super star cluster containing multiple young star clusters and has been extensively studied due to the high concentration of star formation <cit.>. We present the star formation history of this region in the lower left corner of Figure <ref>. This region appears to have constant star formation over the past 6.3-25 Myr with a peak in star formation in around 5-6.3 Myr and drop in star formation in the most recent 5 Myr. Interestingly, despite being a mere 1295 square arcseconds, which is 0.05% of the total size of our coverage area, this tiny region contains 3% of the total mass formed up to 6.3 Myr and 1.8% of the total mass formed up in the past 25 Myr. This region has had a more recent star formation episode than seen for the globally decreasing star formation rate.
Another region of interest is the HII region in tip of the northeast spiral arm. Unlike the Hodge Complex, the star formation history of this region roughly follows the star formation history of galaxy. Similar to the Hodge Complex, this small region of the galaxy contains a significant portion of the recent star formation in the galaxy. Despite it being 0.05% of our total coverage (roughly 1334 square arcseconds), this region contains 5.6% of total mass of NGC 6946 formed in the last 6.3 Myr and 3.9% over the last 25 Myr. There is a large peak in older star formation 20-25 Myr ago relative to the flatter SFH in the past 20 Myr. Since the global SFH has a more gradual decrease in SFR over time, we check that this peak in SFR is not due to a systematic related to completeness. We check SFRs of the locations with the highest stellar density and find 1.5% of SFR in oldest time is attributed to those high density regions, making an insignificant contribution to the high SFR.
Ultimately, these regions of interest are only a small portion of the young star formation in NGC 6946. This points to overall star formation across the galaxy contributing to the peak in the global SFH.
§.§ Density versus Age
Initially, we measured the characteristic age of the population in each cell by randomly sampling their star formation histories 50,000 times. We then obtain 16th, 50th, and 84th percentile time bins. Due to the the double peak distribution of many of the star formation histories, we found no correlation between stellar density and characteristic age for this population.
Thus instead of looking at the characteristic age of the population, we find the youngest age bin in which we detect star formation in each cell, using the following criteria: the SFR of the age bin is at least 3% of the total star formation rate, the age bin contains 1% of the total mass formed, and and the lower bound SFR uncertainty of that time bin must be greater than or equal to zero. In Figure <ref>, we see that the youngest detected population are denser, with the stellar density remaining between 0-2.5 stars/arsec^2 after 12 Myr. A possible explanation is that stars have migrated from their initial site of formation over 12 Myr to populate the field. This is consistent with the timeline of stars emerging for their giant molecular clouds by roughly 8 Myr (see Section <ref>) then populating the field over time. More work will need to be done to confirm that stellar migration is being observed. We plan to model this in our future work.
§ CONCLUSIONS
In this paper, we presented the spatially-resolved star formation history of NGC 6946 in the last 25 Myr, measured using resolved NUV stellar photometry. We implemented a quadtree algorithm to devise a spatial grid, and we measured the SFH independently for each cell using CMD-fitting. We summarize our main findings below.
* We measure the global SFR over the last 25 Myr to be 13.16 +0.91
-0.79 M_⊙/yr.
* 16-25 Myr ago, the SFR was 23.38+2.43
-2.11 M_⊙/yr. The SFR then monotonically decreases between 10-16 Myr, reaching a steady recent SFR in the past 10 Myr of 5.31+0.18
-0.17 M_⊙/yr.
* We present the star formation histories of the Hodge Complex and the HII region at the tip of northeast spiral arm. Both contain a higher amount of recent star formation than expected for regions of their size. The Hodge Complex shows more recent star formation relative to the declining global star formation.
§ ACKNOWLEDGEMENTS
We thank the reviewer for their helpful comments that improved the paper. This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. These observations are associated with program GO-15877.
HST(WFC3/UVIS)
astropy <cit.>, DOLPHOT <cit.>, GeoPandas <cit.>, <cit.>, Matplotlib <cit.>, NumPy <cit.>, Pandas <cit.>.
aasjournal
|
http://arxiv.org/abs/2307.05804v1 | 20230711210047 | Improving Segmentation and Detection of Lesions in CT Scans Using Intensity Distribution Supervision | [
"Seung Yeon Shin",
"Thomas C. Shen",
"Ronald M. Summers"
] | eess.IV | [
"eess.IV",
"cs.CV"
] |
This article has been accepted for publication in Computerized Medical Imaging and Graphics.
DOI: https://doi.org/10.1016/j.compmedimag.2023.102259
inst1]Seung Yeon Shincor1
[cor1]Corresponding author: [email protected]
inst1]Thomas C. Shen
inst1]Ronald M. Summers
[inst1]organization=Imaging Biomarkers and Computer-Aided Diagnosis Laboratory, Radiology and Imaging Sciences,
addressline=Clinical Center, National Institutes of Health,
city=Bethesda,
postcode=20892,
state=MD,
country=USA
We propose a method to incorporate the intensity information of a target lesion on CT scans in training segmentation and detection networks.
We first build an intensity-based lesion probability (ILP) function from an intensity histogram of the target lesion.
It is used to compute the probability of being the lesion for each voxel based on its intensity.
Finally, the computed ILP map of each input CT scan is provided as additional supervision for network training, which aims to inform the network about possible lesion locations in terms of intensity values at no additional labeling cost.
The method was applied to improve the segmentation of three different lesion types, namely, small bowel carcinoid tumor, kidney tumor, and lung nodule. The effectiveness of the proposed method on a detection task was also investigated. We observed improvements of 41.3% → 47.8%, 74.2% → 76.0%, and 26.4% → 32.7% in segmenting small bowel carcinoid tumor, kidney tumor, and lung nodule, respectively, in terms of per case Dice scores. An improvement of 64.6% → 75.5% was achieved in detecting kidney tumors in terms of average precision.
The results of different usages of the ILP map and the effect of varied amount of training data are also presented.
< g r a p h i c s >
* Intensity values in CT scans, i.e., Hounsfield unit, convey important information.
* A method to incorporate the intensity information of a target lesion in training is proposed.
* An intensity distribution of a target lesion is used to define an auxiliary task.
* It informs the network about possible lesion locations based on intensity values.
* A relative improvement of 2.4% (16.9%) was obtained in segmenting (detecting) kidney tumors.
Lesion segmentation lesion detection supervision intensity distribution Hounsfield unit computed tomography carcinoid tumor kidney tumor lung nodule
§ INTRODUCTION
Identification and quantification of abnormalities are the first objectives of medical image acquisition <cit.>. According to the result, immediate treatments can be done, or follow-up studies are triggered for surveillance <cit.>.
Abnormalities can arise at various locations in the body, such as different tissues and organs. When finding a particular type of lesion, it is often coupled with segmentation of an organ/tissue that can contain the lesion, e.g., liver and liver tumor segmentation <cit.>. These two related tasks can be considered either sequentially or jointly. In the first case, organ segmentation can benefit the following lesion identification by restricting the region of interest and thus by enabling more detailed inspection within it <cit.>. Meanwhile, the latter facilitates joint optimization of the relevant tasks and thus can lead to an enhancement of them. In the work of <cit.>, liver and liver tumor segmentations are performed jointly using a single shared network to utilize the correlation between them. Features learned for organ segmentation would be relevant to contained lesions since they reside within the organ.
Nevertheless, it is not always possible to train the organ segmentation together with the target lesion segmentation since it requires additional ground-truth (GT) segmentations of the organ. Indeed, a clinician aims to find and mark abnormalities, but not an entire organ, which hinders the mentioned joint or sequential modeling in most cases.
To boost the lesion segmentation without requiring additional supervision, there have been many attempts <cit.>. <cit.> utilized a multi-task learning network, which performs an image reconstruction task in parallel with tumor segmentation, for brain tumors in magnetic resonance imaging (MRI) scans. By having a shared encoder and separate decoders for each task, training a part of the network, i.e., auto-encoder part, using scans without annotations is enabled. In the work of <cit.>, lesion edge prediction is used as an auxiliary task to help the segmentation of skin lesions. Two tasks can interact with each other within the network and boost each other’s performance in turn. As a noticeable trend, incorporating distance transform maps of GT into segmentation networks has been tried in many different works in past years, which were summarized in the work of <cit.>. Interestingly, most of the benchmark methods showed no or minor improvement against the baseline for tumor segmentation while they performed more preferably for organ segmentation. Challenges of tumor segmentation compared to organ segmentation, such as various locations, shapes, and sizes, are mentioned as a potential reason.
In terms of lesion detection, <cit.> developed the Retina U-Net architecture based on the Retina Net <cit.>. The decoding part of the Retina Net is augmented by additional high-resolution feature levels, and semantic lesion segmentation is performed on top of them to boost the detection task. Despite its effectiveness, it assumes that segmentation annotations are available together with detection annotations, which is not always the case.
In computed tomography (CT) scans, intensity values, i.e., Hounsfield unit (HU), convey important information on the substance of each region, e.g., air, fat, and bone <cit.>. Therefore, they can be used in identifying a particular organ, tissue, or lesion <cit.>. In the work of <cit.>, a HU threshold of 13 or 21 is suggested to discriminate malignant adrenal tumors from benign ones in unenhanced CT scans. In the work of <cit.>, a combined use of lesion morphology and HU values improved the
diagnostic accuracy in differentiating benign and malignant incidental breast lesions on contrast-enhanced chest CT scans. In the work of <cit.>, a specific threshold range of [40, 90] is used to determine areas of hemorrhage on brain CT scans.
In this paper, we propose a method to incorporate the intensity information of a target lesion on CT scans in training segmentation and detection networks. Instead of using hard thresholds as in the previous works <cit.>, an intensity distribution of a target lesion is first built and used to effectively locate regions where the lesions are possibly situated. The intensity distribution can be achieved by investigating intensity values within available GT lesion segmentations or can be provided as prior information.
More specifically, an intensity-based lesion probability (ILP) function constructed from an intensity histogram is used to compute the probability of being lesion for each voxel, and this soft label map is provided for network training as an auxiliary task.
It informs the network about our region of interest, which could contain target lesions, based on the intensity. Compared to the organ segmentation trained jointly with lesion identification tasks, our new task can be understood as a soft and possibly disconnected surrogate of organ segmentation, and it requires no additional annotation cost.
We demonstrate the effectiveness of the proposed method by conducting experiments on three different datasets: 1) an in-house small bowel carcinoid tumor dataset, 2) the KiTS21 dataset <cit.> for kidney tumors, and 3) the LNDb dataset for lung nodules <cit.>. The main contributions of our work are as follows. (1) We extend the idea of our previous paper <cit.> to the segmentation of different lesions at different body locations to verify its generability. (2) We further investigate the effectiveness of the proposed method in several aspects, namely with varied amount of training data, in comparison to the joint organ segmentation, and even on a detection task.
§ DATASETS
§.§ Small Bowel Carcinoid Tumor Dataset
Carcinoid tumor is a rare neoplasm (small bowel neoplasms including carcinoid tumors account for 0.5% of all cancers in the United States <cit.>) and found predominantly within the gastrointestinal tract (50-71.4%) and especially in the small bowel (24-44%) <cit.>. They are often less than a centimeter in size <cit.>.
Our carcinoid tumor dataset is composed of 24 preoperative abdominal CT scans collected at the National Institutes of Health Clinical Center. Each scan is from a unique patient who underwent surgery and had at least one carcinoid tumor within the small bowel. We note that creating a large dataset for small bowel carcinoid tumors is more difficult than for other more prevalent diseases.
All scans are intravenous and oral contrast-enhanced. An oral contrast agent of Volumen was used. Each patient has both arterial and venous phase scans, and either of them was selectively used according to the relevant description in the corresponding radiology report (18 arterial and 6 venous phase scans). They were acquired using 0.5, 1, or 2 mm slice thickness. All scans were cropped manually along the z-axis to include from the diaphragm through the pelvis. We will call this the SBCT dataset.
To achieve GT segmentation of tumors, we used “Segment Editor" module in 3DSlicer <cit.>. The corresponding radiology report and an available ^18F-DOPA PET scan were referred to for help in locating tumors. 88 tumors were annotated in total. We use five-fold cross-validation for this dataset.
§.§ The KiTS21 Dataset
Kidney cancer is the sixth and the ninth most common cancer for men and women, respectively, in the United States <cit.>.
The KiTS21 dataset aims to accelerate the development of automatic segmentation tools for renal tumors and surrounding anatomy. The KiTS21 cohort includes patients who underwent nephrectomy for suspected renal malignancy <cit.>. Preoperative CT scans of these patients were collected to compose the dataset. The official training set comprises 300 CT scans from 300 unique patients who had at least one kidney tumor.
All scans are contrast-enhanced and were acquired in the late arterial phase. Every scan has corresponding GT segmentations of the kidney, tumor, and cyst. In this work, we focus on segmenting tumors while leaving cysts unattended since cysts are benign and clinically less relevant than tumors. We refer the authors to the dataset description paper <cit.> for more information. For experiments, we divide the dataset into training/validation/test sets at a ratio of 7:1:2.
§.§ The LNDb Dataset
Lung cancer is the leading cause of cancer death, which makes up almost 25% of all cancer deaths <cit.>. Being a possible indicator of lung cancer, lung nodules show various shapes and characteristics. Thus, the identification and characterization of them are not trivial and prone to high inter-observer variability <cit.>.
The LNDb dataset includes 294 intravenous contrast-enhanced CT scans from 294 unique patients. Fifty-eight scans among them were withheld by the organizers for the test set and the remaining 236 scans are available. Among all identified lesions (nodule ≥ 3 mm, nodule < 3 mm, non-nodule), only nodules that are greater than or equal to 3 mm were segmented during the annotation process, and they will be segmented in this work. Thirty-five scans have been excluded since they have an empty segmentation map with the above-mentioned reason, resulting in 201 remained scans (236 - 35 = 201).
In this work, we especially focus on improving the segmentation of non-solid nodules since they are more likely to be malignant than solid nodules and more difficult to identify due to their fuzzy appearance and lower incidence <cit.>. We utilized nodule texture ratings (1 - 5) provided in the dataset, where 1 denotes closer to non-solid nodules and 5 denotes closer to solid nodules. According to these ratings, we classified each segmented nodule into two groups, namely, non-solid (≤ 2) or solid (> 2). Nineteen scans were identified to have at least one non-solid nodule after manual inspection. We note that these 19 scans can also contain solid nodules. Finally, they are used for two-fold cross-validation while the remaining 182 (= 201 - 19) scans are included as training images for every fold training.
§ METHODS
§.§ Intensity Distribution Supervision
Figure <ref> presents the intensity histogram of target lesions of each dataset. They were computed by aggregating intensity values within GT lesion segmentations of each dataset. Images were smoothed using anisotropic diffusion <cit.> before the histogram construction.
To make a smooth evaluable function from the discontinuous histogram, we perform kernel density estimation <cit.>. It is a method used to estimate the probability density function based on kernels as basis functions. We used `gaussian_kde' function of SciPy Python library, which uses Gaussian kernels with automatic bandwidth determination.
The resulting function is then rescaled to have the maximum value of 1.
While it could be less precise, the ILP function can be provided also by a user as prior information.
The resulting ILP functions are superimposed with their corresponding histograms in Figure <ref>. They enable faster calculation of the ILP for a large set of voxels (a whole CT scan) than using the histogram. The ILP function, f^ILP, is used to compute the probability of being part of the target lesion for each voxel according to its intensity value. Given an input image volume X = {x_i}_i=1^N, the corresponding ILP volume Y^ILP is defined as:
Y^ILP = {y_i^ILP}_i=1^N = {f^ILP(x_i)}_i=1^N,
where N is the number of voxels. An example of the computed ILP volume is visualized in Figure <ref>.
It is then provided to a network as the label map of an auxiliary task. It informs the network about our region of interest, which could contain target lesions, especially in terms of intensity values. Compared to the organ segmentation trained jointly with lesion identification tasks, our new task can be understood as a soft and possibly disconnected surrogate of organ segmentation, and it requires no additional labeling effort.
§.§ Network Training
§.§.§ Lesion Segmentation Network
The proposed intensity distribution supervision can be easily used for a lesion segmentation network. Figure <ref> visualizes the data used for the network training.
Given an input image volume X, the corresponding ILP volume Y^ILP is generated using the ILP function f^ILP.
Then, it is used as supervision for network training together with the segmentation GT, Y^segm.
For simplicity, we use a network with two output channels, which is similar to the one for joint liver and liver tumor segmentation <cit.>.
However, the second output channel of our network predicts the ILP in place of organ segmentation. The generated GT ILP volume Y^ILP is used as supervision for this channel.
A new loss term for the added task, L^ILP, is incorporated into training accordingly as shown in Figure <ref>.
Cross-entropy loss is used to measure the dissimilarity between the GT and the prediction of the ILP.
Finally, the overall loss function for training the lesion segmentation network is defined as:
L = L^segm + λ L^ILP
where L^segm is the segmentation loss and λ is the relative weight for the ILP loss L^ILP.
We use the generalized Dice loss <cit.> for L^segm.
§.§.§ Lesion Detection Network
Our intensity distribution supervision can be also used to enhance detectors that are based on feature pyramid networks (FPNs) <cit.>, such as the Retina Net <cit.>, with the same philosophy as the Retina U-Net <cit.>. Figure <ref> explains the concept. In the Retina U-Net, to exploit available GT segmentation of lesions, the decoding part of the Retina Net is augmented by additional high-resolution feature levels, and semantic lesion segmentation is performed on top of them. Despite its effectiveness, it is not feasible if the GT segmentation is unavailable.
Our ILP map Y^ILP, which is generated from each input image volume X using the ILP function f^ILP can replace the GT segmentation. The same ILP loss L^ILP is applied as in the segmentation network. Finally, the overall loss function for training the lesion detection network is defined as:
L = L^det + λ L^ILP
where L^det is the typical detection loss for classification and box regression, and λ is the relative weight for L^ILP.
§.§ Evaluation Details
§.§.§ Lesion Segmentation
We first used our own version of the 3D U-Net <cit.> to have more control over the training/test procedures and thus verify the pure impact of using the proposed intensity distribution supervision. Then, we further attempted to combine it with the self-configuring nnU-Net <cit.> to achieve more optimized performance. Within this framework, the `3D' full resolution `U-Net' was used again but with higher complexity in terms of the network size, data augmentation, and test time method. We note that the proposed method of using intensity information can be used for any other segmentation networks.
The ILP functions in Figure <ref> were used for each dataset.
Especially for the LNDb dataset, we used the function of non-solid nodules (Figure <ref>(c)) to emphasize them more during training since our goal is to improve the segmentation of them. Their distribution is wider than that of small bowel carcinoid tumors or kidney tumors because they are located in the lung parenchyma. Nevertheless, it is distinguishable from that of solid nodules (Figure <ref>(d)).
Hyperparameters related to each method and each dataset are summarized in Table <ref>. The learning rates and λ were chosen through the grid search for both methods. While the other values were chosen through the grid search again by ourselves for the 3D U-Net, they were chosen automatically for the self-configuring nnU-Net.
We used the AdamW optimizer <cit.> for the 3D U-Net. The SGD with a momentum of 0.99 was used for the nnU-Net. In all of the implemented networks, 3x3x3 convolution kernels are used except 1x1x1 kernels for the final inference layer.
For data augmentation, various geometric and photometric augmentation methods that are available in their implementation (https://github.com/MIC-DKFZ/nnUNet) were used as is for the nnU-Net. The whole set of photometric augmentations was turned on or off in its entirety to check their relevance in each dataset. Meanwhile, selective sets of augmentations were used for the 3D U-Net after performing an investigation on the effect of each method for each dataset. While only image rotation was used for the SBCT and the LNDb datasets, image scaling and elastic deformations were used as well for the KiTS21 dataset. In test time, a test time augmentation method of image mirroring was used for the nnU-Net.
For evaluation, we use per case and per lesion Dice scores. The per case Dice score denotes an average Dice score per scan. In calculating the per lesion Dice scores, tight local image volumes around each tumor were taken into account. Paired t-tests are conducted to show the statistical significance of the proposed method. We used an NVIDIA Tesla V100 32GB GPU to conduct experiments.
§.§.§ Lesion Detection
For all compared methods, the same backbone FPN <cit.> based on a ResNet50 <cit.> was used. We used the Adam optimizer <cit.> with a learning rate of 10^-4. 0.003 was used for λ of Eq. <ref>. The training was conducted using image patches of size 96 × 96 × 64, which were sampled from scans that have isotropic voxels of 2 × 2 × 2 mm^3. The batch size of 8 was used. For data augmentation, image scaling, rotation, mirroring, and elastic deformations were used.
For experiments, we used the KiTS21 dataset, which has the biggest number of scans. We report average precision (AP) with an intersection over union threshold of 0.1, following the method of <cit.>.
§ RESULTS
§.§ Lesion Segmentation
§.§.§ Experiments on the SBCT Dataset
Quantitative Results
Table <ref> presents quantitative results of segmentation methods, which differ in the ways of using the intensity distribution information. We used the 3D U-Net <cit.> to verify the pure effect of the different usages of the intensity distribution information.
Applying post-processing to the prediction of the segmentation network, where the ILP volume Y^ILP is multiplied with the network predicted probability map, rather worsened the performance (‘3D U-Net + PP’).
This post-processing could oversimply rule out lesions that have intensity values deviating from the built intensity distribution.
We also tried using the ILP volume as an additional input channel instead of as additional supervision (‘3D U-Net + ILP(in)’).
It can be another way to highlight our region of interest at the input level. However, it performed merely on par with the baseline that does not use this additional information.
On the other hand, the proposed method, ‘3D U-Net + ILP’, showed clear improvements for all types of Dice scores when compared to the baseline.
The proposed method of using the intensity distribution supervision does not entail any additional labeling effort.
The ILP function can be constructed and included in training by looking up already available CT scans and corresponding GT tumor segmentation.
We also investigate the effect of having the precise intensity model of a target.
‘3D U-Net + ILP(shifted)’ is the proposed method but uses another ILP function that is +100 shifted from the original one.
The shifted function does not reflect the actual intensity distribution of the target anymore. It performed rather worse than the baseline.
All methods including the proposed method showed higher Dice scores for relatively larger tumors (≥ 125 mm^3, which is approximately ≥ 6 mm diameter) than for all tumors.
Qualitative Results
Figure <ref> presents example segmentation results of small bowel carcinoid tumor. Compared to the baseline method that is trained without the intensity distribution supervision, the proposed method segments more tumors (first and second rows). The last row shows a failure case, where the proposed method missed a blurry small tumor.
§.§.§ Experiments on the KiTS21 Dataset
Quantitative Results
Table <ref> presents quantitative results of different segmentation methods on the KiTS21 dataset. We first used different versions of the 3D U-Net that were augmented using different additional supervision, and again used the nnU-Net for more optimized performance.
We also incorporated the proposed intensity distribution supervision in training the winning method of the KiTS21 challenge <cit.>.
In the 3D U-Net based comparison, the proposed method (`3D U-Net + ILP') outperformed the baseline (`3D U-Net').
We further compare it against a multi-task learning network that performs organ (kidney) segmentation together with lesion (kidney tumor) segmentation, which is `3D U-Net + organ' in Table <ref>.
As our ILP map informs the network about our region of interest, which could contain target lesions, in terms of intensity values, organ segmentation supervision could do the same in a stricter way, i.e., kidney tumors can exist within the kidney.
The proposed method performed on par with the organ-segmentation-augmented method, which requires additional labeling effort while the proposed method does not.
The proposed method (`nnU-Net + ILP') still outperformed the baseline (`nnU-Net') when the nnU-Net was used.
The test time augmentation method of the nnU-Net could decrease the performance gap by benefiting an under-performed method more. We note that the proposed method could be not well harmonized with photometric augmentations since they randomly distort original voxel values and thus can change the physical meaning that each voxel originally has on CT scans. We found in this dataset that photometric augmentations do not really help in improving the performance even for the baseline method (`nnU-Net + photo aug.'). Thus, only geometric augmentations were used. `nnU-Net + organ' showed a better performance than the proposed method, but it used an additional annotation of the kidney.
We also incorporated the proposed intensity distribution supervision in training the winning method of the KiTS21 challenge <cit.>. The method is composed of three steps (networks), which are coarse kidney segmentation, fine kidney segmentation, and tumor segmentation.
Therefore, it uses GT segmentations of the kidney and tumor for training of the first and second networks, and the last network, respectively.
Since there is no publicly available code, we have used our own implementation for the experiment.
When the intensity distribution supervision was incorporated in the last tumor segmentation step at no additional labeling cost, a better performance was again achieved (`<cit.> + ILP').
Figure <ref> shows the segmentation performances on the KiTS21 dataset, depending on the number of training images. Given the original training set of 210 images, 90, 120, 150, or 180 images were randomly sampled to conduct the experiments. The same validation and test sets of 30 and 60 images, respectively, were used for all training set sizes. The proposed method consistently outperformed the baseline for all experiments, with a margin of around 2%.
Qualitative Results
Figure <ref> shows example segmentation results on the KiTS21 dataset. The proposed method segments tumors more precisely (first and second rows) by utilizing the intensity distribution supervision when compared to the baseline.
The last row shows a failure case, where the tumor was missed by both the baseline and proposed methods.
§.§.§ Experiments on the LNDb Dataset
Quantitative Results
Table <ref> presents quantitative segmentation results on the LNDb dataset. As mentioned in Section <ref>, we tried to improve the segmentation of non-solid nodules in this work by incorporating their intensity distribution information into network training.
For both the 3D U-Net and nnU-Net, the inclusion of the intensity distribution supervision (+ ILP in Table <ref>) helped in segmenting non-solid nodules better thus resulting in the performance improvement also for all nodules, except for the per lesion Dice scores of the 3D U-Net.
Per lesion Dice score, by definition, does not take into account FPs that are apart from GT lesions. On the other hand, it focuses on segmentation quality around GT lesions. Therefore, FNs are considered more important than FPs in calculating it. The proposed method with the 3D U-Net reduced FPs but induced FNs, which led to increased per case Dice scores but decreased per lesion Dice scores. Nevertheless, the added intensity distribution supervision on non-solid nodules helped in segmenting them while overcoming their fuzzy appearance and underrepresentation in the dataset.
Qualitative Results
Figure <ref> presents example segmentation results on the LNDb dataset. Compared to the baseline method, the proposed method segments more nodules (first and second rows).
§.§ Lesion Detection
§.§.§ Quantitative Results
Table <ref> presents quantitative results of detection methods that differ in augmenting the baseline Retina Net <cit.> on the KiTS21 dataset. The network architecture of each method is explained in Figure <ref> and the corresponding text. The Retina U-Net <cit.> exploiting lesion segmentation supervision, which is assumed to be available together with detection GTs, outperformed the Retina Net, as suggested in the work of <cit.>.
When the lesion segmentation supervision was replaced with the proposed ILP supervision, it outperformed the baseline Retina Net again and further outperformed the Retina U-Net.
While the ILP function could be less precise, it can be constructed using a small number of GT lesion segmentations or can be even provided by a user as prior information.
Also, while the Retina Net was used as the baseline here, the proposed method can be used to enhance any detectors that are based on FPNs in the same manner.
§.§.§ Qualitative Results
Figure <ref> shows example detection results on the KiTS21 dataset.
The incorporated intensity distribution information helped in locating a tumor (first row) and eliminating a false positive (second row).
The bottom two rows represent failure cases. In the third row, a false positive was detected by the proposed method on the heterogeneous stomach, which resembles a kidney with a tumor in appearance. In the last row, the tumor that has similar intensity values as the rest of the kidney was missed by the proposed method.
§ DISCUSSION
We have presented a method to incorporate the intensity information of a target lesion on CT scans in training segmentation and detection networks.
An ILP function constructed from an intensity histogram of a target lesion is used to effectively locate regions where the lesions are possibly situated. The ILP map of each input CT scan is provided as additional supervision for network training.
It aims to inform the network about our region of interest, which could contain target lesions, especially in terms of intensity values. It requires no additional labeling effort.
The method has been applied to improve the segmentation of three different lesion types, namely, small bowel carcinoid tumors, kidney tumors, and lung nodules. The effectiveness of the proposed method on a detection task has been also investigated for kidney tumors.
Our findings from the experiments are: 1) The proposed method of using the ILP as additional supervision performs better than other usages of it, such as for post-processing and as an additional input channel (Table <ref>).
2) Having a precise and generalizable intensity distribution is important for the success of the method (Table <ref>).
3) It can be effectively used with the nnU-Net for more optimized performance (Table <ref>). 4) It performs favorably against a method that exploits another supervision such as organ segmentation (Table <ref>). 5) Consistent performance gains can be expected over varying training set sizes (Figure <ref>). 6) It can be considered to boost the performance of an underrepresented lesion type (Table <ref>).
7) It can be used to enhance a detector such as the Retina Net <cit.> (Table <ref>).
Carcinoid tumors in our SBCT dataset are small (Figure <ref>). Lung nodules in the LNDb dataset are also small (mostly less than a centimeter <cit.>) as exemplified in Figure <ref>. Even small numbers of false positive and false negative voxels have a big impact on the Dice score of small lesions. Nevertheless, the proposed method showed clear improvements compared to the baseline.
We also note that we segmented nodules from an entire CT scan, whereas the segmentation is conducted when nodule centroids are given for each scan in the LNDb challenge.
Our task is more challenging, which makes achieving high Dice scores difficult again.
For the KiTS21 dataset, we incorporated the proposed intensity distribution supervision also in training the challenge winning method <cit.>. Although the efficacy of the proposed method was verified, our result is not directly comparable with theirs since they used more training images (240 vs. 210). While they divided the dataset into only training and validation sets (a separate test set available for the challenge period), we divided it into training/validation/test sets to enable a strict evaluation within the available data.
Also, we did not use their postprocessing method of counting the number of voxels for each connected component and thresholding them based on their sizes, since that heuristics is not always relevant.
In terms of network training, typical segmentation and detection losses are used together with the ILP loss for the segmentation and detection tasks, respectively (Eq. <ref> and Eq. <ref>).
The proposed method provides an additional opportunity to consider the intensity information of the target lesion in an explicit way while retaining learning about other aspects by the typical loss terms.
A lesion that is not distinct by the ILP model still can be identified by the other aspects.
For example, in the third row of Figure <ref>, the tumor that is not distinct from the kidney by intensity values was still detected by the proposed method.
In this work, for each target lesion, the experiments have been conducted on a single dataset acquired using a particular imaging protocol. The proposed method would be less applicable across datasets that were acquired using different imaging protocols since the intensity distribution of the target lesion can be diffused and incoherent. Also, we took a relatively simple implementation for incorporating the intensity information of the target lesion in the network training. For the same objective, a better approach can be explored.
In future work, we plan to study the effect of incorporating unsupervised images into training since the proposed intensity distribution supervision enables training on them. The proposed method can be further applied to different target lesions.
§ SAMPLE APPENDIX SECTION
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor section <ref> incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
§ DATA AVAILABILITY
The code is available at https://github.com/rsummers11/CADLab/tree/master/intensity_distribution_supervisionhttps://github.com/rsummers11/CADLab/tree
/master/intensity_distribution_supervision
§ ACKNOWLEDGMENT
This research was supported by the Intramural Research Program of the National Institutes of Health, Clinical Center. The research used the high-performance computing facilities of the NIH Biowulf cluster.
§ CONFLICTS OF INTEREST
Potential financial interest: Author Ronald M. Summers receives royalties from iCAD, Philips, Scan Med, PingAn, and Translation Holdings and has received research support from Ping An (CRADA).
37
natexlab#1#1
[#1],#1
[American Cancer Society(2022)]acs_lung_cancer
authorAmerican Cancer Society, year2022.
titleKey statistics for lung cancer.
<https://www.cancer.org/content/dam/CRC/PDF/Public/8703.00.pdf>.
[Ayalew et al.(2021)Ayalew, Fante and Mohammed]ayalew21
authorAyalew, Y.A., authorFante, K.A.,
authorMohammed, M.A., year2021.
titleModified u-net for liver cancer segmentation from
computed tomography images with a new class balancing method.
journalBMC Biomedical Engineering volume3,
pages1–13.
[Black et al.(1998)Black, Sapiro, Marimont and Heeger]black98
authorBlack, M., authorSapiro, G.,
authorMarimont, D., authorHeeger, D.,
year1998.
titleRobust anisotropic diffusion.
journalIEEE Transactions on Image Processing
volume7, pages421–432.
10.1109/83.661192.
[Buzug(2011)]buzug11
authorBuzug, T.M., year2011.
titleComputed tomography, in:
booktitleSpringer handbook of medical technology.
publisherSpringer, pp. pages311–342.
[Cai et al.(2021)Cai, Tang, Yan, Harrison, Xiao, Lin and Lu]cai21
authorCai, J., authorTang, Y., authorYan,
K., authorHarrison, A.P., authorXiao, J.,
authorLin, G., authorLu, L., year2021.
titleDeep lesion tracker: Monitoring lesions in 4d
longitudinal imaging studies, in: booktitle2021 IEEE/CVF
Conference on Computer Vision and Pattern Recognition (CVPR), pp.
pages15154–15164.
10.1109/CVPR46437.2021.01491.
[Cancer.Net(2022)]cancernet_kidney_cancer
authorCancer.Net, year2022.
titleKidney cancer: Statistics.
<https://www.cancer.net/cancer-types/kidney-cancer/statistics>.
[Çiçek et al.(2016)Çiçek, Abdulkadir,
Lienkamp, Brox and Ronneberger]cicek16
authorÇiçek, Ö., authorAbdulkadir, A.,
authorLienkamp, S.S., authorBrox, T.,
authorRonneberger, O., year2016.
title3d u-net: Learning dense volumetric segmentation from
sparse annotation, in: editorOurselin, S.,
editorJoskowicz, L., editorSabuncu, M.R.,
editorUnal, G., editorWells, W. (Eds.),
booktitleMedical Image Computing and Computer-Assisted
Intervention – MICCAI 2016, publisherSpringer International
Publishing, addressCham. pp. pages424–432.
[Diederich(2009)]diederich09
authorDiederich, S., year2009.
titlePulmonary nodules: do we need a separate algorithm
for non-solid lesions?
journalCancer Imaging volume9,
pagesS126.
[Eisenhauer et al.(2009)Eisenhauer, Therasse, Bogaerts, Schwartz,
Sargent, Ford, Dancey, Arbuck, Gwyther, Mooney et al.]eisenhauer09
authorEisenhauer, E.A., authorTherasse, P.,
authorBogaerts, J., authorSchwartz, L.H.,
authorSargent, D., authorFord, R.,
authorDancey, J., authorArbuck, S.,
authorGwyther, S., authorMooney, M., et al.,
year2009.
titleNew response evaluation criteria in solid tumours:
revised recist guideline (version 1.1).
journalEuropean journal of cancer volume45,
pages228–247.
[Fedorov et al.(2012)Fedorov, Beichel, Kalpathy-Cramer, Finet,
Fillion-Robin, Pujol, Bauer, Jennings, Fennessy, Sonka, Buatti, Aylward,
Miller, Pieper and Kikinis]fedorov12
authorFedorov, A., authorBeichel, R.,
authorKalpathy-Cramer, J., authorFinet, J.,
authorFillion-Robin, J.C., authorPujol, S.,
authorBauer, C., authorJennings, D.,
authorFennessy, F., authorSonka, M.,
authorBuatti, J., authorAylward, S.,
authorMiller, J.V., authorPieper, S.,
authorKikinis, R., year2012.
title3d slicer as an image computing platform for the
quantitative imaging network.
journalMagnetic Resonance Imaging volume30,
pages1323 – 1341.
https://doi.org/10.1016/j.mri.2012.05.001.
notequantitative Imaging in Cancer.
[He et al.(2016)He, Zhang, Ren and Sun]he16
authorHe, K., authorZhang, X., authorRen,
S., authorSun, J., year2016.
titleDeep residual learning for image recognition, in:
booktitle2016 IEEE Conference on Computer Vision and Pattern
Recognition (CVPR), pp. pages770–778.
10.1109/CVPR.2016.90.
[Heller et al.(2021)Heller, Isensee, Maier-Hein, Hou, Xie, Li, Nan,
Mu, Lin, Han, Yao, Gao, Zhang, Wang, Hou, Yang, Xiong, Tian, Zhong, Ma,
Rickman, Dean, Stai, Tejpaul, Oestreich, Blake, Kaluzniak, Raza, Rosenberg,
Moore, Walczak, Rengel, Edgerton, Vasdev, Peterson, McSweeney, Peterson,
Kalapara, Sathianathen, Papanikolopoulos and Weight]heller21
authorHeller, N., authorIsensee, F.,
authorMaier-Hein, K.H., authorHou, X.,
authorXie, C., authorLi, F., authorNan,
Y., authorMu, G., authorLin, Z.,
authorHan, M., authorYao, G., authorGao,
Y., authorZhang, Y., authorWang, Y.,
authorHou, F., authorYang, J.,
authorXiong, G., authorTian, J.,
authorZhong, C., authorMa, J.,
authorRickman, J., authorDean, J.,
authorStai, B., authorTejpaul, R.,
authorOestreich, M., authorBlake, P.,
authorKaluzniak, H., authorRaza, S.,
authorRosenberg, J., authorMoore, K.,
authorWalczak, E., authorRengel, Z.,
authorEdgerton, Z., authorVasdev, R.,
authorPeterson, M., authorMcSweeney, S.,
authorPeterson, S., authorKalapara, A.,
authorSathianathen, N., authorPapanikolopoulos, N.,
authorWeight, C., year2021.
titleThe state of the art in kidney and kidney tumor
segmentation in contrast-enhanced ct imaging: Results of the kits19
challenge.
journalMedical Image Analysis volume67,
pages101821.
https://doi.org/10.1016/j.media.2020.101821.
[Hughes et al.(2016)Hughes, Azoury, Assadipour, Straughan, Trivedi,
Lim, Joy, Voellinger, Tang, Venkatesan et al.]hughes16
authorHughes, M.S., authorAzoury, S.C.,
authorAssadipour, Y., authorStraughan, D.M.,
authorTrivedi, A.N., authorLim, R.M.,
authorJoy, G., authorVoellinger, M.T.,
authorTang, D.M., authorVenkatesan, A.M., et al.,
year2016.
titleProspective evaluation and treatment of familial
carcinoid small intestine neuroendocrine tumors (si-nets).
journalSurgery volume159,
pages350–357.
[Isensee et al.(2021)Isensee, Jaeger, Kohl, Petersen and
Maier-Hein]isensee21
authorIsensee, F., authorJaeger, P.F.,
authorKohl, S.A., authorPetersen, J.,
authorMaier-Hein, K.H., year2021.
titlennu-net: a self-configuring method for deep
learning-based biomedical image segmentation.
journalNature methods volume18,
pages203–211.
[Jaeger et al.(2020)Jaeger, Kohl, Bickelhaupt, Isensee, Kuder,
Schlemmer and Maier-Hein]jaeger20
authorJaeger, P.F., authorKohl, S.A.A.,
authorBickelhaupt, S., authorIsensee, F.,
authorKuder, T.A., authorSchlemmer, H.P.,
authorMaier-Hein, K.H., year2020.
titleRetina U-Net: Embarrassingly Simple Exploitation of
Segmentation Supervision for Medical Object Detection, in:
editorDalca, A.V., editorMcDermott, M.B.,
editorAlsentzer, E., editorFinlayson, S.G.,
editorOberst, M., editorFalck, F.,
editorBeaulieu-Jones, B. (Eds.), booktitleProceedings
of the Machine Learning for Health NeurIPS Workshop,
publisherPMLR. pp. pages171–183.
<https://proceedings.mlr.press/v116/jaeger20a.html>.
[Jasti and Carucci(2020)]jasti20
authorJasti, R., authorCarucci, L.R.,
year2020.
titleSmall bowel neoplasms: A pictorial review.
journalRadioGraphics volume40,
pages1020–1038.
<https://doi.org/10.1148/rg.2020200011>,
10.1148/rg.2020200011,
http://arxiv.org/abs/https://doi.org/10.1148/rg.2020200011arXiv:https://doi.org/10.1148/rg.2020200011. notepMID:
32559148.
[Kamble et al.(2020)Kamble, Sahu and Doriya]kamble20
authorKamble, B., authorSahu, S.P.,
authorDoriya, R., year2020.
titleA review on lung and nodule segmentation techniques,
in: editorKolhe, M.L., editorTiwari, S.,
editorTrivedi, M.C., editorMishra, K.K. (Eds.),
booktitleAdvances in Data and Information Sciences,
publisherSpringer Singapore, addressSingapore. pp.
pages555–565.
[Kingma and Ba(2015)]kingma15
authorKingma, D.P., authorBa, J., year2015.
titleAdam: A method for stochastic optimization, in:
editorBengio, Y., editorLeCun, Y. (Eds.),
booktitle3rd International Conference on Learning Representations,
ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track
Proceedings.
<http://arxiv.org/abs/1412.6980>.
[Lin et al.(2017a)Lin, Dollar, Girshick, He, Hariharan and
Belongie]lin17_cvpr
authorLin, T.Y., authorDollar, P.,
authorGirshick, R., authorHe, K.,
authorHariharan, B., authorBelongie, S.,
year2017a.
titleFeature pyramid networks for object detection, in:
booktitleProceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR).
[Lin et al.(2017b)Lin, Goyal, Girshick, He and Dollar]lin17_iccv
authorLin, T.Y., authorGoyal, P.,
authorGirshick, R., authorHe, K.,
authorDollar, P., year2017b.
titleFocal loss for dense object detection, in:
booktitleProceedings of the IEEE International Conference on
Computer Vision (ICCV).
[Lin et al.(2016)Lin, Hsu, Ko, Chu, Chou, Chang and Chang]lin16
authorLin, Y.P., authorHsu, H.H., authorKo,
K.H., authorChu, C.M., authorChou, Y.C.,
authorChang, W.C., authorChang, T.H.,
year2016.
titleDifferentiation of malignant and benign incidental
breast lesions detected by chest multidetector-row computed tomography: Added
value of quantitative enhancement analysis.
journalPLOS ONE volume11,
pages1–11.
<https://doi.org/10.1371/journal.pone.0154569>,
10.1371/journal.pone.0154569.
[Liu et al.(2021)Liu, Tsui and Mandal]liu21
authorLiu, L., authorTsui, Y.Y.,
authorMandal, M., year2021.
titleSkin lesion segmentation using deep learning with
auxiliary task.
journalJournal of Imaging volume7.
<https://www.mdpi.com/2313-433X/7/4/67>,
10.3390/jimaging7040067.
[Loshchilov and Hutter(2019)]loshchilov19
authorLoshchilov, I., authorHutter, F.,
year2019.
titleDecoupled weight decay regularization, in:
booktitleInternational Conference on Learning Representations.
<https://openreview.net/forum?id=Bkg6RiCqY7>.
[Ma et al.(2020)Ma, Wei, Zhang, Wang, Lv, Zhu, Gaoxiang, Liu, Peng,
Wang, Wang and Chen]ma20
authorMa, J., authorWei, Z., authorZhang,
Y., authorWang, Y., authorLv, R.,
authorZhu, C., authorGaoxiang, C.,
authorLiu, J., authorPeng, C., authorWang,
L., authorWang, Y., authorChen, J.,
year2020.
titleHow distance transform maps boost segmentation cnns:
An empirical study, in: editorArbel, T.,
editorBen Ayed, I., editorde Bruijne, M.,
editorDescoteaux, M., editorLombaert, H.,
editorPal, C. (Eds.), booktitleProceedings of the
Third Conference on Medical Imaging with Deep Learning,
publisherPMLR. pp. pages479–492.
<https://proceedings.mlr.press/v121/ma20b.html>.
[Parzen(1962)]parzen62
authorParzen, E., year1962.
titleOn Estimation of a Probability Density Function and
Mode.
journalThe Annals of Mathematical Statistics
volume33, pages1065 – 1076.
10.1214/aoms/1177704472.
[Pedrosa et al.(2019)Pedrosa, Aresta, Ferreira, Rodrigues, Leitão,
Carvalho, Rebelo, Negrão, Ramos, Cunha and Campilho]pedrosa19
authorPedrosa, J., authorAresta, G.,
authorFerreira, C., authorRodrigues, M.,
authorLeitão, P., authorCarvalho, A.S.,
authorRebelo, J., authorNegrão, E.,
authorRamos, I., authorCunha, A.,
authorCampilho, A., year2019.
titleLndb: A lung nodule database on computed tomography.
<https://arxiv.org/abs/1911.08434>,
10.48550/ARXIV.1911.08434.
[Petersenn et al.(2015)Petersenn, Richter, Broemel, Ritter,
Deutschbein, Beil, Allolio, Fassnacht and Group]petersenn15
authorPetersenn, S., authorRichter, P.A.,
authorBroemel, T., authorRitter, C.O.,
authorDeutschbein, T., authorBeil, F.U.,
authorAllolio, B., authorFassnacht, M.,
authorGroup, G.A.S., year2015.
titleComputed tomography criteria for discrimination of
adrenal adenomas and adrenocortical carcinomas: analysis of the german acc
registry.
journalEuropean journal of endocrinology
volume172, pages415–422.
[Phan et al.(2019)Phan, Vo and Phan]phan18
authorPhan, A.C., authorVo, V.Q.,
authorPhan, T.C., year2019.
titleA hounsfield value-based approach for automatic
recognition of brain haemorrhage.
journalJournal of Information and Telecommunication
volume3, pages196–209.
<https://doi.org/10.1080/24751839.2018.1547951>,
10.1080/24751839.2018.1547951.
[Ryu et al.(2021)Ryu, Shin, Lee, Lee, Kang and Yi]ryu21
authorRyu, H., authorShin, S.Y., authorLee,
J.Y., authorLee, K.M., authorKang, H.j.,
authorYi, J., year2021.
titleJoint segmentation and classification of hepatic
lesions in ultrasound images using deep learning.
journalEuropean radiology volume31,
pages8733–8742.
[Shin et al.(2015)Shin, Lee, Yun, Jung, Heo, Kim and Lee]shin15
authorShin, S.Y., authorLee, S., authorYun,
I.D., authorJung, H.Y., authorHeo, Y.S.,
authorKim, S.M., authorLee, K.M.,
year2015.
titleA novel cascade classifier for automatic
microcalcification detection.
journalPLOS ONE volume10,
pages1–22.
<https://doi.org/10.1371/journal.pone.0143725>,
10.1371/journal.pone.0143725.
[Shin et al.(2019)Shin, Lee, Yun, Kim and Lee]shin19
authorShin, S.Y., authorLee, S., authorYun,
I.D., authorKim, S.M., authorLee, K.M.,
year2019.
titleJoint weakly and semi-supervised deep learning for
localization and classification of masses in breast ultrasound images.
journalIEEE Transactions on Medical Imaging
volume38, pages762–774.
10.1109/TMI.2018.2872031.
[Shin et al.(2023)Shin, Shen, Wank and Summers]shin23_spie
authorShin, S.Y., authorShen, T.C.,
authorWank, S.A., authorSummers, R.M.,
year2023.
titleImproving small lesion segmentation in CT scans
using intensity distribution supervision: application to small bowel
carcinoid tumor, in: editorIftekharuddin, K.M.,
editorChen, W. (Eds.), booktitleMedical Imaging 2023:
Computer-Aided Diagnosis, organizationInternational Society for
Optics and Photonics. publisherSPIE. p.
pages124651S.
<https://doi.org/10.1117/12.2651979>,
10.1117/12.2651979.
[Sudre et al.(2017)Sudre, Li, Vercauteren, Ourselin and
Jorge Cardoso]sudre17
authorSudre, C.H., authorLi, W.,
authorVercauteren, T., authorOurselin, S.,
authorJorge Cardoso, M., year2017.
titleGeneralised dice overlap as a deep learning loss
function for highly unbalanced segmentations, in: editorCardoso,
M.J., editorArbel, T., editorCarneiro, G.,
editorSyeda-Mahmood, T., editorTavares, J.M.R.,
editorMoradi, M., editorBradley, A.,
editorGreenspan, H., editorPapa, J.P.,
editorMadabhushi, A., editorNascimento, J.C.,
editorCardoso, J.S., editorBelagiannis, V.,
editorLu, Z. (Eds.), booktitleDeep Learning in Medical
Image Analysis and Multimodal Learning for Clinical Decision Support,
publisherSpringer International Publishing,
addressCham. pp. pages240–248.
[Summers et al.(2006)Summers, Huang, Yao, Campbell, Dempsey, Dwyer,
Franaszek, Brickman, Bitter, Petrick and Hara]summers06
authorSummers, R.M., authorHuang, A.,
authorYao, J., authorCampbell, S.R.,
authorDempsey, J.E., authorDwyer, A.J.,
authorFranaszek, M., authorBrickman, D.S.,
authorBitter, I., authorPetrick, N.,
authorHara, A.K., year2006.
titleAssessment of polyp and mass histopathology by
intravenous contrast–enhanced ct colonography.
journalAcademic Radiology volume13,
pages1490–1495.
https://doi.org/10.1016/j.acra.2006.09.051.
[Tang et al.(2020)Tang, Tang, Zhu, Xiao and Summers]tang20
authorTang, Y., authorTang, Y., authorZhu,
Y., authorXiao, J., authorSummers, R.M.,
year2020.
titleE^2Net: An edge enhanced network for accurate
liver and tumor segmentation on ct scans, in: editorMartel,
A.L., editorAbolmaesumi, P., editorStoyanov, D.,
editorMateus, D., editorZuluaga, M.A.,
editorZhou, S.K., editorRacoceanu, D.,
editorJoskowicz, L. (Eds.), booktitleMedical Image
Computing and Computer Assisted Intervention – MICCAI 2020,
publisherSpringer International Publishing,
addressCham. pp. pages512–522.
[Weninger et al.(2020)Weninger, Liu and Merhof]weninger20
authorWeninger, L., authorLiu, Q.,
authorMerhof, D., year2020.
titleMulti-task learning for brain tumor segmentation,
in: editorCrimi, A., editorBakas, S. (Eds.),
booktitleBrainlesion: Glioma, Multiple Sclerosis, Stroke and
Traumatic Brain Injuries, publisherSpringer International
Publishing, addressCham. pp. pages327–337.
[Zhao et al.(2021)Zhao, Chen and Wang]zhao21
authorZhao, Z., authorChen, H., authorWang,
L., year2021.
titleA coarse-to-fine framework for the 2021 kidney and
kidney tumor segmentation challenge.
<https://openreview.net/forum?id=6Py5BNBKoJt>.
|
http://arxiv.org/abs/2307.04896v1 | 20230710204144 | An abstract formulation of the flat band condition | [
"Jeffrey Galkowski",
"Maciej Zworski"
] | math.AP | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] |
Motivated by the study of flat bands in models of twisted bilayer graphene (TBG), we give abstract conditions which guarantee the existence of a discrete set of parameters for which periodic Hamiltonians exhibit flat bands. As an application, we show that a scalar operator derived from the chiral model of TBG has flat bands for a discrete set of parameters.
Spin-EPR-pair separation by conveyor-mode single electron shuttling in Si/SiGe
Lars R. Schreiber
August 12, 2023
==============================================================================
§ INTRODUCTION
Existence of flat bands for periodic operators (in the sense of Floquet theory) has interesting
physical consequences, especially in the case of nontrivial band topology. A celebrated
recent example is given by the Bistritzer–MacDonald Hamiltonian <cit.> modeling twisted
bilayer graphene
(see <cit.> and <cit.> for its mathematical derivation). A model exhibiting exact flat bands is given by the chiral limit of the Bistritzer–MacDonald
model considered by Tarnopolsky–Kruchkov–Vishwanath <cit.>. Both the Bistritzer–MacDonald model and its chiral limit depend on a parameter corresponding to the angle of twisting between two graphene sheets and, in the chiral model, the perfectly flat bands
appear for a discrete set of values of this parameter.
This follows from a spectral characterization of those magic angles
given by Becker–Embree–Wittsten–Zworski <cit.>. Existence of the first real magic angle
was provided by Watson–Luskin <cit.>, with its simplicity established by
Becker–Humbert–Zworski <cit.>. That paper also showed existence of infinitely many,
possibly complex, magic angles.
The purpose of this note is to provide a simple abstract version of the spectral
characterization of magic angles given in <cit.> (see also <cit.>). In <ref> we
apply this spectral characterization of flat bands in a model to which the argument from <cit.> does not apply.
To formulate our result we consider Banach spaces,
X⊂ Y, and a connected open set Ω⊂ℂ.
The result concerns a holomorphic family of Fredholm operators
of index 0
(see <cit.>):
Q : Ω×ℂ→ℒ ( X, Y ) ,
( α , k ) ↦ Q ( α, k ) .
We make the following assumption: there exists a lattice
Γ^* ⊂ℂ,
and families of invertible operators γ↦ W_∙ (γ ) : ∙→∙,
∙ = X, Y, γ∈Γ^*,
such that
Q ( α , k + γ ) = W_Y ( γ )^-1 Q ( α, k ) W_X ( γ ) , γ∈Γ^* .
A guiding example is given by the chiral model of twisted bilayer graphene (TBG) <cit.>, <cit.>,
<cit.>:
Q ( α, k ) := D ( α ) + k ,
D ( α ) := [ 2 D_z̅ α U ( z ); α U ( - z ) 2 D_z̅ ] ,
Ω = ℂ ,
2D_z̅ = 1 i ( ∂_x_1 + i ∂_x_2 ) , z = x_1 + i x_2 ∈ℂ ,
where U satisfies
U ( z + γ ) = e^ i ⟨γ , K ⟩ U ( z ) , U ( ω z ) = ω U(z) , U ( z̅ ) = - U ( - z ) , ω = e^ 2 π i/3,
γ∈Λ := ωℤ⊕ℤ , ω K ≡ K ≢0 Λ^* , Λ^* := 4 π i/√(3)Λ , ⟨ z , w ⟩ := ( z w̅ ) .
An example of U is given by the Bistritzer–MacDonald potential
U ( z ) = - 4 3 π i ∑_ℓ = 0 ^2 ω^ℓ e^ i ⟨ z , ω^ℓ K ⟩, K = 43 π .
We note that a potential satisfying (<ref>) is periodic with respect to the lattice 3 Λ
and that we can take
Y := L^2 ( ℂ / Γ ; ℂ^2 ) ,
X := H^1 ( ℂ / Γ ; ℂ^2 ) , Γ := 3 Λ .
(For the Fredholm property of D ( α ) + k : X → Y see
<cit.>; the index is equal to 0.) The operators W_∙ ( γ )
are given by multiplication by e^ i ⟨γ, z ⟩, γ∈Γ^*,
with Γ^* the dual lattice to Γ. (The operator is the same but acts on different spaces.)
The self-adjoint Hamiltonian for the chiral model of TBG is given by
H ( α ) := [ 0 D( α )^*; D ( α ) 0 ] ,
and Bloch–Floquet theory means considering the spectrum of
H_k ( α ) := e^ - i ⟨ z, k ⟩ H ( α ) e^ i ⟨ z, k ⟩
: H^1 ( ℂ/Γ ; ℂ^4 ) → L^2 ( ℂ/Γ ; ℂ^4 ) ,
H_k ( α ) = [ 0 Q ( α, k )^*; Q ( α, k ) 0 ] , Q ( α, k ) = D ( α ) + k ,
see <cit.> (we should stress that it is better to consider a modified boundary
condition <cit.> rather than Γ-periodicity but this plays no role in the discussion here).
A flat band at zero energy for the Hamiltonian (<ref>) means that
∀ k ∈ℂ 0 ∈_ L^2 ( ℂ/Γ; ℂ^4 ) H_k ( α )
⟺ ∀ k ∈ℂ _ H^1 ( ℂ/Γ; ℂ^4 ) H_k ( α ) ≠{ 0 }
⟺ ∀ k ∈ℂ _ H^1 ( ℂ/Γ; ℂ^2 ) Q ( k, α ) ≠{ 0 } .
We generalize the result of <cit.> stating that the set of α's for which
(<ref>) holds, which we denote by 𝒜_ch, is a discrete subset of ℂ and that (<ref>) is
equivalent to
∃ k ∈ℂ∖Γ^* _ H^1 ( ℂ/Γ; ℂ^2 ) Q ( k, α ) ≠{ 0 } .
The key property in showing this is the existence of protected states <cit.>, <cit.>:
∀ α∈ℂ ,
k ∈Γ^* _ H^1 ( ℂ/Γ; ℂ^2 ) Q ( k, α )
≥ 2, _ H^1 ( ℂ/Γ; ℂ^2 ) Q ( k , 0 ) = 2 .
This is replaced by the hypothesis (<ref>). We use
_𝒦 to denote the indicator function of 𝒦.
In the notation of (<ref>) and assuming (<ref>), suppose that
there exists a discrete set 𝒦⊂ℂ such that
for some m_0 ∈ℕ and α_0 ∈Ω, we
have,
Q ( α_0,k ) = m_0 _𝒦 ( k ) , Q ( α ,k ) ≥ m_0 _𝒦 ( k ),
k ∈ℂ, α∈Ω .
Then there exists a discrete set 𝒜⊂Ω such that
Q ( α ,k) ≠{ 0 } for α∈𝒜 and k ∈ℂ,
Q ( α,k ) = m_0 _𝒦 ( k ) for
α∈Ω∖𝒜 and k ∈ℂ.
In view of (<ref>) we see that (<ref>) is satisfied for Q given in
(<ref>) with m_0 = 2, α_0 = 0, Ω =ℂ and 𝒦 =
Γ^*. For a direct proof see <cit.> or <cit.>.
Remarks.
Theorem <ref> is valid under a weaker condition than (<ref>).
As seen in <ref>, we need to control the dimension of Q(α,k) for every k using
the dimension of Q ( α, k) for k in some fixed compact set.
That some condition is needed (other than holomorphy and the Fredholm property)
can be seen by considering the simple example of Q(α,k)=1-α k,
X = Y = ℂ. In this case (<ref>) is satisfied with α_0=0 and 𝒦=∅. Nevertheless,
Q(α,k) =0 k≠α^-1
1 k=α^-1
and (<ref>) fails. We opted for the easy to state condition (<ref>) in view
of the motivation from condensed matter physics.
§ PROOF OF THEOREM <REF>
We first fix k_0 ∈ℂ∖𝒦 and define
𝒜_k_0 := ∁{α∈Ω : Q(α, k_0 )^-1:Y→ X exists}.
Since α↦ Q( α , k_0 ) is a holomorphic family of Fredholm operators
of index zero,
and Q( α_0, k_0 ) = { 0 }, we conclude that α↦ Q ( α , k_0 )^-1 is
a meromorphic family of operators and, in particular, 𝒜_k_0 is a discrete set –
see <cit.>. Also, for
α∉𝒜_k_0, k ↦ Q ( α, k )^-1 is a meromorphic
family of operators and the multiplicity
m ( α,k ) := 1/ 2 π∮_∂ D Q( α, ζ ) ^-1∂_ζ Q ( α , ζ ) d ζ ,
is well defined. The integral is over the positively oriented boundary of a disc
D which contains k as the
only possible pole of ζ↦ Q ( α, ζ ). For such D
there exists ε > 0 such that
m ( α,k ) = ∑_ k' ∈ D m( α',k' ) , if |α - α' | < ε.
In particular for a fixed k ∈ℂ, α↦ m ( α ,k ) is upper semicontinuous.
We now define
U := {α∈Ω∖𝒜_k_0 : ∀ k, m ( α,k ) = m_0 _𝒦 ( k ) }.
We note that
α_0∈ U and that Ω∖𝒜_k_0 is connected. Hence U = Ω∖𝒜_k_0 if we show that U is open and closed in the relative topology of
Ω∖𝒜_k_0.
Let α∈ U. We start by showing that for any compact subset K⊂ℂ, there
exists ε_K>0 such that
m(α',k)= m_0 _𝒦(k)=m(α,k) for all k∈ K and |α-α'|<ε_K.
To see this we note that for any fixed k ∈ℂ
there exist D_k = D ( k, δ_k ),
and ε_k>0 such that that (<ref>) holds for |α-α'|<ε_k.
By shrinking D_k (and consequently ε_k) we can assume that (here we use
the discreteness of 𝒦)
D_k ∖{ k }⊂∁𝒦.
Since K is compact, we can find a finite cover
K⊂⋃_i=1^N D_k_i.
Then k_i is the only possible pole for k↦ Q(α,k)^-1 in D_k_i
and for |α-α'|<ε_K:=min_i=1,… Nε_k_i, we have
m(α,k_i)=∑_k∈ D_k_im(α',k).
If k_i∉𝒦 then, as α∈ U, m ( α, k_i ) = 0
and consequently m ( α' , k ) = 0 for k ∈ D_k_i ⊂∁𝒦.
On the other hand, if k_i∈𝒦 then,
m_0=∑_k∈𝒟_k_im(α',k).
and since m(α',k_i)≥ m_0 (by the assumption (<ref>)) we have m(α',k)=0 for k∈ D_k_i∖{ k_i}⊂∁𝒦 and m(α',k_i)=m_0.
Putting those two cases together, we have m(α',k)=m_0_𝒦(k) for k∈ K and |α-α'|<ε_K as claimed in (<ref>).
Now, to complete the proof that U is open, we use (<ref>). Let K⊂ℂ contain the fundamental domain of Γ^* and ε_K as in (<ref>). Then, for all k∈ℂ, there is γ∈Γ^* such that k+γ∈ K. Using (<ref>), we have for |α-α'|<ε_K,
m(α',k+γ)=m(α,k+γ).
But then, by (<ref>)
m(α',k+γ)=m(α',k), m(α,k+γ)=m(α,k),
and hence
m(α',k)=m(α,k)=_𝒦(k).
Since k∈ℂ was arbitrary, this implies α'∈ U.
To show that U is closed suppose that 𝒜_k_0∌α_j →α∉𝒜_k_0
and m ( k, α_j ) = m_0 _𝒦 ( k ). Then, since α∉𝒜_k_0, for every k∈ℂ, there exist ε_k>0 and D_k
such that (<ref>) and (<ref>) hold. In particular, for j large enough (depending on k),
m(α,k)=∑_k'∈ D_k m(α_j,k')=∑_k'∈ D_km_0_𝒦(k')=m_0_𝒦(k).
Hence U is closed and open which means that U = Ω∖𝒜_k_0.
Recalling the definition (<ref>), we proved that
Ω∖𝒜_k_0⊂{α: ∀ k, m ( α,k ) = m_0 _𝒦 ( k ) }⊂Ω∖𝒜_ k_1 ,
for any k_1 ∉𝒦. But this means that 𝒜_k_0 is independent of k_0
and for α∈𝒜 := 𝒜_k_0, Q ( α, k )^-1 does not exist for any k
∈ℂ. Since Q ( α, k ) is a Fredholm operator of index 0, this shows
that Q ( α, k ) ≠{ 0 } for all k.
§ A SCALAR MODEL FOR FLAT BANDS
One of the difficulties of dealing with the model described by (<ref>), (<ref>)
is the fact that D ( α ) acts on ℂ^2-valued functions. Here we propose the following
model in which D ( α ) is replaced by a scalar (albeit second order) operator. This is done
as follows. We first consider P ( α ) : H^2 ( ℂ/Γ ; ℂ^2 )
→ L^2 ( ℂ/Γ ; ℂ^2 ) defined as follows:
P ( α ) := D ( - α ) D ( α ) = Q ( α ) ⊗ I_ℂ^2 + R ( α ),
Q ( α ): = ( 2 D_z̅ )^2 - α^2 V ( z ) ,
R ( α ) := - α[ 0 V_1 ( z ); V_1 ( -z ) 0 ] , V ( z ) := U ( z ) U( -z ) , V_1 ( z ) := 2 D_z̅ U ( z ) .
If we think of P ( α ) as a semiclassical differential system with h = 1/α
(see <cit.>) then Q ( α ) is the quantization of the determinant of
the symbol of D( α ) and R ( α ) is a lower order term.
We lose no information when considering P ( α ) in the characterization
of flat bands (<ref>):
If P ( α, k ) := e^ - i ⟨ z, k ⟩ P ( α ) e^ i ⟨ z, k ⟩
then
_ H^1 ( ℂ/Γ ) ( D ( α ) + k ) ≠{0 } ⟺ _ H^2 ( ℂ/Γ ) P ( α, k ) ≠{ 0 } .
In particular α∈𝒜_ch if and only if
k ∈_ L^2 ( ℂ / Γ ) P ( α, k ) for some
k ∉Γ^* (which then implies this for all k).
We note that P ( α , k ) = ( D ( - α ) + k ) ( D ( α) + k ) and that
D ( - α ) - k = - ℛ ( D ( α ) + k ) ℛ,
ℛ[ u_1; u_2 ] ( z ) = [ u_2 ( -z ); u_1 ( - z ) ]
and hence
_ H^1 ( ℂ/Γ ) ( D ( α ) + k ) =
ℛ_ H^1 ( ℂ/Γ ) ( D ( - α ) - k ) .
Since D ( α ) is elliptic, the elements of the kernels above are in C^∞
( ℂ/Γ ) and hence H^1 can be replaced by H^s for any s
– see <cit.>. Hence if _ H^2 P ( α, k ) ≠{ 0 }
then either _ H^2 ( D ( α ) + k ) = _H^1 ( D ( α) + k ) ≠{ 0 } or _ H^1 ( D ( - α ) + k ) ≠{ 0 }. If k ∉Γ^* then the equivalence of (<ref>) and
(<ref>) gives the conclusion.
We now consider a model in which we drop the matrix terms in
(<ref>), the definition of P ( α ), and have Q ( α ) act on scalar valued functions. The self-adjoint Hamiltonian corresponding to
(<ref>) is now given by
H ( α ) := [ 0 Q ( α )^*; Q ( α ) 0 ], Q ( α ): = ( 2 D_z̅ )^2 - α^2 V ( z ) , V ∈ C^∞ ( ℂ ) ,
V ( x + γ ) = V ( x ) , γ∈Λ:=
ωℤ⊕ℤ , V ( ω x ) = ω̅V ( x ) , ω := e^ 2 π i/3 .
The potential is periodic with respect to Λ, and hence the usual Floquet theory
applies:
H( α , k ) := [ 0 Q( α , k )^*; Q ( α, k ) 0 ], Q ( α , k ): = ( 2 D_z̅ + k )^2 - α^2 V ( z ) ,
_L^2 ( ℂ ) H ( α ) = ⋃_ k ∈ℂ/Λ^* _ L^2 (
ℂ / Λ ) H ( α, k ) ,
where _ L^2 ( ℂ / Λ ) H ( α , k ) is discrete and
is symmetric under E ↦ - E. Just as for the chiral model of
TBG, a flat band at zero for a given α means that
∀ k ∈ℂ 0 ∈_ L^2 (
ℂ / Λ ; ℂ^2 ) H ( α , k ) ⟺ ∀ k ∈ℂ _ H^2 (
ℂ / Λ; ℂ ) Q ( α, k ) ≠{ 0 } .
As in the chiral model, we take W_X(γ)=W_Y(γ)=e^i⟨γ,z⟩, γ∈Λ^*, the dual lattice to obtain (<ref>).
Theorem <ref> shows that as in the case of (<ref>) this happens
for a discrete set of α∈ℂ:
For H and Q given in (<ref>) there exists a discrete set
𝒜_sc⊂ℂ such that
_ H^2 ( ℂ / Λ; ℂ ) Q ( α, k ) ≠{ 0 } for
α∈𝒜_sc, k ∈ℂ,
_ H^2 ( ℂ / Λ; ℂ ) Q ( α, k ) =
_Λ^* ( k ) for α∉𝒜_sc.
This is an immediate consequence of Theorem <ref> once we establish
(<ref>) with m_0 = 1 (and α_0 = 0). The kernel of
Q ( 0 , k ) = 2 ( D_z̅ + k )^2 ,
on H^2 ( ℂ/Λ ) is empty for k ∉Λ^* and
is given by ℂ e^ i ⟨ k , z ⟩, when k ∈Λ^*.
This gives the
first condition in (<ref>). The second one is provided by
For all α∈ℂ and k ∈Λ^*,
_ H^2 (
ℂ / Λ; ℂ ) Q ( α, k ) ≥ 1.
The proof is essentially the same as that of <cit.> and it uses symmetries of
H (α ) in (<ref>): for u ∈ L^2 ( ℂ/Λ; ℂ^2 ),
ℒ_γ u ( z ) := u ( z + γ ) , γ∈Λ , 𝒞 u ( z ) := [ 1 0; 0 ω̅ ] u ( ω z ) , 𝒲 u = [ -1 0; 0 1 ] u ,
ℒ_γ H ( α ) =H ( α ) ℒ_γ, 𝒞H ( α ) = H ( α ) 𝒞, 𝒞ℒ_γ= ℒ_ωγ𝒞,
𝒲 H ( α ) 𝒲 = - H ( α ) , ℒ_γ𝒲 = 𝒲ℒ_γ , 𝒞𝒲 = 𝒲𝒞 .
We introduce two orthogonal subspaces of L^2 ( ℂ/Γ ):
L^2_j := { u ∈ L^2 ( ℂ/Γ ) : ℒ_γ u = u ,γ∈Λ, 𝒞 u = ω̅^j u } , j = 0 , 1 .
Then the standard basis of ℂ^2 satisfies 𝐞_j ∈ L^2_j
and H ( 0 ) 𝐞_j = 0. Using 𝒲 we see that
the spectrum of H ( α ) on L^2_j (with the domain given by H^2 ( ℂ/Γ )
∩ L^2_j) is symmetric under E ↦ - E. Since 0 is a simple eigenvalue
of H( 0 ) |_L^2_j, j = 0, 1 and the eigenvalues of H(α)|_L^2_j are continuous in α, 0 remains an eigenvalue for all α.
That means that _H^2 Q ( α, 0 ) is at least one dimensional. The same argument
applies at all k ∈Λ^* by conjugation with e^ i ⟨ z , k ⟩.
Remarks. 1. The proof of Theorem <ref>
also shows the following spectral characterization of 𝒜_sc: if
T_k := ( 2 D_z̅ + k)^-2 V , k ∉Λ^* ,
then
α∈𝒜_sc ⟺ ∃ k ∉Λ^* α^-2∈_ L^2 ( ℂ/Λ ) T_k
⟺ ∀ k ∉Λ^* α^-2∈_ L^2 ( ℂ/Λ ) T_k ,
Using the methods of <cit.> one can show that for V ( z ) = U ( z ) U ( - z )
with U given by (<ref>) (or for more general classes of potentials described
in <cit.>), T_k^p∈ ( π/√(3) ) ℚ, p ≥ 2. Together
with a calculation for p = 2 (as in <cit.>) this
shows that | 𝒜_sc | =∞. With numerical assistance one can also show
existence of a real α∈𝒜_sc.
2. We can strengthen Proposition <ref> as in <cit.>:
there exists a holomorphic family ℂ∋α↦ u ( α ) ≢0,
such that u ( 0 ) = 1 and Q ( α, 0 ) u ( α ) = 0.
§ NUMERICAL OBSERVATIONS
The spectral characterization (<ref>) allows for an accurate computation
of α's for which (<ref>) exhibits flat bands at energy 0.
For large α's however, pseudospectral effects described in <cit.>
make calculations unreliable. The set (shown as ∙)
𝒜_sc∩{α≥ 0 }
where 𝒜_sc is given in Theorem <ref> looks as follows
(for comparison we show the corresponding set, 𝒜_ch, for the chiral model ∘):
< g r a p h i c s >
The real elements of 𝒜_sc are shown as ∙. They appear to have multiplicity
two. An adaptation of the theta function argument <cit.>, <cit.>, <cit.>, <cit.>
should apply to this case and the evenness of eigenfunctions in Proposition <ref> shows that
they have (at least) two zeros at α∈𝒜_sc. That implies multiplicity of at least 2.
This is illustrated by an animation <https://math.berkeley.edu/ zworski/scalar_magic.mp4> (shown
in the coordinates of <cit.>). When we interpolate between the chiral model and the
scalar model, the multiplicity two real α's split and travel in opposite directions to become
magic α's for the chiral model: see <https://math.berkeley.edu/ zworski/Spec.mp4>.
One of the most striking observations made in <cit.> was a quantization rule for
real elements of 𝒜_ch with the exact potential
(<ref>): if α_1 < α_2 < ⋯α_j < ⋯ is the
sequence of all real α's for which (<ref>) holds, then
α_j+1 - α_j = γ + o ( 1 ) , j → + ∞ , γ≃32.
The more accurate computations made in <cit.> suggests that
γ≃ 1.515.
In the scalar model (<ref>) with V( z ) = U ( z ) U ( -z ) where
U is given by (<ref>) we numerically observe the following rule for real elements of 𝒜_sc:
α_j+1 - α_j = 2 γ + o ( 1 ) , j → + ∞ ,
where γ is the same as in (<ref>).
Acknowledgements
We would like to thank Simon Becker for help with matlab and in particular for
producing the movies referred to above.
JG acknowledges support from EPSRC grants EP/V001760/1 and EP/V051636/1 and
MZ from the NSF grant DMS-1901462
and the Simons Foundation under a “Moiré Materials Magic" grant.
0
[Be*21]suppl S. Becker, M. Embree, J. Wittsten and M. Zworski,
Spectral characterization of magic angles in twisted bilayer graphene, Phys. Rev. B 103, 165113, 2021.
[Be*22]beta S. Becker, M. Embree, J. Wittsten and M. Zworski,
Mathematics of magic angles in a model of twisted bilayer graphene,
Probab. Math. Phys. 3(2022), 69–103.
[BHZ22a]bhz1 S. Becker, T. Humbert and M. Zworski,
Integrability in the chiral model of magic angles, 2208.01620.
[BHZ22b]bhz2 S. Becker, T. Humbert and M. Zworski,
Fine structure of flat bands in a chiral model of magic angles,
2208.01628.
[BiMa11]BM11 R. Bistritzer and A. MacDonald, Moiré bands in twisted double-layer graphene. PNAS, 108, 12233–12237, 2011.
[CGG22]CGG E. Cancès, L. Garrigue, D. Gontier, A simple derivation of moiré-scale continuous models for twisted bilayer graphene. 2206.05685.
[DuNo80]dun B.A. Dubrovin and S.P. Novikov, Ground states in a periodic field. Magnetic Bloch functions and vector
bundles. Soviet Math. Dokl. 22, 1, 240–244, 1980.
[DyZw19]res S. Dyatlov and M. Zworski,
Mathematical Theory of Scattering Resonances,
AMS 2019, <http://math.mit.edu/ dyatlov/res/>
[TKV19]magic
G. Tarnopolsky, A.J. Kruchkov and A. Vishwanath,
Origin of magic angles in twisted bilayer graphene,
Phys. Rev. Lett. 122, 106405, 2019
[Wa^*22]wats A. B. Watson, T. Kong, A. H. MacDonald, and M. Luskin, Bistritzer-MacDonald dynamics in twisted bilayer graphene, 2207.13767.
[WaLa21]lawa A. Watson and M. Luskin, Existence of the first magic angle for the chiral model of bilayer graphene, J. Math. Phys. 62, 091502 (2021).
|
http://arxiv.org/abs/2307.04050v1 | 20230708212820 | Optimization-based Learning for Dynamic Load Planning in Trucking Service Networks | [
"Ritesh Ojha",
"Wenbo Chen",
"Hanyu Zhang",
"Reem Khir",
"Alan Erera",
"Pascal Van Hentenryck"
] | cs.AI | [
"cs.AI",
"cs.LG",
"cs.SY",
"eess.SY"
] |
A Robust and Efficient Optimization Model for Electric Vehicle Charging Stations in Developing Countries under Electricity Uncertainty
[
=========================================================================================================================================
*Co-first authors
The load planning problem is a critical challenge in service network
design for parcel carriers: it decides how many trailers (or loads),
perhaps of different types, to assign for dispatch over time between
pairs of terminals. Another key challenge is to determine a flow plan, which specifies how parcel volumes are assigned to planned loads. This paper considers the Dynamic Load Planning Problem (DLPP) that considers both flow and load planning challenges jointly in order to adjust loads
and flows as the demand forecast changes over time before the
day of operations. The paper aims at developing a decision-support
tool to inform planners making these decisions at terminals across the
network. The paper formulates the DLPP as a MIP and shows that it
admits a large number of symmetries in a network where each commodity
can be routed through primary and alternate paths. As a result, an
optimization solver may return fundamentally different solutions to
closely related problems (i.e., DLPPs with slightly different inputs),
confusing planners and reducing trust in optimization. To remedy this
limitation, the paper proposes a Goal-Directed Optimization (GDO) that
eliminates those symmetries by generating optimal solutions staying
close to a reference plan. The paper also proposes an optimization
proxy to address the computational challenges of the optimization
models. The proxy combines a machine learning model and a
feasibility restoration model and finds solutions that satisfy real-time constraints imposed by planners-in-the-loop. An extensive computational study on industrial instances shows that the optimization proxy is around 10 times faster than the commercial solver in obtaining the same quality solutions and orders of magnitude faster for generating solutions that are consistent with each other. The proposed approach also demonstrates the benefits of the DLPP for load consolidation, and the significant savings obtained from combining machine learning and optimization.
§ INTRODUCTION
The e-commerce market continues to show robust growth and leading
analysts project that today's $3.3 trillion market could grow further
to $5.4 trillion annually by 2026 (<cit.>). Much of e-commerce
relies on home delivery of small packages or parcels and other boxed
freight. Key freight carriers like UPS and FedEx continually seek to
redesign and operate profitable logistic networks that meet e-commerce
customer service expectations. Beyond physical network design
including the location and sizing of various freight processing
terminals, these companies face challenging service network design
problems. A critical service network design challenge for package
carriers are the so-called load planning problems (for
background, see <cit.>). Here, load planning refers to
decisions related to the number of trailers or container loads, perhaps of different types, to plan for dispatch over time between
pairs of terminals. Such planned loads are the transportation capacity
of the network. Flow planning decisions represent another key challenge, where the flow plan specifies how to allocate parcel volumes to planned loads to feasibly and cost-effectively serve network demand. As each package moves from its origin to destination, it is transported by a sequence of planned loads where it is unloaded and sorted at a transfer (hub) terminal between each loaded dispatch. Together, the flow and load plan decisions define a service network that moves package
volume from origins to destinations in order to meet customer service
expectations. The research described in this paper is conducted
directly with a leading global parcel carrier that operates a massive
network moving large volumes of packages each day. Figure <ref> illustrates the load planning operations at an example
terminal. It highlights the planner-in-the-loop environment in which load planning takes place; an important consideration underlying this
research.
Packages at a terminal with the same destination and service class are
referred to as a commodity. A flow plan defines flow
rules for each commodity in the service network; these flow rules
specify how a commodity is routed through the network over time. Since parcel carriers operate massive terminal networks with large numbers of transfer locations, a flow plan may include alternate flow rules that specify loading paths for commodities in addition to the default path specified by the primary flow rules. Both the primary (default) and alternate paths specify how a commodity moves through the network, and these planned paths are service feasible, i.e., they ensure that commodities arrive on time given their service guarantees.
This paper considers the Dynamic Load Planning Problem (DLPP)
faced by the load planner at a terminal as depicted in Figure
<ref> during a short time period (one or two weeks) leading up to the day of operations. The goal of the planner, and thus of the DLPP, is to decide (1) how many loads should be planned for outbound dispatch to other terminals at various times during the day of operations and (2) how to allocate commodity volumes across planned loads respecting the capacity constraints and the primary and alternate flow rules. These two decisions define what is called a load plan in this paper. The objective of the DLPP is to obtain a load plan that
minimizes the number of loads, consolidating the commodities as best
as possible. In practice, the DLPP is solved by planners, who adjust
existing load plans manually to reflect changes in commodity volumes
arriving at the terminal. This process is typically myopic
and creates inefficiencies across the network.
The goal of this research is to develop a decision support tool to
assist planners in solving the DLPP, suggesting load plans that remove
existing inefficiencies. Moreover, for terminals that do not have a
planner, the tool can fully automate the DLPP, bridging the gap
between network design and operations. To develop such a tool, this paper first investigates optimization models for the DLPP. In its general form, the DLPP is strongly NP-hard and its MIP formulation is challenging for state-of-the-art solvers given the size of the instances encountered in practice. Moreover, the natural MIP model exhibits significant symmetries which is highly undesirable for the planner-in-the-loop environment of the industrial partner. Indeed, planners will be extremely confused if small changes in commodities result in completely different load plans. To address this challenge, this paper presents a Goal-Directed Optimization (GDO) that solves a first model to find the optimal solution to the DLPP and uses a second model to find a plan that is as close as possible to a reference plan. GDO is shown to produce consistent plans, i.e., plans that are close for inputs that only differ slightly. Unfortunately, the GDO approach is too time-consuming to be used in planner-in-the-loop environments. To address this final difficulty, this research proposes the use of
optimization proxies that combine a Machine-Learning (ML) model and a
feasibility restoration procedure to obtain near-optimal solutions in
a few seconds, even for the largest terminals. The ML model uses
supervised learning to mimic the GDO approach and predicts the optimal set of planned loads. The feasibility restoration procedure then solves a small MIP model to determine the final allocation of commodity volumes to planned loads, adding extra capacity as needed to ensure feasibility. The proposed approach is practical since it produces high-quality plans that are consistent with each other, where small changes in inputs leads to very similar load plans by virtue of the ML training that mimics the GDO optimization.
The main contributions of the paper can be summarized as follows:
* The paper formalizes the DLPP and develops a natural MIP
formulation to solve it.
* The paper proposes a Goal-Directed Optimization approach to
remedy the limitations of the MIP formulation; it uses a 2-stage
approach to eliminate symmetries and provide optimal load plans that
are close to a reference plan.
* The paper proposes an optimization proxy to address the computational
difficulties of the GDO approach; the optimization proxy uses a
machine learning model to predict the loads and a feasibility
restoration procedure to adjust the predictions to satisfy the problem
constraints and determine the commodity flows. Once trained, the optimization proxy provides high-quality solutions in a few seconds.
* The paper presents extensive computational results on industrial
instances, including some of the largest terminals in the network;
the results demonstrate the significant benefits of optimization and
the ability of the optimization proxy to find high-quality and
consistent solutions in real time. More precisely, the paper shows
that the optimization proxy outperforms a greedy heuristic and the
MIP model solved by a commercial solver both in terms of the
objective function value and consistency metrics. The
optimization proxy is around 10 times faster than the commercial
solver in obtaining solutions with the same objective function value
and orders of magnitude faster in terms of generating solutions that
are consistent with a reference plan. Empirical experiments show
the value of breaking symmetries by GDO, which helps the proxy to
produce high-quality and consistent load plans.
* From a business and sustainability perspective, the experiments
demonstrate the value of having alternate flow paths for the
commodities, in addition to the primary flow paths. The proposed load
plans allocate approximately 17% commodity volume to the alternate
flow paths and reduce the required load capacity by 12%-15%.
The rest of this paper is organized as follows. Section
<ref> summarizes related work. Sections
<ref> and <ref> introduces the DLPP
and its modeling. Sections <ref> and <ref>
present the GDO approach and the optimization proxy. Section
<ref> describes a heuristic that mimic human
planners and serve as a baseline. Section <ref>
describes the computational results. Section <ref>
discusses the benefits of the DLPP formulation, optimization, and
machine learning, quantifying the cost and sustainability benefits and
the important factors driving them.
§ RELATED WORK
Service Network Design.
There is abundant research on network design for the Less-than-truckload (LTL) trucking industry (see
<cit.>). Interested readers
can consult erera2013creating for a detailed description
of LTL operations. <cit.> present a detailed
description of the mathematical models and heuristics for the problems
arising in trucking service network design. The authors describe the
tactical flow and load planning problem which is solved weeks
in advance for “typical” commodity volume (e.g., average daily
origin-destination commodity volume) for a network of terminals. The
goal of the flow and load planning problem is to determine
effective primary flow paths for the commodity volume and the total
trailer capacity required on each flow path in a network of terminals.
Most of these network design problems are formulated over time-space
networks using integer programming models. The flow and load planning
problem with both primary and alternate flow paths for industry-scale
instances can be modeled as large-scale integer programming models
which, unfortunately, cannot be solved directly by commercial
solvers. Therefore, previous work in this area focused mainly on
finding a single cost-effective primary flow path for the
commodities. Exact approaches to solve these problems have been
proposed by <cit.>, <cit.>,
and <cit.>. However, these approaches can only
solve instances with a few thousand packages. For industry-scale
instances, researchers have resorted to various heuristics including
variants of local search heuristic algorithms
(<cit.>,
<cit.>) and greedy algorithms (<cit.>).
Flow and Load Planning with Alternate Paths.
Tactical flow and load planning is typically based on average daily
estimates of origin-destination commodity volume. However, commodity
volume differ substantially from day to day and from week to week
(<cit.>). Hence, planners at a terminal locally
modify the load plans on a daily basis, using the latest estimates of
commodity volume until the day of operations. More specifically, the
planners take advantage of both primary and alternate flow paths to
improve trailer consolidation at their respective
terminals. It is worth highlighting that the primary flow paths come from flow and load planning. Once primary options are available alternate flow paths, that are time feasible, are identified. To the best of our knowledge no paper carefully studies the problem of allocating volume across alternate flow paths in operations. Alternate flow paths are useful to reduce trailers when commodity volume can be split across paths. This is especially useful because of demand uncertainty. <cit.> present a study on the value of
having these alternate flow paths to hedge against demand
uncertainty. They show that it is sufficient to have just one
alternate to contain the impact of most of the fluctuations
in demand; the authors refer to such a load plan as a 2-alt load plan. Subsequently, the authors in <cit.> study the operational decisions that LTL carriers need to make to effectively operate a 2-alt load plan when demand changes dynamically on a day-to-day basis. However, the proposed approach cannot be solved for practical sized instances. This paper proposes a ML-based solution approach for the allocation of volume across both primary and multiple alternate flow paths; the proposed approach is shown to be effective for large scale instances experienced in practice.
Dynamic Load Planning.
Network-wide simultaneous optimization of load planning adjustments is
a daunting challenge due to the scale of the network, number of commodities and the number of transfer hubs for the commodities.
Existing research in the literature may be applicable to the problem of selecting a single primary flow path
(non-splittable) for each commodity at each terminal for each sorting
period in order to minimize the cost of the resulting load plan. Splitting commodity volume across alternate flow paths is likely to improve trailer utilization as it introduces more flexibility in the load planning process. This research considers the DLPP problem at a terminal in which the commodity volume can be split (among primary and alternate flow paths) to promote better trailer utilization, lower transportation cost, and increased sustainability. The flexibility to adjust plans enables terminal planners to better manage daily operations while maintaining service guarantees. This problem is mentioned as an
interesting and useful future research direction by <cit.>.
One paper in the literature, <cit.>, does introduce the problem of re-routing freight volume on alternate flow paths to improve on-time performance of load plans on the day-of-operations; this becomes necessary when the actual volume deviates from the forecasted volume on the day-of-operations. In this work, commodity volume is assigned to exactly one flow path (it is not splittable) such that the total (fixed) trailer capacity is respected and the objective is to minimize the total lateness of shipments. The authors develop MIP models for this problem and propose heuristic algorithms to solve them. Note that a key difference between this approach and the approach proposed in the current paper is that we allow volume to be split across multiple flow paths on the day-of-operations. Furthermore, we also adjust the load plan to identify opportunities to reduce outbound capacity (and improve utilization) as demand forecasts are updated.
The DLPP is also similar to the variable-sized bin packing problem
described by <cit.> where the objective is to
minimize the total space used to pack a set of items into bins
(available in different sizes), such that each item is packed into
exactly one bin. In the DLPP, the packages are the items and trailers
are bins but the key difference is that the DLPP allows for the
splitting of the package volume into compatible trailers in order to
further reduce the transportation cost by promoting better
consolidation or packing.
Machine Learning for Optimization.
In recent years, there has been a notable surge of interest among
researchers in the development of ML surrogates for solving MIPs. This
emerging field has attracted attention due to the potential of ML
techniques to provide efficient approximations for computationally
intensive calculations involved in solving MIPs. We refer the reader
to (<cit.>,<cit.>) for a comprehensive
overview on the topic. The techniques can fall into one of the two
categories. The first category includes methods based on
reinforcement learning
(<cit.>),
where the ML model is trained by interacting with simulation
environments. The second category comprises supervised learning
(<cit.>),
where the ML model imitates the optimization model and replaces
expensive calculations with a quick approximation. This research
focuses on the latter category since the proposed optimization model
could be used as the expert for supervised learning. Optimization
proxies, which combine learning with feasibility restoration, has
emerged from supervised learning. Recent work in this area includes
(<cit.>).
§ PROBLEM DESCRIPTION AND MODELING
Parcel carriers operate massive terminal networks with hundreds of
facilities to move large volumes of parcels each day. Each day
at a terminal is divided into time windows (typically three to four
hours in length), called sort periods or sorts,
during which parcels are sorted. A typical operational day includes
“day”, “twilight”, “night” and “sunrise” sorts that are
non-overlapping in time. All parcels sorted at a terminal
during a given sort with the same service class (e.g., one-day service
or two-day service) and the same destination are referred to as a
commodity. Suppose then that each commodity has a primary flow path and one or more alternate flow paths that each specify a sequence terminals and sorts that parcels will traverse en route from origin to destination. For a specific commodity at a specific terminal at a specific sort, each flow path will determine the next terminal and sort to which packages will be loaded. Typically, shipments are loaded on trailers moving along the primary flow path for the commodity; however, when there are better consolidation opportunities, commodity volume can be split over primary and alternate flow paths, or completely allocated to alternate flow paths. The rest of this section describes the main concepts underlying the DLPP. Section <ref> describes some key terminology and presents examples to illustrate the operations at terminals. Section <ref> describes the DLPP that includes splitting of commodity volume across
primary and alternate flow paths.
§.§ Definitions
Let 𝒢 = (𝒩,𝒮) denote a time-space network. Each node n ∈𝒩 represents a terminal location at a particular time period and is defined by a tuple, i.e. n=(terminal, sort, day). Each arc s ∈𝒮 represents a directed dispatch of loads from one timed node to another. Henceforth in the paper, we refer to each such an arc as a sort pair. Figure <ref> illustrates an example time-space network for terminal A during a single twilight sort period. In this example, three sort pairs are outbound from terminal A on day 1, namely, (A,Twilight,1)→(X,Twilight,2),
(A,Twilight,1)→(Y,Twilight,2), and
(A,Twilight,1)→(Z,Twilight,3). Figure <ref> illustrates another example of terminal B that operates multiple sort periods, i.e., the day, twilight, night sorts on a given day, and seven sort pairs (b_1,b_2,b_3,b_4,b_5,b_6,b_7) outbound from terminal B.
A key objective in load planning is to determine the number of
trailers (possibly of different types) to operate on each sort pair to containerize the total commodity volume allocated to the sort pair. During a sort, each loading door at a terminal builds/loads trailers for a specific sort pair destination. In a single sort facility, as shown in Figure <ref>, if there is commodity volume allocated on each of the three sort pairs, then at least three trailers (one on each sort pair) should be opened at the loading doors corresponding to the sort pair destinations.
In practice, commodities outbound from an origin terminal that arrive over consecutive sorts and that are heading to the same time-space destination can be consolidated together. For that, the concept of load pairs is introduced, where a load pair represents a set of consecutive sort pairs that share the same destination node. Combining sort pairs into load pairs allows better consolidation and trailer utilization, since trailers can be held partially loaded from one sort to the next prior to dispatch to the destination. Figure <ref> illustrates an example of a load pair that is composed of three different sort pairs.
We now relate primary and alternate flow paths to sort pairs. If we consider volume for commodity k ∈𝒦 at some time-space location n, its primary flow path specifies the next (terminal, sort, day) to which it should be loaded. Thus, the primary path identifies a unique outbound sort pair for k at n. Similarly, each alternate flow path identifies a (possibly different) outbound sort pair for k. Recall that primary and alternate flow paths for each k at n are specified in advance, and we assume that loading outbound on any of these options will lead to volume arriving on-time to its destination.
We will define compatible sort pairs for k at n to be the primary path sort pair (the primary sort pair) and any alternate path sort pair (an alternate sort pair). Furthermore, any sort pairs that are in load pairs with compatible sort pairs with an earlier origin sort are also compatible. When volume is assigned to such earlier sort pairs, the decision is to assign volume to trailers that are opened first for loading in those earlier sorts and held for dispatch. Figure <ref> illustrates four compatible sort pairs (outbound from terminal B) for a commodity k sorted in the twilight sort at terminal B.
§.§ Dynamic Load Planning Problem (DLPP)
Parcel carriers typically build a load plan in two phases: (1) the tactical flow and load planning phase specifies an initial plan and provides an input to the scheduling team; and (2) the load plan adjustment allows adjustments to the initial plans up to the day-of-operation. The scheduling and load dispatching teams then execute the
adjusted load plan. Weekly plans that determine the number of loads or
trailers to operate on each sort pair are fixed approximately two
weeks in advance of the operating week. However, due to demand
uncertainty, the volume forecast for commodities may change, and
adjustments to the load plan may be necessary to accommodate actual
volumes. These adjustments may lead to cost decreases when unnecessary
load capacity is removed from the plan.
Consider the following optimization problem during the two weeks
leading into the day-of-operation. Each terminal in the network has
a set of forecasted inbound commodities during some time period (for
example, a single operating day and multiple sorting periods). Each
such commodity arrives during a specific sorting period and has a
destination terminal and service class (specifying a due date at the
destination). Given this information, the fixed flow plan specifies a
primary flow path (next terminal and arriving sorting period) for each
commodity, and possibly also one or more alternate flow paths. Recall
that, if the commodity is assigned to any of these flow paths, then it
will reach its final destination on time according to plan. The
adjustment optimization problem is to assign each commodity to its
primary and/or one of its alternate flow paths while simultaneously
determining how many loads of different types are required for each
proposed flow paths. Note that existing flow and load planning
literature typically assumes that all commodities, arriving
at a terminal during a specific sorting period should be assigned to
the primary flow path. Here, the challenge is different, and is
instead to determine specifically how to split each commodity volume
among its possible compatible flow paths or sort pairs to drive high
load utilization levels and low costs while still meeting service
promises.
Consider the example shown in Figure <ref> with three
commodities (4 units destined to terminal C, 3 units destined to
terminal E, 3 units destined to F) sorted in the twilight sort of day 1 at
terminal B. In this example, we denote each commodity by its
destination terminal name. The commodity destined for terminal F has
three compatible sort pairs: (B,Twilight,1)→ (C,Sunrise,2)
is the primary sort pair, and (B,Twilight,1)→ (E,Sunrise,2)
and (B,Twilight,1)→ (D,Twilight,2) are the alternate sort
pairs. Splitting commodity volume destined to terminal F between the two
alternate sort pairs to C and D yields better consolidation (and
lower transportation cost) as the solution requires one less trailer
on the two arcs: (B,Twilight,1)→ (D,Twilight,2) and
(D,Twilight,2)→ (F,Day,3).
For a
given terminal, define S to be the set of outbound sort pairs and
let K be the set of commodities sorted at the terminal. Each
commodity k ∈ K has a cubic volume of q^k, and a set of
compatible sort pairs S^k. For every outbound sort pair s, there
is a set V_s of trailer types, that can be used to containerize the
total commodity volume allocated to the sort pair. Each sort pair can
have different set of allowed trailer types, i.e., V_s_1 can be
different from V_s_2 for two different sort pairs s_1,s_2 ∈
S. Each trailer type v ∈ V_s has a cubic capacity Q_v and has a
per-unit transportation cost c_v. A solution of the DLPP
determines the number of trailers of each type assigned to each sort
pair, as well as the volume of each commodity allocated to each
trailer. A solution must ensure that all the volume is assigned to
trailers and that the capacities of the trailers are not violated. The
goal of the DLPP is to find a solution that minimizes the costs of the
trailers. Appendix <ref> provides the complexity
results. The DLPP is strongly NP-hard. It becomes weakly NP-hard when
each commodity is compatible with exactly one or with all sort
pairs and there are multiple trailer types. It becomes polynomial when each commodity is compatible with exactly one or with all sort
pairs and there is only one type of trailer.
§.§ A Mixed-Integer Programming Formulation
An optimization model for the DLPP can be defined as follows in Model <ref>:
x,yMinimize ∑_s ∈ S∑_v ∈ V_s c_v y_s,v
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_k ∈ K:s ∈ S^k x^k_s,v≤ Q_v y_s,v, ∀ s ∈ S, v ∈ V_s,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
y_s,v∈ℤ_≥ 0 ∀ s ∈ S, v ∈ V.
It uses a non-negative continuous decision variable x^k_s,v to
represent the volume of commodity k allocated to trailer type v
operating on a sort pair s, and an integer decision variable
y_s,v to determine the number of trailers of type v installed on
sort pair s. The objective (<ref>) minimizes the total cost of creating
loads. In the experiments, c_v = Q_v ∀ v ∈ V, i.e., the
model minimizes the total trailer capacity required to containerize
the total commodity volume in the problem instances. Constraints
(<ref>) ensure that the total volume of each commodity
is assigned to its compatible sort pairs. Constraints
(<ref>) ensure that the total volume on a sort pair
respects the installed trailer capacity on it. Constraints
(<ref>)-(<ref>) define the domain and range of
variables.
§ GOAL-DIRECTED OPTIMIZATION
The optimization model of the DLPP has a large number of
symmetries. Figure <ref> depicts a simple instance with multiple optimal solutions that are operationally different from one another, yet they are equivalent from Model <ref> perspective as they require the same number of trailers of the same type. This is because in Model in <ref>, commodities are indifferent to the
sort pairs they are assigned to, as the volume allocation decisions
(x-variables) do not incur any cost.
Such symmetries are undesirable for many reasons. Paramount among them
are the realities in the field: the model is intended to be used and
validated by planners. If small variations of inputs produce
fundamentally different solutions, planners are unlikely to trust the
model. Indeed, since the model is used multiple times a day, it is
important to ensure that the successive optimal solutions are as
consistent as possible with each other. Fortunately, in practice, a
reference plan is always available and the DLPP should ideally
produce optimal solutions that are as close as possible to the
reference plan.
This section explores how to refine the model presented earlier to
satisfy this requirement, and presents a Goal-Directed Optimization
(GDO) approach to the DLPP. It uses a reference plan
to eliminate symmetries and ensure that the solution is compatible
with the planner-in-the-loop reality in the field. The use of a
reference plan eliminates many symmetries but not all. To break more
symmetries, the GDO approach also adds a flow diversion cost
that captures the cost of using alternate paths instead of the primary
path. For instance, in the example depicted in Figure <ref>,
only the solution shown in Figure <ref> is optimal following our assumptions. The flow
diversion cost is chosen to be proportional to the distance between
the next alternate terminal and the destination of the commodity, as
there is incentive to move commodities as close as possible to their
destination. For example, suppose a commodity k is in
Atlanta and is destined for Chicago. Let the primary
next terminal be Louisville (with flow diversion cost 0),
alternate 1 be Nashville, and alternate 2 be
Memphis. As Nashville is closer to Chicago than
Memphis, the flow diversion cost of allocating volume to
alternate 1 is lower than that of alternate 2. As a result, the
GDO approach has at its disposal a reference plan γ, where
γ_s,v denotes the number of trailers of type v planned to
operate on sort pair s. It also leverages the flow diversion
cost d^k_s that denotes the cost of allocating a per-unit volume
of commodity k ∈ K to a compatible sort pair s ∈ S^k.
The GDO approach first solves Model <ref>
to obtain the optimal objective value Z^*. It then solves a second
MIP Model to bias the trailer decisions so that they are as close as
possible to the reference plan and minimize diversion costs.
The second-stage model is defined as follows:
x,yMinimize ∑_s ∈ S∑_v ∈ V_s| y_s,v - γ_s,v| + ϵ∑_k ∈ K∑_s ∈ S_k∑_v ∈ V_sd^k_s x^k_s,v
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_k ∈ K:s ∈ S^k x^k_s,v≤ Q_v (y_s,v), ∀ s ∈ S, v ∈ V_s,
∑_s ∈ S∑_v ∈ V_s c_v y_s,v≤ Z^*,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
y_s,v∈ℤ_≥ 0 ∀ s ∈ S, v ∈ V.
The objective function (<ref>) minimizes the weighted sum of the Hamming
distance of the trailer decisions from the reference plan γ and
the flow diversion costs. The weight ϵ for the flow diversion cost is sufficiently small such that the cost does not dominate over the Hamming distance term in the objective function. The purpose of the flow diversion cost in (<ref>) is to break the symmetry between solutions with the same Hamming distance; it biases the solution to have more volume allocated to primary sort pairs than alternate sort pairs. Constraints (<ref>),
(<ref>), (<ref>) and (<ref>) are
the same as in Model <ref>. Constraint
(<ref>) ensures that the optimal solution does not use more
trailer capacity than Z^*. Note that the objective function is
non-linear due to the Hamming distance term. It can be linearized by
replacing | y_s,v - γ_s,v| with new variables
w_s,v≥ 0 (s ∈ S, v ∈ V_s) and imposing the following
constraints
y_s,v - γ_s,v≤ w_s,v ∀ s ∈ S, v ∈ V_s,
γ_s,v - y_s,v≤ w_s,v ∀ s ∈ S, v ∈ V_s,
Figure <ref> illustrates the sensitivity of the
trailer decisions (y-variables) subject to increases in the total
commodity volume (∑_k ∈ Kq^k) (x-axis) for the two models:
Model <ref> (red plot) and the GDO approach (blue
plot). As the total commodity volume increases, Model
<ref> exhibits solutions where the trailer decisions
fluctuate dramatically between 1 and 6 trailers for sort pair 1,
and between 1 and 5 trailers for sort pair 2. However, when
using GDO, the trailer decisions in GDO are more consistent and vary
between 1 and 2 trailers on sort pair 1, and is constant at 2 trailers on sort pair
2.
§ LEARNING-BASED OPTIMIZATION PROXIES
The GDO approach produces consistent solutions to the DLPP, but it is
too slow to be used with planners in the loop. This section proposes a
Machine Learning (ML) approach to the DLPP. Its goal is to move some
of the optimization burden offline and produce high-quality solutions
in real time. More precisely, the approach uses the concept of optimization proxies to produce high-quality solutions to an
optimization problem by learning its input/output mapping (see, for
instance,
(<cit.>)
for an overview of this concept and its applications).
The overall methodology underlying optimization proxies is depicted in Figure
<ref>. It consists of two stages,
* an offline stage where an ML model learns the input/output
mapping of the optimization problem;
* an online stage which is used in real time: it receives
an instance, applies the ML model to predict a (possibly infeasible) solution
and uses a repair procedure to deliver a feasible solution.
For the DLPP, the ML model learns the mapping between the (input)
commodity volumes and the (output) trailer decisions; in other words, given the
commodity volumes, the ML model predicts trailer decisions for every
sort pair. The trained ML model may sometimes underestimate
the number of trailers on some sort pairs when executed in real time. To circumvent this issue, the feasibility
restoration step projects the predicted trailer decisions back into the
feasible region; in addition, the feasibility restoration also
computes the volume allocation on the sort pairs. A key element in the
ML training is data augmentation that complements historical
data by generating realistic instances through input perturbations. The ML model formulation is introduced and discussed in more details in what follows.
§.§ The ML Model Formulation
This section defines a machine learning model f, parameterized by
θ, that maps the input parameters, i.e., the commodity volume,
to the optimal trailer decisions:
(<ref>)-(<ref>).
f_θ: ℝ_≥ 0^|K|⟶ℤ_≥ 0^|S| × |V|
𝐩⟼𝐲
The ML inputs are assumed to be taken from a distribution 𝒫
that captures the actual instances.
Given a dataset of input parameters {𝐩_i}_i ∈ N∼𝒫, where N is the set of instances, parametrization θ^* can be obtained by minimizing
the empirical risk shown in (<ref>), where
(<ref>) denotes the optimization problem
solved by Model <ref>, and l denotes the loss function
that measures the L1-distance of the predicted
(f_θ(𝐩)) and optimal (y^*) trailer decisions.
θMinimize 1/N∑_i∈ N l(f_θ(𝐩_i), 𝐲_i^*)
subject to (𝐱_i^*, 𝐲_i^*) = 𝐱, 𝐲∈𝒞(𝐩_i) c(𝐱, 𝐲) ,
It is important to highlight that an ML model could be used to predict
commodity volume allocation on the sort pairs (x-variables) instead
of the trailer decisions (y-variables). This may seem to be a good
approach since, after predicting volume allocation, one can easily
recover the trailer decisions and hence a feasible solution, by
setting y_s,v=⌈∑_k ∈ K:s ∈ S^k
x^k_s,v/Q_v⌉ ∀ s ∈ S, v ∈ V_s.
However, this approach has some shortcomings. First, the output
dimension is significantly larger than the input dimension which makes
it very difficult to develop an effective ML model even for the smallest
instances. Second, recovering trailer decisions is very sensitive to
the predicted volume allocation decisions. Consider an example where
100 cubic volume is allocated to a sort pair which requires two
trailers, each with capacity 50 cubic volume, in the optimal solution.
If the ML model predicts the volume on the sort pair to be 100.5,
then the total number of trailers required is ⌈100.5/50⌉ = 3 which generates a poor solution in terms of the objective function value of Model <ref>.
Experimental results confirmed that it is beneficial to learn the
mapping from input parameters to the trailer decisions rather than the
volume decisions. The trailer decisions 𝐲∈ℤ_≥ 0^|S| × |V| are more aggregated than the
volume allocation decisions ℝ_≥ 0^|K| × |S|
× |V|. The benefits comes from the significant reductions in
output dimensionality and variability. In addition, as presented in
section <ref>, once the trailer
decisions are known, restoring the feasibility of the solution is
relatively easy as the feasibility restoration MIP has a small number
of binary decision variables and therefore, it is easy to solve.
The ML model used in this paper is a deep neural network as illustrated in Figure
<ref>. It uses a Multi-Layer Perceptron (MLP),
where each dense layer is followed with a batch normalization (<cit.>),
a dropout (<cit.>), and a ReLU (Rectified Linear Unit) function.
It maps the input parameter 𝐩
to the flattened trailer decision 𝐲. The
last ReLU guarantees that the output of the neural network is
non-negative. The compatible trailer decisions 𝐲
are then generated by reshaping the flattened decision
𝐲 and masking it with the compatible
trailer mask 𝐦, where m_s, v = 1 indicates that
equipment type v ∈ V is compatible with sort pair s ∈ S. In the training
phase, the loss function is computed by measuring the distance of
predicted compatible trailer decision 𝐲 with the
optimal trailer decisions. Specifically, this work used smooth l_1 loss.
The loss is used to update the parameters of the MLP using stochastic gradient descent (<cit.>)
with back propagation (<cit.>). At inference time (i.e., in real time), the
compatible trailer decisions are rounded to an integer value.
§.§ MIP-based Feasibility Restoration
The proposed ML model predicts the number of trailers
y_s,v for each sort pair s ∈ S and equipment type v
∈ V_s. Let the total trailer capacity installed on each sort pair
s ∈ S be Λ_s = ∑_v ∈ V_sQ_v
(y_s,v). The system of equations
∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s, ∀ s ∈ S,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
is then used to determine the volume of every commodity k ∈ K
allocated to its compatible sort pairs. However, it is possible that
some of the sort pairs do not have sufficient trailer capacity because
the ML model may underestimate the capacity. In that case,
(<ref>) is infeasible. The following linear program
zMinimize ∑_s ∈ S z_s
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v - z_s ≤Λ_s, ∀ s ∈ S,
x^k_s,v,z_s ≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
can be used determine the sort pairs with trailer capacity violations.
Its objective function (<ref>) minimizes the capacity
violations on the sort pairs. Constraints (<ref>)
ensure that total volume of every commodity is assigned to compatible
sort pairs. Constraints (<ref>) determine the sort
pair capacity violations. Constraints (<ref>) define the
domain and range of variables. When Model <ref>
has an optimal objective value equal to 0, it has recovered a
feasible solution to Model <ref>. Otherwise,
additional trailer capacity is required on sort pairs with capacity
violations.
This paper proposes a two-stage MIP-based feasibility restoration
process. In the first stage, Model <ref> is
solved to obtain an optimal solution z^*. Let the set of sort pairs
with trailer capacity violation be S = {s ∈ S: z^*_s >
0}. The feasibility restoration then identifies the cheapest
equipment v to serve the excess volume on sort pair s ∈S. The extra trailer capacity is given by ξ_s =
( ⌈z_s/Q_v⌉ * Q_v ) and the option to add the extra capacity to sort pair s ∈S is
added using a binary decision variable. The second stage
solves the following MIP model:
uMinimize ∑_s ∈S u_s ξ_s
subject to ∑_s ∈ S^k∑_v ∈ V_s x^k_s,v = q^k, ∀ k ∈ K,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s + u_s ξ_s, ∀ s ∈S,
∑_v ∈ V_s∑_k ∈ K:s ∈ S^k x^k_s,v≤Λ_s, ∀ s ∈ S\S,
x^k_s,v≥ 0 ∀ k ∈ K, s ∈ S^k, v ∈ V_s,
u_s ∈{0,1} ∀ s ∈S,
The objective function in (<ref>) minimizes the total
trailer capacity added on each sort pair. Constraints
(<ref>) ensure that the commodity volume is
assigned to the compatible sort pairs. Constraints
(<ref>) and (<ref>) ensure
that commodity volume allocated to each sort pair respects the trailer
capacity. Constraints (<ref>) and
(<ref>) define domain and range of variables. The
number of binary variables in this model is at most the number of sort
pairs in the instance, i.e. | S |. The main difference
between Model <ref> and Model
<ref> is that Model <ref> uses
binary variables u_s instead of continuous variable z_s, to denote
the option of adding extra trailer capacity ξ_s on sort pairs s
∈S. When u_s = 1, extra trailer capacity is added to
sort pair s.
After solving Model (<ref>), adding ξ_s
capacity on every sort pair s ∈S yields a feasible
solution to Model (<ref>). However, the goal is to
use Model (<ref>) to obtain a better feasible
solution. Consider an example with a set of commodities all of which
can be allocated to any of the two sort pairs s_1 and s_2 and
trailer with capacity 2 units. Suppose the optimal solution of Model
<ref> is z_s_1=z_s_2=1. In this case, a
feasible solution to Model <ref> can be recovered by
adding two trailers, one on each sort pair. However, Model
(<ref>) (which has two binary variables) yields a
solution with only one trailer on any one of the two sort pairs. Algorithm 1 provides a summary of the feasibility restoration procedure.
§.§ Value of Symmetry-Breaking Data Generation for Learning
The optimization proxies are trained using the solutions provided by
the GDO which uses the same reference plan for all instances of a given terminal. As
a result, the proxies are consistent by design and do not rely on a
reference plan. The GDO approach is not only critical for
environments with planners in the loop, but it also has an additional
benefit: it makes the learning problem easier. This section provides
theoretical insights about why the data generation using GDO results
in better function approximation than data generation from Model
(<ref>) alone.
Observe that the solution trajectory associated with different
instances can often be effectively approximated by piecewise linear
functions, as depicted in Figure <ref>. This
approximation becomes exact in the case of linear programs and mixed
integer programs when the input reflects incremental changes in the
objective coefficients or right-hand sides of the constraints. This
paper utilizes ReLU-based neural networks to approximate the solutions
of optimization problems. These neural networks are capable of
capturing piecewise linear functions, which makes them well-suited for
this purpose. However, the ability of representing a target piecewise
linear function accurately depends on the model capacity. As the
complexity of the function grows with more pieces, a larger model is
required to obtain a high-quality approximations.
(Model Capacity) (<cit.>)
Let f: ℝ^d →ℝ be a piecewise linear function with p pieces.
If f is represented by a ReLU network with depth k+1, then it must have size at least 1/2kp^1/k-1. Conversely, any piecewise linear function f that is represented by a ReLU network of depth k+1 and size at most s, can have at most (2s/k)^k pieces.
Due to the symmetry in optimal solutions of Model
(<ref>), as shown in
Figure <ref>, the solution trajectory varies
dramatically. Theorem <ref> states that the approximation
of a more volatile solution trajectory (i.e., a piecewise linear
function with more pieces) requires a deep neural network with greater
capacity, which makes the learning task more challenging. In other
words, given a fixed-size ReLU network, higher variability of the
solution trajectory typically results in higher approximation
errors. These errors are bounded by the following theorem.
(Approximation Error) (<cit.>)
Suppose a piecewise linear function f_p', with p' pieces each of width h_k for k ∈ [p'], is used to approximate a piecewise linear f_p with p pieces, where p' ≤ p. Then the approximation error
f_p - f_p'_q ≤1/2h^2_max∑_1≤ k ≤ p|L_k+1 - L_k|,
holds where L_k is the slope of f_p on piece k and h_max is the maximum width of all pieces.
Theorem <ref> relates the approximation error of
a piecewise linear function with the total variation of its slopes.
It implies that the data generated using GDO (which exhibits lower
sensitivity than the data from Model (<ref>))
should facilitate learning and result in lower approximation errors.
§ GREEDY HEURISTIC (GH)
This section proposes a greedy heuristic to construct feasible
solutions for Model (<ref>) and benchmark the quality
of the solution obtained from optimization proxies. This heuristic
iteratively solves linear programs (LP) until all the y-variables
are integers, i.e., they satisfy the integrality tolerance
(10^-5). In each iteration, the algorithm identifies fractional
variables with minimum (⌈y_s,v⌉-y_s,v) value, updates the lower bound of variable y_s,v
to ⌈y_s,v⌉, and re-solves the LP as
shown in Algorithm <ref>. The main idea is that
for a given sort pair s ∈ S and trailer type v ∈ V_s, if
y_s,v has a fractional value very close to an integer ⌈y_s,v⌉, then, this indicates that there is
enough commodity volume to have at least ⌈y_s,v⌉ trailers on the sort pair. GH
greedily adjusts the lower bound of a y-variable in each
iteration till all y-variables can be labelled as integers, in which
case a feasible solution to Model (<ref>) has been
found.
§ COMPUTATIONAL STUDY
This section reports a series of experiments conducted on real-life
instances provided by our industry partner. Section <ref>
presents statistics for the problem instances. Section
<ref> discusses the experimental setup for the
optimization models and proxies. Section
<ref> evaluates the computational
performance of the optimization proxies against the greedy heuristic
(GH) and the optimization models (Model (<ref>) and GDO). Section <ref> evaluates the benefits
of GDO for learning.
§.§ Instances
The experiments are based on industrial instances for three different
terminals in the service network of our industry partner: medium (M),
large (L), and extra-large(XL). Each category
has a reference plan for a
terminal on a particular day as provided by our industry partner. Table <ref> reports the
statistics of the instances: #Arcs denoting the total
number of unique outgoing sort pairs or arcs from the terminal,
#Commodities denoting the number of commodities that are sorted at the terminal and loaded into
outbound trailers (rounded to nearest
multiple of 1,000), and #Loads denoting the number of planned loads in the reference load plan for the
corresponding terminals (rounded to the nearest multiple of 50). Note that, in addition to the planned loads,
small package companies typically operate empty trailers on
the outbound sort pairs for trailer repositioning. This study only
considers trailers that are filled with commodity volume and do not
include empty trailer capacity.
It is worth highlighting that the XL
instance operates more volume and capacity than the M and L instances
combined. Table <ref> reports some statistics for Model
(<ref>) for the three instances: #Integer-Vars and
#Continuous-Vars denoting the number of integer and continuous
decision variables, respectively, and #Constraints denoting the total number
of constraints.
§.§ Experimental Setup
Parameters for GDO
The cost of assigning commodity k to a sort pair s ∈ S^k
(denoted by d^k_s) is defined as
d^k_s =
0, if s is primary flow path for commodity k
(α^k_s + 10*β^k ) otherwise
β_k =
1, if commodity k belongs to one-day service class
2, if commodity k belongs to two-day service class
3, if commodity k belongs to three-day service class
4, otherwise
ϵ = 1/(max_k ∈ K, s ∈ S^k(α^k_s + 10*β^k) )∑_k ∈ K q^k
where α^k_s denotes the distance between the alternate next
terminal and the destination of commodity k ∈ K for sort s, and parameter β_k depends on the commodity service level.
Recall that a commodity k ∈ K is defined as all packages with the
same destination and service class. The term α^k_s ensures that
two commodities with different destinations have different flow
diversion cost. However, two commodities with different service class can
have the same destination. β^k ensures that such commodities have different flow diversion cost for the same
destination. The weight for the flow diversion cost is defined in (<ref>).
Data Generation for ML Model
The dataset is generated by perturbing the input parameters of
real-life instances provided by the industry partner with up to
20,000 commodities. Denote by 𝐩^ref the volume of
different commodities in a given reference plan. The DLPP instances are
generated by perturbing this reference commodity volume. Namely, for
instance i, 𝐩^(i) = γ^(i)×η^(i)×𝐩^ref, where γ^(i)∈ℝ denotes a
global scaling factor and η∈ℝ^|K| is the commodity
level multiplicative white noise. γ is sampled from a uniform
distribution U[80%, 120%], and for every commodity η is
sampled from a normal distribution with mean equals to 1 and standard deviation
of 0.05. For every category, 10,000 instances are generated, and a commercial solver is used to solve the GDO model for each instance. The dataset of
10,000 instances for each category is then split as follows: 80%
for the training set, 10% for the validation set, and 10% for
the test set.
Performance Metrics
The performance metrics in this study are designed to compare the
total trailer cost and the consistency of the
solutions generated by the optimization proxies against the total
trailer cost from Model (<ref>) and then the consistency
of solution from Model (<ref>) of the GDO approach. Given
an instance 𝐩 with optimal trailer decision 𝐲^*
and a feasible trailer decision 𝐲̂, the optimality gap
is defined as
Gap = (Ẑ - Z^*)/|Z^*|,
where Z^* is the optimal trailer cost of Model
(<ref>), and Ẑ is the trailer cost computed
from 𝐲̂. Recall that the total trailer cost does not
increase in Model (<ref>) of the GDO approach due to constraint
(<ref>). If Model (<ref>) cannot be solved
to optimality in 30 minutes, then the best lower bound obtained from
the solver run is used to compute the optimality gap instead of Z^*.
This paper proposes two metrics to quantify the consistency. The
first one is a normalized distance (Δ) between the optimized load plan and the load plan
𝐲, using shifted
geometric means as given by
Δ_s,v =
|y_s,v - y_s,v| if y_s,v = 0
|y_s,v - y_s,v|/y_s,v, otherwise ∀ s ∈ S, v ∈ V_s
Δ = exp(1/|S||V|∑_s∈ S∑_v ∈ V_slog (Δ_s,v + 0.01) ) - 1%.
From a planner perspective, this metric captures the deviation of the
optimized load plan with respect to the reference load plan. As mentioned
in Section <ref>, load plans that are as
close as possible to the reference plan are highly desirable.
The second metric is the total variation of the set of trailer decisions
across a set of N instances (for each terminal). For simplicity,
instances are ordered such that ∑_k ∈ K q^k_i+1≥∑_k
∈ K q^k_i ∀ i ∈{1,2,⋯,N-1}. The goal is to
analyze the variation in trailer decisions on sort pairs when the total
commodity volume is incrementally increased from ∑_k ∈ K
q^k_1 to ∑_k ∈ K q^k_N. Let {𝐲_i}^N_i=1
denote the set of trailer decisions of N instances. The total
variation is defined as:
TV({𝐲_i}^N_i=1) = ∑_i=1^N-1𝐲_i+1 - 𝐲_i_p,
where p=2.
This metric captures the sensitivity of the models, i.e., the impact of changes in total commodity volume on the trailer decisions of different sort pairs. Lower total variation implies that the trailer decisions are less sensitive to changes in total commodity volume. Planners are more amenable to such solutions because fewer (but effective) load plan modifications reduce the solution evaluation effort and is also easier to execute in practice.
The computational efficiency of different models is measured by the
training time of optimization proxies including the data-generation
time and the inference time. Unless specified otherwise, the average
metrics on the test dataset are reported in shifted geometric means:
μ_s(x_1, …, x_n) = exp(1/n∑_i log (x_i + s) ) - s,
where the shift is set as 0.01 for the optimality gap and normalized
distance, 1 second for the inference/solving time, and 1 cube for
the distance between the optimized load plan to the reference load plan.
Implementation Details
All optimization problems are formulated using the Gurobi Python interface,
and solved with Gurobi 9.5 (<cit.>) with 8 CPU threads and
default parameter settings, except for MIPFocus which is set
to a value of 3. All deep learning models are implemented using
PyTorch (<cit.>) and trained using the Adam
optimizer (<cit.>). The ML models are multiple layer
perceptron and are hyperparameter-tuned using a grid search with
learning rate in {10^-1, 10^-2}, number of layers in {3, 4,
5}, and hidden dimension in {128, 256}. For each system, the best
model is selected on the validation set and the performances on the
test set are reported. Experiments are conducted on dual Intel Xeon
[email protected] machines running Linux, on the PACE Phoenix cluster
(<cit.>). The training of ML models is performed on Tesla
V100-PCIE GPUs with 16GBs HBM2 RAM.
§.§ Computational Performance of the Optimization Proxies
This section presents numerical experiments used to assess the
performance of the proposed optimization proxies (Proxies) against the
optimization models (GDO) and the greedy heuristic (GH).
Optimality Gap
Table <ref> presents the optimality gaps of various
approaches, including the results of Model (<ref>)
under various time constraints. In the table, the columns under “Gap
of Model (<ref>)” denote the optimality gaps of the
model under various time limits. Similarly, columns Gap for GH
and Proxies denote optimality gaps for GH and the
optimization proxies. In addition, columns Time(s) denote the solving
times for GH and Proxies.
Recall that Model (<ref>) produces solutions that
exhibit considerable variability when the total commodity volume is
perturbed as detailed in Table <ref> and <ref>. As such, it is unlikely to be practical in
scenarios with planners in the loop. Hence, the table compares the
optimization proxies and the heuristics GH with an “idealized”
benchmark. With this caveat in place, observe the performance of the optimization
proxies under tight time constraints. Proxies generate solutions with
low optimality gaps and may be up to 10 to 50 times faster than GH,
and around 10 times faster than Model (<ref>) solved
with Gurobi. Second, although Model (<ref>)
efficiently produces solutions with low optimality gaps, closing the
optimizality gap proves to be a significant challenge due to the poor
LP relaxation. The performance of GH is also impeded by the
inefficiencies of the LP relaxation, as it solves the LP relaxations
over many iterations; it takes the GH around 30 iterations for
terminal M, 200 iterations for terminal L, and more than 1000
iterations for terminal XL to generate a feasible solution.
Consistency
Tables <ref> and <ref> report
the consistency of solutions obtained from different models in terms
of the normalized distance to the reference load plan and the total
variation of the generated solutions. As GDO requires running Model
(<ref>) and Model (<ref>) sequentially,
these experiments set the same time limits for the two stages. For
example, if a time limit of 30 seconds is set, GDO runs Model
(<ref>) for 30 seconds and subsequently runs Model
(<ref>) using the best upper bound obtained from Model
(<ref>) for another 30 seconds.
The high-level result is that proxies are ideally suited to
produce consistent plans. Table <ref> shows
that the proxies accurately predict, in a few seconds, the results
produced by GDO after an hour. Furthermore, Table <ref> shows that proxies produce solutions that have at
least an order of magnitude smaller total variations in trailer
decisions than both GDO and GH. Proxies produce load plans that
exhibit great stability with changing total commodity volume.
The fact that proxies improve the consistency of the GDO plans is
especially interesting: it means that the optimization proxies, by
virtue of the learning step, avoid oscillations present in the GDO
approach. Of course, it does so at a small loss in objective value
(if, for instance, the GDO model is allowed a minute to run instead of
the 2.5 seconds of the optimizations). But the consistency benefits
are substantial as shown in Table <ref>. The
proxies also provide dramatic improvements over the GH heuristic. Note
also that GDO itself brings significant improvements over Model
(<ref>).
In Table <ref>, observe that the normalized
distance for solution from GDO for the large (L) instance first increases from
0.45% to 11.40%, and then follows the expected
decreasing trend with increase in computational time limit. Recall
that GDO first minimizes the total trailer capacity required in Model
(<ref>) and then solves Model (<ref>) to
minimize the Hamming distance of the solution (and the flow diversion
cost) from the reference load plan. As shows in Table <ref>
the feasible solution obtained from Model <ref> is of
poor quality and is closer to the reference load plan in terms of the
number of trailers. Hence, the normalized distance value is small. As
the computational time limit increases, the feasible solution obtained
from Model (<ref>) exhibits a reduced total trailer
capacity compared to the reference load plan. Hence, the normalized
distance increases as the model tries to find more cost effective solutions. With further increases in computational time, the
normalized distances decrease as the solver finds a better solution
with a smaller Hamming distance using Model <ref>.
It is also interesting to observe that the total trailer capacity
predicted by the ML model, i.e., the capacity provided by all the
trailers predicted to be needed by the ML model, is very close to a
feasible solution. Only a few trailers must be added to recover a
feasible solution. Figure <ref> shows the
distribution of the predicted trailer capacity as a percentage of the
total trailer capacity in the feasible solution generated by the
proxies for each type of terminal. The results show that more than
98% of the trailer capacity is predicted correctly and less than
2% comes from feasibility restoration step generated by algorithm <ref>. More
accurate predictions might even result in a feasibility restoration
model that has fewer decision variables, hence, requiring less
computational time to produce a feasible solution. Appendix <ref> shows that one of the key benefits of the optimization proxies is that it replaces a model with large number of integer decision variables with a prediction model, and requires to solve a relatively simpler feasibility restoration model with small number of binary variables.
§.§ Value of Symmetry-breaking Data Generation
As discussed in section <ref>, the optimal (or near-optimal)
trailer decisions of Model (<ref>) are very sensitive
to changes in total commodity volume due to the presence of symmetries
in the model and the randomized nature of MIP solvers. The solutions
to Model (<ref>) are reported in red in the
plots of Figure <ref>, which illustrates this
behavior. This is not desirable in environments with
planners-in-the-loop, where similar solutions are expected for similar
instances. The GDO approach is much more consistent and its solutions
are shown in orange in the plots of Figure <ref>. The ML component of the optimization proxy uses GDO as
the expert to imitate and learn solution patterns from. As
shown in blue in the plots of Figure <ref>,
the ML model is effective in producing solutions that are close to the
solutions generated by GDO. It should be highlighted that the GDO
approach has two benefits. First, it generates consistent solutions that
are amenable to planner-in-the-loop environments. Second, it makes the
learning problem much more tractable. Designing an ML model for
(<ref>) is really challenging due to the high
sensitivities in small changes: typically an ML model for learning
Model (<ref>) would return an average value.
§ BENEFITS OF DYNAMIC LOAD PLANNING, OPTIMIZATION, AND LEARNING
This section discusses the benefits of the dynamic load planning approach,
optimization, and learning. The load planning methodology studied in
this paper is based on the concepts of primary and alternate flow
paths. With the availability of optimization models, it is possible to
evaluate the benefits of this approach for load consolidation, at
least from a local perspective.
The results in the paper also make it possible to evaluate the
benefits of optimization compared to human planners. During
operations, planners typically assign commodities to the primary flow
paths. If there is no capacity available on the primary flow path,
then planners allocate the remaining volume on the first alternate
flow path and, if there is no capacity on the first alternate, they
turn to the second alternate flow path, and so on. Observe that this
is a greedy strategy of loading commodity volume on trailers, and
hence, it is myopic in nature. A comparison between such a greedy approach
and the optimization models help assess the value of optimization. Of
course, the optimization models are too slow to be used with planners
in the loop. The optimization proxies proposed in the paper are the
technology enabler for translating the benefits of optimization into
practice.
The first results in this section aim at quantifying the value of a
network with alternate flow paths relative to a network with primary
flow options only. Figure <ref> presents some characteristic
of the networks studied in this paper: it shows the distribution of
the number of commodities with a specific number of alternate flow
paths for each instance. It highlights that the network has some
significant flexibility for many of its commodities.
Figure <ref> presents the benefits of the load
planning methodology. It compares the variation in trailer cubic
capacity required to containerize the total commodity volume
(blue curve) and the total volume allocated to alternate flow
paths (green curve) across four different load plans: Primary
Only, Reference Plan, 1-Alt and All-Alt for
the three instance categories. In the Primary Only plan, each
commodity can be assigned only to its primary flow path. The
Reference Plan, referred to as the P-Plan, is the reference load plan from our industry
partner. Note that in the reference plan each commodity can use any number
of compatible alternate flow paths. In the 1-Alt plan, each
commodity can be assigned to either its primary path or the
cheapest alternate path. In the All-Alt plan, each commodity
can be assigned to all the available paths, i.e. splitting is allowed. Observe that the curves are
on different scales: the left scale for the blue curve and the right
scale for the green curve. The P-Plan is produced by the
planners using the greedy approach proposed earlier.
Figure <ref> demonstrates a consistent trend in
cubic capacity required in the four different load plans: the capacity
monotonically decreases and the decreases are significant. Allowing
spittability of commodity volume across primary and alternate flow
paths improves trailer consolidation. These benefits are already
apparent in the P-Plan of the planners, despite the fact that
this is a greedy approach. The optimization model with a single
alternate flow path, i.e., the 1-Alt plan, brings another
step change in consolidation, highlighting the benefits of
optimization. This benefit stems for the fact that a large number of
commodities have at least one alternate flow path in all instances
(see Figure <ref>). Note also that the 1-Alt load
plan requires significantly smaller total trailer capacity than the
P-Plan, although the P-Plan has the flexibility of
using any number of alternate flow paths. The All-Alt plan
brings further benefits but they are rather incremental. Part of the
reasons comes from the fact that a relatively small fraction of the
commodities have more than one alternate flow path. It would be
interesting to study a network with more flexibility as this may bring
further load consolidation benefits.
There is an interesting phenomenon that appears in the medium-sized instance M: the volume
assigned on the alternate flow paths decreased when moving from
1-Alt to an All-Alt plan. This comes from the fact
that this instance has many commodities, with a smaller volume, that
have new alternate flow paths options available in the
All-Alt setting. As a result, commodities with larger volume
are allocated to their primary flow path (as the flow diversion cost
is proportional to the total commodity volume assigned to alternate
flow paths) and the commodities with smaller volume can be allocated
to the alternate flow path that is the primary for the commodities
with larger volume (and not the cheapest alternate path of the
1-Alt setting). Hence, the total volume assigned to the
alternate flow paths reduces, although the total number of commodities
that use alternate flow paths increases.
Figure <ref> compares the percentage of the total
commodity volume that is assigned to the alternate flow paths in the
P-Plan and the All-Alt plan. It is undesirable to
allocate a major proportion of the volume to the alternate flow paths
because the downstream buildings may not be better equipped to handle
or process the large inbound volume. Observe that, on average across
all the instances, the All-Alt plan (resp. P-Plan)
allocates around 17% (resp. 9%) commodity volume on the
alternate flow paths. The All-Alt plan reduces the total
trailer capacity by roughly 12%-15% relative to the
P-Plan. For the XL instance, there is a significant gap
between the P-Plan and All-Alt plan statistics
because most of the commodities in the P-Plan are allocated
to the primary flow paths. This is why the total commodity volume
allocated to the alternate flow paths in the P-Plan and
the Primary Only have a small difference; see Figure
<ref> for XL category.
These results show that optimization proxies can bring substantial
benefits in practice. They provide, in real time, significant
improvements over the existing planning process. Moreover, by virtue
of their training mimicking the GDO optimization, that makes sure that
plans evolve smoothly during the planning process: small changes in
inputs will not result in large changes in the proposed solutions.
These results are eminently practical. One of the challenges
in the operational setting is the need for additional trailers when
the total commodity volume increases. Planners can acquire these
trailers either through empty trailer repositioning or by engaging in
short-term trailer leasing with local companies. Conversely, if the
commodity volume decreases, planners are left with a plan that has low
trailer utilization. The optimization proxies address this issue
directly. Planners can also use the proposed optimization proxies to obtain
recommendations for load plan adjustment in the event of a disruption
(due to uncertainty in commodity volume), even for the largest
terminal, within a matter of seconds. Furthermore, the recommendations
from the optimization proxies are consistent with existing load plans,
which makes it easy for the planners to evaluate and implement the
suggestions. Finally, new terminals in the service network often do
not have dedicated planners to develop load plans and extra capacity
is built in the system to handle the commodity volume in the
worst-case scenario. Optimization proxies can be used as a decision
support tool at such terminals.
§ CONCLUSIONS AND FUTURE WORK
This paper studies the Dynamic Load Planning Problem (DLPP) that
considers both load and flow planning challenges jointly in order to adjust loads and flows
as the demand forecast keeps changing over time before the day of
operations. The paper is motivated by the need of a
decision-support tool to advice planners making these decisions at
terminals across the network. The paper formulates the problem as a
MIP and shows that it admits many symmetries. As a result, the
optimization solver may return fundamentally different solutions to
closely related problems (i.e., DLPPs with slightly different inputs),
confusing planners and reducing trust in optimization. To remedy this
limitation, the paper proposes a Goal-Directed Optimization (GDO) that
eliminates those symmetries by generating optimal solutions staying
close to a reference plan. The paper also proposes an optimization
proxy, combining learning and optimization, to provide
high-quality and consistent load plans. An extensive computational
study on industrial instances shows that the optimization proxy is
around 10 times faster than the commercial solver in obtaining the
same quality solutions and orders of magnitude faster for generating
solutions that are consistent with each other. The proposed approach
also highlights the benefits of the DLPP for load consolidation, and
the significant savings from the combination of machine learning and
optimization.
This research is the first stage of a multi-stage project with our
industry partner (a large parcel carrier) for solving load planning
problems. Future research will extend the proposed approach to clusters of
terminals, taking into account their capacities for processing
commodities. The resulting problem thus requires to determine both
inbound and outbound planning decisions at each terminal, which
significantly complicates the optimization and learning models.
§ ACKNOWLEDGEMENT
This research was partly supported by the NSF AI Institute for Advances in Optimization (Award 2112533).
§ APPENDIX
§.§ Complexity Results
Model <ref> is difficult to solve because in addition to determining the right combination of trailer types to contain volume on each arc, we need to determine the right splits of commodity volume on the given set of compatible arcs. We will analyze the complexity of Model <ref> using the special cases described below.
Case 1: There is only one trailer type available at the terminal, i.e., | V_s | = 1 ∀ s ∈ S. Each commodity k ∈ K is compatible with exactly one sort pair s_k, i.e., S^k = {s_k} ∀ k ∈ K
Case 2: There is only one trailer type available at the terminal, i.e., | V_s | = 1 ∀ s ∈ S. Each commodity k ∈ K is compatible with all sort pairs, i.e., S^k = S ∀ k ∈ K
Cases 1 and 2 are polynomial time solvable
In Case 1, the volume of each commodity k is assigned to its only compatible sort pair, s_k, i.e. x^k_s_k = q^k. Then, the optimal solution has y_s = ⌈∑_k ∈ K: s ∈ S^k x^k_s/Q⌉ = ⌈∑_k ∈ K: s ∈ S^k q^k/Q⌉ ∀ s ∈ S.
In Case 2, the optimal solution is to assign the volume of all commodities on any sort pair s ∈ S and set x^k_s = q^k ∀ k ∈ K, y_s = ⌈∑_k ∈ K q^k/Q⌉, y_s' = 0 ∀ s' ∈ S, s' ≠ s.
Case 3: Same as Case 1, but with more than one trailer type available at the terminal
Case 4: Same as Case 2, but with more than one trailer type available at the terminal
Cases 3 and 4 are weakly NP-Hard
In the optimal solution in Case 3 the volume of each commodity k is assigned to its only compatible sort pair s_k. Thus, it remains to decide the optimal combination of trailer types required to containerize the volume on every sort pair. This is the minimum knapsack problem (see <cit.> for the problem definition) for each sort pair (that has more than one trailer type) as shown in <ref> which is known to be weakly NP-Hard.
For every s ∈ S: yMinimize ∑_v ∈ V_s c_v y_s,v
subject to ∑_k ∈ K: s ∈ S^k q^k≤∑_v ∈ V_sQ_v (y_s,v),
y_s,v∈ℤ_≥ 0 ∀ v ∈ V_s
Similarly, for Case 4 there exists an optimal solution in which the volume of all commodities is assigned to one sort pair s^* ∈ S, i.e. x^k_s^* = q^k ∀ k ∈ K and it remains to solve a minimum knapsack problem for the sort pair s^* due to which Case 4 is weakly NP-Hard.
Case 5: Each commodity k ∈ K is compatible with a subset of sort pairs, i.e., S^k ⊂ S, and has unit volume, q^k = 1. There is only one trailer type with per-unit cost c_s=1 ∀ s ∈ S and capacity Q=max_s ∈ S{∑_k ∈ K1_s ∈ S^k}; hence, y_s ∈{0,1}∀ s ∈ S, as installing one unit of trailer is enough to containerize the total volume that can be assigned to the sort pair. Note that we ignore the index v for trailer because each sort pair has exactly one and the same trailer type.
In the optimal solution of Case 5, each commodity is assigned to exactly one compatible sort pair (i.e. there is no splitting of volume)
We will present a proof by contradiction. WLOG, suppose there exists an optimal solution in which the volume of a commodity k̂ is split between two sort pairs and the volume of all other commodities k ∈ K \{k̂} is assigned to exactly one sort pair s_k. Thus, we have x^k_s_k = q^k ∀ k ∈ K \{k̂} and x^k̂_s_1 + x^k̂_s_2 = q^k. Consider a solution x^k_s =x^k_s ∀ k ∈ K \{k̂} and x^k̂_s_1 = x^k̂_s_1 + ϵ,x^k̂_s_2 = x^k̂_s_2 - ϵ, where ϵ > 0 is a small real number. Note that x^k̂_s_1 + x^k̂_s_2 = q. Consider another solution x̅^k_s = x^k_s ∀ k ∈ K \{k̂} and x̅^k̂_s_1 = x^k̂_s_1 -ϵ,x̅^k̂_s_2 = x^k̂_s_2 + ϵ. Note that both solutions x and x̅ satisfy constraints (<ref>) and are feasible to constraints (<ref>) because we choose Q = max_s ∈ S{∑_k ∈ K1_s ∈ S^k}. The solution x can be written as a convex combination of the solution x and x̅ (x^k_s = 1/2x̅^k_s + 1/2x^k_s ∀ k ∈ K, s ∈ S^k) which contradicts the optimality of the solution.
Case 5 is strongly NP-Hard
We will show that this special case can be solved as a set cover problem which is known to be strongly NP-Hard (<cit.>). An instance of a set cover is given by a ground set U = {x_1, x_2, ⋯, x_n} and a collection of m subsets E_i ⊆ U ∀ i ∈{1,2,⋯,m} of the ground set U. The optimization problem is to find the smallest number of subsets i ∈{1,2,⋯,m} such that ⋃_i ∈{1,2,⋯,m} E_i = U.
From claim <ref> we know that each commodity is assigned to exactly one compatible sort pair in the optimal solution. Let commodity k ∈ K denote element x_k ∈ U, | K | = n and set of sort pairs S = {1,2,⋯,m}. Define K_i = {k ∈ K : x_k ∈ E_i} as the set of commodities or elements that can be covered by selecting sort pairs i ∈{1,2,⋯,m}. Now note that finding the smallest number of sort pairs s ∈ S such that all commodities in K are covered is equivalent to finding the smallest number of subsets i ∈{1,2,⋯,m} to cover all elements in U.
§.§ Additional Experimental Results
Table <ref> compares the the number of integer decision variables in Model <ref> and the average number of binary decision variables in Model <ref> across multiple test instances. The number of integer decision variables remain the same for each instance category because it depends in the number of arcs or sort pairs and trailer types; only the commodity volume changes across different test instances. However, the size of the feasibility restoration model <ref> depends on the predictions of the ML model. Recall that the ML model predicts the value of the integer decision variables of Model <ref>. Hence, if the predictions are accurate, then fewer sort pairs would have capacity violations. Consequently, there would be fewer binary decision variables in Model <ref>; the number of binary decision variables in Model <ref> is equal to the number of sort pairs with capacity violation. As the ML predictions can vary for different test instance with the same set of sort pairs due to different commodity volume, the number of binary variables in Model <ref> can be different for different test instances. This is why the Table <ref> reports fractional values for the average number of binary variables. It is worth highlighting that one of the key benefits of the optimization proxies is that it replaces a model with large number of integer decision variables with a prediction model, and requires to solve a relatively simpler model with small number of binary variables.
|
http://arxiv.org/abs/2307.04226v2 | 20230709163747 | Seismic Data Interpolation based on Denoising Diffusion Implicit Models with Resampling | [
"Xiaoli Wei",
"Chunxia Zhang",
"Hongtao Wang",
"Chengli Tan",
"Deng Xiong",
"Baisong Jiang",
"Jiangshe Zhang",
"Sang-Woon Kim"
] | physics.geo-ph | [
"physics.geo-ph",
"stat.ML"
] |
Journal of Class Files, Vol. 14, No. 8, August 2015
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals
Seismic Data Interpolation based on Denoising Diffusion Implicit Models with Resampling
Xiaoli Wei, Chunxia Zhang, Member, IEEE, Hongtao Wang, Chengli Tan, Deng Xiong, Baisong Jiang, Jiangshe Zhang, Sang-Woon Kim, Life Senior Member, IEEE
Corresponding author: Chunxia Zhang. E-mail: [email protected].
Xiaoli Wei, Chunxia Zhang, Hongtao Wang, Chengli Tan, Baisong Jiang, Jiangshe Zhang are with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China.
Deng Xiong is with the Geophysical Technology Research and Development Center, BGP, Zhuozhou, Hebei, 072751, China
Sang-Woon Kim is with the Department of Computer Engineering, Myongji University, Yongin, 17058, South Korea.
This research was supported by the National Key Research and Development Program of China (No. 2018AAA0102201) and the National Natural Science Foundation of China (No. 61976174).
This work has been submitted to the IEEE TGRS for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.
August 12, 2023
====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
The incompleteness of the seismic data caused by missing traces along the spatial extension
is a common issue in seismic acquisition due to the existence of obstacles and economic constraints, which severely impairs the imaging quality of subsurface geological structures. Recently, deep learning-based seismic interpolation methods have attained promising progress, while achieving stable training of generative adversarial networks is not easy, and performance degradation is usually notable if the missing patterns in the testing and training do not match.
In this paper, we propose a novel seismic denoising diffusion implicit model with resampling.
The model training is established on the denoising diffusion probabilistic model, where U-Net is equipped with the multi-head self-attention to match the noise in each step. The cosine noise schedule, serving as the global noise configuration, promotes the high utilization of known trace information by accelerating the passage of the excessive noise stages.
The model inference utilizes the denoising diffusion implicit model, conditioning on the known traces, to enable high-quality interpolation with fewer diffusion steps. To enhance the coherency between the known traces and the missing traces within each reverse step, the inference process integrates a resampling strategy to achieve an information recap on the former interpolated traces. Extensive experiments conducted on synthetic and field seismic data validate the superiority of our model and its robustness to various missing patterns. In addition, uncertainty quantification and ablation studies are also investigated.
Seismic data interpolation, denoising diffusion model, multi-head self-attention, resampling
§ INTRODUCTION
Seismic exploration interprets geological information and infers subsurface properties by analyzing the pre-stack data collected by geophones planted in the field. Acquisition of high-quality seismic data is a key factor for high-quality seismic data processing and interpretation. However, the collected seismic data is usually severely degraded due to the complex natural environment or limited budget. The degradation of data integrity is typically observed in the form of random or consecutive missing seismic traces, resulting in undersampled or aliased seismic data <cit.>.
Seismic data interpolation has been extensively investigated over the past decades. Initially developed traditional methods often rely on the assumption of global or local linear events to convert the problem into an autoregressive framework <cit.>. Especially, prediction-filter-based methods, combined with the t-x and f-x regularization <cit.>, <cit.>, occupy the research mainstream in this direction. Besides, wave-equation-based methods are able to extrapolate and interpolate wave field <cit.>, whereas they require additional information, e.g., wave velocity.
Two successful categories of model-driven methods involve different constraints to recover seismic data. The first category is the sparsity-based method, which introduces various sparse transforms and sampling functions to interpolate missing data <cit.>. Among these methods, those derived from the projection onto convex sets <cit.> have received more attention due to their relatively high performance. The second category applies the low-rank constraint model to recover data, e.g., using singular value decomposition on block Hankel matrix <cit.>.
While the traditional methods and model-driven methods are capable of achieving interpolation from a theoretical perspective, issues such as manual parameter selection and enormous computation cost cannot be ignored, particularly for massive and high-dimensional field seismic data with advancements in collection technology and efficiency.
With the rapid advancement of deep learning-based generative models, the research focus for seismic data interpolation has shifted towards data-driven methods, which mainly include two categories, i.e., generative neural network and generative adversarial network (GAN).
The preliminary methods in the first category of data-driven models contain the convolutional autoencoder (CAE) <cit.>, <cit.>, U-Net <cit.>, <cit.>, and residual network (ResNets) <cit.>, etc. Liu et al. <cit.> introduce the invertible discrete wavelet transform for replacing the pooling operations in the traditional U-Net model, thereby avoiding the loss of detailed features caused by the downsampling scheme. Some researchers have worked on improving the long-range feature correlation via different attention modules <cit.>, <cit.>, which are critical to maintain the global content consistency, especially under the circumstance of consecutively missing seismic traces <cit.>. Furthermore, regularization terms are important in finding the optimal interpolation function, e.g., spectrum suppression <cit.> and regeneration constraint <cit.>. Some studies also focus on improving the seismic feature extraction ability of neural networks, including the adoption of UNet++ with a nested architecture <cit.> and dynamically updating the valid convolution region <cit.>. However, a standalone neural network is usually insufficient to capture the vast range of dynamic energy in seismic data.
To resolve this issue, the coarse-refine network <cit.> and the multi-stage active learning method <cit.> have been proposed, which exploit the strengths of every sub-network to make the interpolation process more efficient and well-performed.
The second category of data-driven models, GAN-based methods, has achieved impressive results in seismic data interpolation. Kaur et al. <cit.> adopt the framework of CycleGAN to perform self-learning on the seismic features.
The conditional generative adversarial network (CGAN) is introduced to interpolate the seismic data with consecutively missing traces <cit.>. Based on CGAN, the dual-branch interpolation method combining the time and frequency domains improves the smoothness and quality of the reconstructed seismic data <cit.>. The large obstacle is a common trouble in seismic exploration, which leads to big gaps in the collected seismic data and impairs the further data processing. The promising results of conditional Wasserstein generative adversarial networks with gradient penalty (WGAN-GP) have revealed the seismic feature generation capability <cit.>, whose gradient penalty enhances the fidelity of reconstructed signals at large intervals by enforcing the Lipschitz constraint. The coarse-to-fine learning strategy drived by the joint of different loss strengthens the connection between different stages and enables relativistic average least-square generative adversarial network (RaLSGAN) to produce more accurate and realistic signal details <cit.>.
Although the deep learning-based seismic data interpolation method has attracted considerable attention, the instability of GAN training and the complexity of field data still limit its further development. First, while the generator can be implemented with a state-of-the-art generative architecture toward seismic data reconstruction, the demand for training the discriminator cannot be avoided for a GAN-based model, and the optimal solution often lies in a saddle point instead of a local minimum <cit.>. Stable adversarial training requires good initialization and hyperparameter settings. Second, field seismic data usually consist of multiple missing forms due to the influence of ground obstacles and geophone layout conditions, etc. The aforementioned data-driven methods either serve to a specific missing form of seismic data or need retraining when interpolating seismic data with different missing ratios or forms. Since their training is based on a certain mask distribution, the performance of the model may degrade to varying degrees or even fail to achieve the desired effect when transferring to a new scenario.
In this paper, we propose a new seismic denoising diffusion implicit model with resampling (SeisDDIMR) to address the above issues, showing that it only needs to be trained once to complete the reconstruction tasks of different missing rates or missing forms, and it exhibits superior interpolation effects compared to the existing deep learning methods. This denoising diffusion model-based approach retains the strong power of generative neural networks since the backbone can be inherited from state-of-the-art generative architectures. The main contributions of this paper are summarized below:
* Our model's entire training framework is built on denoising diffusion probabilistic models (DDPM) <cit.>, which include two parameterized Markov chains, i.e., a forward diffusion process and a reverse process. The forward diffusion process progressively adds pre-designed Gaussian noise to the initial full seismic data. The reverse process uses variational inference to estimate the noise after a finite time of the forward process under the fixed noise addition mode, and thereby the parameterization estimation of the neural network is completed.
* Our noise matching network follows the U-Net structure equipped with multi-head self-attention (MHSA), which can capture stronger long-range correlations of seismic data.
* The inference process of our model deriving from condition interpolation is accelerated by using denoising diffusion implicit models (DDIM) <cit.>, and we adopt the strategy of resample iterations <cit.> to enhance the consistency of the interpolation content before and after the reverse diffusion step. To make more effective adjustments conditioned on the known seismic traces, we introduce a cosine noise schedule that enables the inverse process to generate meaningful reconstruction signals in the early stages instead of high-noise results under a linear noise schedule. This contributes greatly to the interpolation quality.
* Existing deep learning methods are often limited by the missing forms constructed during training, consequently lacking robustness to effectively interpolate seismic data in cases where the missing patterns do not match or complex missing forms coexist.
Our proposed method breaks through this issue and brings greater flexibility to the application of deep learning interpolation methods in field scenarios.
The remainder of this paper is organized as follows. In Section <ref>, we introduce our SeisDDIMR method including the training, inference, and network architecture. In Section <ref>, experiments with various missing interpolation are performed for both synthetic and field seismic data. The effectiveness of our method is demonstrated by comparing it with popular methods. Furthermore, to indicate the stronger advantages of our model in practical application scenarios, we conduct uncertainty quantification and model robustness validation. Section <ref> presents some ablation studies. Finally, we make conclusions and discussions in Section <ref>.
§ METHODOLOGY
Let x∈ℛ^n_r × n_t as the original complete seismic data, with n_r and n_t as the number of traces and time samples. The degradation process of observed seismic data can be formally expressed as
y=m⊙x, such that m[ i,: ]= J, i is valid
0, else
where ⊙ represents the element-wise multiplication, J is the all-ones matrix, and 0 denotes the zero matrix. The notation m[ i,: ] indicates the missing mask of ith trace data.
Seismic data interpolation aims to learn a function mapping observed seismic data y back to complete data, which is usually implemented by a neural network parameterized by θ. Different from a single neural network model, the diffusion model-based approach incorporates multiple parameterization processes to achieve stepwise approximation. The proposed SeisDDIMR model consists of two main processes, i.e., the training process for estimating the parameters of seismic DDPM and the inference process for interpolating missing seismic data. In Section <ref>, we introduce the key principles of DDPM combined with the background of seismic data interpolation. The following Sections, <ref> and <ref>, provide descriptions of the noise matching network and its corresponding noise schedule. Finally, the inference method, together with its theoretical background, is presented in Section <ref>.
§.§ Seismic Denoising Diffusion Probabilistic Model
Given the complete seismic data samples x_0 ∼ q(x_0), DDPM relies on the generative Markov chain process and the noise matching network to gradually learn the target distribution p_θ(x_0).
The forward diffusion process is a deterministic Markov chain starting from the initial input x_0 and using a pre-specified noise schedule to gradually add Gaussian noise to perturb the data distribution. Given the latent variables x_1, …, x_T derived from the same sample space with x_0, the diffusion process is defined as
q(x_1: T|x_0):=∏_t=1^T q(x_t |x_t-1),
where
q(x_t |x_t-1):=𝒩(x_t ; √(1-β_t)x_t-1, β_t 𝐈).
Here, β_t∈(0,1) is a pre-designed increasing variance schedule of Gaussian noise. The closed form of sampling x_t given by Ho et al. <cit.> reveals the progressive changes during the middle time of the forward process. Letting α_t:=1-β_t and α̅_t:=∏_s=1^t α_s, it can be denoted as
q(x_t |x_0)=𝒩(x_t ; √(α̅_t)x_0,(1-α̅_t) 𝐈).
As t continues to increase, the final data distribution converges to a given prior distribution, i.e., a standard Gaussian for x_0. Correspondingly, the reverse process will gradually denoise for each step of the forward process starting from p(x_T)=𝒩(x_T ; 0, 𝐈) under the Markov chain transition
p_θ(x_0: T):=p(x_T) ∏_t=1^T p_θ(x_t-1|x_t),
where p_θ(x_t-1|x_t):=𝒩(x_t-1 ; μ_θ(x_t, t), Σ_θ(x_t, t)) and the network parameter θ is shared across different reverse stages. This optimization problem of fitting the data distribution q(x_0) can be converted into the minimization of a variational lower bound (VLB) for the negative log likelihood by introducing Jensen’s inequality
L_vlb:=𝔼_q(x_0: T)[logq(x_1: T|x_0)/p_θ(x_0: T)] ≥-𝔼_q(x_0)log p_θ(x_0).
VLB is decomposed into the following KL-divergence form
between two Gaussian distributions by including the Markov property in the denoising diffusion model and the definition form of the forwards process
L_vlb =
𝔼_q[D_KL(q(x_T |x_0) p(x_T))] -𝔼_q[log p_θ(x_0 |x_1)]
+𝔼_q[∑_t=2^T D_KL(q(x_t-1|x_t, x_0) p_θ(x_t-1|x_t))].
According to Ho et al. <cit.>, the Gaussian distribution q(x_t-1|x_t, x_0) can be tractable as
q(x_t-1|x_t, x_0)=𝒩(x_t-1 ; μ̃_t(x_t, x_0), β̃_t 𝐈),
where
μ̃_t(x_t, x_0):=√(α̅_t-1)β_t/1-α̅_tx_0+√(α_t)(1-α̅_t-1)/1-α̅_tx_t
and
β̃_t:=1-α̅_t-1/1-α̅_tβ_t.
Furthermore, the first term in Eq. (<ref>) can be ignored as a constant. The discrete probability density of the second term can be estimated using continuous Gaussian distribution. Combined with the property Eq.
(<ref>), D_KL(q(x_t-1|x_t, x_0) p_θ(x_t-1|x_t)) in the third term of Eq. (<ref>) is simplified to
𝔼[1/2 σ_t^21/√(α_t)(x_t(x_0, ϵ_t)-β_t/√(1-α̅_t)ϵ_t)-μ_θ(x_t(x_0, ϵ_t), t)^2],
where the constant is omitted and ϵ_t ∼𝒩(0, 𝐈). Noting the availability of x_t, Ho et al. <cit.> transfer the predictions about μ_θ to ϵ_θ by the following parameterization
μ_θ(x_t, t)=1/√(α_t)(x_t-β_t/√(1-α̅_t)ϵ_θ(x_t, t)).
Regardless of the coefficients, since they find that removing them benefits sample quality, the popular loss used in DDPM is finally formulated as
L_simple=𝔼_x_0 ∼ q(x_0), ϵ_t ∼𝒩(0, I)[ϵ_t-ϵ_θ(√(α̅_t)x_0+√(1-α̅_t)ϵ_t, t)^2].
Therefore, the network parameters are optimized by the mean squared error (MSE) loss between the Gaussian noise predicted by the network and the real noise for all time nodes of the reverse process except for t=1. Moreover, as discussed in <cit.>, the log-likelihood can be improved in the log domain by parameterizing the variance Σ_θ(x_t, t) = σ_t^2 𝐈 with the following interpolation between β_t and β̃_t
Σ_θ(x_t, t)=exp(v logβ_t+(1-v) logβ̃_t),
where v can be concatenated on another channel of ϵ_θ(x_t, t), serving as the output of the model. Finally, the loss function of our model is set to
L_hybrid=L_simple+λ L_vlb,
where we follow the setting in <cit.> and adopt λ = 0.001 to avoid L_vlb overwhelming L_simple.
Once the training accomplished, sampling x_t-1 from p_θ(x_t-1|x_t) can be conducted with the following iterative update formula
x_t-1=1/√(α_t)(x_t-1-α_t/√(1-α̅_t)ϵ_θ(x_t, t))+σ_t z,
where z∼𝒩(0, 𝐈) (t>1) or z = 0 (t=1).
Fig. <ref> illustrates the detailed stream of the seismic DDPM. The forward process does not require training and directly converts x_0 to the isotropic Gaussian noise. In the reverse process, the denoising model learns to predict the added noise for each time step. When gradually fitting the noise, the estimated value of x_0 can also be obtained at each time step according to
x̂_0=√(1/α̅_t)x_t -√(1-α̅_t/α̅_t)ϵ_θ(x_t, t),
even though it may not be satisfactory during mid-time stamps.
§.§ Noise Matching Network
The noise matching network used in <cit.> is based on the U-Net architecture with self-attention <cit.> and achieves impactful performance. Durall et al. <cit.> adopt this architecture to accomplish seismic data demultiple, denoising, and interpolation.
Different from the aforementioned research works, we use a more appropriate network structure for seismic data generation, whose major stream inherits from the guided-diffusion model <cit.>. It adopts more architecture improvements to attain better generative quality. The overall architecture is displayed in Fig. <ref> using stacked residual blocks (Res Block) and attention blocks (Attn Block and MidAttn Block) for the encoder and decoder of U-Net. x_t is used as the network input for the denoising learning process to obtain predicted noise ϵ_θ(x_t, t), and the accompanying timestamp t is fed to each layer to embed time information by using the following Transformer sinusoidal time embedding (TE) <cit.>
T E_(t, 2 i) =sin(t / 10000^2 i / d)
T E_(t, 2 i+1) =cos(t / 10000^2 i / d),
where d stands for the dimension of embedding vectors, t is the original time, and i is the dimension. Figuratively speaking, it serves for x_t to inform each layer about the current step of reverse diffusion.
Fig. <ref> displays the detailed components of the Res Block, Attn Block, and MidAttn Block from left to right, where N=2 for the encoding process and N=3 for the decoding process. Upsampling and downsampling are executed after Res Block and Attn Block, except for the bottom layer, for a total of four operations. As illustrated in Fig. <ref>, the residual module is implemented with the inclusion of temporal information within. The MHSA module existing in Attn Block and MidAttn Block increases the receptive field of the model so that it can access all of the input seismic signals as introduced in <cit.>. Fig. <ref> makes a detailed illustration of the MHSA module, which receives the feature map as input and conducts three different linear operations W_q, W_k, and W_v to get the query matrix Q, key matrix K, and value matrix V. Each of them is divided into multiple heads, allowing the model to perform parallel computing and capture relevant information from different subspaces to integrate multiple attentions with different focuses. Self-attention is employed on the branches of each head to learn long-range correlations, which are formulated as
Head_i=Attention(Q_i, K_i, V_i)=softmax(Q_i K_i^T/√(d_k)) V_i,
where d_k is the dimension of queries and keys, and i stands for the number index of heads within {1, …, N_head}. We use N_head=4 in the noise matching network. Finally, MHSA is obtained by integrating the attention of each head together as
MHSA(Q, K, V)=Concat(Head_1, …, Head_N_head).
§.§ Cosine Noise Schedule
DDPM <cit.> applies the linear noise schedule for β, where noise increases at a constant rate as the diffusion process proceeds. Since the primary concern in seismic data interpolation is the fidelity of the generated signal, as opposed to diversity, expediting the transition through the stage of high noise can facilitate the reconstruction of unknown areas. We adopt the following cosine schedule <cit.>
α̅_t=f(t)/f(0), f(t)=cos(t / T+s/1+s·π/2)^2,
where the offset s=0.008 is used to prevent β_t from being too small near t=0. The gray and blue dots in Fig. <ref> display the changing trend of α̅_t in the training process. Compared with the linear noise schedule, the cosine noise schedule can decelerate the global rate of information decay. Meanwhile, the gray dots in Fig. <ref> show the changing trend of β_t with respect to diffusion steps during the training process. The reduction of the strong noise states is observable, and it can aid in the interpolation of missing locations. To intuitively observe the differences between the generation processes of different noise schedules, Fig. <ref> illustrates the seismic data interpolation results x̂_0 at some middle timestamps during the reverse diffusion process. The interpolated content at intermediate timestamps under the linear noise schedule may deviate significantly from the ground truth distribution in Fig. <ref>. In contrast, the differences in distribution between each timestamp are much smaller under the cosine noise schedule, as shown in Fig. <ref>. This phenomenon occurs since the cosine noise schedule quickly passes through the high noise phase. Increased availability of known valid information facilitates the generation of missing regions, ensuring consistent alignment between the interpolated content and the ground truth.
§.§ Implicit Conditional Interpolation with Resampling
The trained seismic DDPM operates unconditionally, wherein the inverse diffusion process is generated directly from noise. However, for seismic data interpolation, it is essential to infer unknown signals from known regions. Hence, further refinement of the interpolation process is necessary. Inspired by the RePaint model <cit.>,
we redesign the interpolation process to improve computation feasibility and interpolation quality. Different from the Seismic DDPM used in the training process, the inference process no longer satisfies the Markov assumption, and we adopt the DDIM sampling strategy to mitigate the computation burden existing in the RePaint model.
Intuitively, it seems that the loss function of DDPM ultimately only depends on q(x_t|x_0) and the sampling process is only related to p(x_t-1|x_t), from which Song et al. <cit.> get inspiration for proposing denoising diffusion implict models (DDIM). They introduce the following non-Markovian inference
q_σ(x_1: T|x_0):=q_σ(x_T |x_0) ∏_t=2^T q_σ(x_t-1|x_t, x_0),
with a real vector σ = (σ_1, …, σ_T ) ∈ℝ_≥ 0. They choose
q_σ(x_t-1|x_t, x_0)
= 𝒩(√(α̅_t-1)x_0+√(1-α̅_t-1-σ_t^2)·x_t-√(α̅_t)x_0/√(1-α̅_t), σ_t^2 I)
to ensure q_σ(x_t |x_0) remains consistent with the form in Eq. (<ref>). Under the above definition, the forward process q_σ(x_t |x_t-1, x_0) is still rebuilt as Gaussian and the VLB can then be written as
L_VLB^σ:=𝔼_x_0: T∼ q_σ(x_0: T)[log q_σ(x_1: T|x_0)-log p_θ(x_0: T)]
= 𝔼_x_0: T∼ q_σ(x_0: T)[log q_σ(x_T |x_0)+∑_t=2^T log q_σ(x_t-1|x_t, x_0)]
- 𝔼_x_0: T∼ q_σ(x_0: T)[∑_t=1^T log p_θ^(t)(x_t-1|x_t)-log p_θ(x_T)].
Song et al. <cit.> have proved that the objective function, i.e., Eq. (<ref>), ultimately used by DDPM is a special case of L_VLB^σ under certain conditions, which allows us to directly use the pre-trained DDPM model as a solution for new objectives. With the aforementioned theoretical foundation, sampling from this non-Markovian generative process is focused on constructing σ to improve sample generation and reduce sample steps. Starting from Eq. (<ref>), the sampling operation can be formulated as
x_t-1=√(α̅_t-1)(x_t-√(1-α̅_t)ϵ_θ(x_t, t)/√(α̅_t))
+ √(1-α̅_t-1-σ_t^2)·ϵ_θ(x_t, t)+σ_tz,
where the generative process becomes Markovian and equals DDPM if σ_t=√((1-α̅_t-1) /(1-α̅_t))√(1-α̅_t / α̅_t-1) for all t. Especially, it is reasonable to consider a sampling process of length less than T when q_σ(x_t |x_0) is fixed since the optimization result of DDPM essentially contains its optimization results for arbitrary subsequence parameters. Denoting the increasing time subsequence of the original time sequence [1, …, T] as τ=[τ_1, τ_2, …, τ_m] with of length m (the corresponding changes in α̅_τ_i and β_τ_i are shown in the red points of Figs. <ref> and <ref>, respectively), the σ_τ used in accelerated sampling process follows
σ_τ_i(η)=η√((1-α̅_τ_i-1) /(1-α̅_τ_i))√(1-α̅_τ_i / α̅_τ_i-1),
where η≥ 0. In particular, the generative process is defined as DDIM if η = 0 for all t since the variance σ keeps zero, so that the deterministic forward process becomes an implicit probabilistic model.
Each step of the iterative reverse diffusion stage in the inference process uses the following implicit conditional interpolation formula
x_τ_i-1=m⊙x_τ_i-1^valid+(1-m) ⊙x_τ_i-1^missing,
where x_τ_i-1^valid is directly sampled from the forward diffusion process, i.e., Eq. (<ref>), which adds known information to the reverse process, and x_τ_i-1^missing is obtained by using the DDIM sampling formula Eq. (<ref>). As a result, x_τ_i-1 incorporates information from both known signals and model predicted signals before forwarding it to the next inverse diffusion step. The recovery of missing seismic data is designed as a implicit conditional interpolation process based on valid seismic data.
Merely relying on the known signal as the condition is not adequate. Despite the relationship between the interpolated and known signals, maintaining interpolated signal continuity and consistency with known signals remains challenging. We introduce the resampling strategy <cit.> to enhance the consistency of sampling in the reverse process.
After sampling x_τ_i-1 in the inverse diffusion process, the forward diffusion sampling is performed again to generate x_τ_i, with the difference being that x_τ_i now contains the information from x_τ_i-1^missing, thereby promoting consistency with known signals. Naturally, this kind of resampling operation cannot be performed only once. We define the jump length, denoted as L, to set how many times to backtrack for each resampling process, and we define the jump height, denoted as H, which determines the interval between time steps before and after two different resampling processes.
In a word, our SeisDDIMR model comprises two key processes, i.e., the seismic DDPM training process and the implicit conditional interpolation process with resampling. Algorithm <ref> and Algorithm <ref> list the overview of our training and inference procedure, respectively.
§ EXPERIMENTS
§.§ Evaluation Metrics
We choose three metrics, i.e., MSE, signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR), to compare the fidelity of the interpolated seismic data. MSE between the interpolated seismic data {x̂^j}_j=1^n and the ground truth {x^j_gt}_j=1^n is calculated using
MSE=1/n∑_j=1^n(x̂^j-x^j_gt)^2,
where its value closer to 0 implies a higher fidelity of the interpolation result. The SNR for a single interpolated sample is defined as
SNR=10log_10x_gt^2_F/x_gt-x̂^2_F,
where ·_F represents the Frobenius norm. PSNR is calculated by the following formula as
PSNR=10log_10MAX_x_gt^2/MSE,
where MAX_x_gt refers to the highest value of x_gt. Obviously, larger SNR and PSNR both symbolize higher interpolation fidelity.
The quality of the texture of the interpolation is evaluated using structural similarity (SSIM) <cit.>, which is widely used in the field of image generation following the formula
SSIM(x_gt, x̂) =L(x_gt, x̂) · C(x_gt, x̂) · S(x_gt, x̂).
Separately, L(·), C(·), and S(·) indicate similarities in luminance, contrast, and structure, and they are each defined as
L(x_gt, x̂)=2 μ_x_gtμ_x̂+c_1/μ_x_gt^2+μ_x̂^2+c_1,
C(x_gt, x̂)=2 σ_x_gtσ_x̂+c_2/σ_x_gt^2+σ_x̂^2+c_2,
S(x_gt, x̂)=σ_x_gtx̂+c_3/σ_x_gtσ_x̂+c_3,
where μ_x_gt(μ_x̂), σ_x_gt(σ_x̂), and σ_x_gtx̂ denote the mean value and standard deviation, and covariance, respectively. Constants c_1, c_2, and c_3 are typically set close to zero to prevent numerical instability. Thus, a higher SSIM implies a more similar texture.
§.§ Data Set
We validate our method over one open synthetic dataset provided by the Society of Exploration Geophysicists (SEG) C3
and one field dataset Mobil Avo Viking Graben Line 12 (MAVO). The SEG C3 dataset consists of 45 shots, each with a
201×201 receiver grid, 625 time samples per trace, and a sampling interval of 8 ms. We randomly extract 35,000 128×128 patches, out of which 25,000 patches are utilized for training, 5,000 for validation, and another 5,000 for testing. MAVO dataset comprises a 1001×120 receiver grid with 1500 time samples per trace. It is collected at a time rate of 4 ms and a spatial rate of 25 m. We randomly extract 10,000 256×112 patches, with 6,000 used for training, 2,000 used for validation, and 2,000 used for testing. All seismic patches are firstly normalized within the interval [0,1] by applying min-max normalization.
§.§ Implementation Details
The diffusion step for the Seismic DDPM model is set to 1000.
We train the seismic DDPM model on the training sets of SEG C3 and MAVO separately, as described in Algorithm <ref>, with N iterations of 600,000 and 300,000, respectively. The noise matching network is optimized by AdamW with a learning rate of 1e-4. The batch size is set to 30 for the SEG C3 dataset and 15 for the MAVO dataset. Our SeisDDIMR test is conducted by using Algorithm <ref>, where we adopt diffusion sampling step m=100, jump length L=10, and jump height H=1. We compare our experimental results with 5 currently popular methods, including DD-CGAN <cit.>, cWGAN-GP <cit.>, PConv-UNet <cit.>, ANet <cit.>, and Coarse-to-Fine <cit.>. All of the experiments are implemented using Pytorch 1.12.1 and NVIDIA GeForce RTX 3090 GPU.
§.§ Experimental Results
We conduct Algorithm <ref> to accomplish our model testing. Interpolation reconstructions are performed on three missing categories of seismic data, and the experimental results are displayed below, followed by a comparison to other methods. It worth noting that our SeisDDIMR model is trained only once on each dataset, whereas other comparison methods are trained multiple times according to various trace missing forms, and the details of the training parameters remain consistent with their respective original papers.
§.§.§ Random Missing Traces
For each patch in the test sets of SEG C3 and MAVO, we design random missing phenomena with missing rates ranging from 0.2 to 0.6. The initial values of the missing traces are set to 0. The experimental results of random missing interpolation are listed on the left side of Tab. <ref> and Tab <ref>. Except for being slightly inferior in the SSIM, the other three metrics demonstrate that our model has better fidelity. Fig. <ref> shows the interpolated traces of the random missing MAVO test data. It can be seen that our method achieves the best performance both on amplitudes and phases. As a special case of random missing seismic data, the regular missing scenario will cause a serious aliasing problem. It usually appears as excessive artifacts in the high-frequency band of f-k spectra caused by erroneous estimation or interpolation of the missing data frequency. Fig. <ref> compares the f -k spectra of SEG C3 test data with 70% regular missing traces. Severe aliasing can be noticed in Fig. <ref>. It is obvious that the f-k spectra of the DD-CGAN, cWGAN-G, and ANet are all accompanied by significant high-frequency artifacts. Comparisons between the performance of all methods indicate that our model gains the most consistent f-k spectra with the ground truth.
§.§.§ Consecutive Missing Traces
We randomly create consecutive missing masks, with rates of missing data ranging from 0.1 to 0.4 (not including edge traces), and applied them to the patches in the SEG C3 and MAVO datasets. The value of missing traces is initialized to 0. The interpolation results of the middle four columns of Tab. <ref> and Tab. <ref> indicate that our model consistently surpasses other methods over these two datasets. we provide the comparisons via color plots from the SEG C3 test dataset as in Fig. <ref>. The ground truth data suffers from a consecutive missing of 40% resulting in degenerate missing data. Significant differences in the distribution are visible in the known portions on either side, which hinder the ability of some methods, such as PConv-UNet and ANet, that rely solely on feature similarity to perform the interpolation. DD-CGAN, cWGAN-GP, and Coarse-to-Fine methods based on GAN are still limited in their interpolation ability and tend to smooth small-scale seismic events due to large interval problems. Among these, cWGAN-GP demonstrates a high continuity in strong amplitude regions with biasedly sacrificing the performances on the fidelity of weak amplitudes.
Coarse-to-Fine acquires fine details of weak amplitudes but it still exhibits significant differences from the ground truth data. Our model can consistently improve the performances over both strong and weak amplitudes, and keep anisotropy and spatial continuity of signals.
§.§.§ Multiple Missing Traces
For the SEG C3 and MAVO datasets, we construct multiple missing data scenarios with both consecutive and random missing cases and the range of the total missing rate is [0.2, 0.8]. The missing traces are also initialized with a value of 0. The corresponding quantitative comparison results are listed in the right four columns of Tab. <ref> and Tab. <ref>, where our model consistently outperforms other methods on four metrics. Fig. <ref> exhibits the interpolation results on a multiple missing example with total missing rare 54% from the MAVO test data. Our model produces artifact-free results, while other methods generally result in the ubiquity wide areas of artifacts, especially for DD-CGAN, cWGAN-G, and PConv-UNet, failing to provide reliable recovery. In addition, the amplitudes predicted by our model are more accurate and consistent with the ground truth.
Our model is capable of handling most cases of seismic missing trace reconstruction.
§.§ Model Robustness
In order to study the impact of changes in the missing form on model capability, we evaluate the performance of different methods under the unmatched training and testing mask patterns, as shown in Tab. <ref>.
First, when testing on the unseen consecutive mask pattern, the performance of the models trained on the random mask type has decreased significantly compared to those consecutive missing reconstruction results in Tab. <ref>. Second, although the model trained on the multiple mask form exhibits interpolation capability on different mask types, their results are still worse than those trained on the same mask pattern, as demonstrated in Tab. <ref>.
Third, we can see that the consecutive missing model fails to interpolate random missing data, which is likely due to the significant differences in learning patterns between consecutive missing form and random missing form.
It can be concluded that the effectiveness of generative models, which may be based on GAN or feature similarity, is sensitive to the constructed mask formula in training data. It seems better if the training missing construction can be closer to the missing form of the test data, although there easily exist gaps in the field scenarios. In contrast, our model training does not require rigorous construction of missing scenes and only needs one training to complete interpolation of any missing form while maintaining advantages in performance.
§.§ Uncertainty Quantification
Although various interpolation methods based on deep learning have accomplished promising results in the aforementioned publications, uncertainty quantification of the prediction is still absent subjecting to the fixed inference mode. However, providing measures of uncertainty for the predictions over or under confidence is important to improve the application security and avoid the cost of an error. The uncertainty in deep neural networks is divided into the reducible model uncertainty (also systemic or epistemic uncertainty) and irreducible data uncertainty (also statistical or aleatoric
uncertainty) <cit.>. The model uncertainty
is caused by inadequate models and unsuitable learning patterns, and data uncertainty is an inherent characteristic of data and cannot be reduced or eliminated by improving the subsequent model.
There are multiple random sampling operations in our SeisDDIMR model as stated in Algorithm <ref>, thus we adopt the approach deriving from uncertainty ensemble methods to capture the total uncertainty by calculating the standard deviation of the interpolation results obtained after multiple repetitions of Algorithm <ref>. For a sample x, the uncertainty is computed as
1/n∑_i=1^n(x̂_i-μ̂_i)^2,
where μ̂_i = 1/n∑_i=1^nx̂_i, x̂_i is the interpolation result of single test, and n is the repetition test number. Fig. <ref>-<ref> visualize the uncertainty in the interpolation results of random, consecutive, and multiple missing traces, respectively. The average interpolation results and average residual 1/n∑_i=1^n(x̂_i-x_gt) are also exhibited to provide an intuitive reference.
It seems that unreliable reconstruction results are more likely to occur in the missing areas with patch edges and strong lateral amplitude variations, due to limited information and highly curved events. Besides, areas with high interpolation uncertainty also acquire large residuals.
§ ABLATION STUDY
In this section, we will conduct a series of ablation studies on the key components and hyperparameters from three aspects including the MHSA module, seismic DDPM, and implicit interpolation with resampling strategy.
§.§ MHSA Module
We carry out our model training under different settings in the MHSA module with the total iteration number N = 300,000. The ablation study focuses on the location of MHSA in the network and the number of attention heads. Tab. <ref> lists the interpolation results on the validation set of SEG C3 data with multiple missing traces, where 32, 16, and 8 represent the resolution of the feature map in the noise matching network, respectively, meaning that the MHSA module is placed on the corresponding layer. We list the optimal configuration and its result on the top row. The following several rows show the results with one of the settings changed. It is evident that the best performance is achieved with the settings of attention head number N_head= 4 and attention location = 16, 8.
§.§ Seismic DDPM
The training of the Seismic DDPM is implemented by the process described in Algorithm <ref>. We selected three key components, i.e., diffusion steps T, loss function,
and noise schedule, to validate the superiority of the adopted configuration. Seismic DDPM is trained on the SEG C3 dataset under different settings with the total iteration number N = 300,000, respectively. Tab. <ref> yields the interpolation results on the SEG C3 validation dataset with multiple missing traces. First, the number of diffusion steps T has a significant impact on the diffusion speed of our model. Increasing T refines the model, but also causes additional computational burden. Achieving a balance between computational efficiency and model performance requires a compromise configuration of the diffusion steps. Second, Tab. <ref> indicates that better interpolation results can be achieved by allowing the noise matching network to learn the noise variance σ_t under the hybrid loss L_hybrid. Finally,
training seismic DDPM with different noise schedules indicates that using a linear schedule suffers from significant performance degradation. This finding supports our decision to adopt the cosine schedule, which has demonstrated better performance.
§.§ Implicit Interpolation with Resampling Strategy
To assess the efficacy of our proposed implicit interpolation and resampling strategy, we execute Algorithm <ref> under various configurations on the validation set of the MAVO dataset with multiple missing traces. The interpolation results are presented in Tab.
<ref>. Comparing the interpolation performance of Algorithm <ref> based on DDPM and DDIM, it can be demonstrated that our proposed implicit interpolation significantly enhances the quality of signal recovery with an increase of 0.749 on SNR and PSNR. It is infeasible to explore all potential scenarios for diffusion sampling steps m, jump length L, and jump height H. Therefore, we aim to identify the most feasible options. To select the most suitable hyperparameters, we conduct algorithm <ref> repeatedly, applying various combinations. First, based on the trained DDPM, DDIM conducts m-step sampling. While increasing the number of sampling steps enhances the diffusion effect, it poses a higher computational burden during testing. The comparison of the performance of DDPM without the resampling strategy (last three rows in Tab. <ref>) reveals that a smaller value of m can be selected without significantly sacrificing performance. Consequently, we eventually adopt m = 100. Second, in regard to the values of L and H, it is easily found that an increase in their values results in an improved interpolation performance. However, this is accompanied by an increase in testing time. After considering both factors, L = 10 and H = 1 are ultimately chosen in our model.
§ CONCLUSION
In this paper, we propose the SeisDDIMR method, which tackles the seismic data interpolation problem with a higher model robustness on various missing data scenarios. SeisDDIMR consists of two processes, including the training of seismic DDPM and implicit conditional interpolation with resampling. Seismic DDPM embeds seismic data into a denoising probability model framework. It achieves full-stage parameter sharing using the noise matching network based on the U-Net structure equipped with MHSA. The cosine noise schedule is introduced to speed up the transition during the high noise stage of seismic data. Implicit conditional interpolation with resampling, serving as the inference process of seismic DDPM, achieves flexible interpolation for different missing data scenarios and missing rates by utilizing the existing traces of the seismic data as a condition.
Interpolation experiments on synthetic and field seismic data with multiple patterns of missing data demonstrate that our SeisDDIMR provides superior quality than existing methods and it also has advantages in robustness. Uncertainty quantification is provided to promote practical applications. In addition, a series of ablation experiments verify the rationality and effectiveness of hyperparameters and the design of key model components. In future studies, we will focus on extending our method to 3D or higher-dimensional seismic data interpolation.
§ ACKNOWLEDGMENT
The authors would like to thank the Sandia National Laboratory and Mobil Oil Company for providing open data sets.
IEEEtran
[
< g r a p h i c s >
]Xiaoli Wei is currently pursuing the Ph.D. degree in statistics with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China.
Her research interests include seismic data reconstruction, deep learning, and uncertainty estimation.
[
< g r a p h i c s >
]Chunxia Zhang received her Ph.D degree in Applied Mathematics from Xi'an Jiaotong University, Xi'an, China, in 2010.
Currently, she is a Professor in School of Mathematics and Statistics at Xi'an Jiaotong University. She has authored and coauthored about 30 journal papers on ensemble learning techniques, nonparametric regression, etc. Her main interests are in the area of ensemble learning, variable selection, and deep learning.
[
< g r a p h i c s >
]Hongtao Wang is currently pursuing the Ph.D. degree in statistics with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China. His research interests include Bayesian statistics and deep learning.
[
< g r a p h i c s >
]Chengli Tan
received the B.S. degree in information
and computing science and the M.S. degree in statistics
from Xian Jiaotong University, Xian, China,
in 2014 and 2017, where he is now pursuing the
Ph.D. degree. His current research interests include
adversarial learning, Bayesian nonparametrics, and
stochastic optimization.
[
< g r a p h i c s >
]Deng Xiong is a phD in Geophysics. He currently works for BGP, and serves as a Senior Engineer in R&D Center. He received his PhD from institute of Geology & Geophysics, Chinese Academy of Sciences in 2008. He is interested in near-surface velocity model building and seismic data reconstruction researches in recent years, and presently focuses on some industrial applications of artificial intelligence methods in seismic deblending and regularizations.
[
< g r a p h i c s >
]Baisong Jiang is currently pursuing the Ph.D. degree in statistics with the School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China. His research interests include seismic data reconstruction, deep learning, and image inpainting.
[
< g r a p h i c s >
]Jiangshe Zhang received the M.S. and Ph.D. degrees in applied mathematics from Xi'an Jiaotong University, Xi'an, China, in 1987 and 1993, respectively, where he is currently a Professor with the Department of Statistics. He has authored and co-authored one monograph and over 80 conference and journal publications on optimization, and remote sensing image processing. His current research interests include Bayesian statistics, global optimization, ensemble learning, and deep learning.
[
< g r a p h i c s >
]Sang-Woon Kim received the ME and the PhD degrees from Yonsei University, Seoul, Korea in 1980 and 1988, respectively, both in Electronic Engineering. In 1989, he joined the Department of Computer Engineering at Myongji University. Since 2019, he has continued his research as an Emeritus Professor there. His research interests include Statistical Pattern Recognition, Machine Learning. He is the author or coauthor of 51 regular papers and 13 books. He is a Life Senior Member of the IEEE and a member of the IEEK.
|
http://arxiv.org/abs/2307.07479v1 | 20230714170042 | Rigorous Wilsonian Renormalization Group for impurity models with a spectral gap | [
"Peter Zalom"
] | cond-mat.str-el | [
"cond-mat.str-el",
"cond-mat.mes-hall"
] |
[email protected]
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, CZ-18200 Praha 8, Czech Republic
The Anderson impurity model (AIM) has long served as a cornerstone in the study of correlated electron systems. While numerical renormalization group (RG) offers great flexibility for metallic reservoirs, it becomes impossible in an unbiased way when a spectral gap Δ opens up in the tunneling density of states. The only known exception is provided by the superconducting bath. In this paper, we lift these limitations by a novel numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches which accumulate at the gap edges. This reveals an unusual double scaling which is a shared behavior by the superconducting and the scalar gapped AIMs. Moreover, it requires a special iterative diagonalization procedure with an alternating scheme for discarding states only every second iteration. The discretization and the diagonalization scheme form together, what we refer to as, the log-gap numerical RG. It is successfully applied to the superconducting and to the scalar gapped AIM. Consequently, it reveals that both models belong to the same RG equivalence class which manifests physically in common singlet-doublet quantum phase transitions accompanied by in-gap bound states of given parities. While superconducting AIM is mainly used for benchmarking of the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum, which was recently indirectly considered in Ref. <cit.>. The gapped AIM is then tackled in a first ever exact numerical RG approach and confirms quantitatively assertions based on models with auxiliary metallic leads <cit.>. Moreover, it reveals that calculations performed in Refs. <cit.> are of strictly approximate nature.
Rigorous Wilsonian Renormalization Group for impurity models with a spectral gap
Peter Zalom
August 12, 2023
================================================================================
§ INTRODUCTION
The Anderson impurity model (AIM), proposed by Philip W. Anderson in 1961 <cit.>, has proven to be a fundamental framework for understanding a wide range of phenomena, including heavy fermions <cit.>, quantum dots (QDs) in Coulomb blockade <cit.> or add-atoms on normal surfaces <cit.>. It describes the behavior of localized electrons interacting with a continuum of itinerant states, encapsulating the complex interplay between localized and delocalized degrees of freedom which is beyond the reach of perturbation theory.
The theoretical efforts it stimulated, culminated by the computer aided solution via the Wilsonian Renormalization Group (RG) <cit.>, which is nowadays commonly known as the Numerical RG (NRG). Quite recently, Wilsonian RG has been also applied to the AIM in the analytic framework of the Functional RG techniques <cit.>. We also stress that several non-perturbative but approximate approaches are known to capture some of the aspects of the AIM <cit.>, while at finite temperatures, the purely numeric approach via the Quantum Monte Carlo (QMC) method is routinely used <cit.>.
The main obstacle the RG targets specifically, lies in the formation of mutually interconnected scales in the impurity systems. For metallic itinerant states, as the cascade approaches the Fermi energy, it eventually gives rise to the Kondo effect. However, just by opening a spectral gap Δ, as present in superconducting <cit.> or semiconducting materials <cit.>, we deprive the system of delocalized degrees of freedom at the Fermi energy which severely impedes the energy cascade.
Intuitively, one expects an ordinary scaling of metallic AIM at temperatures much larger then Δ (all energy scales in this paper are measured in units of bandwidth 2B) to be present at least approximately. Formation of local magnetic moments and even Kondo-like screening is then anticipated for temperatures remaining much larger than the Kondo temperature T_K obtained for Δ=0. Contrary, at temperatures much smaller than Δ no states are available to screen the impurity and dramatic changes are expected. However, so-far our understanding is complete only for the superconducting bath <cit.>. As we show here, this is also due to the fortunate nature of the one-lead problem, where the ordinary scaling of the AIM with constant metallic TDOS is preserved.
For general gapped AIMs, no unbiased methods have so-far reached the region of Δ≫ T. While for QMC the restriction is fundamentally due to the computational resources, in the case of NRG the limitations are of methodological character as problems arise already in the first step when Wannier-like states are constructed for the delocalized electrons. Also, as shown here, the low temperature scaling turns out to be unusual and requires crucial modifications to the diagonalization step of NRG.
Up to date, general gapped AIMs are solved only indirectly or approximately with either a small but non-zero TDOS induced into the gap region by adding a weakly coupled metallic lead to the problem <cit.> or Wilson chains were constructed artificially from the corresponding models at Δ=0 <cit.>. While the first approach utilizes rigorous Wilsonian RG on a modified system, in the second approach, truncated Wilsonian chains of the AIM with constant metallic tunneling density of states (TDOS) are postulated to represent the gapped problem up to the energy scale of Δ. The two-scaled nature of the gapped problem is thus completely missed. Such results, as also clearly demonstrated in this paper, should therefore be understood as an approximate attempt. Nevertheless, qualitative conclusions from all currently available methods consensually confirm an unscreened impurity in the doublet ground state (GS) at T=0 for completely particle-hole symmetric scenario. Quantum phase transitions (QPTs) from a doublet to a singlet GS are observed upon further changing the orbital filling or particle-hole symmetry of the band <cit.>.
While similar QPTs constitute basic phenomena in hosts of superconducting nature, the theoretic connection to the isolator/semiconductor problem was recognized only recently in Ref. <cit.>, where the superconducting AIM (SC-AIM) was mapped onto a model with scalar gapped TDOS. Nevertheless, the two-scaled nature was not deciphered as the resulting gapped AIM with specific TDOS was beyond the available NRG techniques and the system was augmented with a weakly coupled metallic lead. Additional studies of broad classes of gapped TDOS functions in Ref. <cit.> only reaffirmed these findings on a more general footing, but the study also suffered from implementing a weakly coupled metallic lead. Consequently, the existence of singlet-doublet QPTs could be so-far asserted only indirectly with the T=0 behavior of truly gapped systems being only extrapolated.
In this paper, we therefore develop NRG technique for general gapped AIMs with the afore-discussed two-scaled nature. We choose the gapped AIM with constant TDOS (defined in Sec. <ref>) and the SC-AIM (defined in Sec. <ref>) for the demonstrations. General restrictions on band discretizations are first given in Sec. <ref> with a general-purpose discretization proposed in Sec. <ref>. The resulting Wilson chains are then finally revealing a two-scaled behavior which is qualitatively the same for both models despite additional divergences appearing in SC-AIM. The kept or discarded scheme of the standard NRG is then modified accordingly in Sec. <ref> to accommodate for the scaling properties. The well-understood two-lead SC-AIM allows then to validate the newly developed NRG technique in Sec. <ref>. Additionally, new insights into the properties of sub-gap states are obtained here. We then proceed to the gapped AIM with piece-wise constant but generally particle-hole asymmetric TDOS in Sec. <ref> and give a first-ever unbiased NRG solution to the problem. In Sec. <ref>, a detailed comparison with existing results on gapped AIM is performed. Here, we show that NRG methods with auxiliary metallic leads provided very good results but techniques using redefined Wilson chains can only be used for qualitative assessment. In Sec. <ref>, the main results of the paper are subsequently briefly summarized.
§ THEORY
§.§ Gapped Anderson impurity model
For later convenience, we consider general AIM with two leads and a QD characterized by the Coulomb repulsion U and the level energy ε_d, which in the most general case is arbitrary, but here we only consider ε_d=-U/2. However, we stress that the herein presented methods do not suffer from any limitations in this regard. The resulting Hamiltonian is then a sum of
H_d
= ∑_σε_d
d^†_σ
d^_σ
+
U
d^†_↑
d^_↑
d^†_↓
d^_↓,
H_α = ∑_𝐤σ ε_𝐤α
c^†_𝐤ασ
c^_𝐤ασ,
H_T,α = ∑_𝐤σ (V^*_𝐤α
c^†_𝐤ασ
d^_σ
+
V_𝐤α
d^†_σ
c^_𝐤ασ),
where c^†_𝐤ασ (c^_𝐤ασ) creates (annihilates) an electron of spin σ∈{↑↓}, quasi-momentum 𝐤 in lead α which takes values L (left) or R (right lead). In analogy, d^†_σ (d^_σ) creates (annihilates) a dot electron of spin σ, ε_𝐤α is an unspecified dispersion relation.
The QD hybridizes with the leads via V_𝐤α, which we leave unspecified and instead prefer to extract the tunneling self-energy via
Σ(ω^+)
=
∑_α∈{L,R}𝐤
V^*_𝐤α(
ω^+ - ε_𝐤α)^-1
V_𝐤α.
and then demand its imaginary part (the tunneling TDOS) to a desired form. For constant but gapped TDOS with potentially particle-hole asymmetric band, we thus require
Σ(ω^+)
=
0
for |ω|<Δ,
[ 1-𝒜 sgn(ω) ] Γ_S
for |ω|≥Δ,
where 𝒜 governs the particle-hole asymmetry of the gapped band. In detail, for 𝒜 = 0 the TDOS remains symmetric as in Refs. <cit.> but for any 𝒜≠ 0 different weights to the hole and electronic parts are ascribed.
§.§ SC-AIM in the scalar representation
An essential clue toward an unified theory of gapped AIMs was provided in Ref. <cit.> by the herein employed mapping procedure of SC-AIM. To briefly summarize the approach, we define SC-AIM with one QD and two superconducting leads. Its Hamiltonian is given as a sum of the QD Hamiltonian H_d according to (<ref>), two tunneling Hamiltonians H_T,α in the form of (<ref>) and two Hamiltonians H_BCS,α describing the left (α=L) and right (α=R) lead via the Bardeen-Cooper-Schrieffer (BCS) theory which gives
H_BCS = ∑_𝐤ασ ε_𝐤
c^†_𝐤ασ
c^_𝐤ασ
- Δ∑_𝐤(e^iφ_α
c^†_𝐤α↑
c^†_-𝐤α↓
+
H.c.),
where Δ is the superconducting gap and φ_α are superconducting order parameters in the left and right lead. For this paper, we choose symmetric hybridizations Γ_L and Γ_R and the gauge φ_L=-φ_R=φ/2 with φ being the phase difference across the SC-AIM. There is however no loss of generality due to the relation between symmetric and asymmetric coupling scenario shown in Ref. <cit.>. It is now customary to employ the Nambu spinors
C^†_α 𝐤 = (
c^†_α 𝐤↑,
c^_α -𝐤↓),
D^† = (
d^†_↑,
d^_↓).
Under the standard BCS assumption of ε_𝐤α=ε_-𝐤α and with a convenient choice of real tunnel couplings V_α𝐤=V^*_α𝐤=V_α-𝐤, the Hamiltonians (<ref>) and (<ref>), apart from possible unimportant constant energy shifts, become
H_α = ∑_𝐤
C_α 𝐤^†𝔼^_α 𝐤
C_α 𝐤^,
H_T,α = ∑_𝐤(
D^†𝕍^_α 𝐤
C_α𝐤^
+
C_α𝐤^†𝕍^_α 𝐤
D^)
with
𝔼_α𝐤 =
-Δ_α C_ασ_x
+Δ_α S_ασ_y
+ ε_𝐤ασ_z,
𝕍_α𝐤 =
V_α𝐤 σ_z,
where σ_i, i∈{x,y,x}, are the Pauli matrices
and C_α≡cosφ_α, S_α≡sinφ_α. The blackboard bold typeface distinguishes matrices from scalars. The corresponding TDOS is then given as
Σ^D(ω^+)
=
∑_α∈{L,R}𝐤𝕍_α𝐤(
ω^+ 1 - 𝔼_α𝐤)^-1𝕍_α𝐤
which yields
Σ^D(ω^+)
=
Γ_S
[
ω1
+ Δcos(φ/2) σ_x
]
F(ω^+)
with
F(ω^+)
≡1/π∫_-B^Bdε/ω^2 - Δ^2 - ε^2 + iη sgn(ω)
=1/π√((ω+ iη)^2 - Δ^2 )ln√((ω+ iη)^2 - Δ^2)+B/√((ω+ iη)^2 - Δ^2)-B.
Taking the η→ 0 limit, we arrive at
F(ω^+)
=
-2/π√(Δ^2-ω^2)arctan(B/√(Δ^2-ω^2)),
for |ω|<Δ
-i sgn(ω)/√(ω^2-Δ^2)
+
ln( B+√(ω^2-Δ^2)/B-√(ω^2-Δ^2))/π√(ω^2-Δ^2),
for Δ<|ω|<B.
The resulting Σ^D(ω^+) has thus a non-zero imaginary part only for |Δ|<ω while all effects of the finite-sized band appear in its real part. However, once the limit B→∞ is taken the real
part out of the gap vanishes too.
To bring the SC-AIM model into a form of a general gapped AIM we perform the unitary transformation 𝕋
𝕋 D
≡
W,
D^†𝕋^†≡
W^†,
where
𝕋
=
1/√(2)(
σ_x - σ_z
)
and
W ^†
=
(
w^_↑
w^†_↓).
The resulting TDOS in basis of W spinors is then
Σ^W(ω^+)
=
Γ_S
[
ω
-
Δcos(φ/2)
]
F(ω^+).
Taking the imaginary part of (<ref>) yields the TDOS for SC-AIM in the scalar representation with typical BCS divergences appearing at the edges while φ modulates the resulting particle-hole asymmetry in the basis of the w fields as shown in Fig. <ref>.
Finally, at half-filling the H_d Hamiltonian is form invariant against the transformation 𝕋 and reads
H_d
= ∑_σε_d
w^†_σ
w^_σ
+
U
w^†_↑
w^_↑
w^†_↓
w^_↓.
§ NRG APPROACH FOR GAPPED TDOS FUNCTIONS
The here presented novel NRG approach generalizes the seminal work <cit.> to the gapped TDOS. We thus start in Sec. <ref> by generalizing the proof given in Ref. <cit.> to the gapped TDOS scenario. While the mathematical proof is relative straightforward, it yields only limited specifications on the discretization procedure itself. Nevertheless, studying discretized self-energy as in Sec. <ref>, allows for a preliminary justification to the here employed log-gap discretization. The resulting Wilson chains incorporate two separate scales in regimes of T ≫Δ and T ≪Δ respectively which require significant modifications to the kept/discarded state scheme of the NRG diagonalizations as discussed in Sec. <ref>. We stress that z-averaging techniques can be potentially incorporated into the presented approach to improve especially the spectral functions. However, in the present paper we focus on subgap spectroscopy and the validation of the developed NRG approach. We therefore set z=1 throughout this work to streamline the presentation.
§.§ Gapped TDOS in approach of R. Bulla
Let us assume that an impurity problem with gapped TDOS is present and its tunneling self-energy is given as
Σ(z)
=
∫_I_Δ∑_μ
dω
V^2(ω) ρ(ω) G_μ(ω)
with G_μ(ω) being the Green's function of the lead with a gapped DOS ρ(ω), while I_Δ denotes an integration interval running from -B to B with a spectral gap of width 2Δ that is centered at the Fermi energy of 0. An example of such a self-energy is provided, for example, by Eq. (<ref>). Our aim is now to construct a corresponding one-channel Hamiltonian
H
-
H_d
= ∫_I_Δ∑_μ
dε
g(ε)
a^†_μ
a_μ
+ ∫_I_Δ∑_μ
dε
h(ε)
d^†_μ
a_μ +H.c.
with g(ε) and h(ε) to be specified in such a way that the tunneling self-energy defined by (<ref>) is identical to (<ref>). We thus first change the integration variable from ε to g and solve for the tunneling self-energy of (<ref>). We obtain
Σ(z)
=
∫_I_Δ∑_μ
dg
h^2[ε(g)] dε/dg
G_μ(g).
Comparing (<ref>) and (<ref>) implies the following relation between h(ε) and g(ε):
V^2(x) ρ(x)
=
h^2[ε(x)] dε/dx,
which holds for every x ∈ I_Δ and has exactly the form known from Ralf Bulla's seminal work <cit.>. This is of crucial importance for gapped TDOS problems as the discretization step can be performed without the introduction of any approximations. We may simply choose, for example, a piecewise constant h(ε) on intervals I^±_n, where + denotes intervals in positive and - in negative frequency domain. To reconstruct the TDOS we can then transfer all required details onto g(ε) via condition (<ref>).
However, Eq. (<ref>) gives us no prescription on how to choose the discretization intervals I^±_n, it just requires them to reside within the interval I_Δ. This leaves us with two main options so n might be either a finite or infinite set of integer numbers. Since the tridiagonalization equations are of the same form as for the ordinary NRG, as applied for example to AIM with metallic bath, a finite number of discretization intervals would leave us with a finite Wilson chain, which will inevitably have some smallest built-in energy scale. Moreover, the concept of approaching a low-temperature fixed point for Wilsonian RG iterations will make no sense for at all.
To avoid such problems, one simply takes the iterative structure of the tridiagonalization equations into account by defining the initial discretization intervals I_0^± at the band edges ± B. As n is increased the intervals patch the remainder of I_Δ and get smaller towards the gap edges at ±Δ. In formal agreement with Ref. <cit.>, we then define the following quantities
ξ_n^+/-
=
∫_I^+/-_n dx x Γ(x)/∫_I^+/-_n dx Γ(x),
( γ_n^+/-)^2
=
∫_I^+/-_n dx Γ(x),
where Γ(x) is the imaginary part of (<ref>). These enter then subsequently the tridiagonalization equations (28)-(31) given in Ref. <cit.>. In detail, each interval I_n^+/- gives rise to operators a^†_n,σ,p for I_n^+ and b^†_n,σ,p for I_n^-. These are connected to a set of orthonormal functions indexed by ±, n and also p, where the last index takes all integer values. Inserting these into the corresponding Hamiltonians and taking piecewise constant approximation of the hybridization term in the energy representation allows only p=0 components, so we can completely drop the p indices and obtain the corresponding discretized Hamiltonian as
H
=
H_imp + ∑_n,σ(
ξ_n^+
a^†_n,σ
a^_n,σ
+
ξ_n^-
b^†_n,σ
b^_n,σ)
+ ∑_n
(
γ_n^+
d^†
a^_n,σ
+
γ_n^-
d^†
b^_n,σ)
+ ∑_n
(
γ_n^+
a^†_n,σ
d^
+
γ_n^-
b^†_n,σ
d^).
§.§ Log-gap discretization for gapped AIMs
To guide us in the choice of the precise form of the discretization intervals I^±_n we solve now for the discretized tunneling self-energy Σ_disc(z) of (<ref>). It reads
Σ_disc(z)
=
∑_n
-( γ_n^+ )^2/ξ_n^+ - z
+
( γ_n^- )^2/-ξ_n^- + z.
The two branch cuts of the continuous Σ(z) that follow the real axis and terminate at ±Δ have thus been replaced by simple isolated poles at positions ξ_n^+>0 and ξ_n^-<0. The coefficients (<ref>) have thus a straightforward meaning of the positions of the poles, while (<ref>) defines the corresponding weights in the discretized TDOS.
Analogous results are present also in other commonly used approaches to NRG. For example, in AIM with metallic reservoir, the continuous self-energy with branch cut over the entire real axis gives rise under the ordinary logarithmic discretization to simple poles that accumulate logarithmically around the Fermi energy. From the perspective of gapped systems, they accumulate exactly where the two branch cuts over positive and negative part of the real axis merge as Δ = 0. Similar pattern is also observed for the superconducting bath, which is the only system with a spectral gap that is currently rigorously treated by NRG and thus closest to the problem advanced here. Once again, using standard two-channel NRG techniques for SC-AIM the discretized tunneling self-energy Σ_disc(z) is obtained in the form of (<ref>) with simple poles emerging along the two branch cuts. Significantly, they are known to accumulate logarithmically at the terminal points z=±Δ of the branch cuts.
Our aim now is therefore to construct a discretization that is not only in line with the requirements of the previous section, but reproduces additionally the logarithmic accumulation of simple poles in the corresponding tunneling self-energy Σ_disc(z) at z=±Δ where the branch cuts of the continuous self-energy terminate. To this end we define the intervals I^±_n via their end points x^±_n as
x_n^±
=
±[
Δ + (1-Δ) Λ^-n],
where Λ is the usual discretization parameter with intervals I^±_n obviously aggregating at the gap edges in a logarithmic way (bandwidth set as 2B ≡ 2). Consequently, we refer to (<ref>) as the log-gap discretization. We stress that it was already considered in Ref. <cit.>, however, the scaling properties of the resulting Wilson chains have been completely missed and the employed standard kept/discarded scheme must have resulted in ill converging solutions at low temperatures, which the authors did not report in the manuscript. As shown in the next section, further modifications to the NRG approach are required to resolve this problem.
Let us, now calculate the pole structure of the discretized self-energy. First, we consider the gapped Anderson model with TDOS (<ref>). We note that the discretized self-energy becomes of form (<ref>) with poles ξ_n^+/- and weights γ_n^+/- obeying
ξ_n^+/- = ±[
Δ + (1-Δ) (1 + Λ)/2Λ^-1 - n],
(
γ_n^+/-)^2
=
(1 ±𝒜) (1-Δ) (Λ-1) Λ^-1 - n.
Setting Δ=𝒜=0, one recovers the standard case of the AIM with constant metallic TDOS, where poles approach the gap edges in a logarithmic fashion. The same holds true also for Δ≠0 at arbitrary 𝒜, only these time the poles are logarithmically approaching the gap edges as in the standard NRG approach to SC-AIM. The log-gap discretization is thus increasingly sensitive to the states which get closer to the Fermi energy, albeit due to the presence of the gap, it can never be reached. Notably, poles (<ref>) are placed symmetrically around the Fermi energy regardless of the values of 𝒜 and/or Δ. Consequently, the particle-hole asymmetry is completely encoded only by the weights γ_n^+/-.
Since the constant gapped TDOS is structureless at the gap edges, let us also corroborate the SC-AIM case in its scalar representation as shown in Fig. <ref>. The BCS-like divergences on the gap edges and a complicated particle-hole asymmetry of (<ref>) leads to the following expression
ξ_n^+/- = ±[
Δ + f^±(φ,Δ,Λ) Λ^-1 - n]
where the prefactors f^±(φ,Δ,Λ) are functions of Δ and Λ and are generally not particle-hole symmetric, i. e. f^+(φ,Δ,Λ) ≠ f^-(φ,Δ,Λ), unless φ=π. Consequently, the overall particle-hole asymmetry of (<ref>) is distributed over poles and weights of the discretized self-energy, causing an effect that needs to be addressed later. Nevertheless, the logarithmic behavior toward the terminal points of the branch cuts is still preserved. Nevertheless, compared to the standard SC-AIM solution the poles are not distributed in a particle-hole asymmetric fashion, which is an effect that needs to be considered later as a potential source of differences.
Next, we will feed the poles and weights into the tridiagonalization equations to obtain the corresponding Wilson chains. Due to the corroborated differences to the only known standard NRG solution we choose the SC-AIM case for these demonstrations. We select Γ=Δ, U=3Δ, ε_d=-U/2 and vary φ. The resulting parameters of Wilson chains for selected values of φ are then shown in the first row of Figs. <ref>(a)-(d), while second and third row demonstrate high- and low-temperature scalings of the hoppings respectively.
As shown in the first row of Figs. <ref>(a)-(d), at large n even hoppings become approximately Δsinφ, while odd hoppings become exponentially suppressed. To decipher the various scalings present in the system, let us look at the second and the third row of Figs. <ref>(a)-(d). Obviously, there is always an initial stage where hoppings scale in an usual Λ^-n/2 way, which however survives down to n→∞ only for φ=0. Otherwise, the scaling is interrupted by a crossover region and only odd hoppings scale exponentially with Λ^-n law.
The behavior of the on-site energies ϵ_n on the other hand is more straightforward as they vary between the approximate values of ±Δ cos(φ). The alternating pattern holds approximately for all model parameters when log-gap NRG approach is applied with only particle-hole symmetric TDOS leading to the vanishing of all on-site energies.
While the presented cases were calculated for SC-AIM, we stress that the alternating feature of the on-site energies and the two-scaled nature of hoppings are observed also for the gapped AIM model with constant TDOS function. However, here the second stage scaling involves odd hoppings to follow the Λ^-n/2 law. So clearly, the presence of the additional BCS-like divergencies in the TDOS of the SC-AIM case modifies the power law of the odd hoppings in the low-temperature regime.
§ ITERATIVE NRG DIAGONALIZATION OF TWO-SCALED WILSON CHAINS
The observed two-scaled nature of the hoppings is an essential feature which needs to be adequately considered during the iterative NRG diagonalization procedure. To begin with, let us however first discuss the special case of φ=0 which exhibits only the ordinary Λ^-n/2 scaling for all hoppings and was first described in the seminal papers on NRG <cit.>. Iterative addition of the sites of the Wilson chains leads to an exponential increase of the total Hilbert space which is dealt with by a truncation strategy as illustrated in Fig. <ref>(a). Thus, already after few NRG iterations high energy states (multiplets) are being discarded systematically with a more or less constant number kept until the algorithm is stopped.
Such a diagonalization procedure works only in conjunction with the Λ^-n/2 scaling of all hoppings, which ensures energy scale separation (high and low eigenenergy states do not mix as the chain grows). While originally established for impurities in metallic baths, the same scaling and consequently energy separation emerges also for one-lead SC-AIM problems <cit.>. Quite surprisingly and without an apparent justification, it was also established for the two-lead SC-AIM, where the addition of complex phases mixes both leads together <cit.>.
In our case, such mixing is also observed once φ≠ 0 and leads to the second stage of scaling, where even hoppings become of the order Δcosφ and odd hopping scale as Λ^-n. In this second stage, we can thus naturally reinterpret the Wilson chain as being composed of serial double QDs interconnected by the exponentially decreasing hopping. If the Hilbert space is never truncated after the odd site is added, splittings of eigenenergies by factor proportional to Λ^-n have to occur at even iterations and energy scale separation can be exploited.
As demonstrated graphically in Fig. <ref>(b), such a kept/discarded strategy is perfectly tailored for the low energy scaling, but it would be difficult to be turned on only after one emerges from the initial scaling. Fortunately, when applied also to the high temperature portion of the Wilson chain, the novel kept/discarded strategy just keeps superfluous high energy states (multiplets) every odd iteration. At even iterations, they give then rise to states which are immediately discarded, so in principle, the novel kept/discarded strategy is just inefficient in the Λ^-n/2 scaling sector but does not break the energy scale separation as demonstrated in the fourth row of Figs. <ref>(a) by the φ=0 case where second stage scaling is not present at all. Here, eigenenergies at odd NRG iterations E_odd (no rescaling applied), clearly converge as n is increased up to 60. Modifications to the kept/discarded strategy in the low-temperature sector are thus not necessary and can be potentially reverted to speed up the numerics, which however requires careful investigations. Such refinements are left for future developments.
For al other values of φ the second stage scaling occurs after a crossover region, but as shown in the fourth row of Figs. <ref>(b)-(d) the here developed kept/discarded strategy indeed ensures energy scale separation. We also stress, that at every NRG iteration new low energy states resurface around the gap edge (not including the subgap states) and get exponentially close to the region right above the gap edge as n is increased. Their appearance does not however alter the other higher lying eigenvalues, which are already converged. This is, of course, a feature well-known also from the standard NRG solution of SC-AIM as discussed in the Appendix of Ref. <cit.>.
The new strategy of keeping or discarding states comes with the price of increased CPU and memory requirements over the traditional scheme of Fig. <ref>(a) since the intermediate Hilbert space used for matrix diagonalizations grows by factor of up to 16, which thus resembles the ordinary two-channel NRG calculations. However, in the scalar basis used here, charge and pseudo-spin quantum numbers label the states (multiplets) and the increased symmetry is crucial for speeding up the diagonalizations when the log-gap NRG approach is employed. Consequently, the here proposed NRG scheme for SC-AIM remains feasible at any modern desktop also when ε≠ -U/2 and even in the presence of non-zero magnetic field, while standard two-channel NRG solution reaches borders of its practical feasibility.
Apparently, the same kept/discarded strategy is also required for the AIM with constant but gapped TDOS, albeit the odd hoppings of the second stage scaling follow the Λ^-n/2 law. Still the even hoppings alternate between positive and negative values. Consequently, the low-temperature portion of the Wilson chain has once again the character of a tight-binding chain composed of serial double QDs connected by exponentially decreasing hoppings. The application of the log-gap NRG with the modified kept/discarded strategy then finally also confirms the working energy scale separation.
While the kept/discarded scheme of Fig. <ref>(b) is crucial for performing unbiased Wilsonian RG calculations for the here discussed models, we would like to briefly explore the outcome when an ordinary kept/discarded scheme according to Fig. <ref>(a) is employed even for Wilson chains of the two-scaled nature. The energy scale separation is then, of course, not ensured and the eigenspectrum of energies become corrupted for φ≠ 0 or φ≠π as demonstrated in Appendix <ref>. Nevertheless, problems occur initially only in its high energy part and the sub-gap spectrum remains largely intact, so intermediate NRG iterations can be used to determine sub-gap properties with only few percent of relative error when compared to the rigorous Wilsonian approach developed here. We will call this approach the approximate log-gap NRG, albeit we stress that it is actually a misnomer as energy scale separation is broken. It should thus strictly be used only for fast scannings and, at least, a portion of the data should always be validated against rigorous methods. Additionally, supragap spectral functions or thermodynamic properties depend upon high-energy states and will thus be susceptible to much larger systematic errors. In the main part of this paper, we will therefore use only the here developed rigorous log-gap NRG and will briefly discuss the veracity and possibilities of its fast but approximate version in Appendix <ref>.
§ RESULTS
§.§ SC-AIM
In the previous sections, a rigorous Wilsonian NRG scheme has been developed to tackle arbitrary problems with QD immersed in a gapped TDOS. Before solving the gapped Anderson model, as an example, we first validate the log-gap NRG against the well-established two-channel standard solution of SC-AIM by using its scalar representation derived in Ref. <cit.>. We will also demonstrate its effectivity in regard to the CPU time by obtaining the full phase diagram and we will exploit the quantum numbers for tracking of in-gap excitations as they leave into the continuum, which was currently speculated upon in Ref. <cit.>.
Starting by obtaining the complete phase diagram for SC-AIM with QD at half-filling in Fig. <ref>, we notice that the resulting singlet-doublet QPT lines divide the Γ/Δ-UΔ plane into the upper doublet GS portion and the lower singlet GS part. Four selected values of φ are shown with φ=π leading to the exclusive doublet GS phase for all parameters as long as ε_d=-U/2. Due to the employed symmetries in the log-gap NRG solution of SC-AIM, the calculations come only at the fraction of the CPU time required by the ordinary NRG and are feasible on any modern desktop. We have thus pre-calculated the data in Fig. <ref> via the log-gap approach and used these as an ansatz for the standard NRG calculations.
The resulting parity transition lines obtained using the log-gap NRG approach with Λ=2 (solid lines) are then compared to the standard two-channel NRG calculations at Λ=4 (points). While only a small and steady difference of ≈ 3% between both methods is observed, we select four cases A, B, C and D at various ratios of U/Γ for a detailed study. The resulting sub-gap spectroscopy including one- and two-particle excitations is then presented in Fig. <ref>. Once again the log-gap NRG approach with Λ=2 (solid and dashed lines for one- and two-particle excitations respectively) and the standard two-channel NRG with Λ=4 (open circles) are generally in a very good agreement, but the difference systematically grows as φ is decreased. This hints toward an increasing importance of the asymmetric pole structure of the self-energy due to the log-gap discretization, while the effect of different Λ appears less significant.
In case A, the ratio U/Γ = 20 drives the system into the doublet GS phase for all possible values of φ. An opposite scenario at small ratio of U/Γ, case D, is then conversely marked by singlet GS phase dominating almost the entire phase evolution. The doublet GS phase is however always present, albeit only in a small region around φ≈π due to the phenomenon of the doublet chimney as explained in Ref. <cit.>. Cases B and C have on the other hand moderate ratios U/Γ, which results in a comparably similar phase regions of singlet and doublet GS phase.
Additionally, the newly developed NRG technique assigns charge and pseudo-spin numbers to the NRG eigenstates and allows thus direct tracking of the in-gap excitations as they cross into the continuum region in cases B, C and D. Only case A is an exception, since the continuous part of the spectrum and the state corresponding to the excitation of interest share similar quantum numbers. The directly observed values have then been used to cross-identify the most likely candidates from the standard two-channel NRG calculations. Our observations are in good accord with indirect observations in Ref. <cit.>.
§.§ Gapped AIM with constant TDOS
Successful benchmarking of the log-gap NRG approach against the standard NRG results allows us to proceed with the gapped AIM problem with constant TDOS and give its full unbiased NRG solution. In this section, we present the observed physical phenomena and emphasize their origin in the shared RG-equivalence class with the SC-AIM. Detailed comparisons to previous attempts on gapped AIM with auxiliary metallic leads or methods with redefined Wilson chains are presented separately in Sec. <ref>. Here, we only state that a very convincing agreement with NRG solutions with auxiliary metallic lead is observed, while the results of Refs. <cit.> are proven as highly approximate.
For now, let us start with the phase diagram of the problem as shown in Fig. <ref>. Using the log-gap discretization, the resulting Wilson chains for gapped AIMs are of one channel nature and can be diagonalized in a standard iterative approach of NRG. The resulting GS parity can be read off directly, unlike in NRG methods with auxiliary metallic leads. The hypothesis about the GS and in-gap parities of Refs. <cit.> are nevertheless finally directly confirmed. Consequently, GS transition lines divide the U/Δ-Γ/Δ plane into an upper doublet and lower singlet portion. Overall, the similarity to the phase diagram of SC-AIM in Fig. <ref> is exemplary.
The particle-hole asymmetry 𝒜 of the gapped band plays thus an analogous role as φ in SC-AIM in its scalar representation. In detail, 𝒜=0 and φ=π correspond to each other and represent the particle-hole symmetric choices for the corresponding models. Consequently, at 𝒜=0 the whole U/Δ-Γ/Δ plane supports only the doublet GS, giving rise to the phenomenon of doublet chimney as discussed in Ref. <cit.> specifically for SC-AIM. Nevertheless, inducing even a small particle-hole asymmetry 𝒜 causes a QPT into the singlet GS at large hybridizations Γ/Δ. The larger the particle-hole asymmetry 𝒜 becomes, the more extended the singlet phase space becomes. All of these findings point then to a deeper connection between SC-AIM and other gapped AIMs as hypothesized already in Ref. <cit.>. In the present approach, the connection is already evident after the discretization step is performed, as the two-scaled Wilson chain emerges for both models.
To further deepen these findings, we select four cases A, B, C and D with exactly the same U/Δ and Γ/Δ as in the SC-AIM case. We then calculate the evolution of the in-gap excitations with respect to the particle-hole asymmetry parameter 𝒜 as shown in Fig. <ref> and use solid (dashed) lines to track the one-particle (two-particle) excitations. We stress that unlike in SC-AIM, in-gap excitations do not come in pairs except of 𝒜=0, but this is a trivial consequence of gapped AIM being directly defined in the basis of (generally) particle-hole asymmetric fermions. Anyways, to make the correspondence to the SC-AIM problem more obvious we also add excitations which appear for the model upon 𝒜→ -𝒜, but we stress that they never appear simultaneously.
In detail, the large Coulomb interaction in case A expels the singlet in-gap states to the edge of the subgap region and ensures doublet GS for all values of 𝒜. At 𝒜=0, the underlying symmetry causes both singlet excited states to overlap, which appears then as two symmetrically placed one-particle excitations in the sub-gap spectral function. At 𝒜≠ 0, one of these singlets is pinned to the gap edge and starts moving closer to the Fermi energy only as 𝒜≈ 0.5. The other singlet crosses then into the continuum part of the spectrum. The only difference to the SC-AIM case lies thus in a different curvature of the observed trajectories, which is of quantitative nature and clearly depends on the shape of the TDOS. The case D shows then in analogy an contrasting outcome to case A. Due to the large hybridization Γ, already a very small particle-hole asymmetry of 𝒜≈ 0.01 suffices to induce a doublet-singlet QPT. Consequently, singlet phase dominates the in-gap states.
The outcome in cases B and C is once again in close analogy to the SC-AIM results with the initial doublet GS being present up to moderate values of 𝒜≈ 0.25. The first excited state at 𝒜=0 is of doublet nature and appears in hole as well as particle part of the spectral function. Increasing then 𝒜 causes both in-gap peaks of the spectral function to move in the same direction towards the gap edge. Together with their symmetric counterparts (thin gray lines), which physically appear together only in SC-AIM, we observe a typical pattern. Once again, up to the missing symmetrization both models behave qualitatively in an analogous way. These findings are thus consistent with the hypothesis of Ref. <cit.>, where both models are stipulated to belong into the same RG universality class.
§.§ Comparison with older results for gapped AIM
In the literature, there are in principle two alternative NRG-based proposals for solving gapped AIM problems. One relies on Wilson chains that follow from corresponding AIM cases with Δ=0 <cit.>. In the other approach, standard NRG methods are used but the gapped system is augmented by a weakly coupled metallic lead as used in Refs. <cit.>.
The main idea for the former approach lies in first constructing a Wilson from the Δ=0 case via the standard logarithmic discretization. Its length is set to M so that the last hopping coefficient equals approximately Δ. The resulting tight-binding chain is then solved by an usual NRG iterative diagonalization and the results are considered to represent the gapped case. In other words, closing the gap just means solving an ordinary Anderson model at ever-growing Wilson chain.
Apparently several problems do arise with such a construction. Taking any finite Δ, we note that the logarithmic discretization (as any other running up to the Fermi energy) inevitably crosses into the gap region at some given N and the Wilson chain should terminate here. But the characteristic energy scale of the smallest hopping is then Λ^-N/2 which is approximately √(Δ) and thus much larger then the desired energy scale, or in other words, the previously chosen length M is actually equal to 2N+1. The problem with such an approach lies therefore not in truly the finiteness of the Wilson chain but in its generalization beyond the length N. In the end, it is thus questionable why a Wilson chain of the ungapped problem with one particular length 2N+1 (which is mathematically not even permissible) should represent the gapped scenario.
On the other hand, the other commonly used method uses only standard NRG techniques and modifies only the model definition by adding a weakly coupled metallic lead fill the gap region with a small but non-zero TDOS. One then calculates a series of spectral functions at decreasing couplings Γ_M of the added metallic lead and analyzes the sub-gap peaks which are present as broadened analogs of the sharp in-gap peaks.
The Γ_M→ 0 limit is then deduced from such a series, which is is time consuming and inaccurate as a non-zero numeric threshold on smallest possible Γ_M exists. Moreover, there is no guarantee for the Γ_M=0 case to adiabatically connect to this series with supporting evidence being only circumstantial for the SC-AIM problem <cit.>. Additionally, this indirect method is not able to give indisputable conclusions about the quantum numbers of the GSs and in-gap states in question. Taking together, both existing methods have to be treated with caution and do not represent a rigorous Wilsonian RG approach to the fully gapped problem.
To proceed, we will thus subject the two approaches and the here developed method to a scrutiny. To this end, we have selected the three cases from Fig. 2 in Ref. <cit.> which correspond to the gapped Anderson impurity model with constant TDOS with parameters according to Table. <ref>. To avoid any inconsistencies in the application of the approach used by the authors, the positions of the in-gap peaks have been read off as indicated in Table. <ref>. Second, using the method of auxiliary metallic lead we have recalculated the spectral functions and determined the peak positions under the consideration. The resulting values are also shown in Table. <ref>. Note that closing the gap at fixed U and Γ causes the in gap states to approach the Fermi energy, where a Kondo peak develops due to the auxiliary metallic lead. The accurate reading of the in-gap peak position becomes therefore increasingly complicated. Nevertheless, such a problem arises mainly for the case of Δ=7.05 × 10^-6B, while all in-gap positions determined in Ref. <cit.> are systematically larger than with the auxiliary lead by a factor of up to 2.
Finally, all three cases can be solved using the here presented log-gap NRG. As already predicted in Fig. <ref>, in-gap peaks are located well inside the gap region, which is sharply separated from the continuous part. The extracted peak positions are then presented in Table. <ref>. Clearly, their correspondence to the method of auxiliary metallic lead is good, while result from Ref. <cit.> are off by a factor of up to 2. The still obvious differences between log-gap NRG and the method using an auxiliary metallic lead grow as Δ is decreased. Here, the method of auxiliary metallic lead becomes increasingly difficult as at the smallest numerically possible value of the coupling Γ_M the in-gap peak was still moving toward the gap edge and thus possibly towards the position determined by the log-gap NRG.
Moreover, in Fig. 2 of Ref. <cit.> one observes the continuous part of the spectral functions to spill over the gap edges into the gap region. We thus conclude that the method used in Ref. <cit.> is highly approximate and inconsistent even on its own. Contrary, in Ref. <cit.> in-gap peaks at 𝒜=0 have always appeared inside of a well-defined gap which is also confirmed after recalculating the three cases of Ref. <cit.> as demonstrated in Fig. <ref>. The same holds true for the results from the log-gap NRG.
We also stress that the mutual correspondence of the log-gap NRG and the method of auxiliary metallic lead is not confined to the three selected 𝒜=0 cases as evident from the result presented in Fig. <ref> with a discussion about the comparison with other methods being postponed up to this moment. Clearly, there are differences perceptible even with a bare eye, but one has to bear in mind that the auxiliary metallic lead method is inherently plagued by several systematic and numeric problems. First, a limit of Γ_M → 0 is hard to reach as a smallest value of numerically allowed Γ_M still broadens the in-gap peaks and reading off the in-gap positions is then associated with errors. Additionally, in-gap peaks close to the gap edges are overlapping with the gap edges. Consequently, most of the 𝒜 point for the case A have not even been accessible with the method of auxiliary metallic lead.
Nevertheless, taking together, with the exception of the case A in Fig. <ref>, all data points are in good agreement with the here presented log-gap NRG approach and we can thus safely conclude, that the generalized Wilson chains according to Refs. <cit.> are not well representing the actual AIM with constant but gapped TDOS but the remaining two approaches do so with the log-gap NRG being, of course, superior not only as the method to solve the truly gapped scenario, but also in terms of practical usability and CPU requirements.
§ CONCLUSIONS
We have reported on a novel NRG scheme that lifts the so-far existing limits on solving impurity problems immersed in scalar but gapped TDOS via unbiased Wilsonian RG techniques. The herein developed method is referred to as the log-gap NRG approach, since it replaces the standard logarithmic discretization that runs up to the Fermi energy with an analogous procedure that accumulates the discretization points logarithmically at the gap edges. While the subsequent tridiagonalization is standard, the resulting Wilson chains contain generally two scalings which require specific diagonalization procedure, where truncation of the Hilbert space is performed only at every even NRG iteration (initial iteration enumerated as 0-th).
Both, the log-gap discretization and the alternating kept/discarded scheme for truncation of the Hilbert space during the iterative solution of the Wilson chain, are inherently connected. The discretization procedure is motivated by the analysis of the poles in the discretized tunneling self-energy in Sec. <ref> and gives rise to two-scaled behavior of the Wilson chain, which requires different truncation strategy of the resulting Hilbert space so that the energy scale separation is not broken. The two-scaled nature of SC-AIM and the gapped AIM hints towards the underlying RG-universality class.
Using the novel NRG technique, first a thorough benchmarking was performed using SC-AIM in its scalar representation as obtained in Ref. <cit.>. The results by the log-gap NRG approach compare in high numeric accord to standard NRG methods, but we stress that the log-gap NRG approach is superior in terms of CPU requirements due to the available symmetries. This holds true also when ε_d≠ -U/2 or non-zero magnetic field on the QD are considered. The generalization to encompass such cases is straightforward and of interest for current experiments where the log-gap NRG method can provide much faster scanning tool in the parameter space of SC-AIM (especially when its approximate version from Appendix <ref> is employed).
The log-gap NRG approach was then finally used in the first ever unbiased Wilsonian RG solution of the AIM with constant but gapped TDOS. The resulting physical quantities are thus valid without any restrictions at arbitrary temperatures corresponding to the given length of the chain. The most general conclusion from these calculations is that the model behaves qualitatively as SC-AIM as they both belong to the same RG-equivalence class. In particular, one observes always a doublet GS at completely particle-hole symmetric scenario with 𝒜=0 and ε_d =-U/2, which corresponds to the φ=π scenario of SC-AIM. Nevertheless, dependent on the values of Γ and U it may eventually undergo a QPT into a singlet GS as 𝒜 is increased, which corresponds to decreasing φ in SC-AIM. A complete phase diagram in half-filled QD scenario is shown in Fig. <ref>, which is highly similar to the corresponding phase diagram of SC-AIM calculated in Fig. <ref>.
In Sec. <ref>, an elaborate comparison has been performed which shows that the existing results from Refs. <cit.> are not only in conflict with the here developed log-gap NRG method, but crucially they show also significant numeric discrepancy when compared to the standard NRG method for a system augmented with an auxiliary metallic lead as demonstrated in fig. <ref>. Contrary, the herein developed log-gap NRG matches very well. Additional reasons for the approach of Refs. <cit.> to be considered with caution is of theoretic origin and was laid down also in Sec. <ref>.
We stress that the method is general and not restricted to SC-AIM or AIM with constant but gapped TDOS. Future generalizations and developments can incorporate the z-averaging techniques to solve for the spectral function, while addition of out-of-half-filling scenario or the presence of magnetic field is almost trivial. Additionally, an asymmetrically placed gap around the Fermi energy, as can appear in realistic TDOS functions, can also easily be incorporated. Another natural avenue for important applications of the here presented method represents the problem of metal-insulator transition, where the herein proposed NRG scheme might be implemented as internal impurity solver in DMFT calculations when the gap is finally open in the system. However, precaution is always required as the presence of the second stage scaling is a necessary prerequisite for meaningful application of RG techniques. Moreover, some additional problems might arise when the gap edge is not sharply defined as already noted in <cit.>. For realistic scenarios, fusion with the adaptive mesh approach according to <cit.> is also necessary.
As a final consequence of the presented approach, we would like to accentuate the missing rigorous RG theory of effective model for SC-AIM. While zero-band width or atomic limit theory are nowadays routinely employed for qualitative analysis of SC-AIM and related experiments <cit.>, we are still not able to fully understand how and why they emerge from the full problem. The two-scaled nature of the Wilson chains, as deciphered here, could however allow to build a rigorous analytic RG theory of fixed points in the spirit of seminal works by K. G. Wilson <cit.>.
Taking together, one of the main limitations of the NRG approach has been successfully eliminated, so an unbiased Wilsonian RG approach for gapped systems is now able to resolve all energy scales at the same footing. Further generalization and development is however necessary to go beyond the basic models implemented here.
We acknowledge discussions with T. Novotný, M. Žonda, K. Wrześniewski and R. Žitko. This work was supported by Grant No. 23-05263K of the Czech Science Foundation.
§ APPROXIMATE VERSION OF THE LOG-GAP NRG APPROACH
In the main text only the results of a rigorous Wilsonian RG approach are used. However, as briefly mentioned in Sec. <ref>, an approximate and a much faster calculation (in terms of the used CPU time) can be used for either preliminary scanning of large parametric spaces or for obtaining subgap properties with relatively small numerical deviations from the actual ones as demonstrated here.
The main idea is to use the ordinary kept/discarded strategy as outlined in Fig. <ref> in conjunction with the log-gap discretization. States are thus discarded after each iteration and the energy scale separation is inevitably broken. Consequently, the eigenspectra are corrupted especially in the high energy sector as shown in Fig. <ref> for SC-AIM with ε_d=-U/2, U=3Δ and Γ=Δ. Notably, φ=0 and φ=π remain exact.
In Fig. <ref>, we then demonstrate that the subgap portion of eigenenergies is nevertheless well usable at least in an approximate manner with a relative error of the order of just few percent when compared to the standard two-channel NRG. We stress however that such calculations might be dangerous without any reference data from exact Wilsonian approaches. Moreover, they should never be used for obtaining spectral functions due to the corruption of the high energy spectrum.
§ TECHNICAL DETAILS OF NRG IMPLEMENTATIONS
In the presented work, all model parameters are measured in units of B, where 2B is the width of the band. A typical value of the gap for SC-AIM as well as the gapped AIM with constant TDOS is set to 0.0005B if not stated otherwise. The effects of finite band width are thus essentially almost completely suppressed. Two standard NRG calculations have been implemented within the open source code of NRG Ljubljana: the standard two-channel NRG for SC-AIM <cit.> and the one-channel calculation for systems augment with metallic leads as described in Refs. <cit.>. The former has been performed at Λ=4 while for the later Λ=2 due to the one-channel nature of the calculation. In both cases at least 1000 states have been kept.
The here developed log-gap NRG algorithm has been implemented in the Flexible DM-NRG Budapest code <cit.> by modifying its kept/discarded routines. Log-gap discretization (<ref>) was performed in a Mathematica script with a subsequent tridiagonalization performed in a stand-alone C++ code. The resulting Wilson chain parameters were then manually fed into the DM-NRG Budapest code. After the discarding step described of the diagonalization scheme, in Sec. <ref>, 500 multiplets have been kept, which has been tested to be sufficient to have no impact on the sub-gap properties of the presented results. Λ=2 was set due to the one-channel nature of the Wilson chain.
To make the general idea of the log-gap NRG method more transparent and to streamline the presentation, the z-averaging was not employed. More specifically, the in-gap positions have been calculated at single value of z=1. This applies to the novel log-gap NRG approach introduced in this paper as well as to the all standard NRG calculations performed within NRG Ljubljana with the exception of supragap function shown in Fig. <ref>.
45
fxundefined [1]
ifx#1
fnum [1]
#1firstoftwo
secondoftwo
fx [1]
#1firstoftwo
secondoftwo
noop [0]secondoftwo
ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]
rl [1]href #1
@bib@innerbibempty
[Pokorný and ŽŽonda(2023)]Pokorny-2023
author author V. Pokorný and author M. ŽŽonda, 10.1103/PhysRevB.107.155111 journal journal
Phys. Rev. B volume 107, pages
155111 (year 2023)NoStop
[Diniz et al.(2020)Diniz,
Diniz, Martins, and Vernek]Diniz-2020
author author G. Diniz, author G. S. Diniz,
author G. B. Martins, and author E. Vernek, 10.1103/PhysRevB.101.125115 journal journal Phys. Rev. B volume 101, pages 125115 (year 2020)NoStop
[Zalom et al.(2021)Zalom,
Pokorný, and Novotný]Zalom-2021
author author P. Zalom, author V. Pokorný, and author T. Novotný, 10.1103/PhysRevB.103.035419 journal journal Phys. Rev. B volume 103, pages 035419 (year 2021)NoStop
[Zalom and Novotný(2021)]Zalom-2021r
author author P. Zalom and author T. Novotný, 10.1103/PhysRevB.104.035437 journal journal Phys. Rev. B volume
104, pages 035437 (year 2021)NoStop
[Zalom and ŽŽonda(2022)]Zalom-2022
author author P. Zalom and author M. ŽŽonda, 10.1103/PhysRevB.105.205412 journal journal
Phys. Rev. B volume 105, pages
205412 (year 2022)NoStop
[Chen and Jayaprakash(1998)]Chen-1998
author author K. Chen and author C. Jayaprakash, 10.1103/PhysRevB.57.5225 journal journal Phys. Rev. B volume
57, pages 5225 (year 1998)NoStop
[Moca and Roman(2010)]Moca-2010
author author C. P. Moca and author A. Roman, 10.1103/PhysRevB.81.235106 journal journal Phys. Rev. B volume 81, pages 235106 (year 2010)NoStop
[Pinto and Frota(2015)]Pinto-2015
author author J. Pinto and author H. Frota, https://doi.org/10.1016/j.jmmm.2014.12.051 journal journal Journal of Magnetism and Magnetic Materials volume 379, pages 217 (year
2015)NoStop
[Anderson(1961)]Anderson-1961
author author P. W. Anderson, http://link.aps.org/doi/10.1103/PhysRev.124.41
journal journal Phys. Rev. volume 124, pages 41 (year 1961)NoStop
[Lee et al.(1986)Lee,
Rice, Serene, Sham, and Wilkins]Lee-1986
author author P. A. Lee, author T. M. Rice,
author J. W. Serene, author L. J. Sham, and author J. W. Wilkins (year
1986)NoStop
[Hewson(1993a)]Hewson-book-1993
author author A. C. Hewson, 10.1017/CBO9780511470752 title The Kondo Problem to Heavy Fermions, Cambridge Studies in
Magnetism (publisher Cambridge University Press, year 1993)NoStop
[Fulton and Dolan(1987)]Fulton-1987
author author T. A. Fulton and author G. J. Dolan, 10.1103/PhysRevLett.59.109 journal
journal Phys. Rev. Lett. volume 59, pages 109 (year 1987)NoStop
[Cronenwett et al.(1998)Cronenwett, Oosterkamp, and Kouwenhoven]Cronenwett-1998
author author S. M. Cronenwett, author T. H. Oosterkamp, and author L. P. Kouwenhoven, 10.1126/science.281.5376.540 journal journal Science volume
281, pages 540 (year 1998)NoStop
[Pustilnik and Glazman(2004)]Pustilnik-2004
author author M. Pustilnik and author L. Glazman, 10.1088/0953-8984/16/16/R01 journal journal Journal of Physics: Condensed Matter volume 16, pages R513 (year
2004)NoStop
[Mitchell et al.(2017)Mitchell, Kim G. L., Hedegard, and Paaske]Mitchell-2017
author author A. K. Mitchell, author P. Kim G. L.,
author P. Hedegard, and author J. Paaske, @noop
journal journal Nat Commun volume 8 (year 2017)NoStop
[Madhavan et al.(1998)Madhavan, Chen, Jamneala, Crommie, and Wingreen]Madhavan-1998
author author V. Madhavan, author W. Chen,
author T. Jamneala, author M. F. Crommie, and author N. S. Wingreen, 10.1126/science.280.5363.567 journal journal
Science volume 280, pages 567
(year 1998)NoStop
[Li et al.(1998)Li,
Schneider, Berndt, and Delley]Li-1998
author author J. Li, author W.-D. Schneider,
author R. Berndt, and author B. Delley, 10.1103/PhysRevLett.80.2893 journal journal
Phys. Rev. Lett. volume 80, pages
2893 (year 1998)NoStop
[Wilson(1975)]Wilson-1975
author author K. G. Wilson, 10.1103/RevModPhys.47.773 journal
journal Rev. Mod. Phys. volume 47, pages 773 (year 1975)NoStop
[Krishna-murthy et al.(1980a)Krishna-murthy, Wilkins, and Wilson]Krishna-1980a
author author H. R. Krishna-murthy, author J. W. Wilkins, and author K. G. Wilson, 10.1103/PhysRevB.21.1003 journal
journal Phys. Rev. B volume 21, pages 1003 (year 1980a)NoStop
[Krishna-murthy et al.(1980b)Krishna-murthy, Wilkins, and Wilson]Krishna-1980b
author author H. R. Krishna-murthy, author J. W. Wilkins, and author K. G. Wilson, 10.1103/PhysRevB.21.1044 journal
journal Phys. Rev. B volume 21, pages 1044 (year 1980b)NoStop
[Kopietz et al.(2010)Kopietz, Bartosch, and Schütz]Kopietz-2010
author author P. Kopietz, author L. Bartosch, and author F. Schütz, DOI=10.1017/CBO9780511470752, title
Introduction to the functional renormalization group, Lect. Notes Phys. (publisher Springer-Verlag Berlin Heidelberg, year
2010)NoStop
[Streib et al.(2013)Streib,
Isidori, and Kopietz]Kopietz-2013
author author S. Streib, author A. Isidori, and author P. Kopietz, 10.1103/PhysRevB.87.201107 journal journal Phys. Rev. B volume 87, pages 201107 (year 2013)NoStop
[Hewson(1993b)]Hewson-1993
author author A. C. Hewson, 10.1103/PhysRevLett.70.4007 journal journal Phys. Rev. Lett. volume 70, pages 4007 (year
1993b)NoStop
[Janiš and Augustinský(2007)]Janis-2007
author author V. Janiš and author P. Augustinský, http://link.aps.org/doi/10.1103/PhysRevB.75.165108 journal
journal Phys. Rev. B volume 75, pages 165108 (year 2007)NoStop
[Jani šš et al.(2019)Jani šš,
Zalom, Pokorný, and Klí čč]Janis-2019
author author V. Jani šš, author P. Zalom, author V. Pokorný, and author A. Klí čč, 10.1103/PhysRevB.100.195114 journal journal Phys. Rev. B volume 100, pages 195114 (year 2019)NoStop
[Takegahara et al.(1992)Takegahara, Shimizu, and Sakai]Takegahara-1992
author author K. Takegahara, author Y. Shimizu,
and author O. Sakai, 10.1143/JPSJ.61.3443 journal journal
Journal of the Physical Society of Japan volume
61, pages 3443 (year 1992)NoStop
[Gull et al.(2011)Gull,
Millis, Lichtenstein, Rubtsov, Troyer, and Werner]Gull-2011
author author E. Gull, author A. J. Millis,
author A. I. Lichtenstein,
author A. N. Rubtsov, author M. Troyer, and author
P. Werner, 10.1103/RevModPhys.83.349 journal journal Rev.
Mod. Phys. volume 83, pages 349
(year 2011)NoStop
[Gubernatis(2016)]Gubernatis-2016
author author J. E. Gubernatis, @noop title Quantum Monte Carlo
methods : algorithms for lattice models (publisher Cambridge
University Press, address New York, NY, year
2016)NoStop
[Hecht et al.(2008)Hecht,
Weichselbaum, von Delft, and Bulla]Hecht-2008
author author T. Hecht, author A. Weichselbaum,
author J. von Delft, and author R. Bulla, http://stacks.iop.org/0953-8984/20/i=27/a=275213 journal
journal J. Phys.: Cond. Mat. volume
20, pages 275213 (year 2008)NoStop
[Heinrich et al.(2018)Heinrich, Pascual, and Franke]Heinrich-2018
author author B. W. Heinrich, author J. I. Pascual, and author K. J. Franke, https://doi.org/10.1016/j.progsurf.2018.01.001
journal journal Progress in Surface Science volume 93, pages 1 (year
2018)NoStop
[Meden(2019)]Meden-2019
author author V. Meden, 10.1088/1361-648x/aafd6a journal
journal Journal of Physics: Condensed Matter volume 31, pages 163001 (year
2019)NoStop
[Galpin and Logan(2008)]Galpin-2008
author author M. R. Galpin and author D. E. Logan, 10.1103/PhysRevB.77.195108 journal
journal Phys. Rev. B volume 77, pages 195108 (year 2008)NoStop
[Moca et al.(2021)Moca,
Weymann, Werner, and Zaránd]Moca-2021
author author C. P. Moca, author I. Weymann,
author M. A. Werner, and author G. Zaránd, 10.1103/PhysRevLett.127.186804 journal journal Phys. Rev. Lett. volume 127, pages 186804 (year 2021)NoStop
[ ŽŽitko et al.(2013) ŽŽitko,
Hansen, Perepelitsky, Mravlje, Georges, and Shastry]Zitko-Mott-2013
author author R. ŽŽitko, author D. Hansen, author E. Perepelitsky, author J. Mravlje, author A. Georges, and author B. S. Shastry, 10.1103/PhysRevB.88.235132 journal journal Phys.
Rev. B volume 88, pages 235132
(year 2013)NoStop
[Kadlecová et al.(2017)Kadlecová, ŽŽonda, and Novotný]Zonda-2017
author author A. Kadlecová, author M. ŽŽonda, and author
T. Novotný, 10.1103/PhysRevB.95.195114 journal journal Phys.
Rev. B volume 95, pages 195114
(year 2017)NoStop
[Bulla et al.(1994)Bulla,
Keller, and Pruschke]Bulla-1994
author author R. Bulla, author J. Keller, and author T. Pruschke, 10.1007/BF01307671 journal journal Z.
Phys. B Condens. Matter volume 94, pages 195 (year 1994)NoStop
[Bulla et al.(2008)Bulla,
Costi, and Pruschke]Bulla-Rev-2008
author author R. Bulla, author T. A. Costi, and author T. Pruschke, http://link.aps.org/doi/10.1103/RevModPhys.80.395 journal
journal Rev. Mod. Phys. volume 80, pages 395 (year 2008)NoStop
[Liu et al.(2016)Liu,
Wang, and Wang]Liu-2016
author author J.-G. Liu, author D. Wang, and author Q.-H. Wang, 10.1103/PhysRevB.93.035102 journal journal Phys. Rev. B volume 93, pages 035102 (year 2016)NoStop
[Satori et al.(1992)Satori,
Shiba, Sakai, and Shimizu]Satori-1992
author author K. Satori, author H. Shiba,
author O. Sakai, and author Y. Shimizu, 10.1143/JPSJ.61.3239 journal journal J. Phys.
Soc. Japan. volume 61, pages 3239
(year 1992)NoStop
[Pavešič et al.(2023)Pavešič, Aguado, and Žitko]Zitko-arxiv
author author L. Pavešič, author R. Aguado, and author R. Žitko, @noop title Quantum dot josephson
junctions in the strong-coupling limit, (year
2023)NoStop
[Žitko(2009)]Zitko-adapt
author author R. Žitko, https://doi.org/10.1016/j.cpc.2009.02.007
journal journal Computer Physics Communications volume 180, pages 1271 (year
2009)NoStop
[Meng et al.(2009)Meng,
Florens, and Simon]Meng-2009
author author T. Meng, author S. Florens, and author P. Simon, http://link.aps.org/doi/10.1103/PhysRevB.79.224521 journal
journal Phys. Rev. B volume 79, pages 224521 (year 2009)NoStop
[Grove-Rasmussen et al.(2018)Grove-Rasmussen, Steffensen, Jellinggaard,
Madsen, Žitko, Paaske, and Nygård]Grove-Rasmussen-2018
author author K. Grove-Rasmussen, author G. Steffensen, author A. Jellinggaard, author M. H. Madsen, author R. Žitko,
author J. Paaske, and author J. Nygård, @noop
journal journal Nat Commun volume 9 (year 2018)NoStop
[Žonda et al.(2023)Žonda, Zalom, Novotný, Loukeris, Bätge, and Pokorný]Zonda-2023
author author M. Žonda, author P. Zalom,
author T. Novotný, author G. Loukeris, author
J. Bätge, and author
V. Pokorný, 10.1103/PhysRevB.107.115407 journal journal
Phys. Rev. B volume 107, pages
115407 (year 2023)NoStop
[Tóth et al.(2008)Tóth,
Moca, Legeza, and Zaránd]Toth-2008
author author A. I. Tóth, author C. P. Moca,
author O. Legeza, and author G. Zaránd, 10.1103/PhysRevB.78.245109 journal journal Phys.
Rev. B volume 78, pages 245109
(year 2008)NoStop
|
http://arxiv.org/abs/2307.04662v2 | 20230710155941 | Baryogenesis and Dark Matter in the Mirror Twin Higgs | [
"Pedro Bittar",
"Gustavo Burdman",
"Larissa Kiriliuk"
] | hep-ph | [
"hep-ph",
"hep-ex"
] |
On the Generalized Uncertainty Principle and Cosmology.
M. Sabido^a
August 12, 2023
=======================================================
§ INTRODUCTION
The standard model (SM) of particle physics is an extremely successful quantum field theory describing the interactions of all known elementary particles.[The only exception, gravity, is non-renormalizable, and its effects can be safely neglected up to extremely high energies.] Nonetheless, there remain many questions that need to be addressed by the SM. Among them is the nature of dark matter, the origin of the baryon asymmetry, and the stability and origin of the only energy scale appearing in the SM. The Mirror Twin Higgs Model (MTH) <cit.>, originally conceived to stabilize the electroweak scale, can be an intriguing source of dark matter candidates. For instance, Refs. <cit.> and <cit.> consider thermal relics in the MTH and fraternal <cit.> twin scenarios, respectively. The possibility of Asymmetric Dark Matter (ADM) <cit.> in TH models is considered in Refs. <cit.> for the fraternal TH, and in Refs. <cit.> for a variety of Twin Higgs scenarios, but most importantly for us, in the context of the MTH.
In the MTH <cit.>, the SM is extended to have a twin SM copy supplemented by a 2 symmetry. The Higgs sector realizes the spontaneous breaking of a global symmetry at some scale f. The SM Higgs is then a pseudo-Nambu-Goldsone boson of this breaking, explaining the stability of the weak scale v, at least up to the energy scale ≃ 4π f. Experimental bounds, mostly from the unobserved invisible Higgs boson decays to the twin sector, impose the need for a soft 2 breaking[In some cases it is even possible to have the MTH with an exact 2 symmetry as shown in <cit.>], resulting in f/v > 1. However, whatever the origin of this soft 2 breaking, this does not reintroduce the hierarchy problem since the 2 breaking is assumed to be valid in the ultra-violet (UV).
In Ref. <cit.>, the MTH was considered to build a model for DM. There it is argued that if a twin baryon is to provide the correct DM abundance, it is necessary to introduce a hard 2 breaking in order to allow for
m_ DM≃ 5 m_N ,
where m_N is the nucleon mass and m_ DM is the mass of the twin baryon. The need for hard 2 breaking results from the fact that just using the soft breaking (i.e., f/v>1) is not enough to obtain the desired value in (<ref>). Its effects in the renormalization group running in the twin sector, appearing through the modification of quark masses and the resulting speed up of the twin QCD running, results in only a mild enhancement of Λ̃_ QCD, much smaller than the factor of 5 needed. Thus, the introduction of hard breaking in the twin QCD coupling. Although it is possible to arrange for the hard breaking to be small enough not to reintroduce the hierarchy problem, this remains an ad hoc aspect of the ADM models in the MTH.
In this paper, we consider a MTH scenario with an additional sector responsible for the baryon and dark matter asymmetries. As we show below, one of the features of the model is that it allows for the correct DM abundance even if (<ref>) is not satisfied therefore vacating the need for hard 2 breaking. This is achieved by showing that we can obtain different baryon and DM number densities without such breaking. In this way, in order to obtain the correct DM to baryon abundance ratio
Ω_ DM/Ω_B = n_
DM/n_B m_DM/m_N,
with m_ DM∼ O(1) m_N, the ratio of number densities must be different. The models we consider require the addition of a sector resulting in baryon number violation on both sides of the MTH. We illustrate this with simple models of baryon number violation with out-of-equilibrium decays. These models are mostly available in the literature as applied to the SM alone. We aim to show the general mechanism that allows ADM models in the context of the MTH to obtain the correct DM abundance without hard 2 breaking. On the other hand, it is interesting that the resulting models address the hierarchy problem, the origin of dark matter, and the baryon asymmetry in a natural way.
The rest of the paper is organized as follows: In the next section, we review the status of ADM models in the context of the MTH. In Section <ref>, we propose mechanisms for generating both the baryon and dark matter number densities needed on both sides of the MTH without incurring hard 2 breaking. Finally, we conclude in Section <ref>.
§ ASYMMETRIC DARK MATTER AND THE MIRROR TWIN HIGGS
The Twin Higgs mechanism <cit.> was originally introduced as a possible solution to the little hierarchy problem. A copy of the SM matter and interactions, supplemented by a 2 symmetry, results in a global symmetry (SU(4)) which is spontaneously broken at a scale f, resulting in a spectrum of Nambu-Goldstone bosons that make up the SM-like Higgs doublet. The SM interactions explicitly break the global symmetry generating a Higgs potential and leading to electroweak symmetry breaking. In the original version, which we call the MTH, all SM particles and interactions are mirrored in the twin sector. However, as it was first pointed out in Ref. <cit.>, the minimum matter content in the twin sector that addresses the little hierarchy problem does not require an entire copy of the SM but just a twin third generation. This case is the so-called fraternal TH (FTH).
The twin Higgs scenario provides several possibilities for DM model building. For instance, models with thermal relics have been considered in the context of the FTH in Refs. <cit.>. In these cases, the twin tau is cosmologically stable due to an accidental U(1) lepton number. The WIMP miracle is recreated since these DM candidates have masses of tens of GeV up to about 100 GeV, and the twin weak interactions determine their thermal relic abundance. Also, in the FTH case, Ref. <cit.> examines asymmetric DM (ADM) models. The preferred scenario involves a light twin b quark with a mass below Λ̃_QCD, asymmetry connected to the SM baryon asymmetry through some UV mechanism, and a cosmologically long-lived twin b baryon. One of the main advantages of the FTH scenarios is that they minimize the new relativistic degrees of freedom, which makes it easier for them to avoid conflicts with the cosmological bounds on N_ eff.. On the other hand, as we will see below, it is more natural to build ADM models in the MTH scenario.
We consider the MTH model with an effective cutoff of Λ≃ 4π f, where f is the spontaneous symmetry breaking scale of the twin Higgs global symmetry. In the limit of exact 2 symmetry, f=v, with v the vacuum expectation value of the Higgs doublet in the SM sector. However, the current experimental bounds from the measurements of the Higgs boson couplings at the LHC <cit.> require the f/v ≳ 3 <cit.>. This requirement can be achieved by assuming a soft breaking of the 2 symmetry, i.e., a breaking occurring in the infrared (IR) by some mechanism that respects the 2 symmetry in the ultraviolet (UV). This soft breaking guarantees that the hierarchy problem is not reintroduced in loops correcting the Higgs potential since the UV 2 symmetry forces the cancellation of contributions quadratically dependent on the cutoff in the Higgs boson two-point function. The soft 2 breaking paradigm can accommodate all known collider phenomenology with minimal tuning <cit.>.
The twin sector of the MTH is particularly well suited to building models of dark matter. In particular, here we consider the scenario where twin baryons, which carry an accidentally conserved global charge just as protons carry baryon number, may constitute all of the observed DM abundance. Thus, we focus on ADM models in the context of the MTH, in which the origin of the baryon and twin baryon asymmetries are related and at the heart of the apparent similarity in the DM and baryon abundances. In particular, it was shown in Ref. <cit.> that the twin neutron in the MTH model is a viable candidate for DM. This results from a scenario where twin neutrinos somehow acquire large masses in order to avoid tight constraints from the cosmological measurements of N_ eff.. However, the twin photon is still in the spectrum. If only twin baryon number B̃ is generated in the twin sector (i.e., no twin lepton number L̃), then charge neutrality of the universe results in the generation of a net twin neutron ñ number after the twin QCD phase transition. Although π̃^± are also stable, their abundance is negligible <cit.>, whereas π̃^0 still decays to twin photons. Finally, nucleosynthesis does not proceed without light twin neutrinos, and we conclude that DM is made entirely of ñ.
On the other hand, Ref. <cit.> also raised a problem with this picture. If the softly broken 2 implies that the number densities of baryon and DM are similar, i.e.
n_B≃ n_DM ,
then (<ref>) implies (<ref>). However, it seems that in order to achieve m_ DM≃ 5 m_B, the 2 symmetry has to be broken in the UV. To see this, we notice that in this scenario
m_ DM∼Λ̃_ QCD ,
where Λ̃_ QCD is the twin sector strong interaction IR scale. But if we only allow for a soft 2 breaking, the only effects raising this scale compared to Λ_ QCD are given by the enhancements of the twin quark masses. This results in a speed-up of the running giving
Λ̃_ QCD≃ 1.4 Λ_ QCD ,
slightly depending on the value of f/v. In this way, with only a soft 2, we have
m_ DM≃ O(1) m_N .
Thus, if the baryon and twin baryon number densities, n_B and n_ DM in (<ref>), were to be equal, we could not obtain the correct DM abundance.
Ref. <cit.> argues that the 2 symmetry forces the number density equality and that, in order to obtain the correct DM abundance, the only way out is a hard breaking of the 2 symmetry, which would be enough to give m_ DM≃ 5 m_N. These masses can be achieved, for instance, by having different values of the QCD and twin QCD couplings at the cutoff Λ. However, this reintroduces two-loop contributions to the Higgs mass squared that are quadratic in Λ. It was then argued that it is possible to introduce enough α̃(Λ) - α(Λ) to obtain the desired value of Λ̃_ QCD≃ 5Λ_ QCD at the same time that a fine-tuning of at the most ≃ 1% is required.
The situation described above, although technically feasible, is far from satisfactory. The MTH remains a natural extension of the SM controlling the Higgs mass UV sensitivity even after the most recent LHC bounds <cit.>, which come mostly from the constraints on the Higgs couplings, but also from the invisible Higgs boson branching ratio <cit.>. Therefore it is desirable to maintain this feature of the model, i.e., to avoid forcing the MTH scenario into a fine-tuned corner of parameter space for the purpose of obtaining the correct DM abundance. Luckily, as we will show below, it is possible to avoid introducing a hard breaking of the 2 and still obtain the observed DM abundance. The key point, of course, is to relax the approximate equality n_ DM≃ n_B to accommodate eqns. (<ref>) and (<ref>) while still using the result (<ref>), i.e. without introducing ad hoc hard 2 breaking.
Although we present a full model in Section <ref> as proof of principle for how this can be achieved, we can sketch the general idea here.
Models of baryogenesis set n_B after annihilation of the symmetric part of the particle-antiparticle plasma, leaving an asymmetric component. The final number density depends on the CP asymmetry, ϵ_CP <cit.>. The details of its computation depends on the specific baryogenesis model under consideration. However, CP violation generically requires ϵ_CP to be proportional to the relative complex phases of the couplings of the theory, which we denote as sinϕ. Because DM is asymmetric, the same structure for the final number density appears in the DM sector of the MTH. The remaining asymmetric component of the DM plasma sets the final DM number density <cit.>. Therefore, the twin ϵ_CP will also be proportional to the complex phase of the couplings of the twin sector, which we generically denote as sinϕ. Due to the 2 symmetry, the baryon asymmetry and DM asymmetry have the same microscopic origin; however, if the baryon and DM phases ϕ and ϕ are different, we can rewrite (<ref>) as
Ω_ DM/Ω_B∼m_ DM/m_N|sinϕ/sinϕ|≃ 5.
From (<ref>), we can see that it is possible to satisfy the DM abundance ratio to baryons if there is an order one misalignment between the phases ϕ in the visible sector and ϕ in the twin sector.
We argue that this can be the case even in the absence of Z_2 breaking in the UV, given that the relative phases in the visible and twin sectors maybe be defined by IR processes that are not necessarily identical and, therefore, could generally come from a soft Z_2 breaking. A simple example is the vacuum alignment leading to the spontaneous breaking of the twin global symmetry at the scale f. This can be compared with the vacuum alignment in the visible sector leading to electroweak symmetry breaking at the scale v. There is no reason why the visible sector vacuum expectation value (VEV) should be real relative to the twin sector VEV.
We can parameterize the vev of the Higgs bi-doublet as
⟨ H⟩ =f
[ 0; sinθ; 0; e^iδcosθ ].
where the Twin Higgs VEV ⟨ H_B ⟩ has a relative phase δ with respect of the SM Higgs VEV ⟨ H_A ⟩. This relative phase propagates to the couplings of visible and twin sector states, e.g. fermions, coupled to the SM and twin Higgses. As a result, the SM Higgs couplings will have relative phases with respect to their twin sector counterparts. For instance, this implies that the CKM phases in the twin sector in the MTH need not be the same as those in the SM. Furthermore, since we are introducing couplings among quarks and new fields in both baryogenesis and darkogenesis, the
overall relative phase of these is potentially receiving additional IR misalignment.
Thus, we conclude that the phases entering in (<ref>) need not be related by a Z_2 transformation in the UV and can differ by order one values, which may result in the correct DM abundance even if the ratio of DM to nuclean masses is still just over unity. In the next section we show an explicit model of ADM in the MTH in which this mechanism is sucessfully implemented.
§ BARYOGENESIS AND DARKOGENESIS
In this section, we specify a model of baryogenesis and its MTH counterpart to exemplify that it is possible to obtain a successful ADM dark matter abundance in this context without introducing hard 2 breaking.
§.§ Baryogenesis
We start by providing a simple and concrete model for baryogenesis. We generate the baryon asymmetry directly at low scales below the sphaleron decoupling temperature. Baryogenesis at low temperatures is appropriate for the twin Higgs since we expect the theory to be completed at the UV scale Λ=4π f ≈ 10. Therefore, we can imagine that the UV completion can play a role in the baryon asymmetry generation.
The model is based on the out-of-equilibrium decays of a singlet fermion N_α that violates baryon number. The need for CP-violation requires at least two flavors of N_1,2, with a mass hierarchy M_N_2>M_N_1, so that the tree-level and loop amplitudes can interfere with different phases. We also require the existence of a colored scalar X in the (3,1)_2/3 representation of the group. We then add the following interactions to the SM sector:
Δℒ_Bgen= λ_iα N_αX̅^a (u_R^ i)_a + ξ_ijϵ^abc X_a (d_R^ i)_b (d_R^ j)_c + h.c.
Here, i,j are the quark generation indices, α=1,2 is the neutral fermion flavor, and a,b,c are color indices. Because of the antisymmetric nature of ϵ^abc, the ξ_ij coupling must be antisymmetric in flavor. This model is often considered in the context of low-temperature baryogenesis <cit.> since it is a simple realization of baryon number violation without proton decay <cit.>.
The baryon asymmetry is generated by the decay of the lightest neutral fermion, N_1. The baryon asymmetry parameter is given by
Y_Δ B=n_N_1/s(Γ(N_1→ B) - Γ(N_1→B̅)/Γ (N_1→tot)) ≡ Y_N_1ϵ_CP^N_1,
where Γ(N→ f) are the decay widths of N_1 to baryon number B=+1 or B=-1 final states, Y_N_1≡n_N_1s is the N_1 yield, s is the entropy density, and ϵ_CP^N_1 is the CP asymmetry. We proceed to compute each piece of (<ref>) separately.
We start with ϵ_CP^N_1. CP violation results from the interference of tree-level and loop amplitudes in N_1 decay as indicated in Fig. <ref>. The decay amplitude can be written as
ℳ = c_0 𝒜_0 + c_1 𝒜_1,
where in c_0 and c_1, we separate all the couplings in the matrix elements. We can then write the CP asymmetry in the generic form <cit.>
ϵ_CP^N_1= Γ (N_1→ Xu_i^c)-Γ (N_1→Xu_i^c)/Γ (N_1→ Xu_i^c)+Γ (N_1→Xu_i^c)= Im{c_0c_1^*}/∑_α|c_0|^22∫ Im{ A_0 A_1^*}δ̃ dΠ_uX/∫ | A_0|^2 δ̃ dΠ_uX ,
where δ̃=(2π)^4δ^4(p_i-p_f) for the initial and final state momenta and dΠ_uX is the final state phase space factor.
From (<ref>) we see that to have a CP asymmetry, there must be a complex phase in the product of the couplings as well as a non-zero relative phase of the two matrix elements, A_0 A_1. To meet this last condition, we need on-shell intermediate states in the loop diagrams so that their matrix elements have a complex phase relative to the tree level one. This, in turn, imposes a lower bound on the mass of N_1 from the mass of the colored scalar X, which should be above the scale due to LHC bounds,
M_N_1>M_X ≳few .
Assuming that the couplings are complex [There is no reason a priori for the couplings to be real. In the following sections, we comment on possible sources for the complex phases.]. Using the diagrams in Fig. <ref>, we obtain
ϵ_CP^N_1 =∑_i,j (λ _i1 λ _i2 ^*λ _j2 ^*λ _j1 )/24π∑_i|λ _i1 |^2 [ 3 ℱ_S ( M_N_2^2/M_N_1^2 )+ℱ_V ( M_N_2^2/M_N_1^2 ) ],
where the functions ℱ_S,V(x) coming from the loop diagrams are defined as
ℱ_S(x) =2√(x)/x-1,ℱ_V(x)=√(x) ( 1+1/x ).
Assuming that there are no flavor hierarchies between the λ_iα couplings, we take λ_iα=λ_uα for all i=1,2,3 as a simplifying approximmation. Then the CP asymmetry is
given by
ϵ_CP^N_1 = 3|λ _u2|^2 /24π [ 3 ℱ_S ( M_N_2^2/M_N_1^2 )+ℱ_V ( M_N_2^2/M_N_1^2 ) ] sinϕ,
where ϕ is the complex phase of the product of the couplings λ _u1λ _u2^*λ _u2^*λ _u1.
Next, we turn our attention to the N_1 yield Y_N_1. Before the decay of N1, the yield will be constant once the early universe processes cease due to the expansion rate. Y_N_1 is set either thermally or non-thermally depending on the physical processes at play and the values of couplings and masses. The important scale that distinguishes the two cases is T_FO, the freeze-out temperature of the processes that change the number density of N_1 in the thermal bath. In this case, the relevant processes are the decay of N_1, its inverse decay, and N_1 annihilations as shown in Fig. <ref>.
Once the freeze-out conditions are satisfied, the inverse decay and annihilation processes will stop happening, leading to the following conditions,
.Γ_u X → N_1/2H|_T^ inv._FO=1, 60mu.Γ_N_1 N_1→ u u/2H|_T^ ann._FO=1.
In these, Γ_i are the process rates for the two reactions , and T^ inv._FO and T^ ann._FO are their respective freeze out temperatures. Since we will be working with small couplings, we can safely assume that the freeze-out temperatures of these reactions will be high, above the mass of N_1. Therefore, N_1 is produced in equilibrium at high energies in the thermal scenario, and the thermal distribution determines its yield. As the inverse decay and annihilations freeze out, only the decay process will change the number density of N_1, mainly after the lifetime of N_1 has elapsed.
As argued before, the out-of-equilibrium decays of N_1 are responsible for the CP and baryon number violation required for baryogenesis. In addition, we need to compute the N_1 lifetime to know when baryogenesis mostly occurs. A long enough lifetime is required to reach the post-sphaleron baryogenesis window, as inverse decays and annihilations usually freeze out at higher temperatures. The lifetime of N_1 is given by
τ_N_1 = 1/Γ_N_1→ u X= 16π^2/3 |λ_u1|^2m_N_1^3/m_N_1^4-m_X^4,
where we included all the decay channels for different flavors of u_i={u,c,t}. Assuming that N_1 and X have masses above the few- region[In this model, N_1 and X can be significantly heavier than the few-s without changing the mechanism and the coupling bounds we derived.], we place a bound on the coupling λ_u1 by requiring the lifetime to be larger than the cosmological time for sphaleron processes on the one hand and lower than the time of BBN. This results in
(τ_BBN≈ 10 s) ≥τ_N_1≥(τ_Sph≈ 10^-12 s) ⇒ 10^-14≲λ_u1≲ 10^-7
To compute the thermal yield, we assume that the early universe processes decouple relativistically as it leads to the largest value of Y_N_1. As we will see shortly, this requirement will minimize the necessary tuning between the different couplings λ_u1 and λ_u2 when fixing the observed value for the baryon asymmetry. For relativistic freeze-out, the equilibrium distribution gives the following yield
Y_N_1=Y_N_1^EQ≃45 ζ(3)/2π^4g_N/g_*,S(T),
where g_N is the N_1 effective number of internal degrees of freedom, and g_*,S is the total entropic effective number of degrees of freedom[The number of effective degrees of freedom is approximately twice the SM since we need to include the twin states as they are coupled to the SM bath through the Higgs portal.].
Finally, we impose the correct baryon asymmetry in order to constrain the parameters of the theory. Using (<ref>), (<ref>) and (<ref>) we have
Y_Δ B=45 ζ(3)/16π^5g_N/g_*,S(T)|λ _u2|^2 [ 3 ℱ_S ( M_N_2^2/M_N_1^2 )+ℱ_V ( M_N_2^2/M_N_1^2 ) ] sinϕ.
The observed baryon abundance measured by the Planck telescope <cit.> is Y_Δ B^exp=(8.75± 0.23)× 10^-11. Then, we can write (<ref>) as
Y_Δ B/8.7× 10^-11=(213.5/g_*,S(T_FO)) ( 3 ℱ_S( 1.5 ) +ℱ_V( 1.5 )/15.3) (|λ _u2|sin^1/2ϕ/2.3× 10^-4)^2 .
We have selected m_N_2^2=1.5 m_N_1^2 as a benchmark point, yet the final abundance only exhibits a weak dependence on the specific choice of mass splitting between the neutral fermions. For larger splittings, the value of the coupling |λ_u2|sin^1/2ϕ is expected become slightly larger.
As we will argue later, we do not have any theoretical information on the origin of ϕ, the relative phase in the couplings. Because of this, we can assume the phases have values uniformly distributed from 0 to 2π[Note that this range means that the phase sinϕ can be negative. However, we can redefine what particle or anti-particle means in this case, thus always making the baryon asymmetry parameter positive.]. If this is the case, the quantity sinϕ is naturally expected to be an order 𝒪(1) parameter. Therefore, we see from (<ref>) that we can fix the observed value for the baryon asymmetry assuming that the coupling of N_2 is of order λ_u2∼ 10^-4. This value means that in this specific model of low-temperature baryogenesis, a coupling hierarchy between λ_u1 and λ_u2 is necessary. The coupling λ_u1 must be smaller in order for N_1 to be long-lived enough to get to post-sphaleron temperatures, and λ_u2 must be larger to reproduce the observed value of the baryon asymmetry, i.e.
10^-14≲ |λ_u1| ≲ 10^-7,
|λ_u2|∼ 10^-4.
On the other hand, this hierarchy of couplings is only necessary if we assume thermal production of N_1. Conversely, N_1 could be non-thermally produced if an additional mechanism was active after the freeze-out temperature of the processes in Fig. <ref>. One possibility is the production via the decay of a new heavy particle, for example, the "reaheaton" τ, a scalar field that induces a reheating period in early cosmology. The reheaton could be originated in different BSM scenarios, like non-thermal DM sectors, inflationary models, or SUSY/string models for the early universe <cit.>. The specific origin of this particle is beyond the scope of our work. The advantage of the N_1 non-thermal production mechanism is that there is no need for a hierarchy between the couplings λ_u1 and λ_u2, such as the one in (<ref>) and (<ref>) for the thermal case. We may then consider, for simplicity, that both couplings are equal in absolute value and that N_1 decays shortly after the reaheton decay. The prompt decay of N_1 is achieved by a larger λ_u1 coupling, of order λ_u1∼ 10^-4.
In this case, the overall baryon asymmetry is given by
Y_Δ B = n_τ/s(Γ(τ→ N_1 → B)-Γ(τ→ N_1 →B)/Γ(τ→tot)) =Y_τ Br_N_1 ϵ_N_1^CP
Where ϵ_N_1^CP is given by (<ref>), Y_τ is the non-thermal yield due to the decay of the reheaton, and Br_N_1 is the branching ratio of the reheaton decay into N_1.
We can estimate the non-thermal yield as a function of the reheating temperature by calculating the number density at this earlier matter-radiation equality epoch. The nonrelativistic energy density of the reheaton-dominated universe is ρ≃ m_τ n_τ. At the radiation epoch, we have the energy density, ρ_τ=π^230g_*(T) T^4. Therefore, we have n_τ∼^4m_τ and we can obtain the reheaton yield as
Y_τ = n_τ/s_RH≃3/4/m_τ.
Here, s_RH=2π^245g_*,S()^3 is the entropy at the reheating temperature. We also used that the effective degrees of freedom g_*(T) and g_*,S(T) are approximately equal at this early epochs. The reheating temperature can be estimated by the freeze out of the reheaton processes in the early universe. The reheaton decay rate can be obtained assuming that since it is a long-lived particle, its decay is possibly mediated by a nonrenormalizable operator. For example, in Refs. <cit.> a dimension five operator was used, resulting in
Γ_τ=α^2/2πm_τ^3/M_*^2
where α is the effective coupling of the processes and M_* is some high scale of the theory. Then, comparing with the Hubble rate at the freeze-out temperature, we have
≃Γ_τ^1/2 M_Pl^1/2/g_*^4(T)=α m_τ^3/2/(2π)^1/2g_*^4(T)M_Pl/M_*^2.
Then, assuming M_*∼ M_Pl, the reheaton yield (<ref>) is
Y_τ≃3/4α/(2π)^1/2g_*^4(T)(m_τ/M_Pl)^1/2
With equation (<ref>) and (<ref>), we see that Y_τ≲ 10^-3.
Notice that the reheating temperatures are below the range of temperatures in which spharaleons processes are active. Because of this, the couplings λ_u1 can be larger. Consequently, N_1 does not need a long lifetime, and baryogenesis will occur at lower temperatures, close to the reheating temperature.
Parametrically, we can write the baryon asymmetry for the non-thermal case as
Y_Δ B/8.7× 10^-11= ( 213.5/g_*(T_) )^4 ( m_τ/100 )^1/2( 3 ℱ_S( 1.5 ) +ℱ_V( 1.5 )/15.3) (α^1/2 |λ _u2|sin^1/2ϕ/1.1× 10^4)^2 .
Notice that in the non-thermal case, the effective degrees of freedom g_*(T_) can assume values with different orders of magnitude depending on the chosen reheating temperature. The reheaton may decay at any time between post-sphaleron and prior to the BBN time. In the numerical example above, we chose a reheating temperature before the SM and Twin QCD phase transition, which yields a total effective degrees of freedom of g_*≃ 213.5.
To conclude this section, we emphasize that our proposal is independent of the specific details of the visible sector baryogenesis model. Reproducing the observed baryon asymmetry is sufficient for the purpose of this work. Our primary focus will be to demonstrate how the 2 symmetry ensures the origin of the dark matter abundance in the twin sector, with a particular focus on the baryon asymmetry dependence on the phase and coupling given either by (<ref>) or (<ref>). What is clear is that a generic baryogenesis model that relies on the out-of-equilibrium decay of some new particle should have a similar dependence on these parameters. As such, our findings have implications beyond the specific model we have presented. We leave the extension of the model to higher temperatures via leptogenesis, the achievement of baryogenesis without hierarchical couplings, and the origin of the reheaton in the non-thermal case for future work that focuses explicitly on baryogenesis.
§.§ Twin Darkogenesis
Now that we have a successful baryogenesis model, we can compute the corresponding DM asymmetry using the twin mechanism discussed in Section 2. The 2 mirror symmetry results in a twin baryon asymmetry that generates the dark matter abundance in the twin sector. Then, to compute the DM abundance, we use the baryogenesis model of Section <ref>. This approach means that, analogously to the SM sector case, we introduce an out-of-equilibrium, CP, and baryon number violating decay. The new twin sector particles are two neutral fermions Ñ_1,2 and a twin-colored scalar X̃ in the (3,1)_2/3 representation of twin QCD, S̃Ũ(3). Then, we can add the following interactions to the twin part of the theory,
Δℒ_Bgen^twin= λ_iαN_αX_a (u_R)^i_a + ξ_ijϵ^abcX_a (d_R)^i_b (d_R)^j_c + h.c.
Here, i,j are the twin-quark generation indices, α=1,2 is the neutral fermion flavor, and a,b,c are twin-color indices. The tilde superscripts indicate that all quantities are associated with the twin sector.
The mechanism for generating the ADM abundance is the result of imposing the 2 symmetry on the baryogenesis mechanism of the previous section. The out-of-equilibrium decay of the twin Ñ_1 violates CP and baryon number and generates a CP asymmetry. As for the case of the N_1, here, the Ñ_1 yield can be obtained thermally or non-thermally. Then, we can write the DM asymmetry as the twin baryon asymmetry parameter as
Y_DM^ thermal/8.7× 10^-11 =(213.5/g_*,S(T_FO)) ( 3 ℱ_S( 1.5 ) +ℱ_V( 1.5 )/15.3) (|λ _u2|sin^1/2ϕ/2.3× 10^-4)^2,
where the expression above corresponds to the thermal determination of the yield. In the non-thermal process, we would have a similar expression for the DM asymmetry, except for the non-thermal yield (<ref>).
Our work does not introduce any new interactions between the visible and twin sectors generating the baryon and DN asymmetries. In this way, we emphasize that the mirror 2 mechanism uniquely gives their common origin without any need for cogenesis and asymmetry transfer between the two sectors. Thus, this method for realizing the ADM idea is similar to previous models of mirror DM <cit.>. One could worry that additional renormalizable interactions could be present or generated in the theory. However, these are very suppressed in our model. Since the particles we introduced in the SM sector do not interact with the Higgs directly and the Higgs portal is the only communication between the twin and SM sectors, any visible-twin interactions happen only at multiple loop order. Lastly, the MTH is expected to be UV completed near the cutoff of the theory. Therefore, one could imagine that both Δℒ_Bgen and Δℒ_Bgen^twin have a common origin approximately at the scale 4π f. In this case, there could be more renormalizable portals beyond the twin Higgs. However, to introduce any new portals, we would need to make assumptions about the structure of the UV theory, which is beyond the scope of this paper. The most important aspect of the baryogenesis extensions we added to both sectors is that they leave the hierarchy problem unaffected, as there are no new interactions with the Higgs.
Since the DM abundance is larger than the baryon abundance, some source of misalignment will be necessary to achieve darkogenesis. The relation between the abundances is
Ω_DM/Ω_B= n_DM/n_Bm_DM/m_B∼ 5.
As discussed in Section <ref>, if there is a process that enforces n_B∼ n_DM, we must have m_DM∼ 5m_B. It is difficult to achieve this mass around 5 with only soft 2 breaking in the MTH. To see this, we observe that given that DM is a twin nucleon, the ratio of the QCD and twin QCD confinement scales can predict the ratio between the DM and nucleon masses. In appendix <ref>, we derive the ratio of the two QCD scales, which is given by
Λ_/Λ _≃ ( f/v )^2/9
This scaling would result in a twin nucleon mass not much above 1. Thus, a hard 2 breaking was usually assumed in QCD running couplings to make the twin nucleon heavier. The central point of this work is to show that it is possible to have a ≃ 1 twin nucleon DM candidate without resorting to hard 2 breaking. To show that, we can rewrite (<ref>) to make explicit the dependence on the baryon and twin baryon asymmetries, Y_Δ B and Y_DM. This results in
Ω_DM/Ω_B=m_p/m_pY_ DM/Y_Δ B(1-r/1-r),
where we defined the baryon and twin baryon fractional asymmetries r and r as
r=n_B/n_B, r=n_B/n_B.
Because we assumed that the SM and twin sectors have the same mechanism to generate the asymmetry, the fractional asymmetries are expected to be the same. Then, using either (<ref>) or (<ref>) and m_p^twin/m_p≃ (f/v)^2/9, we can write (<ref>) as
Ω_DM/Ω_B=( f/v)^2/9(3 ℱ_S ( Δ_N )+ℱ_V ( Δ_N ) / 3 ℱ_S ( Δ_N )+ℱ_V ( Δ_N ) )|λ_u2|^2/|λ_u2|^2|sinϕ/sinϕ|,
where Δ_N=M_N_2^2/M_N_1^2 and Δ_N=M_N_2^2/M_N_1^2.
Since we are not interested in introducing hard 2 breaking, we set |λ_u2|=|λ_u2|. Also, we assume that the mass splittings Δ_N and Δ_Ñ are the same in both sectors. In any case, and as already discussed in Section <ref>, the final abundances only depend weakly on the change of these parameters. Finally, because of the 2 symmetry, the fractional asymmetries in both sectors should be the same. In this way we can rewrite (<ref>) as
Ω_DM/Ω_B≃( f/v)^2/9|sinϕ/sinϕ|≃ 5.
Therefore, it is possible to satisfy the ADM requirement for the DM to baryon abundance if the phases in the visible and twin sectors are misaligned. In Fig. <ref>, we show the allowed phases needed in order to satisfy relation (<ref>).
As we argued in section <ref>, the misalignment of the phases can be viewed as an IR effect and does not qualify as hard 2 breaking. One source of misalignment comes from the relative phase between the SM and twin vevs, δ, defined by (<ref>). Once there is a relative phase difference in the SM and twin masses, the CKM field redefinition introduces different phases on the couplings of (<ref>) and (<ref>).
SM: (Flavor Basis) λ_iα N_αX^a (u_R^ i)_a ⟶λ_iα U^ij_u,R N_αX̅^a (u_R^ j)_a (Mass Basis).
Twin: (Flavor Basis) λ_iα N_αX^a (u_R^ i)_a ⟶λ_iα e^iδ U^ij_u,R N_αX̅^a (u_R^ j)_a (Mass Basis).
Here, U_u,R and e^iδU_u,R are the unitary matrix used to diagonalize the Yukawa terms in the SM and twin sector, respectively. Even if the exact 2 ensures λ_iα=λ_iα in (<ref>) and (<ref>) if the phase is non-zero we expect different imaginary parts of the visible and twin couplings.
λ_iα U^ij_u,R≠λ_iα e^iδU^ij_u,R.
We conclude that having different phases is not a hard breaking of the 2 since it is still a symmetry of the UV theory and does not affect the hierarchy problem in any way. Concerning the UV theory, we are not addressing the specific structure of the twin Higgs model in the UV; we only assume that the 2 can arise as an exact symmetry at those scales. Then, going to IR scales, the phase misalignment mechanism could have other sources beyond the twin Higgs potential. In general, it is difficult to point out all the sources of phase misalignment since this would imply that we have complete knowledge of the flavor sector of the theory[Even in the SM, we do not have information on the origin of the CP phase in the CKM matrix for example.] and all the relaxation mechanisms that took place in the thermal evolution of the model. Therefore, we are justified in treating the phases ϕ and ϕ as unknown parameters and scanning for the values that reproduce the DM to baryon ratio as in Fig. <ref>. Once we reproduce the observed abundances of DM and visible matter, as well as the baryon asymmetry, we can study the phenomenological implications of the model and see how it can be constrained or observed in the future.
§ PHENOMENOLOGY
Once we have obtained the observed baryon asymmetry from the twin ADM model, we can study the phenomenological constraints and signals of the model. Usually, the most important constraints on twin Higgs models come from cosmology. In the original implementation of the MTH, a mirror SM copy in a hidden sector induces significant contributions to dark radiation at Cosmic Microwave Background (CMB) and Big Bang Nucleosynthesis (BBN) epochs. If the twin sector had a thermal history similar to the SM, relativistic twin neutrinos and photons would contribute to the total effective number of relativistic degrees of freedom of the universe.
Many solutions in the literature deal with the potential cosmological problems of Twin Higgs models. One straightforward approach is to decrease the number of relativistic degrees of freedom in the Twin sector with explicit 2 breaking. As discussed in section <ref>, this strategy is used in the Fraternal Twin Higgs model where only the third generation of fermions are kept in the Twin sector <cit.>. This structure is the minimal particle content to address the electroweak Hierarchy Problem. However, it relies on the hard breaking of the 2 symmetry at the UV scale, which does not apply to this work. Another possibility is an asymmetric reheating that injects more energy into the SM sector than the twin sector. This possibility was considered in <cit.>. In asymmetric reheating, a massive long-lived particle freezes out from the thermal bath while still relativistic. As the universe expands, they become non-relativistic and decay in both sectors after they decouple. However, in these mechanisms, the massive particle decays are preferentially arranged to decay into the SM sector. As a consequence, the temperature in the visible sector will be bigger than in the Twin sector, alleviating the Δ_ tension. In principle, we could implement asymmetric reheating in the decays of N_1 of our model. However, since there are preferential decays to the SM, the numerical predictions we found in the last section would change. Instead, we assume a more straightforward solution to the Δ_ tension and leave other implementations for future work.
A simple solution to the cosmological problems of the MTH was introduced by <cit.> and worked out in <cit.>. The idea is to give a large mass to twin neutrinos, making them decouple non-relativistically from the thermal plasma much earlier in cosmic history. Effectively, the twin neutrino contribution to Δ_ is removed. We assume that a seesaw-like mechanism exists and is responsible for generating large twin neutrino masses. The implementation details can be found in previously mentioned literature on the twin neutrinos solution to Δ N_ eff.
Once the twin neutrinos are heavy, our scenario in the MTH has a single viable candidate for DM - twin neutrons. The argument proceeds as in <cit.>. Since Δ N_ eff with heavy twin neutrinos is within experimental bounds, we can keep the twin photon in the spectrum to enforce the 2 symmetry[If the SM prediction of Δ N_ eff remains confirmed with future data, keeping the twin photon in the model can become problematic beyond the 3σ level. If this becomes the case, one could implement asymmetric reheating to avoid the cosmological bounds or have a massive twin photon as in <cit.>.]. Only twin baryon number B̃ is generated with no twin Lepton asymmetry. Therefore, all twin leptons can annihilate, and twin electrons are not DM candidates[If the twin electrons do not annihilate, they could potentially add to the abundance to the point of leading to overclosure.]. The charge neutrality of the universe requires that there is no net production of twin protons since they cannot combine into neutral objects. Therefore, after the twin QCD phase transition a net twin neutron ñ number is generated. Finally, twin nucleosynthesis cannot proceed in the presence of heavy twin neutrinos since there are no protons to combine with neutrons. Because the neutron is stable and the only twin relic, we conclude that dark matter is made entirely of ñ.
Now that we have established that the MTH DM candidate is the twin neutron, we can study the direct detection signals. As previously mentioned, we can estimate the mass of the ñ to be near that of the visible nucleons, corrected by the twin sector scale. The precise relation follows from the definition of the QCD scales in both sectors. The leading order contribution to Λ_ QCD arises from the running of the strong couplings coupling,
α_s(Q^2)=1/b_0(N_f) lnQ^2Λ^2_ QCD
Here, we have defined b_0(N_f)=33-2 N_f and N_f is the number of active quark flavors lighter than the relevant scale m_f<Q. Because of the N_f dependence, there will also be an effect on Λ_ QCD due to the quark mass thresholds. In appendix <ref>, we compute the mass-threshold contributions due to integrating out the heavy quark states f=t,b,c to the QCD scale. If we divide the QCD and twin QCD scales, we obtain the following relation
Λ_ QCD/Λ_ QCD∼(y_t/y_ty_b/y_by_c/y_c)^2/27(f/v)^2/9exp[-2π/9(1/α̃_s-1/α_s)]
Assuming there is no hard 2 breaking, we can set ỹ_f=y_f and α̃_s=α_s. Finally, there is only a soft 2 breaking due to the heavier vev of the twin sector, and we recover equation (<ref>),
Λ̃_/Λ _≃ ( f/v )^2/9.
Since the neutron and twin-neutron masses are proportional to their respective QCD scales, we can write
m_ DM = m_n=(f/v)^2/9 m_ n.
Then, assuming that f/v≳ 3 from the LHC Higgs coupling measurements <cit.> and f/v≲ 10 to limit the fine-tuning of the model, we arrive at a rather narrow range of DM mass in this model:
1.2≲ m_ DM≲ 1.6 .
Next, we estimate the nucleon-DM cross-section. The starting point is understanding the halo's local dark matter profile. Because twin dark matter is twin neutrons, its self-interactions should be of the order of the nucleon cross-sections, around ∼ 1cm^2/g at energies of a few . Because of this value, we observe that twin DM is within or borderline close to the bounds from small-scale structure formation and merging clusters <cit.>. While the suppression of small-scale structure could be a signal of this or other similar ADM models, we leave this part for future work. Several complications are still under debate in the literature regarding the need for suppression in small scales[The reliability of the collisionless cold dark matter simulations to predict small-scale structure suppression and the role of baryonic feedback are some examples of recent discussions in the literature.]. We therefore assume the twin-neutron self-interaction cross-section satisfies the bound. Assuming this, we then expect twin dark matter to have an approximately uniform distribution within the galaxy halo, allowing for the usual dark matter halo profile and velocity distribution.
Direct detection of twin dark matter assumes that the two sectors communicate. This communication can occur either through the Higgs portal or other operators at the UV completion scale of the MTH model. Since we are interested in the scattering of nuclei and twin DM at low energies, we can use the effective theory of light quarks and twin quarks. Generically we can write
ℒ_ eff=c_q q^ij/Λ^2(q_iΓ q_i)(q_jΓq_j),Γ,Γ=1, iγ_5,γ^μ, γ_5γ^μ,σ^μν,
where, i=u,d,s and j=ũ,d̃, s̃. c_q q^ij are the Wilson coefficients of the operator and Λ is some scale high compared to 1. In general, we can write different Lorentz structures, Γ. However, for our purposes, we are only interested in effective quark operators that generate spin-independent interactions that survive the point-like nucleon approximation. Therefore, we only keep Γ,Γ=1,γ^μ since these generate spin independent NR interactions <cit.>.
In the case of (<ref>) being generated by the Higgs portal interaction, we have the following scalar 4-fermion operator,
ℒ_ eff^ higgs=y_i y_j/m_h^2ξ (q_i q_i)(q_jq_j),
where ξ=v^2/f^2 and we used the 2 symmetry to write the twin quark Yukawa coupling ỹ_j to be equal to the visible Yukawa couplings. In this case, we expect this operator to generate a small nucleon cross-section since a double suppression comes from the Yukawa couplings of the light SM and twin quarks.
The other possibility is that the effective operators (<ref>) are generated at the MTH cutoff. Considering this case, we can write the two operators that generate spin-independent non-relativistic interactions, a scalar and a vector operators:
ℒ_ eff^Λ_S=c_S/Λ_S^2(q_i q_i)(q_jq_j), ℒ_ eff^Λ_V=c_V/Λ_V^2(q_iγ^μ q_i)(q_jγ_μq_j).
Furthermore, we absorb the coefficients of the scalar and vector operators into the definition of the cutoff scales Λ_S,V, effectively setting c_S=c_V=1.
The spin-independent cross-section, σ_SI, can be calculated using standard methods as described in appendix <ref>. For the scalar and vector operators, σ_SI is given by
σ_SI^ scalar=μ_Nñ^2/πf_N^2 f_ n^2/Λ_S^4, σ_SI^ vector=μ_Nñ^2/πb_N^2 b_ n^2/Λ_V^4.
where μ_Nñ is the reduced mass of the twin-neutron and nucleon system, and the zero momentum constants are derived from the form factors as
f_N=∑_q f_Tq^(N)≃ 0.3,
f_n=∑_q f_Tq̃^( n)≃ 0.3,
b_N=∑_q F_1^q,N(0)=3,
b_n =∑_qF_1^q,N(0)=3 .
Notice that the vector form factors are ten times larger than the scalar ones at zero momentum. Since the form factor goes with the fourth power in (<ref>), there will be a significant difference in reach for the scales in the vector and scalar operators.
In Figures <ref> and <ref>, we show the spin-independent twin-neutron nucleon scattering cross-section parameter space for the scalar and vector operators, respectively. The green rectangle shows the allowed DM mass given by (<ref>) and the scale of the operator for each cross-section. The different plots highlight the contrasting reach of the scalar and vector scales, with high Λ_S down into the neutrino fog. In the scalar case, the Higgs portal appears at a higher effective scale, around Λ_S ∼ 40, due to the double suppression of the first generation Yukawa couplings. The filled regions are the current exclusion bounds from Darkside 2022 data <cit.>, CRESST-III <cit.> and XENON1T <cit.>. In the case of Darkside, nuclear recoils are subject to quenching effects, which cause a reduction in the energy signal due to various mechanisms whose statistics are not fully understood. Because of these effects, <cit.> considered two models to bound the quenching effect region where quenching fluctuations are suppressed (NQ) or unsuppressed (QF). NQ corresponds to the filled solid pink region (DarkSide50 2022), and QF is the DS50 QF curve in Figs. <ref> and <ref>. While the quenching factor can vary between events, it is typically quantified using calibration sources and simulations. Once these analyses are done, the real exclusion region should lie somewhere in between the NQ and QF curves.
For the neutrino background, we present the Xenon neutrino fog as defined by <cit.>. The index n, the gradient of the DM discovery limit over some exposure measure, labels the different neutrino fog curves and is given by
n=-(dlogσ/dlog N)^-1 ,
where σ is the discovery limit, and N is the number of events. Given a cross-section experimental sensitivity, this definition means that reducing the sensitivity by a factor of x requires increasing the exposure by x^n. Therefore, future experiments can put exclusion bounds inside the neutrino fog region by having sufficient exposure time. The dashed lines correspond to projections by the SuperCDMS <cit.> and SBC <cit.> experiments. A large portion of the parameter space for twin dark matter will likely be probed in the future, especially for the effective vector operators. Due to the smaller scalar form factors, reaching very high cutoff scales is more challenging, and part of the interesting parameter space is down the n>3 neutrino fog region. Promising strategies beyond maximizing exposure could be adopted to probe this region. One of these is using the directionality of the neutrino flux to reduce their background. For a review of direct detection prospects below the neutrino fog, we point out to <cit.>.
We can conclude from the figures above that the interesting scales for the UV completion of the MTH, typically of the order of 10TeV, are beginning to be probed by the Darkside collaboration. This is clearly the case for the vector operator (Figure <ref>). On the other hand, for the scalar operator (Figure <ref>) the suppressed sensitivity resulting from the smaller zero momentum constants in (<ref>) puts this interesting UV completion scale under the neutrino fog, making its detection more challenging. Additionally, the Higgs portal should be always present in the MTH independently of the UV completion. Therefore, reaching the Twin Higgs portal cross section has the potential of excluding or confirming the model. However, due to the double first-generation Yukawa coupling suppression, the signal for direct detection goes deep into the neutrino fog, with difficult experimental prospects. In any case, we see that the direct exploration of the parameter space of this ADM scenario of the MTH is becoming feasible in current and future experiments.
To finish this section, we briefly comment on other possibilities for the phenomenology of the presented model. First, bounds from neutrons oscillation experiments do not apply here since the interactions of X and quarks in (<ref>) is anti-symmetric in flavor. Additionally, the charged X production could be explored at the LHC. This paper assumes that X is heavy enough to be out of reach by collider experiment. However, we are pursuing the collider phenomenology of this low-temperature baryogenesis scenario in a forthcoming publication.
§ CONCLUSIONS
The primary focus of this paper is to present a Mirror Twin Higgs implementation of asymmetric dark matter, as a proof of principle that there is no need to introduce a hard 2 breaking in order to have a consistent dark matter candidate. We focused on adding a particular baryogenesis model to the visible sector of the MTH that successfully generates the baryon asymmetry at low temperatures after sphaleron decoupling. The model relies on the out-of-equilibrium decays of a neutral fermion that violates baryon number and CP symmetry. We showed that the observed baryon asymmetry can be correctly obtained, assuming that the neutral fermion is produced in the early universe, independently of weather this production is thermal or non-thermal.
The 2 symmetry of the MTH model extends the cosmological mechanism responsible for baryogenesis to the twin sector. This mirroring gives rise to an abundance of asymmetric dark matter predominantly composed of twin neutrons. Misalignment of the complex phases between the visible and twin sectors make possible a dark matter abundance consistent with the observed value of Ω_ DM≃ 5 Ω_B. This phase misalignment could arise solely as an IR effect, ensuring that no hard breaking 2 needs to be introduced. A simple example is vacuum misalignment between the two scalar sectors, leading to different phases in their couplings entering the CP asymmetries. This, as well as similar misalignments in the phases of fields entering the singlet couplings in both sectors, can be IR effects and therefore thought of as soft Z_2 breaking, Preserving the UV 2 symmetry of the twin Higgs model is a desirable feature concerning the electroweak stability of the theory, and its protection guarantees that the solution to the little hierarchy problem remains unspoiled, without the need of further tunings.
It is possible to reach the same final DM abundance with different baryogenesis implementations. Because of this, our results are not limited to the specific baryon asymmetry mechanism used in section <ref>. Once a visible baryon asymmetry is achieved, the 2 symmetry and phase misalignment are enough to reproduce the abundance ratio between DM and baryons. Consequently, the mechanism exhibits the potential for generalization to alternative baryogenesis models, such as high-scale leptogenesis or electroweak baryogenesis. These extensions can be explored in future developments of ADM in the MTH framework.
Regardless of the chosen baryogenesis model, implementing ADM in the MTH model without hard 2 breaking predicts that dark matter consists mainly of twin neutrons with masses ranging from 1.2 to 1.6. A significant part of the parameter space is probed assuming an effective interaction of the light quarks and twin quarks. Part of the parameter space is excluded by the data from the Crest-III and Darkside-50 experiments. Promisingly, future experiments such as SuperCDMS and SBC can probe higher effective scales beyond the TeV range. Furthermore, direct detection experiments below the neutrino fog hold significant potential for uncovering the nature of twin asymmetric dark matter. We conclude that the mirror twin Higgs model is a well-motivated BSM approach to address the electroweak stability, the nature of DM, as well as the origin of the baryon asymmetry with the same core concepts. The model presents a compelling candidate for dark matter in the range, requiring extensive exploration through future DM detection experiments.
The authors thank Ivone Albuquerque, Nicolás Bernal, Chee Sheng Fong and Seth Koren for helpful discussions.
They also acknowledge the support of FAPESP grants 2019/04837-9 and 2021/02757-8, and CAPES 88887.816450/2023-00.
§ QCD AND TWIN QCD SCALES
In this appendix we derive the leading order relationship between the SM QCD and twin QCD scales we used throughout the text,
Λ_QCD/Λ_QCD= (f/v)^2/9.
Following <cit.>, the derivation makes use of the quark-mass threshold contributions to Λ_QCD. At leading order, the running coupling α(Q^2,N_f) can be written for momentum greater than the top quark as
α_s(Q^2,6)=1/b(6)log Q^2/Λ_UV^2, Q^2>m_t^2.
Here, N_f=6 is the number of active quark flavors at high energies and b(N_f)=33-2N_f. Crucially, the UV QCD scale defined in this relation is the same between the visible and the twin sector.
The first quark threshold correction appears when we integrate out the top-quark. The coupling below the top quark mass can be written as
α_s(Q^2,5)=1/b(5)log Q^2/Λ_UV^2+c, Q^2>m_b^2,
where c is a constant fixed by requiring the matching between the theory with six and five quarks at the top-quark mass scale, α_s(m_t,6)=α_s(m_t,5). Calculating c we arrive at
1/α_s(Q^2,5)=b(5)logQ^2/m_t^2+b(6)logm_b^2/Λ_5^2,
However, α_s(Q^2,5) also defines the QCD scale for the theory with only 5 active quarks, Λ_5.
1/α_s(Q^2,5)=b(5)logQ^2/Λ_5^2, Q^2>m_b^2.
Comparing (<ref>) to (<ref>), we arrive at a relation for the 5 quark-flavors QCD scale.
Λ_5=Λ_UVm_t^1-b(6)/b(5)/Λ_UV^1-b(6)/b(5).
We can do the same procedure to obtain the quark-mass threshold contributions to the QCD up to the charm-quark. Beyond this point, the theory becomes strongly interacting and we cannot perturbatively integrate out the light quark-flavors since they are below the QCD scale. Thus, the definition for the QCD scale includes threshold contributions from the three heavy states, the top, bottom and charm quarks. We can them write
Λ_QCD≡Λ_3 = Λ_UV^b(6)/b(3) m_t^(1-b(6)b(5))b(5)b(3) m_b^(1-b(5)b(4))b(4)b(3) m_c^(1-b(4)b(3)).
Substituting the values of b(N_f) and using that the mass-Yukawa relation m_q=y_q v, we obtain
Λ_QCD=Λ_UV^7/9 y_t^2/27 y_b^2/27 y_c^2/27 v^2/9.
Similarly, for the twin QCD scale we can write
Λ_QCD=Λ_UV^7/9y_t^2/27y_b^2/27y_c^2/27 f^2/9.
Since there is no 2 breaking in the model, we can write the twin Yukawa couplings as y_q =y_q. Finally, dividing (<ref>) by (<ref>) we obtain the proposed relation,
Λ_QCD/Λ_QCD= (f/v)^2/9.
§ SPIN-INDEPENDENT CROSS-SECTION OF TWIN DM
Now, to find the cross-section, we compute the nucleon-DM scattering matrix. To first order in the perturbative expansion, we have the following nucleon amplitude
ℳ_ N = ⟨n' N'|ℒ_ eff|n N⟩ = 1/Λ^2⟨ N'| q_iΓ q_i |N ⟩⟨n'| q_jΓq_j |ñ⟩.
We use N' and n' to denote the nucleon and twin-neutron final states, respectively. The nucleon-spinor bilinears can parameterize each matrix element,
⟨ N'| q_iΓ q_i |N ⟩ = ∑_i=u,d,s F_i^(N)(q^2) u_N'Γ u_N,
where F_l^(N)(q^2) are the hadronic form factors associated to the nucleons N=p,n. For direct detection of twin DM, it is sufficient to use the hadronic form factors at zero transferred momentum since their variation is negligible compared to the recoil energies considered. In this limit, we can relate the scalar form factors with the fraction of the nucleon mass carried by the light quarks,
⟨ N| qq |N⟩ =m_N/m_qF_i^(N)(0)u_N u_N = f_Tq^(N)u_N u_N.
The vector form factors are related to the conserved flavor singlet vector current associated with the baryon number.
⟨ N| qγ^μ q |N ⟩ = F_i^q,N(0)u_N γ^μ u_N
The form factors at zero momentum can be obtained by perturbative and lattice calculations or by experiment. We used the values from <cit.>.
Finally, we can calculate the spin-independent cross-section.
σ_SI^ scalar=μ_Nñ^2/πf_N^2 f_ n^2/Λ_S^4, σ_SI^ vector=μ_Nñ^2/πb_N^2 b_ n^2/Λ_V^4.
where μ_Nñ is the reduced mass of twin-neutron and nucleon, and we have defined the constants
f_N=∑_q f_Tq^(N)≃ 0.3,
f_n=∑_q f_Tq̃^( n)≃ 0.3,
b_N=∑_q F_1^q,N(0)=3,
b_n =∑_qF_1^q,N(0)=3.
Notice that the vector form factors are ten times larger than the scalar ones at zero momentum. Since the form factor goes with the fourth power in (<ref>), there will be a significant difference in reach for the vector and scalar probes to the scale of the operator.
JHEP
|
http://arxiv.org/abs/2307.05999v2 | 20230712082627 | Flexible and Fully Quantized Ultra-Lightweight TinyissimoYOLO for Ultra-Low-Power Edge Systems | [
"Julian Moosmann",
"Hanna Mueller",
"Nicky Zimmerman",
"Georg Rutishauser",
"Luca Benini",
"Michele Magno"
] | cs.CV | [
"cs.CV",
"eess.IV"
] |
All three authors contributed equally to this research.
1234-5678-9012
ETH Zurich
Zurich
Switzerland
[email protected]
[1]
ETH Zurich
Zurich
Switzerland
[email protected]
[1]
University of Lugano
Lugano
Switzerland
[email protected]
ETH Zurich
Zurich
Switzerland
[email protected]
ETH Zurich/University of Bologna
Zurich/Bologna
Switzerland/Italy
[email protected]
ETH Zurich
Zurich
Switzerland
[email protected]
Tiny-issimo-YOLO
This paper deploys and explores variants of TinyissimoYOLO, a highly flexible and fully quantized ultra-lightweight object detection network designed for edge systems with a power envelope of a few milliwatts. With experimental measurements, we present a comprehensive characterization of the network's detection performance, exploring the impact of various parameters, including input resolution, number of object classes, and hidden layer adjustments. We deploy variants of TinyissimoYOLO on state-of-the-art ultra-low-power extreme edge platforms, presenting an in-depth a comparison on latency, energy efficiency, and their ability to efficiently parallelize the workload. In particular, the paper presents a comparison between a novel RISC-V-based parallel processor (GAP9 from Greenwaves) with and without use of its on-chip hardware accelerator, an ARM Cortex-M7 core (STM32H7 from ST Microelectronics), two ARM Cortex-M4 cores (STM32L4 from STM and Apollo4b from Ambiq), and a multi-core platform aimed at edge AI applications with a CNN hardware accelerator (Analog Devices MAX78000). Experimental results show that the GAP9's hardware accelerator achieves the lowest inference latency and energy at 2.12 and 150 respectively, which is around 2x faster and 20% more efficient than the next best platform, the MAX78000. The hardware accelerator of GAP9 can even run an increased resolution version of TinyissimoYOLO with 112×112 pixels and 10 detection classes within 3.2ms, consuming 245. To showcase the competitiveness of a versatile general-purpose system we also deployed and profiled a multi-core implementation on GAP9 at different operating points, achieving 11.3 with the lowest-latency and 490 with the most energy-efficient configuration. With this paper, we demonstrate the flexibility of TinyissimoYOLO and prove its detection accuracy detection datasets. We demonstrate its suitability for real-time ultra-low-power edge inference by benchmarking its performance.
< g r a p h i c s >
TinyissimoYOLO as deployed on GAP9 (both the 8-core cluster and the hardware accelerator) by this paper.
Flexible and Fully Quantized Ultra-Lightweight TinyissimoYOLO for Ultra-Low-Power Edge Systems
Michele Magno
August 12, 2023
===============================================================================================
§ INTRODUCTION
With the widespread adoption of edge devices, particularly mcu based nodes, the iot is revolutionizing various domains including health monitoring, smart manufacturing, and home automation <cit.>. These low-power devices enable increased automation, cost reduction, bandwidth optimization, and enhanced privacy by processing collected data on the edge, i.e., directly on the sensor node. However, the limited memory and computing resources of mcu pose a significant challenge when it comes to deploying advanced machine learning models.
Overcoming this challenge and enabling efficient machine learning on resource-constrained devices is a crucial area of research in embedded networked sensor systems, as illustrated by <cit.>.
A foundational element of many iot applications is the extraction of semantic information about the environment with image sensors. Specifically, object detection <cit.>, the vital task of identifying and precisely localizing objects within a given image, plays a pivotal role in a wide range of systems. For instance, in the realm of autonomous mobile agents, object detection enables obstacle avoidance and tracking, path planning, and scene understanding, contributing to safe and efficient navigation. Similarly, in augmented and virtual reality devices, such as head-mounted displays, lightweight and energy-efficient object detection algorithms are crucial to enable real-time visual recognition without exceeding the limited power resources of the wearable device.
A majority of recent research on object detection focuses on perfecting detection accuracy. State-of-the-art models have on the order of 10^8 or more parameters, requiring power-hungry hardware such as GPUs <cit.> for inference. Therefore, state-of-the-art models like the yolo series of networks <cit.> cannot be directly ported to low power edge processors due to the memory and compute constraints of mcu.
Consequently, there is a growing demand for approaches to enable semantic understanding on edge ultra-low power, constrained hardware. This shift has led to a surge of interest in various research areas, including architecture search, quantization techniques, and advanced inference engines tailored for resource-constrained devices <cit.>. mcu are now being equipped with novel open-source energy-efficient cores, such as RISC-V cores, parallel processing engines, dedicated hardware accelerators, and specialized co-processors aimed at enabling efficient execution of complex machine learning tasks <cit.>.
By combining these advancements in parallel processing, hardware accelerators, and quantization techniques, mcu are now executing quite sophisticated ml models.
However, deploying machine learning on mcu is still far from trivial and mapping networks like yolo for advanced visual tasks, beyond simple classification, is still at the boundary of feasibility.
This work presents a cutting-edge yolo-based network designed to push the boundaries and to achieve accurate and real-time object detection on mcu at sub-millijoule inference energy consumption. By leveraging novel low-power processors, we move from TinyissimoYOLO <cit.> and present an improved, highly flexible network architecture – flexible TinyissimoYOLO, shown in <ref>. Our approach capitalizes on the unique capabilities of the latest generation of mcu platforms, harnessing their computational capabilities to set a new state of the art in energy-efficiency object detection, with a memory footprint of below 500. Through extensive evaluation and performance analysis, we demonstrate the potential of our network in enabling energy-efficient computer vision applications. This research contributes to the advancement of energy-efficient processing on resource-constrained devices, opening up new possibilities for a wide range of real-world applications. To support our contribution, we evaluate a complete pipeline for deploying accurate, lightweight quantized object detection on several novel mcu, bridging the gap between state-of-the-art models and on-edge execution. We provide a detailed description for each part of our pipeline, relying as much as possible on open-source tools, with the hope of enabling others to successfully deploy object detection models on ultra-low-power edge systems.
To quantify the benefits of the proposed approach, we evaluate the performance and energy efficiency of our object detection pipeline running on several mcu, executing different TinyissimoYOLO networks on one or multiple general-purpose cores as well as specialized cnn accelerators.
We investigate and expand the TinyissimoYOLO network proposed by Moosmann et al. <cit.>, a lightweight general multi-object detection network optimized for a single processor, by extending its flexibility. Depending on the target platform's capabilities, TinyissimoYOLO can be scaled to make the best use of the target hardware and offer optimal performance. We deploy TinyissimoYOLO on multiple microcontroller architectures: the GAP9 multi-core RISC-V mcu (Greenwaves), which features a hardware accelerator for cnn, MAX78000 (Analog Devices), a multi-core platform with an energy efficient cnn accelerator, Apollo4b (Ambiq), the most power-efficient ARM Cortex-M4 core currently available, and STM32L4R9 and STM32H7A3 from STMicroelectronics to establish a baseline with the most popular ARM cortex-M4F and ARM cortex-M7 cores.
Furthermore, for GAP9, we map a Pareto front of operating points (supply voltage and operating frequency) to evaluate the trade-off between latency and inference energy. In addition, we evaluate the per-layer inference power consumption and quantify the efficiency enhancements enabled by the integrated NE16 cnn accelerator.
Additionally, we explore several variations of TinyissimoYOLO, highlighting the trade-offs between prediction accuracy and resource consumption. We evaluate the detection performance for a set of image resolutions and different kernel sizes applied in the first layer. We report the size of the network for each modification and the detection accuracy. Our investigation demonstrates that, as expected, increased input resolution contributes to higher detection accuracy, but also emphasized the increasing memory consumption.
The rest of this paper is organized as follows: <Ref> provides an overview of the works focusing on cnn optimization and deployment on microcontrollers. <Ref> details our deployment pipeline, from choosing the architecture and training, to quantization and porting to target hardware. <Ref> specifies the experimental setup and reports the results, presenting an in-depth analysis of the trade-off between performance and energy efficiency. Furthermore, we compare our GAP9-deployed model against state-of-the-art deployments on different mcu, focusing on latency, energy efficiency and inference efficiency (MACs/cycle). Lastly, <Ref> concludes our work.
§ RELATED WORK
In the past decade, deep learning approaches have revolutionized the field of image-based scene understanding, through object detection <cit.>and semantic segmentation <cit.>. Semantic understanding of objects in the environment is an essential capability for autonomous agents, for tasks such as localization <cit.>, mapping <cit.>, and navigation <cit.>. You Only Look Once or commonly abbreviated as YOLO is one of the most popular and optimized deep learning algorithms used to perform real-time detection <cit.>. To effectively detect and track objects, YOLO uses a repurposed classifier or localization which is a model applied to an image at several locations and scales <cit.>. However, the majority of works require power-hungry hardware such as GPUs and are not suitable for deployment on low-power edge devices. Even some of the more resource-conscious approaches (<cit.>) still require powerful hardware consuming multiple watts and requires several tens megabytes of memory to run inference at sensor-rate. Based on the recent literature, we observed that existing YOLO approaches tend to have high memory requirements, limiting their applicability on resource-constrained devices. To address this limitation, in this paper, we evaluate and optimize a novel and flexible lightweight algorithm inspired by YOLOv1. The algorithm is specifically designed to achieve optimal performance in terms of accuracy while keeping the memory requirements below 500. This breakthrough in memory efficiency makes our algorithm well-suited for low-power mcu, which serve as the processing units for a wide range of extreme edge devices. By significantly reducing memory demands without compromising accuracy, our approach opens new avenues for deploying efficient and accurate object detection on resource-constrained devices.
The deployment of image classification and object detection models on mcu has garnered significant attention in recent years. Canepa et al. <cit.> propose a method for detecting specific objects in surveillance video frames using deep neural networks on an STM32 mcu. Although they achieve high prediction accuracy, their slow inference rate (0.03) limits its suitability for real-time applications, and their power consumption of approximately 400 is relatively high. To enable semantic understanding on edge devices and small autonomous agents, the models must be small and efficient enough to be executed on low-memory (<1) and ultra-low-power platforms (<100) at significantly higher inference rates.
This task requires a specialized workflow, composed of three main strategies.
Firstly, exploiting the benefits of resource-aware neural architecture search <cit.>. Secondly, pruning and quantization strategies <cit.>, as offered by commonly used deep learning frameworks such as PyTorch. Notable quantization frameworks targeting ultra-low-power hardware include TensorFlow Lite <cit.> and Microsoft NNI <cit.>, and other academic platforms.
And lastly, deployment on lightweight hardware using inference engines, which aim to improve data locality, memory usage, and spatiotemporal execution.
TinyML software suites<cit.>, including the open-sourced TensorFlow Lite Micro<cit.>, EdgeML<cit.> and CMSIS-NN <cit.>, allow for deploying neural networks on mcu and are mainly designed for ARM Cortex-M and as such less attractive for RISC-V based processors.
Similarly, Wulfert et al.<cit.> present an object detection method for resource-limited systems, performing camera-based human detection directly on a small ESP32 mcu. While they achieve a high inference rate of 12, their approach is limited to detecting a single class. Likewise, Palossi et al.<cit.> demonstrate real-time human tracking on a nano drone, mounted with GAP8, a RISC-V parallel platform from Greenwaves, but are constrained to a single class. Lamberti et al. <cit.> propose a specialized low-power Automatic License Plate Recognition system executed on GAP8 at an approximate frequency of 1.
In contrast to these prior works, our paper makes several novel contributions targeting a variety of promising platforms. Firstly, our proposed flexible lightweight algorithm ensures outstanding performance in terms of accuracy while keeping the memory requirements low enough for deployment on mcu. Secondly, we target specifically different low-power mcu, which serve as the processing units for a wide range of edge devices. This diverse platform evaluation further highlights the versatility and robustness of our approach. Overall, this work addresses the limitations of existing methods and introduces a memory-efficient algorithm suitable for resource-constrained devices, opening up new possibilities for efficient and accurate object detection in real-world applications. Additionally, the evaluation of our proposed algorithm on different platforms, including ARM, RISC-V cores and hardware accelerators, provides valuable insights into the benefits and trade-offs associated with each hardware architecture. This comparative analysis allows us to identify the strengths and weaknesses of each platform, enabling us to make informed decisions based on the specific requirements of the application at hand.
This analysis provides a comprehensive view of the various approaches, highlighting the benefits of parallel processing in the multi-core RISC-V processor and the efficiency gains achieved through hardware accelerators. Furthermore, it offers a valuable perspective on power consumption, latency, and scalability, that influence the choice of hardware for object detection tasks on resource-constrained devices.
Giordano et al. <cit.> benchmark a single architecture for image classification on several different platforms. Moss et al. <cit.> evaluate different image classification architectures on a single platform, MAX78000. Unlike these works, we describe the full deployment pipeline in the context of object detection, from architecture exploration to quantization and hardware-optimized implementation.
In the realm of efficient neural architecture design for mcu, MCUNet <cit.> presents a framework that combines the lightweight inference engine (TinyEngine) with the efficient neural architecture (TinyNAS), enabling ImageNet-scale inference on mcu. Building upon this, MCUNetV2 <cit.> introduces memory-efficient patch-based inference, further enhancing memory performance for image classification and object detection. However, while these works report peak memory consumption and MACs, they lack comprehensive power consumption measures or run-time evaluations. By exploring various variations of the TinyissimoYOLO <cit.> network and optimizing it to be deployable in different platforms, we provide a detailed analysis of the trade-offs between prediction accuracy, run-time, and power consumption across multiple deployment platforms. Notably, we demonstrate the feasibility of reducing the MACs to approximately 3 million, representing a 10-50 times reduction compared to MCUNetV2. In contrast to the previously published TinyissimoYOLO paper, which evaluated the network's performance on a restricted subset of the PascalVOC <cit.> dataset and only on single-core processors, this work investigates the network's performance without any restrictions on the dataset and explore and benchmark the flexibility and the hardware overall energy efficiency. This allows us to evaluate the performance of our flexible TinyissimoYOLO version across the entire PascalVOC dataset, considering all classes and unrestricted object counts within each image. By combining advancements in architecture design, memory-efficient inference, quantization-aware training, and a comprehensive evaluation across various deployment platforms, our work makes significant contributions to the field. We showcase the remarkable reduction in MACs achieved by our proposed approach, surpassing the state-of-the-art MCUNetV2. Furthermore, our thorough evaluation on the complete PascalVOC dataset demonstrates the robustness and scalability of our flexible TinyissimoYOLO network. Overall, our work pushes the boundaries of energy-efficient object detection on mcu, providing valuable insights and paving the way for further advancements in the field.
§ BACKGROUND AND IMPLEMENTATION
In this section, we describe TinyissimoYOLO, which we used as the basis for our explorations, training, and dataset, the MCU platforms we compare to and deploy on, and the deployment tools.
§.§ TinyissimoYOLO
TinyissimoYOLO is a general multi-object detection network, designed to enable fast and accurate detection on microcontroller platforms. Its architecture is shown in <Ref>.
The original network uses an input resolution of 88× 88 pixels and produces an output vector of dimension ( S× S( B× 5+C)), where S× S is the grid of prediction cells, B is the number of boxes predicted per cell and C is the number of classes. This architecture is highly flexible: increasing the input resolution trades improved detection performance for increased computational load and a proportional increase in the number of parameters of the last layer. The number of detected classes C can also be varied, affecting only the parameter count and computational volume in the last layer.
The cnn backbone used for feature extraction is small in comparison to state-of-the-art object detection networks. The original yolov1 network has 20 GMAC and 45M parameters, while the more recent yolov7 scales from 1.75 GMAC to 420 GMAC and 6.2M to 151M parameters. In comparison, the deployed TinyissimoYOLO can be scaled from 32 MMAC to 57 MMAC and from 441K to 888K parameters. Using 8-bit quantization further reduces the model size and memory footprint by a factor of 4 when compared to equivalent 32-bit floating-point models, making TinyissimoYOLO ideally suited for real-time inference on resource-constrained mcu platforms.
In this work, we investigate the influence of network parametrization on detection performance. We train TinyissimoYOLO with different input resolutions (88× 88, 112×112 and 224× 224 pixels), different numbers of output classes C (3, 10 and 20) and different kernel sizes in the first layer (3× 3 and 7× 7).
§.§ Training and Dataset
For training, testing and validation of the TinyissimoYOLO variants, we used the PascalVOC dataset <cit.>. 90 of the PascalVOC training dataset was used to train the network, with the remaining 10 serving as the validation set. The training data was augmented with geometric operations such as cropping, scaling, and shifting, as well as photometric operations including blurring, and modifying the brightness, contrast, saturation, and hue.
We used the SGD <cit.> optimizer with the cosine annealing scheduler proposed in <cit.>. The hyperparameters used depend on the input size and the chosen number of classes, and are detailed for the GAP9-deployed networks in <ref>.
§.§ Network Quantization
We used QuantLab[<https://github.com/pulp-platform/quantlab>] to train and quantize the networks deployed to GAP9's cluster. QuantLab is a modular, PyTorch-based framework for qat, offering experiment management facilities, support for various quantization algorithms, and automated model conversion functionality from full-precision to fake-quantized and fully integerized models. We quantized TinyissimoYOLO networks to 8-bit weight and activation precision using the tqt algorithm <cit.> <ref>. A QuantLab experiment starts from a standard full-precision PyTorch network, which is converted to its trainable, fake-quantized version according to a user-defined configuration. The quantized network is then trained according to the experiment specification. Finally, the trained fake-quantized model can be automatically converted to an integer-only model. In the integerized model, normalization, rescaling and activation layers are merged into requantization layers. A requantization layer consists of channel-wise integer multiplication, channel-wise addition, logical right shift and clipping, effectively executing an affine transformation and clipping in fixed-point arithmetic. This approach has been described multiple times in literature and has been variously termed "integer channel normalization" <cit.> or "dyadic quantization" <cit.>. Finally, the integerized model is exported as a backend- and hardware-agnostic ONNX model, where the exported ONNX operators are annotated with precision information, allowing the deployment backend to select the correct kernels. Both the fake-quantized TinyissimoYOLO models trained in QuantLab and the deployable integer-only models generated from them exhibited no accuracy drop compared to their full-precision counterparts.
For the models mapped to the MAX78000 platform, we use the same training and quantization procedures as in <cit.>.
Since the only purpose of deploying the network on the two ARM Cortex-M4 and the ARM Cortex-M7 is to assess the power and performance metrics of those single-core processors running the network inference, the network was implemented in TensorFlow, trained for a few epochs and quantized with TensorFlow-Lite. Similarly, the model deployed to GAP9's NE16 accelerator was quantized with the ptq flow integrated in Greenwaves' NNTool, as no NE16 deployment backend for QuantLab-integerized models was available at the time of our experiments. We did not evaluate the resulting model's accuracy, as the principal purpose of the NE16 deployment was to evaluate the hardware's performance and efficiency. However, it is important to note that NE16 is fully compatible with the integerized models produced by QuantLab, including the quantized TinyissimoYOLO models whose performance we report in <Ref>.
§.§ MCU Platforms
We compare deployments on different mcu platforms, which we introduce here.
§.§.§ ARM Cortex-M4 and Cortex-M7
The used mcu from STMicroelectronics (STM32H7A3 and STM32L4R9) each feature an ARM Cortex-M single-core processor. The H7A3 and L4R9 mcu use a Cortex-M7 and a Cortex-M4 operating at up to 280MHz and 120MHz respectively, with core voltages of up to 1.3V.
Apollo4b also uses an ARM Cortex-M4 processor running up to 192MHz with a nominal core voltage of 0.65V. Apollo4b's main distinguishing characteristic is its utilization of Ambiq's proprietary subthreshold power-optimized technology platform, designed to offer maximum power efficiency for edge applications.
MAX78000 features an ARM Cortex-M4, a built-in cnn accelerator which has 64 specialized processors with built-in convolutional engine, pooling unit and dedicated 442 weight memory. In addition, a 32-bit RISC-V coprocessor supports ultra-low-power signal processing.
§.§.§ GAP9
GAP9 features 10 RISC-V cores. One core acts as a fabric controller, orchestrating system operation, while a pulp cluster of 9 cores implementing custom instruction set extensions is available for efficient, high-performance execution of compute-intensive tasks. Additionally, it includes NE16, a dedicated on-chip hardware accelerator for cnn inference. The GAP9 architecture is based on the open-source soc Vega <cit.>. The cores' maximum operating frequency is 370 for both the 9-core cluster and the fabric controller. For additional flexibility, GAP9 is provisioned for dynamic voltage and frequency scaling, allowing users to trade-off between latency and energy efficiency. The compute cluster, consisting of 9 cores, one for orchestration and 8 workers, offers general-purpose compute power at extreme energy efficiency while the cnn hardware accelerator NE16 (based on RBE <cit.>) is specialized for highly efficient MAC operations. NE16 features 9x9x16 8x1bit MAC units, which are optimally used in 3x3 convolutions, but it also offers support for 1x1 and 3x3 depth-wise convolutions and fully connected layers. GAP9 has a hierarchical memory layout, with 128 of high-bandwidth, single-cycle-accessible L1 scratchpad memory in the cluster, 1.5 of interleaved L2 memory for data and code as well as 2 of on-chip flash memory. GAP9 also offers a rich set of peripherals for connecting to external memory, sensors and standard interfaces such as UART.
§.§ MCU Deployment
Different mcu platforms require different deployment tools, which we introduce in this section. Note that we deploy networks to two compute domains on GAP9 (the RISC-V cluster and the NE16 cnn accelerator) using two different flows as described below.
§.§.§ General-Purpose Processor Deployment
For deployment to GAP9's cluster, we use the DORY framework <cit.>. DORY is an automated deployment utility for ultra-low-power edge platforms with hierarchical memory layouts. It takes a precision-annotated ONNX file as the input and generates ANSI C code which implements the specified network on the target platform. Tiling between up to three hierarchical memory levels (L1 scratchpad memory, L2 main on-chip memory and L3 off-chip memory) is automatically performed with an ilp-based tiling algorithm which takes into account the hardware-specific constraints (i.e., the memory size of each hierarchical level) and various heuristics.
For deployment on the ARM Cortex-M4/M7 platforms, we use TensorFlow-Lite Micro <cit.>. As such, the deployment on the single-core ARM Cortex-M4 and M7 is performed by quantizing the network weights to 8-bits, generating the C++ code using TensorFlow-Lite Micro, and compiling the code for the corresponding microcontroller.
§.§.§ CNN Accelerator Deployments
The deployment on MAX78000's cnn accelerator was performed by using Analog Devices' training framework called ai8x-training [<https://github.com/MaximIntegratedAI/ai8x-training/>] and deployment framework called ai8x-synthesis [<https://github.com/MaximIntegratedAI/ai8x-synthesis/>]. After qat with ai8x-training, the ai8x-synthesis framework is used to quantize the network weights, activations and input using a forked version of the Neural Network Distiller by Intel AI Lab [<https://nervanasystems.github.io/distiller>]. Finally, the C code used to deploy the network on the MAX78000's cnn accelerator is generated by the "izer" tool, which converts the quantized trained model into C code. The complete network fits the accelerator and all the weights can be stored inside the weight memory of the accelerator.
Deployment of the network on GAP9's NE16 neural engine was done in collaboration with Greenwaves Technologies. To deploy networks on the GAP9 microcontroller, Greenwaves Technologies distributes a deployment framework called NNTool as part of the GAP SDK [https://github.com/GreenWaves-Technologies/gap_sdk]. NNTool is used for [label=(*),,font=]
* post-training network quantization
* network evaluation for activation and parameter sizing and
* code generation for deployment. Analogous to DORY, it calculates a tiling of the model's individual layers such that the data for each tile fits into the L1 scratchpad.
§.§.§ End-to-End evaluation
To perform an accurate measurement including the image acquisition, we implemented the complete sensing pipeline consisting of the microcontroller platforms running TinyissimoYOLO and attached RGB cameras. In this way, the edge inference can be performed with real-world data, without relying on synthetic data from the dataset. We attached an OV5647 RGB CMOS camera from Omnivision to the GAP9 and attached the OVM7692 CameraCubeChip
to the MAX78000. As such we are able to demonstrate the functionality of the trained networks on two different devices. We performed our measurements on the GAP9 evaluation Kit and on a custom-designed PCB for the MAX78000, shown in <ref>.
Further, <ref> shows some examples of images captured on GAP9 after processing with the TinyissimoYOLO network, while <ref> shows some other example images of a demo running image capturing and inference on the MAX78000, while streaming the image (after adding the detected boxes on the MAX78000 itself) live, via UART, to an attached PC.
§ EXPERIMENTAL RESULTS
To evaluate the trade-off between detection performance and energy efficiency, we trained fifteen different network variants on the PascalVOC dataset and report the map for each class in <Ref>.
All network versions shown can be deployed on general purpose mcu such as GAP9. To compare the performance of GAP9 with the reported performance on the paper <cit.>, we deployed TinyissimoYOLO (TY:3-3-88) on GAP9. First, we deployed it on single-core mcu only[Despite GAP9 being multi-core, the network deployment was done such that the inference runs only on one of the eight compute cores] to fairly compare to other single-core implementations such as STM32H7A3 featuring an ARM Cortex-M7, STM32L4R9 with an ARM Cortex-M4 and the sub-threshold computing microcontroller Apollo4b from Ambiq, also with a single-core ARM Cortex-M4. Then we also deployed a parallelized implementation, for the best possible performance on the eight-core RISC-V cluster of GAP9 without making use of the built-in neural engine hardware accelerator, showing the advantages of a parallel platform.
Using GAP9's capability to set the core voltage and frequency, we provide an analysis of different operating points, mapping a Pareto front of the latency-efficiency trade-off.
As a last step, the network is deployed on the NE16 neural engine of GAP9.
TinyissimoYOLO was limited by the 442 in accelerator weight-memory of the MAX78000, which restricted it to a network with only 3 of 20 available classes being trained, with an input resolution of 88×88 pixels to avoid relying on a specialized input streaming mode. As GAP9 features a bigger (1.5) on-chip memory and can even use external memory, we also chose a second variant of the network, based on the results gathered. The second version features an input resolution of 112×112 pixels and is trained to detect 10 detection classes and has been deployed on GAP9 (TY:10-3-112) in a single-core, multi-core and neural engine accelerated version. We compare our implementations on GAP9 with the deployment on the MAX78000 mcu.
The single-core, multi-core, and neural engine deployment's performances are provided. We measured the power consumption of the whole soc which we supplied with 1.8 and toggled a GPIO to detect the start and end of the cnn execution.
§.§ Network Architecture Variations
The 15 trained networks vary in terms of network image input resolution, the first layer's kernel size, and the number of classes the network is trained for object detection. Notably, increasing the input resolution yields an increase of the input to the last fully connected layer, while increasing the number of classes yields an increase of the output of the last fully connected layer. In particular, by changing the input or the number of classes to detect, the number of network weight parameters will change accordingly. Therefore, <ref> lists all the trained variants of TinyissimoYOLO and reports the number of parameters and the memory required to store all the network weights in quantized 8-bits.
<ref> reports the map for each detection class as well as the network's overall map.
We varied the network input resolution between 88×88 pixels, 112×112 pixels, and 224×224 pixels.
Even though we report training the network with an input resolution of 224×224 pixels, the network learning rate starting with 0.001 is unstable at the beginning leading the network to not get trained properly. However, by setting the initial learning rate to 0.0001, the network training is stable again, even though it takes more epochs to be trained.
When varying the number of object classes to predict, from 3 classes to 20 classes, for 88×88 pixel resolution the number of parameters scales by a factor of 1.3x, while for 224×224 pixel resolution, the scaling factor is 2x.
Notably, changing the network's first kernel to a kernel size of 3×3 to 7×7, we unsurprisingly note a constant increase of 1920 parameters.
Comparing the performance of the various TinyissimoYOLO networks, we first report the change of the first layer's kernel size from 3×3 to 7×7, does not increase the performance. In particular, it decreases overall performance constantly and rarely, for general poor performing classes, such as bottle or sofa, an increase of map on a per class comparison can be reported, as for example TY:20-3-112 with 13% map for sofa while TY:20-7-112 with 16% map.
Increasing the input resolution, constantly increase the map performance in overall network performance as well as in every class-to-class comparison. Notably, not surprisingly, decreasing the number of classes the network gets learned to detect increases the overall map performance of the network and of each class itself.
§.§ GAP9 - RISC-V MCU performance
We deployed two different cnn, the original TinyissimoYOLO (TY:3-3-88) for a fair comparison and an adapted network (TY:10-3-112) for more classes and higher accuracy.
Single-core performance
Here we report the single-core performance of both networks deployed on GAP9.
TY:3-3-88:
Single-core execution on GAP9 results in [group-separator=,]26 Mcycles, so an equivalent of 69.77 at the maximum frequency of 370 while we reach 1.25 MAC/cycles. The average power consumption is 26.14, which gives us an energy consumption of 1738. In <ref> we show the power consumption of the single-core implementation, showing a stable power consumption with a ripple, possibly from the internal voltage regulator as we measure the whole soc power. In <ref> we show the number of cycles by layer in blue - we see that the convolutional layers are the most computationally expensive, especially the first one.
112x112 input 10 classes:
The single-core execution time of this network is 114.15 while consuming 2990 per inference. We show the cycles per layer in <ref> in green, seeing again that the first layer consumes most cycles and max-pooling is insignificant with only 3% of the overall number of cycles.
GAP9 Multi-Core mcu performance
We deployed two different cnn, first the original TinyissimoYOLO for a fair comparison and then an adapted network for more classes and higher accuracy. Here we report the multi-core performance of both networks.
TY:3-3-88: In <ref> we show the speedup for execution on 8 versus 1 core, for parallelizing by columns and, for the convolutional layers, by output channels.
We first parallelized by columns, which gives us good results for the first layers, however, leads to low speedup on small spatial dimensions of feature maps. Therefore we switched to parallelizing by output channels for layers 11 and 12, gaining 245k cycles. Layer 13 can not be parallelized by output channels, as it requires a higher stack size than what we can allocate.
Overall we achieve a speedup of 6.14x running on 8 versus 1 core, only needing [group-separator=,]4.4 MCycles.
GAP9 can run at frequencies up to 370 on both the fabric controller and the cluster while maintaining an inference efficiency of 7.73MAC/cycle on the multi-core cluster. We measured energy consumption and latency for different operating points between 50 and 370, always choosing the minimum core voltage at which the system is still able to operate (in 50 steps). The MAC/cycle is not dependent on the frequency, meaning the latency scales linearly with the frequency.
In <ref> we show our results, marking Pareto front points in green. We reach the most energy-efficient point at 150, as this allows operation at the minimum voltage of 0.65. At this operating point, GAP9 only consumes 490.21 per inference and exhibits a latency of 27.9.
Peak performance is reached at the maximum operating frequency of 370, with only 11.3 latency and an inference energy of 721.
In <ref>, we show the power consumption per layer for the most energy-efficient operating point (150 at 0.65). We see the resulting latency, 27.87, which corresponds to an energy per inference of 490.
In <ref> we show the power consumption per layer for the least latency operating point, the aforementioned 370 at 0.8. We see the resulting latency, 11.3, which with the measured energy efficiency of 162/ corresponds to an energy per inference of 721.
In both <ref> and <ref> we visualize the execution times of the different layers by different colors. We can note a tendency that layers that parallelize better consume more power - which is expected, as the cluster then is fully used.
TY:10-3-112:
We achieve an overall speedup of 6.77 when parallelizing on 8 cores. In <ref> we show the speedup per layer, as before earlier convolutional layers parallelize better due to the higher number of columns - however, in this network, there is no need to parallelize by output channels as the speedup of the last convolutional layers is still above 6. At peak performance (370) we achieve a latency of 16.87 while consuming 1057 per inference. Optimizing for energy-efficiency and running at 150, latency increases to 41.62 while consuming 765.
Comparison to ARM
In <ref> we compare our deployment on GAP9 RISC-V cores (single-core as well as multi-core) against deployments on the H7A3, L4R9, and Apollo. Comparing the single-core versions, GAP9 clearly outperforms the other architectures in terms of latency (by a factor of more than 10 to the next best architecture) and energy per inference (by a factor of almost 50). In inference efficiency, we can also outperform the other architectures, as we can execute vectorized 8-bit operations on GAP9 contrary to the other architectures.
For our multi-core implementation, we show the most energy efficient (150) and least latency (370) operating points of GAP9. Compared to the peak performance single-core implementation we can either achieve similar latency and around 3x reduced energy per inference or reduce latency by a factor of around 2.5 but only be around 2x more energy-efficient.
§.§ GAP9 Hardware Accelerator performance
We also deployed the proposed two networks on the Hardware accelerator on GAP9.
TY:3-3-88:
In <ref> we show the cycles and MAC/cycle per layer. We have an array of 9×9 NE16 engines that can handle 16 multiplications at a time, so are ideal for a multiple of 16 input channels. Note that max-pool layers can not be executed on the accelerator, but have to be computed on the cluster. Those two factors compromise the MAC/cycle number for the first layer. On average we achieved 41.22MAC/cycle.
In <ref> (150, the most energy-efficient operating point) and <ref> (370, the least latency operating point) we show the power consumption for the network running on the GAP9 hardware accelerator. We observe that while the hardware accelerator is active the power consumption is higher than when only the cluster is active (on max-pooling operations, which is executed tiled in two parts on the first layer).
At (370 we achieved a latency of 2.12 and an energy per inference of 149, which is a 5.3x speedup compared to only using the general purpose cores. At 150 the latency is 5.24 while the energy per inference is 105, reducing the energy by 79% compared to the multi-core implementation.
TY:10-3-112:
Deploying the 112×112 input network on the HW accelerator at 370 we reached 42.84MAC/cycle and a latency of 3.46, while consuming only 245. Compared to the multi-core implementation this is a 4.9x speedup. Running at 150 we can improve the energy efficiency to 177 per inference while increasing the latency to 8.54, reducing the energy per inference by 77% compared to the multi-core implementation.
Comparison to MAX78000
In <ref> we compare the most energy efficient (150) and least latency (370) operating points of GAP9 to the implementation on the cnn accelerated MAX78000 mcu.
Experimental results show the MAX78000 outperforms the single-core and multi-core implementations on GAP9 in terms of latency, inference efficiency, and energy per inference.
However, the network implementation on the neural engine of GAP9 outperforms the inference latency and energy per inference of the MAX78000 because of the high clock frequency available. Even though the inference efficiency is 2.47x times less with 43.2MAC/cycle, GAP9 with the NE16 reaches a latency of only 2.12 and energy consumption of 149 at peak performance, being 2.6x faster and 1.3x more energy efficient than the MAX78000. At the most energy efficient frequency GAP9 still reaches a slightly lower latency than the MAX78000, but can even reduce the energy consumption by a factor of 1.8.
By using the GAP9's multi-core processor or the neural engine, we gain flexibility, as it is a multi-purpose architecture that allows to deploy of arbitrary networks while the MAX78000 is limited to a specific set of layers and only internal memory. In section <ref> we showed that we can gain accuracy and train for more classes if we can use bigger networks that can be deployed on GAP9 but not on the MAX78000.
Comparison TY:10-3-112:
We also deployed a more general-purpose TinyissimoYOLO network on GAP9, which has a higher input resolution and 10 detection classes. In <ref> we note the network has 700k parameters. Furthermore, the input of 112×112 pixels RGB image consumes another 100 of memory while the in-between network calculations need at most approximately 375 of memory. As such, a microcontroller that needs to run such a network requires at least 1 of Flash while having 512 of RAM. Furthermore, this network clearly can not fit the MAX78000 anymore. We, therefore, deployed the network on GAP9 only. In particular, we deployed the network on one single core, on all eight cores and on the neural engine itself. <ref> shows the comparison of the performances achieved. We note, despite the sheer size of the network, it runs within 3.5 on the neural engine, while being executed 32x and 3x slower on the single-core and multi-core implementation, respectively.
§ CONCLUSION
This work provides a comprehensive evaluation of various network adjustments for TinyissimoYOLO for edge processors with a 100s-of- memory budget and in a 10s-of-milliwatt-range power envelope. We demonstrate the versatility of TinyissimoYOLO by training the network to detect up to 20 classes. Despite its small size, TinyissimoYOLO achieves remarkably high detection accuracy, coming close to the performance of YOLOv1 when trained on the entire PascalVOC dataset. Particularly noteworthy is the fact that with an input resolution of 112×112, TinyissimoYOLO outperforms ten individual classes of the original YOLOv1 network within the 20 classes network.
Additionally, we present an exhaustive investigation into the network's deployability with a fair benchmark and discussion of single-core microcontrollers and the benefit of parallelization. Furthermore, the novel RISC-V-based multi-core GAP9 processor is compared with the MAX78000 accelerator and the GAP9's neural engine (NE16). On NE16, we find that inference at the maximum clock frequency of 370 only takes 2.12, with an energy consumption per inference of 105, nearly half the energy consumption compared to the MAX78000 platform for a 3-class network with 88× 88 input resolution. This network's architecture and size are largely dictated by the MAX78000 accelerator's limitations, which do not apply to GAP9's heterogeneous architecture - layers not supported by NE16 can still be efficiently mapped to the multi-core RISC-V cluster. We conclude that multi-core, general-purpose platforms are essential to achieving acceptable performance and efficiency levels. Heterogeneous systems incorporating domain-specific accelerators provide an efficiency boost in the accelerated applications, but the presence of tightly coupled general-purpose processors is essential to maintain flexibility.
This is illustrated by the deployment of a larger, more powerful 10-class TinyissimoYOLO network using a larger input resolution of 112×112 pixels to GAP9's cluster and to NE16. Even with this more powerful network, GAP9 can perform 285 inferences per second, with the object detection update rate ultimately restricted by the exposure time of low-power cameras rather than inference latency. The remaining time between frames could be used for additional processing of the inference results, e.g. for object tracking. In conclusion, the energy efficiency and real-time capabilities of TinyissimoYOLO make it well-suited for low-power processors and applications such as always-on smart cameras, where it can perform object detection efficiently.
To Marco Fariselli, for his help in deploying TinyissimoYOLO on the neural engine of GAP9, to Thorir Mar Ingolfsson, for his help with DORY on GAP9, and to Jakub Mandula for proofreading this paper.
Further, the authors would like to thank armasuisse Science & Technology for funding this research.
ACM-Reference-Format
|
http://arxiv.org/abs/2307.06933v1 | 20230712170428 | FDAPT: Federated Domain-adaptive Pre-training for Language Models | [
"Lekang Jiang",
"Filip Svoboda",
"Nicholas D. Lane"
] | cs.LG | [
"cs.LG",
"cs.AI",
"cs.DC"
] |
^†University of Cambridge^♢Flower Labs
Combining Domain-adaptive Pre-training (DAPT) with Federated Learning (FL) can enhance model adaptation by leveraging more sensitive and distributed data while preserving data privacy. However, few studies have focused on this method. Therefore, we conduct the first comprehensive empirical study to evaluate the performance of Federated Domain-adaptive Pre-training (FDAPT). We demonstrate that FDAPT can maintain competitive downstream task performance to the centralized baseline in both IID and non-IID situations. Furthermore, we propose a novel algorithm, Frozen Federated Domain-adaptive Pre-training (FFDAPT). FFDAPT improves the computational efficiency by 12.1% on average and exhibits similar downstream task performance to standard FDAPT, with general performance fluctuations remaining less than 1%. Finally, through a critical evaluation of our work, we identify promising future research directions for this new research area.
<ccs2012>
<concept>
<concept_id>10010147.10010178.10010219</concept_id>
<concept_desc>Computing methodologies Distributed artificial intelligence</concept_desc>
<concept_significance>500</concept_significance>
</concept>
</ccs2012>
[500]Computing methodologies Distributed artificial intelligence
FDAPT: Federated Domain-adaptive Pre-training for Language Models
Lekang Jiang^†, Filip Svoboda^†, Nicholas D. Lane^†♢
August 12, 2023
====================================================================
§ INTRODUCTION
Although Pre-trained Language Models (PLMs) have reached state-of-the-art (SOTA) performance in many general natural language processing (NLP) tasks <cit.>, they did not demonstrate satisfactory results on tasks within a specific domain <cit.>. To further improve the model performance, researchers proposed a second-phase pre-training in domains, which is also known as Domain-adaptive Pre-training (DAPT) <cit.>. The aim of DAPT is to adapt original PLMs to new domains by continuing the pre-training task on unlabelled text data in the target domain. Consequently, the domain-specific PLMs can achieve higher performance than original models, such as Clinical BERT <cit.> and BioBERT <cit.> in clinical and biomedical domains.
The DAPT process requires extensive training data in the domain to optimize model adaptation. However, data in sensitive domains, such as patients’ information in hospitals, are usually private and restricted from direct sharing. Hence, a practical challenge of DAPT is to develop enhanced domain-specific PLMs by leveraging private and distributed data while preserving data privacy.
Federated Learning (FL) <cit.> is a decentralized approach, which allows multiple clients to collaboratively train a joint model without exchanging raw data from each client. Therefore, combining DAPT with FL can offer the following advantages: 1) Privacy guarantee. Sensitive data can be utilized without direct sharing with other clients or the server to protect data privacy. 2) Enhanced performance. More enhanced and powerful domain-specific PLMs can be developed by training on extensive private and distributed data. 3) Cost saving. Only raw text data are needed, saving the substantial costs and expenses of data labelling. 4) Wide applicability. Domain-specific PLMs can be fine-tuned on any downstream tasks in the domain for improvement.
Despite the potential advantages of incorporating FL into DAPT, only a proof-of-concept study has investigated this method. The previous work conducted simple experiments with a fixed number of clients and limited experimental situations, showing that pre-training in federated manners is applicable with some decline in accuracy <cit.>. This initial study leaves substantial research gaps in thoroughly assessing the performance and developing novel algorithms to improve the results.
The main contributions of this research are: 1) We derive the formal definition of non-IIDness (non-independent and identically distributed) in the context of FDAPT. 2) We design a comprehensive empirical study to evaluate the performance of standard FDAPT, and conduct extensive experiments to obtain valuable results. 3) We propose Frozen Federated Domain-adaptive Pre-training (FFDAPT), a straightforward but effective algorithm, which improves the computation efficiency by 12.1% and remains similar performance to the vanilla FDAPT. 4) Through a critical evaluation of our work, we identify promising future research directions for this new research area.
§ RELATED WORK
§.§ Domain-adaptive Pre-training
Language models are often pre-trained on heterogeneous corpora, such as Wikipedia, to capture general knowledge of languages. However, these models are task-agnostic and lack domain awareness. For example, linguistic characteristics between general texts and clinical narratives are different <cit.>. Clinical narratives contain specialized medical terminology and abbreviations that are not commonly found in general text. These differences downgrade the performance of PLMs on tasks in specific domains. Hence, researchers propose the DAPT methods to adapt language models to target domains and tasks <cit.>.
There are two types of DAPT approaches for PLMs. The first method is to continue the pre-training task on abundant unlabeled domain-specific texts without changing the loss functions and model structures. Studies demonstrated the effectiveness of this straightforward approach in various domains. For example, an empirical study <cit.> conducted experiments in 4 domains, including biomedical, computer science publications, news, and reviews. It illustrated that DAPT can result in performance increases under both high- and low-resource settings. In addition, domain-specific models trained by DAPT, such as Clinical BERT <cit.> and BioBERT <cit.>, can significantly outperform the original model on tasks in those domains. The alternative DAPT method is to add domain-distinguish pre-training tasks to the original training objective. This approach is more complicated but can be more effective for domain adaptation. For example, studies showed that incorporating adversarial domain discrimination into DAPT can enhance domain-invariant features <cit.>, which further improves the model performance.
§.§ Federated Pre-training
Researchers trained Word2Vec <cit.> from scratch using FL on Wikipedia and compared the performance with centralized Word2Vec <cit.>. The results showed that combining Word2Vec and FL can achieve enhanced word representations, with better similarity, analogy, and categorization outcomes. Both model quality and convergence time in FL settings are comparable to centralized training.
FedBERT <cit.> incorporated Split Learning <cit.> with FL to resolve computational constraints in cross-device settings. It showed that the transformer layer is computationally costly to train on edge devices. Thus, it adopted Split Learning to update the transformer layer on the powerful server and conducted other layers on edge devices. The results demonstrated that Federated Split Learning can reach the same performance compared to standard FL.
While the above work focused on pre-training in the general domain, a proof-of-concept study conducted experiments in the DAPT process <cit.>. It continued pre-training BERT <cit.> using FL with 5 clients on the clinical dataset and tested on 2 downstream tasks. Notably, different situations of data distributions are not considered in the experiments. The outcome indicated that pre-training and fine-tuning of BERT are applicable to FL with some decline in accuracy.
§ METHODOLOGY
§.§ Experimental Setups
Training and Evaluation. In the FDAPT process, we initialize each client's model with the weights of a PLM and continue training on domain-specific datasets using the same pre-training tasks under various federated settings. Specifically, we adopt the Federated Averaging (FedAvg) <cit.> algorithm for our experiments. To evaluate domain-specific PLMs, we fine-tune them on different downstream tasks within the domain and compare their performance.
Model Architecture and Framework. In our experiments, we use DistilBERT <cit.> model due to its efficiency and decent performance. Although GPT series models <cit.> have achieved incredible performance recently, their enormous model sizes will limit the scale of our experiments. The purpose of this empirical study is to demonstrate the effectiveness of FDAPT in comprehensive federated settings, rather than developing new SOTA models. Therefore, we select the DistilBERT model for simulations to minimize experimental costs while obtaining sensible results. In addition, we use Flower <cit.>, a user-friendly FL framework, for experimental simulations. Flower provides a flexible environment to easily customize FL configurations, simplifying the implementation and execution of FL experiments.
Datasets. We conduct FDAPT on the PubMed dataset, which contains abstracts and full-text research articles in the biomedical domain. The PubMed dataset adopted in our experiments <cit.> is smaller than the version used for BioBERT <cit.>, which minimizes the experimental costs while maintaining pronounced results. Domain-specific PLMs are evaluated on 9 publicly available downstream tasks in the biomedical domain, including 6 Named Entity Recognition (NER), 2 Relation Extraction (RE) and 1 Question Answering (QA) datasets, as shown in Table <ref>.
§.§ Non-IIDness in Federated Pre-training
Existing research in federated pre-training methods only focused on IID situations and ignored the non-IID issue, which commonly exists in practical FL applications and can cause performance drops. Non-IIDness in supervised learning is usually related to the distribution of data labels and features <cit.>. However, datasets used for pre-training only contain raw texts, which do not have pre-defined labels or features. Therefore, we define three types of non-IIDness in the federated pre-training process, including quantity skew, sentence length distribution skew and vocabulary distribution skew.
Quantity skew refers to the imbalance in the number of training data across different clients. In the context of federated pre-training, we define it as the imbalance of the number of raw texts among all clients. Additionally, sentence length and the number of vocabulary are important features of text data. Therefore, we argue that federated pre-training with imbalanced average sentence length or the number of vocabularies may affect the final results. We define the sentence length distribution skew as the imbalance of the average sentence length across multiple clients, and vocabulary distribution skew as the imbalance of the number of unique words across all clients. When creating these skews, the aim is to maximize a single metric discrepancy among all clients, while keeping other metrics almost the same.
§.§ Frozen Federated Domain-adaptive Pre-training
The pre-training process often requires extensive computational resources. For example, it took 23 days to train the domain-specific BioBERT model on eight V100 GPUs <cit.>. Hence, mitigating computation costs is important in the FDAPT process.
Freezing specific layers or parameters of a model during fine-tuning or transfer learning aims to preserve the pre-trained knowledge encoded within those layers, while allowing other parts of the model to adapt to new tasks or domains. Studies showed that fine-tuning only a fourth of model layers can achieve 90% of the original model quality <cit.>. Therefore, we propose FFDAPT, a simple but effective method that incorporates freezing methods into FDAPT to improve efficiency, as shown in Algorithm <ref>.
For each client, a portion of the consecutive layers is frozen, and the number of frozen layers is determined by the size of the dataset. We set a scaling hyper-parameter γ to control the actual number of frozen layers and improve flexibility. Since freezing all layers are meaningless during training, ε is set to be the maximum number of frozen layers. Through our method, each client freezes some layers during training to improve the computation efficiency.
§ RESULTS
We report all the experimental results in Table <ref>. We demonstrate that FDAPT achieves competitive performance on downstream tasks against the centralized baseline in both IID and non-IID situations. Moreover, FFDAPT improves the training efficiency by approximately 12.1% on average, while maintaining similar downstream tasks performance compared to vanilla FDAPT.
§.§ FDAPT
We make the following observations regarding FDAPT.
The performance drops of federated models compared to the centralized baseline are acceptable. While federated models exhibit a slight performance decrease compared to the centralized model on some tasks, all models pre-trained with FDAPT surpass the original model on all tasks. These datasets contain both word-level and sentence-level tasks, underscoring the effectiveness of FDAPT. On the other hand, the performance of federated models decreases by less than 1% on almost all datasets compared to the centralized baseline. Hence, we argue that this level of performance reduction is tolerable to preserve data privacy.
Federated models occasionally outperform the centralized baseline. Federated models trained in specific settings can outperform the centralized approach on NCBI, LINNAEUS, Species-800, GAD, EU-ADR and BioASQ 7b, including both token-level and sentence-level tasks. Notably, FDAPT with 2 clients under the IID setting increases the F1 scores on LINNAEUS by 0.8%. Meanwhile, on the EU-ADR dataset, FDAPT with 8 clients under non-IID settings of sentence length skew, achieves the F1 score of 81.7%, which is 1.1% higher than the centralized approach. Furthermore, federated models outperform the centralized model on BioASQ 7b with increases up to 2.4% in terms of strict accuracy.
§.§ FFDAPT
The aim of FFDAPT is to improve computational efficiency. The improvement of training efficiency is calculated by the following equation.
I = T-T_F/T_F· 100%
where T and T_F are the round time for standard FDAPT and FFDAPT respectively. We calculate the improvement of efficiency for each experimental scenario, and report the average value of 12.1% as the final result of improvement.
In terms of downstream tasks performance, FFDAPT can achieve similar outcomes compared to standard FDAPT in all situations. The performance variations are no greater than 1% for most of the cases. Remarkably, FFDAPT leads to higher performance than vanilla FDAPT in some specific scenarios. For example, the F1 scores on the Species-800 dataset increase by 0.8% for models trained with 8 clients under the IID setting. Additionally, FFDAPT with 8 clients improves the F1 scores by 1.4% on the EU-ADR dataset in non-IID settings of vocabulary distribution skew. Moreover, FFDAPT with 2 clients results in a 1.6% performance enhancement on the QA task in the situation of vocabulary distribution skew. Although there are performance decreases in some situations, they still remain better than the original model.
§ DISCUSSION AND FUTURE WORK
This study makes significant contributions to FDAPT, but we acknowledge some limitations and point out promising research directions to stimulate future studies.
More real-world simulations. Due to limited data and computational resources, our experiments focus on some common settings to obtain meaningful results while minimizing experimental costs. Future research can involve more large-scale simulations and different model structures to represent potential real-world scenarios.
Improve computation and communication efficiency. We demonstrate the efficacy of our FFDAPT algorithm, but further efficiency improvements may be possible. For instance, module adapters <cit.> can be used to mitigate computational costs, and communication-efficient algorithms, such as FedPCL <cit.>, can be considered to improve communication efficiency.
Other challenges in FL. Our research adopts FL to address the concerns of data privacy. Therefore, although other challenges related to FL are not directly related to our experiments, they are also worth investigating to improve the effectiveness of FDAPT. For example, addressing system heterogeneity, developing specific federated strategies and integrating privacy-enhancing approaches are all promising research directions.
Domain-related challenges. This research is based on the biomedical domain and suggests that FDAPT can have similar performance compared to centralized pre-training. In contrast, a previous study <cit.> showed that FDAPT is worse than centralized pre-training in the clinical domain. Our results are more comprehensive and conclusive because we evaluate models on 9 downstream datasets, in contrast to their assessment on only 2 tasks of the same type. Nonetheless, it is still worth justifying whether the inconsistency is caused by domain differences. Additionally, another promising research direction is to develop domain-distinguish methods for better domain adaptation. For example, adversarial training can be added for FDAPT to improve the robustness of the domain-specific model.
§ CONCLUSION
In this paper, we present the first comprehensive empirical investigation of FDAPT for NLP tasks in cross-silo settings. We continue training the DistilBERT model on biomedical corpora using FDAPT, and test the model performance on 9 different datasets, including both token-level and sentence-level tasks, to make a comprehensive evaluation. We formulate 3 types of non-IIDness in the context of FDAPT, and simulate both IID and non-IID situations. The results show that FDAPT can retain competitive downstream task performance to the centralized baseline in all IID or non-IID scenarios. The performance drops of FDAPT are less than 1% most of the time, and it can occasionally outperform the centralized approach. In all experimental situations, models trained by FDAPT surpass the original DistilBERT model, demonstrating the effectiveness of FDAPT.
Furthermore, we propose an innovative strategy, FFDAPT, to enhance computational efficiency. We incorporate the freezing methods into standard FDAPT, which increases the training speed by approximately 12.1% on average. Additionally, FFDAPT exhibits similar downstream task performance to vanilla FDAPT, with general performance fluctuations remaining less than 1%. In specific circumstances, FFDAPT can improve the performance by up to 1.6%.
Finally, we point out promising future research directions for this new research area. It is crucial to continue investigating in this field to advance the understanding and release the full potential of FDAPT.
ACM-Reference-Format
|
http://arxiv.org/abs/2307.05848v1 | 20230711234727 | Empirical radius formulas for canonical neutron stars from bidirectionally selecting EOS features in extended Bayesian analyses of observational data | [
"Jake Richter",
"Bao-An Li"
] | nucl-th | [
"nucl-th",
"astro-ph.HE",
"nucl-ex"
] |
shapes.geometric, arrows
startstop = [rectangle, rounded corners, minimum width=3cm, minimum height=1cm,text centered, draw=black, fill=red!30]
io = [trapezium, trapezium left angle=70, trapezium right angle=110, minimum width=3cm, minimum height=1cm, text centered, draw=black, fill=blue!30]
process = [rectangle, minimum width=3cm, minimum height=1cm, text centered, text width = 4cm, draw=black, fill=orange!30]
decision = [diamond, minimum width=3cm, minimum height=1cm, text centered, text width = 4cm, draw=black, fill=green!30]
arrow = [thick,->,>=stealth]
E_ sym(ρ)
0E_ sym(ρ_0)
U_ sym(ρ,k)
∂ U/∂ k
0U_ sym(ρ_0,k_F)
L(ρ)
η
ł0L(ρ_0)
|
http://arxiv.org/abs/2307.05576v1 | 20230710121906 | Bulk viscous universe with cosmological constant | [
"Athira Sasidharan",
"Titus K Mathew"
] | gr-qc | [
"gr-qc"
] |
Bulk viscous universe with cosmological constant
Athira Sasidharan^* and Titus K Mathew^+
e-mail:[email protected], [email protected]
^*Department of
Physics, NSS Hindu College, Changanacherry, Kerala, India
^+Department of
Physics, Cochin University of Science and Technology, India,.
In this paper we consider dissipative effects in ΛCDM model, i.e., we consider a universe with cosmological constant having viscous matter. We assume the most general form for bulk
viscous coefficient,
ζ=ζ_0+ζ_1ȧ/a+ζ_2ä/ȧ and obtained various constrains for ζ's . We also studied the background study of the model with ζ=ζ_0 and ζ=ζ_1ȧ/a. Extracted the value of ζ_1 using the Pantheon data and also obtained its thermodynamic evolution and the age.
§ INTRODUCTION
Since the discovery of accelerating universe <cit.>, active research is been taking place looking for the cause producing the acceleration and also for a model that would incorporate this acceleration. To the present, there are many models that fits this acceleration. Of these, the simplest and the most successful is the interpretation of dark energy as the cosmological constant. However, the discrepancy between the observed and calculated values of dark energy density, known as cosmological constant problem<cit.>, and unexplained coincidence of two dark sectors-dark energy and dark matter, known as Cosmic Coincidence problem <cit.>, make rooms for other models in explaining the current acceleration. Some of these models include quintessence
<cit.>, k-essence <cit.> and perfect fluid
models (like Chaplygin gas model) <cit.>, f(R) gravity <cit.>, f(T)gravity <cit.>, Gauss-Bonnet theory <cit.>, Lovelock
gravity <cit.>, Horava-Lifshitz gravity <cit.>,
scalar-tensor theories <cit.>, braneworld models
<cit.> etc.
A less complicated unified dark energy model is the bulk viscous models. In <cit.>, bulk viscous matter dominated universe is considered and was found that this viscosity alone can produce acceleration in the expansion of the universe. Phase space analysis of this model indicates that only viscosity with constant bulk viscous coefficinet predicts all the conventional phases of the universe i.e., a prior radiation dominated phase, followed by a decelerated matter dominated phase and then finally evolving to a de Sitter type universe <cit.>. A bayesian analysis of this model shows that it is not so superior over the ΛCDM model, but have only a slight advantage over it<cit.>. However Maartens <cit.> has pointed out that these viscous models violates near equilibrium condition (NEC),
Π/P≪ 1
There are works <cit.> showing that Λ is an inevitable content of the universe. The matter content of the universe has disspations so it is worth full to consider a universe filled wth viscous matter having a cosmological constant <cit.>. Also recent papers <cit.> showed that introducing Λ with viscosity can attain this NEC. We neglect the other dissipative phenomena like shear viscosity as it is inconsistent with the isotropic nature of the universe. So the only viscosity component to be considered is the bulk viscosity.
In this paper we first analyse the basic formalism of the bulk viscous matter dominated universe with cosmological constant. We consider the general form for the bulk viscous coefficient and using Eckart formalism, obtain the expression for Hubble Parameter and the scale factor. We also analyse the equation of state parameter and the deceleration parameter and from the behavior of these parameters, constrains on the viscous parameters was obtained. In section <ref>, we did the background study of the model for constant viscosity and constrains on the parameter is also obtained. We also analysed the age, thermodynamic behavior and asymptotic behavior of the model. In section <ref>,we consider the viscous coefficient as a function of Hubble parameter, i.e., ζ=ζ_1 H, extracted the value of ζ_1 and studied the background evolution and cosmological parameters and the age of the universe. In section <ref>, the results and conclusion are discussed.
§ VISCOUS MATTER WITH COSMOLOGICAL CONSTANT
We consider a spatially flat universe described by FLRW metric. We assume that the universe contains viscous matter (both dark and baryonic) and cosmological constant as dark energy. We neglect the radiation component since its percentage composition is very small and also we are dealing with the late time acceleration. Eckart formalism <cit.> is used for the bulk viscous pressure and is given by,
P^*=P-3ζ H
where P is the normal pressure, which we assume as zero for the whole matter component of the universe (both dark and baryonic) and ζ is the coefficient of bulk viscosity. So the effective pressure will only be that from the bulk viscosity. The coefficient ζ is basically a transport coefficient, hence it would depend on the dynamics of the cosmic fluid. Since the exact form of ζ is unknown, we consider the most general form for the bulk viscous coefficient ζ <cit.>, which is a linear combination of the three terms as,
ζ=ζ_0+ζ_1ȧ/a+ζ_2ä/ȧ
The first term is a constant ζ_0, the second term is proportional to the Hubble
parameter, which characterizes the dependence of the bulk viscosity on velocity, and the third is proportional to ä/ȧ, characterizing the effect of acceleration on the bulk viscosity.In terms of Hubble parameter
H=ȧ/a, this can be written as,
ζ=ζ_0+ζ_1H+ζ_2(Ḣ/H+H)
The Friedmann equations governing the bulk viscous universe with cosmological constant are given as,
H^2=ρ_m+ρ_Λ/3
2ä/a+(ȧ/a)^2=ρ_Λ-P^*
where we have taken 8π G = 1, ρ_m and ρ_Λ=Λ/8π G are the densities of matter and cosmological constant Λ, respectively and overdot represents the derivative with respect to cosmic time t. We consider separate conservation equations for matter and dark energy and are given below,
ρ̇_m+3H(ρ_m+P^*)=0.
ρ̇_Λ=0
where we have assumed a constant equation of state for Λ, ω_Λ=-1. Using the Friedmann equations (<ref>) and (<ref>) and using equations. (<ref>) and (<ref>), we get the differential equation for the Hubble parameter as,
Ḣ=1/2-ζ̃_2(ζ̃_0 HH_0+(ζ̃_1+ζ̃_2-3)H^2+3H_0^2 Ω_Λ 0)
where we have defined the dimensionless bulk viscous parameters ζ̃_0, ζ̃_1, ζ̃_2 as,
ζ̃_0=3ζ_0/H_0, ζ̃_1=3ζ_1, ζ̃_2=3ζ_2
H_0 is the
present value of the Hubble parameter and Ω_Λ 0 is the present density parameter of dark energy. Integrating equation (<ref>) we can get the expression for the Hubble parameter as,
H=H_0[(y+ζ̃_0)(y-2(ζ̃_1+ζ̃_2-3)-ζ̃_0)e^H_0(t-t_0)y/2-ζ̃_2-(y-ζ̃_0)(y+2(ζ̃_1+ζ̃_2-3)+ζ̃_0)/2(ζ̃_1+ζ̃_2-3)(e^H_0(t-t_0)y/2-ζ̃_2(-y+2(ζ̃_1+ζ̃_2-3)+ζ̃_0)-(y+2(ζ̃_1+ζ̃_2-3)+ζ̃_0))]
where y=√(ζ̃_0^2-12Ω_Λ 0(ζ̃_1+ζ̃_2-3)) and t_0 is the present cosmic time. As t-t_0→∞, H→ H_0[y+ζ̃_0/2(ζ̃_1+ζ̃_2-3)], a constant provided ζ̃_2<2. When t-t_0 is small, H evolves as H_0[2(2-ζ̃_2)+H_0(t-t_0)(ζ̃_0+6Ω_Λ 0+y)/2(2-ζ̃_2)+H_0(t-t_0)(y-2(ζ̃_1+ζ̃_2-3)-ζ̃_0)].
Using the definition of the Hubble parameter, we could obtain the expression for the scale factor from equation (<ref>) as,
a=e^H_0(t-t_0)(y-ζ̃_0)/2(ζ̃_1+ζ̃_2-3)[y+2(ζ̃_1+ζ̃_2-3)+ζ̃_0+e^H_0(t-t_0)y/2-ζ̃_2(y-2(ζ̃_1+ζ̃_2-3)-ζ̃_0)/2y]^ζ̃_2-2/ζ̃_1+ζ̃_2-3
When Ω_Λ 0=0, the scale factor reduces to
a(t)=[(ζ̃_0+ζ̃_12-3/ζ̃_0)+(3-ζ̃_12/ζ̃_0)
e^ζ̃_0/2-ζ̃_2H_0(t-t_0)]^2-ζ̃_2/3-ζ̃_12
which is the expression obtained in <cit.>. When t-t_0 is small, the scale factor evolves as
a∼[1+H_0(t-t_0)(y-ζ̃_0)/2(ζ̃_1+ζ̃_2-3)][1+H_0(t-t_0)/2-ζ̃_2(y-2(ζ̃_1+ζ̃_2-3)-ζ̃_0)]^ζ̃_2-2/ζ̃_1+ζ̃_2-3
When t-t_0 is very large, from the expression of scale factor we see that it will increases exponentially.
§.§ Equation of state and Deceleration parameter
The equation of state parameter ω and the deceleration parameter q can be obtained using the following relation,
ω=-1-2/3Ḣ/H^2
q=-1-Ḣ/H^2
Using the expression (<ref>) and (<ref>), we get the expressions for ω and q as,
ω=-1+2y^2 (ζ̃_0+ζ̃_1+ζ̃_2-3+3 Ω _Λ 0) /3 (ζ̃_2-2) (Sinh[H_0 (t-t_0) y/2 (2-ζ̃_2)] (ζ̃_0+6 Ω _Λ 0)+Cosh[H_0 (t-t_0) y/2 (2-ζ̃_2)] y)^2
q=-1+y^2(ζ̃_0+ζ̃_1+ζ̃_2-3+3 Ω _Λ 0)/(ζ̃_2-2) (Sinh[H_0 (t-t_0) y/2 (2-ζ̃_2)] (ζ̃_0+6 Ω _Λ 0)+Cosh[H_0 (t-t_0) y/2 (2-ζ̃_2)] y)^2
The present value of ω and q can be obtained by putting t=t_0 and are,
ω_0=2ζ̃_0+2ζ̃_1-ζ̃_2+6Ω_Λ 0/3(ζ̃_2-2)
q_0=ζ̃_0+ζ̃_1-1+3Ω_Λ 0/ζ̃_2-2
The present universe will be accelerating only if 3ω_0+1<0 and q_0<0 and for the universe to be in quintessence region and to avoid big rip, it should satisfy the relation q_0>-1. Using these conditions and from the behaviour of the Hubble parameter and the scale factor, for a universe to begin from the big bang and then entering it to decelerated epoch and then making a transition to the accelerated epoch in the past, a set of conditions has to be satisfied by the ζ̃'s. These conditions are,
* ζ̃_0>0, ζ̃_2<2, ζ̃_0+ζ̃_1>1-3Ω_Λ 0, ζ̃_1+ζ̃_2<3, ζ̃_0+ζ̃_1+ζ̃_2<3-3Ω_Λ 0
* ζ̃_0<0, ζ̃_2>2, ζ̃_0+ζ̃_1<1-3Ω_Λ 0, ζ̃_1+ζ̃_2>3, ζ̃_0+ζ̃_1+ζ̃_2>3-3Ω_Λ 0
If we neglect the cosmological constant i.e., Ω_Λ 0=0, then these would reduce to the conditions obtained in the reference <cit.>.
§ WITH CONSTANT BULK VISCOSITY
Let us consider the case when bulk viscous coefficient is a constant, i.e., when ζ=ζ_0 . The expression for Hubble parameter becomes,
H=H_0y-ζ̃_0-6 Ω _Λ0+e^1/2 H_0 (t-t_0) y(y+ζ̃ _0+6 Ω _Λ0)/y+ζ̃ _0-6+e^1/2 H_0 (t-t_0) y(y-ζ̃ _0+6)
where y=√(ζ̃_0^2+36Ω_Λ 0)
Similarly, one could obtained the expression for scale factor for constant ζ as,
a=e^1/6 H_0(t-t_0) (ζ̃ _0-y)((y+ζ̃ _0-6)+e^H_0(t-t_0)y/2( y-ζ̃ _0+6)/2 y)^2/3
Similarly, the corresponding equation of state and the deceleration parameter for constant viscosity becomes,
ω=(-1-(ζ̃ _0-3+3 Ω _Λ 0) y^2/3(y Cosh[1/4 H_0 (t-t_0) y]+(ζ̃ _0+6 Ω _Λ 0) Sinh[1/4 H_0 (t-t_0) y])^2)
q=(-1-(ζ̃ _0-3+3 Ω _Λ 0) y^2/2(y Cosh[1/4 H_0 (t-t_0)y]+(ζ̃ _0+6 Ω _Λ 0) Sinh[1/4H_0 (t-t_0)y])^2)
As mentioned before, for an accelerating universe, the present value of equation of state ω_0<-1/3 and the present value of the deceleration parameter q_0<0. To avoid big rip, the equation of state parameter ω_0>-1, above the phantom limit. These conditions help us to constrain the value of ζ̃_0 as
1-3Ω_Λ 0<ζ̃_0<3(1-Ω_Λ 0).
From observation Ω_Λ is constrained in the range 0.65-0.75 <cit.>. This constrains the ζ̃_0 in between -1.25<ζ̃_0<1.05.
§.§ Age of the universe
Age of the universe in this case can be obtained by equating a=1 in the equation (<ref>) and is
found to be,
Age≡(2/H_0 y)Log[1-2 y/6+y-ζ̃_0].
The plot of age of the universe for different values of (ζ̃_0,Ω_Λ) subjected to
the constrain (<ref>) are shown in the figure (<ref>).
The age plot shows reasonably good agreement for (ζ_0,Ω_Λ)=(-0.5,0.7) but the agreement with respect
(ζ_0,Ω_Λ)=(0.1,0.68) is slightly less and for the third choice it is not in nice agreement.
But corresponding to the best agreement pair the viscosity is negative. Whether is physically feasible or not may evident from
the further considerations of the entropy evolution and dynamical system behaviour.
§.§ Thermodynamics
We now check the validity of the Generalized second law and maximization of entropy condition in this case.
Assuming apparent horizon as the boundary of the universe and obtaining the horizon entropy using the
Bekenstein relation and matter entropy using the Gibbs equation, we calculated the expression for the first derivative and second derivative of the total
entropy with respect to time. The relation obtained are as follows:
Ṡ=64 π ^2e^t' ỹb^2 ỹ^4 (ỹ-6+ζ̃_0 +e^1/2 t' ỹ (ỹ+6-ζ̃_0))/H_0 (ỹ-ζ̃_0 -6 Ω_Λ +e^1/2 t' ỹ (6 Ω_Λ +ỹ+ζ̃_0))^5,
S̈=-384 π ^2 b^2 ỹ^5 e^3/2t' ỹ(b ỹ+2 (1+Ω_Λ) ỹCosh[1/2t' ỹ]+2 d Sinh[1/2t'ỹ])/((-1+e^1/2t'ỹ)ζ̃_0 -6 Ω_Λ+ỹ+e^1/2t'ỹ (6 Ω_Λ +ỹ))^6,
where b=ζ̃_0+3Ω_Λ 0-3, d=ζ̃_0+12 Ω_Λ-ζ̃_0Ω_Λ and t'=H_0(t-t_0).
The evolution of Ṡ and S̈ with respect to the scale factor for different values of Ω_Λ and ζ̃_0 subjected to the constrain (<ref>) are plotted and are shown in figures (<ref>) and (<ref>) respectively
From the figures, it is clear that GSL and maximization of entropy condition is valid for the model.
§.§ Phase space analysis
We also try to study the asymptotic behavior of the model. We chose u and v as the phase space variables defined as
u =Ω_m=ρ_m/3H^2,
v =1/H_0/H+1,
which varies in the range 0≤
u≤1 and 0≤ v≤1.
Using the conservation equation and differential equation for Hubble parameter, we can obtained the autonomous equations for u and v as,
u' =(1-v)/v^2(v(1-u)ζ̃_0 -3Ω_Λ u (1-v)),
v' =(1-v)/2 v(3Ω_Λ(1-v)^2+ζ̃_0 v(1-v)-3v^2).
There are three critical points for the above autonomous equation and the corresponding eigen values are listed in the Table <ref>.
Inorder to represent a universe with unstable matter dominated phase and a stable, physically feasible accelerated
phase we see that ζ̃_0 must be positive subjected to the constrain (<ref>).
In determining the age corresponding to this model we have noted that, the best fit have arised both with negative value
of ζ_0 and also with positive value (the black line in the age plot) of ζ_0. But the asymptotic analysis presented
here, however supports only a positive value for ζ_0. Earlier in the analysis without cosmological constant also we conclude
that, the case with ζ=ζ_0 is preferred over other cases. Thus even though the age
prediction has been changed slightly, the present model is also predicting a conventional evolution of the universe with
constant viscosity as in the case of the model without cosmological constant.
§ WITH Ζ=Ζ_1H
Let us consider another special case of ζ=ζ_1 H. So here ζ depends only on the velocity component of the expansion of the universe. The expression for the Hubble Parameter and the scale factor are as follows,
H=-√(3) H_0Ω_Λ 0(6-2 ζ̃ _1-2√(3(3-ζ̃ _1)Ω_Λ 0)+2e^ H_0(t-t_0) √(3(3-ζ̃ _1)Ω_Λ 0)(3-ζ̃ _1+√(3(3-ζ̃ _1)Ω_Λ 0)))/√((3-ζ̃ _1)Ω_Λ 0)(6-2 ζ̃ _1-2√(3(3-ζ̃ _1)Ω_Λ 0)-2 e^ H_0(t-t_0) √(3(3-ζ̃_1)Ω_Λ 0) (3-ζ̃ _1+ √(3(3-ζ̃ _1)Ω_Λ 0)))
ł
a=12^1/ζ̃_1-3e^-√(3) H_0 (t-t_0)Ω_Λ 0/√((3-ζ̃_1)Ω_Λ 0)(ζ̃_1-3+√(3(3-ζ̃_1)Ω_Λ 0)+e^ H_0(t-t_0)√(3(3-ζ̃_1)Ω_Λ 0)(3-ζ̃_1+√(3(3-ζ̃_1)Ω_Λ 0))/√((3-ζ̃_1)Ω_Λ 0))^2/3-ζ̃_1
From the expression of Hubble parameter and the scale factor, we see that inorder to represent the conventional behavior of the universe, ζ̃_1 should be less than 3. In this case one could obtain the expression for the Hubble parameter in terms of the scale factor a. And it is found to be,
H=H_0√([a^ζ̃_1-3(ζ̃_1-3+3Ω_Λ 0)-3Ω_Λ 0/ζ̃_1-3])
Since a direct relation between the Hubble parameter H and the scale factor a is found out, it is possible to extract the value of ζ_1.
§.§ Extraction of ζ̃_1
To extract the value of ζ̃_1, we use the latest Pantheon Type Ia Supernova data consisting of 1048 data points.. The method used is the χ^2 minimization technique and is defined as,
χ^2≡∑^n_k=1[μ_t-μ_k]^2/σ_k^2,
where μ_k is the observational distance modulus for the k-th
Supernova (obtained from the data) with red shift z_k, σ_k^2 is the variance of the measurement,
n is the total number of data and μ_t is the theoretical distance modulus for the
k-th Supernova with the same redshift z_k, which is given as
μ_t=m-M=5log_10[d_L/Mpc]+25
where, m and M are the apparent and absolute magnitudes of the
SNe respectively. d_L is the luminosity distance and is defined as
d_L=c(1+z)∫_0^zdz'/H,
where c is the speed of light. Using the expression for H from equation (<ref>), we construct the χ^2 function.
We extract the values of Ω_Λ 0 and H_0 along with ζ̃_1. The values are given in the table below <ref>.
§.§ Evolution of equation of state parameter and deceleration parameter
The expression for the equation of state parameter and the deceleration parameter for this model can be obtained by making ζ̃_0=ζ̃_2=0 in the equations (<ref>) and (<ref>) respectively.
ω=-1-(ζ̃_1-3)(ζ̃_1-3+3 Ω_Λ 0)/(√(3(ζ̃_1-3))Cos[1/2 H_0 (t-t_0) √(3Ω_Λ 0(ζ̃_1-3))] +3√(Ω_Λ 0)Sin[1/2H_0 (t-t_0) √(3Ω_Λ 0(ζ̃_1-3))] )^2
q=-1-3 (ζ̃_1-3)(ζ̃_1-3+3 Ω _Λ 0)/2 (√(3(ζ̃_1-3))Cos[1/2H_0(t-t_0)√(3Ω _Λ 0(ζ̃_1-3))] +3 √(Ω_Λ 0)Sin[1/2√(3) H_0 (t-t_0) √(ζ̃_1-3)√(Ω_Λ 0)] )^2
The equation of state parameter ω and the deceleration parameter q, in terms of scale factor are given as,
ω=9 a^3 Ω_Λ 0 -a^ζ̃_1ζ̃_1 (ζ̃_1 -3+3 Ω_Λ 0)/-9 a^3 Ω_Λ 0 +3 a^ζ̃_1 (ζ̃_1 -3+3 Ω_Λ 0),
q=-1-a^ζ̃_1 (-3+ζ̃_1) (ζ̃_1 -3+3 Ω_Λ )/-6 a^3 Ω_Λ +2 a^ζ̃_1 (ζ̃_1-3+3 Ω_Λ ).
The plot of ω and q for the best estimated values of ζ̃_1 and Ω_Λ are shown
in the figures <ref> and <ref> respectively.
The equation of state is zero in the recent past. It decreases to the
negative values and finally saturated at ω=-1 corresponding to a de Sitter epoch in the extreme future. The evolution of the deceleration parameter starts from around q ∼ 0.5 in the past, which corresponds
to decelerated epoch and decreasing as the universe expands. It saturates at q=-1 corresponding the future de Sitter phase.
The present value of ω and q can be obtained by putting a=1 in the expressions given by equation (<ref>) and (<ref>), respectively and are obtained as,
ω_0=-ζ̃_1 /3-Ω_Λ,
q_0=1/2 (1-ζ̃_1 -3Ω_Λ ).
Using the best estimated values of ζ̃_1 and Ω_Λ, we get ω_0=-0.867033 and q_0=-0.80055, which is near to concordance value obtained by WMAP observation.
§.§ Age of the universe
The age of the universe in this model can be obtained by equating the scale factor (equation (<ref>)) to one and is found to be
Age≡Log[3-ζ̃_1-√(3)√((3-ζ̃_1) Ω _Λ 0)/3-ζ̃_1+√(3)√((3-ζ̃_1) Ω _Λ 0)]/√(3) p √(-(-3+ζ̃_1) Ω _Λ 0).
Using the best estimated values for ζ̃_1 and Ω_Λ, the age is found to 18.44Gyr and is matching with
the concordance value of the age of the universe obtained from the oldest globular observations. In this way the model is promising
in predicting the age.
§ CONCLUSION
We analyse a universe with a cosmological constant and bulk viscous matter. By considering the general form for ζ=ζ_0+ζ_1ȧ/a+ζ_2ä/ȧ, we obtain the constrains of the viscous parameters by finding the evolution of Hubble parameter, scale factor and cosmological parameters.
Two special cases for the viscous coefficient ζ, ζ=ζ_0, a constant and ζ=ζ_1 H, depending on the velocity of the expanding universe are considered. For ζ=ζ_0, for the constrain is -1.25<ζ̃_0<1.05. It is also found out that under this constrain the age of the universe is in accordance with the galactic observations. GSL and maximization of entropy condition are also found to be valid for the model.
For ζ=ζ_1H, the value on ζ_1 is extracted using pantheon data and is found to be 0.351. The present value of deceleration parameter and equation of state is found to be q_0=-0.80055 and ω_0=-0.867033, respectively, which is near to concordance value obtained by WMAP observation. The age is found to 18.44Gyr and is matching with the observations.
The addition of cosmological constant in the bulk viscous matter dominated universe improves age of the universe as well as other cosmological parameters.
Riess1
A. G. Riess et al., Observational Evidence from Supernovae for
an Accelerating Universe and a Cosmological Constant, Astron.
J., 116, 1009 (1998).
Perl1
S. Perlmutter et al., Measurements of Ω and Λ
from 42 High-Redshift Supernovae, Astrophys. J., 517, 565 (1999).
Bennet1
C. L. Bennett et al., First-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic
Results, Astrophys. J. Suppl. Ser., 148, 1 (2003).
Tegmark1
Tegmark et al., Cosmological parameters from SDSS and WMAP, Phys. Rev. D, 69, 103501 (2004).
Seljak
Seljak et al., Cosmological parameter analysis including SDSS
Ly forest and galaxy bias: Constraints on the
primordial spectrum of fluctuations, neutrino mass, and dark
energy, Phys. Rev. D, 71, 103515 (2005).
Komatsu1
E. Komatsu et al., Seven-year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Cosmological Interpretation,
Astrophys. J. Suppl. Ser., 192, 18 (2011).
Weinberg
S Weinberg., The cosmological constant ,
Rev. Mod. Phys., 61, 1 (1989).
Carroll
S M Carroll, ,
Living Rev. Rel. , 4, 1 (2001).
Zlatev
Zlatev, L. Wang and P. J. Steinhardt, Quintessence, Cosmic Coincidence and the Cosmological constant ,
Phys. Rev. Lett, 82, 896 (1999).
fujii
Yasunori Fujii, Origin of the gravitational constant and
particle masses in a scale-invariant scalar-tensor theory,
Phys. Rev. D, 26, 2580 (1982).
carroll
Sean M. Carroll, Quintessence and the Rest of the World:
Suppressing Long-Range Interactions, Phys. Rev. Lett., 81, 3067 (1998).
chiba1
Takeshi Chiba, Takahiro Okabe and Masahide Yamaguchi,
Kinetically driven quintessence, Phys. Rev. D, 62, 023511
(2000).
kamen1
Alexander Kamenshchik, Ugo Moschella and Vincent Pasquier, An
alternative to quintessence, Phys. Lett. B, 511 265 (2001).
capo1
Salvatore Capozziello, Curvature quintessence, Int. J.
Mod Phys D, 11 483 (2002).
ferraro1
R. Ferraro and F. Fiorini, Modified teleparallel gravity:
Inflation without an inflaton, Phys. Rev. D, 75 084031 (2007).
nojiri
Shin'ichi Nojiri, Sergei D. Odintsov, and Misao Sasaki,
Gauss-Bonnet dark energy, Phys. Rev. D, 71 123509 (2005).
pad2
T. Padmanabhan and D. Kothawala, Lanczos-Lovelock models of
gravity, Phys. Rep., 531 115 (2013).
horava1
Petr Hořřava, Quantum gravity
at a Lifshitz point, Phys. Rev. D, 79 084008 (2009).
amendola1
Luca Amendola, Scaling solutions in general nonminimal
coupling theories, Phys. Rev. D, 60 043501 (1999).
dvali1
Gia Dvali, Gregory Gabadadze and Massimo Porrati, 4D gravity
on a brane in 5D Minkowski space, Phys. Lett. B, 485 208
(2000).
fabris1
J. C. Fabris, and S. V. B. Gonçalves, and R.de Sá Ribeiro,
Bulk viscosity driving the acceleration of the Universe,
Gen. Relat. Gravit., 38 495 (2006).
li1
Baojiu Li and John D. Barrow, Does bulk viscosity create a
viable unified dark matter model?, Phys. Rev. D, 79 103521
(2009).
Hiplito1
W. S. Hipólito-Ricaldi and H. E. S. Velten, and W. Zimdahl,
Viscous dark fluid universe, Phys. Rev. D, 82 063507
(2010).
av1
Arturo Avelino and Ulises Nucamendi, Can a matter-dominated
model with constant bulk viscosity drive the accelerated expansion
of the universe?, JCAP, 04 006 (2009).
av2
Arturo Avelino and Ulises Nucamendi, Exploring a
matter-dominated model with bulk viscosity to drive the accelerated
expansion of the Universe, JCAP, 08 009 (2010).
Athira1
Athira Sasidharan and Titus K. Mathew, Bulk viscous matter and
recent acceleration of the universe, Eur. Phys. J. C, 75, 348 (2015).
Jerin1
N D Jerin Mohan, Athira Sasidharan and Titus K. Mathew,Bulk viscous matter and recent acceleration of the universe based on causal viscous theory, Eur. Phys. J. C, 77, 849 (2017).
Athira2
Athira Sasidharan and Titus K. Mathew, Phase space analysis of bulk viscous matter
dominated universe, JHEP, 06, 138 (2016).
Athira3
Athira Sasidharan, N. D. Jerin Mohan, Moncy V. John and Titus K. Mathew, Bayesian analysis of bulk viscous matter dominated universe, Eur. Phys. J. C, 78, 628 (2018).
Maartens
R Maartens, Dissipative Cosmology, Classical and Quantum Gravity, 12, 1455 (1995).
Gron
N. Mostafapoor and O Gron, Viscous ΛCDM universe models, Astrophys. Space Sci, 333, 357-368 (2011).
Cruz
N Cruz, E Gonzalez and J Jovel,
Study of a viscous ΛWDM model : Near-Equilibrium Condition, Entropy Production and Cosmological constrains,
Symmetry, 14 1866 (2022).
Cruz1
N Cruz, E Gonzalez and J Jovel,
Singularities and soft- Big Bang in a viscous ΛCDM model,
Phys. Rev. D, 105 024047 (2022).
Eckart1
Carl Eckart, The Thermodynamics of Irreversible Processes.
III. Relativistic Theory of the Simple Fluid, Phys. Rev.
58 (1940) 919.
weinberg2 S. Weinberg, Gravitation and
cosmology: principles and applications of the general theory of
relativity, John Wiley & sons Inc., New york U.S.A. (1972).
ren1
Jie Ren and Xin-He Meng, Cosmological model with viscosity
media (dark fluid) described by an effective equation of state,
Phys. Lett. B, 633 1 (2006).
Singh J.P. Singh, Pratibha Singh, Raj Bali, Bulk viscosity and decaying vacuum density in Friedmann universe, Int J Theor Phys, 51 3828 (2012).
Avelino A. Avelino et.al, Bulk Viscous Matter-dominated Universes: Asymptotic Properties,
JCAP, 1308 12 (2013).
|
http://arxiv.org/abs/2307.06153v1 | 20230712132307 | Biofilm.jl: a fast solver for one-dimensional biofilm chemistry and ecology | [
"Mark Owkes",
"Kai Coblentza",
"Austen Erikssona",
"Takumi Kammerzella",
"Philip S. Stewart"
] | physics.bio-ph | [
"physics.bio-ph"
] |
a]Mark Owkesauthor
a]Kai Coblentz
a]Austen Eriksson
a]Takumi Kammerzell
b]Philip S. Stewart
[author] Corresponding author.
E-mail address: [email protected]
[a]Mechanical & Industrial Engineering, Montana State University
[b]Chemical & Biological Engineering, Montana State University
Biofilms are communities of microorganisms that grow on virtually all surfaces with sufficient nutrients including aquatic and industrial water systems and medical devices. Biofilms are complex, structured communities where the interplay of growth, metabolism, and competition between species interact with physical processes of diffusion, convection, attachment, and detachment. This work describes a model of a one-dimensional biofilm in a stirred tank reactor that incorporates these complexities. The model is implemented in the modern Julia programming language providing an efficient tool for studying a large variety of biofilms and the intricate communities the microorganisms create. Details of the new software, known as Biofilm.jl, including the mathematical model and organization and execution of the code, are provided. Examples of biofilms modeled using Biofilm.jl are presented such as a single heterotroph, a sulfide-oxidizing bacteria (SOB) and sulfate-reducing bacteria (SRB), and a phototroph. Postprocessing tools are described that allow a Biofilm.jl user to make plots and extract specific values from the solution and explore the simulated biofilm results.
modelsoftwarereaction-diffusionmicroorganismaccumulationJulia
PROGRAM SUMMARY
Program Title: Biofilm.jl
CPC Library link to program files: (to be added by Technical Editor)
Developer's repository link: https://github.com/markowkes/Biofilm.jl
Code Ocean capsule: (to be added by Technical Editor)
Licensing provisions(please choose one): MIT
Programming language: Julia (Developed on v1.8, tested on v1.7)
Supplementary material: https://markowkes.github.io/Biofilm.jl
Nature of problem:
This software solves for the temporal and spatial dynamics of an arbitrary number of substrates (or products) and particulates (biomass species) in a one-dimensional biofilm. The model includes the growth of particulates and the associated substrate consumption (or production), the biofilm thickness dynamics due to growth within and detachment from the top of the biofilm. Additionally, source terms can be used to model the death of biomass or other effects. Discontinuous inputs, such as the diurnal cycle or periodic dosing, can be included.
Solution method:
The software solves for the temporal dynamics of particulates, substrates, and biofilm thickness, which are described by differential equations. These equations are discretized using a finite volume method and organized into a single system of ordinary differential equations that are solved using the DifferentialEquations.jl library. The software includes a collection of postprocessing tools to assist the user with exploring the simulation results.
§ INTRODUCTION
Biofilms are communities of microorganisms that are attached to a surface <cit.> and predominate in almost all wet environments with sufficient nutrients <cit.>. Biofilms can form on a variety of surfaces including, for example, natural aquatic systems, teeth (dental plaque), and water system piping <cit.>. Furthermore, biofilms are not confined to solid-liquid surfaces and can be found at solid-air or liquid-liquid interfaces <cit.>. Biofilms have been shown to be associated with many human diseases and grow on a variety of medical devices and implanted biomaterials <cit.>.
Biofilm microorganisms, known as particulates, grow and consume or produce substrates. For example, an aerobe is a particulate that will consume the substrate oxygen when it grows. A biofilm may consist of a single particulate or many particulates. When there are many particulates, interactions with consumed or produced substrates and other physical phenomena can lead to highly structured multispecies communities <cit.>. Interestingly, these multispecies biofilms have been demonstrated to be profoundly different from planktonic (drifting or floating) cells with increased growth rates, enhanced tolerance to antibacterial agents, and changes in gene expression <cit.>.
Mathematical models have played an important role in the biofilm field because the complexity of these systems - in which biological phenomena such as growth, metabolism, and competition between species interact with physical phenomena such as diffusion, convection, attachment and detachment - leads to non-intuitive emergent properties. These behaviors can be accessed rigorously through computational approaches. Many different model structures from one-dimensional flat slabs to fully three-dimensional particle-based simulations have been derived <cit.>
. Several comprehensive reviews describe and discuss the variety of models in detail
<cit.>.
One of the earliest model structures was a one-dimensional version in which the biofilm is treated as a uniformly thick flat slab
<cit.>.
Solute transport into and out of the film is assumed to be governed by Fick’s Law. The competition for space within the biofilm by different microbial species and abiotic particles is analyzed by differential material balances on each particulate. The dynamics of biofilm thickness are controlled by the net balance of integrated growth and detachment from the top layer of the biofilm.
Though relatively simplistic, these types of models successfully capture such phenomena as: coexistence of aerobes and anaerobes in close proximity within the same biofilm and stratification of these two types of microorganisms
<cit.>, activity of biofilm-based wastewater treatment processes
<cit.>,
slow growth and dormancy in some regions of the biofilm
<cit.>,
reduced antimicrobial susceptibility in the biofilm mode of growth
<cit.>,
and plasmid transfer
<cit.>
.
Here we recapitulate the classic 1D biofilm model with an updated code and solver. This has the potential to be used as an educational and research tool, replacing outdated code developed at the Montana State University Center for Biofilm Engineering. This legacy code, now three decades old, is relatively slow and is not easily modified to include custom kinetic expressions or time-dependent boundary conditions. The updated code, known as Biofilm.jl, is written in the modern Julia programming language <cit.> which is designed to be fast while being a relatively easy language to write, run, and process results. Within Julia, Biofilm.jl leverages the state-of-the-art differential equation solvers <cit.> that help Biofilm.jl achieve fast and stable solutions to many biofilm problems.
The manuscript provides a detailed overview of the biofilm model and governing equations in Section <ref> and the numerical methods and implementation are presented in Section <ref>. Specifics on how the code is organized and can be executed are in Section <ref>. Finally, results on applying the code to five different test cases are provided in Section <ref>.
§ GOVERNING EQUATIONS
Biofilm.jl simulates a one-dimensional biofilm within a stirred tank reactor. More specifically, the concentrations of particulates X_t and substrates S_t within the tank are solved along with the particulate volume fractions P_b and substrate concentrations S_b within the biofilm. The thickness of the biofilm L_f is also computed. The code is written to solve for an arbitrary number of particulates and substrates, N_x and N_s, respectively.
§.§ Tank Equations
The temporal dynamics of particulate concentrations in the tank environment are described by
d X_t:j/dt = μ_j(S_t) X_t:j - Q X_t:j/V + v_det A X_b:j(L_f)/V + src_X:j
for j=1,…,N_x. The terms on the right-hand-side (RHS) are
1) the growth of the particulate in the tank,
2) transport due to flow out of the tank,
3) transfer to the tank due to the detachment of particulates from the biofilm, and
4) a source term of particulates.
The variables in the equation are as follows, t is time,
μ_j(S_t) is the growth rate of the j^th particulate and is dependent on the substrate concentrations within the tank S_t,
Q is the flowrate into and out of the tank,
V is the volume of the tank,
A is the area of the biofilm,
X_b:j(L_f) = ρ_j P_b:j(L_f) is the j^th particulate concentration at the top of the biofilm that can be computed from the particulate density ρ_j and volume fraction at the top of the biofilm P_b:j(L_f), and
src_X:j is the source term for the j^th particulate.
The detachment velocity v_det is modeled using
v_det≡ K_det L_f^2.
The substrate concentrations in the tank environment are described by
d S_t:k/dt = -∑_j=1^N_xμ_j(S_t) X_t:j/Y_j,k + Q S_in:k/V - Q S_t:k/V - A S_ t:k^flux/V + src_S:k
for k=1,…,N_s, where Y_j,k is the biomass yield coefficient on the substrate,
S_in:k is the influent substrate concentration,
S_t:k^flux is the flux of substrates into the biofilm from the tank due to diffusion, and
src_S:k is the source term for the k^th substrate.
The terms on the RHS are
1) consumption of substrates due to the growth of the particulate in the tank,
2) transport due to flow into the tank,
3) transport due to flow out of the tank,
4) transfer of substrates into the biofilm due to diffusion, and
5) a source term of substrates.
§.§ Biofilm Equations
The particulates within the biofilm are described using their volume fractions P_b and are described with
∂ P_b:j/∂ t =
μ_j(S_b) P_b:j
- ∂ v(z) P_b:j/∂ z
+ src_X:j/ρ_j
for j=1,…,N_x and
P_b:j(t,z) is the j^th particulate within the biofilm and varies with time and location within the biofilm.
The terms on the RHS are
1) the growth of the particulate in the biofilm,
2) transport through the biofilm due to the growth velocity v_i, and
3) source term of particulate at i^th location in biofilm.
The growth velocity v(z) is the vertical velocity of the biofilm due to the growth and sources of particulates deeper within the biofilm. In general, this velocity can be written as an integral that captures growth and sources from the bottom of the biofilm (z=0) to some location z within the biofilm, ,
v(z)= ∫_z'=0^z∑_j=1^N_x1/P_tot(μ_j(S_b(z')) P_b:j + src_X:j/ρ_j) dz'
where P_tot=∑_j=1^N_xP_b:j
The dynamics of substrates within the biofilm are described by
∂ S_b:k/∂ t =
D_b:k∂^2 S_b:k/∂ z^2
- ∑_j=1^N_xμ_j(S_b) X_b:j/Y_j,k
+ src_S:k
for k=1,…,N_s, where D_b:k is the effective diffusion coefficient of the k^th substrate within the biofilm.
The terms on the RHS are
1) diffusion of substrates in the biofilm,
2) consumption of substrates due to the growth of the particulate in the biofilm, and
3) source term of substrates. Note that the top boundary condition of the diffusion term must match the flux of substrate into the biofilm S_t:k^flux and provides a connection between the tank and biofilm dynamics.
§.§.§ Biofilm Thickness
The thickness of the biofilm L_f is described by
d L_f/dt = v(L_f) - v_det
where the first term on the RHS is the growth velocity at the top of the biofilm (Eq. <ref>) and the second term is the detachment velocity (Eq. <ref>).
§ DISCRETIZATION AND NUMERICAL METHODS
The model solves for the particulate and substrate concentrations within the tank and the particulate volume fractions and substrate concentrations in the biofilm along with the biofilm thickness. The tank is assumed to be a stirred tank reactor and thus the tank concentrations are only a function of time. The variations within the biofilm are, however, important and a one-dimensional grid is used with N_z grid cells. The variables are stored at the cell centers denoted z_m,i for i = 1,…,N_z. Cell faces are denoted z_i for i = 1,…,N_z+1.
The governing equations consist of five time-dependent differential equations (Eqs. <ref>, <ref>, <ref>, <ref>, and <ref>) that are solved at the same time using built-in ODE solvers in Julia's DifferentialEquations.jl library <cit.>. To use the solvers the dependent variables are organized in a single vector such that the system of equations can be written as
d 𝐬𝐨𝐥/dt=𝐫𝐡𝐬
where
𝐬𝐨𝐥=[X_t,S_t,P_b,S_b,Lf]^𝖳
where [·]^𝖳 indicates transpose. The bold terms on the RHS of the previous equation contain the different particulates and substrates and can be written as
X = [X_t:1,X_t:2,…,X_t:N_x]^𝖳,
S = [S_t:1,S_t:2,…,S_t:N_s]^𝖳,
P_b = [P_b:1,P_b:2,…,P_b:N_x]^𝖳, and
S_b = [S_b:1,S_b:2,…,S_b:N_s]^𝖳.
These last two expressions contain the particulate and substrates within the biofilm. The biofilm is discretized with N_z grid points and therefore each of these variables should be written as a vector of values at each grid location, ,
P_b:j=[P_b:j,1,P_b:j,2,…,P_b:j,N_z]^𝖳
S_b:j=[S_b:j,1,S_b:j,2,…,S_b:j,N_z]^𝖳
All together 𝐬𝐨𝐥 is a vector with
N_var = N_x + N_s + N_x· N_z + N_s· N_z + 1
variables.
Similarly, 𝐫𝐡𝐬 is a vector of length N_var defined with
𝐫𝐡𝐬=[d X/dt,d S/dt,d P_b/dt,d S_b/dt,d Lf/dt]^𝖳
where the right-hand sides are defined using the equations provided in Section <ref>. The RHS of the ordinary differential equations, Eq. <ref>, <ref>, and <ref>, can be used as written. However, the partial differential equations (Eq. <ref> and <ref>) that describe the particulate volume fractions and substrate concentrations within the biofilm need to be discretized before they can be used with Julia's ODE solvers. Biofilm.jl uses a finite volume discretization to approximate the spatial derivatives.
To discretize Eq. <ref> for the particulate volume fractions in the biofilm, the equation is written at each grid cell, ,
∂ P_b:j,i/∂ t =
μ_j(S_b:i) P_b:j,i
- ∂ v_i P_b:j,i/∂ z
+ src_X:j,i/ρ_j
for j=1,…,N_x and i=1,…,N_z.
The spatial derivative is discretized as
∂ v_i P_b:j,i/∂ z = P_b:j,i+1^flux - P_b:j,i^flux/Δ z
where Δ z is size of the grid cell and the flux on the i^th face (z_i location) is defined as
P_b:j,i^flux = v(z_i) P_b:j,i-1.
for j=1,…,N_x and i=2,…,Nz+1.
This expression approximates the volume fraction at the cell face using the volume fraction in the cell below the face introducing upwinding that stabilizes the solution. Note that a zero-flux condition is used at the bottom of the biofilm, P_b:j,1^flux=0. In the previous equation v(z_i) is computed using the discrete form of Eq. <ref> which is
v(z_i)= ∑_n=1^i-1∑_j=1^N_x1/P_tot(μ_j(S_b:i) P_b:j,i + src_X:j,i/ρ_j) Δ z
and includes contributions from growth and sources in all cells below the i^th face.
Similarly, to discretize Eq. <ref> for the substrate concentrations in the biofilm, the equation is written at each grid cell, ,
∂ S_b:k,i/∂ t =
D_b:k∂^2 S_b:k,i/∂ z^2
- ∑_j=1^N_xμ_j(S_b:i) X_b:j,i/Y_j,k
+ src_S:k,i
for k=1,…,N_s and i=1,…,N_z.
The diffusion term with a second-order derivative is discretized using
D_b:k∂^2 S_b:k,i/∂ z^2
= S_b:k,i+1^flux - S_b:k,i^flux/Δ z
and the flux on the i^th face (z_i location) is defined using a second-order, central finite difference operator that can be written as
S_b:k,i^flux = D_b:kS_b:k,i - S_b:k,i-1/Δ z
for k=1,…,N_s and i=2,…,N_z
The flux requires boundary conditions at the top and bottom of the biofilm. A zero-flux condition is used at the bottom of the biofilm, , S_b:k,1^flux=0. For the boundary condition at the top of the biofilm, the diffusive flux from the tank is matched with the diffusive flux at the top of the biofilm, i.e.,
S_t:k^flux = S_b:k,N_z+1^flux
D_t:kd S_t:k/dz = D_b:kd S_b:k,N_z/dz
where D_t is the diffusion coefficient in the tank. A boundary layer within the tank at the top of the biofilm of thickness L_L is introduced to evaluate the derivative in the tank. Discretizing these derivatives requires introducing the substrate concentration at the top of the biofilm S_top, which when combined with finite differences, leads to
D_t:kS_t:k - S_top/L_L = D_b:kS_top - S_b:k,N_z/Δ z/2.
Rearranging provides an expression for S_top, ,
S_top = D_t:k (Δ z/2) S_t:k + D_b:k L_L S_b:k,N_z/ D_t:k (Δ z/2) + D_b:k L_L .
With S_top defined, either the LHS or RHS of Eq. <ref> can be used to set the flux at the top of the biofilm.
Note that the previous expression is well-posed if the tank boundary layer is set to zero, , L_L=0 and the expression reduces to S_top=S_t:k.
In summary, the model consists of solving Eq. <ref>, which contains N_var (Eq. <ref>) ODEs. The RHSs of these ODEs are defined with Eqs. <ref>, <ref>, <ref>, <ref>, and <ref>. The next section describes the code organization that solves these equations including details on the inputs and outputs of the model.
§ ORGANIZATION AND EXECUTION OF THE CODE
The model is written entirely Julia. It was developed in Julia v1.8 and tested on v1.7. To use the model, a user will, typically, create a case file that defines the input parameters, calls the solver, and does any post-processing such as making plots or animations. Each of these is described in more detail below. Unit tests that compare simulated results to analytic solutions and a series of simulations compared to precomputed solutions are included and described in <ref>. These unit tests ensure the accuracy of the code base.
§.§ Installation
Installation instructions are provided in the README file with additional detailed instructions provided at
https://markowkes.github.io/Biofilm.jl/installation.
§.§ Input Parameters
The input parameters describe the biofilm to be simulated and provide code options. All the parameters are added to a dictionary (key:value structure). The dictionary is initially created with
# Create empty dictionary to hold parameters
d = createDict()
Parameters are then added by calling addParam!. For example, adding the simulation title and time to the dictionary can be done with, ,
addParam!(d, "Title", "Single Substrate and Particulate Case")
addParam!(d, "tFinal", 1.0)
This process of calling addParam! is repeated until all the parameters are added to the dictionary.
The dictionary is now checked to make sure all the parameters are provided and packaged into a structure p that the solver is expecting by calling
# Package and check parameters
p = packageCheckParam(d)
The structure p fully describes the case and is the only variable passed to the solver. It might seem redundant to have the parameters entered into the dictionary d and then packaged into the struct p. While you can directly create the struct p, creating the dictionary first and then the struct is recommended because when building the dictionary much more helpful error messages can be produced to help the user identify and fix issues with the parameters, then packaging the parameters into the struct allows the solver to run more efficiently.
Below is a complete list of each parameter. Additional examples of setting the parameters can be found in the examples (<https://markowkes.github.io/Biofilm.jl/examples/>).
§.§.§ Simulation Parameters
* Title : Description of the case, used, e.g., on the title of plots.
* tFinal : Simulation is performed from t=0 to t=tFinal days.
* tol : Tolerance used for differential equation solver. The solution will have an error less than tol.
* outPeriod : Period in days between text outputs during simulation. Note that the solver will take smaller timesteps than outPeriod to achieve the specified tolerance tol and the entire solution will be available when the solver completes.
§.§.§ Particulate Parameters
* XNames : Vector of strings with the name of each particulate used on text output and plots.
* Xto : Vector of initial particulate concentration(s) in the tank in g/m^3.
* Pbo : Vector of initial particulate volume fractions(s) in the biofilm. This value is used to set the volume fraction at all the grid cells.
* rho : Vector of particulate densities in g/m^3.
* Kdet : Detachment coefficient of particulates from the top of the biofilm in 1/m-d.
* mu : Vector of functions that provide the particulate growth rates in 1/d. For example, in the example in Section <ref> there are two particulates, “Live” and “Dead”, and one substrate. The growth rates are defined as
addParam!(d, "mu", [(S,X,Lf,t,z,p) -> (mumax * S[1]) / (KM + S[1])
(S,X,Lf,t,z,p) -> 0.0 ] )
which sets the growth rate of the first particulate to the Monod equation <cit.>
μ_1=μ_Live = μ_maxS_1/K_M + S_1
and the second growth rate to μ_2=μ_Dead = 0. The growth rates can be other functions such as a double Monod equation, inhibition model, or any other function of the substrate concentrations S, particulate concentrations X, biofilm thickness Lf, time t, height in biofilm z, or other parameters within the struct p.
* srcX : Vector of functions that provide the sources of particulate concentrations in g/m^3-d.
§.§.§ Substrate Parameters
* SNames : Vector of strings with the name of each substrate used on text output and plots.
* Sto : Vector of initial substrate concentration(s) in the tank in g/m^3.
* Sbo : Vector of initial substrate concentration(s) in the biofilm in g/m^3. This concentration is used to set the concentration at all the grid cells.
* Sin : Vector of functions that provide the influent substrate concentrations in g/m^3.
* srcS : Vector of functions that provide the sources of substrates concentrations in g/m^3-d.
* Yxs : Matrix of biomass yield coefficients on the substrate. The matrix has size N_x × N_s. Within the matrix, the value in the j^th row and k^th column, , Yxs[j,k], provides the number of grams of the j^th particulate grown for each gram of the k^th substrate consumed. Positive values of Yxs indicate substrate consumption and negative values indicate substrate generation by particulate growth. If the particulate does not depend on the substrate the yield coefficient should be infinite. However, to make this input more user-friendly, a value of 0 or Inf can be entered and the code automatically converts a 0 to Inf.
* Db : Vector of effective substrate diffusion coefficients through the biofilm in m^2/day.
* Dt : Vector of aqueous substrate diffusion coefficients in the fluid environment within the tank in m^2/day.
§.§.§ Tank Parameters
* V : Volume of tank in m^3.
* A : Area of biofilm within the tank in m^2.
* Q : Flowrate into and out of tank in m^3/day.
§.§.§ Biofilm Parameters
* Nz : Number of grid points used to discretize the biofilm.
* LL : Thickness of boundary layer within the tank on the surface of biofilm in m.
* Lfo : Initial thickness of the biofilm in m.
§.§.§ Optional Parameters
* plotPeriod [default=outPeriod] : Period in days between plot renderings during the simulation. For long simulations, you can increase plotPeriod to reduce the number of plots that are created and speed up the simulation. Note: plotPeriod is required to be a multiple of outPeriod.
* plotSize [default=(1600,1000)] : Size of plots in pixels.
* makePlots [default=true] : Boolean that controls if code should produce plots while running.
* optionalPlot [default="growthrate"] : While running, the code displays six plots. The sixth plot defaults to plotting the particulate growth rate versus location in biofilm. Setting optionalPlot = "source" will change this plot to the particulate source term versus location in biofilm.
When defining the parameters it is often useful to reference other variables or functions. For example, when defining the growth rate in Eq. <ref> the variables μ_max and K_m are used. In the code, these variables can be defined before growthrate is added to the dictionary. Similarly, functions can be defined and used when defining parameters, see Example <ref>.
§.§ Overview of Solver
The solver consists of a collection of Julia (.jl) files. Each file is described in additional detail below.
Biofilm.jl : is the main file of the code base and defines the Biofilm module, exported functions and objects, and includes all the other files. Biofilm.jl is used by first including the module
julia> using Biofilm
and then calling one of the functions described below.
solver.jl : contains the function BiofilmSolver(p::param), which runs a simulation of a biofilm using the parameters provided in the struct p that includes all the inputs described in Section <ref>. This function prepares the solver, runs the solver, and formats and returns the solution to be easily interpreted by the user.
The workhorse within BiofilmSolver is a call to the function solve(), which is part of the DifferentialEquations.jl package <cit.> This package contains many state-of-the-art methods for solving differential equations and is key to the speed of Biofilm.jl. The ODE solver computes the solution of Eq. <ref>.
rhs.jl : contains biofilmRHS!(), which computes the RHS of the differential equations in Eq. <ref>. The first part of this function splits the vector of dependent variables sol into X_t, S_t, P_b, S_b, and L_f. Next, a number of intermediate variables are computed which are terms that appear multiple times in the RHSs by calling functions that are defined in computes.jl. Finally the RHS of Eqs. <ref>, <ref>, <ref>, <ref>, and <ref> are computed and organized in a single vector.
computes.jl : contains functions that compute intermediate variables that appear multiple times in the evaluation of the differential equation RHSs. By computing these variables once the results can be reused to improve the speed of the code.
outputs.jl : produces a text output to the terminal and plots the solution.
structs.jl : defines the composite types or structs used by the code. These structs hold a number of parameters or variables together to reduce the number of variables that need to be passed into functions. At the top of each function, any variables from the structures are accessed with the @unpack macro.
parameters.jl : defines the functions createDict(), addParam!(d,name,value), and packageCheckParam(d) that are used to create a dictionary of parameters and package and check the parameters. Additionally, this file provides the function printDict(d) that can be used to print all the values in the a dictionary.
tools.jl : provides a variety of helper functions.
postprocess.jl : contains functions to help a user analyze the results of a simulation. Additional details are provided in the next section.
§.§ Postprocessing Output
A simulation is run by calling
julia> t,zm,Xt,St,Pb,Sb,Lf,sol = BiofilmSolver(p) # Run solver
The results are returned in the following output variables.
t is a vector of the solution times.
zm is a vector of grid cell centers used to discretize the biofilm. Since the biofilm thickness (and thus the grid) changes with time, only the grid at the final time is provided as an output.
Xt and St are arrays of the particulate and substrate concentrations in the tank as a function of times t.
The variables Pb and Sb are arrays of the biofilm particulate volume fractions and substrate concentrations at the final simulation time as a function of the grid point locations zm.
Finally, sol is an object that is created by DifferentialEquations.jl that contains the complete solution of the ODE solver. Although it contains the entire solution, it is difficult to process since it contains N_var (Eq. <ref>) dependent variables.
These outputs can be analyzed to provide additional information on the biofilm that has been simulated using the following functions.
biofilm_plot(sol,p) : can be used to produce the standard plots of the simulation as shown in, , Fig. <ref>. This is useful to make a plot of the solution at the end of the simulation, especially if plots are turned off during the simulation using the optional parameter makePlots=false.
This function calls a plotting recipe and allows for considerable customization of the plot using the inputs of the Plots.jl library and solution processing form DifferentialEquations.jl. The examples in Section <ref> demonstrate some plots that can be created with this function.
biofilm_analyze(sol,p,times [,makePlot=true]) : will display the dependent variables at the specified times t. For example, after running the example described in Section <ref>, the solution could be analyzed to check the solution at t=[0,0.25,0.5,0.75,1.0], ,
julia> times = 0:0.25:1.0
julia> biofilm_analyze( sol, p, times)
Analyzing Single Substrate and Particulate Case
Time | Bug | Oxygen | min,max( Bug) | min,max( Oxygen) | Lf
0.000 | 10 | 10 | 0.08, 0.08 | 0, 0 | 10
0.250 | 102 | 51.7 | 0.08, 0.08 | 29.1, 51.3 | 545
0.500 | 256 | 2.94 | 0.08, 0.08 | 0.568, 2.87 | 348
0.750 | 257 | 2.93 | 0.08, 0.08 | 0.745, 2.87 | 312
1.000 | 257 | 2.93 | 0.08, 0.08 | 0.761, 2.87 | 309
where the first column is time, the second and third columns are the particulate and substrate concentrations in the tank, the fourth and fifth columns are the minimum and maximum of the particulate volume fractions and substrate concentration within the biofilm, and the last column is the biofilm thickness. This function provides the outputs Xt, St, and Lf at the specified times for further postprocessing. For example, this is used to process the example in Section <ref> to create Fig. <ref>.
Including the optional parameter makePlots=true will create a plot of the solution within the biofilm at each specified time. This is used in the processing of the example in Section <ref>.
biofilm_movie(sol, p, times [, filename="anim.gif", fps=20)] : can be used to make an animation of the solution at the specified times. The optional parameters filename and fps can be included to change the filename or frame rate of the movie. This function uses the @animate macro and gif() function from the Plots.jl package.
biofilm_sol2csv( sol, p [, filename="biofilm.csv"]) : takes the output of a simulation and writes the results as a CSV file. Each row in the file corresponds to a different output time and the column contains the time, the tank particulate and substrate concentrations, biofilm particulate volume fractions and substrate concentrations, and finally the biofilm thickness. This function can be useful for users that prefer looking at results in a spreadsheet.
§ EXAMPLES
This section provides a series of example cases that highlight the capabilities of Biofilm.jl. These examples are the same as those provided with the code, however, additional postprocessing has been done for some of the cases. The code used to create the results in this manuscript is available in a branch called CodePaper accessible at <https://github.com/markowkes/Biofilm.jl/tree/CodePaper/examples>
§.§ Case 1: Single Substrate and Particulate
This is a simple example of a single particulate named "Heterotroph" that consumes the substrate "Nutrient" to grow using the Monod equation <cit.>.
The case is run by starting Julia, installing Biofilm.jl, and executing
julia> include("examples/Case1.jl")
The output from the program is shown in Fig. <ref>. Note that the size of the plot was adjusted by creating the plot with the command
julia> biofilm_plot(sol,p,size=(900,600))
which is run after the simulation is complete. The top row shows how the particulate concentration (left), substrate concentration (center), and biofilm thickness (right) vary with time. The bottom row shows the particulate volume fraction (left), substrate concentration (center), and particulate growth rate (right) versus location within the biofilm at the end of the simulation (1.00 days for this case).
The heterotroph concentration in the tank starts at the initial condition of Xto=10 g/m^3 and increases to a steady-state value of 257 g/m^3 as seen in the top-left plot. The nutrient concentration in the tank starts at an initial concentration of Sto=10 g/m^3, increases quickly due to the flow into the tank with a concentration of Sin=100 g/m^3, then starts decreasing as the heterotroph concentration increases and more of the nutrient is consumed until a steady-state of 2.92 g/m^3 is reached as shown in the top-center plot. The biofilm thickness increases from a value of Lfo=1.0E-5 μm quickly due to growth in the initial system with an abundance of the nutrient. Eventually, the thickness decreases to a steady thickness of 309 μm as observed in the top-right plot.
Within the biofilm at the final time of t=1.0 days, the bottom-left plot shows that the heterotroph volume fraction is 0.08 everywhere. This value comes from the initial condition of Pbo=0.08 and the fact that there is only one particulate in this simple system. The nutrient concentration, in the bottom-center plot, is highest at the top of the biofilm and decreases within the biofilm as diffusion of the nutrient into the biofilm is balanced by the consumption by the heterotroph. The concentration of the nutrient at the top of the biofilm has a value of Sb[Nz]=2.87 g/m^3, which is close to the steady-state tank concentration of 2.92 g/m^3. The difference is due to the diffusion rate through the boundary layer and very top of biofilm (Eq. <ref>) The heterotroph's growth rate, shown in the bottom-right plot, is highest near the top of the biofilm where the nutrient concentration is highest and decreases further into the biofilm.
§.§ Case 2: Multiple Substrates
This example is based on the acid stress response in Zhang et al. <cit.> and features biomass X that consumes glucose s and produces lactate p. Here the variables s and p stand for substrate and product, respectively. Note that both s and p are modeled with Eqns. <ref> and <ref> with S=[s,p]^𝖳.
The presence of lactate can inhibit the growth of biomass. This appears as a reduction in growth rate, as p increases, until a maximum value of p_max above which the growth ceases, ,
μ = μ_max s (1 - p/p_max) p ≤ p_max
0.0 p> p_max
where the term in parentheses captures inhibition. In the code, this is done with two steps. The first is to define a function that computes the growth rate for a single combination of s and p, ,
# Define mu function
function mu(s,p)
if p < p_max
mu = mu_max*s*(1-p/p_max)
else
mu = 0.0
end
return mu
end
Next, this function is provided to the growth rate variable in the parameter dictionary (see Section <ref>), ,
# Growthrate: call mu(s,p) for S = [s,p]
addParam!(d, "mu", [(S,X,Lf,t,z,p) -> mu(S[1],S[2])])
which calls mu(s,p) function using the notation S = [S[1],S[2]] = [s,p].
To see the impact of the inhibition, two simulations are run with the inhibition turned on and off, respectively. The results from both simulations are shown in Fig. <ref> which was generated with the code
# Run with inhibition
p_max = 400
t_in,zm_in,Xt_in,St_in,Pb_in,Sb_in,Lf_in,sol_in = BiofilmSolver(p)
plt = biofilm_plot(sol_in,p,"Inhibition",
size=(900,600), line=(:blue,[:solid :dash]))
# Run without inhibition add to plot
p_max = Inf
t_no,zm_no,Xt_no,St_no,Pb_no,Sb_no,Lf_no,sol_no = BiofilmSolver(p)
plt = biofilm_plot!(plt,sol_no,p,"No Inhibition",
size=(900,600), line=(:red,[:solid :dash]))
This code is run after running a standard Case 3 simulation which sets the case parameters. The code above runs two simulations, the first with the inhibition turned on (p_max = 400) and plots the results in blue. Next a second simulation is run without inhibition (p_max = Inf and the result is added to the previous plot and colored in red. Note that the second plot command uses biofilm_plot! (note the exclamation point), which adds to the plot instead of creating a new plot.
The results in Fig. <ref> show the output of both simulations. In the top row, the biomass (left) and substrate (center) concentrations along with the biofilm thickness (right) are shown versus time. After roughly 2.5 days all the quantities reach a steady state. The bottom-left plot shows the biomass volume fraction, which since there is only one particulate remains at the input value of 1/6 throughout the biofilm. The bottom-center plot shows the glucose and lactate concentrations throughout the biofilm. The glucose (solid lines) is diffusing from the tank into the biofilm and is consumed by the biomass, which lowers the concentration at lower heights. The lactate (dashed lines) is produced by biomass and has high concentrations in the lower part of the biofilm. Diffusion moves the lactate toward the surface of the biofilm where it enters the tank. The biomass growth rate is shown in the bottom-right figure. Growth is highest at the top of the biofilm where high glucose concentrations exists and decreases at lower heights in the biofilm.
Comparing the inhibited (blue) results to the case without inhibition (red) shows that without inhibition the biomass concentrations in the tank (top-left) is over 2 times larger and the biofilm is roughly 25% thicker (top-right). For both cases the growth of biomass occurs at the top of the biofilm and ceases at lower heights within the biofilm, however the limit to growth is different in the two cases. Without inhibition, the biomass consumes all of the glucose that is diffusing through the biofilm and the glucose concentration goes to zero, limiting growth, at lower heights in the biofilm (bottom-center). With inhibition, growth is limited by the inhibition due to the higher concentrations of lactate lower in the biofilm (bottom-center) and some glucose exists even at the bottom of the biofilm (bottom-center).
§.§ Case 3: Multiple Particulates
This example demonstrates a simulation with multiple particulates, namely "Living Bug" and "Dead Bug". The living bug consumes the substrate and grows using the Monod equation. The living bugs die and turn into dead bugs, this is captured in the model with the source terms
src_Living = - b X_Living, and
src_Dead = b X_Living.
The results at the end of the 100-day simulation are shown in Fig. <ref>. Of note for this case are the variations within the biofilm. Near the surface of the biofilm, the substrate diffuses from the tank (bottom-center plot) and is consumed by the living bugs causing the living bug volume fraction to be larger in roughly the top half of the biofilm. Deeper in the biofilm, the living bugs do not receive substrate and die causing the dead bug volume fraction to increase. For this case, the parameter optionalPlot="source" is used by adding it to the parameter dictionary with
addParam!(d, "optionalPlot", "source")
With this parameter, the bottom-right figure shows the source term that transfers living bugs to dead bugs. The source has the largest magnitude near the top of the biofilm where the living bug concentration is the highest.
§.§ Case 4: Multiple Substrates and Particulates
This case demonstrates how to model a biofilm with three substrates and three particulates. The substrates are oxygen, sulfate, and hydrogen sulfide and the particulates are sulfide-oxidizing bacteria (SOB), sulfate-reducing bacteria (SRB), and dead bacteria.
The system requires interactions between the bacteria and substrates to function.
Oxygen and sulfate enter the tank with the influent.
The SRB uses the sulfate and produces sulfide, but this growth is inhibited by oxygen.
The SOB can then use the sulfide, produced by the SRB, and oxygen to grow.
The SRB and SOB slowly die creating the dead bacteria.
These dynamics appear in the growth rates which can be written as
μ_SOB = μ_max:B( S_1/K_m:B_1 + S_1·S_3/K_m:B_3 + S_3)
μ_SRB = μ_max:C( S_2/K_m:C_2 + S_2·1.0/1.0 + S_1/K_I)
μ_Dead = 0.0
where S_1, S_2, and S_3 correspond to oxygen, sulfate, and hydrogen sulfide, respectively.
The parameters μ_max:B, μ_max:C, K_m:B_1, K_m:B_3, K_m:C_2, and K_I are constants and set in the case file.
The source terms can be written as
src_SOB = - D_SOB X_1
src_SRB = - D_SRB X_2 - D_O:SRB S_1
src_Dead = + D_SOB X_1 + D_SRB X_2 + D_O:SRB S_1
where X_1, X_2, and X_3 correspond to SOB, SRB, and dead biomass, respectively. The constants D_SOB and D_SRB describe the death rate of SOB and SRB, respectively. D_O:SOB is the death rate of SRB due to the precense of oxygen.
These expressions are coded as
addParam!(d, "mu", [
(S,X,Lf,t,z,p) -> mumaxB*(S[1]./(KmB1.+S[1])).*(S[3]./(KmB3.+S[3])), # SOB
(S,X,Lf,t,z,p) -> mumaxC*(S[2]./(KmC2.+S[2])).*(1.0./(1.0.+S[1]/KI)) , # SRB
(S,X,Lf,t,z,p) -> 0.0 ] ) # Dead
and
addParam!(d, "srcX", [
(S,X,Lf,t,z,p) -> - D_SOB*X[1] , # SOB
(S,X,Lf,t,z,p) -> - D_SRB*X[2] - D_O_SRB*S[1] , # SRB
(S,X,Lf,t,z,p) -> + D_SRB*X[2] + D_SOB*X[1] + D_O_SRB*S[1] ]) # Dead
The results after 100 days are shown in Fig. <ref>. The top row shows the SOB, SRB, and dead bacteria concentrations within the tank (left), substrate concentrations (center), and biofilm thickness (right). After roughly 50 days the quantities reach a steady state. The bottom row shows the particulate volume fractions (left), substrate concentrations (center), and particulate growth rates (right). In the bottom of the biofilm, a layer of dead cells collects as little sulfate (needed by SRB) or oxygen (needed by SOB) reaches this layer. In the middle region, the SOB lives and has access to the sulfide from the SRB and oxygen diffusing from the tank. In the top layer, the SRB thrives as the sulfate concentration is very high.
§.§ Case 5: Phototroph
This example demonstrates how a particulate that grows in light with a diurnal cycle can be simulated. The particulate is a phototroph with a growth rate that depends on light intensity I(t,z), ,
μ_phototroph = μ_max I(t,z).
As the phototroph grows it generates the substrate, oxygen, through the biomass yield coefficient Yxs=[-0.52].
The light within the tank turns on and off throughout each day. Mathematically, the intensity of light within the tank is 1.0 on each day after 0.25 days and then 0.0 at 0.75 days creating the on/off cycle. In the code, this is achieved by defining the intensity function as
# Define smoothed Heaviside function
smoothHeaviside(t,t0)=0.5*tanh.(100*(t.-t0).-0.5).+0.5
# Define light intensity as function of time
intensity(t) = 1.0 - (smoothHeaviside(mod(t,1),0.25)
-smoothHeaviside(mod(t,1),0.75))
where the smoothHeaviside function spreads out the discontinuity in intensity over a finite time.
Furthermore, the light dissipates as it travels through the biofilm and the dissipation is defined with
diss=2000; # Dissipation rate [1/m]
dissipation(z,Lf) = max.(0.0,1.0.-(Lf.-z)*diss)
Finally, the light intensity at a specified time and position in the biofilm can be computed with
# Define light intensity as a function of
# time and location within biofilm
light(t,z,Lf) = intensity(t)*dissipation(z,Lf)
These functions are defined before the input parameters and are called when the growth rate is defined as
mumax = 0.4;
addParam!(d, "mu", [(S,X,Lf,t,z,p) -> mumax*light(t,z,Lf)])
To check the light intensity function, we can make a plot of light intensity over a few days within the tank, ,
# Define constants
Lf = 600e-6; z = Lf;
# Define times to plot
t = range(0.0,3.0,1000)
# Make plot
plot(t,map(t -> light(t,z,Lf),t))
which produces the plot shown in Fig. <ref>. Figure <ref> shows how the light dissipates as it travels through the biofilm and it is largest at the top of the biofilm (600 μm) and decreases linearly until reaching a zero intensity.
Due to the diurnal discontinuities in the light intensity, it is important to make sure the solver reduces the timestep near the discontinuity. This is achieved by setting discontinuityPeriod=0.25 using
# Let the solver know when discontinuities (changes in light) occur
addParam!(d, "discontinuityPeriod",0.25)
which makes the solver look for discontinuities every 0.25 days. Without this parameter, periods when the light is turned on or off are missed since the solver uses large timesteps.
The output of this example is shown in Fig. <ref>. The top row shows the phototroph concentration (top-left), oxygen concentration (top-center), and biofilm thickness (top-right) all as a function of time. The results vary during each day as the light turns on and off. The bottom row displays the variations within the biofilm at the final time of 50 days when the light is off. With the light off, no growth is occurring and the quantities do not vary throughout the biofilm.
The quantities within the biofilm can be analyzed at different times. In particular, it will be useful to analyze the results when the light is on. This can be done by running the biofilm_analyze() postprocessing function with the solution of Case 5 when the light is on (, 49.5 days) which is done by running
tout=49.5 # Time when the light is on
biofilm_analyze(sol,p,tout,makePlot=true)
This code analyzes the results and produces the plots shown in Fig. <ref>. When the light is on, the phototroph is growing (right) and the oxygen concentration is high within the biofilm as it is produced by the phototroph and decreases towards the top of the biofilm due to diffusion (center). Since there is only one particulate, the figure of the volume fraction (left) remains the same throughout the entire simulation.
Due to the high-frequency changes in the results, this example can be post-processed to observe additional features in the solution. To plot just the last two days of the simulation, the biofilm_analyze() function can be used again, but now with multiple times, ,
# Times to analyze solution
tout = 48.0:0.01:50.0;
# Get solution at certain times
Xtout,Stout,Lfout = biofilm_analyze(sol,p,tout)
# Plot tank particulate concentration versus time
plot(tout,Xtout')
which produces a simpler version of the left plot in Fig. <ref>. The complete code to produce the three plots in Fig. <ref> is available in <ref>.
With these results, it is clear how the quantities vary with time. When the light is on the phototroph (particulate) concentration increases and then decreases when the light is off. The oxygen concentration is highest in the tank when the light is on and oxygen is generated by the growing phototroph. When the light is off, the oxygen concentration decreases to 8.6 which matches the specified influent substrate concentration Sin. The biofilm thickness increases when the light is on due to the phototroph growth in the biofilm and then decreases when the light is off.
§ CONCLUSIONS
A modern implementation of a one-dimensional biofilm model is presented. The software, known as Biofilm.jl, solves for the temporal dynamics of particulates (biomass) and substrates which are effected by biological phenomena including growth, metabolism, and competition as well as physical processes such as diffusion, convection, attachment and detachment. The model is implemented in Julia <cit.> and leverages the state-of-the-art differential equations solvers provided by DiffereentialEquations.jl library <cit.>.
Example biofilms are provided to demonstrate the capabilities of Biofilm.jl. The first example is of a single heterotroph growing in an environment with a nutrient. The second example demonstrates an acid stress response and has a biomass that consumes glucose and produces lactate which inhibits the biomass growth. The third example demonstratess a single biomass species that grows and then dies. The fourth example demonstrates a complex interplay of sufide-oxidizing bacteria (SOB), sulfate-reducing bacteria (SRB), and dead bacteria with the substrates oxygen, sulfate, and hydrogen sulfide. Finally, a fifth example is of a phototroph in a diurnal cycle. The examples highlight the variety of biofilms that Biofilm.jl can simulate as well as provide templates for postprocessing simulation results.
Biofilm.jl is provided with an open-source license and is hosted on github. The repository contains detailed user manual with installation, execution, and postprocessing instructions as well as an overview of the governing equations. Coding and Julia best practices, including documentation, unit tests, and plot recipes, are implemented.
§ CODE UNIT TESTS
Tests have been implemented to ensure the code is running correctly. The tests consist of 1) a series of unit tests that check specific parts of the code and compare the results to analytic solutions and 2) examples that are compared with previously computed answers to ensure a code modification does not introduce a bug. Below are details on the unit tests and the analytic solutions.
The tests can be run by cloning the GitHub repository, activating the package, entering the package manager by typing "]" at the Julia prompt, and then running test. In addition to manual tests, all the tests are executed automatically each time a new commit is added to the GitHub repository.
§.§ Zero L_L
Having the code behave robustly when the boundary layer within the tank is set to zero is a nice code feature. This is achieved through Eq. <ref> which avoids dividing by zero. The unit test test_zeroLL runs a simulation with LL = 0.0 and checks that the substrate concentration at the top of the biofilm S_top equals the substrate concentration at the topmost grid cell within the biofilm. The test shows the condition is met to machine precision.
§.§ Flux matching
At the top of the biofilm the flux of substrate through the tank boundary layer matches the flux through the top of the biofilm to the first grid point, , Eq. <ref>. This test ensures that the calculation of S_top is correct and the fluxes do match computing S_top then computing the two fluxes and ensuring they are equal. The test shows the condition is met to machine precision.
§.§ Diffusion test
The distribution of substrates within the biofilm is mainly controlled by diffusion and consumption due to the growth of particulates. In this test, the steady-state distribution of substrate within the biofilm is compared with the analytic solution for a biofilm with one substrate and one particulate. The particulate has a growth rate of μ = μ_max S /K_M. This biofilm, at steady-state, has an analytic solution of <cit.>
S_b:1(z) = S_t:1coshϕ z / Lf/coshϕ
where
ϕ = √(μ_maxρ P_b:1 L_f^2/D_b K_m Y_1,1)
The unit test consists of running a series of simulations until the biofilm reaches a steady state with different numbers of grid cells within the biofilm. For each simulation, the error, which is the maximum difference between the analytic and computed solution, is computed. Finally, the order of the method is computed by fitting a line to the error versus the number of grid points in log-log space. It is expected the methods are second-order accurate and that the test shows an order of 1.9 for the grids tested.
§.§ Steady-State with Large Diffusivity
If the diffusion coefficients D_b and D_t are considered to be very large, a steady-state solution to the biofilm model can be found since the substrate concentration can be assumed to be uniform within the biofilm and also equal to the tank concentration. Note that in the following derivation of the analytic solution, subscripts on the substrate and particulate have been dropped as there is only one of each.
The steady-state solution can be found by setting the time derivatives equal to zero in the differential equations. For example, the
thickness of the biofilm is described by Eq. <ref> and at steady-state this reduces to
v(z_N_z+1) = v_det,
which means that the vertical velocity due to growth in the biofilm (LHS) matches the detachment velocity (RHS). The LHS can be computed using Eq. <ref> which when simplified, by setting src_X = 0 and letting S_b = S where S is the uniform substrate concentration in the biofilm and tank, becomes v(z_N_z) = μ(S) L_f. The RHS can be evaluated with the definition of the detachment velocity in Eq. <ref>. Combining these simplifications and solving for L_f leads to
L_f = μ(S)/K_det
which provides the steady-state biofilm thickness.
The particulate concentration in the tank is described by Eq. <ref>, which can be simplified by assuming steady-state, no source term, and using v_det = K_det L_f^2 = μ(S) L_f (Eq. <ref>) leads to
0 = μ(S) X_t - Q X_t/V + μ(S) L_f A X_b:N_z/V.
The steady-state substrate concentration can be simplified by taking Eq. <ref> and letting the time derivative equal zero and setting the source term to zero. Furthermore, the flux of substrate into the biofilm can be found by assuming S_b=S and then integrating Eq. <ref> over the biofilm and using the divergence theorem which leads to
S^flux_t = μ(S) X_b/Y_1,1 L_f
and states that the flux of substrate into the biofilm is equal to the consumption used by the growth of particulates within the biofilm.
All together these simplifications change Eq. <ref> into
0 = -μ(S) X_t/Y_1,1 + Q S_in/V - Q S/V - μ(S) X_b L_f A/Y_1,1 V
which can be rearranged to provide an expression for X_t(S), ,
X_t = Y_1,1Q/μ(S) V(S_in - S)
- X_b L_f A/V
To solve for the steady-state solution, an initial guess for the substrate concentration S was made. With this guess, the biofilm thickness L_f can be computed with Eq. <ref>, tank particulate concentration with Eq. <ref>, and then the residual of Eq. <ref> is evaluated. Based on this residual, the guess for S is updated and the procedure is repeated until the residual becomes less than a tolerance.
Running this test shows that the simulated value of the substrate concentration is found to be within 0.004% of the analytic value.
§.§ Time Integration
This test ensures the time integration is implemented correctly by checking the substrate concentration in the tank with an inflow and outflow. There is a particulate in the simulation, but the growth rate is set to zero so that it does not consume (or produce) substrate. The diffusion coefficients are also set to very small numbers to limit the impacts of substrate entering or leaving the biofilm. With these assumptions, Eq. <ref> simplifies to
d S_t/dt = Q/V(S_in - S_t)
which has a solution
S_t(t) = S_in + (S_t^0 - S_in) e^-Q/V t
where S_t^0 is the initial condition.
Comparing the computed solution to the analytic solution shows that the maximum relative error in the solution is roughly 1e-6.
§ CASE 5 POSTPROCESSING CODE
This code was used to produce the plots in Figs. <ref> and <ref> in Section <ref>. The code can be executed after the simulation for Case 5 is run.
# Biofilm quantities
tout = 49.5 # Time when light is on
biofilm_analyze(sol,p,tout,makePlot=true,plotSize=(900,325))
# Could also make plot by directly calling recipe
# biofilm_plot_film(sol([0,tout]),p,size=(900,325))
savefig("Case5_lighton.pdf")
# Times to analyze solution
tout = 48.0:0.01:50.0;
# Get solution at certain times
Xtout,Stout,Lfout = biofilm_analyze(sol,p,tout)
# Function to plot when the light is on
rectangle(w, h, x, y) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])
function plot_light()
plot(rectangle(0.5,1000.0,48.25,0.0),
opacity=.2,
fillcolor=:yellow,
linecolor=:yellow,
)
plot!(rectangle(0.5,1000.0,49.25,0.0),
opacity=.2,
fillcolor=:yellow,
linecolor=:yellow,
)
end
# Plot tank particulate concentration versus time
plot_light()
plot!(tout,Xtout',
linecolor=:blue,
xlabel=("Time [days]"),
ylabel=("Phototroph Conc. [g/m3]"),
legend=false,
size=(300,300),
ylims=(minimum(Xtout)-0.001,maximum(Xtout)+0.001),
)
savefig("Case5_Xt.pdf")
# Plot tank substrate concentration versus time
plot_light()
plot!(tout,Stout',
linecolor=:blue,
xlabel=("Time [days]"),
ylabel=("Oxygen Conc. [g/m3]"),
legend=false,
size=(300,300),
ylims=(minimum(Stout)-0.1,maximum(Stout)+0.1),
)
savefig("Case5_St.pdf")
# Plot biofilm thickness versus time
plot_light()
plot!(tout,1e6.*Lfout,
linecolor=:blue,
xlabel=("Time [days]"),
ylabel=("Biofilm Thickness [um]"),
legend=false,
size=(300,300),
ylims=(minimum(1e6*Lfout)-2,maximum(1e6*Lfout)+2),
)
savefig("Case5_Lf.pdf")
elsarticle-num
|
http://arxiv.org/abs/2307.07475v2 | 20230714165525 | Analysis of Unified Galaxy Power Spectrum Multipole Measurements | [
"Jamie Donald-McCann",
"Rafaela Gsponer",
"Ruiyang Zhao",
"Kazuya Koyama",
"Florian Beutler"
] | astro-ph.CO | [
"astro-ph.CO"
] |
firstpage–lastpage
Low-Scale Leptogenesis with Low-Energy Dirac CP-Violation
Serguey T. Petcov
August 12, 2023
===========================================================
We present a series of full-shape analyses of galaxy power spectrum multipole measurements from the 6dFGS, BOSS, and eBOSS galaxy surveys. We use an emulated effective field theory of large-scale structure (EFTofLSS) model to conduct these analyses. We exploit the accelerated prediction speed of the neural-network-based emulator to explore various analysis setups for our cosmological inference pipeline. Via a set of mock full-shape analyses of synthetic power spectrum multipoles, designed to approximate measurements from the surveys above, we demonstrate that the use of alternative priors on nuisance parameters and restricted model complexity reduces many of the biases previously observed in marginalised cosmological constraints coming from EFTofLSS analyses. The alternative priors take the form of a Jeffreys prior <cit.>; a non-informative prior that can mitigate against biases induced by marginalising over poorly constrained nuisance parameters. When performing a joint analysis of all synthetic multipoles, we see an improvement in the level of agreement between the marginalised ln(10^10A_s) constraints and the truth; from ∼2.0σ to ∼0.42σ. Using our pipeline to analyse the measured multipoles, we find an improvement in the level of agreement with cosmic microwave background (CMB) results; from ∼2.4σ to ∼0.5σ. Therefore, we conclude that the spectroscopic galaxy survey datasets listed above are consistent with constraints obtained from the CMB.
large-scale structure of the Universe – methods: data analysis – cosmology: cosmological parameters
§ INTRODUCTION
Conducting full-shape analyses of galaxy clustering statistics <cit.>, such as the power spectrum, is becoming a standard approach to complement analyses that focus of specific features like the baryon acoustic oscillations (BAO). To run one of these full-shape analyses, we require a theoretical model that allows us to make a prediction for the clustering statistic of interest for a given set of cosmological parameters θ. There are two possible routes here: 1.) use a simulation-based model, 2.) use an analytical model. A simulation-based model will likely be more accurate on small, nonlinear, scales. Comparisons of dark matter only N-body simulation codes have shown agreement in predictions of the dark matter power spectrum for scales k ≲ 1 h Mpc^-1 <cit.>. However, developing a simulation-based model requires many simulations with different sets of cosmological parameters sampling from the parameter space of interest. These suites of simulations <cit.> require huge computational cost to produce, and this cost can prohibit the use of such models. An analytic model may be less accurate on nonlinear scales <cit.>, but using such a model will incur a significantly lower computational cost.
One such analytical model that is gaining in popularity when conducting full-shape analyses is the effective field theory of large-scale structure <cit.>. This perturbation-theory based model maps predictions for the dark matter clustering to that of galaxies via a series of nuisance parameters ϕ, that are marginalised over when putting constraints on the cosmological parameters θ. Two popular examples of EFTofLSS code implementations are PyBird <cit.> and CLASS-PT <cit.>. Predictions for the galaxy power spectrum multipoles can be made with PyBird in 𝒪(1 s)[This is a processor dependant statement. In <cit.> the prediction speed was reported as 1.01 s± 13.1 ms. Based on 100 predictions made on a laptop with an Intel i5 2.50 GHz dual-core processor with four threads and 8 GB of RAM. Table 1 of <cit.> reports prediction speeds from CLASS-PT. In default mode, the performance appears similar to PyBird.]. This is significantly faster than a numerical simulation, but running an MCMC with PyBird still requires a non-negligible amount of computational resources. This cost can limit the exploration of the analysis setup when using this model to carry out parameter inference.
The idea of emulation to reduce computational cost is being used more and more frequently for cosmological inference problems and is now used to accelerate inference pipelines that are based on analytic theory models <cit.> as well as those with simulation-based models <cit.>. These emulators consist of nonlinear interpolators that are fitted to (or trained with) a set of input and output pairs {θ, Y(θ)}, with Y(θ) being the function of interest. The nonlinear interpolation scheme generally takes the form of a machine learning algorithm like a Gaussian process or neural network (NN). In <cit.>, the NN-based EFTEMU was added to the matryoshka suite of emulators <cit.>. The EFTEMU was developed to reduce the cost of EFTofLSS model evaluations and increased the prediction speed of the galaxy power spectrum multipoles by over three orders of magnitude. This increase in prediction speed opens up the opportunity to test more analysis setup choices when using the EFTofLSS model.
In this paper, we exploit the increased prediction speed from the EFTEMU to perform full-shape analyses of galaxy power spectrum multipole measurements from several completed galaxy surveys. We also examine how the analysis setup impacts the inferred cosmology. Through a series of mock full-shape analyses, we validate our cosmological inference pipeline. We then demonstrate that using alternative priors and more restrictive sets of nuisance parameters can alleviate some of the biases in the inferred cosmological parameters that can be seen when conducting full-shape analyses with the EFTofLSS. We find that using these alternative priors can alleviate some of the slight tensions in the marginalised cosmological parameter constraints when comparing with results from cosmic microwave background (CMB) analyses. The paper is organised as follows. In section <ref> we introduce the galaxy surveys considered for this work, along with the multipole measurements used. In section <ref>, we further introduce the EFTofLSS and discuss any changes made to the EFTEMU for this work. In section <ref>, we present a series of mock analyses designed to test our inference pipeline. In section <ref>, we present results from the analysis of the multipole measurements introduced in section <ref>. We conclude in section <ref>.
§ DATA
There have now been several large-scale spectroscopic redshift surveys that have run to completion; combining to provide detailed maps of the universe covering a wide redshift range. For this work, we focus on three surveys that cover distinct redshift ranges: the 6dF galaxy survey <cit.>, the baryon oscillation spectroscopic survey <cit.>, and the extended baryon oscillation spectroscopic survey <cit.>. The redshift catalogues from each of these surveys are now publicly available such that galaxy clustering measurements can be made for each of them. <cit.> presents measurements of the power spectrum multipoles from each of these surveys, along with wide-angle and window function matrices. These matrices allow wide-angle effects and the survey window function to be included in theory predictions of the galaxy power spectrum multipoles via two simple matrix multiplications. All measurements have 40 k-bins over the range 0 < k < 0.4 h Mpc^-1. The BOSS and eBOSS samples are split into subsamples for the northern and southern galactic cap (NGC and SGC) and, in the case of BOSS, two redshift bins (BOSSz1 and BOSSz3). This results in seven sets of multipoles with four effective redshifts z_eff=[0.096, 0.38, 0.61, 1.52]. We refer the reader to Table 1 in <cit.> for more details about each sample.
§.§ Mocks
When exploring analysis setups, we need to examine if a particular setup leads to more or less bias in the inferred cosmological parameters than another. Mock multipoles were published alongside the measurements in <cit.>. These mocks are those used to calculate covariance matrices and contain survey geometry and systematics to match their associated measurements. It is helpful to have sets of mock multipoles for which we know the true cosmology as well as the "true" values for the nuisance parameters of the EFTofLSS model (bias parameters and counterterms, see section <ref>). To that end, we produce a set of mock multipoles using PyBird with the cosmology set to the TT,TE,EE+lowE+lensing+BAO ΛCDM best-fit values from table 2 in <cit.>. The nuisance parameters are fit to the mean of the mock multipole measurements published in <cit.> for each sample. We refer to the resulting multipoles as the "PyBird mocks".
The nuisance parameters for the PyBird mocks are determined by finding the maximum a posteriori (MAP) estimate for four bias parameters and six counterterms. This is done by finding the minimum of the negative log-likelihood (see section <ref> for likelihood definition) with a wide uniform prior on all bias parameters and counterterms. Except for the linear bias, this prior ranges from -50 < b_i < 50. The linear bias prior is truncated at zero to allow for positive values only. The nuisance parameters are fit to the mean of the mock multipoles on scales 0 < k < 0.2 h Mpc^-1, and the covariance is rescaled by a factor of 10. Figure <ref> shows the PyBird mock multipoles alongside the multipole measurements and mocks from <cit.> for the z=0.61 NGC sample. The bottom panel shows the residuals normalised by the rescaled covariance Δ(k)/(σ(k)/10). We can see that the agreement of the PyBird mock multipoles and the mocks of <cit.> is within 1σ. It should be noted that the agreement is better still when considering the unscaled covariance. Plots showing the PyBird mocks for the other samples all exhibit similar results.
§ MODEL
As alluded to in section <ref>, there are two general routes to modelling the galaxy power spectrum. The first is to use numerical simulations; providing accurate small-scale predictions but coming at a high computational cost. The second is to develop an analytic model; producing computationally efficient predictions (in comparison to numerical simulations) but being less accurate on small scales.
Probing the small, nonlinear, scales of the galaxy power spectrum can improve the constraints on the cosmological parameters. For a given survey, we will have a larger number of galaxy-galaxy pairs with small separations than large separations; thus, the statistical error on small scales will be lower than on large scales. The EFTofLSS was developed to extend the scales of validity of analytic predictions, allowing us to probe smaller scales and exploit the reduced statistical error.
§.§ EFTofLSS
Standard perturbation theory (SPT) models the dark matter overdensity field as a perfect fluid. Although successful on large scales, where the density perturbations are small, its description starts to break down when entering nonlinear scales <cit.>. In recent years considerable effort has been put into an effective description which extends the range of SPT into a mildly nonlinear regime.
EFTofLSS introduces a cut-off scale which acts as an effective low-pass filter, leading to the fluid equations now being solved in terms of long-wavelength overdensity and velocity fields. Furthermore, an effective stress-energy tensor is introduced, which captures the effects of the small scales physics on the larger scales. At a given order n, the effect of these small scales and their backreaction onto the long wavelength field can be captured by a finite number of so-called "counterterms" c_i. These counterterms are free parameters that must be fitted to data or calibrated with simulations. Including a nonlinear bias scheme, mapping the underlying dark matter field as described above to the observed galaxy densities, the 2D redshift-space galaxy power spectrum in terms of scale k and cosine of angle to the line-of-sight μ, can be written as
P_g (k, μ) = Z_1 (μ)^2 P_11(k)
+ 2 ∫d^3 q/(2 π)^3 Z_2(q,k - q, μ)^2 P_11(|k - q|) P_11(q)
+ 6 Z_1(μ) P_11(k) ∫d^3 q/(2 π)^3 Z_3( q,-q,k ,μ) P_11(q)
+ 2 Z_1(μ) P_11(k) ( c_ctk^2/k_M^2 + c_r,1μ^2 k^2/k_M^2 + c_r,2μ^4 k^2/k_M^2)
+ 1/n̅_g ( c_ϵ, 1 + c_mono.k^2/k^2_M + 3/2 c_quad. ( μ^2 - 1/3 ) ) k^2/k^2_M) .
In the above Z_i are the redshift-space galaxy density kernels (for their exact form, see ), n̅_g is the mean galaxy density[For the analyses of this work we use values of 4×10^-4 h^3 Mpc^-3 for the 6dFGS and BOSS samples. For the eBOSS QSO samples we use 1.5×10^-5 h^3 Mpc^-3.], and k_M^-1 is a normalisation scale[More recent papers that use the PyBird EFTofLSS model have an additional normalisation scale k_R. For this work, we neglect k_R, as such k_R=k_M. Throughout we set k_M=0.7 Mpc^-1.]. Overall the 1-loop EFTofLSS introduces ten nuisance parameters. Four parameters (b_1-4) are introduced in the expansion of the galaxy density and velocity field in terms of the underlying dark matter field. These parameters are found in the galaxy kernels Z_i. It has been noted that b_2 and b_4 are highly degenerate <cit.>. It is common to reparameterise such that
c_2 = (b_2+b_4) / √(()2) ,
c_4 = (b_2-b_4) / √(()2) .
There are three stochastic parameters (c_ϵ, 1,c_mono.,c_quad.) that are introduced to capture the difference between the actual observed galaxy field and its expected value. Finally, three counterterms that encapsulate the impact of UV physics: the effective sound speed of the dark matter field c_ct, and c_r,1 and c_r,2 which control the impact of small scales on redshift space distortion.
§.§ Alcock-Paczyński effect
A reference cosmology is required to measure the galaxy power spectrum from redshift catalogues provided by surveys like those introduced in section <ref>. Any differences between the true underlying cosmology and the reference cosmology lead to distortions of distances parallel and perpendicular to the line of sight. This is the so-called Alcock-Paczyński (AP) effect <cit.>. The distortion parallel and perpendicular to the line of sight is given by the distortion parameters q_∥ and q_, respectively. These parameters are defined as
q_∥ = D_A(z)H(z=0)/D_A^ref.(z)H(z=0) ,
q_ = H^ref.(z)H(z=0)/H(z)H^ref.(z=0) ,
with H(z) and D_A(z) being the Hubble parameter and angular-diameter distance as a function of redshift, respectively. The superscript ref. in the above equations indicates quantities calculated at the reference cosmology. The AP distortion is applied to the scales and angles as k'=q_^-1 B k^ref. and μ'=F^-1 B^-1μ^ref.. With F=q_∥ / q_, and B given by
B = [1+(μ^ref.)^2(F^-2-1)]^1/2 .
The 2D power spectrum can then be decomposed into multipoles via
P_l(k) = 2l+1/2q_∥ q_^2∫_-1^1 P(k', μ')ℒ_l(μ^ref.)dμ^ref ,
with ℒ_l being the l-th order Legendre polynomial.
The EFTEMU (and PyBird) make predictions for the power spectrum multipoles rather than the 2D power spectrum. To include the AP effect, via Equation <ref>, we need to reconstruct the 2D power spectrum from the multipoles. We do this via
P(k, μ) = ∑_l=0 P_l(k)ℒ_l(μ) .
The EFTEMU (as trained for this work) makes predictions for the first two even multipoles. Reconstructing the 2D power spectrum from only the first two even multipoles will result in systematic errors when including the AP effect via Equation <ref>. These errors are expected to be small compared to the error associated to the multipole measurements discussed in <ref>. It should be noted that the PyBird mocks introduced in section <ref> were constructed including the hexadecapole P_4(k). As such, the mock analyses of section <ref> will test if these systematic errors from the 2D power spectrum reconstruction impact the inferred cosmology.
§.§ Emulator
The EFTofLSS model described above (as implemented in PyBird) takes 𝒪(1 s) to produce predictions for a given set of cosmological parameters at a given redshift. Although efficient enough for direct use when conducting cosmological inference, this prediction time does prohibit the exploration of analysis setups (such as prior choice, scale cuts, and fixed parameters). If running a typical MCMC using this model requires 𝒪(10^5 – 10^6) model evaluations, then 𝒪(days) would be required to reach convergence. In <cit.>, the EFTEMU was added to the matryoshka <cit.> suite of emulators. The EFTEMU was developed to accelerate EFTofLSS predictions by several orders of magnitude by replacing the direct calculation of the kernels P_n,l of the EFTofLSS model with predictions from simple NNs.
The EFTEMU was originally trained with data drawn from a five-dimensional ΛCDM parameter space, approximately centred on the Planck 2018 best-fit cosmology. Despite being wide, this training space is too restrictive to constrain some of the ΛCDM parameters much beyond this when using the large-scale structure data considered for this work. With this in mind, we re-train the EFTEMU for this work. The width of the prior on ω_c, h, and ln(10^10A_s) was increased significantly, and the spectral index n_s was fixed as we do not expect to get any meaningful constraint on n_s from our analyses. Table <ref> compares the prior for the original EFTEMU to that used in this work. The larger training space required a change in the training procedure compared to that in <cit.>. The increased width of the cosmological prior, particularly for ln(10^10A_s), increases the dynamic range of the kernels P_n,l. The original preprocessing procedure involved rescaling all P_n,l such that at every k-value their magnitude was in the range [0,1]. We modify this procedure by first taking the log of the P_n,l before rescaling into the range [0,1]. Figure <ref> shows the kernels for the PyBird mocks at z=0.61 for the first three even multipoles on scales 0.001 ≤ k ≤ 0.3 h Mpc^-1. There are 21 kernels for each multipole, and these 21 kernels can be split into three groups. The first group (P_n,l^11) contains the linear terms, the second group (P_n,l^loop) contains the loop terms, and the third group (P_n,l^ct.) contains the counterterms. These three groups also represent the grouping used for the EFTEMU; each component of the EFTEMU emulates a different group <cit.>. It can be seen from Figure <ref> that some of the P^loop_l and P^ct._l kernels are exclusively negative or have a zero crossing. To allow us to take the log of these kernels, we include either a simple sign change or the addition of a constant to the kernel preprocessing. Taking the log results in a reduced dynamic range in the training data and leads to higher prediction accuracy. We also significantly increase the number of samples generated for training and testing from 10,000 to 50,000. Only 40,000 are used for training; the remaining 10,000 are used for testing.
Figure <ref> shows the prediction error on the monopole of the power spectrum when producing predictions with the re-trained EFTEMU. Each row shows the prediction error at a different redshift, and each column shows the prediction error computed with different sets of nuisance parameters. The orange shaded regions show the 68% and 95% credible intervals (CIs) of the prediction error as a function of k. The solid coloured lines show the inverse signal-to-noise ratio (SNR) for the monopole measurements considered for this work at their respective redshifts. The shaded regions have been calculated from predictions for 10,000 unseen cosmologies. For the left column, the 10,000 cosmologies have been combined with sets of nuisance parameters that produce "reasonable" predictions for the monopole. We take random draws from a very wide uniform prior[0<b_1<10, -10<{b_2, b_4}<10, -500<{b_3, c_ct, c_r,1, c_r,2}<500.] on the nuisance parameters and calculate the multipoles for each set of cosmological and nuisance parameters. We define "reasonable" predictions as those which the monopole is strictly positive and those which can be said to remain perturbative[See appendix <ref> for our perturbative condition.]. Any sets of parameters that do not meet these criteria are rejected, and the nuisance parameters resampled from the prior. This is repeated until we have nuisance parameters for all 10,000 cosmologies. For the right column, samples from the posterior resulting from full-shape analysis of the 6dFGS-like PyBird mock (see section <ref>) are used to inform the nuisance parameters for the unseen cosmologies. For each unseen test cosmology, the posterior sample with the closest cosmology[The nearest neighbour in the 4D cosmological parameter space. With the Euclidean distance as the distance metric.] is selected, and its nuisance parameters are associated to that test cosmology. The two columns of Figure <ref> show two different aspects of the prediction accuracy: the left column represents the prediction accuracy across the entire theoretically viable parameter space, the right column represents the prediction accuracy for power spectra that look more similar to something that has been previously observed. We can see from the right column that for all redshifts considered and for all k<0.25 h Mpc^-1, the prediction error from the emulator is less than the error on the data at the 68% level at each respective redshift. However, from the left column, we can see that for z=0.38, 0.61 when considering the entire theoretically viable prior space, the prediction error can be greater than the error on the data on small scales (k≳ 0.17 h Mpc^-1). In practice, we find that the level of prediction accuracy from the re-trained EFTEMU does not induce any significant bias to the cosmological parameters when performing inference, as shown in section <ref>.
§ MOCK ANALYSES
In this section, we present the results from a series of analyses of the PyBird mocks (described in section <ref>). These mock analyses aim to verify that our cosmological inference pipeline does not induce biases in the cosmological parameter constraints. In addition, we explore how various analysis setups impact the results. In all cases, to put constraints on cosmological parameters, we sample from the posterior distribution via Preconditioned Monte Carlo <cit.>; as implemented in pocoMC[Various parameters control the efficiency of the sampling with pocoMC. We use the default values for all of these.] <cit.>. Precondition Monte Carlo utilises Normalising Flows <cit.> and Sequential Monte Carlo <cit.> to efficiently sample from posterior distributions even when they have a very complex shape. We use a Gaussian likelihood of the form
ln[ℒ(P|θ,ϕ)] = -1/2(P-P̃)^TC^-1(P-P̃) ,
with P being a concatenation of the multipole measurements considered P=[P_0, P_2], P̃ being the multipole predictions from the model P̃=[P̃_0, P̃_2] for a given set of cosmological parameters θ and nuisance parameters ϕ, and C being the covariance matrix.
Many of the nuisance parameters of the EFTofLSS model appear linearly as multiplicative factors for the kernels. This allows us to marginalise over these parameters analytically rather than sampling from them. This is standard practice when conducting parameter inference with the EFTofLSS <cit.>. Carrying out the analytic marginalisation reduces dimensionality and thus leads to a more efficient inference of the cosmological parameters. Although it is more efficient to analytically marginalise the linearly appearing parameters, the prediction speed of the EFTEMU means that fully sampling the parameter space is tractable. We explore the use of both the marginalised and full likelihood in the results below.
§.§ Fiducial Results
We start by presenting results from an analysis with a fiducial setup. For this fiducial setup, we analyse the power spectrum monopole and quadrupole on scales 0.01 < k < 0.15 h Mpc^-1. Figure <ref> shows that the nearest neighbour prediction error on these scales is considerably lower than the error associated to the mocks at all redshifts for which the EFTEMU is trained. We fix three out of the ten nuisance parameters to zero, those parameters being c_4, c_r,2, c_mono.. These parameters are commonly set to zero in analyses of the monopole and quadrupole with PyBird <cit.>. The priors on ω_c, h, and ln(10^10A_s) are those that define the emulator training space (given in Table <ref>). For ω_b, we use a truncated normal distribution as the prior, with a mean of 0.02235 and a standard deviation of 0.00049[This is motivated by BBN <cit.> and are the same values as those used in <cit.>.]. The hard bounds of this prior are given by the emulator training space as with the other cosmological parameters. The priors on the nuisance parameters are given in Table <ref>. We refer to the prior of Table <ref> as the "classic" prior. A majority of the EFTofLSS works cited in this paper use a prior of a similar form. Note that the prior on c_ϵ,1 is defined independent of n̅_g. For n̅_g=4×10^-4 h^3 Mpc^-3 the prior width is 400, which is in line with other works that use the PyBird EFTofLSS model.
Figure <ref> shows the resulting marginalised 1D and 2D posteriors from the analysis of the PyBird mocks with the fiducial setup and the full likelihood[Throughout this work, plots showing marginalised posterior distributions have been produced directly or with the assistance of GetDist <cit.>.]. The two contour levels in the off-diagonal panels are 1σ and 2σ, and the grey dashed lines indicate the location of the true values used to generate the mocks. Along with the sampled parameters ω_c, h, and ln(10^10A_s) we also plot the marginalised posterior distributions on two derived parameters: Ω_m=(ω_c+ω_b)h^-2, and Ã=b_1^2A_s10^8. For the purposes of this plot, the derived à posterior samples have had the truth subtracted, such that the 1D marginalised posterior should peak exactly at zero if unbiased. This normalisation of à allows us to compare the distributions calculated for each sample as they all have different b_1 values. Looking at Figure <ref>, it is clear that for PyBird mocks with a higher SNR (BOSSz1 and BOSSz3 NGC), the agreement with the truth is very good for all parameters. For PyBird mocks with a lower SNR (6dFGS and eBOSS QSO SGC), we observe some significant shifts from the truth in many of the 1D and 2D projections. A likely cause for these shifts is the volume effect <cit.>; these shifts are (at least partially) a result of marginalisation. In previous works, it has been shown that ln(10^10A_s) is particularly susceptible to volume effects <cit.>, and indeed it is the parameter in Figure <ref> that shows the most significant observed shift. See appendix <ref> for more discussion on the volume effect with a toy example.
The shifts induced in marginalised posteriors are reduced when the constraining power from the data is higher. Figure <ref> shows, with dashed coloured lines, the 2σ region of the 2D marginalised posterior distributions on b_1 and ln(10^10A_s) resulting from analysis of the PyBird mocks for various samples with the fiducial setup described above. Also plotted in Figure <ref>, with coloured
shaded regions, is the 2σ region of the 2D marginalised posteriors obtained from analysis of the PyBird mocks with covariance matrices rescaled by a factor of 1 / 50. It can be seen that although there is agreement with the truth (represented with dotted grey lines) at the 2σ level in both cases for all the data samples plotted, the agreement is significantly better when the covariance has been rescaled. The posteriors have shrunk and remained consistent with the truth. If it were the case that the biases observed in Figure <ref> were resulting from anything other than marginalisation, we would not see this behaviour. We also note from Figure <ref> that the shift in posteriors and median values (shown with coloured squares and points) resulting from rescaling the covariance is along a line of constant à (shown with grey solid lines). Giving a compelling argument for using à as a diagnostic quantity when understanding if observed biases in ln(10^10A_s) are a result of a true systemic bias from the analysis pipeline or a result of volume effects. Finally, we note that rescaling the covariance in this way does not only resolve the observed bias in ln(10^10A_s), but in all parameters shown in Figure <ref>.
§.§ Exploration of Analysis Setups
The results from the previous section have shown that the analysis pipeline developed for this work can return unbiased constraints on cosmological parameters of interest for a typical EFTofLSS analysis setup. We can exploit the increased prediction speed of the EFTEMU to explore various analysis setups and observe their impact on the constrained cosmology.
§.§.§ Scale Cuts
We start by exploring different scale cuts. It can be seen from the solid coloured lines in Figure <ref> that there is clear scale dependence in the inverse SNR for all the data samples considered for this work. There is also a clear scale dependence in the emulator prediction error. As mentioned in section <ref>, when analysing LSS data, there is a general expectation that the SNR increases when pushing to smaller scales. However, this is only true if the scales are not dominated by shot noise. If we combine this with a higher modelling error on smaller scales, although the expectation might be that including smaller scales will improve the constraints, this might not be the case.
Figure <ref> shows the peak posterior values and 68% CIs of 1D marginalised posteriors (with coloured squares and lines, respectively) on the cosmological parameters Ω_m, h, and ln(10^10A_s) resulting from analysis of the PyBird mocks with k_max.=0.150,0.175,0.200 h Mpc^-1 and the full likelihood. The results from the analysis of the BOSS-like mocks all show the same general trend; including smaller scales shrinks the 68% CI, reduces the observed bias in the peak posterior value, or both. The results for 6dFGS show a slightly tighter constraint on Ω_m and h when including smaller scales but the constraint on ln(10^10A_s) remains almost constant. This is likely because the constraint on ln(10^10A_s) from 6dFGS is completely dominated by volume effects. We can also see that including smaller scales worsens the agreement with the truth for the eBOSS-like mocks; the 68% CI shrinks, the peak posterior shifts away from the truth, or both. As can be seen from Figure <ref>, the emulator error is always significantly lower than the error associated with the eBOSS-like mocks; thus, the cause for the behaviour of the eBOSS-like results is more likely to be a result of the worsening SNR rather than emulator error. It can also be seen from Figure <ref> that the smaller-scale modes have larger errors, thus including them worsens the volume effect.
Table <ref> quantifies the level of agreement between the true cosmological parameters of the PyBird mocks and the 1D marginalised posteriors resulting from analysis of these mocks with k_max.=0.15 h Mpc^-1 and k_max.=0.2 h Mpc^-1. For the purposes of this paper, we quantify the agreement as the number of σ separating the peak posterior values of two given marginalised distributions. We define the agreement N_σ as
N_σ = | μ_0 - μ_i |/√(σ_0^2+σ_i^2) ,
with μ_i and σ_i being the mean and 1σ error calculated from the 1D marginalised posterior, and μ_0 and σ_0 being the mean and 1σ error of the reference (when calculating N_σ for the PyBird mocks σ_0=0.). In the case of asymmetric distributions, if the residual μ_0 - μ_i is positive, we use the 1σ error to the right of the peak posterior. If the residual is negative, we use the 1σ error to the left of the peak posterior. We note that for all apart from the eBOSS-like mocks, the level of agreement does not significantly change and is at the ≲ 0.5σ level for Ω_m and h when comparing the results obtained with the two k_max. values. For the BOSS-like mocks, the level of agreement improves to <1σ for ln(10^10A_s) when including smaller scales. It is also worth noting that although the analyses with k_max.=0.2 h Mpc^-1 include scales at which the observed emulator error from Figure <ref> is at a similar level to the data error, we find no significant bias in the constrained cosmology for those samples least susceptible to volume effects (BOSSz1 NGC, BOSSz3 NGC).
§.§.§ Bayesian Model Comparison
pocoMC allows us to easily calculate the Bayesian evidence for each posterior distribution. We use these evidence calculations to compare EFTofLSS sub-models. We define the full model as the PyBird EFTofLSS model with all nuisance parameters free, and a sub-model as any model that results from fixing any single nuisance parameter or combination of parameters to zero.
The first sub-model we consider (ℳ_1) is that of the fiducial setup; with c_4, c_r,2, and c_mono. all set to zero. Figure <ref> shows the natural log of the Bayes factor ln(B_i) resulting from analysis of the PyBird mocks with k_max.=0.2 h Mpc^-1 and the full likelihood. With ln(B_i) given by
ln(B_i)=ln𝒵(ℳ_i) - ln𝒵(ℳ_0) .
In the above equation, 𝒵(ℳ_0) is the evidence calculated for the full model, and 𝒵(ℳ_i) is the evidence calculated for the sub-model being tested. We can see that although ln(B_i) is positive for all data samples, indicating that the sub-model is preferred, the preference is weak for all samples apart from the two eBOSS-like samples.
The next sub-model we consider (ℳ_3) is chosen by observing the level constraint beyond the prior for each of the bias parameters and counterterms when analysing the PyBird mocks with ℳ_0 and k_max.=0.2 h Mpc^-1 and the full likelihood. Figure <ref> shows the ratio of the prior standard deviation to the 1D marginalised posterior standard deviation for each bias parameter and counterterm. We can see that the only parameters to have a significant constraint beyond the prior (ratio > 1) are b_1, c_2, and c_r,1. As such, we define sub-model ℳ_3 to be that with b_1, c_2, and c_r,1 as the only free nuisance parameters, and all others fixed to zero. The results of Figure <ref> are clearly prior dependent; a reduction in the prior width for c_r,1 will result in the ratio in Figure <ref> being lower for this parameter. These results represent the case in which we are limited to the classic prior defined in Table <ref>. We calculate the Bayes factor for each sample in the same way as for sub-model ℳ_1. These Bayes factors are also plotted in Figure <ref>. We can see that sub-model ℳ_3 is preferred over the full model ℳ_0 at a similar level to ℳ_1 for all the BOSS-like samples and the 6dFGS-like sample. However, the preference for sub-model ℳ_3 over the full model for the eBOSS-like samples is much stronger than sub-model ℳ_1. This stronger preference for the more restrictive sub-model ℳ_3 is likely because of the SNR of the eBOSS-like samples, as discussed in previous sections (shot noise leads to a worse SNR on small scales compared to other samples). As the parameters set to zero primarily impact small scales, and the small scales of the eBOSS-like samples are much noisier than the other samples, the data provides very little evidence for these parameters.
Table <ref> shows the same as table <ref> for analyses of the PyBird mocks with sub-model ℳ_3. We can see that generally, the agreement is of a similar level or better than that from the results obtained with sub-model ℳ_1. For the eBOSS-like mocks, the level of agreement is significantly better, and the evolution with k_max. is now similar to that of the results from the BOSS-like mocks when considering ln(10^10A_s). These results show that we can reduce the parameter space significantly without biasing the constrained cosmology and, in some cases, can alleviate biases likely caused by volume effects.
§.§.§ Priors on Nuisance Parameters
The choice of prior for the nuisance parameters can have a significant impact on the constraint on the cosmological parameters <cit.>, however physically motivating priors on these parameters is challenging. The EFTofLSS is a perturbative model, and as such, if the contribution to the model from the loop corrections becomes too large, the model breaks down; this has led to priors on the nuisance parameters restricting values to be 𝒪(1). In this section, we explore using a Jeffreys prior <cit.> as an alternative to the zero-centred Gaussian priors commonly used in the literature. We explore the use of a Jeffreys prior because it is non-informative. This is a desirable property as it means we are not favouring any particular region of the parameter space a priori. <cit.> shows that the use of the Jeffreys prior on nuisance parameters can resolve volume effects like those observed in the results presented in previous sections. The Jeffreys prior is defined as
J(θ) = √(|F(θ)|) ,
with F(θ) being the Fisher information matrix, which for a Gaussian likelihood with covariance independent of model parameters θ can be written as
F_ij(θ) = ∂ M(θ)/∂θ_iC^-1∂ M(θ)/∂θ_j^T .
From the equations above, we can see that partial derivatives of the model with respect to the model parameters are needed to evaluate the Jeffreys prior. These partial derivatives are trivial for the nuisance parameters that appear linearly in the model. They are simple sums of relevant kernels that are predicted by the EFTEMU (or PyBird) for a given set of cosmological parameters. For this work, we only impose the Jeffreys prior on these linearly appearing nuisance parameters. This means that volume effects related to these parameters should be mitigated. However, any volume effects related to marginalisation over the remaining nuisance parameters (b_1, c_2, and c_4) and the cosmological parameters will still remain. In practice, we impose hard bounds at -100 and 100 on the linear nuisance parameters in addition to the Jeffreys prior when using the Jeffreys prior with the full likelihood, and we impose additional Gaussian priors with σ=200 when using the Jeffreys prior with the marginalised likelihood. These additional priors are chosen relatively arbitrarily and are motivated by the practicalities of our inference pipeline[pocoMC requires prior samples as starting positions for particles. This means we must define a prior that we can sample from when using the full likelihood, hence the imposition of the hard bounds at -100 and 100.]. For the mock analyses presented below, the linearly appearing parameters are constrained well within the additional uniform prior when using the full likelihood. We also test setting σ=1000 when using the Jeffreys prior and see no significant difference when comparing to posteriors calculated with σ=200.
Figure <ref> shows 1D marginalised posteriors for the cosmological parameters obtained from analysis of the PyBird mocks with sub-models ℳ_1 and ℳ_3 (defined in section <ref>), the Jeffreys prior, and the full likelihood (these setups will henceforth be referred to as JP1 and JP3, respectively). Also plotted are the results obtained with ℳ_1 and ℳ_3, the classic prior, and the full likelihood (henceforth be referred to as CP1 and CP3, respectively). We start by considering the results obtained with CP1 and JP1. We can see that for all samples, the agreement with the truth is better when using the Jeffreys prior; this is particularly noticeable for ln(10^10A_s). When using the classic prior, the ln(10^10A_s) peak posterior values shift significantly depending on the SNR of the sample. When using the Jeffreys prior, these peak posterior values are more consistently located around the true value. We expect consistency when examining the results obtained from analysis of the PyBird mocks as they are sample variance free. We can visualise the consistency of the results by calculating the agreement between the results obtained from each sample with Equation <ref> and plotting this as a matrix in Figure <ref>. We can see that for the results obtained with the Jeffreys prior, with the exception of 6dFGS, there is good agreement between the results from each other sample; however, the results obtained with the classic prior show some inconsistency. We can quantify the level of consistency by averaging the lower triangle of the matrices in figure <ref>. This results in 0.30σ and 0.94σ for the Jeffreys prior and classic prior, respectively.
From Figure <ref>, we can also see that using the Jeffreys prior results in an increase in the width of the 68% CIs of the marginalised 1D posteriors. This should be expected, as many of the nuisance parameters converge to the prior when using the classic prior. These parameters have some degeneracy with the cosmological parameters; expanding the space that these parameters can explore inevitably leads to some degradation of the constraints on the cosmological parameters. If we examine the results obtained with JP3, we see that for the low SNR BOSS-like mocks (BOSSz1 and BOSSz3 SGC), we still have a reduction in bias in the ln(10^10A_s) constraint whilst at the same time maintaining a CI that is competitive with the classic prior. We note that for the eBOSS-like mocks, although the ln(10^10A_s) bias is reduced, it is not reduced to the same degree as with JP1. We also note that a greater bias observed in the Ω_m constraints when using the JP3 compared to the CP3.
§.§ Joint Analyses
So far, we have considered each sample individually, which can give interesting insights into how the specifics of each sample (such as redshift and sample selection) impact the results. However, we would ultimately like to analyse multiple samples simultaneously to improve constraining power on the cosmological parameters. To do this we treat each sample as being independent, and as such define the joint likelihood as ln[ℒ_joint(θ | ϕ_joint)] = ∑_i ln[ℒ(θ | ϕ_i)], with θ being the shared cosmological parameters, ϕ_joint being the complete set of nuisance parameters ϕ_joint=[ϕ_1, ϕ_2, …, ϕ_n], and ln[ℒ(θ | ϕ_i)] being defined in Equation <ref>. Unless explicitly stated, the joint analyses of mocks and data measurements in this work are done with the marginalised likelihood. We exclusively use the marginalised likelihood for these kinds of analyses as the joint parameter space can become very large when considering multiple samples. The analytic marginalisation keeps the dimensionality low, thus keeping the joint analyses tractable [It is feasible to sample the parameter space for these joint analyses fully. However, we find that the number of particles for the sampler needs to be increased as suggested in the pocoMC documentation; <https://pocomc.readthedocs.io/en/latest/>. These extra particles mean extra likelihood evaluations are required for each iteration. This adds to the computational cost for each analysis that is already increased by expanding the dimensionality.].
Figure <ref> shows the posterior distributions resulting from analysis of all the BOSS-like mocks (BOSSz1 NGC, BOSSz1 SGC, BOSSz3 NGC, BOSSz3 SGC) with sub-model ℳ_1, the classic prior, k_max.=0.2 h Mpc^-1, and the marginalised likelihood. We note that biases can be observed in the marginalised posteriors. To verify that our joint inference pipeline does not cause these biases, we also analyse the BOSS-like mocks with the covariance rescaled by a factor of 50. These results are also plotted in Figure <ref>. We can see that the ∼ 1σ shift from the truth when considering ln(10^10A_s) has been completely resolved. It can be seen that there is still a slight shift when considering ω_c and h. These biases are now more likely a result of the analysis setup, emulator error, or both rather than volume effects. We do not explore this further, as in all projections of the posterior resulting from analysis with the rescaled covariance, the truth is contained within 1σ. Appendix <ref> compares results obtained with the inference pipeline of this work with those obtained with the pipeline of <cit.>.
Figure <ref> summarises the marginalised 1D posteriors for the cosmological parameters of interest resulting from analyses of various combinations of the PyBird mocks, with various analysis setups. All analyses were conducted with k_max.=0.2 h Mpc^-1 and the marginalised likelihood. Results obtained with sub-models ℳ_1 and ℳ_3 and the classic prior (as before referred to as CP1 and CP3) are represented with blue and orange points and lines, respectively. Results obtained with sub-models ℳ_1 and ℳ_3 and the Jeffreys prior (JP1 and JP3) are represented with green and red points and lines, respectively. Much of what can be seen from Figure <ref> is in line with that from Figure <ref>. That being; when limited to the classic prior, the results obtained using CP3 are less biased than those obtained with CP1, and when considering alternative priors, the results obtained with JP1 are less biased compared to those from CP1 and CP3 at the cost of wider error bars, and although JP3 reduces the bias in the ln(10^10A_s) constraints compared to CP1 these results are more biased than those from CP3 when considering Ω_m.
As mentioned above, the 68%CIs are considerably wider when using JP1. This raises the question, is it even worth combing high SNR data with low SNR data if the Jeffreys prior is needed to mitigate against bias? To answer this, we look at the ratio of the 68% CIs resulting from the joint analysis of the BOSSz1 NGC and BOSSz3 NGC PyBird mocks with CP1 to the 68% CIs resulting from joint analysis of all the PyBird mocks with JP1. For Ω_m, h, and ln(10^10A_s), this ratio is 0.81, 0.92, and 0.99, respectively. We can see that the use of the Jeffreys prior in JP1 has degraded the constraint in such a way that it is better to simply combine the two samples that have negligible volume effects rather than combine all samples. If we instead look at the ratio of the 68% CIs obtained from analysis of BOSSz1 NGC and BOSSz3 NGC with CP1 to those obtained from the analysis of all the mocks with JP3, it is 1.3, 1.3, and 1.7 for Ω_m, h, and ln(10^10A_s), respectively. In this case, there is a significant benefit from doing the joint analysis of all the samples even if the Jeffreys prior is required. It is important to note that when using JP3, we see a ∼ 1σ shift from the truth when considering Ω_m. This is no worse than the bias in Ω_m seen in the results of the joint analysis of all the PyBird mocks with CP1 but is worse than that from the joint analysis with JP1.
§ MAIN RESULTS
In this section, we present the main results of this work; constraints on cosmological parameters from analysis of the unified power spectrum multipole measurements discussed in section <ref>. We repeat many of the analyses discussed in section <ref>, replacing the mock multipoles with those measured from the 6dFGS, BOSS, and eBOSS redshift surveys.
§.§ Individual Constraints
We start by presenting the cosmological parameter constraints obtained via analysis of each sample individually. Figure <ref> shows the peak posterior values and 68% CIs for the cosmological parameters Ω_m, h, and ln(10^10A_s) resulting from analysis of the galaxy power spectrum multipole measurements with four different setups. The first (shown with blue points and lines) being sub-model ℳ_1 (c_4, c_r,2, and c_mono. set to zero; see section <ref>) with the classic prior (see table <ref> and section <ref>), the next (shown with green) being sub-model ℳ_1 with the Jeffreys prior described in section <ref>, the third (shown with orange) being sub-model ℳ_3 (all nuisance parameters set to zero except b_1, c_2, and c_r,1) with the classic prior, and the last being sub-model ℳ_3 with the Jeffreys prior. We refer to these four setups as CP1, JP1, CP3, and JP3, respectively. The black points and lines, and grey shaded regions, show the 99% CI of the Planck 2018 ΛCDM TT, TE, EE+low ℓ+lowE+lensing+BAO results[The 99% CIs have been plotted for Planck to make them more visible for comparison.]. The results shown in figure <ref> are also summarised in table <ref>.
The first thing to note is the strange appearance of the CIs resulting from the analyses of BOSSz1 and BOSSz3 SGC with JP1. The marginalised 1D posteriors on Ω_m and h are multimodal in these cases. The second modes of these distributions correspond to chain samples with extreme nuisance parameters. This could indicate a breakdown of the model. Further discussion on these results can be found in appendix <ref>.
With the exception of these results, we see good agreement between the results obtained with both sub-models and prior choices. Each given sample and parameter has agreement within 1σ for all analysis setups. However, we do note that although <1σ, there are more differences between the analysis setups when considering ln(10^10A_s).
Table <ref> quantifies the average level of agreement[N_σ is calculated for each sample and the comparison via Equation <ref>, then these values are averaged.] between the results presented in Figure <ref> and the Planck 2018 results. When we compare the level of agreement between the results obtained with CP1 and CP3 and the Planck 2018 results, we find that they are similar for both setups. For Ω_m and h there is very little difference between the results obtained with the two setups for a majority of the samples. When considering the results from the eBOSS samples, we see more differences in the Ω_m and h constraints when comparing the two setups. However, as there is a shift from an Ω_m that is lower than that from Planck 2018 to one that is higher, the average level of agreement does not change significantly. As mentioned above, the differences between the results obtained with CP1 and CP3 are clearer when considering ln(10^10A_s). For a majority of the samples, there is a shift in the peak posterior ln(10^10A_s) value towards the Planck result. This is combined with an ∼10% reduction in the width of the 68% CIs. However, Table <ref> shows a similar level of agreement when using CP1 and CP3. This is because of the results from the eBOSS QSO NGC analysis. We can see from Figure <ref> that the ln(10^10A_s) posterior when using CP1 is higher than that from Planck 2018. When using CP3, this shifts to even higher values. If we exclude this result, the average level of agreement in the ln(10^10A_s) constraints between the results obtained with CP1 and CP3 and the Planck results is now 1.06 and 0.860, respectively. We now consider the average level of agreement between the results obtained with JP1 and JP3 and the Planck 2018 results. We see that there is better agreement with the Planck 2018 ln(10^10A_s) constraint when compared to results obtained with CP1 for a majority of samples. As discussed in previous sections, using J1 widens the 68% CIs. Some of the improvement in agreement with the Planck results will be because of this. However, the shifts in the peak posterior values towards the Planck 2018 results that can be seen in Figure <ref> will also result in better agreement.
These results show that reducing model complexity going from CP1 to CP3 does not induce any statistically significant bias when considering analyses of the same sample with the two sub-models. They also show that using the reduced sub-model results in ln(10^10A_s) peak posterior values that are closer to that from Planck 2018 for the majority of data samples. We can also see that using the Jeffreys prior allows us to obtain results consistent with those obtained with the classic prior whilst being more agnostic to the form of the nuisance parameter prior. The Jeffreys prior can also increase the level of agreement with CMB results for ln(10^10A_s). However, this can come with the possible probing of unphysical regions of the parameter space.
§.§ Joint Constraints
Figure <ref> is the same as figure <ref> but summarises the 1D marginalised posterior distributions on the cosmological parameters of interest resulting from joint analyses of the unified multipole measurements. These results are also summarised in <ref>. Also plotted are the Planck 2018 results and relevant results from <cit.>. These works use the EFTofLSS to constrain ΛCDM parameters from analysis of galaxy power spectrum multipoles measured from different datasets. <cit.> uses PyBird to analyse the same eBOSS QSO multipoles used for this work. As with sub-model ℳ_1 of this work c_4, c_r,2, and c_mono. are fixed to zero. <cit.> uses PyBird to perform joint analysis of 6dFGS, BOSS, eBOSS QSO multipole measurements. The BOSS samples used in <cit.> are slightly different from those used in this work; we refer the reader to Table 1 in <cit.> for details. The analysis of <cit.> also differs in that the hexadecapole P_4(k) is included in the data vector in addition to P_0(k) and P_2(k). Additionally, fewer nuisance parameters are fixed to zero than in either of the sub-models of this work. <cit.> only fixes c_4 to zero. <cit.> uses an independent modelling pipeline for the EFTofLSS to analyse BOSS multipole measurements. Again the BOSS measurements used in <cit.> are slightly different from those used in this work; we refer the reader to section 2.1 of <cit.> for details. The <cit.> analysis also differs in that n_s is free, and the data vector includes P_4(k). The form of the nuisance parameters in the <cit.> pipeline differs from that of this work <cit.>. None of these nuisance parameters are fixed in the <cit.> analysis. Each of the three joint analysis results presented in Figure <ref> approximates one of the EFTofLSS works above in the sense that the same kind of data is used. The eBOSS analysis is comparable to <cit.>, the BOSS analysis is comparable to <cit.>, and the ALL analysis is comparable to <cit.>.
Table <ref> quantifies the level of agreement between the results of the joint analyses presented in Figure <ref> and the EFTofLSS literature results and the Planck 2018 ΛCDM results. We first note the good agreement between the results of each joint analysis with their respective EFTofLSS literature results. With the exception of the constraint on h from the ALL analysis, we see the results of this work agree with the literature results within ≲1σ. The results of the joint eBOSS analysis show a more significant dependence on the analysis setup for all parameters compared to the BOSS and ALL analyses. Unlike with the analyses of the PyBird mocks, it is more difficult to determine if these shifts in the results are because of volume effects (resulting from a given analysis setup), sample variance, or errors in the modelling. From the mock analysis results presented in Figure <ref>, we see a slight shift towards the truth when using CP3. From Figure <ref>, we see that using CP3 shifts the results towards those of <cit.>. However, this shift is away from the other EFTofLSS literature results and the Planck 2018 results. If we look again at Figure <ref>, we see that using JP1 shifts the ln(10^10A_s) results even closer to the truth. Comparing to the equivalent result in Figure <ref>, we see that using JP1 shifts the ln(10^10A_s) back toward the results obtained with CP1. We note that the à posteriors obtained with both sub-models agree very well with each other and with those from <cit.>. The linear bias values obtained with CP1 are 2.4±0.3 and 2.3±0.3 for the NGC and SGC, respectively. This aligns with the linear bias obtained via analysis of the eBOSS QSO samples with non-EFTofLSS models <cit.>. These linear bias values are significantly lower when using sub-model CP3 at 2.1±0.2 for both the NGC and SGC.
The results of the BOSS and ALL analyses show less dramatic shifts in the parameters compared to those from the eBOSS analysis and behave more like the results of the mock analyses. We see that for Ω_m and h, there is very little difference between the analysis setups. There is slightly better agreement with the EFTofLSS literature results and Planck 2018 for these parameters when using JP1 for both the BOSS and the ALL analysis. This results from the increased width of the 68% CI in addition to a slight shift in the peak posterior values. The width of the 68% and 95% CIs appear wider from the <cit.> results than those from this work. This is most likely a result of the differences in the analysis setup mentioned above; for example, allowing n_s to vary. <cit.> shows an increase in the CIs of all relevant cosmological parameters when including n_s as a free parameter. When we examine ln(10^10A_s), we observe that the BOSS and ALL joint analyses with CP1 display a level of agreement with the Planck 2018 results that is at the ∼2.5σ level. For the results from CP3, this is at the ∼2.4σ and ∼1.9σ levels for the BOSS and ALL analyses, respectively. The results from JP1 improve the level of agreement with the Planck 2018 results for both the BOSS and ALL joint analyses to <1σ. The results from JP3 also show improved agreement with the Planck 2018 results. However, this is still >1σ for the results from the BOSS analysis. Although the peak posterior agrees with that of the JP1 analysis, the 68% CI is tighter and results in a >1σ difference.
§ CONCLUSIONS
We have presented results from multiple cosmological inference analyses of mock galaxy power spectrum multipoles designed to determine how choices about the analysis setup impact the inferred cosmological parameters. To minimise the computational cost of these mock analyses, we use the neural-network-based EFTEMU to predict the power spectrum multiples. The training procedure of the EFTEMU has been improved beyond that in <cit.> to allow for accurate predictions to be made on a much larger cosmological prior space. The main analysis setup choices we explore are the choice of prior on the nuisance parameters of the EFTofLSS model and which parameters to include in our analyses. The classic EFTofLSS prior takes the form of zero-centred Gaussian distributions with narrow widths on the majority of the nuisance parameters. We compare the Bayesian evidence calculated from analyses of the mock multipoles with different sets of nuisance parameters fixed at zero and the classic prior. Fixing different sets of nuisance parameters to zero results in different EFTofLSS sub-models. The first sub-model we consider (ℳ_1) is constructed by fixing the parameters c_4, c_r,2, and c_mono. to zero. This is a typical choice in the EFTofLSS literature. The next sub-model we consider (ℳ_3) is constructed by fixing all nuisance parameters but b_1, c_2, and c_r,1 to zero. There is a significant preference for sub-model ℳ_3 over ℳ_1 for the eBOSS-like mocks constructed for this work. The results of the mock analyses show less bias in the inferred cosmology when using sub-model ℳ_3 instead of ℳ_1 to analyse the eBOSS-like mocks with the classic prior.
The classic prior is broadly motivated by the idea of keeping the nuisance parameters small in order for the EFTofLSS model to remain perturbative. However, many of the parameters are very weakly constrained and introduce volume effects that bias the cosmological parameters upon marginalisation. We explore the use of a Jeffreys prior, a non-informative prior that can mitigate against these volume effects. Results from mock analyses with the Jeffreys prior show a significant reduction in bias in the ln(10^10A_s) constraint. Considering the joint analysis of all the mocks, we find the shift from the true ln(10^10A_s) is reduced from 2.0σ to 0.42σ comparing results obtained with sub-model ℳ_1 and the classic prior and Jeffreys prior respectively. The use of the Jeffreys prior comes at the cost of widening the marginalised posteriors on the cosmological parameters. This comes as a consequence of allowing the nuisance parameters to explore a much large prior volume. For example, we find that the ratio of the width of the 68% CI to the peak posterior value for the Ω_m marginalised posterior is ∼7.9% when analysing only the mocks with negligible volume effects, with the classic prior. If we instead analyse all of the mocks with the Jeffreys prior, this ratio is ∼9.5%. This represents a ∼20% weakening of the constraint on Ω_m even though much more data has been used. We can reduce this degradation of the constraint by using the Jeffreys prior with sub-model ℳ_3. The more restrictive parameter space leads to a less significant widening of the 1D marginalised posteriors for the cosmological parameters when combing ℳ_1 with the Jeffreys prior. If we compute the ratio again, it is now ∼6.4%, representing a ∼20% tightening of the constraint. Again see a benefit from analysing all the data, including those samples that are susceptible to volume effects.
From the results of the mock analyses, we expect that when the analysis setup uses sub-model ℳ_3 with the classic prior, we see better agreement with the truth compared to sub-model ℳ_1 for the eBOSS samples, and similar levels of agreement for all other samples. We also expect that when using the Jeffreys prior with both sub-models, we will see better agreement with the truth compared to analysis with sub-model ℳ_1 and the classic prior. Upon joint analysis of the unified multipole measurements provided in <cit.>, we find that analysis with sub-model ℳ_3 and the classic prior leads to better agreement with Planck 2018 LCDM results compared to results from the same analysis with ℳ_1. The level of agreement is improved from 2.4σ to 1.9σ for ln(10^10A_s). Analysing all of the multipoles with the Jeffreys prior and both sub-models leads to better levels of agreement again. This is now 0.54σ and 0.70σ for ℳ_1 and ℳ_3 respectively. These results indicate that some of the slight tensions between results obtained via analysis with the EFTofLSS and those obtained via analysis of the CMB are a result of analysis setup.
When using the Jeffreys prior, the nuisance parameters can take extreme values. From analysis of the individual data samples, some cases indicate that using the Jeffreys prior allows for probing regions of the parameters space in which the EFTofLSS model is no longer valid. This presents as multimodal distributions in the cosmological parameters. As mentioned above, we also see a degradation of the cosmological parameter constraint when using the Jeffreys prior. Both of these issues can be addressed by the inclusion of the classic prior within the Jeffreys prior. This limits how extreme the nuisance parameters can be, removing potentially unphysical regions of the parameter space and preventing degradation of the cosmological parameter constraints. This is explored in <cit.>. It is shown that the resolution of volume effects can be achieved without severe degradation of the cosmological parameter constraints. We do not explore this in this work as the values of the prior widths for the nuisance parameters represent extra hyperparameters of the analysis that need to be motivated. The use of the Jeffreys prior without this additional tight prior represents an agnostic approach to the nuisance parameters, whether all regions of the parameter space are physically valid or not.
§ ACKNOWLEDGEMENTS
The authors would like to thank the authors of <cit.> for sharing their MCMC chains. The authors would also like to thank the ICG LSS journal club members, who have provided valuable comments and feedback during various stages of this work. JD-M and RG were supported by STFC studentships. RZ is supported by NSFC grants 11925303 and 11890691, and is also supported by the Chinese Scholarship Council (CSC) and the University of Portsmouth. KK is supported by the STFC grant ST/W001225/1. FB is a University Research Fellow and has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant
agreement 853291). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
§ DATA AVAILABILITY
Multipole measurements, covariance matrices, window function matrices, and wide-angle matrices can be found at <https://fbeutler.github.io/hub/deconv_paper.html>. The Planck 2018 ΛCDM chains used for comparison in this work are available at <https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Cosmological_Parameters>. The PyBird mocks, posterior samples, and inference pipeline will be made publicly available upon acceptance.
mnras
§ REMAINING PERTURBATIVE
One of the main arguments for using zero-centred Gaussian priors on the parameters that appear linearly in the EFTofLSS model is that if these parameters become too large, the model is no longer perturbative. We define a simple check for the model remaining perturbative by evaluating two conditions,
Condition 1: |∑_i P_i,l^11(k)b_i^11| > |∑_i P_i,l^loop(k)b_i^loop| ,
Condition 2: |∑_i P_i,l^11(k)b_i^11| > |∑_i P_i,l^ct.(k)b_i^ct.| ,
with P_i,l^11(k), P_i,l^loop(k), and P_i,l^ct.(k) being the kernels associated to linear, loop, and counterterm contributions respective, b_i^11, b_i^loop, and b_i^ct. being the bias parameters, or counterterms, or combination associated to each kernel. If either of the above conditions fails, we say the model is no longer perturbative. Whilst finalising this work, the authors were made aware of <cit.>, in which a "perturbativity prior" is introduced. We note some similarity as this perturbativity prior sets a Gaussian prior on the overall loop term. However, the perturbative condition above was developed independently of <cit.> and is different in that a uniform probability is given to models that pass the perturbative condition.
Figure <ref> shows the marginalised 1D posteriors for the cosmological parameters of interest obtained with different analysis setups when analysing the BOSSz1 SGC multipole measurements. In black are the results obtained when using the Jeffreys prior. We note the unusual shape of the marginalised posteriors for Ω_m and h; there are second modes of the distributions very far from the Planck 2018 results (plotted in Figure <ref> in purple) and at the extremes of the prior-space. The nuisance parameters associated to the cosmological parameters of these second modes tend to have more extreme values that those in the central mode. This is potentially indicates a breakdown in the EFTofLSS model in these regions. We can test this by imposing the perturbative condition defined by the equations above when sampling[In practice, this involves heavily penalising any sample that breaks this condition, rather than imposing a hard bound]. The red lines in Figure <ref> show the results obtained when doing this. We can see that the extreme second modes in the Ω_m and h posteriors have been removed. However, this also results in a shift in ln(10^10A_s) to lower values. These results appear to show that imposing the perturbative condition when sampling negates the effects of the Jeffreys prior when in comes to resolving the volume effects. A possible cause for this is the way that the perturbative condition has been included. Imposing the condition reduces the prior volume. However, no change has been made to the Jeffreys prior. So the prior volume corrected for with the inclusion of the Jeffreys prior is not the prior volume explored. It should be note that these results are obtained with a prior that is more still agnostic than the classic style prior. We have made no choice on how large the nuisance parameters can be. We also note that the condition defined by the equations above is by no means exact. If we use a slightly more relaxed version of the condition,
Condition (relaxed): |∑_i P_i,l^11(k)b_i^11|
> |∑_i P_i,l^ct.(k)b_i^ct.+∑_i P_i,l^loop(k)b_i^loop| ,
then we obtain the results shown with blue lines in Figure <ref>. In this case the ln(10^10A_s) constraint agrees with that obtained with the classic style prior (shown with green), but again is a more agnostic prior than the classic style prior.
§ TOY MODEL
The purpose of this appendix is to give some intuition for the volume effect discussed frequently in the main text. All of what follows is based on <cit.>.
<cit.> shows that the marginalised χ^2 can be written as
χ^2_m(Ω) ≃χ^2_∗(Ω) + log{[ℱ_∗(Ω)] } + const. ,
with χ^2_∗(Ω) = χ^2(Ω, n_∗), where Ω and n_∗ are the model parameters of interest and n_∗ being the best-fit nuisance parameters, respectively, and ℱ_∗(Ω) is given by
ℱ_∗,ij(Ω) = 1/2∂^2χ^2/∂ n_i∂ n_j|_n_∗ .
The two terms in Equation <ref> are referred to as the profile and Laplace terms, respectively. The Laplace term is responsible for the volume effect that induces biases in the marginalised posterior on the parameters of interest Ω when marginalising over the nuisance parameters n.
To aid in understanding, we present results from analysis with a toy model. This toy model is the same as that discussed in section 2.3 of <cit.>; it is a simple power law of the form n x^Ω. We start by defining true values for n and Ω. For the example here, we use 50 and 1.5, respectively. We then generate 100 x-values as random draws from a uniform distribution 𝒰(1,5). We compute the model prediction for these 100 x-values and true parameters and associate the same uncertainty σ to each of them (this corresponds to a covariance matrix with a constant diagonal and zero off-diagonals). This synthetic data is then analysed with the same kind of inference pipeline used for the work presented in the main text. When σ=25 the inferred Ω after marginalisation over n is 1.500^+0.031_-0.033; very good agreement with the true Ω. However, when σ=800 the inferred Ω is 0.73^+0.93_-0.91; we now have a shift in the peak posterior. This shift in the peak posterior has come from the Laplace term.
Figure <ref> compares the profile and Laplace terms for the two values of σ. The top panel shows the results with σ=25, and the bottom panel shows the results with σ=800. The blue lines show the profile term χ_∗^2(Ω), and the dashed orange lines show the sum of the profile term and the Laplace term log{[ℱ_∗(Ω)] }. The results have been normalised such that the minimum has a value of zero. The grey solid line shows the location of the true Ω. We can see that when σ=25, both minima are in agreement with the truth; the Laplace term has a negligible impact when the constraining power is high. When σ=800, we see that the minimum of the profile term is still located at the truth. However, the sum of the Laplace and profile terms is shifted toward lower values of Ω.
As discussed in section <ref> <cit.> show that a Jeffreys prior can be used to mitigate against the volume effect. Figure <ref> also shows the peak posterior for Ω when carrying out inference with a Jeffreys prior with a red dashed line. For this toy example, we only have one nuisance parameter n. As such, the Fisher matrix needed to evaluate the Jeffreys prior is a single value. Given by
F = ∂(n x^Ω)/∂ nC^-1∂(n x^Ω)/∂ n^T = x^ΩC^-1(x^Ω)^T .
We can see from Figure <ref> that the marginalised peak posterior is now in good agreement with the truth and minimum of the profile term.
We can relate this toy example to the work of the main text by considering the form of the toy model. We can think of n as being one of the linearly appearing nuisance parameters of the EFTofLSS model and x^Ω as the kernel or combination of kernels relevant for that nuisance parameter.
§ COMPARISON WITH PYBIRD
Figure <ref> compares posterior distributions from the joint analysis of the BOSS-like PyBird mocks with two different inference pipelines. The first is the pipeline of this work, with model predictions from the EFTEMU and sampling with pocoMC. The second pipeline is a slightly modified version of that used in <cit.>, with model predictions being made with PyBird and the sampling done with Cobaya <cit.>. Both analyses were conducted with the marginalised likelihood, k_max.=0.2 h Mpc^-1, sub-model ℳ_1, and the classic prior. The percentage difference in the posterior means is less than 0.5% for all parameters in Figure <ref>. The percentage difference in the width of the 68% CIs is, at worst, 5%.
§ RESULTS TABLES
Table <ref> summarises the cosmological constraints resulting from the unified multipole measurements discussed in section <ref>. Table <ref> summarises the cosmological constraints resulting from the joint analyses discussed in section <ref>.
|
http://arxiv.org/abs/2307.04085v1 | 20230709023446 | Vector Commitments with Efficient Updates | [
"Ertem Nusret Tas",
"Dan Boneh"
] | cs.CR | [
"cs.CR"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
====================================================================================
Dynamic vector commitments that enable local updates of opening proofs
have applications ranging from verifiable databases with membership changes to stateless clients on blockchains.
In these applications, each user maintains a relevant subset of the committed messages and the corresponding opening proofs with the goal of ensuring a succinct global state.
When the messages are updated, users are given some global update information and update their opening proofs to match the new vector commitment.
We investigate the relation between the size of the update information and the runtime complexity needed to update an individual opening proof.
Existing vector commitment schemes require that either the information size or the runtime scale linearly in the number k of updated state elements.
We construct a vector commitment scheme that asymptotically achieves both length and runtime that is sublinear in k, namely k^ν and k^1-ν for any ν∈ (0,1).
We prove an information-theoretic lower bound on the relation between the update information size and runtime complexity that shows the asymptotic optimality of our scheme.
While in practice, the construction is not yet competitive with Verkle commitments,
our approach may point the way towards more performant vector commitments.
§ INTRODUCTION
A Vector Commitment (VC) scheme <cit.> enables a committer to succinctly commit to a vector of elements.
Later, the committer can generate an opening proof to prove that
a particular position in the committed vector is equal to a certain value.
VCs have found many applications in databases and blockchains <cit.> as they enable
a storage system to only store a commitment to the vector instead of the entire vector.
The data itself can be stored elsewhere along with opening proofs.
In a multiuser system, every user might store only one position of the vector along with the opening proof for that position.
Dynamic VCs <cit.> are vector commitments that support updates to the vector.
Suppose the committed vector is of length N and some k < N positions in the vector are updated,
so that a new vector commitment is published.
Then, every user in the system will need to update their local opening proof to match the updated commitment,
and this is done with the help of some global update information U that is broadcast to all users.
This information is typically generated and published by a manager who maintains the entire vector.
Applications of dynamic VCs include verifiable databases, zero-knowledge sets with frequent updates <cit.> and stateless clients for blockchains <cit.>.
The challenge is to design a VC scheme that minimizes the size of the update information U
as well as the computation work by each user to update their local opening proof.
For example, consider stateless clients on a blockchain as an important application for dynamic VCs.
The state of the chain can be represented as a vector of length N, where position i corresponds to the state of account number i.
Every user will locally maintain its own state (corresponding to some position in the vector)
along with an opening proof that enables the user to convince a third party
as to its current state.
Whenever a new block is published, the state of the chain changes.
In particular, suppose k out of the N positions in the vector need to be updated.
The block proposer will publish the update information U along with the new block,
and every user will update their opening proof to match the new committed state of the chain.
Thus, users can ensure that their opening proofs are up to date with respect to the latest committed state of the chain.
We stress that in this application, the data being updated, namely the updated positions and diffs, is published as part of the block.
The update information U only contains additional information that is needed to update the opening proofs.
When we refer to the size of U, we refer to its size, excluding the updated data (i.e., excluding the updated positions and diffs).
In this paper, we investigate the trade-off between the length |U| of the update information and the time complexity of proof updates.
Dynamic VCs can be grouped into two categories in terms of these parameters (Table <ref>).
Tree-based VCs <cit.> enable users to update their proofs in time O(N).
Each opening proof typically consists of (N) inner nodes,
and the update information U contains the changes in the inner nodes affected by the message updates.
Each user calculates its new opening proof by downloading the relevant inner nodes published as part of U.
When k positions are updated, a total of O(k log(N)) inner nodes in the tree are affected in the worst case.
Thus, when each inner node has length Θ(λ), proportional to the security parameter λ,
the update information consists of O(k log(N)λ) bits.
In contrast, algebraic VCs <cit.> enable users to update their opening proofs with only
knowledge of the updated data.
They do not require any additional update information U to be published beyond the indices and the `diffs' of the updated data.
Thus, the length of the update information needed to update the opening proofs is O(1).
However, algebraic VCs typically require each user to read all of the changed messages and incorporate the effect of these changes on their proofs, resulting in Θ(k) work per proof update.
To summarize, while tree-based VCs support efficient calculation of the new opening proofs by publishing a large amount of update information, linear in k,
algebraic VCs do not require any additional update information beyond the updated data, but suffer from a large runtime for proof updates, linear in k.
We formalize the dichotomy of VCs in Section <ref>.
§.§ Our Results
We propose a family of VCs that can support sublinear update, where both the length |U| of the update information and the complexity of proof updates are sublinear in k.
More specifically, our VCs can attain |U| = Θ(k^νλ), ν∈ (0,1), with a proof update complexity of Θ(k^1-ν) operations.
Our candidate construction with sublinear update is a homomorphic Merkle tree, first developed by <cit.>, where each inner node can be expressed as a sum of the partial digests of the messages underneath (Section <ref>).
The algebraic structure of these trees enable each user to calculate the effect of a message update on any inner node without reading other inner nodes or messages.
We identify homomorphic Merkle tree constructions based on lattices, from the literature <cit.>.
In Section <ref>, we provide the update algorithms (Alg. <ref>) for homomorphic Merkle trees, parameterized by ν∈ (0,1).
Our algorithm identifies a special subset of size Θ(k^ν) of the inner nodes affected by the message updates, and publish their new values as U; so that the users need not calculate these values.
These inner nodes are selected carefully to ensure that any inner node outside of U is affected by at most Θ(k^1-ν) updated messages.
Thus, to modify its opening proof, each user has to calculate the partial digests of at most Θ(k^1-ν) updated messages per inner node within its proof (that consists of Θ(log(N)) inner nodes).
Moreover, to calculate these partial digests, the user only needs the `diffs' of the updated messages.
This brings the asymptotic complexity of proof updates to Θ(k^1-ν) operations, while achieving an update information size of Θ(k^νλ) as opposed to Θ(kλ) on Merkle trees using SHA256.
In Section <ref>, we prove an information theoretic lower bound on the size of the update information given an upper bound on the runtime complexity of proof updates.
The bound implies the asymptotic optimality of our scheme with sublinear update.
Its proof is based on the observation that if the runtime complexity is bounded by O(k^1-ν), a user that wants to update its proof cannot read beyond O(k^1-ν) updated messages.
Then, to calculate the effect of the remaining k-O(k^1-ν) messages on its opening proof, the user has to download parts of the structured update information U.
Finally, to obtain the lower bound on |U|, we use Shannon entropy and lower bound the number of bits, namely O(k^νλ), required to capture the total information that will be downloaded by the users; while maintaining the security of the VC with parameter λ.
§.§ Applications
We identify three main applications for VCs with sublinear update.
§.§.§ Stateless clients for PoS Ethereum
Ethereum is the largest decentralized general purpose computation platform by market cap.
Ethereum state (, user accounts) is currently stored in the form of a Merkle tree <cit.> and grows approximately by half every year <cit.>.
Stateless clients <cit.> were proposed to mitigate the problem of state bloat and prevent the state storage and maintenance from becoming a bottleneck for decentralization.
Stateless clients maintain an opening proof to their account balances within the Ethereum state, thus can effortlessly prove the inclusion of their accounts within the latest state.
This enables the other Ethereum clients to verify the transactions that come with opening proofs without having to download the full state and check the validity of the claimed account balances.
Since block verification now requires downloading the proofs for the relevant state elements, Verkle trees <cit.> were proposed as a replacement for Merkle trees due to their short proof size.
Each new Ethereum block contains transactions that update the state elements and their opening proofs.
Archival nodes and block producers still maintain the full state so that they can inform the stateless clients about their new opening proofs.
For this purpose, block producers must broadcast enough information to the clients over the peer-to-peer gossip network of Ethereum.
As minimizing the proof size was paramount to decentralizing verification for blocks, minimizing the update information size becomes necessary for decentralizing the role of the block producer who has to disseminate this information.
However, reducing the length of the update information must not compromise the low overhead of stateless clients by requiring larger number of operations per proof update.
Therefore, the ideal VC scheme for stateless clients must strike a delicate balance between the size of the update information and the runtime complexity of proof updates.
In Section <ref>, we provide the update algorithms (Algs. <ref> and <ref>) for Verkle trees.
We observe that Verkle trees do not support sublinear update, and fall under the same category as tree-based VCs with update information length Θ(k λ).
Despite this fact, Verkle trees are highly practical in terms of updates.
In Section <ref>, we estimate that the update information size after a typical Ethereum block does not exceed |U| ≈ 100 kBytes (compared to the typical block size of <125 kBytes).
Moreover, each Verkle proof can be updated within approximately less than a second on commodity hardware.
In contrast, even the most efficient homomorphic Merkle tree construction <cit.> requires an update information size of 110.88 MBytes and an update time of 32.6 seconds when the trade-off parameter ν is 1/2, despite its asymptotic optimality (Section <ref>).
The large update information size is due to the lattice-based construction of these VCs.
Designing dynamic VCs that are both asymptotically optimal and practically efficient remains an open problem.
§.§.§ Databases with frequent membership changes
VCs with sublinear update can support databases with frequent membership changes.
When a user first registers, a message is updated to record the membership of the user.
The user receives this record and its opening proof, using which it can later anonymously prove its membership.
When the user leaves the system, the message is once again updated to delete the record.
In all these steps, membership changes result in updates to the opening proofs of other members.
When these changes are frequent, it becomes infeasible to distribute new proofs after each change.
VCs with sublinear update offer an alternative and efficient way to update the opening proofs of the users in the event of such changes.
§.§ Related Work
There are many VC constructions, each with different guarantees regarding the proof, commitment and public parameter sizes, verification time, updatability and support for subvector openings <cit.> (cf <cit.> for an SoK of VCs).
First formalized by <cit.>, almost all VCs allow some degree of updatability.
Whereas <cit.> enable updating the commitment and the opening proofs with only the knowledge of the old and the new messages, most VCs require some structured update information beyond the messages when the users do not have access to the internal data structures.
Among the lattice-based accumulators, vector commitments and functional commitments <cit.>, constructions amenable to sublinear update are presented in <cit.>.
Homomorphic Merkle trees were formalized and instantiated by <cit.> in the context of streaming authenticated data structures and parallel online memory checking.
The construction presented in <cit.> offers an alternative VC with sublinear update as it is not a Merkle tree, yet has the property that each inner node can be expressed as a sum of the partial digests of individual messages.
For dynamic accumulators that support additions, deletions and membership proofs, Camacho and Hevia proved that after k messages are deleted, Ω(k) bits of data must be published to update the proofs of the messages in the initial accumulated set <cit.>.
Their lower bound is information-theoretic and follows from a compression argument (Appendix <ref>).
Christ and Bonneau subsequently used a similar method to prove a lower bound on the global state size of a revocable proof system abstraction <cit.>.
As revocable proof systems can be implemented by dynamic accumulators and vector commitments, their lower bound generalizes to these primitives, , after k messages are updated in a dynamic VC, at least Ω(k) bits of data must be published to update the opening proofs (Appendix <ref> for the proof).
They conclude that a stateless commitment scheme must either have a global state with linear size in the number of accounts, or require a near-linear rate of local proof updates.
In our work, we already assume a linear rate of local proof updates, , after every Ethereum block or k messages in our parameterization, and that the message updates are publicized by the blockchain.
We instead focus on the trade-off between the global structured update information size (beyond the published messages) and the runtime complexity of proof updates.
§ PRELIMINARIES
§.§ Notation
We denote the security parameter by λ.
An event is said to happen with negligible probability, if its probability, as a function of λ, is o(1/λ^d) for all d>0.
An event happens with overwhelming probability if it happens except with negligible probability.
We denote the set {0,1,2,…,N-1} by [N].
When y = O(h(x) (x)), we use the shorthand y=O(h(x)) (similarly for Θ(.) and Θ(.)).
The function H(.) ℳ→{0,1}^λ represents a collision-resistant hash function.
We denote the binary decomposition of an integer x by (x), and for c>2, its base c decomposition by _c(x).
A vector of N elements (n_0, …, n_N-1) is shown as (n_i)_i.
The notation 𝐱[i:j] denotes the substring starting at the i^th index and ending at the j^th index within the sequence 𝐱.
In the subsequent sections, k will be used to denote the number of updated messages.
For a prime p, let 𝔽_p denote a finite field of size p.
We use 𝔾 to denote a cyclic group of prime order p with generator g.
The Lagrange basis polynomial for a given x ∈𝔽_p is denoted as L_x(X):
C
L_x(X) = ∏_i ∈𝔽_p
i ≠x X-i/x-i
We will use |G| and |H| to denote the maximum size of the bit representation of a single group element and a single hash value respectively.
We will use T_G and T_f to denote the time complexity of a single group operation and a single function evaluation for the hash functions in Section <ref>.
§.§ Vector Commitments
A vector commitment (VC) represents a sequence of messages such that each message can be proven to be the one at its index via an opening proof.
A dynamic vector commitment allows updating the commitment and the opening proofs with the help of an update information when the committed messages are changed.
Dynamic (updateable) vector commitments can be described by the following algorithms:
KeyGen(1^λ, N) → pp Given the security parameter λ and the size N=(λ) of the committed vector, the key generation algorithm outputs public parameters pp, which implicitly define the message space ℳ.
Commit_pp(m_0, …, m_N-1) → (C, ) Given a sequence of N messages in ℳ and the public parameters pp, the commitment algorithm outputs a commitment string C and the data required to produce the opening proofs for the messages.
Here, contains enough information about the current state of the VC's data structure (, the current list of committed messages) to help generate the opening proofs.
Open_pp(m, i, ) →π_i The opening algorithm is run by the committer to produce a proof π_i that m is the i^th committed message.
Verify_pp(C, m, i, π_i) →{0,1} The verification algorithm accepts (, outputs 1) or rejects a proof.
The security definition will require that π_i is accepted only if C is a commitment to some (m_0, …, m_N-1) such that m = m_i.
Update_pp(C, (i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N], ) → (C', U, ') The algorithm is run by the committer to update the commitment C when the messages (m_i_j)_j ∈ [k] at indices (i_j)_j ∈ [k] are changed to (m'_i_j)_j ∈ [k]. The other messages in the vector are unchanged. It takes as input the old and the new messages, their indices and the data variable . It outputs a new commitment C', update information U and the new data variable '.
ProofUpdate_pp(C, p((i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N]), π_j , m', i, U) →π_j'
The proof update algorithm can be run by any user who holds a proof π_j for some message at index j and a (possibly) new message m' at that index.
It allows the user to compute an updated proof π'_j (and the updated commitment C') such that π'_j is valid with respect to C', which contains m'_i, i ∈ N, as the new messages at the indices i ∈ N (and m' as the new message at index i).
Here, p(.) specifies what portion of the old and the new messages is sufficient to update the opening proof.
For instance, the proof update algorithm often does not need the old and the new messages in the open; but can carry out the proof update using only their differences.
In this case, p((i, m_i)_i ∈ [N], (i, m'_i)_i ∈ [N]) = (i, m'_i-m_i)_i ∈ N.
Correctness of a VC requires that ∀ N = (λ), for all honestly generated parameters pp KeyGen(1^λ, N), given a commitment C to a vector of messages (m_0, …, m_N-1) ∈ℳ^N, generated by Commit_pp (and possibly followed by a sequence of updates), and an opening proof π_i for a message at index i, generated by Open_pp or ProofUpdate_pp, it holds that Verify_pp(C, m_i, i, π_i)=1 with overwhelming probability.
Security of a VC is expressed by the position-binding property:
A VC satisfies position-binding if ∀ i ∈ [N] and for every PPT adversary 𝒜, the following probability is negligible in λ:
C
[Verify_pp(C, m, i, π_i) = 1
Verify_pp(C, m', i, π'_i) = 1 m ≠m' pp KeyGen(1^λ, N)
(C, m, m', π_i, π'_i) 𝒜(pp)]
We relax the succinctness assumption of <cit.> and denote a value to be succinct in x if it is (x).
§.§ KZG Polynomial Commitments
The KZG commitment scheme <cit.> commits to polynomials of degree bounded by ℓ using the following algorithms:
KeyGen(1^λ, ℓ) → pp outputs pp = (g, g^τ, g^(τ^2), …, g^(τ^ℓ)) as the public parameters, where g is the generator of the cyclic group 𝔾 and τ is a trapdoor (pp[i] = g^τ^i).
Commit(pp, ϕ(X)) → (C, )
The commitment to a polynomial ϕ(X) = ∑_i=0^ℓ-1 a_i X^i is denoted by [ϕ(X)],
and is computed as [ϕ(X)] = ∏_i=0^ℓ (pp[i])^a_i.
The commitment algorithm outputs C = [ϕ(X)] and = ϕ(X).
Open_pp(m, i, ) →π:
outputs the opening proof π_i that ϕ(i) = m, calculated as the commitment to the quotient polynomial (ϕ(X)-ϕ(i)) / (X-i).
Verify(C, m, i, π) accepts if the pairing check
e(C/g^m, g) = e(π, pp[1]/g^i )
holds.
We refer to <cit.> for the security analysis of this scheme.
§.§ Merkle Trees
Merkle Tree is a vector commitment using a collision-resistant hash function.
In a Merkle tree, hashes of the committed messages constitute the leaves of a c-ary tree of height h = log_c(N), where each inner node is found by hashing its children.
The depth of the root is set to be 0 and the depth of the leaves is ⌈log_c(N) ⌉.
The commitment function outputs the Merkle root as the commitment C and the Merkle tree as .
The opening proof for a message m_x at some index x is the sequence of h(c-1) hashes consisting of the siblings of the inner nodes on the path from the root to the hash of the message m_x.
We hereafter consider binary Merkle trees (c=2) and assume N=c^h = 2^h unless stated otherwise.
Let u_b_0, b_1,…,b_i-1, b_j ∈{0,1}, j ∈ [i], denote an inner node at depth i-1 that is reached from the root by choosing the left child at depth j if b_j=0 and the right child at depth j if b_j=1 (b_0= and u_ is the root).
By definition, for a message m_x at index x, H(m_x) = u_,(x).
§.§ Verkle Trees
A Verkle tree <cit.> is similar to a Merkle tree except that each inner node is calculated as the hash of the KZG polynomial commitment to its children.
Let b_j ∈ [c], j=1, …, h, denote the indices of the inner nodes on the path from the root to a leaf at index x, _c(x) = (b_1, …, b_h), relative to their siblings.
Define f_b_0,…,b_j, j ∈ [h], as the polynomials determined by the children of the inner nodes on the path from the root to the leaf, where f_b_0=f_ is the polynomial determined by the children of the root.
Let C_b_0,…,b_j = [f_b_0,…,b_j], j ∈ [h], denote the KZG commitments to these polynomials.
By definition, u_b_0,…,b_j = H(C_b_0,…,b_j), and the value of the polynomial f_b_0,…,b_j at index b_j+1 is u_b_0,…,b_j+1 for each j ∈ [h].
Here, u_b_0 = H(C_b_0) is the root of the tree, and u_b_0,…,b_h equals the hash H(m_x) of the message at index x.
For consistency, we define C_b_0,…,b_h as m_x.
For example, given h = 3 and c = 4, the inner nodes from the root to the message m_14 have the indices b_0 = 0, b_1 = 3 and b_2 = 2, and they are committed by the polynomials f_, f_,0 and f_,0,3 respectively.
The commitment function Commit_pp(m_0, …, m_N-1) outputs the root u_b_0 as the commitment C and the Verkle tree itself as .
The Verkle opening proof for the message m_x, (x) = (b_1, …, b_h), consists of two parts: (i) the KZG commitments (C_b_0,b_1, …, C_b_0,…, b_h-1) on the path from the root to the message, and (ii) a Verkle multiproof.
The goal of the Verkle multiproof is to show that the following evaluations hold for the inner nodes from the root to the message: f_b_0,…,b_j(b_j+1)=u_b_0,…,b_j+1 = H(C_b_0,…,b_j+1), j ∈ [h].
It has two components: (i) the commitment [g(X)] and (ii) the opening proof π' for the polynomial h(X)-g(X) at the point t=H(r,[g(X)]), where
C
g(X)=∑_j=0^h-1 r^j f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1, h(X)=∑_j=0^h-1 r^j f_b_0,…,b_j(X)/t-b_j+1,
and r=H(C_b_0,..,C_b_0,…,b_h-1,u_b_0,b_1,..,u_b_0,…,b_h,b_1,..,b_h).
Thus, Open_pp(m, i, ) outputs ((C_b_0,b_1, …, C_b_0,…,b_h-1), ([g(X)], π')).
To verify a Verkle proof π = ((C_b_0,b_1, …, C_b_0,…,b_h), (D,π')), the algorithm Verify_pp(C, m, x, π) first computes r and t using u_b_0,…,b_j = H(C_b_0,…,b_j), j ∈ [h], and u_b_0,…,b_h = H(m).
Then, given the indices (x) = (b_1, …, b_h) and the commitments (C_b_0,b_1, …, C_b_0,…,b_h), it calculates
C
y = ∑_j=0^h-1 r^j C_b_0,…,b_j/t-b_j+1 E = ∑_j=0^h-1 r^j/t-b_j+1 C_b_0,…,b_j.
Finally, it returns true if the pairing check e(E-D-[g(X)],[1]) = e(π', [X-t]) is satisfied.
As the degree c of a Verkle tree increases, size of the opening proofs and the runtime of the verification function decreases in proportion to the height h = log_cN of the tree.
This enables Verkle trees to achieve a short opeining proof size for large number of messages (as in the case of the Ethereum state trie) by adopting a large degree (, c=256).
In comparison, each Merkle proof consists of (c-1) log_cN inner nodes, which grows linearly as c increases.
§ FORMALIZING THE DICHOTOMY OF VCS
We first analyze the trade-off between the number of operations required by proof updates and the size of the update information U by inspecting different types of dynamic VCs.
Recall that the number of updated messages is k ≤ N.
§.§ Updating KZG Commitments and Opening Proofs
In the subsequent sections, we assume that each user has access to a dictionary of KZG commitments to the Lagrange basis polynomials L_i(X), i ∈𝔽_p, and for each polynomial, its opening proofs at each point j ∈𝔽_p, j < N.
With the help of this table, one can instantiate a KZG based VC to the messages (m_i)_i ∈ [N], by treating them as the values of the degree N polynomial ϕ(X) at inputs i ∈𝔽_p, i<N.
We next analyze the complexity of the update information and the proof updates in this VC.
The update and proof update algorithms are described by Alg. <ref> in Appendix <ref>.
§.§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is updated at some index i such that m'_i m_i + δ for some δ∈𝔽_p.
Then, the polynomial ϕ(X) representing the vector is replaced by ϕ'(X) such that ϕ'(X) = ϕ(X) if X ≠ i, and ϕ'(i) = ϕ(i) + δ at X = i.
Thus, the new KZG commitment C' to ϕ'(X) is constructed from the commitment C to ϕ(X) as follows:
rCl
C' = [ϕ'(X)] = [ϕ(X)+δL_i(X)]
= [ϕ(X)][L_i(X)]^δ = C ·[L_i(X)]^δ = C ·[L_i(X)]^m'_i-m_i.
If the vector is modified at k different indices i_1,...,i_k from message m_i_j to m'_i_j, j ∈ [k], then
the new commitment C' = [ϕ'(X)] becomes
rCl
[ϕ(X)+∑_j=1^k (m'_i_j-m_i_j) L_x_i_j(X)]
= [ϕ(X)] ∏_j=1^k[L_i_j(X)]^(m'_i_j-m_i_j) = C ∏_j=1^k[L_i_j(X)]^(m'_i_j-m_i_j).
Thus, the commitment can updated given only the old and the new messages at the updated indices, besides the table.
§.§.§ Proof Update
Let π_x denote the opening proof of a polynomial ϕ(X) at a point (x,m_x).
When k messages are updated, the new opening proof π'_x can be found as a function of the old proof π_x and the opening proofs π_i_j,x of the Lagrange basis polynomials L_i_j(X), j ∈ [k], at the index x (m'_x = m_x+∑_j=1^k (m'_i_j-m_i_j) · 1_x=i_j is the new value of m_x after the k updates):
rCl
π'_x = [ϕ'(X)-m_x-∑_j=1^k δ_j ·1_x=i_j/X-x]
= π_x ∏_j=1^k [L_i_j(X)-L_i_j(x)/X-x]^m'_i_j-m_i_j = π_x ∏_j=1^k π^m'_i_j-m_i_j_i_j,x
Thus, the proof can updated given only the old and the new messages at the updated indices, besides the table.
The update information is set to be the empty set, , U = ∅.
§.§.§ Complexity
The size of the update information is constant, , Θ(1).
Each user can update its proof after k accesses to the dictionary, and in the worst case, Θ(k log|ℳ|) = Θ(k) group operations as log(m'_i-m_i)≤log|ℳ| for all i ∈ [N].
§.§ Updating Merkle Trees and Opening Proofs
We next consider a Merkle tree and analyze the complexity of the update information size and the runtime for proof updates.
A simple update scheme would be recalculating the new Merkle tree given all of the old messages or the old inner nodes of the Merkle tree, and the message updates.
However, this implies a large complexity for the runtime of the proof update algorithm that scales as Ω(k) when users keep track of the inner nodes, and as Ω(N) when the users recalculate the tree from scratch at each batch of updates.
Moreover, in many applications, the users do not have access to any messages or inner nodes besides those that are part of the Merkle proof held by the user.
Hence, in the following sections, we describe update and proof update algorithms
that reduce the runtime complexity of the proof updates at the expanse of larger update information (Alg. <ref> in Appendix <ref>).
§.§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is updated at some index x, (b_1,…,b_h) = (x), to m'_x.
Then, the root C=u_b_0 and the inner nodes (u_b_0,b_1, …, u_b_0,b_1,…,b_h), (b_1,…,b_h) = (i), must be updated to reflect the change at that index.
Given the old inner nodes, the new values for the root and these inner nodes, denoted by C'=u'_b_0 and (u'_b_0,b_1, …, u'_b_0,b_1,…,b_h), are calculated recursively as follows:
rCl
u'_b_0,b_1,…,b_h H(m'_x),
u'_b_0,b_1,…,b_j ^
H(u'_b_0,b_1,…,b_j,0, u_b_0,b_1,…,b_j,1) if b_j+1 = 0, j<h
H(u_b_0,b_1,…,b_j,0, u'_b_0,b_1,…,b_j,1) if b_j+1 = 1, j<h
When the messages are modified at k different points i_j, j ∈ [k], the calculation above is repeated k times for each update.
As the updated inner nodes are parts of the Merkle proofs, the update information consists of the new values at the inner nodes listed from the smallest to the largest depth in the canonical left to right order.
For instance,
U = ((, u'_), (0, u'_0), (1, u'_1), (00, u'_00), (10, u'_10), …)
implies that the root u_ and the inner nodes u_0, u_1, u_00 and u_10 were updated after k messages were modified at the leaves of the Merkle tree.
We reference the updated inner nodes using their indices (, U[b_0, b_1 … b_j] = v, when (b_1 … b_j, v) ∈ U).
§.§.§ Proof Update
The Merkle proof π_x for a message at index
x, (b_1, …, b_h) = (x), is the sequence (u_b_1, u_b_1,b_2, …, u_b_1,b_2,…,b_h).
When k messages are updated, some of the inner nodes within the proof might have changed.
A user holding the Merkle proof for index x can find the new values of these inner nodes by querying the update information with their indices.
§.§.§ Complexity
Upon receiving the update information U, each user can update its proof in Θ(log^2(N)+|H| log(N)) = Θ(1) time by running a binary search algorithm to find the updated inner nodes within U that are part of its Merkle proof, and reading the new values at these nodes.
Since modifying each new message results in h = log(N) updates at the inner nodes and some of the updates overlap, |U| = Θ(k log(N/k) (log(N)+|H|)) = Θ(k)|H|, as each updated inner node is represented by its index of size Θ(log(N)) and its new value of size |H| in U.
§.§ Dichotomy of VCs
In the case of KZG commitments, |U| = Θ(1), and there is no information overhead on top of the message updates.
For Merkle trees with an efficient proof update algorithm, |U| = Θ(k)|H|,
thus there is an extra term scaling in Θ(k)|H| = Θ(k)λ, since |H| = Ω(λ) for collision-resistant hash functions.
In contrast, for KZG commitments, each user has to do Θ(k) group operations to update its opening proof; whereas in Merkle trees, each user can update its proof in Θ(1) time, which does not depend on k.
Hence, KZG commitments outperform Merkle trees in terms of the update information size, whereas Merkle trees outperform KZG commitments in terms of the time complexity of proof updates.
Table <ref> generalizes this observation to a dichotomy between algebraic VC schemes and tree-based ones favoring shorter runtimes for proof updates.
The algebraic and tree-based ones outperform each other in terms of the update information size and runtime complexity respectively.
§ VECTOR COMMITMENTS WITH SUBLINEAR UPDATE
We would like to resolve the separation in Table <ref> and obtain a vector commitment, where both the size of the update information and the complexity of proof updates have a sublinear dependence on k.
In particular, |U| = Θ(g_1(k)λ) in the worst case,
and the proof update algorithm requires at most Θ(g_2(k)) operations,
where both g_1(k) and g_2(k) are o(k).
We say that such a VC supports sublinear update.
In this section, we describe a family of VCs with sublinear update, parameterized by the values ν∈ (0,1) and characterized by the functions (g_1,g_2) = (k^ν, k^1-ν).
§.§ Homomorphic Merkle Trees
We first introduce homomorphic Merkle trees where messages placed in the leaves take values in a set ℳ.
We will use two collision-resistant hash functions f̃𝒟×𝒟→ℛ and f ℳ→ℛ,
where both ℳ and 𝒟 are vector spaces over some field 𝔽,
and ℛ is an arbitrary finite set.
We will also need an injective mapping g: ℛ→𝒟, which need not be efficiently computable.
We use g^-1: 𝒟→ℛ to denote the inverse of g,
meaning that g^-1(g(x)) = x for all x ∈ℛ.
We require that g^-1 be efficiently computable.
Now, for j ∈ [h], where h is the height of the tree, every node u_b_0,…,b_j∈𝒟 of the homomorphic Merkle tree is characterized by the following expressions:
llCl
a leaf node: g^-1(u_b_0,(i)) = f(m_i)
an internal node: g^-1(u_b_0,…,b_j) = f̃(u_b_0,…,b_j,0, u_b_0,…,b_j,1) for j < h
The homomorphic property of the Merkle tree refers to the fact that
there are efficiently computable functions
h_i,j: 𝒟→𝒟 for i ∈ [N] and j ∈ [h],
such that every inner node u_b_0,…,b_j∈𝒟 can be expressed as
rCl
u_b_0 = ∑_i ∈[N] h_i,0(m_i)
u_b_0,…,b_j = ∑_i(i)[0:j-1]=(b_1,…,b_j) h_i,j(m_i).
We refer to the function h_i,j as a partial digest function
and refer to h_i,j(m_i) as the partial digest of m_i.
In a homomorphic Merkle tree, every internal node is the sum of the partial digests of the leaves under that node.
We will show in Section <ref> that each function h_i,j can be expressed
as an iterated composition of the functions f and f̃.
Evaluating h_i,j requires evaluating the functions f and f̃ exactly h-j times.
Opening proof for a message consists of both children of the internal nodes on the path from the message to the root (as opposed to Merkle opening proofs that contain only the siblings of the internal nodes on the path).
For instance, the opening proof for the message m_i at leaf index i, with (i) = (b_1,…,b_h), is (i, (u_b_0,…,b_j,0,u_b_0,…,b_j,1)_j=0,…,h-1).
Opening proofs are verified using the functions f and f̃ (not by using the functions h_i,j).
To verify an opening proof (i, (u_b_0,…,b_j,0,u_b_0,…,b_j,1)_j=0,…,h-1) for a message m_i with respect to the root u_b_0, the verifier checks if the following equalities hold:
llCl
for the leaf: g^-1(u_b_0,(i)) = f(m_i)
for the internal nodes: g^-1(u_b_0,…,b_j) = f̃(u_b_0,…,b_j,0, u_b_0,…,b_j,1) for j = h-1, …, 0.
If so, it accepts the proof, and otherwise it outputs reject.
As an example, consider a homomorphic Merkle tree that commits to four messsages m_0,m_1,m_2,m_3.
Then, its root u_ and inner nodes u_,0, u_,1, u_,0,0, u_,0,1, u_,1,0, u_,1,1 can be calculated as follows:
rClrCl
u_ = h_0,0(m_0) + h_1,0(m_1) + h_2,0(m_2) + h_3,0(m_3) ; u_,0,0 = h_0,2(m_0)
u_,0 = h_0,1(m_0) + h_1,1(m_1) ; u_,0,1 = h_1,2(m_1)
u_,1 = h_2,1(m_2) + h_3,1(m_3) ; u_,1,0 = h_2,2(m_2)
u_,1,1 = h_3,2(m_3)
The opening proof for m_3 is given by (3, ((u_,0, u_,1), (u_,1,0, u_,1,1))), and verified by checking the following equations:
llCl
for u_,1,1: g^-1(u_,1,1) = f(m_i)
for u_,1: g^-1(u_,1) = f̃(u_,1,0, u_,1,1)
for u_: g^-1(u_) = f̃(u_,0, u_,1)
It now follows that
when a message m_i is updated to m'_i, each inner node on the path from the leaf to the root can be updated from u_b_0,…,b_j to u'_b_0,…,b_j using the functions h_i,j as follows:
u'_b_0,…,b_j =
h_i,j(m'_i) + ∑_x ≠ i
(x)[0:j-1]= (b_1,…,b_j) h_x,j(m_x) =
u_b_0,…,b_j + h_i,j(m'_i) - h_i,j(m_i)
When the partial digest functions are linear in their input, the expression h_i,j(m'_i) - h_i,j(m_i) can be written as
h_i,j(m'_i) - h_i,j(m_i) = sign(m'_i-m_i)h_i,j(|m'_i-m_i|).
This lets us calculate the updated internal node using only the knowledge of the message diff m_i'-m_i.
We provide examples of homomorphic Merkle tree constructions in Section <ref> with linear partial digest functions h_i,j.
Homomorphic Merkle proofs in these constructions consist of the two siblings of the inner nodes on the path from the proven message to the root (Section <ref>).
Unlike in Section <ref>, homomorphic Merkle trees enable calculating the new inner nodes after message updates using only the new and the old updated messages, in particular using only their difference.
Hence, we can construct a tree that achieves the same complexity for the update information size as algebraic VCs, albeit at the expanse of the proof update complexity, without requiring the users to keep track of the old messages or to calculate the tree from scratch given all messages (Appendix <ref> for further discussion).
This is in contrast to Merkle trees based on SHA256.
The update and proof update algorithms of such a homomorphic Merkle tree with no structured update information and the same asymptotic complexity as algebraic VCs is described in Appendix <ref>.
Since the homomorphic Merkle trees can achieve both extremes in terms of update information size and update runtime (Table <ref>), with a smart structuring of the update information, they can support sublinear update.
We show how in the next subsection.
§.§ Structuring the Update Information
We now describe the new update and proof update algorithms that enable homomorphic Merkle trees to achieve sublinear complexity as a function of the parameter ν (Alg. <ref>).
§.§.§ Update Information
When the messages (i_j, m_i_j)_j ∈ [k] change to (i_j, m'_i_j)_j ∈ [k], the update information U is generated recursively using the following algorithm:
* Start at the root u_b_0. Terminate the recursion at an inner node if there are k^1-ν or less updated messages under that node.
* If there are more than k^1-ν updated messages with indices ≥ N/2, , under the right child, then publish the new right child of the root as part of U, and apply the same algorithm to the subtree rooted at the right child, with u_b_0 and N replaced by u_b_0,1 and N/2 respectively.
* If there are more than k^1-ν updated messages with indices less than N/2, , under the left child, then publish the new left child of the root as part of U, and apply the same algorithm to the subtree rooted at the left child, with u_b_0 and N replaced by u_b_0,0 and N/2 respectively.
The new values of the inner nodes included in U are again listed from the smallest to the largest depth in the canonical left to right order.
§.§.§ Proof Update
When the messages (i_j, m_i_j)_j ∈ [k] are updated to (i_j, m'_i_j)_j ∈ [k], a user first retrieves the inner nodes within its Merkle proof that are published as part of the update information.
It then calculates the non-published inner nodes within the proof using the partial digests.
For instance, consider a user with the proof (u_b_1, u_b_1,b_2, …, u_b_1,b_2,…,b_h) for some message m_x, (b_1, …, b_h) = (x).
To update the proof, the user first checks the update information U and replaces the inner nodes whose new values are provided by U:
u'_b_1,…,b_d U[b_1 …b_d], d ∈ [h], if U[b_1 …b_d] ≠.
Otherwise, the user finds the new values at the nodes u_b_1,…,b_d, d ∈ [h], using the functions h_x,d:
rCl
u'_b_1, …, b_d-1,b_d = u_b_1, …, b_d-1,b_d
+ ∑_j ∈[k] 1_(i_j)[:d] = (b_1, …, b_d-1,b_d) (sign(m'_i_j-m_i_j)h_i_j,d(|m'_i_j -m_i_j|)))
§.§.§ Complexity
Finally, we prove bounds on the complexity given by these algorithms:
Complexity of the update information size and the runtime of proof updates are as follows: g_1(k) = k^ν and g_2(k) = k^1-ν.
We finally show that this VC publishes O(k^ν) new inner nodes in the worst case.
Let 𝒰 denote the subset of the inner nodes published by the algorithm as part of U such that no child of a node u ∈𝒰 is published.
Then, there must be over k^1-ν updated messages within the subtree rooted at each node u ∈𝒰.
Since there are k updated messages, and by definition of 𝒰, the subtrees rooted at the nodes in 𝒰 do not intersect at any node, there must be less than k/k^1-ν = k^ν inner nodes in 𝒰.
Since the total number of published inner nodes is given by 𝒰 and the nodes on the path from the root to each node u ∈𝒰, this number is bounded by k^νlog(N) = Θ(k^ν).
Hence,
|U| = Θ(k^νlog(N)(log(N)+|H|)) = Θ(k^ν)|H| = Θ(k^ν) λ,
which implies g_1(k) = k^ν.
For each inner node in its Merkle proof, the user can check if a new value for the node was provided as part of U, and replace the node if that is the case, in at most Θ(log(N)+|H|) time by running a binary search algorithm over U.
On the other hand, if the new value of a node in the proof is not given by U, the user can calculate the new value after at most k^1-νlog(N) function evaluations.
This is because there can be at most k^1-ν updated messages within the subtree rooted at an inner node, whose new value was not published as part of U.
This makes the total time complexity of a proof update at most
C
Θ(log(N)(log(N)+|H|+k^1-νlog(N)T_f)) = Θ(k^1-ν) T_f,
which implies g_2(k) = k^1-ν.
§.§ Constructions for Homomorphic Merkle Trees
Homomorphic Merkle trees were proposed by <cit.>.
These hash functions are lattice-based, and their collision-resistance is proven by reduction to the hardness of the gap version of the shortest vector problem (𝖦𝖠𝖯𝖲𝖵𝖯_γ), which itself follows from the hardness of the small integer solution problem.
We next describe the construction introduced by <cit.>, which is similar to those proposed by later works <cit.> (an alternative construction is provided in Appendix <ref>).
Its correctness and security follow from <cit.>.
Let L(𝐌) denote the lattice defined by the basis vectors 𝐌⊂ℤ^k × m_q for appropriately selected parameters k,m,q, where m = 2 k log q.
Consider vectors u ∈{0, …, t}^k log q, where t is a small integer.
The homomorphic hash functions f ℤ^k log q→ L(𝐌) and f̃ℤ^k log q×ℤ^k log q→ L(𝐌) used by <cit.> are defined as f(x) = 𝐌x and f̃(x,y) = 𝐌𝐔 x + 𝐌𝐃 y respectively.
Here, 𝐔 and 𝐃 are special matrices that double the dimension of the multiplied vector and shift it up or down respectively.
The remaining entries are set to zero.
For convenience, we define 𝐋 = 𝐌𝐔 and 𝐑 = 𝐌𝐃.
Since the domain and range of the hash functions are different, to ensure the Merkle tree's homomorphism, authors define a special mapping g ℤ^k_q →ℤ^k logq_q from the range of the hash functions to their domain.
Here, g(.) takes a vector 𝐯∈ℤ_q as input and outputs a radix-2 representation for 𝐯.
However, as there can be many radix-2 representations of a vector, to help choose a representation that yields itself to homomorphism, authors prove the following result: for any x_1, x_2, …, x_t ∈ℤ_q, there exists a short radix-2 representation g(.) such that g(x_1 + x_2 + … + x_t q) = b(x_1) + b(x_2) + … + b(x_t) q ∈{0, …, t}^k log q, where the function b ℤ^k_q →{0,1}^klogq returns the binary representation of the input vector.
This equality enables the mapping g(.) to preserve the hash functions' original homomorphic property.
Then, given an inner node u_b_0,…,b_j as input, the homomorphic Merkle tree uses the short radix-2 representation g(.) that enforces the following equality: g(u_b_0,…,b_j) = g(𝐋 u_b_0,…,b_j,0 + 𝐑 u_b_0,…,b_j,1 q) = b(𝐋 u_b_0,…,b_j,0) + b(𝐑 u_b_0,…,b_j,1) q.
Finally, this enables calculating the value of each inner node as a sum of the partial digests h_i,j(.) of the messages m_i under the node u_b_0,…,b_j (, (m_i)_(i)[0:j] = (b_0,…,b_j)) as outlined in Section <ref>, i.e., u_b_0,…,b_j equals
rCl
𝐋g(u_b_0,b_1,…,b_j,0) + 𝐑g(u_b_0,b_1,…,b_j,1)
= 𝐋g(𝐋g(u_b_0,…,b_j,0,0) + 𝐑g(u_b_0,…,b_j,0,1)) + 𝐑g(𝐋g(u_b_0,…,b_j,1,0) + 𝐑g(u_b_0,…,b_j,1,1))
= 𝐋b(𝐋g(u_b_0,…,b_j,0,0)) + 𝐋b(𝐑g(u_b_0,…,b_j,0,1))
+ 𝐑b(𝐋g(u_b_0,…,b_j,1,0)) + 𝐑b(𝐑g(u_b_0,…,b_j,1,1))
= ∑_i(i)[0:j-1]=(b_1,…,b_j) h_i,j(m_i),
where h_i,j(.) is expressed in terms of the bits (i)[j:h-1] = (b'_1, …, b'_h-j):
C
h_i,j(m_i) = f_b'_1(f_b'_2(…f_b'_h-j(f(m_i))))
Here, f_0(.) and f_1(.) are defined as 𝐋b(.) and 𝐑b(.) respectively.
Since b(.), binary expansion, is a linear operation and matrix multiplication is linear, h_i,j(.) is linear in its input.
Opening proof of a message m consists of its index and g(α_i) and g(β_i), i ∈ [h], h = log(N), where α_i and β_i are the children of the inner nodes on the path from m to the root.
The proof can be verified in log(N) time by iteratively checking if f(m) = g^-1(α_h) (or = g^-1(β_h)) and f̃(g(α_i),g(β_i)) = g^-1(α_i-1) (or =g^-1(β_i-1) depending on the message index), where g^-1 returns a number given its radix-2 representation <cit.>.
Note that both f and f̃ are homomorphic hash functions <cit.>.
Other examples of homomorphic hash functions include Pedersen hashes and KZG commitments.
However, the homomorphic property of the hash function is not sufficient for constructing a homomorphic Merkle tree when the function is combined with the output of other functions in a serial manner as in Merkle trees.
For the lattice-based function, this was possible because of repeated linearity <cit.>, which refers to the existence of a linear mapping g(.) from the range to the domain of the hash function.
This mapping enabled the iterative hashing within the Merkle tree to preserve the linearity of the hash function.
Such repeated linearity does not exist for Pedersen hashes and KZG commitments as a linear mapping from the range to the domain would imply the violation of the discrete log assumption.
That is why Verkle trees based on KZG commitments are not homomorphic and cannot support sublinear update.
§.§ A Concrete Evaluation
Suppose the Ethereum state is persisted using the homomorphic Merkle tree construction of <cit.> with the trade-off parameter ν = 1/2.
We next estimate the size of the update information and the proof update time after observing an Ethereum block with ERC20 token transfers.
Suppose the block has the target size of 15 million gas <cit.>, and each token transfer updates the balance of two distinct accounts stored at separate leaves of the homomorphic Merkle tree.
Since each ERC20 token transfer consumes approximately 65,000 gas, there are ∼ 230 such transactions in the block, and the block updates k = 460 accounts.
Suppose the homomorphic Merkle tree has degree 2 and commits to N = 256^3 = 2^24 accounts.
For comparison, 256^3 ≈ 16.7 million, matching in magnitude the total number of cumulative unique Ethereum addresses, which is 200 million as of 2023 <cit.>.
Each opening proof consists of 2log(N) = 48 inner nodes.
When 460 accounts are updated, in the worst case, the update information consists of ⌈√(k)⌉log(N) = 528 inner nodes.
To evaluate its size, we use the parameters calculated by <cit.> for secure instantiations of the homomorphic Merkle trees from both their paper and <cit.>.
Since the parameters for <cit.> result in a large inner node size on the order of hundreds of MBs, our evaluation takes the size of an inner node as that of <cit.>, namely |H| = 0.21 MB (which is equal to the key size in <cit.>).
This implies an update information size of |U| = 110.88 MBytes and an opening proof size of |π| = 10.08 MBytes.
As for update time, in the worst case, each user has to calculate the partial digests of 44 updated messages at each height of the homomorphic Merkle tree, , the effect of these updated messages on each inner node of its opening proof.
Calculating the partial digest of a message at height h measured from the leaves requires h evaluations of the hash function.
This implies a proof update complexity of 2 ∑_i=0^logN-1 i min(⌈√(k)⌉, 2^i) = 11,900 hash evaluations.
To find numerical upper bounds for the update time, we use the hash function evaluation times, namely T_f = 26.84 and T_f = 2.74 ms, published by <cit.> for both the hash function in <cit.> and their new and more performant function (these times are for commodity hardware; <cit.> for the details).
This gives an upper bound of 319.4 and 32.6 seconds for the update time using the hash functions in <cit.> and <cit.> respectively.
Based on the benchmarks for the practical hash function introduced in <cit.>, Table <ref> compares the number of published inner nodes ⌈ k^ν⌉log(N), the total update information size ⌈ k^ν⌉log(N) |H| (assuming that the size of each inner node is |H| upper bounded by 0.21 MBytes), the number of hash function evaluations per proof update 2 ∑_i=0^logN-1 i min(⌈ k^1-ν⌉, 2^i) and the proof update time 2 ∑_i=0^logN-1 i min(⌈ k^1-ν⌉, 2^i) T_f (assuming that each hash evaluation takes less than T_f = 2.74 ms) at ν = 0, 1/4, 1/2, 3/4, 1.
The degree of the homomorphic Merkle tree and the opening proof size are fixed at 2 and 48 inner nodes (|π| = 10.08) respectively.
§ UPDATING VERKLE TREES AND OPENING PROOFS
We now describe the update and proof update functions for Verkle trees (Algs. <ref> and <ref> respectively).
Since Verkle trees were proposed to support stateless clients,
we describe an update scheme that minimizes the runtime complexity of proof updates and does not require the users to download the updated messages or have access to old inner nodes.
As Verkle trees do not support sublinear update, we numerically estimate the size of the update information and the complexity of proof updates in Section <ref>.
§.§ Update Information
Suppose the vector (i, m_i)_i ∈ [N] is modified at some index x, (b_1, …, b_h) = (x) to be m'_x.
Since each inner node is the hash of a KZG commitment, the new inner nodes u'_b_0,…,b_j = H(C'_b_0,…,b_j), j ∈ [h], can be found as a function of the old commitments at the nodes and the powers of the Lagrange basis polynomials as described in Section <ref>:
C
C'_b_0,…,b_h m'_x, C'_b_0,…,b_j C_b_0,…,b_j [L_b_j+1]^(u'_b_0,…,b_j+1-u_b_0,…,b_j+1)
When k messages are updated, the above calculation is repeated k times for each update.
Update information U consists of the new values of the KZG commitments on the path from the updated messages to the Verkle root akin to the Merkle trees, ordered in the canonical top-to-bottom and left-to-right order.
§.§ Verkle Proofs
Let π_x denote the Verkle proof of some message m_x at index x, (b_1,…,b_h) = (x): π_x = ((C_b_0,b_1, …, C_b_0,…,b_h-1), ([g(X)], π)).
We define π^f_x as the opening proof for index x within polynomial f.
We observe that the commitment [g(X)] and the proof π can be expressed as functions of the opening proofs of the inner nodes u_b_0,b_1, …, u_b_0,…,b_h at the indices b_1,…,b_h within the polynomials f_b_0, …, f_b_0,…,b_h-1, respectively:
rCl
[g(X)] = [∑_j=0^h-1 r^j f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]
= ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j = ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j.
Similarly, the opening proof π=π^(h-g)_t for index t within the polynomial h(X)-g(X) can be expressed as follows (Appendix <ref>):
rCl
[h(X)-g(X)-(h(t)-g(t))/X-t] = ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j/t-b_j+1
= ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j/t-b_j+1
We assume that each user holding the Verkle proof π_x for some index x, (b_1,…,b_h) = (x), also holds the opening proofs π^f_b_0,…,b_j_b_j+1, j ∈ [h], in memory.
As we will see in the next section, the user also holds the KZG commitments at the children of the inner nodes on the path from the root to the message m_x, C_b_0,…,b_j,i for all j ∈ [h] and i ∈ [c] in memory.
These opening proofs and KZG commitments are not broadcast as part of any proof; however, they are needed for the user to locally update its Verkle proof after message updates.
§.§ Proof Update
When the messages (i_j, m_i_j)_j ∈ [k] are updated to (i_j, m'_i_j)_j ∈ [k], to calculate the new Verkle proof π'_x, the user must obtain the new commitments C'_b_0, …, C'_b_0,…,b_h-1 on the path from the root to message m_x, the new commitment [g'(X)] and the new opening proof π' for the polynomial h'(X)-g'(X) at index t'= H(r',[g'(X)]).
Message updates change the commitments at the inner nodes, which in turn results in new polynomials f_b_0,…,b_j, j ∈ [h].
Suppose each polynomial f_b_0,…,b_j, j ∈ [h], is updated so that
C
f'_b_0,…,b_j(X) = f_b_0,…,b_j(X) + ∑_i=0^c-1(f'_b_0,…,b_j(i)-f_b_0,…,b_j(i)) L_i(X),
where, by definition, f'_b_0,…,b_j(i)-f_b_0,…,b_j(i) = u'_b_0,…,b_j,i-u_b_0,…,b_j,i = H(C'_b_0,…,b_j,i)-H(C_b_0,…,b_j,i).
Then, given the new and the old commitments (C_b_0,…,b_j,i,C'_b_0,…,b_j,i) for i ∈ [c] and j ∈ [h], the table of Lagrange basis polynomials, and using the technique in Section <ref>, the new opening proofs π̃^f_b_0,…,b_j_b_j+1 after the message updates can be computed as follows for j ∈ [h]:
C
π̃^f_b_0,…,b_j_b_j+1 = π^f_b_0,…,b_j_b_j+1 ∏_i=0^c-1[L_i(X)-L_i(b_j+1)/X-b_j+1]^(H(C'_b_0,…,b_j,i)-H(C_b_0,…,b_j,i)),
where [L_i(X)-L_i(b_j+1)/X-b_j+1] is the opening proof of the Lagrange basis polynomial L_i(X) at index b_j+1.
Once the new opening proofs are found, the new commitment [g'(X)] and the new proof π' become
C
[g'(X)] = ∏_j=0^h-1 (π̃^f_b_0,…,b_j_b_j+1)^r'^j, π' = ∏_j=0^h-1 (π̃^f_b_0,…,b_j_b_j+1)^r'^j/t'-b_j+1
where r'=H(C'_b_0,b_1,..,C'_b_0,…,b_h-1,u'_b_0,b_1,..,u'_b_0,…,b_h,b_1,..,b_h) and
t'=H(r',[g'(X)]).
Note that both r' and t' can be calculated by the user given the new KZG commitments C'_b_0,…,b_j,i for all i ∈ [c] and j ∈ [h].
Finally, to retrieve the new KZG commitments C'_b_0,…,b_j,i for all i ∈ [c] and j ∈ [h], the user inspects the commitments published as part of the update information U:
C'_b_0,b_1,…,b_j-1,i U[b_0,b_1,…,b_j-1,i] if U[b_0,b_1,…,b_j-1,i] ≠ and C'_b_0,b_1,…,b_j-1,i C_b_0,b_1,…,b_j-1,i otherwise, for all i ∈ [c] and j ∈ [h].
In Verkle trees, the user cannot calculate the effect of an updated message on an arbitrary inner node without the knowledge of the inner nodes on the path from the message to the target node.
For instance, suppose U[b_0,b_1,…,b_j-1,i] = for some i ∈ [c] and j ∈ [h], and the user wants to calculate the effect of an update from m_x to m'_x on C'_b_0,…,b_j-1,i,b̃_j+1,…,b̃_h, (x) = (b_1,…,b_j-1,i,b̃_j+1,…,b̃_h) and b̃_j = i.
Then, for each ℓ∈{j,…,h-1}, the user have to find
rCl
C'_b_0,…,b̃_j,…,b̃_h m'_x
C'_b_0,…,b̃_j,…,b̃_ℓ C_b_0,…,b̃_j,…,b̃_ℓ [L_b̃_ℓ+1]^(u'_b_0,…,b̃_j,…,b̃_ℓ+1-u_b_0,…,b̃_j,…,b̃_ℓ+1),
where C'_b_0,…,b̃_j,…,b̃_ℓ are the commitments on the path from the target commitment C_b_0,b_1,…,b_j-1,i to the message m_x.
Hence, the user has to know the original commitments on the path from the message to the target commitment, , keep track of inner nodes, which contradicts with the idea of stateless clients.
This shows the necessity of publishing all of the updated inner nodes as part of the update information.
§.§ Complexity
Suppose each KZG commitment is of size |G| and each hash H(C) of a KZG commitment, each inner node, has size |H|.
Then, updating a single message results in one update at each level of the Verkle tree and requires Θ(h|H|) group operations.
Thus, when k messages are updated, the new Verkle root can be found after Θ(kh|H|) group operations.
As U consists of the published KZG commitments at the inner nodes and their indices, |U| = Θ(k log_c(N)(log(N)+|G|)) = Θ(k)|G|, which implies g_1(k) = k.
The user can replace each KZG commitment at the children of the inner nodes from the root to its message in Θ(log(N)+|G|) time by running a binary search algorithm over U.
Since there are ch such commitments to be updated, , C_b_0,…,b_j,i, i ∈ [c] and j ∈ [h], updating these commitments takes Θ(c h (log(N)+|G|)) = Θ(1) time.
Upon obtaining the new commitments C'_b_0,…,b_j-1,i, i ∈ [c], j ∈ [h], with access to the table of Lagrange basis polynomials, the user can update each opening proof π_b_j+1 (for the function f_b_0,…,b_j),
j ∈ [h], with Θ(c|H|) group operations.
Since there are h such proofs, updating them all requires Θ(c h |H|) group operations.
Given the new proofs, computing the new commitment [g'(X)] and proof π' requires Θ(h |H|) group operations.
This makes the total complexity of updating a Verkle proof Θ(c h + 2 h) |H| T_G + Θ(c h (log_c(N)+|G|)).
For a constant c and h = log_c(N), this implies a worst-case time complexity of Θ(1) |H| T_G for Verkle proof updates, , g_2(k) = 1.
§.§ A Concrete Evaluation
We now estimate the size of the update information and the number of group operations to update an opening proof after observing an Ethereum block consisting of ERC20 token transfers.
As in Section <ref>, suppose the block has the target size of 15 million gas <cit.>, and each token transfer updates the balance of two distinct accounts stored at separate leaves of the Verkle tree.
Then, there are ∼ 230 such transactions in the block, and the block updates k = 460 accounts.
We assume that the Verkle tree has degree 256 ( <cit.>) and commits to 256^3 accounts as in Section <ref>.
Then, each proof consists of 2 KZG commitments, C_,b_1 and C_,b_1,b_2 and a multiproof consisting of the commitment [g(X)] and proof π'.
These components are elements of the pairing-friendly elliptic curve BLS12_381 and consist of |G| = 48 bytes <cit.>.
This implies a proof size of (log_c(N)+1)|G| = 192 bytes (excluding the message at the leaf and its hash value; adding those makes it 272 bytes).
When 460 accounts are updated, in the worst-case, the update information has to contain k log_c(N) (log(N)+|G|) = 460 × 3 × (24+48) Bytes, , 99.4 kBytes.
This is comparable to the size of the Ethereum blocks, which are typically below 125 kBytes <cit.>.
Hence, even though the update information of Verkle trees is linear in k, it does not introduce a large overhead beyond the block data.
Note that the runtime of the proof updates are constant and do not scale in the number of updated messages k, or the Ethereum block size.
On the other hand, in the worst case, an opening proof can be updated after c log(c) |H| + 2 log_c(N) |H| group operations.
Then, with |H|=256, the number of bits output by SHA256, as many as c log_c(N) |H| + 2 log_c(N) |H| = (c + 2) log_c(N) |H| = 774 × 2256 ≈ 200,000 elliptic curve multiplications might have to be made.
Following the benchmarks published in <cit.> for the specified curve, these operations can take up to (c + 2) log_c(N) 0.000665471 ns = 0.52 seconds on commodity hardware, given a runtime of 665471 nanoseconds per exponentiation of a group element with a message hash value.
This is again comparable to the 12 second inter-arrival time of Ethereum blocks.
Table <ref> compares the Verkle proof size |π| = (log_c(N)+1) |G|, update information size |U| = k log_c(N) (log_cN+|G|), the upper bound (c + 2) log_cN |H| on the number of group operations needed for a single proof update and the estimated time it takes to do these operations on a commodity hardware for different values of c, the Verkle tree degree, while keeping the number of accounts and the updated accounts fixed at 2^24 and 460 respectively.
The table shows the trade-off between the Verkle proof and update information size on one size and update complexity on the other.
Comparing Table <ref> with Table <ref> shows that the Verkle tree with any given degree c, 1 < c ≤ 256, significantly outperforms the existing homomorphic Merkle trees in Section <ref> in terms of almost all of proof size, update information size and proof update time.
§ LOWER BOUND
Finally, we prove the optimality of our VC scheme with sublinear update by proving a lower bound on the size of the update information given an upper bound on the complexity of proof updates.
The lower bound is shown for VCs that satisfy the following property.
It formalizes the observation that for many dynamic VCs (, Merkle trees <cit.>, Verkle trees <cit.>, KZG commitments <cit.>, RSA based VCs <cit.>), the opening proof for a message at some index can often act as a commitment to the vector of the remaining messages.
A VC scheme is said to be if the following probability is negligible in λ for all PPT adversaries 𝒜:
C
[Verify_pp(C, m_i^*, i^*, π) = 1
Verify_pp(C', m_i^*, i^*, π) = 1 pp KeyGen(1^λ, N);
π, m_i^*, (m_0, …, m_i^*-1, m_i^*+1, …, m_N-1),
(m'_0, …, m'_i^*-1, m'_i^*+1, …, m'_N-1) 𝒜(pp);
(m_0, …, m_i^*-1, m_i^*+1, …, m_N-1)
≠(m'_0, …, m'_i^*-1, m'_i^*+1, …, m'_N-1);
Commit_pp(m_0, …, m_i^*-1, m_i^*, m_i^*+1, …, m_N-1) = C;
Commit_pp(m'_0, …, m'_i^*-1, m_i^*, m'_i^*+1, …, m'_N-1) = C' ]
To prove the lower bound, we first show that implies that (i^*, m_i^*, π) is a binding commitment to the rest of the vector.
Consider a dynamic and VC, where π is the correctly generated opening proof for the message m_i at some index i.
Then, for any i ∈ [N], it holds that the tuple (i, m_i, π) is a binding commitment to the vector of messages m_j, j ∈ [N], j ≠ i.
Since the VC is , with overwhelming probability, no PPT adversary 𝒜 can find an opening proof π^*, an index i^*, a message m^* and two sequences of messages
such that
C
(m_1, …, m_i^*-1, m_i^*+1, …, m_N-1) ≠(m'_1, …, m'_i^*-1, m'_i^*+1, …, m'_N-1)
and Verify_pp(C, m_i^*, i^*, π) = Verify_pp(C', m_i^*, i^*, π) = 1,
where C and C' are commitments to the message sequences (m_1, …, m_i^*-1, m_i^*, m_i^*+1, …, m_N-1) and (m'_1, …, m'_i^*-1, m_i^*, m'_i^*+1, …, m'_N-1).
Thus, it holds that the tuple (i, m_i, π) is a binding commitment to the vector of messages m_j, j ∈ [N], j ≠ i, with the following new commitment function:
C
NewCommit_pp((m_j)_j ∈[N], j ≠i) = (i, m_i, Open_pp(m_i, i, )),
where = Commit_pp(m_0, …, m_N-1)..
The following lemma shows that all randomized VCs can be derandomized to obtain a deterministic and secure VC as we do not use hiding commitments in this work.
Consider a VC , where the commitment is a random function of the public parameters pp and the committed messages.
Let ' denote the VC that is the same as , except that the randomness is fixed.
Then, ' is a correct and secure VC with at most the same upper bound on the error probability.
Let R denote the sequence of bits sampled uniformly at random from the set ℛ to instantiate the VC .
Since is binding, no PPT adversary 𝒜 can find two different sequences of messages 𝐦 and 𝐦' such that (𝐦, R) = (𝐦', R') for some R,R' ∈ℛ, except with negligible probability.
This implies that for any fixed R^* ∈ℛ, no PPT adversary 𝒜 can find two different sequences of messages 𝐦 and 𝐦' such that (𝐦, R^*) = (𝐦', R^*), except with negligible probability.
Hence, the commitment scheme '(.) = (., R^*) is a position-binding, , secure VC.
Its correctness follows from the correctness of .
Finally, equipped with Lemmas <ref> and <ref>, we can prove the following lower bound for dynamic and VCs.
Consider a dynamic and VC such that for every PPT adversary 𝒜, it holds that
C
[Verify_pp(C, m, i, π_i) = 1
Verify_pp(C, m', i, π'_i) = 1 m ≠m' pp KeyGen(1^λ, N)
(C, m, m', π_i, π'_i) 𝒜(pp)] ≤e^-Ω(λ).
Then, for this VC, if g_2(k) = O(k^1-ν), then g_1 = Ω(k^ν) for all ν∈ (0,1).
Suppose the messages m_i_j, j ∈ [k], are updated to m'_i_j.
Define 𝒮 as the sequence (m'_i_j)_j ∈ [k], and let m'_i = m_i for i ∉{i_j j ∈ [k]}.
Let 𝒫_i, i ∈ [N], denote the user that holds the opening proof π_i for the message m_i at index i, and aims to calculate the new proof π'_i for the message m'_i using π_i, the update information U and the old and the new sequences of messages m_i, m'_i, i ∈ [N].
Suppose g_2 = O(k^1-ν).
Then, there exists a constant α such that each user can read at most α k^1-ν of the updated messages while updating its opening proof.
Let 𝒮_i ⊆ (m'_i_j)_j ∈ [k] denote the sequence of updated messages and their indices, which were not observed by 𝒫_i, and 𝒮_i = 𝒮∖𝒮_i denote the sequence read by 𝒫_i.
Here, |𝒮| denotes the number of messages within the sequence 𝒮.
Since 𝒫_i is assumed to know m'_i, it must be that m'_i ∈𝒮_i.
We next show that each user 𝒫_i that successfully updates its opening proof must download enough bits of U to generate a binding, deterministic commitment to the set 𝒮_i.
By Lemma <ref>, the tuple (i, m'_i, π'_i) is a binding commitment to the sequence of messages (m'_j)_j ∈ [N], j ≠ i.
This implies that the tuple (i, 𝒮_i, π'_i) is a binding commitment to the sequence 𝒮_i.
By Lemma <ref>, the commitment (i, 𝒮_i, π'_i) can be de-randomized to obtain a deterministic commitment C_i to the sequence 𝒮_i (with at most the same upper bound on the error probability).
Let denote the deterministic VC scheme such that C_i = (𝒮_i).
Since is a deterministic function given the public parameters, and the updated messages are sampled independently and uniformly at random, then I(𝒮_i;{m_i}_i ∈ N,𝒮_i|pp) = 0, where I(.;.) is the mutual information.
Moreover, as π_i is a function of the old messages {m_i}_i ∈ N and the randomness of the original VC, I(C_i; π_i|pp) = 0.
Hence, C_i = f(U, i, {m_i}_i ∈ N, π) is a deterministic function of the update information U.
For all i ∈ [k], it holds that |𝒮_i| ≥ k - α k^1-ν and m'_i ∉𝒮_i.
Given these constraints, the minimum number of distinct sequences 𝒮_i is k/α k^1-ν = k^ν/α.
For an appropriately selected β that will be defined later, without loss of generality, let 𝒮_0, …, 𝒮_M-1 denote the first
C
M = min(⌊k^ν/β - α/β - λ/βk^1-ν ⌋, k^ν/α)
distinct sequences.
Since C_i is a deterministic function of U for all i ∈ N, it holds that the Shannon entropy H(.) of U satisfies the following expression:
C
H(U) ≥H(C_0, …, C_M-1)
≥H(C_0) + ∑_i=1^M-1 H(C_i | C_0, …, C_i-1)
As g_2(k) = O(k^1-ν), there exists a constant β such that each user can download at most β k^1-ν bits of data from U.
Then, for all i ∈ [k], it must be that H(C_i) ≤ H(U) ≤β k^1-ν since C_i is a deterministic function of U for each i ∈ [N].
Finally, we show that H(C_0), H(C_i | C_0, …, C_i-1) = Ω(λ) for all i=1, …, M-1.
Towards contradiction, suppose ∃ i^* H(C_i^* | C_0, …, C_i^*-1) = o(λ).
Note that
rCl
H(C_0, …, C_i^*-1) ≤ ∑_i=0^M-1 H(C_i)
≤ min(k^ν/β - α/β - λ/βk^1-ν, k^ν/α) βk^1-ν ≤k-αk^1-ν-λ.
Now, consider an adversary 𝒜 that tries to break the binding property of the VC scheme .
Due to the upper bound on the entropy of (C_0, …, C_i^*-1), it holds that
H(𝒮_i^* | C_0, …, C_i^*-1) ≥λ;
since H(𝒮_i^*) ≥ k-α k^1-ν, and
rCl
H(𝒮_i^*) - H(𝒮_i^* | C_0, …, C_i^*-1) = I(𝒮_i^*; (C_0, …, C_i^*-1))
≤ H(C_0, …, C_i^*-1) ≤k-αk^1-ν-λ.
However, when H(C_i^* | C_0, …, C_i^*-1) = o(λ), for sufficiently large λ, given (C_0, …, C_i^*-1), the adversary can find a collision such that (𝒮_i^*)=(𝒮'_i^*) for two 𝒮_i^*≠𝒮'_i^*, with probability 2^-o(λ).
As this is a contradiction, it must be that H(C_0) and H(C_i | C_0, …, C_i-1) = Ω(λ) for all i < M, and thus, H(U) = Ω(k^νλ) and g_1(k) = Ω(k^ν).
Theorem <ref> shows that the update information length scales as Θ(k^νλ) when the runtime complexity for proof updates is Θ(k^1-ν) and the error probability for the security of the VC is e^-Ω(λ) for a PPT adversary.
When the error probability is just stated to be negligible in λ, then the same proof can be used to show that the update information length must scale as Ω(k^ν(λ)) for any polynomial function of log(λ).
§ CONCLUSION
Dynamic VCs with sublinear update are the key to reducing the size of the global update information while minimizing the runtime of clients synchronizing with the latest commitment.
In this work, we propose a construction that can achieve an update information size of Θ(k^ν) and a proof update time of Θ(k^1-ν) in the number of changed messages k.
Our construction combines a novel update algorithm (Alg. <ref>) with homomorphic Merkle trees <cit.> that allow each inner node to be expressed as a linear function of the underlying messages.
It achieves the smallest asymptotic complexity for the update information size and proof update time.
We also provide update algorithms for the Verkle trees proposed for stateless clients on Ethereum.
The existing instantiations of homomorphic Merkle trees are based on lattices and require relatively large parameters for security.
Consequently, despite the appealing asymptotic complexity of our construction,
its performance for concrete parameters is dominated by Verkle trees.
As such, designing asymptotically optimal and practically efficient dynamic VCs remains an open problem.
An interesting direction is to design a more preferment homomorphic Merkle tree system.
Acknowledgments.
This work was partially funded by NSF, DARPA, the Simons Foundation, and NTT Research.
Additional support was provided by the Stanford Center for Blockchain Research.
Opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not
necessarily reflect the views of DARPA.
plain
§ LOWER BOUND ON THE SIZE OF THE UPDATE INFORMATION
Consider a dynamic accumulator, where k out of N messages m_i,j are updated to m'_i,j≠ m_i_j, j ∈ [k].
Suppose |M| = (λ).
Then, Ω(k (log(N|ℳ|))) bits of information must be published to enable updating the opening proofs after these k updates.
The proof idea is very similar to those presented in <cit.>.
Namely, the update information must contain a minimum amount of bits for the VC to remain correct and secure after the update.
Consider a game between a platform 𝒫 maintaining the data structures of the VC and an adversary 𝒜.
The platform 𝒫 updates k out of N messages m_i,j to m'_i,j≠ m_i_j, j ∈ [k], in a way not known to user 𝒜, and publishes the update information U along with the new commitment value C' (let m'_i = m_i for i ∉{i_j j ∈ [k]}).
Before receiving the update information, 𝒜 knows the old sequence of messages m_i, i ∈ [N], and their opening proofs π_i.
Upon receiving the update information, 𝒜 updates the opening proofs for each message to π'_i.
Then, it must be that for all j ∈ [k], Verify_pp(C', m'_i_j, i_j, π'_i_j) = 1, and for all i ∉{i_j j ∈ [k]}, Verify_pp(C', m_i, i, π'_i) = 1.
Otherwise there would be messages among m'_i, i ∈ [N], for which an updated witness cannot be computed, violating correctness.
Similarly, for all j ∈ [k], Verify_pp(C', m̃, i_j, π'_i_j) = 0 for any m̃≠ m'_i_j; as otherwise the position binding property, thus security, would be violated.
Hence, by calling the function Verify_pp(C', m_i, i, π'_i_j) for each index, 𝒜 can figure out the indices i_j, j ∈ [k], where the messages were updated.
Similarly, by evaluating the function Verify_pp(C', m̃, i_j, π'_i_j) for the |ℳ| possible messages m̃ for each j ∈ [k], 𝒜 can identify the new value m'_i_j of the message at each such index i_j.
Hence, the adversary can recover the sequence (i_j, m'_i_j)_j ∈ [k].
As there are N!/(N-k)!log^k|ℳ| possible sequences (i_j, m'_i_j)_j ∈ [k], it holds that
|U| ≥log(N!/(N-k)!log^k|ℳ|) = Ω(k (logN+log|ℳ|)).
When |M| = Ω((λ)), the minimum number of bits to be published depends on the error probability for the security of the PPT adversary.
As in Remark <ref>, if |ℳ| = Θ(2^λ) and the error probability is e^-Ω(λ), then the same proof can be used to show that Ω(kλ) bits of information must be published.
When the error probability is just stated to be negligible in λ, then the number of bits must scale as Ω(kλ) for any polynomial function of log(λ).
§ HOMOMORPHIC MERKLE TREES WITH NO UPDATE INFORMATION
§.§ Update Information
When k messages are updated, the new commitment, the new Merkle root can be calculated just like any other inner node, by incorporating the effect of the old and the new messages, (i_j, m'_i_j-m_i_j)}_j ∈ [k]:
rCl
C' = u'_b_0 = u_b_0 + ∑_j ∈[k] (h_i_j,0(m'_i_j) - h_i_j,0(m_i_j))
= C + ∑_j ∈[k] sign(m'_i_j-m_i_j)h_i_j,0(|m'_i_j - m_i_j|)
As in KZG commitments, the update information is U=∅.
§.§ Proof Update
When the messages are modified at k points, each user holding a Merkle proof π_x for index x can calculate the new values of the inner nodes within the proof using the old and the new messages and modify the proof respectively.
§.§ Complexity
Calculating each partial digest h_x,j takes at most logN function evaluations.
Then, each user can update each inner node within its Merkle proof after at most klog(N) operations, making the total number of operations Θ(klog^2N) = Θ(k) in the worst case.
The size of the update information U is Θ(1).
Hence, this scheme matches the algebraic VCs in terms of complexity.
§ WHY ARE HOMOMORPHIC MERKLE TREES NEEDED?
Merkle trees based on SHA256 can also achieve complexity sublinear in k, for both the update information and the runtime of proof updates, if the users have access to the old messages and inner nodes of the Merkle tree.
In this case, homomorphism is not needed since the nodes can find the effect of the updated messages on the inner nodes within their Merkle proofs by hashing these messages together with the old inner nodes.
However, this is possible for only a single batch of updates.
Indeed, if this scheme is to be repeated, the assumption of having access to the old inner nodes requires the users to keep track of changes throughout the Merkle tree, by calculating the effect of all updated messages on all inner nodes.
This implies a runtime linear in k per proof updates.
In contrast, homomorphic Merkle trees can maintain a sublinear complexity for future proof updates since they do not require access to the old messages and inner nodes for finding the partial digests of the updated messages.
§ AN ALTERNATIVE CONSTRUCTION
An alternative tree-based VC is proposed by <cit.>, where each inner node is itself a lattice-based VC to its children (akin to Verkle trees <cit.>).
Opening proof for a message consists of the inner nodes (commitments) on the path from the message to the root, along with the opening proofs for these inner nodes with respect to their parent nodes.
The construction again enables expressing each inner node as a sum of partial digests of the messages underneath.
Using the public parameters and the updated inner nodes, users can then derive their updated opening proofs at different heights of the tree.
This construction supports trees of large degrees c without a linear increase in the proof size as would be the case for Merkle trees; this however comes at the cost of a larger runtime complexity for proof updates, proportional to the degree.
Section <ref> describes similar steps in the context of Verkle trees, and exposes the dependence of the runtime complexity of proof updates on the tree degree c.
§ DERIVATION OF THE OPENING PROOF Π^(𝐡-𝐠)_𝐭
Since
rCl
h(X)-g(X) = ∑_j=0^h-1 r^j (f_b_0,…,b_j(X)/t-b_j+1 - f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1)
= ∑_j=0^h-1 r^j (X-t)f_b_0,…,b_j(X)+u_b_0,…,b_j+1(t-b_j+1)/(t-b_j+1)(X-b_j+1),
the opening proof π=π^(h-g)_t for index t within the polynomial h(X)-g(X) is
rCl
[h(X)-g(X)-(h(t)-g(t))/X-t]
= [∑_j=0^h-1 r^j/X-t ((X-t)f_b_0,…,b_j(X)+u_b_0,…,b_j+1(t-b_j+1)/(t-b_j+1)(X-b_j+1)-u_b_0,…,b_j+1/t-b_j+1) ]
= [∑_j=0^h-1 r^j/t-b_j+1 f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]
= ∏_j=0^h-1 [f_b_0,…,b_j(X)-u_b_0,…,b_j+1/X-b_j+1]^r^j/t-b_j+1 = ∏_j=0^h-1 (π^f_b_0,…,b_j_b_j+1)^r^j/t-b_j+1
§ UPDATE AND PROOF UPDATE ALGORITHMS FOR KZG COMMITMENTS AND MERKLE TREES
|
http://arxiv.org/abs/2307.06279v1 | 20230709050025 | SpreadNUTS -- Moderate Dynamic Extension of Paths for No-U-Turn Sampling & Partitioning Visited Regions | [
"Fareed Sheriff"
] | stat.CO | [
"stat.CO",
"cs.LG"
] |
— Moderate Dynamic Extension of Paths for No-U-Turn Sampling & Partitioning Visited Regions
Fareed Sheriff
May 17, 2023
============================================================================================
§ INTRODUCTION & PRIOR WORK
Markov chain Monte Carlo (MCMC) methods have existed for a long time and the field is well-explored. The purpose of MCMC methods is to approximate a distribution through repeated sampling; most MCMC algorithms exhibit asymptotically optimal behavior in that they converge to the true distribution at the limit. However, what differentiates these algorithms are their practical convergence guarantees and efficiency. While a sampler may eventually approximate a distribution well, because it is used in the real world it is necessary that the point at which the sampler yields a good estimate of the distribution is reachable in a reasonable amount of time. Similarly, if it is computationally difficult or intractable to produce good samples from a distribution for use in estimation, then there is no real-world utility afforded by the sampler. Thus, most MCMC methods these days focus on improving efficiency and speeding up convergence.
We present a cursory overview of popular MCMC techniques. Random-walk Metropolis-Hastings is a rudimentary algorithm for sampling from a distribution by inducing a Markov chain on repeated samples: the next sample is chosen through a draw from the sampling distribution that takes the current sample as a parameter. However, as the name suggests, this exhibits strong random walk behavior, making it undesirable practically due to the possibly long burn-in period and large number of samples needed to thoroughly explore the distribution space. In fact, many MCMC algorithms suffer from random walk behavior and often only mitigate such behavior as outright erasing random walks is difficult. Hamiltonian Monte Carlo (HMC) is a class of MCMC methods that theoretically exhibit no random walk behavior because of properties related to Hamiltonian dynamics. This paper introduces modifications to a specific HMC algorithm known as the no-U-turn sampler (NUTS) that aims to explore the sample space faster than NUTS, yielding a sampler that has faster convergence to the true distribution than NUTS.
§.§ Hamiltonian/Hybrid Monte Carlo
[This subsection summarizes relevant parts of <cit.>]
Hamiltonian dynamics work on a system of position-momentum pairs (p,q) subject to Hamilton's equations
dq_i/dt = ∂ H/∂ p_i, dp_i/dt = -∂ H/∂ q_i
where p,q are vector-valued functions of time over a d-dimensional space and H(q,p) is the Hamiltonian, which represents the system's total energy. We assume for HMC that the Hamiltonian expresses the system's potential and kinetic energies H(q,p) = U(q)+K(p). We also define for HMC U(q) to be the negative of the log density of q up to a constant and K(p) = 12p^TM^-1p to be the negative of the log density of the Gaussian with zero mean and covariance matrix M (often, the Gaussians will be uncorrelated, so M will be diagonal), also up to a constant. We thus rewrite Hamilton's equations to be
dq_i/dt = (M^-1p)_i, dp_i/dt = - ∂ U/∂ q_i
As with MCMC methods as a whole, the Hamiltonian is (time-)reversible and is invariant under Hamilton's equations, meaning the acceptance probability is 1. In practice, it is close to 1 because we cannot practically make the Hamiltonian invariant when solving Hamilton's equations due to error accumulated when solving the PDEs numerically.
To numerically solve the PDEs, we use a symplectic integrator, which preserves the Hamiltonian's invariance under integration of Hamilton's equations. A commonly-used symplectic integrator is the leapfrog integrator, which makes use of a "halfstep" in the integration process to better inform the estimate of the Hamiltonian in the next timestep. The equations that govern the leapfrog integrator are as follows with stepsize :
p_i(t+2) = p_i(t)- /2∂ U/∂ q_iq(t)
q_i(t+) = q_i(t) + p_i(t+2)/m_i
p_i(t+) = p_i(t+2) - /2∂ U/∂ q_i q(t+)
In effect, we compute an estimate of p at t+2, estimate q using this estiamte of p, then again estimate p using the estimate of q at t+, thus taking into account the estimate of p at t+2 and p's relationship with q.
HMC samples from continuous distributions on ^d with well-defined densities and partials of the log densities. We define the joint distribution P of (p,q) on the Hamiltonian H to be
P(q,p) = 1/Ze^-1/TH(q,p)
for any positive constant Z and T. Then,
H(q,p) = U(q)+K(p) → P(q,p) = 1/Ze^-U(q)/Te^-K(p)/T
We choose U(q) to be -logπ(q) for the distribution π from which we are trying to sample. The distribution of K(p) is independent of q, but it is common to use a quadratic like K(p) = p^TM^-1p/2. For diagonal M, this yields K(p) = ∑_ip^2_i/2m_i.
HMC works in two steps. The first step draws a value for momentum p using the zero-centered Gaussian with covariance matrix M. The second step conducts a Metropolis update using the Hamiltonian. Using a stepsize of for L steps, a trajectory of samples is calculated, which is accepted with probability
min(1,exp(U(q)-U(q^*)+K(p)-K(p^*)_H(q,p)-H(q^*,p^*)))
which works exactly because the Hamiltonian is time-reversible.
Practical considerations to take into account when implementing HMC include varying ,L. Note, however, that HMC requires adjustment/setting of the parameters , L.
§ NO-U-TURN SAMPLING
One of the few and biggest problems with HMC<cit.> is the necessity to tune ,L — without proper tuning, we lose many of the efficiency guarantees of HMC. No-U-turn sampling (NUTS)<cit.>] aims to alleviate some of these problems. NUTS is a type of HMC algorithm that does not calculate the trajectory for constant L steps and instead stops the trajectory when sufficient error or space explored has been accumulated. Furthermore, it tunes dynamically to make NUTS an effectively parameterless version of HMC.
NUTS replaces a constant L by stopping the trajectory once some condition has been triggered. This condition is checking that the distance between the proposal q^* and the initial q will not continue to increase. We can check this by taking the product of the momentum and the difference between the sampled proposal and initial proposal (q^*-q)· p^* (the U-turn condition), noting that if it is negative, then the direction of our next step will be toward already-sampled points. Because this does not maintain time-reversibility, NUTS runs the Hamiltonian both forward and backward with equal probability and calculates the U-turn condition between the endpoints of the extension of the trajectory generated in the current iteration, checking that it is nonnegative. NUTS generates the trajectory through a doubling scheme that randomly chooses a direction (forward or backward in time), then on the ith iteration of generating this trajectory takes 2^i timesteps in the chosen direction, adding the calculated points to the current trajectory. A point is chosen as a sample from this trajectory in the following manner: once the trajectory is generated first by sampling some rejection energy threshold u uniformly from [0,P(q,p)] = [0,e^-H(q,p)], extending the point forward and backward in time repeatedly, then uniformly randomly selecting a point from this "tree" of points (trajectory).
§ MODERATE DYNAMIC EXTENSION OF PATHS
We consider two additions to the NUTS scheme: relaxing the U-turn condition checks on the induced binary tree of the generated trajectory with, and increasing the size of the trajectory by more than double every iteration. Our reasoning behind both of these ideas is that the number of U-turn condition checks on the subtrees of the subtrajectory created by the doubling process in NUTS adds excessive (and potentially underjustified) overhead when checking that the U-turn condition is not violated between the two leaves on the edge of each sutree. This overhead is linear in the number of generated points. While it is stated that "except for very simple models with very little data, the costs of these inner products should be negligible compared to the cost of computing gradients" <cit.> (in reference to the inner products calculated when evaluating the U-turn condition), such a rigorous check can in and of itself be counterproductive and could risk cutting off the trajectory being generated before it has sufficiently explored the space around it. This is because while the U-turn condition checks whether the trajectory turns back on itself, if we check for violation between many pairs of points, adjacent or not, this degenerates into a check that the trajectory is always pointing in the direction of unexplored space.
However, this is not a very useful condition to force because we could have a trajectory that moves backward a tiny bit but later continues to move away from previously-explored points, thus exhibiting a general trend toward unexplored space. While we agree that checking that no violation of the U-turn condition should occur between the first few points on the path, we note that as long as the general trend of the path does not violate the U-turn condition, the path contributes to exploring space. We thus strike a compromise: we relax the U-turn condition checks on the balanced tree built on each iteration's points by continuing to check that the U-turn condition is not violated between the leaves on the edge of each subtree of the tree built on each iteration's point, but now build a k-ary tree on the calculated points instead of a binary tree where k is the iteration number. This both decreases the number of U-turn condition checks and iteratively relaxes the strictness of the U-turn violation penalty as more points are generated.
Specifically, instead of doubling the tree by adding 2^k points to the end of our path in direction d∼{-1,1}, we add k^k points and check the U-turn condition fewer times on these points: where we would check the U-turn condition around 2^klog_2k time on these k^k points, we now check the condition k^k-1/k-1≈ k^k-1=2^(k-1)log_2k, which is less than 2^klog_2k by a multiplicative factor of k (which grows asymptotically).
§ PARTITIONING VISITED REGIONS
To prevent ourselves from exploring parts of the distribution that we have already explored, when sampling from the generated trajectory, we bias our selection toward points the space around which we have not already explored. This still satisfies detailed balance because the probability of having already chosen a point from some subspace of the distribution is uniform across all subspaces. Thus, we still have the same convergence guarantees as NUTS. However, we attempt to sample the distribution in a more "spread out" manner by exploring unexplored parts of the trajectory (which itself maintains the invariant of a fixed density) so in the end we still sample in accordance with the distribution's density but with regularization that enforces exploring unexplored parts of the space.
We can keep track of how much space we have explored close to a datapoint using any type of querying data structure that allows us to calculate some measure of how explored the space around a given point is (for example, a multidimensional Gaussian convoluted with all previously-sampled points). For sake of example and efficiency, we consider a k-dimensional binary search tree T on all sampled points that allows us to find the closest point in average-case Ø(logn) time with insertion also taking Ø(logn).
Our metric d_p for how much space has been explored near a given point p will be the squared L_2 norm of p with the closest neighbor in T (sum of squares of difference of coordinates). We then define the probability of choosing p to be proportional to d_p and the metric on all other points of the trajectory so that the probability we select p from trajectory t = (p_0,⋯, p_k) equals
m_p/∑_p_i∈ tm_p_i
We can then choose a point by allocating some proportion of a uniform r.v. to each point and sampling from this uniform to select the point. This is efficient and so the entire procedure allows us to regularize toward sampling the distribution thoroughly while maintaining sampling by density with the cost of a multiplicative Ø(logn) factor to the sampling process.
§ RESULTS
We discuss our testing regime in more detail: we randomly generate mixtures of multivariate Gaussians, which we use to compare how well regular NUTS samples compared to the modified NUTS algorithm presented in this paper by comparing the empirical distributions of each algorithm with the true distribution of the mixtures using a sort of discretized total variation metric. We refer to our algorithm as "SpreadNUTS" because it attempts to spread NUTS trajectories over the sample space to better leave less of the sample space unexplored.[Our code for SpreadNUTS is based on the code at <cit.>, and we test SpreadNUTS against this implementation of NUTS]
§.§ Testing Regime
We randomly select k Gaussian distributions where k is distributed over a discrete uniform that takes values from 1 to 4 (the choice of 5 is arbitrary). We choose the means of the distributions uniformly randomly from the interval [-⃗2⃗0⃗, 2⃗0⃗] (this choice is also arbitrary); we choose the covariance matrix by generating a matrix whose entries are uniformly random over [0,1], multiplying it by its transpose (generating a valid correlation matrix), then multiplying by a draw from a uniform over interval [0,4] (also arbitrary). This ensures the covariance matrix is positive semidefinite (and is also diagonally dominant). We also uniformly randomly choose a dimension for the Gaussians from 1 to 5. Finally, we generate mixture probabilities p⃗ such that the elementwise sum is 1 and each value is nonnegative by generating [0,1] entries, then dividing by the sum of these entries. While this does not yield a uniform distribution (the distribution is biased toward 1⃗D⃗ where D is the dimension and is chosen uniformly from 1 to 3 — the low upper bound on dimension is because for dimensions 4 or higher, regular NUTS tends to perform very slowly and it takes too much time to generate samples), this is okay for our purposes because we desire mixtures biased toward uniformly sampling from each vertex so there is sufficient density for sampling algorithms to actually sample from the Gaussians. This randomly generates Gaussian mixtures. Our choice of using Gaussian mixtures was arbitrary and based primarily on convenience of sampling through methods other than Monte Carlo.
We define our discretized total variation metric by randomly sampling from the Gaussian mixture (which we do by randomly sampling from each Gaussian, then choosing a number of samples from each collection of samples proportional to the probability of the Gaussian relative to the rest of the mixture). We then generate a relative empirical pdf by discretizing the interval from -⃗2⃗0⃗ to 2⃗0⃗ into 0.1-unit squares, calculating the proportion of samples in each square. Our discretized total variation metric m_TV is calculated by taking the absolute difference between the relative empirical pdfs of the samples generated from each algorithm and the relative empirical pdf generated by sampling directly from Gaussians weighted by the relative empirical pdf of the Gaussians. Our comparison between the two algorithms is done by both looking at both the ratio and actual values of m_TV between the algorithms and the mixture samples over choice of dimension. We also compare this with the m_TV between the Gaussian mixtures resampled again in order to obtain a means of roughly evaluating how well our algorithm performs both relative to NUTS and relative to a true sampler.
§.§ Results & Conclusion
We compare the m_TV metric between NUTS and SpreadNUTS by plotting them against each other and samples resampled from the mixture as well as by plotting the log of the m_TV ratio between NUTS and SpreadNUTS as well as between each algorithm and samples resampled from the mixture. In the first plot, the lower the m_TV, the better. In the second plot, the close to 0 the score the better; specifically, the log of the ratio between the algorithm and resampled mixture should ideally be close to 0 because this indicates it performs as well as samples from the mixture. We then discuss trends we noticed and provide examples of plots to compare NUTS to SpreadNUTS visually.
The following is a plot of m_TV vs. dimension for NUTS, our algorithm, and samples from a Gaussian mixture all compared against samples from a Gaussian mixture. Note that we compare two distinct draws from a Gaussian mixture with each other when calculating the m_TV to estimate how much of the m_TV of the algorithms is due to randomness attributed to relatively small sample size (we sample 10000 points per mixture and discard the first 500 as burn-in). Alongside it is a comparison of ratios between NUTS m_TV and our algorithm's m_TV with the mixture m_TV vs. dimension to see how close to a random sample the two algorithms get to m_TV.
The following are plots of m_TV ratio with the mixture m_TV for varying values of k (the number of Gaussians in the mixture) after fixing dimension.
The above shows that for dimension 1, NUTS performs better than SpreadNUTS; however, for higher dimensions, SpreadNUTS gets closer and closer to Gaussian sampling, suggesting that it handles density islands better than NUTS.
We note some interesting idiosyncracies of SpreadNUTS: in spite of the fact that it tends to perform better than NUTS in higher dimensions, what might actually be going on is that when the distance between "islands" of density in a distribution is sufficiently small enough for classical NUTS to feasibly leap across islands, SpreadNUTS simply makes it more likely that we will actually leap across islands. However, when the distance between these islands is too large for classical NUTS to reasonably travel between islands, SpreadNUTS cannot increase a low probability of traversing these islands enough for it to happen often. Thus, we conclude that while SpreadNUTS may increase the probability of traversing relatively high-density portions of the distribution relative to classical NUTS, it only attempts to "smooth" sampling across parts of the sample space that classical NUTS explores — it cannot explore parts of the sample space that classical NUTS does not explore. We examine two examples that showcase this trend: a 2d Gaussian mixture consisting of two distributions (μ,I_2),(-μ, I_2) with equal weight on both. In the first figure, μ = ⟨2.5,2.5⟩; in the second figure μ = ⟨5,5⟩. We compare SpreadNUTS to NUTS and see that while SpreadNUTS increases the probability of traversing these islands relative to classical NUTS, SpreadNUTS does not traverse the islands when classical NUTS does not. Furthermore, looking at the above figures, we can see that on the whole, SpreadNUTS m_TV gets closer to Gaussian sampling as dimension increases while NUTS first increases at dimension 2, then decreases at dimension 3 but still with significantly greater m_TV than either Gaussian sampling or SpreadNUTS sampling. We note that the number of dimensions used was small (3) and the number of Gaussians in the mixture was from 1 to 4; furthermore, the number of samples was 9.5K for each sampling method. Some error may have been introduced in the relatively small number of samples. A bigger point of contention is that the number of dimensions was too small to make any concrete claims about the efficacy of NUTS vs. SpreadNUTS and the use of Gaussian mixtures as our sample distribution may have introduced some bias that helps SpreadNUTS sample better than NUTS. There is more testing to be done, but we tentatively conclude that SpreadNUTS alleviates to some degree the lack of sample space exploration present in NUTS.
unsrt
§ APPENDIX
We derive the gradient and log-likelihood of Gaussian mixture M ∼∑^Nπ_i(μ_i, Σ_i). The likelihood (for a single datapoint x) is
p_M(x|π,μ⃗,Σ⃗) = ∑_i=1^Nπ_i(x|μ_i,Σ_i)
and the log-likelihood is
lnp_M(x|π, μ⃗,Σ⃗) = ln(∑_i=1^Nπ_i(x|μ_i,Σ_i))
For a single Gaussian, this devolves to c -0.5 (μ- x)^TΣ^-1(μ-x) for extra constant c = -0.5ln(|Σ^-1|(2π)^k).
Then, the gradient of the log-likelihood w.r.t. μ⃗ is
∂ln(p_M(x|π, μ⃗, Σ⃗))/∂μ⃗ = 1/∑_iπ_i(x|μ_i,Σ_i)·∂ p(x|π,μ⃗,Σ⃗)/∂μ⃗
∂ p(x|π,μ⃗,Σ⃗)/∂μ⃗ = ∑_i∂π_i(x|μ_i,Σ_i)/∂μ_i
∂π_i(x|μ_i,Σ_i)/∂μ_i = ∂/∂μ_i(π_i√(|Σ^-1_i|(2π)^-k)exp(-1/2(μ_i-x)^TΣ^-1_i(μ_i-x))) = Σ^-1(x-μ_i)π_i(x|μ_i,Σ_i)
∂ln(p_M(x|π, μ⃗, Σ⃗))/∂μ⃗ = ∑_iΣ^-1(x-μ_i)π_i(x|μ_i,Σ_i)/∑_iπ_i(x|μ_i,Σ_i)
For a single Gaussian, this simplifies to Σ^-1(x-μ).
As an aside, our testing regime experiences compounding rounding errors when exponentiating and taking logs, specifically when we take the log of the exponential of a number close to 0, which rounds to 0. We attempt to alleviate this problem by expressing the proportions of the normal likelihoods π_i(x|μ_i,Σ_i) to the sum of the normal likelihoods as the exponential of the difference of the log likelihood and the log of the sum of likelihoods, where we calculate the log of the sum of likelihoods by summing logs as below:
log(x+y) = log(x(1+yx)) = logx + log(1+yx) = logx + log(1+e^logy-logx)
log∑_ix_i = log(x_1(1+1/x_1∑_i=2^kx_i)) = logx_1 + log(1+e^log∑_i>1x_i-logx_1)
log∑_i>1x_i = logx_2 + log(1+e^log∑_i>2x_i-logx_2)
x_i/∑x_i = exp(logx_i-log∑x_i)
Thus, we can recursively express the log of sums as the sum of log sums (in practice, we sort the Gaussian pdfs when evaluating logs to minimize error at each step, yielding a technique known as LogSumExp or LSE). This helps decrease error accumulated when summing likelihoods because of the error introduced when summing exponentials.
|
http://arxiv.org/abs/2307.05590v1 | 20230710174756 | Improved Efficiency and Accuracy of the Magnetic Polarizability Tensor Spectral Signature Object Characterisation for Metal Detection | [
"James Elgy",
"Paul D. Ledger"
] | math.NA | [
"math.NA",
"cs.NA",
"65N30, 35R30, 35B30"
] |
Improved Efficiency and Accuracy of the Magnetic Polarizability Tensor Spectral Signature Object Characterisation for Metal Detection
J. Elgy and P.D. Ledger
School of Computer Science and Mathematics, Keele University
Keele, Staffordshire U.K
corresponding author: [email protected]
=============================================================================================================================================================
§ ABSTRACT
Magnetic polarizability tensors (MPTs) provide an economical characterisation of conducting metallic objects and can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict, and security screening at event venues and transport hubs. Previous work has established explicit formulae for their coefficients and a rigorous mathematical theory for the characterisation they provide.
In order to assist with efficient computation of MPT spectral signatures of different objects to enable the construction of large dictionaries of characterisations for classification approaches, this work proposes a new, highly-efficient, strategy for predicting MPT coefficients.
This is achieved by solving an eddy current type problem using hp–finite elements in combination with a proper orthogonal decomposition reduced order modelling (ROM) methodology and offers considerable computational savings over our previous approach.
Furthermore, an adaptive approach is described for generating new frequency snapshots to further improve the accuracy of the ROM. To improve the resolution of highly conducting and magnetic objects, a recipe is proposed to choose the number and thicknesses of prismatic boundary layers for accurate resolution of thin skin depths in such problems. The paper includes a series of challenging examples to demonstrate the success of the proposed methodologies.
keywords: Metal detection, magnetic polarizability tensor, eddy current problems, reduced order models, thin skin depth effects.
Conflict of Interest Statement
The authors have no conflicts of interests to declare.
Data Availability Statement
The authors' software used throughout this paper is publicly available at <https://github.com/MPT-Calculator/MPT-Calculator/> and datasets will be made public upon acceptance of the manuscript.
Funding Information
The authors are grateful for the financial support received from the Engineering and Physical Science Research Council (EPSRC, U.K.) through the research grant EP/V009028/1.
Practitioner Points
The main contributions of this paper are as follows:
* We have developed a new more efficient framework for computing the coefficients of the Magnetic Polarizability Tensor from reduced order model predictions using Proper Orthogonal Decomposition (POD). We demonstrate, with practical real world numerical examples, that this has lead to extremely significant time savings over our previous approach.
* We have developed a simple recipe for designing boundary layer discretisations for problems with thin electromagnetic skin depths.
* We demonstrate a Greedy algorithm for adaptively choosing new POD snapshot parameters and the performance benefit compared to a non–adaptive strategy.
§ INTRODUCTION
Metal detection uses low frequency magnetic induction to locate and identify highly conducting magnetic objects within the imaging region. Traditional metal detectors rely on simple thresholding and measured field perturbations are mapped to a simple audible tone to aid with detection. For this reason, hobbyist metal detectors wear headphones, listen for changes in pitch and volume, and have become trained in recognising the different signals that a detector receives for targets made of different materials (such as coins and other buried treasure) buried at different depths.
While risks are relatively low for the hobbyist metal detector, small objects close to the surface may give rise to similar signals to that of larger objects buried at depth and false positives are common. there are other safety critical applications of metal detection technology, such as in the location and identification of hidden unexploded ordnance (UXO) in areas of former conflict in order to allow the ground to be returned safely to civilian use, where minimising the number of false positives and false negatives is vital. In addition, at transport hubs, public events, and increasingly in some schools, the early identification of potential threat objects may offer significant improvements to ensuring the safety of travellers, participants and attendees, respectively. However, traditional walk-through metal detection methods, where people are expected to remove all metallic objects and are screened individually, can be slow and lead to long queues. In all these applications, improving the accuracy of metal detection technology has the potential to bring significant societal benefits. This paper contributes to this vision by providing an improved computational tool that can aid with accurately and efficiently characterising highly conducting and magnetic objects.
The signals measured by metal detectors contain considerably more information about the size, shape, material and location of hidden targets than the simple audible alarm may suggest. These signals are directly related to the perturbed magnetic field due to the presence of a highly conducting magnetic object.
In the case of a highly conducting spherical magnetic object, an analytical solution is available for the perturbation in magnetic field
(H⃗_α- H⃗_0)(x⃗) at positions x⃗ away from the object for low-frequency time harmonic fields in the eddy current regime. This includes results for where H⃗_0 is a uniform background field <cit.> and also where H⃗_0 is the background field due a dipole source <cit.>.
Related analytical and semi-analytical solutions have been obtained for highly conducting and magnetic spheroids <cit.> and ellipsoids <cit.>. More generally, approximate dipole models suggest that field perturbation due to the presence of a highly conducting magnetic object has a magnetic dipole moment that can be expressed in terms of a complex symmetric magnetic polarizbility tensor (MPT) and a background magnetic field at the position of the object <cit.> and MPTs have been approximately computed for objects with homogeneous conductivity and permeability by a variety of different schemes eg. <cit.>.
However, the dipole model only provides an approximation to (H⃗_α- H⃗_0)(x⃗). Its accuracy can be assessed by comparing it to a rigorously described asymptotic expansion of the perturbed magnetic field due to the presence of a highly conducting magnetic object B_α <cit.>
(H⃗_α - H⃗_0)(x⃗)_i = (D⃗^2 G(x⃗,z⃗))_ij (ℳ)_jk (H⃗_0 (z⃗))_k + (R⃗(x⃗))_i ,
which holds for x⃗ away from B_α, where | R⃗(x⃗)| ≤ C α^4 H⃗_0 _W^2,∞(B_α) is a residual with C being a constant independent of α, G(x⃗,z⃗):= 1/ ( 4 π | x⃗-z⃗|) is the free space Green's Laplace function, ℳ = (ℳ)_ije⃗_i ⊗e⃗_j is the complex symmetric MPT, which is shown to be independent of position <cit.>, e⃗_i is the ith orthonormal unit vector and Einstein summation convention has been applied. In the above, the object is described B_α = α B+z⃗, which means it can be thought of as a unit sized object B with the same shape as B_α, but placed at the origin, scaled by α and then translated by z⃗. This comparison was performed in <cit.> for the situation where H⃗_0 is a uniform background field and also when H⃗_0 is the background field due to a dipole source.
The advantages of the asymptotic expansion include that it provides a measure of accuracy of the approximation and has explicit expressions for computing the MPT coefficients <cit.>. Furthermore, these explicit expressions hold for the MPT characterisation of inhomogeneous objects <cit.> and multiple objects <cit.>
where
the electrical conductivity 0 ≪σ_*< ∞ and magnetic permeability 0 < μ_* < ∞ in the object are isotropic and independent of frequency ω, but not necessarily homogeneous. In general, the conductivity and magnetic permeability are described in the object and the surrounding region B_α^c= ℝ^3 ∖B_α by
σ_α = {[ σ_* in B_α; 0 in B_α^c = ℝ^3 ∖B_α ] ., μ_α = {[ μ_* in B_α; μ_0 in B_α^c ] . ,
and μ_0=4π× 10^-7 H/m the permeability of free space. It is also convenient to define μ_r: = μ_* /μ_0 as the (position dependent) relative magnetic permeability.
There are considerable benefits to exploiting the MPT's spectral signature <cit.>
(the variation of the MPT coefficients as a function of exciting frequency) compared to characterising an object by an MPT at a fixed frequency, which only characterises the object's shape and materials up to the best fitting ellipsoid. Improved object characterisations can also be obtained by using high order generalised magnetic polarizability tensors (GMPT) <cit.>, which provide additional information about the object. Both the MPT <cit.> and the GMPT <cit.> spectral signatures of different objects (including MPT characterisations of objects with inhomogeneous materials) have been experimentally verified using laboratory measurements and exhibit excellent agreement with theory.
A reduced order methodology using proper orthogonal decomposition (POD) has been developed for computing MPT spectral signatures <cit.> and this has been applied to computing dictionaries of objects <cit.> that have in turn been used for training machine learning classifiers to identify hidden objects <cit.>.
Through the addition of prismatic boundary layers, the MPT characterisation has been enhanced to consider highly magnetic objects <cit.>.
Practical applications of MPTs and related technology includes at transport hubs <cit.>, location and identification of landmines and UXOs <cit.>, food safety <cit.> and scrap sorting <cit.>.
While a reduced order modelling approach had been developed for computing MPT spectral signatures, our previous approach still requires large computational demands for complex realistic geometries, which impede its application to the characterisation of complex inhomogeneous realistic targets. Furthermore, it was not clear how best to choose the number or thickness of prismatic boundary layers in order to achieve accurate results nor was clear how best to choose the number of snapshots in order to achieve a reduced order model that is in close agreement to the underlying full order model. This work addresses these important shortcomings through the following novelties:
* A new computational formulation for efficiently computing MPT coefficients from POD predictions leading to very significant computational savings compared to our previous approach.
* An adaptive algorithm for choosing new snapshot solutions for improving the POD reduced order model.
* A new efficient strategy for designing boundary layer discretisations for capturing problems with thin skin depths.
* Application of our latest developments to a range of challenging realistic examples.
The paper is organised as follows: We begin with some brief comments on notation in Section <ref>. In Section <ref> we review the explicit formulae for computing the MPT coefficients, which includes, in Section <ref>, new discrete formulae for computing the MPT coefficients in terms of finite element matrices. Then, in Section <ref>, we briefly review the off-line and on-line stages of the POD projected (PODP) approach and explain how the calculation of the MPT coefficients can be considerably accelerated by combining the approach described in Section <ref> with a PODP reduced order description. In Section <ref>, we also outline an adaptive algorithm for computation of new snapshot frequencies to improve the accuracy of the POD approach. In Section <ref> we describe the computational resources used for the computational experiments conducted in this work and provide details of the specific versions of libraries used for simulations and where the open-access software, which accompanies this work, can be accessed from. Then, in Section <ref>, we describe a recipe for choosing the number and thicknesses of boundary layer discretisations in order to resolve thin skin depths associated with highly conducting and highly magnetic objects. Section <ref> presents a series of challenging examples to demonstrate the success of the proposed methodologies and the paper closes with some concluding remarks in Section <ref>.
§ NOTATION
We use calligraphic symbols e.g. ℳ for rank 2 tensors and denote their coefficients by (ℳ)_ij. By e⃗_i we denote the ith orthonormal basis vector and use bold face italics e.g. ξ for vector fields. We denote the components of vector fields by (ξ)_i = e_i ·ξ, which should be distinguished from e.g. θ_i^(0), which refers to the ith θ^(0) vector field. We use bold face upper case Roman font for linear algebra matrices e.g. 𝐀 and bold face lower case for linear algebra vectors 𝐛 and denote their coefficients by (𝐀)_ij and (𝐛)_i, respectively.
§ MAGNETIC POLARIZABILITY TENSOR OBJECT CHARACTERISATION
Recall that the complex symmetric MPT ℳ =(ℳ)_ije⃗_i ⊗e⃗_j has 6 complex coefficients (ℳ)_ij and admits the additive decomposition (ℳ)_ij:=(ℛ̃)_ij+( ℐ)_ij=(𝒩^0)_ij+(ℛ)_ij+( ℐ)_ij <cit.> where
(𝒩^0[ α B,μ_r] )_ij :=α^3δ_ij∫_B(1-μ̃_r^-1)ξ+α^3/4∫_B∪ B^cμ̃_r^-1∇×θ̃_i^(0)·∇×θ̃_j^(0)ξ,
(ℛ[α B, ω,σ_*,μ_r])_ij :=-α^3/4∫_B∪ B^cμ̃_r^-1∇×θ_i^(1)·∇×θ_j^(1)ξ,
(ℐ[α B, ω,σ_*,μ_r])_ij :=α^3/4∫_Bν(θ_i^(1)+(θ̃_i^(0)+e_i×ξ))·(θ_j^(1)+(θ̃_j^(0)+e_j×ξ))ξ,
are each the coefficients of real symmetric rank 2 tensors and where μ̃_r (ξ⃗) = μ_r(ξ⃗) for ξ⃗∈ B and μ_r (ξ⃗)=1 for ξ⃗∈ B^c:= ℝ^3 ∖B has been introduced.
In the above, δ_ij is the Kronecker delta, ν (ξ⃗) := α^2 ωμ_0 σ_*(ξ⃗), the overbar denotes the complex conjugate and θ⃗_i^(1)∈ℂ^3 is the solution of the vectorial transmission problem
∇×μ_r^-1∇×θ⃗_i^(1) -νθ⃗^(1) = νθ⃗_i^(0) in B,
∇×∇×θ⃗_i^(1) = 0⃗ in B^c ,
∇·θ⃗_i^(1) = 0 in B^c,
[n⃗×θ⃗_i^(1)]_Γ =0⃗, [n⃗×μ̃_r^-1∇×θ⃗_i^(1)]_Γ =0⃗ on Γ:=∂ B,
θ⃗_i^(1) = O ( |ξ⃗|^-1 ) as | ξ⃗ | →∞,
where ξ⃗ is measured from the origin, which lies inside B, [· ]_Γ denotes the jump over Γ and n⃗ is the unit outward normal. Note that θ⃗_i^(0) = θ̃⃗̃_i^(0) + e⃗_i ×ξ⃗∈ℝ^3 satisfies a simpler vectorial transmission problem that is independent of ν, but still dependent on μ_r <cit.>. The problem (<ref>) is set on an unbounded domain and, for the purposes of approximate computation, it is replaced by a problem on a bounded domain Ω with truncation in the form of a convex outer boundary ∂Ω placed sufficiently far from the object of interest B and n⃗×θ⃗_i=0⃗ applied on ∂Ω as an approximation to (<ref>e) leading to
∇×μ_r^-1∇×θ⃗_i^(1) -νθ⃗^(1) = νθ⃗_i^(0) in B,
∇×∇×θ⃗_i^(1) = 0⃗ in Ω∖B ,
∇·θ⃗_i^(1) = 0 in Ω∖B,
[n⃗×θ⃗_i^(1)]_Γ =0⃗, [n⃗×μ̃_r^-1∇×θ⃗_i^(1)]_Γ =0⃗ on Γ,
n⃗×θ⃗_i^(1) = 0⃗ on ∂Ω.
§.§ Finite Element Discretisation
As discussed in <cit.>, we employ a high order H⃗(curl) conforming finite element method (FEM) approximation on unstructured tetrahedral meshes of variable size h with elements of uniform order p of the form
θ_i^(1,hp)(ξ,ω) =∑_k=1^N_dN^(k)(ξ)(𝐪_i(ω))_k,
where N^(k) is a typical H(curl) conforming basis function and N_d are the number of degrees of freedom. The FEM approximation of (<ref>) for the ith direction then corresponds to the solution of the linear system of equations
A(ω)q_i(ω)=r(θ_i^(0,hp) , ω),
for the solution coefficients q_i(ω)∈ℂ^N_d in which the Coulomb gauge (<ref>c) is replaced by regularisation <cit.> and ω denotes the parameters of interest. In the above, A(ω)= K(μ_r) - ω C(α^2 μ_0 σ_*) +ε M∈ℂ^N_d× N_d is a large parameter dependent complex symmetric sparse matrix, where K is a curl-curl stiffness contribution, C a damping contribution associated with the conducting region B and M a mass contribution with small regularisation parameter ε to circumvent the Coulomb gauge in Ω∖B. In addition, r(θ_i^(0,hp), ω)∈ℂ^N_d is a known source term <cit.>[Equation (17)]. Compared to <cit.>, we additionally allow for the possibility to include prismatic layers to model thin skin depths, which we discuss further in Section <ref>. Following <cit.>, and noting the convention that p=0 elements have constant tangential components on edges, but consist of vector valued linear basis functions, we ensure that integration of element integrals is approximated by Gaussian quadrature of sufficient order so that it can integrate degree 2(p+1) polynomials exactly, independently of the geometry order c used to represent curved boundary and transmission faces. The linear system (<ref>) is solved to a relative tolerance TOL using a conjugate gradient solver and a balancing domain decomposition by constraints (BDDC) preconditioner <cit.>. The MPT coefficients follow by post-processing where attention must also be paid to using a Gaussian quadrature scheme of sufficient order (especially on elements where the curved boundary differs significantly from a linear approximation in addition to taking in to account the order of elements p).
§.§ Improved Efficiency for the Calculation of the MPT Coefficients
We focus on the improved efficiency for computing (ℛ[α B, ω,σ_*,μ_r])_ij and (ℐ[α B, ω,σ_*,μ_r])_ij that are functions of problem parameters ω. In the following, we focus on ω= ω although similar efficiencies could also be applied to other parameters of interest. First, for (ℛ[α B, ω,σ_*,μ_r])_ij, using (<ref>) and (<ref>), we have
(ℛ[α B, ω,σ_*,μ_r])_ij = - α^3/4∑_k=1^N_d∑_ℓ=1^N_d(𝐪_i)_k ∫_Ωμ̃_r^-1∇×N⃗^(k)·∇×N⃗^(ℓ)ξ⃗(𝐪_j)_ℓ
= - α^3/4𝐪_i^T 𝐊𝐪_j ,
where 𝐊∈ℝ^N_d × N_d and
(𝐊)_kℓ:=∫_Ωμ̃_r^-1∇×N⃗^(k)·∇×N⃗^(ℓ)ξ⃗,
is independent of ω.
Next, for (ℐ[α B, ω,σ_*,μ_r])_ij, we have
(ℐ[α B, ω,σ_*,μ_r])_ij = α^3/4 ( ∫_B νθ⃗_j^(1)·θ⃗_i ^(1)ξ⃗ + ∫_B νθ̃⃗̃_j^(0)·θ̃⃗̃_i^(0)ξ⃗ + ∫_B νe⃗_i ×ξ⃗·e⃗_j ×ξ⃗ξ⃗ + ∫_B νθ⃗_j^(1)·θ̃⃗̃_i^(0)ξ⃗ .
.
+ ∫_B νθ⃗_j^(1)·e⃗_ i ×ξ⃗ξ⃗ + ∫_B νθ̃⃗̃_j^(0)·θ⃗_i^(1)ξ⃗ + ∫_B νθ̃⃗̃_j^(0)·e⃗_i ×ξ⃗ξ⃗ .
.
+ ∫_B νe⃗_j ×ξ⃗·θ⃗_i^(1)ξ⃗ + ∫_B νe⃗_j ×ξ⃗·θ̃⃗̃_i^(0)ξ⃗ ),
and, since we know (ℐ[α B, ω,σ_*,μ_r])_ij = (ℐ[α B, ω,σ_*,μ_r])_ji∈ℝ, then
(ℐ[α B, ω,σ_*,μ_r])_ij = α^3/4 ( ∫_B νθ̃⃗̃_i^(0)·θ̃⃗̃_j^(0)ξ⃗ + ∫_B νe⃗_i ×ξ⃗·e⃗_j ×ξ⃗ξ⃗ + ∫_B νθ̃⃗̃_j^(0)·e⃗_i ×ξ⃗ξ⃗ .
+∫_B νθ̃⃗̃_i^(0)·e⃗_j ×ξ⃗ξ⃗+ Re (
∫_B νθ⃗_j^(1)·θ⃗_i ^(1)ξ⃗ + ∫_B νθ⃗_j^(1)·θ̃⃗̃_i^(0)ξ⃗ .
. . + ∫_B νθ⃗_i^(1)·θ̃⃗̃_j^(0)ξ⃗
+ ∫_B νθ⃗_j^(1)·e⃗_ i ×ξ⃗ξ⃗ + ∫_B νθ⃗_i^(1)·e⃗_ j ×ξ⃗ξ⃗ )
) .
Writing θ̃_i^(0,hp)(ξ,ω)=∑_k=1^M_dÑ^(k)(ξ)o_k,i, recalling ν (ξ⃗)= ωα^2 μ_0 σ_*(ξ⃗) = ων̃(ξ⃗), and following a similar procedure to above then
(ℐ[α B, ω,σ_*,μ_r])_ij = ωα^3/4 ( 𝐨_i^T 𝐂^(1)𝐨_j + c_ij + 𝐬_i^T 𝐨_j +
𝐬_j^T 𝐨_i +
Re (
𝐪_i^T 𝐂𝐪_j + 𝐨_i^T 𝐂^(2)𝐪_j+
𝐨_j^T 𝐂^(2)𝐪_i . .
. . +
𝐭_i^T 𝐪_j + 𝐭_j^T 𝐪_i
) ),
where
(𝐂 )_k ℓ := ∫_B ν̃N⃗^(k)·N⃗^(ℓ)ξ⃗,
(𝐂^(1))_k ℓ := ∫_B ν̃Ñ⃗̃^(k)·Ñ⃗̃^(ℓ)ξ⃗, (𝐂^(2))_k ℓ := ∫_B ν̃Ñ⃗̃^(k)·N⃗^(ℓ)ξ⃗,
(𝐬_i)_k := ∫_B ν̃e⃗_i ×ξ⃗·Ñ⃗̃^(k)ξ⃗, (𝐭_i)_k: = ∫_B ν̃e⃗_i ×ξ⃗·N⃗^(k)ξ⃗, c_ij := ∫_B ν̃e⃗_i ×ξ⃗·e⃗_j ×ξ⃗ξ⃗,
and 𝐂∈ℝ^N_d × N_d, 𝐂^(1)∈ℝ^M_d ×M_d, 𝐂^(2)∈ℝ^M_d ×N_d, 𝐬_i ∈ℝ^M_d, 𝐭_i ∈ℝ^N_d, c_ij∈ℝ are independent of frequency. Hence, we have reduced the computation of (ℛ[α B, ω,σ_*,μ_r])_ij and (ℐ[α B, ω,σ_*,μ_r])_ij to matrix vector products where the matrices can be precomputed and stored as they are all independent of ω.
The number of matrices that need to be formed when computing (<ref>) can be reduced from 6 to 3 by using the same basis functions for both the θ̃^(0)_i and θ^(1)_i problems. In this case N⃗^(k) = Ñ⃗̃^(k) and 𝐂 = 𝐂^(1) = 𝐂^(2)∈ℝ^N_d × N_d and t⃗_i = s⃗_i ∈ℝ^N_d. Using the same basis functions in both problems requires that basis functions that are gradient terms in the basis <cit.> remain in B_α for the θ̃^(0)_i problem and these must be removed via postprocessing <cit.>.
§ PODP APPROACH
We begin by briefly reviewing the off-line and on-line stages of the PODP approach and show the calculation of the MPT coefficients can be considerably accelerated by combining the approach <ref> with a PODP reduced order description.
Then, we consider how the error bounds derived in Lemma 1 of <cit.> can be used to drive an adaptive procedure for improving the spectral signature.
§.§ Off-line Stage
Following the solution of (<ref>) for 𝐪_i(ω) for different values of the set of parameters, ω, we construct matrices 𝐃_i∈ℂ^N_d× N, i=1,2,3, each with the vector of solution coefficients as its columns in the form
𝐃_i =[𝐪_i(ω_1),𝐪_i(ω_2),...,𝐪_i(ω_N)],
where N≪ N_d denotes the number of such snapshots. Application of a singular value decomposition (SVD) e.g. <cit.> gives
𝐃_i=𝐔_i Σ_i 𝐕_i^H,
where 𝐔_i∈ℂ^N_d× N_d and 𝐕_i ∈ℂ^N× N are unitary matrices and Σ_i∈ℝ^N_d× N is a diagonal matrix enlarged by zeros so that it becomes rectangular. In the above, 𝐕_i^H= 𝐕_i^T is the Hermitian of 𝐕_i.
The diagonal entries (Σ_i)_jj=s_j are the singular values of 𝐃_i and they are arranged as s_1>s_2>...>s_N, which decay rapidly towards zero, motivating the introduction of the truncated SVD (TSVD) e.g. <cit.>
𝐃_i≈𝐃_i^M = 𝐔_i^MΣ_i^M(𝐕_i^M)^H,
where 𝐔_i^M∈ℂ^N_d× M are the first M columns of 𝐔_i, Σ_i^M∈ℝ^M× M is a diagonal matrix containing the first M singular values and (𝐕_i^M)^H∈ℂ^M× N are the first M rows of 𝐕_i^H where M follows from retaining singular values s_1,…,s_M where s_M is the largest singular value such that s_M /s_1 < TOL_Σ.
§.§ On-line Stage
In the on-line stage of PODP, 𝐪_i^PODP ( ω) ≈𝐪_i(ω) is obtained as a linear combination of the columns of U_i^M where the coefficients of this projection are contained in the vector p_i^M ( ω) ∈ℂ^M so that
θ_i^(1,hp)(ξ,ω) ≈(θ_i^(1,hp))^PODP (ξ, ω) := N(ξ)
𝐪_i^PODP ( ω) =
N(ξ) 𝐔_i^Mp_i^M( ω) ∈ Y^(PODP) ,
where Y^(PODP)⊂ Y ∩ W^(hp) <cit.>. To obtain p_i^M ( ω), we solve the reduced linear system
𝐀_i^M(ω)𝐩_i^M( ω)=𝐫_i^M(θ_i^(0,hp), ω),
which is obtained by a Galerkin projection and is of size M× M where 𝐀_i^M(ω)=(𝐔_i^M)^H𝐀(ω)𝐔_i^M and 𝐫_i^M(θ^(0,hp) , ω)=(𝐔_i^M)^H𝐫 ( θ_i^(0,hp), ω). Note, since M<N ≪ N_d, the size of (<ref>) is significantly smaller than (<ref>) and, therefore, substantially computationally cheaper to solve. After solving this reduced system, and obtaining 𝐩_i^M(ω), we obtain an approximate solution for θ_i^(1,hp)(ξ,ω) using (<ref>). Moreover, the matrix 𝐀_i^M(ω) and right hand side 𝐫_i^M(θ^(0,hp) , ω) can be computed efficiently for new ω <cit.>.
§.§ Improved Efficiency for the PODP Prediction of MPT Coefficients
We outline how the efficiency of the PODP prediction of MPT coefficients can be significantly improved for the case where ω = ω and note that similar efficiencies can be gained when using PODP for other problem parameters. Since 𝐪_i^PODP ( ω) = 𝐔_i^Mp_i^M( ω), we can obtain the PODP prediction of the MPT coefficients as
(ℛ^PODP [α B, ω,σ_*,μ_r])_ij = - α^3/4𝐩_i^M^T 𝐊_ij^M 𝐩_j^M ,
where 𝐊_ij^M :=( 𝐔_i^M)^H 𝐊𝐔_j^M ∈ℂ^M × M can be precomputed once. Finally, for each new value of ω, the coefficients (ℛ^PODP [α B, ω,σ_*,μ_r ])_ij can be obtained from vector–matrix–vector products of small dimension M.
Similarly, we obtain
(ℐ^PODP [α B, ω,σ_*,μ_r])_ij = ωα^3/4 ( 𝐨_i^T 𝐂^(1)𝐨_j + c_ij +
𝐬_i^T 𝐨_j +
𝐬_j^T 𝐨_i+
Re (
𝐩_i^M^T 𝐂^M 𝐩_j^M + 𝐨_i^T 𝐂^(2),M𝐩_j^M+ . .
. .
𝐨_j^T 𝐂^(2),M 𝐩_i^M +
(𝐭_i^M)^T 𝐩_j^M + (𝐭_j^M)^T 𝐩_i^M ) ),
where 𝐂^M_ij : = ( 𝐔_i^M)^H 𝐂𝐔_j^M ∈ℂ^M × M , 𝐂^(2),M_j := 𝐂^(2)𝐔_j^M ∈ℂ^M_d × M, 𝐭_i^M =𝐭_i 𝐔_j^M ∈ℂ^M and further efficiencies are made by precomputing 𝐨_i^T 𝐂^(1)𝐨_j + c_ij +
𝐬_i^T 𝐨_j +
𝐬_j^T 𝐨_i and 𝐨_i^T 𝐂^(2),M.
Once 𝐊_ij^M, 𝐂^M_ij, 𝐭_i^M and 𝐨_i^T 𝐂^(1)𝐨_j + c_ij +2 𝐬_i^T 𝐨_j and 𝐨_i^T 𝐂^(3),M_j have been precomputed, the cost of computing (ℛ^PODP [α B, ω,σ_*,μ_r])_ij and (ℐ^PODP [α B, ω,σ_*,μ_r])_ij is at most that of computing several matrix vector products with 𝐩_i^M. The size of these matrices are independent of the mesh size and polynomial order used and are all either small square M × M matrices or vectors of length M and so the cost of evaluation is O(M^2) and the cost of solving (<ref>) is at most O(M^3). As M is small, this is considerably cheaper than the repeated solution of (<ref>) for new parameters, which is done iteratively and each iteration involves a matrix vector product requiring O(nz) operations where nz is the number of non-zeros of 𝐀. Furthermore, the aforementioned matrices and vectors needed for the PODP prediction can be computed once for all frequencies of interest. Further efficiencies can also be made by choosing N⃗^(k) = Ñ⃗̃^(k) as per Remark <ref>.
§.§ Adaptive Selection of Frequency Snapshots
The a–posteriori error estimate derived in <cit.> is restated below
An a-posteriori error estimate for the tensor coefficients computed using PODP is
| (ℛ[α B, ω,σ_*,μ_r ])_ij - (ℛ^PODP[α B, ω,σ_*,μ_r ])_ij |≤ (Δ[ω])_ij ,
| (ℐ[α B, ω,σ_*,μ_r ])_ij - (ℐ^PODP[α B, ω,σ_*,μ_r ])_ij | ≤ (Δ[ω])_ij,
where
(Δ[ω])_ij: = α^3/8α_LB ( r̂_i (ω) _Y^(hp)^2 + r̂_j (ω) _Y^(hp)^2 + r̂_i (ω) - r̂_j (ω) _Y^(hp)^2
) ,
and α_LB is a lower bound on a stability constant.
Note that the above error estimate also applies to both | (ℛ[α B, ω,σ_*,μ_r ])_ij - (ℛ^PODP[α B, ω,σ_*,μ_r ])_ij | and
| (ℛ̃[α B, ω,σ_*,μ_r ])_ij - (ℛ̃^PODP[α B, ω,σ_*,μ_r ])_ij | since (𝒩^0,PODP[α B,μ_r])_ij= (𝒩^0[α B,μ_r])_ij is computed once and independently of the POD.
To further improve the PODP technique, and overcome the potential issues of not choosing enough (or the best) snapshot frequencies, we use the adaptive procedure in Algorithm <ref>, following a Greedy approach where new snapshots are selected corresponding to the maximum error computed for that iteration <cit.>.
Here, 0<ϑ≤ 1 controls how many additional snapshots are generated in each adaption. Typically ϑ is chosen so that at most only 2-3 additional snapshots are included at each iteration.
§ COMPUTATIONAL RESOURCES AND SOFTWARE
The computational resources were used to perform the simulations in this paper correspond to workstations with the following specifications
* Workstation 1: Intel Core i5-10600 CPU with a clock speed of 3.30 GHz and 64GB of DDR4 RAM with a speed of 3200 MT/s
* Workstation 2: Intel Xeon W-2265 CPU with a clock speed of 3.50 GHz and 128GB of DDR4 RAM with a speed of 3200 MT/s
In the case of timings, all timings where performed using wall clock times with the package (version 0.61.0) and version 3.10. Finite element simulations were performed using and <cit.> versions 6.2204 and 6.2203, respectively, using 1.23.3. These libraries are called from the open source [ is publically available at <https://github.com/MPT-Calculator/MPT-Calculator/>] (April 2023 release) <cit.>.
In the off-line stage, two different forms of parallelism are applied. The assembly of the matrices and the underlying iterative solution of (<ref>), which requires repeated matrix-vector products in the conjugate gradient solver, is accelerated by using the shared memory parallelism across multiple computational threads as these operations are trivially parallelisable within . Importantly, this does not lead to further memory usage. Provided sufficient memory resources are available, the computation of the full order solutions for different ω is further accelerated by using multi-processing with different cases being considered simultaneously, which leads to higher memory demands.
In the on-line stage, the computation of the solution 𝐩_i^M to (<ref>) and the computation of
(ℛ^PODP [α B, ω,σ_*,μ_r])_ij and (ℐ^PODP [α B, ω,σ_*,μ_r])_ij using (<ref>) and (<ref>), respectively, which has already been reduced to small matrix vector products, is further accelerated by multi-processing in .
§ RECIPE FOR NUMBER AND THICKNESSES OF BOUNDARY LAYERS
The depth at which the amplitude of the electromagnetic field decays to 1/e of its surface value is known as the skin-depth and, for a homogeneous isotropic conductor, is commonly approximated by <cit.>
δ(ω, σ_*, μ_r) ≈√(2/ωσ_* μ_0μ_r),
which, for high σ_*, ω and μ_r, can become very small compared to the object dimensions.
Combining prismatic boundary layer elements with unstructured tetrahedral meshes with p–refinement of the elements has previously been shown to offer advantages over purely tetrahedral meshes for capturing the high field gradients associated with thin skin depths for θ⃗_i^(1) in the case of high σ_*, ω and μ_r <cit.>. Similar performance benefits have also been reported in other applications, such as the Maxwell eigenvalue problem <cit.> and singularly perturbed elliptic boundary value problems <cit.>. Here, the prismatic layers allow h–refinement to be achieved in a direction normal to the surface of the conductor, while leaving the tangential element spacing unchanged, which is ideal for addressing the high field gradients in the normal direction, but without resulting in a large increase in the number of degrees of freedom. On the other hand, using p–refinement alone on a traditional unstructured meshes is sub–optimal and converges only at an algebraic rate due to the small skin depths while attempting to do h–refinement of the unstructured tetrahedral mesh leads to an excessive N_d. For the PODP approach, the same FEM discretisation is needed for all ω snapshots and, in order to ensure that this is be accurate for the complete signature,
we fix a maximum target ω of interest and, for a given μ_r, we set τ:=√(2/ (μ_rν))= δ/α to be the smallest non-dimensional skin-depth
that is to be resolved.
Our previous work did not have a recipe for choosing the number of layers or indeed their thicknesses, which is essential for their practical application with a new geometry and/or new material parameters. Instead, the thickness of the layers were often chosen to model thin coatings (such as in some denominations of British coins). While offering considerable benefits, the inclusion of prismatic layers must nonetheless also be weighed up against the increase in computational resources (including both run time and memory usage). Our goal in this section is to determine a simple recipe for choosing the number and thicknesses of boundary layers that can be used to achieve a high level of accuracy at a reasonable computational cost.
To this end, we begin by considering a conducting homogeneous sphere of radius α=1× 10^-3 m and set σ_* = 1× 10^6 S/m while considering different cases of μ_r=1,16,64. For the approximate solution of θ⃗_i^(1) and, hence the MPT coefficients, we construct a computational domain Ω consisting of a unit radius sphere B placed centrally in a [-1000, 1000]^3 box and generate an unstructured mesh of 21 151 tetrahedra and represent the surface of the sphere by curved elements of degree c=5, which we use throughout.
The mesh, is augmented by the additional of prismatic layers placed just inside ∂ B. Three schemes for defining the structure of L prismatic layers are considered:
* “Uniform” refinement, where the total thickness of the layers is equal to τ and each layer of prismatic elements has thickness t_ℓ=τ/L, ℓ=1,…,L with t_1 being the closest to the conductor's surface and the layers numbered consecutively towards the inside of the conductor.
* “Geometric decreasing” refinement, where the total thickness is limited to τ and the thickness of each layer is defined by the geometric series t_ℓ+1 = 2t_ℓ with ∑_l=1^L t_ℓ = t_1 ( 1-2^L)/(1-2)= τ.
* “Geometric increasing” refinement, where the thickness of the layers are defined as t_ℓ+1 = 2t_ℓ with t_1 = τ. Recall that for highly magnetic objects, τ is small so for small L, the prismatic boundary layer elements are still thin compared to the size of the object.
An illustration of each refinement strategy is shown in Figure <ref> showing the thicknesses for L=3 layers of elements in terms of the non-dimensional skin depth τ for the three strategies.
We note that in each of these schemes, the total number of prismatic elements remains constant and, in the case of L=1, the uniform, geometric decreasing and geometric increasing strategies all lead to identical discretisations. This means that for a given mesh, order of elements and number of layers the number of degrees of freedom remains the same for all three strategies.
Figure <ref> shows the relative error between the approximated MPT and the exact solution for the sphere, E = ‖ℳ^hp - ℳ‖_F / ‖ℳ‖_F under p–refinement for each of the different approaches. L=1,2,3,4 layers and μ_r= 1, 16, 64, in turn, for the fixed target frequency ω = 1× 10^8 rad/s.
Here, and unless otherwise stated for subsequent simulations, the regularisation was set as ε = 1× 10^-10 and the iterative solver relative tolerance was set at TOL = 1× 10^-8.
For each value of μ_r, the convergence of E with respect to number of degrees of freedom, N_d, and E with respect to computational time (showing overall time using 2 cores on the workstation 1 described in Section <ref>) are shown.
In each case, the points on each curve correspond to different polynomial orders.
In the case of μ_r=1, and the chosen α, σ_* and ω and chosen initial mesh, the resulting skin depth can be resolved well by all schemes. With the exception of L=3 and the geometric increasing strategy, all schemes lead to similar convergence curves in terms of both N_d and time. A small benefit is observed for L=2 using the geometric increasing scheme over the other schemes. Indeed, for this case, using the initial tetrahedral mesh alone is already able to achieve exponential convergence when p–refinement is applied.
For μ_r>1 the uniform and geometric decreasing strategies are seen to produce similar results for all values of L while there is a considerable benefit in accuracy by using the geometric increasing scheme with L≥ 2 both with respect to N_d and computational time. As μ_r increases, further benefits in accuracy with respect to N_d and computational time are offered by using L≥ 3 and, by changing the scale, exponential convergence with respect to N_d^1/3, is obtained using sufficiently large L, which is the expected behaviour for this smooth problem. However, while combining p–refinement with L≥ 3 achieves very high accuracy, for practical problems, a relative error of E=1× 10^-3, is sufficient given the accuracy to which MPT coefficients can be measured and the ability to which materials and practical geometries are known. This level of accuracy can already be achieved using L=2, the geometric increasing scheme and p–refinement and, therefore, in the later practical computations, this is what we will employ. These findings are also consistent with the theory of <cit.>, which would suggest a first layer of thickness O((p+1)τ) if their findings for their one-dimensional problem are extrapolated to our three-diemsional problem.
We have also tested the same strategy for the same conducting sphere, but instead with μ_r=100,200,400, 800 and also observed that the same strategy of L=2 layers with geometric increasing refinement leads to a relative error of E=1× 10^-3 for the conducting sphere and a target frequency of ω = 1× 10^8 rad/s using order p=3 elements. The strategy has also been applied to objects with larger α, which represent more challenging problems, and the scheme has also performed well.
§ NUMERICAL EXAMPLES
In this section, we consider a range of numerical examples to illustrate the improvements in accuracy and speedup for calculation of the MPT tensor coefficients using PODP, the use of adaption to choose new frequency snapshots and the geometric increasing recipe proposed for the construction of prismatic layers in Section <ref>.
§.§ Conducting Sphere
We consider the conducing sphere described in Section <ref> for the particular case where μ_r=32. The computational domain Ω is discretised by 21 151 unstructured tetrahedra and L=2 layers of prismatic elements following the geometric increasing strategy resulting in 1275 prisms. Using TOL_Σ = 1× 10^-6, a total of N=13 solutions at logarithmically spaced snapshot (SS) frequencies 1× 10^1≤ω≤1× 10^8 rad/s are computed using order p=3 elements (leading to M=11) and we compare the MPT coefficients obtained using our previous approach <cit.> with the results obtained using the new accelerated computation described in Section <ref>. In Figure <ref>, we show a comparison between the MPT coefficients obtained using the previous approach (called the Integral method (IM)) and the new accelerated approach (called the Matrix method (MM), see Section <ref>) where we observe excellent agreement for all frequencies. Similar agreement can be found for spheres using other values of μ_r.
Timings were performed using workstation 1, as described in Section <ref>, for IM and MM methods for the MPT coefficient computation in the POD scheme for the this problem. When accelerated with 2 multiprocessing cores, and the use of multi-threading, as previously described in Section <ref>, we observe that the MM computation time is reduced from 12 164 seconds to 18 seconds giving an overall time of just 629 seconds. The breakdown in computational time will be expanded further for a more challenging example in Section <ref>.
The adaptive procedure outlined in Algorithm <ref> is demonstrated for the same discretisation in Figure <ref>, which shows the spectral signature for (ℛ̃[α B, ω,σ_*,μ_r ])_ij, that we subsequently refer to as (ℛ̃)_ij, including the a-posteriori error certificates (ℛ̃±Δ)_ij obtained at different iterations where (Δ)_ij
reduces as new SS are adaptively chosen. In this example TOL_Δ = 1× 10^-3, leading to the 4 graphs shown. Note that due to object symmetries the MPT is multiple of identity for this case.
Importantly, while the effectivity indices of the error certificates are large, they are computed at negligible additional cost during the on–line stage and, as the figures show, provide an effective way to choose new SS to reduce (Δ)_ij.
Only the behaviour for (ℛ̃)_ij are shown here, since the error certificates are the same for both (ℛ̃)_ij and (ℐ)_ij. To illustrate the performance of the adaptive POD, compared to non-adaptive logarithmically spaced SS, Figure <ref> shows the maximum error, Λ, against N. The figure shows the significant improvement associated with the adaptive POD compared to the non-adaptive scheme. Nevertheless, using a logarithmic SS with N=13 still provides a very good starting point for an initial choice of frequencies from which adaption can then be performed.
§.§ Conducting and Magnetic Disks
In this section, we consider the MPT characterisation of thin conducting and magnetic disks with their circular face in the x_1-x_3 plane.
We consider a disk with physical dimensions radius r=10 × 10^-3 m and thickness h=1 ×10^-3 m and start with μ_r = 32 and σ_*=1× 10^6 S/m. The computational domain Ω consists of a dimensionless disk B of radius r/α, and thickness h/α with α =1× 10^-3 m placed centrally in the box [-1000,1000]^3. The geometric increasing methodology from Section <ref> is applied to construct boundary layers for different values of μ_r at a target frequency of ω = 1 × 10^8 rad/s. Then, N=13 solutions are computed at logarithmically spaced SS frequencies in the range 1× 10^1≤ω≤1× 10^8 rad/s using TOL_Σ=1 × 10^-6, resulting in M=11. This process results in a discretisation of 24 748 tetrahedra and 2995 prisms with p=3 giving converged solutions at the snapshots.
Due to the symmetries of the disk, which, in addition to mirror symmetries, is rotationally symmetric around e_2, the non-zero independent tensor coefficients associated with the object reduce to (ℳ)_11 = (ℳ)_33 and (ℳ)_22 <cit.>. For this reason, in Figures <ref> and <ref> we show only (ℛ̃)_11 = (ℛ̃)_33, (ℛ̃)_22, (ℛ̃)_12 = (ℛ̃)_23 = (ℛ̃)_13 = 0 and (ℐ)_11 = (ℐ)_33, (ℐ)_22, (ℐ)_12 = (ℐ)_23 = (ℐ)_13 = 0. The figures show excellent agreement between the IM and MM methodologies.
To illustrate the adaptive procedure described by Algorithm <ref>, we show the spectral signature for (ℛ̃)_ij including the a-posteriori error certificates (ℛ̃±Δ)_ij obtained at different iterations for same magnetic disk in Figure <ref>. Starting with the setup used in Figure <ref> and a stopping tolerance of TOL_Δ = 1× 10^-3, we show the first 4 iterations of the adaptive algorithm resulting in N= 15, 17, 19 non-logarithmically spaced snapshots, respectively, in the subsequent 3 iterations in a similar manner to the earlier sphere example. The convergence behaviour for certificates for (ℐ +Δ)_ij is very similar and, therefore, not shown.
To illustrate the performance of the adaptive POD, compared to non-adaptive logarithmically spaced SS, Figure <ref> shows the maximum error, Λ, against N, which, like the earlier sphere example, shows significant benefits of the adaptive scheme over logarithmically spaced SS.
Considering the limiting case of an infinitely thin conducting non-magnetic disk in the x_1-x_3 plane and the corresponding limiting case when the disk is magnetic, based on measurements observations for finitely thick disks <cit.> (which is rotated to a disk in the x_1-x_2 plane for our situation), it has been proposed that the form of the MPT (when its coefficients are displayed as a matrix) undergo the transition
ℳ (α B, ω, σ_*,1)= [ 0 0 0; 0 c 0; 0 0 0 ]→ℳ (α B, ω, σ_*,μ_r →∞ ) = [ m 0 0; 0 m 0; 0 0 0 ]
as μ_r becomes large.
We would like to investigate the form this transition takes numerically for a finitely thick disk and wish to
understand the impact of changing the μ_r value. Thus, we consider the same conducting permeable disk as in Figure <ref>, but now with μ_r =1, 8,16, 64 in turn. In each case, different thickness prismatic layers were constructed according to the recipe in Section <ref> for a target value of ω =1 × 10^8 rad/s and then the MPT coefficients obtained by using a PODP method for N=13 snapshot solutions at logarithmically spaced frequencies in the range 1× 10^1≤ω≤1× 10^8 rad/s and TOL_Σ=1× 10^-6. We show the independent non-zero diagonal coefficients of ℛ̃ and ℐ. Interestingly, despite the relative simplicity of the geometry and its homogeneous materials, we observe the presence of multiple local maxima in (ℐ)_ij for the cases of μ_r =16 and μ_r=64. This can be explained by the spectral theory of MPT spectral signatures <cit.> where, in this case, the first term in the expansions in Lemma 8.5 is no longer dominant and multiple terms play an important role.
§.§ Inhomogeneous Bomblet
One potential practical application is the MPT characterisation of unexploded ordnance, which may assist in allowing the land to be safely released for civilian use in areas of former conflict. A common type of ordnance is the near spherical bomblet (e.g. a BLU-26 submunition <cit.>). This commonly consists of a die-cast aluminium shell, an explosive payload and fuze, aerodynamic flutes used to induce a rotation in the bomblet as it falls as part of the arming process, multiple (typically hundreds) small steel fragmentation balls, and a steel clamp ring <cit.>. As an example, Figure <ref> shows photographs of a recovered bomblet shell, showing the metallic ring, remnants of the fragmentation balls that are cast in the shell, and flutes. See also Figure 1 in <cit.>.
In the following we consider idealised models of the bomblet, first without fragmentation balls and then with them included. Throughout the following we make assumptions based on a sample part a BLU-26 bomblet and our understanding from the limited information openly available.
§.§.§ No Fragmentation Balls
While the flutes (which appear to be made of the same material as the shell <cit.>) are geometrically interesting (having a mirror symmetry and a 4-fold rotation symmetry <cit.>), with the steel clamp ring orientated in the same plane as the mirror symmetry, which we take to be x_1-x_2, the symmetries of this object imply the MPT will be diagonal with (ℳ)_11 = (ℳ)_22 and (ℳ)_33 as independent non-zero coefficients. But, given that the flutes make up only a small fraction of the overall volume, and removing them does not change which coefficients are independent, they are omitted. In this section, we consider two simplified idealised models: The first has a 3.14 cm radius a solid spherical aluminium ball with a steel clamp ring attached around the outside; the second is the same as the first but has a hollow spherical region of radius 2.4 cm.
To compute the MPT spectral signature of the first model, a computational domain Ω consisting of a dimensionless object B made up of a sphere of radius 3.14 cm joined to a clamp ring, which is modelled with radius 3.44 cm and height 0.31 cm, is centrally placed in a [-100, 100]^3 non-conducting region. The physical object B_α is obtained from the non-dimensional object B using a scaling of α =0.01 m. The material parameters are chosen as μ_r = 1 and σ_*=3× 10^7 S/m for the aluminium sphere and μ_r = 200, σ_*=6× 10^6 S/m for the steel ring.
Based on these materials, the resulting τ = δ / α at a maximum target frequency of ω=1× 10^8 rad/s for both materials is much smaller than those previously considered, making this problem considerably more challenging.
For this reason, boundary layer elements are added to both the non-magnetic sphere, and the magnetic ring with thickness chosen according to τ for each material. In particular, the geometric increasing refinement strategy with L=2 layers is used resulting in a mesh with 29 141 unstructured tetrahedra, and 6355 prisms, which was found to be converged at SS frequencies using uniform p=4 elements, resulting in a problem with N_d ≈ 1.23 × 10^6.
Due to the complexity and size of this object, we have increased the regularisation to ε = 1× 10^-8 and the iterative solver tolerance to TOL=1× 10^-7. For the POD, N=13 logarithmically spaced frequency SS was employed using TOL_Σ=1× 10^-6 and the resulting MPT signature obtained by applying the adaptive Algorithm <ref> after 4 iterations, where M=19 and TOL_Δ=10^-3, are compared in Figure <ref>.
The figure shows that the initial logarithmically spaced snapshots are in good agreement with the additional adaptively introduced snapshots and that the addition of new full order solutions does not significantly change the approximate tensor coefficients obtained by the POD method. The MPT spectral signature of the hollow bomblet, which is also shown, was obtained in a similar way using a discretisation with 30 379 tetrahedra and 8115 prisms and the same settings as before. As shown in the figure, The solid bomblet provides a good approximation for the hollow example. This agreement improves with frequency due to the decreasing skin depth and reduced contribution from the cavity.
We show contours of the computed | Re(θ_2^(1,hp))| (normalised to [0,1] to ease comparison) that are obtained as part of the solution process at the fixed frequencies ω = 1 × 10^2, 1 × 10^4, and 1 × 10^6 rad/s in Figure <ref>. The figure shows the contours on a plane chosen to be perpendicular to e_3, highlighting the decay of the fields inside the object as skin depth decreases. We use the same mesh, as the earlier discussion, however, we have increased the order to p=5 in order ensure a fine resolution of the field, which was not necessary for the MPT spectral signature due to the averaging through volume integration. The results highlight the extremely small skin depths associated with the higher frequencies, and that the fields concentrate along the magnetic ring, although in the case of ω= 1× 10^2 rad/s the wavelength is too large to induce significant eddy currents in the ring.
§.§.§ With Fragmentation Balls
Next we consider the case where steel fragmentation balls are included in the hollow bomblet. We believe that they are cast in to the aluminium shell, although we are not certain of this. We also do not know the number of balls, their size, or their location inside the bomblet. For our model, we have assumed that they are 2mm in radius and are placed approximately equidistant from each other throughout a layer adjacent to the interior of the shell. The properties of the balls are also unknown, but they are likely to be made of steel. We assume that they have material parameters μ_r =200, σ_*=6 × 10^6 S/m as per the assumed steel for the clamp ring. Importantly, introducing these fragmentation balls breaks the mirror and rotational symmetries of the bomblet, and so the MPT now has 6 independent coefficients as a function of ω.
We consider models with 100, 200, and 400 fragmentation balls which, given their size relative to the bomblet diameter, cannot be easily resolved without introducing an extremely large number of additional elements in the mesh or using a special geometrical techniques such as a NURB enhanced finite elements <cit.>. Given the aforementioned assumptions and approximations, we instead use an L_2 projection to project the expected varying distribution of μ_r and σ_* on to piecewise constant materials for a mesh consisting of 33 410 tetrahedra and 9 063 prisms.
To show the effect of increasing the number of fragmentation balls, we show a comparison between the hollow bomblet used for Figure <ref>, the addition of a steel layer in the interior shell and alternatively the addition of a layer with 100, 200, and 400 fragmentation balls. The effect of increasing the number of fragmentation balls is shown in Figure <ref> where (ℛ̃)_11, (ℐ)_11, (ℛ̃)_12, and (ℐ)_12 are shown as a sample of the 6 independent coefficients. The figure shows an increase in magnitude from the hollow bomblet solution towards the layered bomblet solution for the on-diagonal entries and the generation of off-diagonal terms in the MPT when the fragmental balls are added due to the breaks in the previous symmetries for the hollow and layered bomblets. The inclusion of the balls has important effects on the MPT signature and these differences may be useful when undertaking object classification.
§.§.§ Timings
As an illustration of timings for the challenging bomblet example, we consider the solid bomblet from Section <ref> and note the computations savings reported here also carry over to the other bomblet models and other challenging examples in general. Timings were performed using workstation 2 described in Section <ref> with comparisons made for IM and MM for the MPT coefficient computation in the POD scheme. In Figure <ref>, we show the wall clock timing break down for the aforementioned setup, accelerated with the use of multi-threading as previously described in Section <ref> and 2 multiprocessing cores. The timing is broken down in to the off-line stage of generating the mesh, computing the solution coefficients for the θ_i^(0,hp), computing the solution coefficients for the θ_i^(1,hp) snapshots, and computing the ROM (in which the TSVDs are obtained), and an on-line stage consisting of solving the smaller linear systems (<ref>) and computing the tensor coefficients. Comparing the IM and MM approaches, we see a significant saving in the final stage of computing the MPT coefficients, which, for this particular discretisation, required approximately 6× 10^4 seconds using the IM. In comparison, the computation of the MPT coefficients for the MM becomes negligible and reduces the overall computation to around 2× 10^4 seconds. There is an additional memory overhead in the MM approach, which is associated with the building of the larger matrices
𝐊, 𝐂, 𝐂^(1) and 𝐂^(2) of dimension N_d × N_d if M_d =N_d, which can be incorporated to the off-line stage of the ROM if desired. The on-line stage of the ROM only requires the smaller dimension matrices of size at most M× M with M ≪ N_d and the larger matrices can be disposed off once 𝐔_i^M are available. Nevertheless, this additional memory overhead is still less than the peak memory requirements during the computation of the solution coefficients for the θ_i^(1,hp) snapshots and so it is not of concern.
§ CONCLUSIONS
In this paper, we developed a new computational formulation for efficiently computing MPT coefficients from POD predictions. This, in turn,
has led to significant computational savings associated with obtaining MPT spectral signature characterisations of complex and highly magnetic conducting objects including magnetic disks of varying magnetic permeability and an idealised bomblet geometry. We have included timings to demonstrate the improvement in performance.
The paper has proposed a significant enhancement to our previous POD scheme by incorporating adaptivity where we choose new snapshot frequencies based on an a–posteriori error estimate computed, which is obtained at negligible computational cost. This adaptive method has been shown to provide an efficient way of choosing new frequency snapshots that leads in smaller a–posteriori error estimates compared to using the same number of logarithmically spaced frequency snapshots. The adaptive scheme is particularly useful for further refining the number and location of frequency snapshots given an initial set of logarithmically spaced snapshots.
In addition, we have significantly extended our earlier work <cit.> and provided a simple recipe to determine the number and thicknesses of prismatic boundary layers
so as to achieve accurate solutions under p–refinement. By considering a magnetic sphere (for which an exact solution is known), we have shown that by choosing suitable thicknesses of 2 layers of prismatic elements and p–refinement was sufficient to achieve a relative error E<1× 10^-3 for the MPT over a wide range of skin depths associated with materials and frequency excitations.
We have also included challenging realistic numerical examples that show the importance of using prismatic layers and the increased efficiency and improved accuracy of our new MPT calculation formulation.
We expect the procedures presented in this work to be invaluable for constructing large dictionaries of MPT characterisations of complex in-homogeneous realistic metallic objects. We also expect that the presented formulations for the adaptive POD, boundary layer construction, and the efficient post-processing to be transferable to the computation of GMPT object characterisations.
§ ACKNOWLEDGEMENTS
The authors would like to thank Prof. Peyton, Prof. Lionheart and Dr Davidson for their helpful discussions and comments on polarizability tensors.
The authors are grateful
for the financial support received from the Engineering and Physical Science Research Council (EPSRC, U.K.) through the research grant EP/V009028/1.
Both authors would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for their support and hospitality during the programme Rich and Non-linear Tomography - a multidisciplinary approach where work on this paper was undertaken and related support from EPSRC grant EP/R014604/1.
acm
|
http://arxiv.org/abs/2307.05438v1 | 20230711170356 | Tensor gradiometry with a diamond magnetometer | [
"A. J. Newman",
"S. M. Graham",
"A. M. Edmonds",
"D. J. Twitchen",
"M. L. Markham",
"G. W. Morley"
] | cond-mat.mtrl-sci | [
"cond-mat.mtrl-sci",
"physics.ins-det",
"physics.optics",
"quant-ph"
] |
Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United Kingdom
EPSRC Centre for Doctoral Training in Diamond Science and Technology, University of Warwick, Coventry CV4 7AL, United Kingdom
Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot OX11 0QR Oxfordshire, United Kingdom
Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United Kingdom
EPSRC Centre for Doctoral Training in Diamond Science and Technology, University of Warwick, Coventry CV4 7AL, United Kingdom
Vector magnetometry provides more information than scalar measurements for magnetic surveys utilized in space, defense, medical, geological and industrial applications. These areas would benefit from a mobile vector magnetometer that can operate in extreme conditions. Here we present a scanning fiber-coupled nitrogen vacancy (NV) center vector magnetometer. Feedback control of the microwave excitation frequency is employed to improve dynamic range and maintain sensitivity during movement of the sensor head. Tracking of the excitation frequency shifts for all four orientations of the NV center allow us to image the vector magnetic field of a damaged steel plate. We calculate the magnetic tensor gradiometry images in real time, and they allow us to detect smaller damage than is possible with vector or scalar imaging.
Tensor gradiometry with a diamond magnetometer
G. W. Morley
August 12, 2023
==============================================
Tensor gradiometry calculates the gradient of all three components of a vector in all three directions <cit.>. At each point in space, magnetic tensor gradiometry measures
[ B_xx B_yx B_zx; B_xy B_yy B_zy; B_xz B_yz B_zz ]
=
[ ∂B_x/∂ x ∂B_x/∂ y ∂B_x/∂ z; ∂B_y/∂ x ∂B_y/∂ y ∂B_y/∂ z; ∂B_z/∂ x ∂B_z/∂ y ∂B_z/∂ z; ].
This technique, also called full tensor gradiometry, can provide benefits over vector measurements including improvements to spatial resolution <cit.>, directional filtering <cit.>, common mode noise rejection <cit.> and decoupling the contributions from homogeneous background fields <cit.>. Gradients due to local anomalies are larger than homogeneous background fields and therefore smaller features can be resolved where normally they would be dominated by the larger background.
Measuring the magnetic field components around an ensemble of NV centers in diamond can provide vector information instead of just the scalar projection along one of the four possible NV center axes in the diamond <cit.>. NV vector magnetometry takes advantage of using the magnetic sensitivity of all four possible orientations of the defect rather than just a single orientation, increasing the probe population by four. Vector measurements are important in applications such as magnetic navigation <cit.>, battery monitoring <cit.>, unexploded ordnance detection <cit.>, space missions, and geological surveys <cit.> and could be applied to areas such as archaeology, non-destructive testing (NDT) <cit.>, magneto-cardiography (MCG) <cit.>, magneto-encephalography (MEG) <cit.> and nuclear magnetic resonance spectroscopy <cit.>. Magnetic imaging of damage in steel is of great interest to industry <cit.>.
Magnetic sensing is possible with NV centers in diamond through optically detected magnetic resonance (ODMR). Magnetometry with NV centers provides high dynamic range and the ability to perform measurements in a wide temperature range, in chemically harsh environments and under high irradiation. Ensembles of NV centers provide high magnetic sensitivities, with small sensing volumes e.g. 1 mm^3 or less <cit.>. NV center magnetometers have shown a range of applications including sensing induced eddy currents <cit.>, single-neuron action potentials <cit.> and magnetic nanoparticle sensing in biomedical tissue <cit.>.
Vector magnetometry with NV centers uses excitation of the ground state spin level transition m_s = 0 → m_s = ± 1 for at least three of the four possible orientations of the NV centers in diamond. This enables vector magnetometry with NV centers using a single sensor head, unlike other magnetometers such as fluxgates which require multiple sensors. This means that low non-linearity and non-orthogonality are built in. A majority of high-sensitivity NV center magnetometers are tabletop setups which are immobile preventing them from being used for scanning over a sample of interest.
Our fiber-coupled NV magnetometer setup can take vector measurements while scanning through three-dimensional space, rather than being fixed at a single point in space, allowing magnetic tensor gradiometry images to be recorded. A small magnet injects magnetic flux into the sample to be imaged, and distortions of this flux by the sample are imaged by the diamond magnetometer <cit.>.
Here we present a scanning fiber-coupled NV magnetometer that is capable of both vector magnetometry and magnetic tensor gradiometry with a moving sensor head, independent of the main optoelectronics housed in a mobile equipment rack. Changes in the ODMR peak frequencies from magnetic field perturbations are tracked to calculate magnetic field vectors and gradients, while simultaneously moving the sensor head through the environment. As a demonstration of its capabilities, the magnetic vector measurements around a 316 stainless steel plate (Fig. <ref>) are made. The plate has holes drilled in at various positions with a range of sizes to simulate damage. We use no magnetic shielding.
The fiber coupled scanning setup is shown in Fig. <ref>. A Laser Quantum 532 nm GEM laser is used for excitation of the NV ensemble of the diamond, inside the sensor head. A laser power of 1 W is used with approximately 0.6 W of power measured at the diamond. The optoelectronics are kept on a mobile rack which allows for more portability. The fiber is a custom ordered FG910UEC 3 m fiber with a core diameter of 0.91 mm with steel ferrule-connector-physical-contact (FCPC) connectors on both ends. The fiber is directly coupled to the diamond on the sensor head via a SM1FC2 fiber adapter which is then screwed down onto the antenna substrate to make contact with the diamond. The fiber is secured to reduce modal noise from movement of the fiber during scanning of the sensor head. Fluorescence from the sensor head is sent back through the same fiber to be focused onto one of the two photodiodes on a Thorlabs PD450A balanced detector. The other photodiode takes a reference beam directly from the green laser which allows for cancellation of common mode noise, mainly laser noise. When the sensor head is on-resonance, the reference and fluorescence signals are balanced to get the best cancellation performance. The subtracted signal from the output of the balanced detector is then digitized and demodulated via a Zurich MFLI DC-500 kHz lock-in amplifier (LIA).
Microwaves are provided by an Agilent E8257D microwave source and amplified via a 43-dB Mini-Circuits ZHL-16W-43-S+. Microwaves are delivered to the diamond by a 2 m coaxial cable which connects to an SMA adapter soldered directly to a co-planar waveguide on an aluminum nitride ceramic printed circuit board. The waveguide then leads to a 3.0 mm diameter loop antenna with the diamond situated in the center. The aluminum nitride provides high thermal conductivity to act as a good heat sink, pulling heat from the diamond as it is heated by both the laser and microwaves. The diamond is a 1 mm cube. It is a low-strain diamond grown with chemical vapor deposition by Element Six, having (100) polished faces and 99.995% ^12C isotopic purification <cit.>.
A permanent Nd-Fe-B magnet is attached to the side of the sensor head using a custom 3D printed mount which allows 360 degree rotation around the azimuthal angle and 180 degree rotation in the polar angle. The magnet is orientated until the bias field projection differs for all four crystallographic directions of the NV center, to provide all eight separate resonance peaks in the ODMR spectrum, as seen in Fig. <ref>. Parameter optimization was performed to achieve the best sensitivity for vector magnetometry, following the process outlined in <cit.>. With a low-pass filter (LPF) with a 150 Hz 3-dB point, the optimum parameters for microwave power after the amplifier, modulation frequency and modulation amplitude are found to be 30 dBm, 3.0 kHz and 3.0 MHz, respectively. The microwaves are sine wave frequency modulated. To determine the sensitivity during optimisation, a linear fit is made to the fourth ODMR peak (<ref>) and then fast Fourier transforms (FFT) of thirty 1 s time traces were taken and averaged, when on the central frequency of that peak. The mean sensitivity is taken as the average value between 10 and 150 Hz, excluding the 50 Hz peak due to the mains electricity.
A 3D printer stage is used to scan the sensor head in three axes. The stage allows for control over the step size and speed of the scan as well as the lift-off distance in the z axis. The step size used for the x and y movement was 0.5 mm.
Changes to the external magnetic field are measured by looking at the change in peak resonance frequency due to Zeeman shift of the NV center m_s = - 1 spin level. The shift in resonance frequency causes the fluorescence intensity to change which results in a change in the LIA voltage output. When scanning over a magnetic object, like the steel plate, the change in external field is caused by the difference in the amount of material near the sensor head. The changes in voltage are used to provide a process variable for a proportional linear feedback system which adjusts the microwave frequency to continually follow the central frequencies of the four chosen peaks. Using the peak's central resonance frequency as a set-point, any changes to the continually monitored voltage while at that central frequency can be used to determine the shift in resonance frequency, by using the known gradient of the ODMR peak. The calculated shift in resonance is used as an error signal which is sent to the microwave source to change the microwave frequency until the monitored voltage returns back to its set-point value, at the center of the peak. This is then expanded to cover four ODMR peaks by sequentially looping over each one, approximately every 70 ms, to measure their individual shifts, as shown in Fig. <ref>. Tracking the four peaks can then be used to determine the magnetic field components along each of the four NV center axes which ultimately allows for calculation of the B_x, B_y, and B_z components of the field, relative to a coordinate frame defined around the sensor head. Another advantage to this resonance tracking technique is that it provides a much higher dynamic range of the magnetometer as the sensitive range is no longer limited to the linewidth, or linear region, of the ODMR peak.
The vector components of an applied field are reconstructed using four ODMR line centers which each represent the four possible orientations of the NV center. The first four ODMR line centers used represent the m_s = 0 to m_s = -1 transitions of the four orientations (Fig. <ref>). The method to calculate the vector components was taken from the work done by Schloss et al. <cit.>. The linearized NV center ground-state Hamiltonian allows magnetic field components to be determined from the frequency shifts of the four peaks. The result is
[ B_x; B_y; B_z; ]
= 𝐀^-1[ Δ f_1; Δ f_2; Δ f_3; Δ f_4; ]
,
𝐀 =
[ ∂ f_1/∂B_x ∂ f_1/∂B_y ∂ f_1/∂B_z; ∂ f_2/∂B_x ∂ f_2/∂B_y ∂ f_2/∂B_z; ∂ f_3/∂B_x ∂ f_3/∂B_y ∂ f_3/∂B_z; ∂ f_4/∂B_x ∂ f_4/∂B_y ∂ f_4/∂B_z ]
,
where Δ f_i is the shift in frequency of peak i = 1,2,3,4 and 𝐀^-1 is the psuedo-inverse of matrix 𝐀 where B_x, B_y and B_z are the magnetic field components in the frame defined around the sensor head.
To calculate matrix 𝐀, first the sensor head was placed inside a Helmholtz coil with a calibration of 78.2 μ T / A between 0.1 and 1 A. The calibration was performed with a Bartington Instruments Mag-03MS100 three-axis fluxgate magnetometer. The coil was aligned along the defined x axis relative to the sensor head and the peak shifts were tracked as the current was increased in steps of 0.1 A. This was repeated for the y and z axes. A linear fit for all four peaks for each axis orientation was made to determine the change in peak central frequency with the increase in applied field, as shown in Fig. <ref>. The matrix 𝐀, from the twelve fits, was found to be
𝐀 (MHz/uT) =
[ 0.12 0.04 -0.13; -0.02 -0.10 -0.09; 0.03 0.11 -0.19; -0.09 -0.03 -0.14 ]
.
With this calibration matrix it is then possible to track the peak shifts and calculate the vector components of the field which caused said shift. We do this in real-time.
A 160 mm x 160 mm scan was done across the 150 mm x 150 mm 316 stainless steel plate, with a step size of 0.5 mm for both the x and y directions and with a separation distance of approximately 1 mm between the surface of the steel sample and the base of the antenna. The antenna is 1 mm thick so the sensing diamond is 2 mm away from the sample. A total magnetic intensity (TMI) map is produced by using |B| = √(B_x^2 + B_y^2 + B_z^2).
Both the large and small holes are successfully imaged in all three vector components; an example of B_x is shown in Fig. <ref>. The features are also clearly visible in the TMI map, Fig. <ref>, where the change in field magnitude indicates a lack of steel. With the addition of vector magnetometry we are also able to plot the magnetic field vectors which can help with locating and analyzing defects, as shown in Fig. <ref>. The magnetic field preferentially flows through the higher permeability steel, as opposed to the air and this can be visualized with the field vectors as they move away and around the hole feature in the map to stay flowing through the steel. This can be useful for detecting defects not directly visible by giving information on whether the features seen in a magnetic field map are defects due to loss of material or are just regions of lower magnetic field intensity, where you would see the field vectors pass unperturbed.
Two horizontal lines of evenly spaced quadrupole shapes at the top and bottom of the plate indicate the locations of the large holes in the steel, with the grid of nine holes in the center. The quadrupole field shape coming from the holes (Fig. <ref> and Fig. <ref>) is found to be due to the orientation of the small bias magnet relative to the hole features. Rotating the magnet 45 degrees, between having its square face facing the diamond and having it edge-on to the diamond, produces different field component and TMI images, as seen in Fig. <ref>, where the B_x field component now has a more dipole-like shape. By performing COMSOL simulations, it was verified that the rotation of the small bias magnet was the cause of the change in field pattern seen. Simulations of a 316 stainless steel plate with a 6 mm hole and a 1 mm cube 0.1 T permanent magnet placed 1 mm above the hole matches well with what is measured when the magnet is not rotated (Fig. <ref>) and with a rotation of 45 degrees (Fig. <ref>).
Scanning vector magnetometry makes it possible to perform tensor gradiometry. We have three matrices each containing the B_x, B_y and B_z field components at different adjacent x, y and z positions so we can take the gradient of the field components with respect to the three Cartesian directions and generate nine tensor components for each point in space.
An example of this can be seen in Fig. <ref> where the B_xy tensor component from Eq. <ref> has been plotted and it is clear that some features, for example the three lowest holes in the smaller grid of nine 2 mm holes in the center of the plate, are shown more clearly when compared to the TMI map (Fig. <ref>) and B_x map (Fig. <ref>) where the three lowest of the smaller central holes are virtually impossible to resolve. The tensor components for the change in z direction were also calculated but did not provide any useful information, for more on this see the appendix. To compare this with the B_x map, a horizontal line profile on the B_xy map (Fig. <ref>) is taken across the small holes at the same y positions as on the B_x map.
In conclusion, magnetic tensor gradiometry was shown to improve the imaging of small holes in steel, making them easier to characterize.
The authors thank Jeanette Chattaway, Lance Fawcett and Matty Mills of the Warwick Physics mechanical workshop. We are grateful for insightful discussions with Rajesh Patel throughout this work. Alex Newman's Ph.D studentship is funded by an EPSRC iCASE award to NNL (the National Nuclear Laboratory). Stuart Graham's Ph.D studentship is funded by DSTL (the Defence Science and Technology Laboratory). This work is supported by the UK Networked Quantum Information Technologies (NQIT) Hub, and the UK Hub in Quantum Computing and Simulation, part of the UK National Quantum Technologies Programme, with funding from UKRI EPSRC grants EP/M013243/1 and No. EP/T001062/1 respectively. This work is also supported by Innovate UK grant 10003146, EPSRC grant EP/V056778/1 and an EPSRC Impact Acceleration Account (IAA) award. GWM is supported by the Royal Society.
55
fxundefined [1]
ifx#1
fnum [1]
#1firstoftwo
secondoftwo
fx [1]
#1firstoftwo
secondoftwo
noop [0]secondoftwo
ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]
rl [1]href #1
@bib@innerbibempty
[Schiffler et al.(2014)Schiffler, Queitsch, Stolz, Chwala, Krech, Meyer, and Kukowski]Schiffler_Tensor_Grad
author author M. Schiffler, author M. Queitsch,
author R. Stolz, author A. Chwala, author
W. Krech, author H.-G. Meyer, and author N. Kukowski, 10.1093/gji/ggu173
journal journal Geophys. J. Int. volume 198, pages 954 (year
2014)NoStop
[Luheshi(2021)]luheshi_2021
author author M. Luheshi, title Full tensor gradiometry, in 10.1017/9781108913256.005 booktitle
Integration of Geophysical Technologies in the Petroleum Industry, editor edited by editor H. Wilson, editor K. Nunn, and editor M. Luheshi (publisher Cambridge University
Press, year 2021) p. pages
124–168NoStop
[Kebede and Mammo(2021)]KEBEDE2021e06872
author author B. Kebede and author T. Mammo, https://doi.org/10.1016/j.heliyon.2021.e06872 journal journal Heliyon volume 7, pages e06872 (year 2021)NoStop
[Stolz et al.(2021)Stolz,
Schiffler, Zakosarenko, Larnier, Rudd, Chubak, Polomé, Pitts, Schneider, Schulz, Meyer, and Campbell]gradiometry_spatail_res_improvments
author author R. Stolz, author M. Schiffler,
author V. Zakosarenko, author H. Larnier, author
J. Rudd, author G. Chubak, author L. Polomé, author B. Pitts, author M. Schneider, author M. Schulz, author M. Meyer, and author K. Campbell, title Status and
future perspectives of airborne magnetic gradiometry, in 10.1190/segam2021-w6-05.1 booktitle First International
Meeting for Applied Geoscience; Energy Expanded Abstracts (year 2021) pp. pages 3554–3561NoStop
[Pedersen and Rasmussen(1990)]magnetic_tensor_gradiometetry_small_scale_fields
author author L. B. Pedersen and author T. M. Rasmussen, 10.1190/1.1442807 journal
journal Geophys volume 55, pages 1558 (year 1990)NoStop
[Foss(2006)]foss_improvments_to_source_resolution
author author C. Foss, 10.1190/1.2164761 journal journal Lead. Edge volume 25, pages
81 (year 2006)NoStop
[Chwala et al.(2012)Chwala,
Stolz, Zakosarenko, Fritzsch,
Schulz, Rompel, Polome,
Meyer, and G Meyer]directional_filtering_gradiometry
author author A. Chwala, author R. Stolz,
author V. Zakosarenko, author L. Fritzsch, author
M. Schulz, author A. Rompel, author L. Polome, author M. Meyer, and author H. G Meyer, 10.1071/ASEG2012ab296
journal journal ASEG volume 2012, pages 1 (year 2012)NoStop
[Wood et al.(2015)Wood,
Bennie, Duong, Jasperse,
Turner, and Anderson]magnetic_gradiometry_common_mode_noise_rejection_example
author author A. A. Wood, author L. M. Bennie,
author A. Duong, author M. Jasperse, author
L. D. Turner, and author
R. P. Anderson, 10.1103/PhysRevA.92.053604 journal journal Phys.
Rev. A volume 92, pages 053604
(year 2015)NoStop
[Keenan et al.(2022)Keenan,
Clark, and Leslie]magnetic_gradiometry_common_mode_noise_rejection_example_2
author author S. T. Keenan, author D. A. Clark, and author K. E. Leslie, 10.1088/1361-6668/ac5016 journal journal Supercond. Sci. Technol. volume 35, pages 045005 (year 2022)NoStop
[Schmidt and Clark(2006)]Schmidt2006TheCharacterization
author author P. W. Schmidt and author D. A. Clark, 10.1190/1.2164759 journal
journal Lead. Edge volume 25, pages 75 (year 2006)NoStop
[Schmidt et al.(2004)Schmidt, Clark, Leslie, Bick, Tilbrook, and Foley]Schmidt2004GETMAGExploration
author author P. Schmidt, author D. Clark,
author K. Leslie, author M. Bick, author
D. Tilbrook, and author
C. Foley, 10.1071/EG04297
journal journal Explor. Geophys. volume 35, pages 297 (year
2004)NoStop
[Murphy and Brewster(2007)]Murphy2007TargetData
author author C. A. Murphy and author J. Brewster, 10.1071/ASEG2007ab096 journal
journal ASEG Extended Abstracts volume
2007, pages 1 (year 2007)NoStop
[Schloss et al.(2018)Schloss, Barry, Turner, and Walsworth]Schloss2018SimultaneousSpins
author author J. M. Schloss, author J. F. Barry,
author M. J. Turner, and author R. L. Walsworth, 10.1103/PhysRevApplied.10.034044 journal journal Phys. Rev. Appl. volume 10, pages 34044 (year 2018)NoStop
[Luo et al.(2015)Luo,
Wu, Wang, Duan, Liu, Wang, and An]Luo2015FullApplication
author author Y. Luo, author M.-P. Wu,
author P. Wang, author
S.-L. Duan, author
H.-J. Liu, author J.-L. Wang, and author Z.-F. An, 10.1007/s11770-015-0508-y
journal journal Appl. Geophys. volume 12, pages 283 (year
2015)NoStop
[Kubota et al.(2023)Kubota,
Hatano, Kainuma, Shin,
Nishitani, Shinei, Taniguchi,
Teraji, Onoda, Ohshima,
Iwasaki, and Hatano]battery_monitoring
author author K. Kubota, author Y. Hatano,
author Y. Kainuma, author J. Shin, author
D. Nishitani, author
C. Shinei, author T. Taniguchi, author T. Teraji, author S. Onoda, author T. Ohshima, author T. Iwasaki, and author M. Hatano, https://doi.org/10.1016/j.diamond.2023.109853 journal
journal Diam. Relat. Mater. volume
135, pages 109853 (year 2023)NoStop
[Hatano et al.(2021)Hatano,
Shin, Nishitani, Iwatsuka,
Masuyama, Sugiyama, Ishii,
Onoda, Ohshima, Arai,
Iwasaki, and Hatano]battery_monitoring_2
author author Y. Hatano, author J. Shin,
author D. Nishitani, author H. Iwatsuka, author
Y. Masuyama, author
H. Sugiyama, author
M. Ishii, author S. Onoda, author T. Ohshima, author K. Arai, author T. Iwasaki, and author M. Hatano, 10.1063/5.0031502 journal
journal Appl. Phys. Lett. volume
118, pages 034001 (year 2021)NoStop
[Zuo et al.(2017)Zuo,
Wang, and Chen]ordance_detection
author author B. Zuo, author L. Wang, and author W. Chen, 10.3390/s17091976 journal journal Sensors volume 17 (year 2017), 10.3390/s17091976NoStop
[Gang et al.(2015)Gang,
Yingtang, Hongbo, GuoQuan, and Zhining]Gang2015IntegratedSystem
author author Y. Gang, author Z. Yingtang,
author F. Hongbo, author R. GuoQuan, and author L. Zhining, https://doi.org/10.1016/j.jmmm.2014.08.022 journal journal J. Magn. Magn. Mater. volume 374, pages 289 (year 2015)NoStop
[Gang et al.(2016)Gang,
Yingtang, Zhining, Hongbo, and Guoquan]Gang2016DetectionSystem
author author Y. I. N. Gang, author Z. Yingtang, author L. Zhining,
author F. Hongbo, and author R. Guoquan, https://doi.org/10.1016/j.jmmm.2015.11.034 journal journal J. Magn. Magn. Mater. volume 402, pages 1 (year 2016)NoStop
[Young and Clark(2010)]Young2010MagneticEnvironment
author author J. A. Young and author D. A. Clark, in 10.1109/ICEAA.2010.5650898 booktitle 2010 International Conference on Electromagnetics in Advanced
Applications (year 2010) pp. pages
701–704NoStop
[Zhou et al.(2021)Zhou,
Patel, Frangeskou, Nikitin,
Green, Breeze, Onoda,
Isoya, and Morley]Zhou2021ImagingMagnetometer
author author L. Q. Zhou, author R. L. Patel,
author A. C. Frangeskou,
author A. Nikitin, author B. L. Green, author
B. G. Breeze, author
S. Onoda, author J. Isoya, and author G. W. Morley, 10.1103/PhysRevApplied.15.024015
journal journal Phys. Rev. Appl. volume 15, pages 24015 (year
2021)NoStop
[Dale and Morley(2017)]gavin_morley_mcg
author author M. W. Dale and author G. W. Morley, @noop title Medical applications
of diamond magnetometry: commercial viability, (year
2017), http://arxiv.org/abs/1705.01994 arXiv:1705.01994
NoStop
[Fagaly(2006)]Fagaly2006SuperconductingApplications
author author R. L. Fagaly, 10.1063/1.2354545 journal
journal Rev. Sci. Instrum. volume
77, pages 101101 (year 2006)NoStop
[Körber et al.(2016)Körber, Storm, Seton, Mäkelä, Paetau, Parkkonen,
Pfeiffer, Riaz, Schneiderman,
Dong, Hwang, You,
Inglis, Clarke, Espy,
Ilmoniemi, Magnelind, Matlashov, Nieminen, Volegov, Zevenhoven, Höfner, Burghoff,
Enpuku, Yang, Chieh,
Knuutila, Laine, and Nenonen]Korber2016SQUIDsHealthcare
author author R. Körber, author J.-H. Storm, author H. Seton,
author J. P. Mäkelä,
author R. Paetau, author L. Parkkonen, author
C. Pfeiffer, author
B. Riaz, author J. F. Schneiderman, author H. Dong, author S.-m. Hwang, author L. You, author B. Inglis,
author J. Clarke, author M. A. Espy, author
R. J. Ilmoniemi, author
P. E. Magnelind, author
A. N. Matlashov, author
J. O. Nieminen, author
P. L. Volegov, author
K. C. J. Zevenhoven, author
N. Höfner, author
M. Burghoff, author
K. Enpuku, author S. Y. Yang, author J.-J. Chieh, author J. Knuutila, author P. Laine, and author J. Nenonen, 10.1088/0953-2048/29/11/113001 journal journal Supercond. Sci. Technol volume 29, pages 113001 (year 2016)NoStop
[Acuña(2002)]AcunaMSpaceMag
author author M. H. Acuña, 10.1063/1.1510570 journal
journal Rev. Sci. Instrum. volume
73, pages 3717 (year 2002)NoStop
[Cochrane et al.(2016)Cochrane, Blacksberg, Anders, and Lenahan]Cochrane2016
author author C. J. Cochrane, author J. Blacksberg, author M. A. Anders, and author P. M. Lenahan, 10.1038/srep37077 journal
journal Sci. Rep. volume 6, pages 37077 (year 2016)NoStop
[Boto et al.(2018)Boto,
Holmes, Leggett, Roberts,
Shah, Meyer, Muñoz,
Mullinger, Tierney, Bestmann,
Barnes, Bowtell, and Brookes]Boto2018
author author E. Boto, author N. Holmes,
author J. Leggett, author G. Roberts, author
V. Shah, author S. S. Meyer, author L. D. Muñoz, author K. J. Mullinger, author T. M. Tierney, author S. Bestmann, author G. R. Barnes, author R. Bowtell, and author M. J. Brookes, 10.1038/nature26147 journal journal Nature volume 555, pages
657 (year 2018)NoStop
[Jiang et al.(2019)Jiang,
Frutos, Wu, Blanchard,
Peng, and Budker]Jiang2019MagneticResonance
author author M. Jiang, author R. P. Frutos,
author T. Wu, author
J. W. Blanchard, author
X. Peng, and author
D. Budker, 10.1103/PhysRevApplied.11.024005 journal journal
Phys. Rev. Appl. volume 11, pages
024005 (year 2019)NoStop
[Hoult et al.(2014)Hoult,
Ekim, and Regier]Hoult2014
author author N. A. Hoult, author O. Ekim, and author R. Regier, 10.1061/(ASCE)EM.1943-7889.0000812 journal journal J. Eng. Mech. volume 140, pages 4014097 (year 2014)NoStop
[Paulraj et al.(2013)Paulraj, Yaacob, Majid, Kazim, and Krishnan]Paulraj2013
author author M. P. Paulraj, author S. Yaacob,
author M. S. A. Majid, author M. N. F. M. Kazim, and author P. Krishnan, https://doi.org/10.1016/j.proeng.2013.02.049 journal journal Procedia Eng. volume 53, pages 376 (year 2013), note malaysian
Technical Universities Conference on Engineering & Technology 2012, MUCET
2012NoStop
[Hu et al.(2022)Hu,
He, Zhou, Shu, and Yang]Hu2022
author author M. Hu, author J. He, author C. Zhou, author
Z. Shu, and author
W. Yang, https://doi.org/10.1016/j.measurement.2021.110364 journal
journal Measurement volume 187, pages 110364 (year 2022)NoStop
[Suresh et al.(2017)Suresh,
Abudhahir, and Daniel]Suresh2017DevelopmentTube
author author V. Suresh, author A. Abudhahir, and author J. Daniel, https://doi.org/10.1016/j.measurement.2016.10.015 journal journal Measurement volume
95, pages 273 (year 2017)NoStop
[Kikuchi et al.(2010)Kikuchi, Kurisawa, Ara, Kamada, and Kobayashi]Kikuchi2010FeasibilityTube
author author H. Kikuchi, author Y. Kurisawa,
author K. Ara, author
Y. Kamada, and author
S. Kobayashi, 10.3233/JAE-2010-1224 journal journal Int. J.
Appl. Electromagn volume 33, pages
1087 (year 2010)NoStop
[Kikuchi et al.(2011)Kikuchi, Sato, Shimizu, Kamada, and Kobayashi]Kikuchi2011FeasibilityPlants
author author H. Kikuchi, author K. Sato,
author I. Shimizu, author Y. Kamada, and author
S. Kobayashi, 10.1109/TMAG.2011.2147296 journal journal IEEE
Trans. Magn volume 47, pages 3963
(year 2011)NoStop
[Peng et al.(2020)Peng,
Anyaoha, Liu, and Tsukada]Peng2020AnalysisAssessment
author author X. Peng, author U. Anyaoha,
author Z. Liu, and author K. Tsukada, 10.1109/TMAG.2020.2981450 journal journal IEEE
Trans. Magn volume 56, pages 1
(year 2020)NoStop
[Liu et al.(2017)Liu,
He, Zhang, Cao, and Fernandes]Liu2017TheMethod
author author B. Liu, author L.-y. He,
author H. Zhang, author Y. Cao, and author
H. Fernandes, https://doi.org/10.1016/j.measurement.2017.02.051 journal
journal Measurement volume 103, pages 275 (year 2017)NoStop
[Barry et al.(2020)Barry,
Schloss, Bauch, Turner,
Hart, Pham, and Walsworth]Barry2020SensitivityMagnetometryb
author author J. F. Barry, author J. M. Schloss,
author E. Bauch, author M. J. Turner, author
C. A. Hart, author
L. M. Pham, and author
R. L. Walsworth, 10.1103/RevModPhys.92.015004 journal journal
Rev. Mod. Phys. volume 92, pages
15004 (year 2020)NoStop
[Taylor et al.(2008)Taylor,
Cappellaro, Childress, Jiang,
Budker, Hemmer, Yacoby,
Walsworth, and Lukin]Taylor2008High-sensitivityResolution
author author J. M. Taylor, author P. Cappellaro,
author L. Childress, author L. Jiang, author
D. Budker, author P. R. Hemmer, author A. Yacoby, author R. Walsworth, and author M. D. Lukin, 10.1038/nphys1075
journal journal Nat. Phys volume 4, pages 810 (year 2008)NoStop
[Acosta et al.(2009)Acosta,
Bauch, Ledbetter, Santori,
Fu, Barclay, Beausoleil,
Linget, Roch, Treussart,
Chemerisov, Gawlik, and Budker]Acosta2009DiamondsApplications
author author V. M. Acosta, author E. Bauch,
author M. P. Ledbetter, author C. Santori, author
K.-M. C. Fu, author
P. E. Barclay, author
R. G. Beausoleil, author
H. Linget, author J. F. Roch, author F. Treussart, author S. Chemerisov, author W. Gawlik, and author D. Budker, 10.1103/PhysRevB.80.115202
journal journal Phys. Rev. B. volume 80, pages 115202 (year
2009)NoStop
[Plakhotnik et al.(2014)Plakhotnik, Doherty, Cole, Chapman, and Manson]Plakhotnik2014All-OpticalNanodiamond
author author T. Plakhotnik, author M. W. Doherty, author J. H. Cole,
author R. Chapman, and author N. B. Manson, 10.1021/nl501841d journal journal Nano
Lett. volume 14, pages 4989 (year 2014)NoStop
[Toyli et al.(2012)Toyli,
Christle, Alkauskas, Buckley,
Van de Walle, and Awschalom]Toyli2012MeasurementK
author author D. M. Toyli, author D. J. Christle,
author A. Alkauskas, author B. B. Buckley, author
C. G. Van de Walle, and author D. D. Awschalom, 10.1103/PhysRevX.2.031001 journal journal Phys.
Rev. X volume 2, pages 31001
(year 2012)NoStop
[Toyli et al.(2013)Toyli,
de las Casas, Christle, Dobrovitski, and Awschalom]Toyli2013FluorescenceDiamond
author author D. M. Toyli, author C. F. de las
Casas, author D. J. Christle,
author V. V. Dobrovitski, and author D. D. Awschalom, 10.1073/pnas.1306825110 journal journal
Proc. Natl. Acad. Sci. U.S.A. volume 110, pages 8417 (year 2013)NoStop
[Doherty et al.(2013)Doherty, Manson, Delaney, Jelezko, Wrachtrup, and Hollenberg]Doherty2013TheDiamond
author author M. W. Doherty, author N. B. Manson,
author P. Delaney, author F. Jelezko, author
J. Wrachtrup, and author
L. C. L. Hollenberg, https://doi.org/10.1016/j.physrep.2013.02.001 journal
journal Phys. Rep. volume 528, pages 1 (year 2013)NoStop
[Hsieh et al.(2019)Hsieh,
Bhattacharyya, Zu, Mittiga,
Smart, Machado, Kobrin,
Höhn, Rui, Kamrani,
Chatterjee, Choi, Zaletel,
Struzhkin, Moore, Levitas,
Jeanloz, and Yao]Hsieh2019ImagingSensor
author author S. Hsieh, author P. Bhattacharyya, author C. Zu,
author T. Mittiga, author T. J. Smart, author
F. Machado, author B. Kobrin, author T. O. Höhn, author N. Z. Rui, author M. Kamrani, author S. Chatterjee,
author S. Choi, author
M. Zaletel, author V. V. Struzhkin, author J. E. Moore, author V. I. Levitas, author R. Jeanloz, and author N. Y. Yao, 10.1126/science.aaw4352 journal journal
Science volume 366, pages 1349
(year 2019)NoStop
[Ivády et al.(2014)Ivády, Simon, Maze, Abrikosov, and Gali]Ivady2014PressureStudy
author author V. Ivády, author T. Simon,
author J. R. Maze, author I. A. Abrikosov, and author A. Gali, 10.1103/PhysRevB.90.235205 journal journal Phys.
Rev. B. volume 90, pages 235205
(year 2014)NoStop
[Graham et al.(2023)Graham,
Rahman, Munn, Patel,
Newman, Stephen, Colston,
Nikitin, Edmonds, Twitchen,
Markham, and Morley]stuarts_sensitivity
author author S. M. Graham, author A. T. M. A. Rahman, author L. Munn,
author R. L. Patel, author A. J. Newman, author
C. J. Stephen, author
G. Colston, author A. Nikitin, author A. M. Edmonds, author D. J. Twitchen, author M. L. Markham, and author G. W. Morley, 10.1103/PhysRevApplied.19.044042
journal journal Phys. Rev. Appl. volume 19, pages 044042 (year
2023)NoStop
[Zhang et al.(2022)Zhang,
Lin, Dong, Du, Gao, Yu, Feng, Chen,
Guo, and Sun]Zhang_sensitivity
author author S.-C. Zhang, author H.-B. Lin,
author Y. Dong, author
B. Du, author X.-D. Gao, author C. Yu, author Z.-H. Feng, author X.-D. Chen, author G.-C. Guo, and author F.-W. Sun, 10.1364/PRJ.462851 journal journal
Photon. Res. volume 10, pages 2191
(year 2022)NoStop
[Chatzidrosos et al.(2019)Chatzidrosos, Wickenbrock, Bougas,
Zheng, Tretiak, Yang, and Budker]Chatzidrosos2019Eddy-CurrentDiamond
author author G. Chatzidrosos, author A. Wickenbrock, author L. Bougas,
author H. Zheng, author O. Tretiak, author
Y. Yang, and author
D. Budker, 10.1103/PhysRevApplied.11.014060 journal journal
Phys. Rev. Appl. volume 11, pages
14060 (year 2019)NoStop
[Ibrahim et al.(2018)Ibrahim, Foy, Kim, Englund, and Han]Ibrahim2018Room-TemperatureMagnetometry
author author M. I. Ibrahim, author C. Foy,
author D. Kim, author
D. R. Englund, and author
R. Han, in 10.1109/VLSIC.2018.8502329 booktitle 2018 IEEE Symposium
on VLSI Circuits (year 2018) pp. pages
249–250NoStop
[Xu et al.(2022)Xu,
Zhang, Yuan, Bian,
Fan, Li, and Liu]Xu2022BurstMagnetometry
author author C. Xu, author J. Zhang, author H. Yuan, author
G. Bian, author P. Fan, author M. Li, and author T. Liu, 10.1063/5.0078426 journal journal Appl. Phys. Lett volume 120, pages 084104 (year 2022)NoStop
[Barry et al.(2016)Barry,
Turner, Schloss, Glenn,
Song, Lukin, Park, and Walsworth]Barry2016OpticalDiamondb
author author J. F. Barry, author M. J. Turner,
author J. M. Schloss, author D. R. Glenn, author
Y. Song, author M. D. Lukin, author H. Park, and author R. L. Walsworth, 10.1073/pnas.1601513113
journal journal Proc. Natl. Acad. Sci. U.S.A. volume 113, pages 14133 (year 2016)NoStop
[Kuwahata et al.(2020)Kuwahata, Kitaizumi, Saichi, Sato, Igarashi, Ohshima, Masuyama, Iwasaki, Hatano, Jelezko, Kusakabe, Yatsui, and Sekino]Kuwahata2020MagnetometerApplications
author author A. Kuwahata, author T. Kitaizumi,
author K. Saichi, author T. Sato, author
R. Igarashi, author
T. Ohshima, author Y. Masuyama, author T. Iwasaki, author M. Hatano, author F. Jelezko, author M. Kusakabe, author T. Yatsui, and author M. Sekino, 10.1038/s41598-020-59064-6
journal journal Sci. Rep. volume 10, pages 2483 (year
2020)NoStop
[Davis et al.(2018)Davis,
Ramesh, Bhatnagar, Lee-Gosselin, Barry, Glenn, Walsworth, and Shapiro]Davis2018MappingMagnetometry
author author H. C. Davis, author P. Ramesh,
author A. Bhatnagar, author A. Lee-Gosselin, author
J. F. Barry, author
D. R. Glenn, author
R. L. Walsworth, and author
M. G. Shapiro, 10.1038/s41467-017-02471-7 journal journal Nat.
Commun. volume 9, pages 131 (year 2018)NoStop
[Edmonds et al.(2021)Edmonds, Hart, Turner, Colard, Schloss, Olsson, Trubko, Markham, Rathmill, Horne-Smith, Lew, Manickam, Bruce, Kaup, Russo, DiMario, South, Hansen, Twitchen, and Walsworth]diamond_properties_edmonds_walsworth
author author A. M. Edmonds, author C. A. Hart,
author M. J. Turner, author P.-O. Colard, author
J. M. Schloss, author
K. S. Olsson, author
R. Trubko, author M. L. Markham, author A. Rathmill, author B. Horne-Smith, author W. Lew, author A. Manickam, author S. Bruce,
author P. G. Kaup, author J. C. Russo, author
M. J. DiMario, author
J. T. South, author
J. T. Hansen, author
D. J. Twitchen, and author
R. L. Walsworth, 10.1088/2633-4356/abd88a journal journal Mater.
Quantum Technol. volume 1, pages
025001 (year 2021)NoStop
[Patel et al.(2020)Patel,
Zhou, Frangeskou, Stimpson,
Breeze, Nikitin, Dale,
Nichols, Thornley, Green,
Newton, Edmonds, Markham,
Twitchen, and Morley]Patel2020SubnanoteslaSensor
author author R. L. Patel, author L. Q. Zhou,
author A. C. Frangeskou,
author G. A. Stimpson, author B. G. Breeze, author
A. Nikitin, author M. W. Dale, author E. C. Nichols, author W. Thornley, author B. L. Green,
author M. E. Newton, author A. M. Edmonds, author
M. L. Markham, author
D. J. Twitchen, and author
G. W. Morley, 10.1103/PhysRevApplied.14.044058 journal journal
Phys. Rev. Appl. volume 14, pages
44058 (year 2020)NoStop
|
http://arxiv.org/abs/2307.04675v2 | 20230710162105 | LINFA: a Python library for variational inference with normalizing flow and annealing | [
"Yu Wang",
"Emma R. Cobian",
"Jubilee Lee",
"Fang Liu",
"Jonathan D. Hauenstein",
"Daniele E. Schiavazzi"
] | cs.LG | [
"cs.LG",
"stat.CO"
] |
]Yu Wang
]Emma R. Cobian
]Jubilee Lee
]Fang Liu
]Jonathan D. Hauenstein
]Daniele E. Schiavazzi
[]Department of Applied and Computational Mathematics and Statistics
University of Notre Dame, Notre Dame, IN, USA
LINFA: a Python library for variational inference with normalizing flow and annealing
[
=====================================================================================
Variational inference is an increasingly popular method in statistics and machine learning for approximating probability distributions.
We developed LINFA (Library for Inference with Normalizing Flow and Annealing), a Python library for variational inference to accommodate computationally expensive models and difficult-to-sample distributions with dependent parameters.
We discuss the theoretical background, capabilities, and performance of LINFA in various benchmarks. LINFA is publicly available on GitHub at <https://github.com/desResLab/LINFA>.
§ INTRODUCTION
Generating samples from a posterior distribution is a fundamental task in Bayesian inference.
The development of sampling-based algorithms from the Markov chain Monte Carlo family <cit.> has made solving Bayesian inverse problems accessible to a wide audience of both researchers and practitioners.
However, the number of samples required by these approaches is typically significant and the convergence of Markov chains to their stationary distribution can be slow especially in high-dimensions. Additionally, satisfactory convergence may not be always easy to quantify, even if a number of metrics have been proposed in the literature over the years.
More recent paradigms have been proposed in the context of variational inference <cit.>, where an optimization problem is formulated to determine the optimal member of a parametric family of distributions that can approximate a target posterior density.
In addition, flexible approaches to parametrize variational distributions through a composition of transformations (closely related to the concept of trasport maps, see, e.g., <cit.>) have reached popularity under the name of normalizing flows <cit.>.
The combination of variational inference and normalizing flow has received significant recent interest in the context of general algorithms for solving inverse problems <cit.>.
However, cases where the computational cost of evaluating the underlying probability distribution is significant occur quite often in engineering and applied sciences, for example when such evaluation requires the solution of an ordinary or partial differential equation.
In such cases, inference can easily become intractable. Additionally, strong and nonlinear dependence between model parameters may results in difficult-to-sample posterior distributions characterized by features at multiple scales or by multiple modes.
The LINFA library is specifically designed for cases where the model evaluation is computationally expensive. In such cases, the construction of an adaptively trained surrogate model is key to reducing the computational cost of inference <cit.>.
In addition, LINFA provides an adaptive annealing scheduler, where temperature increments are automatically determined based on the available variational approximant of the posterior distribution. Thus, adaptive annealing makes it easier to sample from complicated densities <cit.>.
This paper is organized as follows. The main features of the LINFA library are discussed in Section <ref>, followed by a brief outline of a few selected numerical tests in Section <ref>. Conclusions and future work are finally discussed in Section <ref>. The paper is completed by a brief description of the background theory and reference to the relevant papers in Appendix <ref>, a detailed presentation of a four benchmarks in Appendix <ref>, and a list of all the relevant hyperparameters in Appendix <ref>.
§ CAPABILITIES
LINFA is designed as a general inference engine and allows the user to define custom input transformations, computational models, surrogates, and likelihood functions.
1 - User-defined input parameter transformations - Input transformations may reduce the complexity of inference and surrogate model construction in situations where the ranges of the input variables differ substantially or when the input parameters are bounded. A number of pre-defined univariate transformations are provided, i.e, , , , and . These transformations are independently defined for each input variable, using four parameters (a,b,c,d), providing a nonlinear transformation between the normalized interval [a,b] and the physical interval [c,d]. Additional transformations can be defined by implementing the following member functions.
* - It evaluates the transformation from the normalized to the physical space. One transformation needs to be defined for each input. For example, the list of lists
defines a hyperbolic tangent transformation for the first two variables and an exponential transformation for the third.
* - This is the log Jacobian of the transformation that needs to be included in the computation of the log posterior density to account for the additional change in volume.
2 - User-defined computational models - LINFA can accommodate any type of models from analytically defined posteriors with the gradient computed through automatic differentiation to legacy computational solvers for which the solution gradient is not available nor easy to compute. New models are created by implementing the methods below.
-3pt
* - This is a pre-processing function used to generate synthetic observations. It computes the model output corresponding to the default parameter values (usually defined as part of the model) and adds noise with a user-specified distribution. Observations will be stored in a file and are typically assigned to so they are available for computing the log posterior.
* - This function solves the model for multiple values of the physical input parameters specified in a matrix format (with one sample for each row and one column for each input parameter dimension).
3 - User-defined surrogate models - For computational models that are too expensive for online inference, LINFA provides functionalities to create, train, and fine-tune a surrogate model. The class implements the following functionalities:
-3pt
* A new surrogate model can be created using the constructor.
* The (i.e. upper and lower bounds) are stored as a list of lists using the format .
* A pre-grid is defined as an a priori selected point cloud created inside the hyper-rectangle defined by . The pre-grid can be either of type (tensor product grid) where the grid order (number of points in each dimension) is defined through the argument , or of type , in which case the variable defines the total number of samples.
* Surrogate model Input/Output. The two functions and are provided to save a snapshot of a given surrogate or to read it from a file.
* The function is provided to perform an initial training of the surrogate model on the pre-grid. In addition, the function is also available to re-train the model once additional training examples are available.
* The function evaluates the surrogate model at multiple input realizations. If a transformation is defined, the surrogate should always be specified in the normalized domain with limits coincident with the normalized intervals.
4 - User-defined likelihood - A user-defined likelihood function can be defined by passing the parameters, the model, the surrogate and a coordinate transformation using
and then assigning it as a member function of the class using:
.
5 - Linear and adaptive annealing schedulers - LINFA provides two annealing schedulers by default. The first is the scheduler with constant increments. The second is the adaptive scheduler
<cit.> with hyperparameters reported in Table <ref>. For the AdaAnn scheduler, the user can also specify a different number of parameter updates to be performed at the initial temperature t_0, final temperature t_1, and for any temperature t_0<t<1. Finally, the batch size (number of samples used to evaluate the expectations in the loss function) can also be differentiated for t=1 and t<1.
6 - User-defined hyperparameters - A complete list of hyperparameters with a description of their functionality can be found in Appendix <ref>.
§ NUMERICAL BENCHMARKS
We tested LINFA on multiple problems. These include inference on unimodal and multi-modal posterior distributions specified in closed form, ordinary differential models and dynamical systems with gradients directly computed through automatic differentiation in PyTorch, identifiable and non-identifiable physics-based models with fixed and adaptive surrogates, and high-dimensional statistical models. Some of the above tests are included with the library and systematically tested when pushing the master branch on GitHub. A detailed discussion of these test cases is provided in Appendix <ref>. LINFA can be installed through the Python Package Index (PyPI) typing
To run the tests type
where is the name of the test case, either , , , , or .
§ CONCLUSION AND FUTURE WORK
In this paper, we have introduced the LINFA library for variational inference, briefly discussed the relevant background, its capabilities, and report its performance on a number of test cases. Some interesting directions for future work are mentioned below.
Future versions will support user-defined privacy-preserving synthetic data generation and variational inference through differentially private gradient descent algorithms. This will allow the user to perform inference tasks while preserving a pre-defined privacy budget, as discussed in <cit.>.
LINFA will also be extended to handle multiple models. This will open new possibilities to solve inverse problems combining variational inference and multi-fidelity surrogates <cit.>.
In addition, for inverse problems with significant dependence among the parameters, it is often possible to simplify the inference task by operating on manifolds of reduced dimensionality <cit.>. New modules for dimensionality reduction will be developed and integrated with the LINFA library.
Finally, the ELBO loss typically used in variational inference has known limitations, some of which are related to its close connection with the KL divergence. Future versions of LINFA will provide the option to use alternative losses.
§ ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support from the NSF Big Data Science & Engineering grant #1918692 and the computational resources provided through the Center for Research Computing at the University of Notre Dame. DES also acknowledges support from NSF CAREER grant #1942662.
§ BACKGROUND THEORY
§.§ Variational inference with normalizing flow
Consider the problem of estimating (in a Bayesian sense) the parameters z∈𝒵 of a physics-based or statistical model
x = f(z) + ε,
from the observations x∈𝒳 and a known statistical characterization of the error ε.
We tackle this problem with variational inference and normalizing flow. A normalizing flow (NF) is a nonlinear transformation F:ℝ^d×Λ→ℝ^d designed to map an easy-to-sample base distribution q_0(z_0) into a close approximation q_K(z_K) of a desired target posterior density p(z|x).
This transformation can be determined by composing K bijections
z_K = F(z_0) = F_K∘ F_K-1∘⋯∘ F_k∘⋯∘ F_1(z_0),
and evaluating the transformed density through the change of variable formula <cit.>.
In the context of variational inference, we seek to determine an optimal set of parameters λ∈Λ so that q_K(z_K)≈ p(z|x). Given observations x∈𝒳, a likelihood function l_z(x) (informed by the distribution of the error ε) and prior p(z), a NF-based approximation q_K(z) of the posterior distribution p(z|x) can be computed by maximizing the lower bound to the log marginal likelihood log p(x) (the so-called evidence lower bound or ELBO), or, equivalently, by minimizing a free energy bound <cit.>.
ℱ( x) = 𝔼_q_K( z_K)[log q_K( z_K) - log p( x, z_K)]
= 𝔼_q_0( z_0)[log q_0( z_0)] - 𝔼_q_0( z_0)[log p( x, z_K)] - 𝔼_q_0( z_0)[∑_k=1^K log|∂ z_k/∂ z_k-1|].
For computational convenience, normalizing flow transformations are selected to be easily invertible and their Jacobian determinant can be computed with a cost that grows linearly with the problem dimensionality.
Approaches in the literature include RealNVP <cit.>, GLOW <cit.>, and autoregressive transformations such as MAF <cit.> and IAF <cit.>.
§.§ MAF and RealNVP
LINFA implements two widely used normalizing flow formulations, MAF <cit.> and RealNVP <cit.>.
MAF belongs to the class of autoregressive normalizing flows. Given the latent variable z = (z_1,z_2,…,z_d), it assumes p(z_i|z_1,…,z_i-1) = ϕ[(z_i - μ_i) / e^α_i], where ϕ is the standard normal distribution, μ_i = f_μ_i(z_1,…,z_i-1), α_i = f_α_i(z_1,…,z_i-1), i=1,2,…,d, and f_μ_i and f_α_i are masked autoencoder neural networks <cit.>.
In a MADE autoencoder the network connectivities are multiplied by Boolean masks so the input-output relation maintains a lower triangular structure, making the computation of the Jacobian determinant particularly simple.
MAF transformations are then composed of multiple MADE layers, possibly interleaved by batch normalization layers <cit.>, typically used to add stability during training and increase network accuracy <cit.>.
RealNVP is another widely used flow where, at each layer the first d' variables are left unaltered while the remaining d-d' are subject to an affine transformation of the form z_d'+1:d = z_d'+1:d ⊙ e^α + μ, where μ = f_μ(z_1:d') and α = f_α(z_d'+1:d) are MADE autoencoders.
In this context, MAF could be seen as a generalization of RealNVP by setting μ_i=α_i=0 for i≤ d' <cit.>.
§.§ Normalizing flow with adaptive surrogate (NoFAS)
LINFA is designed to accommodate black-box models f: 𝒵→𝒳 between the random inputs z = (z_1, z_2, ⋯, z_d)^T ∈𝒵 and the outputs (x_1, x_2,⋯,x_m)^T ∈𝒳, and assumes n observations x = { x_i}_i=1^n ⊂𝒳 to be available.
Our goal is to infer z and to quantify its uncertainty given x.
We employ a variational Bayesian paradigm and sample from the posterior distribution p( z| x)∝ℓ_ z( x,f) p( z), with prior p( z) via normalizing flows.
This requires the evaluation of the gradient of the ELBO (<ref>) with respect to the NF parameters λ, replacing p( x, z_K) with p( x| z_K) p( z) =ℓ_ z_K(x,f) p( z), and approximating the expectations with their MC estimates.
However, the likelihood function needs to be evaluated at every MC realization, which can be costly if the model f(z) is computationally expensive. In addition, automatic differentiation through a legacy (e.g. physics-based) solver may be an impractical, time-consuming, or require the development of an adjoint solver.
Our solution is to replace the model f with a computationally inexpensive surrogate f: 𝒵×𝒲→𝒳 parameterized by the weigths w∈𝒲, whose derivatives can be obtained at a relatively low computational cost, but intrinsic bias in the selected surrogate formulation, a limited number of training examples, and locally optimal w can compromise the accuracy of f.
To resolve these issues, LINFA implements NoFAS, which updates the surrogate model adaptively by smartly weighting the samples of z from NF thanks to a memory-aware loss function.
Once a newly updated surrogate is obtained, the likelihood function is updated, leading to a new posterior distribution that will be approximated by VI-NF, producing, in turn, new samples for the next surrogate model update, and so on.
Additional details can be found in <cit.>.
§.§ Adaptive Annealing
Annealing is a technique to parametrically smooth a target density to improve sampling efficiency and accuracy during inference.
In the discrete case, this is achieved by incrementing an inverse temperature t_k and setting p_k(z,x) = p^t_k(z,x), for k=0,…,K, where 0 < t_0 < ⋯ < t_K≤ 1.
The result of exponentiation produces a smooth unimodal distribution for a sufficiently small t_0, recovering the target density as t_k approaches 1. In other words, annealing provides a continuous deformation from an easier to approximate unimodal distribution to a desired target density.
A linear annealing scheduler <cit.> with fixed temperature increments is often used in practice, where for with constant increments .
Intuitively, small temperature changes are desirable to carefully explore the parameter spaces at the beginning of the annealing process, whereas larger changes can be taken as t_k increases, after annealing has helped to capture important features of the target distribution (e.g., locating all the relevant modes).
The AdaAnn scheduler determines the increment ϵ_k that approximately produces a pre-defined change in the KL divergence between two distributions annealed at t_k and t_k+1=t_k+ϵ_k, respectively.
Letting the KL divergence equal a constant τ^2/2, where τ is referred to as the KL tolerance, the step size ϵ_k becomes
ϵ_k = τ/ √(𝕍_p^t_k[log p( z,x)]).
The denominator is large when the support of the annealed distribution p^t_k(z,x) is wider than the support of the target p(z,x), and progressively reduces with increasing t_k.
Further detail on the derivation of the expression for ϵ_k can be found in <cit.>.
§ DETAILED NUMERICAL BENCHMARKS
§.§ Simple two-dimensional map with Gaussian likelihood
A model f:ℝ^2→ℝ^2 is chosen in this experiment having the closed-form expression
f( z) = f(z_1,z_2) = (z_1^3 / 10 + exp(z_2 / 3), z_1^3 / 10 - exp(z_2 / 3))^T.
Observations x are generated as
x = x^* + 0.05 |x^*| ⊙x_0,
where x_0∼𝒩(0, I_2) and ⊙ is the Hadamard product.
We set the true model parameters at z^* = (3, 5)^T, with output x^* = f( z^*)=(7.99, -2.59)^T, and simulate 50 sets of observations from (<ref>). The likelihood of z given x is assumed Gaussian and we adopt a noninformative uniform prior p( z).
We allocate a budget of 4×4=16 model solutions to the pre-grid and use the rest to adaptively calibrate f using 2 samples every 1000 normalizing flow iterations.
Results in terms of loss profile, variational approximation, and posterior predictive distribution are shown in Figure <ref>.
§.§ High-dimensional example
We consider a map f: ℝ^5→ℝ^4 expressed as
f(z) = A g(e^z),
where g_i(r) = (2· |2 a_i - 1| + r_i) / (1 + r_i) with r_i > 0 for i=1,…,5 is the Sobol function <cit.> and A is a 4×5 matrix. We also set
a = (0.084, 0.229, 0.913, 0.152, 0.826)^T A = 1/√(2)[ 1 1 0 0 0; 0 1 1 0 0; 0 0 1 1 0; 0 0 0 1 1; ].
The true parameter vector is set at z^* = (2.75, -1.5, 0.25, -2.5, 1.75)^T. While the Sobol function is bijective and analytic, f is over-parameterized and non identifiabile.
This is also confirmed by the fact that the curve segment γ(t) = g^-1(g( z^*) + v t)∈ Z gives the same model solution as x^* = f(z^*) = f(γ(t)) ≈ (1.4910, 1.6650, 1.8715, 1.7011)^T for t ∈ (-0.0153, 0.0686], where v = (1,-1,1,-1,1)^T.
This is consistent with the one-dimensional null-space of the matrix A.
We also generate synthetic observations from the Gaussian distribution
x = x^* + 0.01· |x^*| ⊙x_0, and x_0∼𝒩(0, I_5).
Results are shown in Figure <ref>.
§.§ Two-element Windkessel Model
The two-element Windkessel model (often referred to as the RC model) is the simplest representation of the human systemic circulation and requires two parameters, i.e., a resistance R ∈ [100, 1500] Barye· s/ml and a capacitance C ∈ [1× 10^-5, 1 × 10^-2] ml/Barye.
We provide a periodic time history of the aortic flow (see <cit.> for additional details) and use the RC model to predict the time history of the proximal pressure P_p(t), specifically its maximum, minimum, and average values over a typical heart cycle, while assuming the distal resistance P_d(t) as a constant in time, equal to 55 mmHg.
In our experiment, we set the true resistance and capacitance as z_K,1^*=R^* = 1000 Barye· s/ml and z_K,2^*=C^* = 5× 10^-5 ml/Barye, and determine P_p(t) from a RK4 numerical solution of the following algebraic-differential system
Q_d = P_p-P_d/R, d P_p/d t = Q_p - Q_d/C,
where Q_p is the flow entering the RC system and Q_d is the distal flow.
Synthetic observations are generated by adding Gaussian noise to the true model solution x^*=(x^*_1,x^*_2,x^*_3)=(P_p,min, P_p,max, P_p,avg)= (78.28, 101.12, 85.75), i.e., x follows a multivariate Gaussian distribution with mean x^* and a diagonal covariance matrix with entries 0.05 x_i^*, where i=1,2,3 corresponds to the maximum, minimum, and average pressures, respectively.
The aim is to quantify the uncertainty in the RC model parameters given 50 repeated pressure measurements. We imposed a non-informative prior on R and C. Results are shown in Figure <ref>.
§.§ Three-element Wndkessel Circulatory Model (NoFAS)
The three-parameter Windkessel or RCR model is characterized by proximal and distal resistance parameters R_p, R_d∈ [100, 1500] Barye·s/ml, and one capacitance parameter C ∈ [1× 10^-5, 1× 10^-2] ml/Barye.
This model is not identifiable. The average distal pressure is only affected by the total system resistance, i.e. the sum R_p+R_d, leading to a negative correlation between these two parameters. Thus, an increment in the proximal resistance is compensated by a reduction in the distal resistance (so the average distal pressure remains the same) which, in turn, reduces the friction encountered by the flow exiting the capacitor. An increase in the value of C is finally needed to restore the average, minimum and maximum pressure. This leads to a positive correlation between C and R_d.
The output consists of the maximum, minimum, and average values of the proximal pressure P_p(t), i.e., (P_p,min, P_p,max, P_p,avg) over one heart cycle.
The true parameters are z^*_K,1 = R^*_p = 1000 Barye·s/ml, z^*_K,2=R^*_d = 1000 Barye·s/ml, and C^* = 5× 10^-5 ml/Barye. The proximal pressure is computed from the solution of the algebraic-differential system
Q_p = P_p - P_c/R_p, Q_d = P_c-P_d/R_d, d P_c/d t = Q_p-Q_d/C,
where the distal pressure is set to P_d=55 mmHg.
Synthetic observations are generated from N(μ, Σ), where μ=(f_1(z^*),f_2(z^*),f_3(z^*))^T = (P_p,min, P_p,max, P_p,ave)^T = (100.96, 148.02, 116.50)^T and Σ is a diagonal matrix with entries (5.05, 7.40, 5.83)^T. The budgeted number of true model solutions is 216; the fixed surrogate model is evaluated on a 6× 6× 6 = 216 pre-grid while the adaptive surrogate is evaluated with a pre-grid of size 4× 4× 4 = 64 and the other 152 evaluations are adaptively selected.
This example also demonstrates how NoFAS can be combined with annealing for improved convergence. The results in Figure <ref> are generated using the AdaAnn adaptive annealing scheduler with intial inverse temperature t_0=0.05, KL tolerance τ=0.01 and a batch size of 100 samples. The number of parameter updates is set to 500, 5000 and 5 for t_0, t_1 and t_0<t<t_1, respectively and 1000 Monte Carlo realizations are used to evaluate the denominator in equation (<ref>). The posterior samples capture well the nonlinear correlation among the parameters and generate a fairly accurate posterior predictive distribution that overlaps with the observations. Additional details can be found in <cit.>.
§.§ Friedman 1 model (AdaAnn)
We consider a modified version of the Friedman 1 dataset <cit.> to examine the performance of our adaptive annealing scheduler in a high-dimensional context.
According to the original model in <cit.>, the data are generated as
y_i = μ_i(β)+ ϵ_i, μ_i(β)=β_1sin(π x_i,1x_i,2)+ β_2(x_i,3-β_3)^2+∑_j=4^10β_jx_i,j,
where ϵ_i∼𝒩(0,1).
We made a slight modification to the model in (<ref>) as
μ_i(β) = β_1sin(π x_i,1x_i,2)+ β_2^2(x_i,3-β_3)^2+∑_j=4^10β_jx_i,j,
and set the true parameter combination to β=(β_1,…,β_10)=(10,±√(20), 0.5, 10, 5, 0, 0, 0, 0, 0). Note that both (<ref>) and (<ref>) contain linear, nonlinear, and interaction terms of the input variables X_1 to X_10, five of which (X_6 to X_10) are irrelevant to Y. Each X is drawn independently from 𝒰(0,1). We used R package <cit.> to generate a Friedman 1 dataset with a sample size of n=1000.
We impose a non-informative uniform prior p(β) and, unlike the original modal, we now expect a bimodal posterior distribution of β. Results in terms of marginal statistics and their convergence for the mode with positive z_K,2 are illustrated in Table <ref> and Figure <ref>.
[b]0.4
.8!
2in]l c c c c
True 2cMode 1
Value Post. Mean Post. SD
β_1 = 10 10.0285 0.1000
β_2 = ±√(20) 4.2187 0.1719
β_3 = 0.5 0.4854 0.0004
β_4 = 10 10.0987 0.0491
β_5 = 5 5.0182 0.1142
β_6 = 0 0.1113 0.0785
β_7 = 0 0.0707 0.0043
β_8 = 0 -0.1315 0.1008
β_9 = 0 0.0976 0.0387
β_10 = 0 0.1192 0.0463
tablePosterior mean and standard deviation for positive mode in the modified Friedman test case.
[b]0.58
< g r a p h i c s >
< g r a p h i c s >
figureLoss profile (left) and posterior marginal statistics (right) for positive mode in the modified Friedman test case.
§ HYPERPARAMETERS IN LINFA
This section contains the list of all hyperparameters in the library, their default values, and a description of the functionalities they control.
General hyperparameters are listed in Table <ref>, those related to the optimization process in Table <ref>, and to the output folder and files in Table <ref>. Hyperparameters for the proposed NoFAS and AdaAnn approaches are listed in Table <ref> and <ref>, respectively.
Finally, a hyperparameter used to select the hardware device is described in Table <ref>.
0.2in
unsrtnat
|
http://arxiv.org/abs/2307.03885v1 | 20230708033002 | Hot QCD Phase Diagram From Holographic Einstein-Maxwell-Dilaton Models | [
"Romulo Rougemont",
"Joaquin Grefa",
"Mauricio Hippert",
"Jorge Noronha",
"Jacquelyn Noronha-Hostler",
"Israel Portillo",
"Claudia Ratti"
] | nucl-th | [
"nucl-th",
"hep-ph",
"hep-th"
] |
1staddress]Romulo Rougemontmycorrespondingauthor
[mycorrespondingauthor]Corresponding author
[email protected]
2ndaddress]Joaquin Grefa
3rdaddress]Mauricio Hippert
3rdaddress]Jorge Noronha
3rdaddress]Jacquelyn Noronha-Hostler
2ndaddress]Israel Portillo
2ndaddress]Claudia Ratti
[1staddress]Instituto de Física, Universidade Federal de Goiás, Av. Esperança - Campus Samambaia, CEP 74690-900, Goiânia, Goiás, Brazil
[2ndaddress]Physics Department, University of Houston, Houston TX 77204, USA
[3rdaddress]Illinois Center for Advanced Studies of the Universe, Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
In this review, we provide an up-to-date account of quantitative bottom-up holographic descriptions of the strongly coupled quark-gluon plasma (QGP) produced in relativistic heavy-ion collisions, based on the class of gauge-gravity Einstein-Maxwell-Dilaton (EMD) effective models.
The holographic approach is employed to tentatively map the QCD phase diagram at finite temperature onto a dual theory of charged, asymptotically Anti-de Sitter (AdS) black holes living in five dimensions. With a quantitative focus on the hot QCD phase diagram, the nonconformal holographic EMD models reviewed here are adjusted to describe first-principles lattice results for the finite-temperature QCD equation of state, with 2+1 flavors and physical quark masses, at zero chemical potential and vanishing electromagnetic fields.
We review the evolution of such effective models and the corresponding improvements produced in quantitative holographic descriptions of the deconfined hot QGP phase of QCD.
The predictive power of holographic EMD models is tested by quantitatively comparing their predictions for the hot QCD equation of state at nonzero baryon density and the corresponding state-of-the-art lattice QCD results. Hydrodynamic transport coefficients such as the shear and bulk viscosities predicted by these EMD constructions are also compared to the corresponding profiles favored by the latest phenomenological multistage models simultaneously describing different types of heavy-ion data.
We briefly report preliminary results from a Bayesian analysis using EMD models, which provide systematic evidence that lattice QCD results at finite temperature and zero baryon density strongly constrains the free parameters of such bottom-up holographic constructions. Remarkably, the set of parameters constrained by lattice results at vanishing chemical potential turns out to produce EMD models in quantitative agreement with lattice QCD results also at finite baryon density.
We also review
results for equilibrium and transport properties from anisotropic EMD models, which effectively describe the hot and magnetized QGP at finite temperatures and magnetic fields with zero chemical potentials.
Finally, we provide a critical assessment of the main limitations and drawbacks of the holographic models reviewed in the present work, and point out some perspectives we believe are of fundamental importance for future developments.
QCD phase diagram critical point quark-gluon plasma gauge-gravity duality equations of state
empty
§ INTRODUCTION
Quantum chromodynamics (QCD) is the quantum field theory (QFT)
responsible for the
sector of the standard model of particle physics associated with the strong interaction. At the most fundamental level, it comprises quarks and gluons (collectively called partons) as particles of the corresponding fermionic and non-Abelian gauge vector fields, respectively <cit.>.
A rich and complex diversity of phases and regimes is possible for QCD matter, depending on the conditions to which partons are subjected <cit.>. These different regimes have been intensively investigated in the last five decades, conjuring simultaneous efforts from theory, experiments, astrophysical observations, and large computational simulations <cit.>.
At the microscopic level, QCD is fundamentally responsible for two of the most important aspects of ordinary baryonic matter in our universe, namely: i) the stability of nuclei due to the effective exchange of pions binding the nucleons (protons and neutrons), with the most fundamental interaction between the composite hadronic particles being mediated via gluon exchange between quarks; ii)
most of its mass, thus generating the vast majority of the mass of ordinary matter in our universe, as a result of the dynamical breaking of chiral symmetry at low energies — for instance, at low temperatures compared to the typical scale T_c∼ 150 MeV of the QCD deconfinement crossover transition at zero baryon density <cit.>. In fact, about ≳ 98% of the mass of the nucleons (and, consequently, also the mass of atoms and the ordinary macroscopic structures of the universe built upon them) comes from strong interactions,
with the tiny rest being actually due to the current quark masses generated by the Higgs mechanism <cit.>. Intrinsically related to the two aforementioned facts, QCD also presents what is called color confinement, which generically refers to the fact that quarks and gluons, as degrees of freedom carrying color charge under the non-Abelian gauge group SU(N_c=3) of QCD, are never observed in isolation as asymptotic states in experiments, being confined inside color-neutral hadrons <cit.>.
Relying on various properties of QCD, we can determine its degrees of freedom at specific energy scales.
Due to the number of colors, N_c=3, and quark flavors, N_f=6, QCD is an asymptotically free non-Abelian gauge theory <cit.>. That is, the β-function for the QCD coupling constant is negative, implying that it is a decreasing function of the renormalization group energy scale, vanishing at asymptotically high energies. Conversely, QCD becomes a strongly coupled non-perturbative QFT at energy scales below or around the QCD dimensional transmutation scale, Λ_QCD∼ 200 MeV, indicating the failure of perturbative QFT methods when applied to low energy QCD phenomena (e.g. quark confinement). Indeed, due to quark confinement, one expects a hadron gas resonance (HRG) phase at low energies and temperatures,
while, due to asymptotic freedom, a deconfined phase of quarks and gluons
called the quark-gluon plasma (QGP) is expected at high energies. Because of its asymptotic freedom, the latter could naively be expected to be a weakly interacting medium. In fact, at high enough temperatures, as attained in the quark epoch (where the cosmic background radiation temperature varied from hundreds of GeV to hundreds of MeV within a time window of microseconds), and before the QCD phase transition in the early universe, the QGP was a weakly coupled fluid. As a clear comparison, hard thermal loop (HTL) perturbation theory in QCD seems to provide a reasonable description of some thermodynamic observables computed non-perturbatively in lattice QCD (LQCD) simulations for temperatures T≳ 300 MeV <cit.>. However, at temperatures below that approximate threshold, the agreement between perturbative QCD (pQCD) and non-perturbative LQCD results is generally lost, which approximately sets the temperature window T_c ∼ 150 MeV < T < 2T_c ∼ 300 MeV (at zero baryon density) for which the QGP is a strongly coupled fluid <cit.>. This is just within the range of temperatures probed by relativistic heavy-ion collision experiments conducted e.g. at the Relativistic Heavy Ion Collider (RHIC) <cit.> and at the Large Hadron Collider (LHC) <cit.>.
§.§ Some phenomenological results from heavy-ion collisions
The strongly coupled nature of the QGP produced in heavy-ion collisions is not only deduced from thermodynamic observables but also from hydrodynamic transport coefficients. These coefficients are typically inferred from the analysis of phenomenological models simultaneously describing several types of heavy-ion data <cit.>.
The hot and dense medium produced in relativistic heavy-ion collisions is commonly believed to pass through several different stages during its space and time evolution, as sketched in Fig. <ref>. Initially, two heavy ions are accelerated to speeds close to the speed of light, and at very high energies, the gluon density inside those nuclei grows until reaching a saturation value, forming the so-called color glass condensate (CGC) <cit.>, which is a typical source of initial conditions for the medium produced after the collision. For a characteristic time interval ≲ 1 fm/c after the collision[Notice that 1 fm/c ≈ 3.33564× 10^-24 s, so that the characteristic time scales involved in heavy-ion collisions are extremely short.], in the pre-equilibrium stage, the system is expected to be described by a turbulent medium composed by highly coherent gluons. Therefore, this stage is dominated by the dynamics of classical chromodynamic fields forming the so-called glasma, a reference to the fact that this is an intermediate stage between the color glass condensate and the quark-gluon plasma <cit.>.
As the glasma expands and cools, it begins to decohere towards a state of QCD matter which possesses an effective description in terms of relativistic viscous hydrodynamics <cit.> and whose physically relevant degrees of freedom correspond to deconfined, but still strongly interacting quarks and gluons formed around ≳ 1 fm/c after the collision.
As the QGP keeps expanding and cooling, it eventually hadronizes by entering into the QGP-HRG crossover region of the QCD phase diagram <cit.>. The next stage of the space and time evolution of the system comprise the so-called chemical freeze-out <cit.>, when inelastic collisions between the hadrons cease and the relative ratio between the different kinds of particles in the hadron gas is kept fixed. Afterwards, there is the thermal or kinetic freeze-out, when the average distance between the hadrons is large enough to make the short-range residual strong nuclear interaction between them effectively negligible. This fixes the momentum distribution of the hadrons. After that, the produced hadrons are almost free and the particles resulting from their decays reach the experimental detectors, providing information on the previous stages in the evolution of the system.
Of particular relevance for the topics to be approached in the present review are the shear, η, and bulk viscosities, ζ. These hydrodynamic transport coefficients cannot be directly measured in heavy-ion collision experiments and are typically employed as free functions (of temperature and eventually also of other possible variables, such as chemical potentials and/or electromagnetic fields) in phenomenological hydrodynamic models, which are then fixed by comparison to heavy-ion data (for example, using Bayesian inference methods <cit.>).
From such an approach, it is generally found that, around the QGP-HRG crossover region at zero baryon density in the QCD phase diagram, η/s (where s is the entropy density of the medium) should be of the same order of magnitude (in natural units with c=ħ=k_B=1) of 1/4π (which, as we shall discuss in section <ref>, is a benchmark value for strongly coupled quantum fluids coming from a very broad class of holographic models <cit.>), being at least one order of magnitude smaller than perturbative calculations <cit.>. The small value of the shear viscosity to entropy density ratio, η/s, inferred for the QGP produced in heavy-ion collisions is physically interpreted as a clear manifestation of its nearly-perfect fluidity, as sketched in Fig. <ref>.
As a reference, in the QGP-HRG crossover window, where the QGP temperature is low enough to make the medium hadronize, T_c∼ 150 MeV ∼ 1.72× 10^12 K ∼ 10^5 T_center of sun (see e.g. https://solarscience.msfc.nasa.gov/interior.shtmlNASA/Marshall Solar Physics). In heavy-ion collisions realized in particle accelerators, the QGP attains temperatures at most 2 - 3 times T_c while much higher temperatures were achieved in the early universe.
Besides η/s, also the bulk viscosity to entropy density ratio ζ/s plays a prominent role in the phenomenological description of heavy-ion data <cit.>. For instance, in Ref. <cit.> the JETSCAPE Collaboration developed a state-of-the-art phenomenological multistage model for heavy-ion collisions, which was employed to simultaneously describe several hadronic measurements from different experiments at RHIC and LHC. Their results favor the temperature-dependent profiles (at zero baryon density) for ζ/s and η/s shown in Fig. <ref>. These phenomenological results for the hydrodynamic viscosities will be compared to quantitative microscopic holographic calculations and predictions in section <ref>.
By varying the conditions under which heavy-ion collisions take place in particle accelerators, it is possible to experimentally probe some aspects and regions of the QCD phase diagram at finite temperature and nonzero baryon density. For instance, for heavy-ion collisions at the LHC operating at the center of mass energies of √(s_NN) = 2.76 - 5.02 TeV, the energy of the collisions is so large that average effects due to a nonzero baryon chemical potential μ_B become negligible (note that fluctuations of conserved charges do still play a role at these energies <cit.>). On the other hand, the Beam Energy Scan (BES) program at RHIC scans out lower collision energies spanning the interval √(s_NN) = 7.7 - 200 GeV <cit.>, where the baryon chemical potential reached within the QGP is of the same order of magnitude of the temperature, allowing experimental access to some regions of the QCD phase diagram at nonzero μ_B. Furthermore, fixed-target experiments at RHIC <cit.>, and also experiments with lower collision energies at HADES <cit.>, and FAIR <cit.>, aim at experimentally probing the structure of the QCD phase diagram in the (T,μ_B)-plane at higher baryon densities. One of the main purposes of such experiments is to determine the location of the conjectured critical endpoint (CEP) of the line of first-order phase transition which, from several different model calculations, is expected to exist in the QCD phase diagram at high-baryon densities <cit.>.
§.§ Lattice QCD results
An important limitation of phenomenological multistage models is that
several physical inputs are not calculated from self-consistent microscopic models or systematic effective field theories.
As mentioned above, these inputs can be constrained by experimental data (and some underlying phenomenological model assumptions). However, such a phenomenological approach cannot explain why and how certain transport and equilibrium properties arise from QCD.
The strongly coupled nature of QCD at low energies renders the systematic methods of pQCD not applicable to describe a wide range of physically relevant phenomena that can be probed by experiments in high-energy particle accelerators
and also by astrophysical observations.
However, at vanishing or small chemical potentials μ_B,
another first-principles method for investigating equilibrium phenomena (such as the behavior of several thermodynamic observables) in QCD is available, namely, LQCD simulations.
The general reasoning behind this method, originally developed by Kenneth Wilson <cit.>, amounts to discretizing the Euclidean, imaginary-time version of the background spacetime. Matter fields, such as the fermion fields of the quarks, are defined at the sites of the resulting discretized grid, while gauge fields, such as the gluons, are treated as link variables connecting neighboring sites <cit.>. The Euclidean path integral, defined in the imaginary-time Matsubara formalism for finite-temperature statistical systems, can then be performed using Monte Carlo methods. Continuum QCD can formally be recovered by taking the limit in which the lattice spacing between neighboring sites goes to zero. In practice, due to the large increase in the computational cost of numerical simulations with decreasing lattice spacing, the formal continuum limit is approached by extrapolating a sequence of calculations with progressively decreasing lattice spacings, which are nonetheless still large enough to be computationally manageable <cit.>.
Some very remarkable achievements of LQCD relevant to this review include the first principles calculation of light hadron masses, like pions and nucleons, compatible with experimental measurements <cit.>, and also the determination of the nature of the transition between the HRG and QGP phases of QCD at zero baryon density, which turns out to be a broad continuous crossover <cit.>.
However, despite its notable successes, LQCD calculations also feature some important limitations, in particular: i) the difficulties in performing numerical simulations at nonzero baryon density, due to the so-called sign problem of lattice field theory <cit.>, and ii) the issues in calculating non-equilibrium transport observables associated with the real-time dynamics of the system. The former is an algorithmic issue that arises from the fermion determinant of the quarks becoming a complex quantity at real nonzero μ_B, which implies that it cannot be employed to define a probabilistic measure to be used in importance sampling — thus spoiling the direct evaluation of the LQCD path integral by means of Monte Carlo methods. The latter is due to difficulties in analytically continuing the Euclidean correlators calculated in the lattice at imaginary times to real-time intervals in a spacetime with Minkowski signature <cit.>.
Nonetheless, in recent years several different techniques have been developed and applied to calculate in LQCD the equation of state at finite temperature and moderate values of baryon chemical potential, and also to estimate the behavior of some transport coefficients at finite temperature and zero baryon density, as reviewed in Refs. <cit.>. In fact, state-of-the-art lattice simulations for the continuum-extrapolated QCD equation of state with 2+1 flavors and physical values of the quark masses are now available up to μ_B/T≤ 3.5 <cit.> from a novel expansion scheme, and up to μ_B/T≤ 3 from a traditional Taylor expansion <cit.>. Some of these LQCD results for thermodynamic observables at finite (T,μ_B) will be compared to quantitative microscopic holographic calculations and predictions in section <ref>.
§.§ Some basic aspects of the holographic gauge-gravity duality
The limitations of present-day lattice simulations mentioned above prevent first-principles QCD calculations to be employed in the investigation of strongly interacting QCD matter at higher baryon densities, where an actual phase transition
between confining hadronic and deconfined partonic degrees of freedom may exist, as depicted in the sketch displayed in Fig. <ref>. Also, LQCD simulations of QCD transport properties are considerably difficult already at μ_B=0 <cit.>, let alone at finite baryon density. In such cases, it is customary to resort to effective models and other alternative theoretical approaches to obtain some qualitative insight and even some quantitative predictions for the behavior of QCD matter under such extreme conditions.
One such alternative approach, which is the theoretical tool considered in the present review, is what is broadly called the holographic gauge-gravity duality (also known, under more restricted conditions, as the AdS-CFT correspondence) <cit.>. The holographic gauge-gravity duality is motivated by the framework of string theory, which originally had an old and curious relationship with the strong interaction. Indeed, (non-supersymmetric) string theory was originally developed as an S-matrix theory for the strong nuclear force between hadrons, which were empirically known to fall into linear Regge trajectories relating their total angular momentum J to their mass squared m^2, in what is known as the Chew-Frautschi plots <cit.>. By modeling a meson as a relativistic open string spinning around its center, it is possible to reproduce the observed Chew-Frautschi relation, J=α_0+α'm^2, where the relativistic string tension is given in terms of the measured slope of the linear Regge trajectory, σ=(2πα')^-1≈(440 MeV)^2 <cit.>. The slope is approximately the same for the different Regge trajectories defined by the different measured values of the Regge intercept, α_0 (which is known to depend on the flavor quantum numbers of the hadrons considered — hadrons with the same flavor quantum numbers fall into the same Regge trajectory, and can be viewed as resonances of this trajectory with different values of mass and angular momentum). However, since this simple string model also predicts results in striking contradiction with hadronic experiments (e.g. a wrong, soft exponential falloff for the associated Veneziano scattering amplitude in the high energy limit of hard scattering for hadrons at fixed angles), it has been abandoned as a model for hadrons, being superseded by the advent of QCD, with its theoretical and experimental successes as the fundamental description of the strong interaction.
Later, the theoretical interest in string theory greatly resurfaced, although within a very different context, with the so-called first and second superstring revolutions, which correspond, respectively: 1) to the discovery of five different consistent supersymmetric quantum string theories in 10 spacetime dimensions (superstring theories of Type I, Type IIA, Type IIB, Heterotic SO(32) and Heterotic E_8⊗ E_8); and also, 2) the latter discovery that these five superstring theories in 10 dimensions are related through a web of duality transformations, besides being also related to a theory of membranes defined in 11 spacetime dimensions called M-theory, whose low energy limit corresponds to a unique 11-dimensional theory of supergravity. A remarkable common feature of all superstring theories is that all of them possess a tensorial spin 2 massless particle in their spectrum, which is the graviton, the hypothetic vibrational string mode responsible for mediating the gravitational interaction at the quantum level. Due to that reason, and also due to the fundamental fact that at low energies superstring reduces to supergravity, therefore containing general relativity as the low energy, classical description of gravity, superstring theory is an interesting candidate for a theory of quantum gravity <cit.>. There is also some expectation that the standard model would emerge as a low-energy sector in string theory with 6 of its 10 dimensions compactified in some appropriate manifold, which should be chosen in a very specific way in order to generate the observed phenomenology of particle physics in our universe. This way, string theory could be seen as a “theory of everything”, in the sense of possibly describing all the particles and fundamental interactions in nature.
Regardless of whether string theory is the unifying theory of all the fundamental interactions of nature <cit.> or not, it is undeniable that new effective approaches and applications, directly inspired by string theory and aimed towards the strong interaction, flourished with the advent of the holographic gauge-gravity duality. Before discussing some of their phenomenological aspects in regard to the physics of the hot and baryon dense strongly-coupled QGP in section <ref>, we discuss below some basic general aspects of the holographic correspondence.
The original formulation of the so-called AdS-CFT correspondence <cit.>, relates Type IIB superstring theory defined on the product manifold between a 5-dimensional Anti-de Sitter (AdS) spacetime and a 5-dimensional sphere, AdS_5⊗ S^5, to a conformal quantum field theory (CFT) corresponding to 𝒩=4 Supersymmetric Yang-Mills (SYM) theory with gauge group SU(N_c),[𝒩=4 refers to the number of different supersymmetries of the theory.] defined on the conformally flat 4-dimensional boundary of AdS_5. Two other early realizations of the AdS-CFT duality comprise also the relation between M-theory defined on AdS_4⊗ S^7 and the Aharony-Bergman-Jafferis-Maldacena (ABJM) superconformal field theory defined on the 3-dimensional boundary of AdS_4, besides the relation between M-theory defined on AdS_7⊗ S^4 and the so-called 6D (2,0) superconformal field theory defined on the 6-dimensional boundary of AdS_7. In a very naive and imprecise way, one could in principle think of the first example of the 𝒩=4 SYM theory as a “toy model” for QCD, while the second example regarding the ABJM theory could be taken as a “toy model” for low-dimensional condensed matter systems. However, this is inadequate from a realistic phenomenological perspective, both at the quantitative and qualitative levels, as we shall discuss in section <ref>.
Before doing that, let us first comment a little bit more on the original proposal (see e.g. the discussion in section 3 of the standard review <cit.>, and also other works such as <cit.> for details). We take for definiteness the example relating Type IIB superstring theory compactified on AdS_5⊗ S^5 and 𝒩=4 SYM theory living on the boundary of AdS_5. One first considers Type IIB string theory in flat ℝ^1,9 Minkowski spacetime and a collection of N_c coincident parallel D3-branes in this background.[An endpoint of an open string must satisfy either Dirichlet or Neumann boundary conditions. If one considers Neumann boundary conditions on p spatial dimensions plus time, then the remaining D-p-1 dimensions must satisfy Dirichlet boundary conditions. Since for Dirichlet boundary conditions a string endpoint is fixed in space, while for Neumann boundary conditions it must move at the speed of light, then with Neumann boundary conditions on p+1 dimensions, the open string endpoints are constrained to move within a (p+1)-dimensional hypersurface, which is a dynamical object called Dp-brane. Dp-branes are shown to be related to black p-branes <cit.>, which are solutions of higher dimensional (super)gravity which generalize the concept of black holes by having extended event horizons which are translationally invariant through p spatial dimensions. They actually provide different descriptions of a single object, which in a perturbative string regime is accurately described by Dp-branes not backreacting on the background spacetime, while at low energies (corresponding to take α'≡ l_s^2 to be small, where l_s is the fundamental string length, so that massive string states can be neglected) and large gravitational fields, the backreaction of the Dp-branes on the background produces a black p-brane geometry <cit.>.] The perturbative string theory excitations in this system correspond to vibrational modes of both, closed strings, and also open strings with their ends attached to the D3-branes. If we consider the system defined at low energies compared to the characteristic string scale, (α')^-1/2≡(l_s)^-1, only massless string modes can be excited which, for closed strings give a gravity supermultiplet and, for the open strings with their ends attached to the (3+1)-dimensional worldvolume of the N_c coincident D3-branes, give a 𝒩=4 vector supermultiplet with gauge group SU(N_c). A low energy effective action for these massless string excitations in the background considered can be schematically written by integrating out the massive string modes,
S_eff = S_ℝ^1,9 bulk + S_ℝ^1,3 brane + S_int,
where S_ℝ^1,9 bulk is the low energy action for the gravity supermultiplet, corresponding to Type IIB supergravity (SUGRA) in ℝ^1,9 plus higher order derivative corrections coming from the integration of the string massive modes; S_ℝ^1,3 brane is the low energy action for the 𝒩=4 vector supermultiplet living on the ℝ^1,3 worldvolume of the N_c coincident D3-branes, corresponding to 𝒩=4 SYM theory with gauge group SU(N_c) plus higher order derivative corrections coming from the integration of the string massive modes; and S_int is an interaction term between the bulk and brane modes.
The higher order derivative corrections for the bulk and brane actions coming from the integration of massive string modes are proportional to positive powers of α', while the interaction action is proportional to positive powers of the square root of the 10D Newton's gravitational constant, κ_10≡√(8π G_10)∼ g_sα' ^2, where g_s is the string coupling, so that by considering the so-called decoupling limit where α'≡ l_s^2→ 0 with fixed N_c,g_s, one has S_ℝ^1,9 bulk→ S_ℝ^1,9 IIB SUGRA, S_ℝ^1,3 brane→ S_ℝ^1,3 𝒩=4 SYM, and S_int→ 0, so that we end up with two decoupled actions,
lim_α'→ 0 (fixed N_c,g_s) S_eff = S_ℝ^1,9 IIB SUGRA + S_ℝ^1,3 𝒩=4 SYM.
For a given number N_c of coincident D3-branes, the `t Hooft coupling effectively controlling the strength of the interactions in the 𝒩=4 SYM SU(N_c) gauge theory is given by λ_t≡ N_c g_SYM^2= N_c g_s.[The relation g_SYM^2= g_s can be inferred from the fact that a closed string, governed by the g_s coupling, can be formed from the collision between the endpoints of two open strings moving on the D3-branes, with g_SYM being the coupling of the non-Abelian gauge field corresponding to the massless mode of the open strings on these branes <cit.>.] This picture holds for any value of λ_t (and since the SYM theory is a CFT, its `t Hooft coupling remains constant for any value of energy so that one actually has infinitely many different SYM theories, each one of them defined at some given value of λ_t).
Another perspective for the same system can be considered as follows. The effective gravitational field generated by the collection of N_c coincident D3-branes is ∼ N_c g_s (l_s/r)^4 <cit.>, and by considering a very large N_c such that λ_t = N_c g_s≫ 1 even for small values of g_s (so that one can ignore quantum string loop contributions in the bulk), very close to the D3-branes for r→ 0 the gravitational field is very intense and its backreaction on the background spacetime highly distorts its geometry, producing a curved manifold. In this limit it is necessary to replace the perturbative string description of D3-branes in flat Minkowski spacetime with the associated black 3-brane supergravity solution, whose near-horizon (i.e. near-black brane) geometry approaches precisely that of AdS_5(L)⊗ S^5(L), with the same curvature radius L for the AdS_5 and S^5 manifolds.[For the other two early examples of the AdS-CFT correspondence mentioned before, one obtains: AdS_4(L/2)⊗ S^7(L) and AdS_7(2L)⊗ S^4(L) (see e.g. <cit.>).] On the other hand, far away from the black brane the background geometry is still that of Minkowski ℝ^1,9. In both regions (near and far from the black brane), since we considered that the string coupling g_s is small (so that string loops may be discarded), by taking the decoupling limit as before, with l_s→ 0 and fixed N_c,g_s, the bulk spacetime is inhabited only by Type IIB SUGRA fields.
By comparing the two perspectives above for the same system, when defined in the same regime corresponding to low energies, low string coupling, large N_c, and strong `t Hooft coupling (α'≡ l_s^2→ 0 with fixed N_c,g_s, but such that g_s is small, N_c is large and λ_t = N_c g_SYM^2 = N_c g_s≫ 1), one notices that in both views there is a common element, which is Type IIB SUGRA defined on ℝ^1,9, and it is then conjectured that the remaining pieces in each perspective should be dual to each other: strongly coupled, large N_c, 𝒩=4 SYM theory with gauge group SU(N_c), defined on ℝ^1,3 (which is equivalent, up to a conformal factor, to the boundary of AdS_5), and classical, weakly coupled Type IIB SUGRA defined on AdS_5(L)⊗ S^5(L). The duality involved in this comparison actually conveys a detailed mathematical dictionary translating the evaluation of physical observables in a classical SUGRA theory defined at weak coupling on top of a background given by the product of an AdS spacetime and a compact manifold, to the calculation of other observables in a different, conformal quantum gauge field theory defined at strong coupling and with a large number of colors on top of the conformally flat boundary of the AdS manifold. Then, the notion of the hologram comprised in the AdS-CFT duality refers to the fact that the gravitational information of a higher dimensional bulk spacetime can be encoded in its boundary.
This is the weakest form of the holographic AdS-CFT correspondence, and a particular case of the broader gauge-gravity duality, being largely supported by a plethora of independent consistency checks (see e.g. <cit.>). The strongest version of the AdS-CFT conjecture, corresponding to a particular case of the so-called gauge-string duality (which is more general than the gauge-gravity duality, which can be seen as a low-energy limit of the latter), proposes that the duality should be valid for all values of g_s and N_c, therefore relating 𝒩=4 SYM theory on ℝ^1,3 with arbitrary `t Hooft coupling and an arbitrary number of colors for the gauge group SU(N_c), and full quantum Type IIB superstring theory generally formulated in a nonperturbative way on AdS_5(L)⊗ S^5(L) (instead of just its classical low energy limit corresponding to Type IIB SUGRA). It is also posited that high derivative/curvature corrections in the bulk correspond to the inverse of `t Hooft coupling corrections in the dual CFT, since according to the detailed holographic dictionary, α'/L^2={l_s/[l_s (N_c g_s)^1/4]}^2 =1/√(λ_t), and that quantum string loop corrections in the bulk correspond to the inverse of N_c corrections in the dual CFT, since, g_s (l_s/L)^4 = g_s (l_s/[l_s (N_c g_s)^1/4])^4 = 1/N_c.
The conjectured holographic AdS-CFT duality has a very clear attractive feature, which is the fact that complicated nonperturbative calculations in a strongly coupled quantum CFT can be translated, through the detailed mathematical holographic dictionary, into much simpler (although not necessarily easy) calculations involving weakly coupled classical gravity in higher dimensions.
More generally, the broader holographic gauge-gravity duality[The even broader gauge-string duality is very difficult to handle in practice, due to the present lack of a detailed and fully nonperturbative definition of string theory on asymptotically AdS spacetimes. Consequently, we focus in this review only on its low-energy manifestation corresponding to the gauge-gravity duality, which is the framework where the vast majority of the calculations are done in the literature regarding the holographic correspondence.] is not restricted to bulk AdS spacetimes and dual boundary CFTs. Indeed, for instance, by considering the backreaction of effective 5D massive fields living on AdS_5, which are associated with the Kaluza-Klein (KK) reduction on S^5 of the originally 10D massless modes of SUGRA, the background AdS_5 metric is generally deformed within the bulk, and the effective 5D bulk spacetime geometry becomes just asymptotically AdS, with the metric of AdS_5 being recovered asymptotically near the boundary of the bulk spacetime. Generally, there is also a corresponding deformation of the dual QFT theory at the boundary of the asymptotically AdS spacetime induced by the consideration of relevant or marginal operators, which may break conformal symmetry and supersymmetry and whose scaling dimension is associated through the holographic dictionary to the masses of the effective 5D bulk fields. In this sense, one has a broader holographic gauge-gravity duality relating a strongly coupled QFT (not necessarily conformal or supersymmetric) living at the boundary of a higher dimensional asymptotically AdS spacetime, whose geometry is dynamically determined by a classical gravity theory interacting with different matter fields in the bulk. In the holographic gauge-gravity duality, the extra dimension connecting the bulk asymptotically AdS spacetime to its boundary plays the role of a geometrization of the energy scale of the renormalization group flow in the QFT living at the boundary <cit.>, with low/high energy processes in the QFT being mapped into the deep interior/near-boundary regions of the bulk spacetime, respectively.
Since its original proposal by Maldacena in 1997 <cit.>, the holographic gauge-gravity duality has established itself as one of the major breakthroughs in theoretical physics in the last few decades, being applied to obtain several insights into the nonperturbative physics of different strongly coupled quantum systems, comprising studies in the context of the strong interaction <cit.>, condensed matter systems <cit.> and, more recently, also quantum entanglement and information theory <cit.>.
§.§ Main purpose of this review
Holographic gauge-gravity models are generally classified as being either i) top-down constructions when the bulk supergravity action comes from known low-energy solutions of superstrings and the associated holographic dual at the boundary is precisely determined, ii) or bottom-up constructions when the bulk effective action is generally constructed by using phenomenological inputs and considerations with the purpose of obtaining a closer description of different aspects of some real-world physical systems, but the exact holographic dual, in this case, is not precisely known. Actually, for bottom-up holographic models, one assumes or conjectures that the main aspects of the gauge-gravity dictionary inferred from top-down constructions remain valid under general circumstances, such that for a given asymptotically AdS solution of Einstein field equations coupled to other fields in the bulk, some definite holographic dual QFT state at the boundary should exist.[This putative bottom-up holographic dual does not need to (and generally will not) coincide with the exact QFT taken as a target to be described in the real world. Instead, one will generally obtain some holographic dual of a QFT which is close to some aspects of the target QFT, but which differs from the latter in many other regards. In a general sense, this is not different, for instance, from the reasoning employed to construct several non-holographic effective models for QCD, where a given effective model is used to produce approximate results for some but not all aspects of QCD. In fact, if an exact holographic dual of real-word QCD (with gauge group SU(3), 6 flavors and physical values of the quark masses) does exist, its dual bulk formulation will likely comprise not merely a gravity dual, but instead some complicated nonperturbative full string dual whose formulation is currently unknown.] In order to be useful in practice for different phenomenological purposes, such an assumption for bottom-up holographic models should provide explicit examples where the target phenomenology is indeed well reproduced by the considered bulk gravity actions, which should furthermore be able to provide new and testable predictions. In fact, as we are going to discuss in this review, one can construct holographic bottom-up models which are able to provide quantitative results and predictions in compatibility with first principles LQCD simulations and with some phenomenological outputs inferred from heavy-ion collisions, besides providing new predictions for thermodynamic and transport quantities in regions of the QCD phase diagram currently not amenable to first principles analysis due to the limitations discussed in the preceding sections.
Let us first analyze thermal SYM theory[That the SYM theory is completely inadequate as a holographic model for the confined phase of QCD is immediately obvious from e.g. the fact that SYM is a CFT and QCD is not. Even if one considers a comparison of SYM with just pure YM theory (i.e. the pure gluon sector of QCD without dynamical quarks), issues remain since YM features linear confinement between static, infinitely heavy probe quarks (corresponding to an area law for the Wilson loop <cit.>) and a mass gap in the spectrum.] as a possible “proxy” for the strongly coupled deconfined QGP, as it has been commonly considered within a considerable part of the holographic literature for years. It is often said that SYM theory has some qualitative features in common with QCD at the typical temperatures attained by the QGP in heavy-ion collisions, namely: within the considered temperature window, both theories are strongly coupled, deconfined, with non-Abelian vector fields corresponding to gluons transforming in the adjoint representation of the gauge group, and their η/s have comparable magnitude.
Although the points above are true, they are insufficient to establish a reliable connection between SYM and QCD. Indeed, there are infinitely many different holographic theories with the same properties listed above. In fact, all gauge-gravity duals are strongly coupled and all isotropic and translationally invariant Einstein's[That is, with the kinetic term for the metric field in the bulk action given by the usual Einstein-Hilbert term with two derivatives.] gauge-gravity duals have a specific shear viscosity given by the “(quasi)universal holographic” result η/s=1/4π <cit.>, which is actually a clear indication that even for nonconformal gauge-gravity duals with running coupling (which is not the case of SYM theory, since it is a CFT), the effective coupling of the holographic theory remains large at all temperature scales. Consequently, classical gauge-gravity duals lack asymptotic freedom, featuring instead a strongly coupled ultraviolet fixed point, being asymptotic safe but not asymptotic free. Moreover, there are infinitely many different holographic duals with deconfined phases at high temperatures. In the face of this infinite degeneracy of holographic gauge-gravity duals with the very same generic features often employed to “justify” the use of SYM theory as a “proxy” for the QGP, one may be led to conclude that such a choice is not well-defined. One may argue that this choice is more related to the fact that SYM theory is the most well-known and one of the simplest examples of gauge-gravity duality, than to any realistic phenomenological connection between the SYM plasma and the real-world QGP.
In order to take steps towards lifting the infinite degeneracy of holographic models to describe (some aspects of) the actual QGP, one needs to look at the behavior of more physical observables than just η/s. In this regard, the SYM plasma is easily discarded as a viable phenomenological holographic model for the QGP due to several reasons, among which we mention mainly the following. The SYM plasma is a CFT, while the QGP is highly nonconformal within the window of temperatures probed by heavy-ion collisions, and this fact makes the equation of state for the SYM plasma completely different from the one obtained for the QGP in LQCD simulations, not only quantitatively, but also qualitatively <cit.>. Indeed, dimensionless ratios for thermodynamic observables such as the normalized pressure (P/T^4), energy density (ϵ/T^4), entropy density (s/T^3), the speed of sound squared (c_s^2), and the trace anomaly (I/T^4=(ϵ-3P)/T^4, which is identically zero for a CFT), are all given by constants in the SYM plasma, while they display nontrivial behavior as functions of the temperature in the QGP. Furthermore, the bulk viscosity vanishes for the conformal SYM plasma, while it is expected to possess nontrivial behavior as a function of the temperature in the QGP, playing an important role in the description of heavy-ion data, as inferred from phenomenological multistage models (see the discussion in section <ref> and Fig. <ref>). Therefore, when considering thermodynamic equilibrium observables and transport coefficients, the SYM plasma is not a realistic model for the QGP both at the quantitative and qualitative levels.
On the other hand, the holographic duality can be indeed employed to construct effective gauge-gravity models which make it possible to actually calculate several thermodynamic and transport observables, displaying remarkable quantitative agreement with state-of-the-art LQCD simulations at zero and finite baryon density, while simultaneously possessing transport properties very close to those inferred in state-of-the-art phenomenological multistage models for heavy-ion collisions. Additionally, such holographic models also provide quantitative predictions for the QGP in regions of the QCD phase diagram which are currently out of the reach of first-principles calculations. The main purpose of the present paper is to review these results, mainly obtained through specific bottom-up constructions engineered within the so-called Einstein-Maxwell-Dilaton class of holographic models, discussing the main reasoning involved in their formulation, and also pointing out their phenomenological limitations and drawbacks, in addition to their successful achievements. This will be done in the course of the next sections, with holographic applications to the hot and baryon dense strongly coupled QGP being discussed in section <ref>. We will also review some applications to the hot and magnetized QGP (at zero chemical potential) in section <ref>. In the concluding section <ref>, we provide an overview of the main points discussed through this review and list important perspectives for the future of phenomenological holographic model applications to the physics of the QGP.
In this review, unless otherwise stated, we make use of natural units where c=ħ=k_B=1, and adopt a mostly plus metric signature.
§ HOLOGRAPHIC MODELS FOR THE HOT AND BARYON DENSE QUARK-GLUON PLASMA
In this section, we review the construction and the main results obtained from phenomenologically-oriented bottom-up holographic models aimed at a quantitative description of the strongly coupled QGP at finite temperature and baryon density. We focus on a class of holographic constructions called Einstein-Maxwell-Dilaton (EMD) gauge-gravity models, which has provided up to now the best quantitative holographic models for describing equilibrium thermodynamic and hydrodynamic transport properties of the hot and baryon dense QGP produced in heavy-ion collisions. We also discuss different predictions for the structure of the QCD phase diagram, comprising at high baryon chemical potential a line of first-order phase transition ending at a CEP, which separates the phase transition line from the smooth crossover observed at low baryon densities.
§.§ Holographic Einstein-Maxwell-Dilaton models
In order to possibly obtain a quantitative holographic model for the QGP (and also quantitative holographic constructions for other strongly coupled physical systems in the real world), one necessarily needs to break conformal symmetry in the holographic setting. Breaking conformal symmetry alone is not sufficient to reproduce QCD, since one needs to obtain a holographic modeling of specific phenomenological properties, and not just an arbitrary or generic nonconformal model. Therefore, the conformal symmetry-breaking pattern needs to be driven in a phenomenologically-oriented fashion.
One possible approach to obtain a non-conformal system is a bottom-up holographic construction where the free parameters of the model are constrained by existing results from LQCD in some specific regime. Once the parameters are fixed, one can then use this model to make predictions.
Of course, as in any effective theory construction, the functional form of the bulk action and also the ansatze for the bulk fields must be previously chosen based on some symmetry and other relevant considerations, taking into account a given set of observables from the target phenomenology and the basic rules for evaluating these observables using holography.
The seminal works of <cit.> laid down a remarkably simple and efficient way of constructing quantitative holographic models for the strongly coupled QGP in equilibrium. The general reasoning originally developed in these works may be schematically structured as follows:
* The focus is on constructing an approximate holographic dual or emulator for the equation of state of the strongly coupled QGP in the deconfined regime of QCD, without trying to implement confinement (e.g. Regge trajectories for hadrons), chiral symmetry breaking at low temperatures, asymptotic freedom at asymptotically high temperatures, nor an explicit embedding into string theory.
In this construction, the QCD equation of state (and the second-order baryon susceptibility for the case of finite baryon densities, see <ref>) is used to fix the free parameters at finite temperature and vanishing chemical potentials. Note that only these specific LQCD data are used to fix the free parameters of the models. All other resulting thermodynamic quantities or transport coefficients are then predictions of the model;
* The dynamical field content and the general functional form of the bulk gravity action is taken to be the simplest possible in order to accomplish the above.
One considers a bulk metric field (holographically dual to the boundary QFT energy-momentum tensor) plus a Maxwell field with the boundary value of its time component providing the chemical potential at the dual QFT. Additionally, a real scalar field (called the dilaton) is used to break conformal symmetry in the holographic setting, emulating the QGP equation of state at zero chemical potential. The dilaton field also relates string and Einstein frames, as used e.g. in the holographic calculation of parton energy loss (some results in this regard will be briefly reviewed in section <ref>);
* The general functional form for the bulk action constructed with the dynamical field content features at most two derivatives of the fields. The bulk action includes the Einstein-Hilbert term with a negative cosmological constant (associated with asymptotically AdS_5 spacetimes) for the metric field g_μν, the kinetic terms for the Abelian gauge field A_μ and the dilaton field ϕ, an almost arbitrary potential (free function) V(ϕ) for the dilaton, and an interaction term between the Maxwell and the dilaton fields, which features another free function of the dilaton field, f(ϕ).
The free functions, V(ϕ) and f(ϕ), the effective 5D Newton's constant, G_5, and the characteristic energy scale of the nonconformal model, Λ∝ L^-1, need to be dynamically fixed by holographically matching the specific set of LQCD results mentioned in the first item above. Note that these parameters comprise the entire set of free parameters of the bottom-up EMD construction.
* The effects of the dynamical quarks in the medium are assumed to be effectively encoded in the form of the bottom-up model parameters fixed to holographically match the QCD equation of state and second-order baryon susceptibility obtained from LQCD simulations at zero chemical potential (no explicit flavor-branes are employed for this purpose in the holographic EMD models reviewed in the present paper).
More details on the procedure mentioned above will be discussed in section <ref>. Let us now comment on the main limitations of such an approach, some of which are fairly general and refer to all classical gauge-gravity models.
First, gauge-gravity models such as the one mentioned above lack asymptotic freedom. This is expected from the original AdS-CFT correspondence since classical gravity in the bulk lacks the contributions coming both from massive string states and quantum string loops. By discarding such contributions in the bulk, one obtains a strongly coupled dual QFT at the boundary with a large number of degrees of freedom (large N_c).
The consideration of deformations of the bulk geometry given by asymptotic (but not strictly) AdS solutions of classical gravity does not seem enough to claim that such deformations could in principle describe asymptotic freedom in the dual gauge theory at the boundary.
The fact that η/s=1/4π for any value of temperature (and chemical potentials) in isotropic and translationally invariant gauge-gravity models with two derivatives of the metric field, conformal or not, is a clear indication that such models are strongly coupled at all energy scales. Therefore, these models miss asymptotic freedom in the ultraviolet regime.
It is then clear that the ultraviolet regime of such models is in striking contradiction with perturbative QCD (excepted to be relevant at high temperatures), where η/s is an order of magnitude larger than 1/4π.
One possible way of improving this situation has been discussed in Ref. <cit.>.
There they consider the effects of higher curvature corrections to the metric field in the bulk (i.e., higher derivative corrections to Einstein's gravity) in the presence of a dilaton field, which allows for a temperature-dependent η/s.
Higher derivative corrections for the bulk action are associated with contributions coming from massive string states, which are expected to lead to a reduction of the effective coupling of the boundary QFT theory. However, consistently including higher derivative curvature corrections for an EMD model, taking into account the full dynamical backreaction of the higher curvature terms into the background geometry, is a very challenging task that has yet to be done.
Another general limitation of gauge-gravity models for QCD is that a realistic holographic description of thermodynamic and hydrodynamic observables in the HRG confining phase is unfeasible. Standard gauge-gravity models describe large N_c systems. However, the pressure of the QCD medium in the confining hadronic phase goes as ∼ N_c^0 = 𝒪(1), while in the deconfined QGP phase it goes as ∼ N_c^2. Therefore, the pressure in hadron thermodynamics is N_c^-2 suppressed relative to the pressure in the QGP phase in a large N_c expansion. Formally, the hadron phase requires string loop corrections in the bulk in order to have a feasible holographic dual description at the boundary. Such a quantum string loop corrected holographic dual would be more complicated than simple classical gauge-gravity models.
The two above limitations are common to all gauge-gravity models aimed at realistically describing QCD. Further limitations are related to the EMD constructions reviewed here. We have already alluded to the fact that such models are not intended to describe chiral symmetry breaking, confinement, and thus, hadron spectroscopy. These points, together with the intrinsic limitations of gauge-gravity models regarding the description of hadron thermodynamics and asymptotic freedom, clearly restrict the target phenomenology of such EMD models to be the hot deconfined phase of QCD matter corresponding to the strongly coupled QGP produced in heavy-ion collisions.
Another phenomenological limitation of EMD models is that they only describe a single conserved charge (i.e. only one finite chemical potential is possible). Typically, finite baryon chemical potential, μ_B is considered (see Sec. <ref>).
However, the hot and baryon dense QGP produced in relativistic heavy-ion collisions at low energies actually comprises three chemical potentials (μ_B, the electric charge chemical potential, μ_Q, and the strangeness chemical potential, μ_S). In equilibrium, these chemical potentials can be related to each other through the global strangeness neutrality condition realized in such collisions, due to the fact that the colliding nuclei do not carry net strangeness. The strangeness neutrality condition is
⟨ S⟩ = ⟨ N_S̅-N_S⟩ = VT^3χ̂_1^S = 0
where N_S is the number of strange quarks, N_S̅ is the number of strange antiquarks, and χ̂_1^S≡∂(P/T^4)/∂(μ_S/T) is the reduced strangeness density.
Additionally, μ_Q can also be constrained by the charge to baryon number ratio of the colliding nuclei.
There is a small isospin imbalance for lead-lead (Pb+Pb) collisions at the LHC and gold-gold (Au+Au) collisions at RHIC,
⟨ Q⟩/⟨ B⟩ = ⟨ N_Q - N_Q̅⟩/⟨ N_B - N_B̅⟩ = χ̂_1^Q / χ̂_1^B = Z/A ≈ 0.4
where Z is the atomic number and A is the mass number of the colliding nuclei. Thus, between strangeness neutrality and charge conservation, we can then determine μ_Q=μ_Q(T,μ_B) and μ_S=μ_S(T,μ_B) from (T,μ_B) <cit.>. These phenomenological constraints from heavy-ion collisions are not implemented in the holographic EMD constructions reviewed here, where one simply sets μ_Q=μ_S=0.
We finish these introductory comments on phenomenological bottom-up holographic EMD models for the QGP by remarking that these models are partially inspired by, but not actually derived from string theory. Therefore, the actual applicability of the holographic dictionary for such constructions, and more generally, for any bottom-up gauge-gravity model, may be questioned. Indeed, the phenomenological viability of bottom-up holographic models can be checked by direct comparison with the results of the target phenomenology.
The degree of agreement between holographic EMD results and several first principles LQCD calculations as well as hydrodynamic viscosities inferred from phenomenological multistage models describing several heavy-ion data, provides compelling evidence that the holographic dictionary works in practice for these models.
The general reasoning outlined above may be systematically adapted to successfully describe different aspects of phenomenology, indicating that at least some of the entries in the holographic dictionary may have a broad range of validity.
For instance, one could consider using gauge-gravity models to describe pure YM theory without dynamical quarks.
Bottom-up dilatonic gauge-gravity models with specific functional forms for the dilaton potential may be engineered to quantitatively describe the thermodynamics of a deconfined pure gluon plasma with a first-order phase transition (although the thermodynamics of the confining phase corresponding to a gas of glueballs cannot be described by classical gauge-gravity models), besides describing also glueball spectroscopy <cit.>.
§.§.§ Holographic equations of state
A gauge-gravity model is usually defined by its action on the classical gravity side of the holographic duality, while different dynamic situations for its dual QFT, living at the boundary of the asymptotically AdS bulk spacetime, are related to different ansatze and boundary conditions for the bulk fields. For instance, given some bulk action, the vacuum state in the dual QFT is associated with solutions of the bulk equations of motion with no event horizon, which is accomplished by an ansatz for the metric field with no blackening function. Thermal states in equilibrium for the same dual QFT are often associated with equilibrium black hole (or more generally, black brane) solutions of the bulk equations of motion, which now require a blackening function in the ansatz for the metric field. Hydrodynamic transport coefficients and characteristic equilibration time scales may be evaluated from the spectra of quasinormal modes <cit.> of these black hole solutions slightly disturbed out of thermal equilibrium, while different far-from-equilibrium dynamics may be simulated by taking into account boundary conditions and ansatze for the bulk fields with nontrivial dependence on spacetime directions parallel to the boundary <cit.>.
The main bottom-up holographic models reviewed in the present manuscript are specified by actions of the EMD class, whose general form in the bulk is given below <cit.>,
S=∫_ℳ_5 d^5x ℒ = 1/2κ_5^2∫_ℳ_5 d^5x √(-g)[R-(∂_μϕ)^2/2 -V(ϕ) -f(ϕ)F_μν^2/4],
where κ_5^2≡ 8π G_5 is the 5D gravitational constant. The bulk action (<ref>) is supplemented by two boundary terms: i) the Gibbons-Hawking-York (GHY) boundary action <cit.>, which in a manifold ℳ_5 with a boundary (as in the case of asymptotically AdS spacetimes) is required in the formulation of a well-defined variational problem with a Dirichlet boundary condition for the metric field,[By the variational principle, the variation of the gravity action must vanish for arbitrary variations δ g_μν of the metric field in the bulk. In the case of spacetime manifolds with a boundary, in calculating the variation of the metric tensor in the bulk, integration by parts in directions transverse to the boundary leads to a boundary term that is nonvanishing even by imposing the Dirichlet boundary condition that the metric is held fixed at the boundary, δ g_μν|_∂ℳ_5=0. This boundary term is exactly canceled out by the variation of the GHY action (see e.g. chapter 4 of <cit.>), allowing for the variation of the total gravity action to vanish in compatibility with Einstein's equations in a bulk spacetime with a boundary.] and ii) a boundary counterterm action employed to remove the ultraviolet divergences of the on-shell action by means of the holographic renormalization procedure <cit.>. Since these two boundary actions do not contribute to the bulk equations of motion while being required in order to write the full holographic renormalized on-shell action, which will not be needed in the calculations reviewed in the present work, we do not write their explicit form here.[The holographic renormalized on-shell action is employed in the evaluation of the pressure of the medium defined in the dual QFT at the boundary, also for the calculation of hydrodynamic transport coefficients extracted from perturbations of the bulk fields, and for the analysis of far-from-equilibrium dynamics. However, here we will not consider far-from-equilibrium calculations. Regarding the equilibrium pressure of the medium, its calculation can also be done by integrating the entropy evaluated through the Bekenstein-Hawking relation for black hole thermodynamics <cit.> over the temperature, which does not require holographic renormalization. Moreover, for the holographic calculation of the specific hydrodynamic transport coefficients reviewed in this work, which are related through Kubo formulas to the imaginary part of thermal retarded correlators of the relevant dual QFT operators, holographic renormalization can also be bypassed through the use of radially conserved fluxes extracted from the equations of motion for the relevant bulk perturbations — see <cit.> and also <cit.>.]
The set of free parameters and functions {G_5,Λ,V(ϕ),f(ϕ)} comprised in the bottom-up EMD setup can be fixed by taking as phenomenological inputs some adequate lattice results on QCD thermodynamics at finite temperature and zero chemical potentials (and vanishing electromagnetic fields), where Λ is a characteristic energy scale of the nonconformal holographic model employed to express in powers of MeV dimensionful observables in the dual QFT, which are calculated in the gravity side of the holographic correspondence in powers of the inverse of the asymptotic AdS radius L. In practice, we simply set L=1 and trade it off as a free parameter by the energy scale Λ, without changing the number of free parameters of the model <cit.>. The set {G_5,Λ,V(ϕ)} can be fixed by the LQCD equation of state evaluated at vanishing chemical potential, while f(ϕ) may be fixed, up to its overall normalization, by the LQCD second order baryon susceptibility, also evaluated at zero chemical potential <cit.>.[However, as we are going to discuss afterward in this section, and more deeply in section <ref>, available LQCD results cannot constrain the set of free parameters of the EMD model to be fixed in a unique way.]
In order to do this, one first needs to specify the adequate ansatze for the bulk EMD fields such as to describe isotropic and translationally invariant thermal states at the dual boundary quantum gauge theory (as in LQCD simulations). Since we are going to consider, in general, also the description of thermal states at finite baryon chemical potential, we take the form below for the bulk fields corresponding to isotropic and translationally invariant charged EMD black hole backgrounds in equilibrium <cit.>,
ds^2 = g_μνdx^μ dx^ν = e^2A(r)[-h(r)dt^2+dx⃗^2]+dr^2/h(r), ϕ = ϕ(r), A = A_μdx^μ=Φ(r)dt,
where r is the holographic radial coordinate, with the boundary at r→∞ and the black hole horizon at r=r_H, and r_H being the largest root of the blackening function, h(r_H)=0. The set of general EMD equations of motion obtained by extremizing the bulk action (<ref>) with respect to the EMD fields can be written in the following form <cit.>,
R_μν-g_μν/3[V(ϕ)-f(ϕ)/4F_αβ^2]-1/2∂_μϕ∂_νϕ-f(ϕ)/2g^αβF_μαF_νβ =0,
∂_μ(√(-g)f(ϕ)g^μαg^νβF_αβ) =0,
1/√(-g)∂_μ(√(-g)g^μν∂_νϕ)-∂ V(ϕ)/∂ϕ-F_μν^2/4∂ f(ϕ)/∂ϕ =0,
which, for the isotropic ansatze for the EMD fields in equilibrium given in Eqs. (<ref>), reduce to the following set of coupled ordinary differential equations of motion,
ϕ”(r)+[h'(r)/h(r)+4A'(r)]ϕ'(r)-1/h(r)[∂ V(ϕ)/∂ϕ-e^-2A(r)Φ'(r)^2/2∂ f(ϕ)/∂ϕ] =0,
Φ”(r)+[2A'(r)+d[lnf(ϕ)]/dϕϕ'(r)]Φ'(r) =0,
A”(r)+ϕ'(r)^2/6 =0,
h”(r)+4A'(r)h'(r)-e^-2A(r)f(ϕ)Φ'(r)^2 =0,
h(r)[24A'(r)^2-ϕ'(r)^2]+6A'(r)h'(r)+2V(ϕ)+e^-2A(r)f(ϕ)Φ'(r)^2 =0,
where Eq. (<ref>) is a constraint. These equations of motion are discussed in detail in Refs. <cit.>. They must be solved numerically, and different algorithms have been developed through the years to accomplish this task with increasing levels of refinement <cit.>. Two different sets of coordinates are used in this endeavor: the so-called standard coordinates (denoted with a tilde), in which the blackening function goes to unity at the boundary, h̃(r̃→∞)=1, and also Ã(r̃→∞)→r̃, such that holographic formulas for the physical observables are expressed in standard form; and the so-called numerical coordinates (denoted without a tilde), corresponding to rescalings of the standard coordinates used to specify definite numerical values for the radial location of the black hole horizon and also for some of the initially undetermined infrared expansion coefficients of the background bulk fields close to the black hole horizon, which is required to start the numerical integration of the bulk equations of motion from the black hole horizon up to the boundary.[Notice that the part of the bulk geometry within the interior of the black hole horizon is causally disconnected from observers at the boundary.] In fact, with such rescalings, all the infrared coefficients are determined in terms of just two initially undetermined coefficients, ϕ_0 and Φ_1, which are taken as the “initial conditions” (in the holographic radial coordinate, r) for the system of differential equations of motion. Those correspond, respectively, to the value of the dilaton field and the value of the radial derivative of the Maxwell field evaluated at the black hole horizon.
For the holographic calculation of physical observables at the boundary QFT, one also needs to obtain the ultraviolet expansion coefficients of the bulk fields near the boundary, far from the horizon. For the evaluation of the observables reviewed in this paper, it suffices to determine four ultraviolet expansion coefficients of the bulk fields, namely, h_0^far coming from the blackening function h(r) of the metric field, Φ_0^far and Φ_2^far coming from the nontrivial component of the Maxwell field Φ(r), and ϕ_A coming from the dilaton field ϕ(r), with the functional forms of the ultraviolet expansions being derived by solving the asymptotic forms of the equations of motion near the boundary <cit.>. In order to determine the numerical values of the ultraviolet coefficients for a given numerical solution generated by a given choice of the pair of initial conditions (ϕ_0,Φ_1), one matches the full numerical solution for the bulk fields to the functional forms of their corresponding ultraviolet expansions near the boundary. While the values of h_0^far, Φ_0^far and Φ_2^far can be easily obtained, the evaluation of ϕ_A is much more subtle and delicate due to the exponential decay of the dilaton close to the boundary <cit.>. In Refs. <cit.>, different algorithms were proposed to extract ϕ_A in a reliable and numerically stable way from the near-boundary analysis of the numerical solutions for the dilaton field, with progressively increasing levels of accuracy and precision. Moreover, in Ref. <cit.>, a new algorithm for choosing the grid of initial conditions (ϕ_0,Φ_1) was devised in order to cover the phase diagram of the dual QFT in the (T,μ_B)-plane in a much more efficient and broader way than in earlier works, like e.g. <cit.>. Together with more precise fittings to LQCD results at zero chemical potential, which led to the construction of an improved version of the EMD model at finite temperature and baryon density in Ref. <cit.>, all the algorithmic upgrades mentioned above allowed to obtain predictions from this improved EMD model not only for the location of the CEP <cit.>, but also for the location of the line of first-order phase transition and the calculation of several thermodynamic <cit.> and transport <cit.> observables in a broad region of the (T,μ_B)-plane, including the phase transition regions, where the numerical calculations are particularly difficult to perform due to the coexistence of competing branches of black hole solutions and the manifestation of significant noise in the numerical solutions.
Before comparing some thermodynamic results from some different versions of the EMD model in the literature, displaying the aforementioned improvements and discussing some of their consequences for the holographic predictions regarding the structure of the QCD phase diagram in the (T,μ_B)-plane, we provide below the relevant formulas for their calculation on the gravity side of the holographic duality. The numerical solutions for the EMD fields in thermal equilibrium generated by solving the bulk equations of motion for different pairs of initial conditions (ϕ_0,Φ_1) are associated through the holographic dictionary with definite thermal states at the boundary QFT, where the temperature T, the baryon chemical potential μ_B, the entropy density s, and the baryon charge density ρ_B of the medium are given by <cit.>,[We provide the formulas in the standard coordinates (with a tilde) and in the numerical coordinates (in terms of which the numerical solutions are obtained and the relevant ultraviolet coefficients are evaluated). It is worth mentioning that <cit.> introduced three extra free parameters in the holographic model, corresponding to different energy scaling parameters for μ_B, s, and ρ_B, besides the one for T. These parameters are unnecessary as they artificially augment the number of free parameters of the bottom-up construction without a clear physical motivation. In the holographic formulas reviewed in this paper there is just a single energy scale Λ associated with the nonconformal nature of the EMD model <cit.>, as mentioned above. In this context, if an observable has energy dimension p, its formula in the gravity side of the holographic duality gets multiplied by Λ^p in order to express the corresponding result in the dual QFT at the boundary in physical units of MeV^p.]
T = .√(-g'_t̃t̃g^r̃r̃ ')/4π|_r̃=r̃_HΛ=e^Ã(r̃_H)/4π|h̃'(r̃_H)|Λ = 1/4πϕ_A^1/ν√(h_0^far)Λ,
μ_B = lim_r̃→∞Φ̃(r̃)Λ = Φ_0^far/ϕ_A^1/ν√(h_0^far)Λ,
s = S/VΛ^3=A_H/4G_5VΛ^3=2π/κ_5^2e^3Ã(r̃_H)Λ^3 = 2π/κ_5^2ϕ_A^3/νΛ^3,
ρ_B = lim_r̃→∞∂ℒ/∂(∂_r̃Φ̃)Λ^3 = -Φ_2^far/κ_5^2ϕ_A^3/ν√(h_0^far)Λ^3,
where A_H is the area of the black hole event horizon, the prime denotes radial derivative, and ν≡ d-Δ, with d=4 being the number of spacetime dimensions of the boundary and with Δ=(d+√(d^2+4m^2L^2))/2 being the scaling dimension of the (relevant) QFT operator dual to the bulk dilaton field ϕ(r), which has a mass m obtained from the form of the dilaton potential V(ϕ), to be discussed in a moment.
The dimensionless ratio
χ̂_2^B≡χ_2^B/T^2≡∂^2(P/T^4)/∂(μ_B/T)^2
corresponds to the reduced second order baryon susceptibility. When evaluated at μ_B=0, χ̂_2^B has an integral expression given by <cit.>
χ̂_2^B(T,μ_B=0)=1/16π^2s/T^31/f(0)∫_r_H^∞dr e^-2A(r)f(ϕ(r))^-1,
which is to be evaluated over EMD backgrounds generated with the initial condition Φ_1=0.[Although the holographic mapping (ϕ_0,Φ_1)↦(T,μ_B,s,ρ_B) is highly nontrivial <cit.>, choosing Φ_1=0 automatically provides only EMD backgrounds with μ_B=0.] In numerical calculations <cit.>, one actually takes the following substitutions in Eq. (<ref>), r_H→ r_start and ∞→ r_max, where r_start is some small number (typically r_start∼ 10^-8) employed to avoid the singular point of the EMD equations of motion at the rescaled numerical horizon r_H=0, and r_max is a numerical parametrization of the radial position of the boundary, which is ideally at r→∞. Of course, it is not possible to use infinity in numerical calculations, and in practice, r_max∼ 2 - 10 is typically enough for the numerical EMD backgrounds to reach, within a small numerical tolerance, the ultraviolet fixed point of the holographic renormalization group flow associated with the AdS_5 geometry. It must be also emphasized that Eq. (<ref>) is not valid at μ_B≠ 0. In fact, to calculate the second order baryon susceptibility at finite μ_B, we take in practice
χ̂_2^B=∂(ρ_B/T^3)/∂(μ_B/T)
where ρ_B is the baryon density.
The pressure of the dual QFT fluid can be approximated as follows (for fixed values of μ_B),
P(T, μ_B)≈∫_T_low^T dT s(T,μ_B),
where T_low is the lowest value of temperature available for all solutions with different values of μ_B within the set of EMD black hole backgrounds generated with the grid of initial conditions considered. Eq. (<ref>) ceases to be a good approximation for the pressure for values of T∼ T_low.[The reason for taking a finite T_low instead of zero as the lower limit in the temperature integral of the entropy density in Eq. (<ref>) is that it is numerically difficult to obtain solutions of the EMD equations of motion at very low temperatures. For instance, T_low=2 MeV for the calculations done in Ref. <cit.>. By varying the value of T_low it is possible to numerically check the window of values for which the approximate results for the pressure remain stable within a given numerical tolerance.] The first law of thermodynamics then allows the calculation of the energy density of the medium according to
ϵ(s,ρ_B) = Ts(T,μ_B)-P(T,μ_B)+μ_Bρ_B(T,μ_B),
and the trace anomaly of the energy-momentum tensor (also known as the interaction measure) of the dual QFT at the boundary is
I(T,μ_B) = ϵ(T,μ_B)-3P(T,μ_B).
The square of the speed of sound in the medium is defined as c_s^2=(d P/d ϵ)_s/ρ_B, which can be calculated along different trajectories of constant entropy over baryon number in the (T,μ_B)-plane. For phenomenological applications in the context of heavy-ion collisions, one can rewrite this c_s^2 in terms of derivatives of T,μ_B <cit.>,
[c_s^2(T,μ_B)]_s/ρ_B=ρ_B^2∂_T^2P-2sρ_B∂_T∂_μ_BP +s^2∂_μ_B^2P/(ϵ+P)[∂_T^2P∂_μ_B^2P-(∂_T∂_μ_BP)^2]
that provides a much more convenient formula since most equations of state use T,μ_B as the free variables.
The above expressions allow the calculation of the main thermodynamic observables characterizing the equilibrium state of the QGP. Particularly, in order to fix the free parameters of the EMD model, we take as phenomenological inputs state-of-the-art continuum extrapolated results from first principles LQCD simulations with 2+1 flavors and physical values of the quarks masses, regarding the QCD equation of state <cit.> and the second order baryon susceptibility <cit.>, both evaluated at finite temperature and zero chemical potential. In fact, the choice of an adequate susceptibility is what seeds the bottom-up EMD model with phenomenological information concerning the nature of the controlling state variable(s) of the medium besides the temperature.[For instance, while the baryon susceptibility is used in the present section, the magnetic susceptibility will be employed in section <ref> within the context of the anisotropic EMD model at finite temperature and magnetic field, but with zero chemical potential.] In this way, it was constructed in Ref. <cit.>, and latter also used in Refs. <cit.>, a second-generation improved version of the EMD model (relative to previous constructions in the literature, namely, the original one in Refs. <cit.>, and the first generation improved EMD model of Refs. <cit.>), which is defined by the bulk action (<ref>) with the following set of holographically fixed bottom-up parameters and functions,
V(ϕ) = -12cosh(0.63 ϕ)+0.65 ϕ^2-0.05 ϕ^4+0.003 ϕ^6, κ_5^2 = 8π G_5=8π(0.46), Λ=1058.83 MeV,
f(ϕ) = (-0.27 ϕ+0.4 ϕ^2)+1.7 (100 ϕ)/2.7.
A number of observations are in order concerning the forms fixed above for the dilaton potential V(ϕ) and the Maxwell-dilaton coupling function f(ϕ).
First, regarding the dilaton potential, since from the ultraviolet asymptotic expansions for the EMD fields the dilaton is known to vanish at the boundary for relevant QFT deformations <cit.>, the boundary value V(0)=-12 =̇ 2Λ_AdS_5 is required in order to recover the value of the negative cosmological constant of AdS_5 in the ultraviolet regime, as Λ_AdS_d+1=-d(d-1)/2L^2 is equal to -6 for d=4 and L=1 (recall that we set here the asymptotic AdS radius to unity).[We remark that, in spite of the similar notation, the cosmological constant Λ_AdS_5=-6 has no relation with the nonconformal energy scale Λ in (<ref>).] One notices from (<ref>) that for this EMD model, the dilaton field has a mass squared given by m^2=∂_ϕ^2V(0)≈ -3.4628, which satisfies the Breitenlohner-Freedman (BF) stability bound <cit.> for massive scalar fields in asymptotically AdS backgrounds, m^2 > m^2_BF = -d^2/4L^2 = -4. Also, since the scaling dimension of the QFT operator dual to the dilaton is Δ=(d+√(d^2+4m^2L^2))/2≈ 2.73294 < d = 4 (which implies that ν≡ d-Δ≈ 1.26706), as anticipated, this is a relevant operator triggering a renormalization group flow from the AdS_5 ultraviolet fixed point towards a nonconformal state as one moves from the ultraviolet to the infrared regime of the dual QFT, or correspondingly, as one moves from the near-boundary to the interior of the bulk in the gravity side of the holographic duality. In fact, if one wishes to introduce a relevant deformation in the dual QFT away from the conformal regime asymptotically attained in the ultraviolet, and simultaneously satisfy the BF stability bound, then one should engineer the dilaton potential such as to have Δ_BF = 2 < Δ < d = 4, or equivalently, m^2_BF = -4 < m^2 < 0. Moreover, the dilaton potential in (<ref>) monotonically decreases from its maximum at the boundary to the deep infrared of the bulk geometry, such that there are no singular points (associated with local extrema of the potential) in the bulk equations of motion between the boundary and the black hole horizon, and also, Gubser's criterion for admissible classical gravitational singularities <cit.>, V(ϕ(r_H))≤ V(ϕ(r→∞)=0)=-12, is satisfied.
Second, concerning the Maxwell-dilaton coupling function, one should note from Eq. (<ref>) that the baryon susceptibility calculated at zero chemical potential cannot fix the overall normalization of f(ϕ). In (<ref>) this overall normalization was chosen such that f(0)=1, as originally proposed in <cit.>.[In practice, this choice for the overall normalization of f(ϕ) can be motivated by the fact that it allows a quantitative description of LQCD results at nonzero μ_B, as we are going to see later in this review.] Moreover, by also following <cit.>, we choose f(ϕ) such that it asymptotically goes to zero for large ϕ(r), in the infrared regime of the theory. However, differently from <cit.>, in order to obtain a quantitative description of this observable at zero chemical potential one seems to be forced to engineer a functional form for f(ϕ) such that it presents a very fast variation close to the boundary (i.e., for ϕ(r→∞)→ 0).[This is the practical reason for the term ∼(100 ϕ) in (<ref>) (the numerical factor of 100 can be substituted by some other `large number' without considerably affecting the results).] This peculiar feature has been also observed in other bottom-up EMD constructions with different functional forms for f(ϕ) and which had been proved to quantitatively describe χ̂_2^B(T,μ_B=0) from LQCD simulations with 2+1 flavors and physical values of the quark masses <cit.>.
In Fig. <ref>, we display the improvements in the holographic fits, from three different EMD models in the literature, taking as the target data to be described the LQCD results for the reduced second-order baryon susceptibility at vanishing chemical potential — one can also notice the improvements in the lattice results (see the figure caption for the details). The profile for the Maxwell-dilaton coupling f(ϕ) in Eq. (<ref>) was engineered to produce the result in the bottom panel of this figure, by using Eq. (<ref>) evaluated over the zero chemical potential, finite temperature EMD backgrounds. Those backgrounds, in turn, are generated with the choices of the EMD parameters in Eq. (<ref>), which were fixed in order to produce the results shown in Fig. <ref> for the holographic equation of state at μ_B=0. In Fig. <ref>, the full set of LQCD results shown were used as input for the model. In particular, the holographic trace anomaly seems very difficult to quantitatively reproduce to the corresponding LQCD results over the entire temperature interval considered.
With the bottom-up EMD parameters fixed in Eq. (<ref>) for the μ_B=0 portion of the equation of state by the results displayed in Fig. <ref> and the parameters fixed in Eq. (<ref>) relvant for the μ_B>0 portion of the equation of state by the results displayed in bottom panel of Fig. <ref> (i.e. χ̂_B^2 at μ_B=0), one can proceed to make holographic predictions for several observables relevant for the physics of the strongly coupled QGP.
Aside from the specific set of LQCD results at μ_B = 0 used to fix the free parameters of the EMD model, any other calculation follows as a legitimate prediction of the holographic setup considered.
In order to populate the phase diagram of the model, several EMD black hole solutions are numerically generated with a set of initial conditions (ϕ_0,Φ_1/Φ_1^max) chosen as indicated in the two top panels of Fig. <ref> <cit.>, where Φ_1^max = √(-2V(ϕ_0)/f(ϕ_0)) is a bound on the maximum value of Φ_1, given some ϕ_0>0 (which produces only positive values for the dilaton field), such as to have asymptotically AdS_5 solutions <cit.>. The corresponding holographic EMD predictions for the QCD equation of state at finite temperature and baryon chemical potential are also shown in Fig. <ref> and compared to state-of-the-art LQCD results at finite baryon density (with μ_Q=μ_S=0, as in the holographic model) <cit.>. One notices a good quantitative agreement between the EMD holographic predictions and the lattice results for the QCD equation of state at finite (T,μ_B), except for the baryon charge density for T≳ 190 MeV with μ_B/T≳ 2. It is important to emphasize that the holographic predictions shown in Fig. <ref> were obtained from the holographic EMD model of Ref. <cit.>, which was constructed in 2017, 4 years before the publication of the lattice results of Ref. <cit.>. As far as we know, this was the first model in the literature, holographic or not, to correctly predict at the quantitative level the behavior of this state-of-the-art lattice QCD equation of state at finite temperature and baryon chemical potential.
In this regard, it is also important to point out that in the same 2017 paper <cit.>, holographic predictions were put forward for higher-order baryon susceptibilities at zero chemical potential, which were quantitatively confirmed one year later by the LQCD simulations of Ref. <cit.>, as depicted in the top panel of Fig. <ref>.
A broad scanning of the phase diagram of the EMD model of Ref. <cit.>, comprising not only the crossover region and the CEP originally reported in this paper, but also the line of first-order phase transition ending at the CEP, was finally obtained in Ref. <cit.>, thanks to the significant algorithmic and numerical improvements achieved in that work, which also allowed the calculation of physical observables over the phase transitions regions in the phase diagram of the model. The EMD model prediction for the QCD phase diagram in the (T,μ_B)-plane is displayed in the bottom panel of Fig. <ref>, with the predicted CEP location lying around (T,μ_B)_CEP^[1706.00455]≈(89,724) MeV. The different curves characterizing the crossover region refer to characteristic points (extrema or inflections) of different equilibrium and transport observables that evolve with μ_B such that they merge at the CEP <cit.>. The CEP location also coincides with the end of the coexistence region with multiple black hole solutions with the same values of (T,μ_B) in the phase diagram of the model, as displayed in Fig. <ref> (b). Within this coexistence region, the thermodynamically stable branch of black hole solutions refers to the backgrounds with the largest pressure (or, equivalently, the smallest free energy). In Ref. <cit.>, also the discontinuity gaps for all the considered thermodynamic observables were calculated across the first-order phase transition line.
Before discussing transport coefficients results from holography, we close the present section with an important observation that will be further discussed in section <ref>. The functional forms of V(ϕ) and f(ϕ) are not uniquely fixed by current lattice QCD results. The very same set of LQCD results at μ_B=0 <cit.>, which was used to fix the dilaton potential and the Maxwell-dilaton coupling function for the EMD model of Refs. <cit.>, was also employed to fix different functional forms for V(ϕ) and f(ϕ) in the EMD model proposed in Ref. <cit.>. They also found a good quantitative fit to those set of LQCD results, and a very close result to that of <cit.> (Δ≈ 2.73294) for the scaling dimension of the QFT operator dual to the bulk dilaton field, namely Δ≈ 2.769. Although the EMD model of Ref. <cit.> had not been compared to LQCD results at finite μ_B, it predicts a CEP in a different location in the phase diagram, (T,μ_B)_CEP^[1702.06731]=(111.5± 0.5,611.5± 0.5) MeV.
More recently, another competing EMD model was proposed in Ref. <cit.> that employed the LQCD results for the equation of state at finite temperature and μ_B=0 from the HotQCD collaboration <cit.> to fix V(ϕ). For the baryon susceptibility they used the Wuppertal-Budapest results from <cit.> to fix f(ϕ), also imposing by construction Δ=3 for the scaling dimension of the QFT operator dual to the dilaton. Up until this work from 2022 <cit.>, only the Wuppertal Budapest LQCD results were used. While the Wuppertal Budapest and HotQCD collaboration results predominately agree, there are still quantitative differences at large temperatures and the error bars from HotQCD are slightly larger.
Then, the results from <cit.> also produced a holographic equation of state in good quantitative agreement with the state-of-the-art LQCD results at finite temperature and baryon chemical potential from Ref. <cit.>, besides a good agreement with lattice results on higher order baryon susceptibilities <cit.>, but with yet another different location for the CEP, (T,μ_B)_CEP^[2201.02004]≈(105,555) MeV.
In Fig. <ref>, we display the holographic predictions for the QCD CEP from the three competing bottom-up EMD models mentioned above, which were shown to be in quantitative agreement with available state-of-the-art LQCD results, while presenting different predictions for regions of the QCD phase diagram still out of the reach of first principles LQCD simulations. These three competing EMD models present a very fast variation in the behavior of the Maxwell-dilaton coupling function f(ϕ) near the boundary, which seems to be a rather robust feature connected to the holographic EMD description of LQCD results with 2+1 flavors and physical values of the quark masses. These results motivated the need for a more systematic approach to investigate, in a quantitative way, the structure of the different EMD predictions for the location of the QCD critical endpoint. This can be accomplished through a Bayesian analysis of holographic EMD models, and initial results will be briefly mentioned in section <ref>.
§.§.§ Holographic transport coefficients
One of the most attractive features of the holographic gauge-gravity duality, when applied to the strongly coupled QGP, is that, besides the evaluation of thermodynamic observables at finite temperature and baryon density, it also allows for the calculation of transport coefficients entering as microscopic inputs into hydrodynamic calculations and also the evaluation of other microscopic properties such as partonic energy loss. These transport observables, which are of fundamental relevance for the phenomenology of the QGP produced in relativistic heavy-ion collisions, are generally determined through the holographic duality by employing two kinds of approaches, namely,
* Hydrodynamic coefficients (such as the first-order shear and bulk viscosity transport coefficients <cit.> and coefficients associated with higher-order derivative expansions of the energy-momentum tensor of the boundary QFT <cit.>, besides different conductivities and diffusion coefficients associated with the transport of conserved charges <cit.>), and also the thermal production rates of photons and dileptons within the medium <cit.>, may be evaluated through the use of holographic Kubo formulas obtained via linear response theory. The Kubo formulas relate transport coefficients to the expectation values of retarded thermal correlators of gauge invariant operators at the dual QFT, which can be calculated by solving with some adequate boundary conditions linearized equations of motion for quadratic perturbations of the bulk fields defined at the level of the bulk action, with these linearized equations of motion for the perturbations being evaluated over the equilibrium background geometries holographically associated with definite thermal states at the boundary QFT;[Alternatively, some of these hydrodynamic transport coefficients can also be calculated from the spectra of quasinormal modes in different channels of holographic gauge-gravity models, see e.g. <cit.>.]
* Observables associated with momentum transport and the energy loss of partons within the strongly coupled quantum fluid are generally evaluated by employing the Nambu-Goto action for strings within different setups (which may be holographically associated with probe partons traversing the medium described by the background black hole solutions) <cit.> (see also <cit.>).
Let us first review some relevant EMD predictions for a few hydrodynamic transport coefficients, namely the shear viscosity, bulk viscosity, and baryon conductivity. Afterwards, we shall also briefly review some EMD results for transport observables associated with partonic energy loss.
Here we will consider the calculation of homogeneous hydrodynamic transport coefficients of the hot and baryon dense quantum fluid holographically dual to the EMD model close to thermal equilibrium. The SO(3) rotation symmetry of the isotropic medium classifies into different irreducible representations (also called “channels”) the gauge and diffeomorphism invariant combinations of the linearized plane-wave EMD field perturbations at the level of the equations of motion, evaluated at zero spatial momentum <cit.>. The bulk viscosity of the boundary QFT is holographically dual to the diffeomorphism and gauge invariant bulk EMD perturbation transforming under the singlet (scalar) representation of SO(3). The baryon conductivity is dual to the EMD perturbations transforming under the triplet (vector) representation, and the shear viscosity is dual to the EMD perturbations transforming under the quintuplet (tensor) representation of the SO(3) rotation symmetry group of the isotropic medium. Indeed, due to the fact that these gauge and diffeomorphism invariant EMD perturbations transform under different irreducible representations of SO(3), they do not mix at the linearized level and, consequently, one obtains a single decoupled equation of motion for each of these bulk perturbations <cit.>.
The tensor components of the isotropic EMD SO(3) quintuplet graviton perturbation are given by five independent combinations of components of the bulk metric field perturbation sourcing the piece of the boundary energy-momentum tensor which is traceless and transverse to the fluid flow. These components satisfy the same differential equation, corresponding to the equation of motion for a massless scalar perturbation over the background geometry considered. The equation of motion has the same form in the standard and in the numerical coordinates (as a consequence of the diffeomorphism invariance of these perturbations) <cit.>.
Then, it was shown <cit.> that the shear viscosity satisfies η/s = 1/4π ∀ T>0, μ_B≥ 0, as expected since the isotropic EMD model fits into the very broad class of holographic gauge-gravity models which are translationally and rotationally invariant, besides having two derivatives of the metric field in the bulk action <cit.>. However, the natural dimensionless ratio for the shear viscosity at finite baryon densities is no longer simply η/s, but rather η T/(ϵ+P) <cit.>. This dimensionless ratio reduces to η/s when evaluated at μ_B=0, developing a nontrivial behavior as a function of (T,μ_B) at nonzero baryon densities. η T/(ϵ+P) has been analyzed in detail across the phase diagram of the EMD model in Ref. <cit.>, where it was shown that η T/(ϵ+P) decreases with increasing values of μ_B. In that work, η T/(ϵ+P) developed an inflection point and a minimum, with the former evolving toward the CEP of the model, where it acquires an infinite slope.
For larger values of the baryon chemical potential and lower temperatures, η T/(ϵ+P) develops a discontinuity gap across the first order phase transition line of the model, as depicted in Fig. <ref> (a). With the overall reduction in the value of η T/(ϵ+P) with increasing μ_B, the EMD model predicts that the QGP becomes even closer to the perfect fluid limit in its baryon-dense regime.
The three vector components of the EMD SO(3) triplet perturbation are associated with the spatial components of the perturbation of the bulk Maxwell field sourcing the baryon vector current at the dual boundary QFT. Again, due to the spatial isotropy of the medium, these vector components satisfy a single decoupled equation of motion.
One may consider the bulk spatial Maxwell perturbation, a≡ a_i, i∈{x,y,z}, to calculate in holography the baryon conductivity, which gives the same result in any direction. The equation of motion for the vector perturbation <cit.> is
a”(r,ω)+[2A'(r)+h'(r)/h(r)+∂_ϕ f(ϕ)/f(ϕ)ϕ'(r)]a'(r,ω)+e^-2A(r)/h(r)[ω^2/h(r)-f(ϕ)Φ'(r)^2]a(r,ω)=0,
where ω is the frequency of the plane-wave ansatz for the Maxwell perturbation and the prime denotes the radial derivative. One must solve Eq. (<ref>) imposing the infalling wave boundary condition for the Maxwell perturbation at the background black hole horizon. In holography, this is equivalent to solving for the retarded thermal correlator of the boundary baryon vector current operator with the further requirement that the Maxwell perturbation is normalized to unity at the boundary <cit.>. These two boundary conditions may be systematically implemented by writing the Maxwell perturbation as follows <cit.>,
a(r,ω)≡r^-iωP(r,ω)/r_max^-iωP(r_max,ω),
where r_max is a numerical parametrization of the boundary (see below Eq. (<ref>)), and P(r,ω) is a regular function at the black hole horizon, whose equation of motion is obtained by substituting (<ref>) into (<ref>).
The holographic Kubo formula for the baryon conductivity in the EMD model in physical units of MeV <cit.> is given by
σ_B(T,μ_B)=-1/2κ_5^2ϕ_A^1/νlim_ω→01/ω(e^2A(r)h(r)f(ϕ)Im[a^*(r,ω)a'(r,ω)])|_on-shell Λ [MeV],
where the term between brackets in Eq. (<ref>) is a radially conserved flux that can be calculated at any value of the radial coordinate.
The details regarding the numerical procedure are discussed in <cit.>.
The dimensionless ratio σ_B/T has been analyzed in detail in Ref. <cit.> where it was shown that it generically increases with the temperature, as displayed in Fig. <ref> (b). For σ_B/T there is a temperature window from T∼ 150-180 MeV where the different curves at fixed values of μ_B approximately cross. For values of temperature above this crossing window, T>180 MeV, σ_B/T decreases with increasing μ_B, whereas the opposite behavior is observed for temperatures below the crossing window T<150. One also notices that at the CEP of the model, the baryon conductivity is finite and develops an infinite slope, with a small discontinuity gap being observed across the first-order phase transition line at larger values of μ_B and lower values of T. In Ref. <cit.>, they also calculated the second-order baryon susceptibility, χ_2^B, and the baryon diffusion coefficient, D_B across the phase diagram of the EMD model. It was found that χ_2^B diverges at the critical point (a universal feature of all critical points) whereas D_B→ 0 at the CEP since D_B=σ_B/χ_2^B.
The traceful and transverse piece of the boundary energy-momentum tensor T^μν is associated with the bulk viscous pressure of the medium. Note that Tr[T^μν]≠ 0 in nonconformal boundary QFTs, where the trace anomaly of T^μν is related to the bulk dilaton field. The dilaton field is introduced in the bulk action to break the conformal symmetry of the dual gauge theory at the boundary. The scalar EMD SO(3) singlet perturbation is composed by the spatial trace of the graviton and the dilaton perturbation. The singlet perturbation sources the traceful part of T^μν, being holographically related to the bulk viscosity.
Denoting the singlet perturbation by ℋ, its equation of motion <cit.> is shown to be given by
ℋ”+[4A'+h'/h+2ϕ”/ϕ-2A”/A']ℋ'+[e^-2Aω^2/h^2+h'/h(A”/A'-ϕ”/ϕ')+e^-2A/hϕ'(3A'∂_ϕ f(ϕ)-f(ϕ)ϕ')Φ'^2]ℋ=0,
which must be solved with infalling boundary condition at the background black hole horizon and normalized to unity at the boundary. In practice this is implemented by setting,
ℋ(r,ω)≡r^-iωF(r,ω)/r_max^-iωF(r_max,ω),
where F(r,ω) is a regular function at the black hole horizon, whose equation of motion is obtained by substituting (<ref>) into (<ref>).
The holographic Kubo formula for the bulk viscosity in the EMD model <cit.> is
ζ/s(T,μ_B)=-1/36πlim_ω→01/ω(e^4A(r)h(r)ϕ'(r)^2Im[ℋ^*(r,ω)ℋ'(r,ω)]/A'(r)^2)|_on-shell,
where the term between brackets in Eq. (<ref>) is a radially conserved flux that may be evaluated at any value of the radial coordinate. The details concerning the numerical calculations are discussed in <cit.>. At μ_B=0, the numerical results obtained using this holographic formula were checked to be the same as the holographic formula provided in <cit.> by following a different approach based on the r=ϕ gauge. The latter approach, however, does not seem to be extensible to finite μ_B calculations.
Similarly to shear viscosity at μ_B>0, one can no longer use ζ/s as the natural hydrodynamic expression, but instead the dimensionless combination ζ T/(ϵ+P) that reduces to ζ/s at μ_B=0. ζ T/(ϵ+P) was analyzed in detail in Ref. <cit.>, where it was shown that ζ T/(ϵ+P) develops a peak in the crossover region at μ_B=0. In contrast to older versions of the EMD model from Refs. <cit.>, this peak does not move toward the CEP of the model as one increases μ_B.
Instead in new versions of the EMD model, the location of the peak in ζ T/(ϵ+P) moves to slightly higher values of T as the baryon density increases.
While in the original EMD construction of Ref. <cit.> the height of the peak of ζ T/(ϵ+P) remains approximately constant as μ_B increases toward the CEP, both in the second generation improved EMD model of Ref. <cit.> (see Fig. <ref> (c) ) and in the first generation improved model of Ref. <cit.> the magnitude of the peak of ζ T/(ϵ+P) reduces as one increases the value of μ_B. Therefore, the behavior of the peak of ζ T/(ϵ+P) is clearly model dependent within the class of holographic EMD constructions.
In Fig. <ref> (c) at different values of μ_B, ζ T/(ϵ+P) starts to develop both an inflection point and a minimum as a function of T, with both characteristic points evolving toward the CEP location as the baryon density of the medium is increased (see also the bottom panel in Fig. <ref>). At the CEP, ζ T/(ϵ+P) acquires an infinite slope, while further developing discontinuity gaps across the first-order phase transition line of the EMD model. Similarly to what happens with the shear viscosity, the magnitude of the bulk viscosity is also suppressed with increasing values of μ_B. This overall suppression of viscous effects within the strongly coupled medium maybe constitutes a robust property of holographic EMD models seeded with lattice QCD inputs, since this same qualitative behavior has been also observed in the older versions of the EMD model of Refs. <cit.>.
In Fig. <ref> (d), we show the comparison between the EMD prediction for [ζ/s](T) at μ_B=0 to extracted values of [ζ/s](T) from recent Bayesian analyses <cit.> that simultaneously describe several experimental heavy-ion data. The holographic EMD prediction for [ζ/s](T) is in the ballpark of values favored by state-of-the-art phenomenological models. Considering that η/s (for any holographic model) is in the correct magnitude for extracted η/s from experimental data and that there is quantitative agreement for the equation of state as well (see Figs. <ref> and <ref>) between the EMD predictions and the QCD equation of state and susceptibilities at finite (T,μ_B), there is reasonable evidence for the practical and quantitative applicability of bottom-up EMD holography as an effective modeling of the strongly coupled QGP produced in heavy-ion collisions. This argument will be further strengthened in section <ref>, when we will discuss the applicability of the anisotropic version of the EMD model at finite temperature and magnetic fields to the physics of the hot and magnetized QGP.
The fact that at the CEP of the EMD model the baryon conductivity and also the shear and bulk viscosities remain finite indicates that the EMD model is compatible with the model B dynamical universality class <cit.>. This seems to be a common feature of large N_c gauge theories (as in any holographic gauge-gravity model) <cit.>, and it is different from general expectations for N_c=3 QCD, where these three observables are expected to diverge at the CEP <cit.>, in compatibility with the model H dynamical universality class <cit.>.
It is also informative to briefly comment on some results obtained from the calculation of the spectra of homogeneous quasinormal modes (QNMs) in the SO(3) quintuplet, triplet, and singlet channels of the EMD model <cit.>. In fact, the QNMs of asymptotically AdS black holes <cit.> encode a wide range of physical information concerning the holographic dual QFT linearly perturbed out of thermal equilibrium.
The near-boundary expansions of the perturbed bulk fields typically feature a leading order non-normalizable mode and a subleading normalizable mode for each field perturbation. The leading modes source the corresponding local and gauge invariant operators at the dual boundary QFT, while the subleading modes are associated with the expectation values of these operators. If one sets the subleading modes to zero at the boundary and imposes the infalling wave condition at the black hole horizon, the corresponding solutions to the linearized equations of motions for the bulk perturbations can be used to evaluate the on-shell action and obtain the retarded thermal correlators of the dual QFT, which are associated through Kubo formulas to transport coefficients of the strongly coupled quantum fluid. For transport coefficients extracted from the imaginary part of the Green's functions, this procedure is physically equivalent to the calculation of transport coefficients through the use of radially conserved fluxes, which has been discussed before.
On the other hand, since the retarded thermal correlators of the dual QFT are given by minus the ratio between the subleading and the leading modes of the bulk perturbations <cit.>, by setting these leading modes to zero at the boundary and imposing the causal infalling wave condition at the black hole horizon, one gets the poles of these Green's functions. Since the frequency eigenvalue problem for QNMs defined on asymptotically AdS spacetimes is precisely defined by the Dirichlet boundary condition corresponding to the vanishing of these leading modes at the boundary <cit.>,[Notice this is different from the calculation of transport coefficients discussed before, where these leading modes for the on-shell perturbations of the bulk fields were normalized to unity at the boundary.] one sees that the QNMs describing the exponential decay of linear perturbations of asymptotically AdS black holes holographically correspond to the poles of retarded thermal Green's functions at the dual QFT. These, in turn, describe hydrodynamic and non-hydrodynamic dispersion relations of collective excitations in the strongly coupled quantum fluid, in terms of which it is possible to calculate, respectively, some hydrodynamic transport coefficients <cit.> (in an alternative way to the more direct method of holographic Kubo formulas previously discussed) and also some upper values for characteristic equilibration times of the dual QFT linearly perturbed out of equilibrium.
Indeed, as discussed in <cit.>, the non-hydrodynamic QNMs[Non-hydrodynamic QNMs are associated with collective excitations of the medium with nonvanishing frequencies even in the homogeneous regime of perturbations with zero wavenumber.] with the lowest absolute value of its imaginary part, corresponding to the longest-lived non-hydrodynamic excitations of the system, give upper bounds for different equilibration times of the medium close to thermal equilibrium. From the lowest homogeneous non-hydrodynamic QNMs in the SO(3) quintuplet, triplet, and singlet channels of the EMD model of Ref. <cit.>, it has been shown that the equilibration times in these different channels are very close to each other at high temperatures while developing a pronounced separation at the CEP. This result indicates that the energy-momentum tensor dual to the bulk metric field, the baryon current dual to the bulk Maxwell field, and the scalar condensate dual to the bulk dilaton field, equilibrate at considerably different rates in the critical regime of the EMD model, with the baryon current taking the longest time to approach thermal equilibrium, while the energy-momentum tensor generally equilibrates faster than the other observables, also within the regions of the phase diagram far from the criticality. Moreover, in most cases, the characteristic equilibration times of the medium decrease with increasing values of the baryon chemical potential, while strongly increasing with decreasing values of temperature.
There have been various holographic calculations of transport coefficients associated with partonic energy loss within the strongly coupled quantum fluid, such as the energy loss of heavy quarks due to the heavy quark drag force <cit.>, the Langevin momentum diffusion coefficients for heavy quarks <cit.>, and the jet quenching parameter associated with the energy loss of light partons moving at the speed of light <cit.>.
These energy loss transport coefficients are evaluated by considering different calculations done with a probe Nambu-Goto (NG) action for a classical string defined over the background solutions for the bulk fields.
The NG action depends on √(λ_t), where the `t Hooft coupling is typically considered in holographic calculations as an extra free parameter. In principle, this parameter may be fixed in different ways by considering holographic observables calculated with the NG action compared to different kinds of phenomenological data (see e.g. Refs. <cit.>). For the class of isotropic EMD models at finite temperature and baryon density, the holographic formulas for these partonic energy loss observables were derived in Ref. <cit.>, and in Ref. <cit.>. The corresponding results for the improved EMD model were also numerically calculated across its phase diagram, including the regions with the CEP and the line of first-order phase transition. It was found that the heavy quark drag force and energy loss, the Langevin momentum diffusion, and the jet quenching parameter are all enhanced by increasing the baryon density of the medium toward the critical region of the phase diagram. In fact, faster partons are more sensitive to the temperature and baryon chemical potential of the medium. Those results indicate that there is more jet suppression and partonic energy loss in the baryon-dense regime of the fluid. All of these observables developed an infinite slope at the CEP, while displaying large discontinuity gaps across the line of first-order phase transition. In the bottom panel of Fig. <ref> some crossover characteristic curves (made of sequences of inflection points or extrema) of these observables converging to a single location corresponding to the CEP are displayed with other characteristic curves for different observables of the model — see Ref. <cit.> for details.
§.§.§ Holographic Bayesian analysis
The results for the EMD model discussed above rely on the choice of holographic potentials V(ϕ) and f(ϕ). That is, calculations require that suitable functional forms are provided, along with the corresponding parameters. As discussed above, several competing parametrizations for these functions can be found in the literature, but no systematic comparison between them has been performed thus far.
A pressing question regarding any particular parametrization of the EMD model concerns how much of its predictions are informed by lattice QCD results used to fit the different parameters, and how robust they are against uncertainties in these results.
Such issues can only be addressed by quantifying uncertainties in V(ϕ) and f(ϕ) and systematically comparing different parametrizations.
The tools required for a systematic analysis of parameter sensitivity and uncertainty quantification in modeling the QCD equation of state can be found in the framework of Bayesian statistical inference <cit.>.
In recent years, Bayesian statistics have become the state-of-the-art tool for systematically assessing models and hypotheses across high-energy physics, including neutron-star <cit.> and heavy-ion physics <cit.>.
The core tenet of Bayesian inference resides in Bayes' theorem:
P(M^(θ)| D) = P(D| M^( θ)) × P(M^(θ))/P(D),
where D represents the data and M^(θ) is a given model with parameters θ. Equation (<ref>) follows from expressing the joint probability P(D∩ M^(θ)) in terms of the associated conditional probabilities P(M^(θ)| D) and P(D| M^( θ)). The conditional distribution P(M^(θ)| D) is called the posterior and can be used to discriminate between different parameter sets θ. It is the product of the likelihood P(D| M^( θ)), quantifying agreement between model and data, and the prior P(M^(θ)), which assigns a priori weights to the different parameter sets to reflect prior knowledge.
The denominator P(D) on the right-hand side of Eq. (<ref>) is known as the evidence and can be obtained as a normalization constant.
Recently, an improved numerical implementation of the EMD model developed within the MUSES Collaboration has enabled a Bayesian analysis over lattice QCD results for the zero-density equation of state.
In Eqs. (<ref>) and (<ref>), the very nonlinear character of the potentials over ϕ make functional forms, such as seen in Fig. <ref>, highly sensitive to precise parameter values.
A complete Bayesian analysis is underway and will be published shortly. Here, we briefly highlight and explain the results obtained from an initial analysis.
New parametric ansatze for the free functions V(ϕ) and f(ϕ) of the holographic EMD action (<ref>) are introduced to reproduce qualitative features of Eqs. (<ref>) and (<ref>) in a way that depends more transparently on parameter values:
V(ϕ) = -12cosh[(γ_1 Δϕ_V^2 + γ_2 ϕ^2/Δϕ_V^2 + ϕ^2) ϕ],
f(ϕ) = 1 - (1-A_1) [1/2 + 1/2tanh(ϕ - ϕ_1/δϕ_1)]
- A_1[1/2 + 1/2tanh(ϕ - ϕ_2/δϕ_2)].
Equation (<ref>) interpolates between two different exponential slopes, γ_1 and γ_2, for ϕ≪Δϕ_V and ϕ≫Δϕ_V, respectively. Equation (<ref>), on the other hand, goes from f(ϕ)≈ 1, for ϕ_1-ϕ≪δϕ_1 to a plateau of height f(ϕ)≈ A, for ϕ in the range ϕ_1-ϕ_2, before finally going to f(ϕ)≈ 0, for ϕ-ϕ_2≫δϕ_2.
The prior distribution for parameter values was taken to be uniform within designated ranges, shown in Table <ref>, on the left.
Random samples from this prior distribution were then fed into a Markov Chain Monte Carlo (MCMC) algorithm <cit.>. This MCMC implements random changes to parameters such that the equilibrium probability distribution, to be reached after a sufficiently large number of iterations, coincides with the posterior distribution given by Eq. (<ref>). This algorithm can then be reiterated to generate a large sample of parameter sets from the posterior.
Parameters are fit based on both the baryon susceptibility and the entropy density from lattice QCD at μ_B=0 <cit.>.
The agreement between model and lattice results is quantified by the likelihood P(D| M^( θ)), chosen to be Gaussian.
The corresponding covariance matrix is chosen according to the lattice QCD error bars while implementing auto-correlation between neighboring points. An extra parameter is introduced to gauge these correlations and is also estimated within the Bayesian inference <cit.>.
The 95% confidence interval obtained from lattice QCD results in this fashion is shown in Table <ref>, on the right.
Finally, parameter sets from the posterior can be used to compute predictions. The statistical distribution of predictions can then be used to quantify uncertainties stemming from the lattice QCD errorbars, as well as the sensitivity to different model parameters. As a check that these predictions are compatible with lattice QCD results, Fig. <ref> compares predictions for different values of μ_B/T, shown as thin semitransparent lines, to the finite-density lattice QCD equation of state from <cit.>, shown as wide bands with the same color scheme. While it is not apparent at first sight, thousands of lines are shown over each band in Fig. <ref>.
Remarkably, the zero-density equation of state constrains the model parameters so tightly that these lines accumulate in what appears to be a very thin band.
Constraining the model with input from lattice QCD in this fashion, one is able to extract predictions at higher densities, and even around the QCD phase transition.
Because it generates a large set of model realizations, this Bayesian analysis of the EMD model will also enable the investigation of the role of each different model parameter, both in predictions and in fitting lattice results.
Perhaps even more importantly, this kind of analysis provides the possibility of assigning probabilities to predictions and hypotheses.
In principle, Bayesian model selection can also be used to discriminate between different models.
Overall, the combination of bottom-up holographic models with Bayesian tools thus provides a promising tool for extrapolating knowledge on the low-density and high-temperature QCD equation of state to higher densities in a partially systematic way.
Because of its ability to capture the physics of the strongly coupled QGP in the crossover region, the EMD model is a particularly fitting candidate for this task.
§.§ Other holographic models
Although the focus of the present review is on the results from bottom-up holographic EMD models for the hot and baryon-dense QCD phase diagram, in this section, we briefly mention some results obtained from other kinds of holographic constructions.
Within the broad class of bottom-up Einstein-Dilaton constructions, but without considering the effects of flavor dynamics effectively enclosed in the form of the dilaton potential matched to the corresponding LQCD results, as originally proposed in Refs. <cit.>, there is the so-called class of “Improved Holographic QCD” (ihQCD) models originally devised in Refs. <cit.>, and further reviewed in <cit.>. Due to the fact that flavor dynamics are not taken into account in those ihQCD models, such a class of bottom-up holographic constructions actually refers to effective models for pure Yang-Mills systems, instead of QCD. In a pure YM system at T=0 at large color-charge separations, there is a linear confining potential for infinitely heavy probe quarks as well as a mass gap featured in the physical spectrum of glueball excitations, which are both well described by ihQCD models.
In contrast to the deconfinement crossover observed in actual QCD with 2+1 dynamical quark flavors, pure YM theory has a first-order phase transition between a confining gas of glueballs and a deconfined phase corresponding to a pure gluon plasma.
At finite temperature the ihQCD models are able to achieve this first-order phase transition, just like what is seen in pure YM theory.
However, η/s=1/4π in these ihQCD models that demonstrates the theory is strongly coupled at all energy scales and, therefore, misses crucial properties related to asymptotic freedom in the ultraviolet.
As explicitly shown in Ref. <cit.>, higher curvature corrections to ihQCD models can provide a nontrivial temperature dependence for [η/s](T), allowing this observable to acquire a similar profile to what is expected for pure YM and also QCD matter where [η/s](T) is expected to largely increase with the temperature of the medium in the ultraviolet regime due to asymptotic freedom.
Simple Einstein's gauge-gravity models with two derivatives of the metric field in the bulk gravity action lack asymptotic freedom, while the consideration of higher curvature corrections for the bulk action is associated with corrections that reduce the value of the effective `t Hooft coupling of the dual QFT at the boundary.
Generalizations of the original ihQCD constructions for pure YM systems that consider a very large number N_f of quark flavors where the ratio x≡ N_f/N_c remains finite in the holographic setup[The number of colors N_c is always very large] are known in the literature as V-QCD models <cit.>.
The letter “V” stands for the so-called Veneziano limit of large N_c, N_f, with fixed x=N_f/N_c.
In such bottom-up models, the flavor dynamics are taken into account by considering the full backreaction of tachyonic flavor D-branes on the gluonic backgrounds.
The V-QCD models have been employed to calculate a large number of physical observables, ranging from spectroscopy <cit.> to thermodynamic quantities <cit.> and transport coefficients <cit.>, and have been also used in some far-from-equilibrium calculations, see e.g. <cit.>.
Most of these V-QCD models have been mainly applied in the literature to study the physics involving neutron stars and QCD matter at high densities, see also the recent review <cit.>.
The class of EMD models reviewed here may be viewed as Taylor-expanded versions of the more general class of V-QCD models with vanishing tachyon field (see discussion in section 3.2 of Ref. <cit.>).
However, it is important to stress that the details involved in the holographic constructions may lead to considerably different results.
By comparing the fitting results for the EMD model of Refs. <cit.> with the LQCD results in Fig. <ref> and in the bottom panel of Fig. <ref>, one can see that the EMD model provides a better description of first principles lattice results on the QCD thermodynamics than the several different V-QCD models considered in Fig. <ref>. In particular, for the trace anomaly of the energy-momentum tensor, one notices in Fig. <ref> (a) that the different V-QCD constructions miss even qualitatively the correct LQCD behavior for this observable below the pseudocritical temperature. Indeed, while in actual QCD with 2+1 flavors there is no phase transition at μ_B=0 between the hadron gas and QGP regimes, but just an analytical crossover <cit.>, in the holographic V-QCD approach there is a first-order phase transition <cit.>, which is reminiscent from the ihQCD backgrounds embedded in such constructions. Therefore, keeping in mind the limitations and shortcomings stated in section <ref>, it is fair to say that the EMD class of holographic models discussed in this review remains the leading description to provide a quantitative description of lattice results on actual QCD thermodynamics with 2+1 dynamical flavors with physical quark masses, both at zero and finite baryon density.
Another class of holographic models, but of top-down nature, which has been extensively studied in the literature, mainly connected to spectroscopic properties of QCD, is the so-called Witten-Sakai-Sugimoto model <cit.> — see also <cit.> for a review.[This top-down holographic construction stems from Type IIA instead of Type IIB superstring theory. Contrary to most gauge-gravity models, the background geometries in the Witten-Sakai-Sugimoto model are not asymptotically AdS and feature a dilaton field that diverges at the boundary, consequently, the Witten-Sakai-Sugimoto model has no ultraviolet fixed point <cit.>.] This kind of holographic model has not been shown to be able to provide an accurate quantitative description of first principles lattice results of QCD thermodynamics with dynamical quark flavors. In Ref. <cit.> the Witten-Sakai-Sugimoto approach has been employed to provide a phenomenologically realistic description of cold and dense nuclear matter at zero temperature, which is in good agreement with some known theoretical and observational constraints regarding the physics of neutron stars. See also the recent review <cit.> for a broad discussion on the holographic modeling of compact stars.
§ HOLOGRAPHIC MODELS FOR THE HOT AND MAGNETIZED QUARK-GLUON PLASMA
The QCD phase diagram is not just a function of (T,μ_B) but is also dependent on the chemical potentials for strangeness (μ_S) and electric charge (μ_Q), electromagnetic fields, the number of flavors relevant for a given environment, etc. By varying the centrality class of heavy-ion collisions it is possible, to investigate the phase diagram of QCD in the plane of temperature and magnetic field, (T,eB). The most intense magnetic fields ever created by humankind are reached in high-energy peripheral heavy-ion collisions at RHIC (eB_max∼ 5 m_π^2∼ 0.09 GeV^2 for Au+Au collisions at center of mass energies of √(s_NN)=200 GeV with an impact parameter of b∼ 12 fm) and at the LHC (eB_max∼ 70 m_π^2∼ 1.3 GeV^2 for Pb+Pb collisions at center of mass energies of √(s_NN)=2.76 TeV with an impact parameter of b∼ 13 fm) [We note that eB = 1 GeV^2⇒ B ≈ 1.69× 10^20 G.] — see e.g. Fig. 2 in <cit.>; see also Refs. <cit.>. The study of the QCD matter under the influence of strong magnetic fields is also relevant in the context of the physics of magnetars <cit.> and of the early universe <cit.>, making it a very active research field in recent years see, e.g., <cit.>.
Even though very intense magnetic fields are produced in the early stages of noncentral heavy-ion collisions, being therefore important in those initial stages, due to the receding spectator hadrons fastly leaving the collision zone, one generally expects the magnitude of such strong magnetic fields to have significantly decayed by the time the QGP is formed.
Very intense magnetic fields are produced in the early stages of noncentral heavy-ion collisions such that they should play an important role in the initial conditions.
However, it is generally expected that the strength of the magnetic fields decay significantly by the time the QGP if formed because the spectator nucleons quickly leave the collision zone (and the protons, in particular, carry electric charge) <cit.>.
Early papers argued that by considering effects due to the electric conductivity induced in the medium <cit.> and the quantum nature of the sources of such fields <cit.>, the decay of the magnetic field may be considerably delayed within the medium. More recently in <cit.> it was argued that an incomplete electromagnetic response of the medium to the decaying external magnetic field that is associated with an induced electric current that is lower than expected by Ohm's law, leads to a strong suppression in the magnitude of the induced magnetic field in the medium (two orders below previous estimates in the literature done by assuming the validity of Ohm's law). This argument may help to explain the consequences of the recent STAR isobar run <cit.> where it was originally thought that strong magnetic fields would lead to the chiral magnetic effect.
Nonetheless, it is interesting to investigate the structure of the QCD phase diagram in the (T,eB)-plane from a theoretical perspective. At low temperatures the magnitude of the chiral condensate is enhanced with increasing magnetic fields constituting the so-called magnetic catalysis phenomenon <cit.>. However, for higher temperatures slightly above the QCD crossover region, the inverse effect is observed with a reduction in the magnitude of the chiral condensate and a decreasing pseudocritical crossover temperature for increasing values of the magnetic field, known as inverse magnetic catalysis (or magnetic inhibition) as found in the first principles lattice QCD simulations of Refs. <cit.>, see also <cit.>. There is also a prediction <cit.> that a first-order phase transition line ending at a critical point exists in the (T,eB)-plane of the QCD phase diagram for very high values of the magnetic field, eB∼ 4 - 10(2) GeV^2 <cit.>, although current lattice simulations <cit.> for the QCD equation of state with 2+1 flavors and physical values of the quark masses only found an analytic deconfinement crossover for values of 110 MeV < T < 300 MeV and eB ≲ 0.7 GeV^2.
Various holographic models have been proposed in the literature to study different aspects of strongly coupled quantum systems under the influence of external magnetic fields, with either a more qualitative view towards different physical observables calculated from holographic methods — see e.g. <cit.>, or with a more quantitative perspective aimed towards direct comparisons with results from first principles LQCD calculations — see, for instance, <cit.>.
In the present section, we focus on quantitative holographic EMD predictions for some thermodynamic and transport observables of the hot and magnetized strongly coupled QGP.
§.§ Anisotropic Einstein-Maxwell-Dilaton models
The first phenomenological anisotropic holographic EMD model at finite temperature with a constant external magnetic field (and μ_B=0) was <cit.>. This model generalized the isotropic approach considered in the previous section to anisotropic EMD backgrounds with the SO(3) rotation symmetry broken down to SO(2) in the transverse plane to the magnetic field.
The general form of the bulk action in this case is the same as in Eq. (<ref>), but the Maxwell-dilaton coupling function f(ϕ) must be different from the isotropic case at finite temperature and baryon chemical potential.
In the isotropic EMD model at finite (T,μ_B), f(ϕ) effectively represents the coupling associated with the conserved baryon current, with this coupling being dynamically fixed in the holographic setup by matching the LQCD baryon susceptibility evaluated at finite temperature and zero chemical potential, as discussed in section <ref>.
In the case of the anisotropic EMD model at finite (T,eB) the coupling must be associated with the electric sector, instead of the baryon sector of the dual QFT at the boundary. Then, instead of the baryon susceptibility the phenomenological input seeded to the holographic model to “teach” the asymptotically AdS black hole backgrounds to behave as a hot and magnetized QGP, is the LQCD magnetic susceptibility evaluated at finite temperature and zero magnetic field.[In principle, one could also choose to use the electric susceptibility, instead of the magnetic susceptibility, in order to fix the Maxwell-dilaton coupling f(ϕ) for the electric sector of the dual QFT at the boundary. However, as discussed in Appendix A of Ref. <cit.>, a simple EMD model is not versatile enough to adequately cover the entire electromagnetic sector of the QGP, in the sense that by fixing f(ϕ) by matching the LQCD electric susceptibility, one obtains a holographic prediction for the magnetic susceptibility in disagreement with the corresponding LQCD result, and vice-versa. Therefore, it seems unfeasible to obtain a simultaneously good description of QCD magnetic and electric response functions using a single EMD model. Consequently, in order to describe magnetic field-related phenomena, one chooses the magnetic susceptibility as a phenomenological input to fix f(ϕ) within the holographic EMD approach.]
We shall review the main aspects of this endeavor in the next section, but before that, paralleling the discussion made in section <ref> for the improvements done through the years regarding the isotropic EMD model at finite (T,μ_B), we briefly comment below on the improvements done also in the construction of the anisotropic EMD model at finite (T,eB).
The original construction at finite (T,eB) presented in Ref. <cit.> has the same set of free parameters {G_5,Λ,V(ϕ)} of the first generation improved isotropic EMD model of Refs. <cit.>, meaning that both models represent the same system at finite temperature when the baryon chemical potential and the magnetic field are turned off. On the other hand, as already mentioned, the Maxwell-dilaton coupling f(ϕ) for the anisotropic EMD model is different from the isotropic case. In Ref. <cit.> it was constructed an improved version of the anisotropic EMD model (with this improved version being also used in Refs. <cit.>), where the set of free parameters and functions {G_5,Λ,V(ϕ),f(ϕ)} was updated by performing a better matching procedure to more recent lattice results on the QCD equation of state and magnetic susceptibility at finite temperature and zero magnetic fields. The set of improved free parameters {G_5,Λ,V(ϕ)}, originally obtained in Ref. <cit.> for the improved anisotropic EMD model, was later employed also in the second generation improved isotropic EMD model of Refs. <cit.>. In what follows, we mainly review the results for physical observables calculated with the improved version of the anisotropic EMD model at finite (T,eB) from Refs. <cit.>.
§.§.§ Anisotropic holographic thermodynamics
The general EMD equations of motion obtained from the bulk action (<ref>) are given by Eqs. (<ref>) — (<ref>). The presence of a constant external magnetic field, which we take to be directed along the z-axis, breaks the SO(3) rotation symmetry of the dual QFT down to SO(2) rotations around the direction of the magnetic field. This symmetry breaking implies that the ansatz for the bulk metric field must be anisotropic when the magnetic field is turned on.
Thus, for the description of a hot and magnetized fluid in thermodynamic equilibrium, we take the following anisotropic and translationally invariant charged black hole ansatze for the bulk EMD fields <cit.>,
ds^2 = g_μνdx^μ dx^ν= e^2a(r)[-h(r)dt^2+dz^2]+e^2c(r)(dx^2+dy^2)+dr^2/h(r),
ϕ =ϕ(r), A=A_μ dx^μ=ℬxdy ⇒ F=dA=ℬdx∧ dy,
where ℬ is the constant magnetic field expressed in the numerical coordinates. By substituting the ansatze (<ref>) into the general EMD field equations (<ref>) — (<ref>), one obtains the following set of coupled ordinary differential equations of motion <cit.>,
ϕ”+(2a'+2c'+h'/h)ϕ'-1/h(∂ V(ϕ)/∂ϕ+ℬ^2e^-4c/2∂ f(ϕ)/∂ϕ) =0,
a”+(14/3c'+4/3h'/h)a' +8/3a'^2+2/3c'^2+2/3h'/hc'
+2/3h V(ϕ)-1/6ϕ'^2 =0,
c”-(10/3a'+1/3h'/h)c' +2/3c'^2-4/3a'^2-2/3h'/ha'
-1/3h V(ϕ)+1/3ϕ'^2 =0,
h”+(2a'+2c')h' =0,
a'^2+c'^2-1/4ϕ'^2+(a'/2+c')h'/h+4a'c'
+1/2h(V(ϕ)+ℬ^2e^-4c/2f(ϕ)) =0,
where Eq. (<ref>) is a constraint. The steps used to numerically solve the above equations of motion for a given pair of initial conditions (ϕ_0,ℬ) are discussed in detail in Refs. <cit.> (with algorithmic and numerical improvements regarding the original approach devised in <cit.>).
Similarly to the isotropic EMD model at finite temperature and baryon density discussed in section <ref>, one extracts the following set of ultraviolet expansion coefficients required for the holographic calculation of several thermodynamic observables: {h_0^far,a_0^far,c_0^far,ϕ_A} from the numerical solutions for the background anisotropic EMD fields at finite temperature and magnetic field evaluated near the boundary.
From these ultraviolet coefficients one can write down the following holographic formulas for the temperature T, the electric charge e times the constant external magnetic field B at the boundary (expressed in standard coordinates), and the entropy density s (measured, respectively, in units of MeV, MeV^2, and MeV^3) <cit.>,
T=1/4πϕ_A^1/ν√(h_0^far)Λ,
eB=e^2(a_0^far-c_0^far)ℬ/ϕ_A^2/νΛ^2,
s=2π e^2(a_0^far-c_0^far)/κ_5^2 ϕ_A^3/νΛ^3,
where the energy scale Λ, as well as the 5D Newton's constant and the dilaton potential are the same as given in Eq. (<ref>).
In order to fix the Maxwell-dilaton coupling function f(ϕ) for the anisotropic EMD model at finite temperature and magnetic field, one needs to dynamically match the holographic magnetic susceptibility at finite temperature and zero magnetic field with the corresponding LQCD result.
As discussed in <cit.>, the holographic EMD formula for the regularized magnetic susceptibility evaluated at finite temperature and zero magnetic field may be written as follows in the numerical coordinates,[One should ideally take T_low=0, however, due to numerical difficulties in reaching exactly the vacuum geometry in the EMD model, we numerically subtract a zero magnetic field background geometry with a small but nonzero temperature, similarly to what was done in Eq. (<ref>) for the calculation of the pressure.]
χ(T,B=0)=χ_bare(T,B=0)-χ_bare(T_low,B=0)=-1/2κ_5^2[(1/√(h_0^far)∫_r_start^r^var_max dr f(ϕ(r)))|_T,B=0-(same)|_T_low,B=0]_on-shell,
where r^var_max≡√(h_0^far)[r̃^fixed_max- a_0^far+ln(ϕ_A^1/ν)], with r̃^fixed_max being a fixed ultraviolet cutoff in standard coordinates which must be chosen such that the upper limits of integration in Eq. (<ref>) satisfy r_conformal≤ r^var_max≤ r_max for all the background geometries under consideration. We remark that r_conformal is a value of the radial coordinate[Typically, r_conformal∼ 2.] where the background geometry already reached the conformal AdS_5 ultraviolet fixed point (within some numerical tolerance), and r_max≥ r_conformal is the maximum value of the radial coordinate up to which we perform the numerical integration of the bulk equations of motion. By taking as phenomenological input the LQCD magnetic susceptibility at finite temperature and zero magnetic field with 2+1 flavors and physical values of the quark masses from Ref. <cit.>, one may fix the form of the Maxwell-dilaton coupling function as follows <cit.>,
f(ϕ)=0.95 sech(0.22ϕ^2-0.15ϕ-0.32),
with the result displayed in Fig. <ref> (a).
Also in Fig. <ref>, we show the predictions from the anisotropic EMD model at finite (T,eB) <cit.> compared to the LQCD results from <cit.> for (b) the pressure difference,[Similarly to what was done in Eq. (<ref>), one may evaluate the pressure as the temperature integral of the entropy density in (<ref>), calculated with the magnetic field held fixed. As discussed in detail in Section 2 of <cit.>, this gives the isotropic pressure in the so-called “B-scheme”, where the magnetic field is held fixed during compression, with the pressure being the response function of the system to such a compression. Correspondingly, this also gives the anisotropic longitudinal pressure in the direction of the magnetic field in the so-called “Φ-scheme”, where it is the magnetic flux that is held fixed during a compression. In the Φ-scheme, the transverse pressures (to the direction of the magnetic field) depend on the magnetization of the medium, which requires holographic renormalization of the bulk action to be evaluated through the gauge-gravity duality, and that has not been calculated in Refs. <cit.>.] Δ p(T,eB)≡ p(T,eB)-p(T=125MeV,eB), (c) the normalized entropy density s/T^3 (we also show the LQCD results from <cit.> at B=0), and (d) the crossover temperature as a function of the magnetic field, as extracted from the inflection of s/T^3. For the values of the magnetic field considered there is no actual phase transition between the hadronic and partonic regimes of the hot and magnetized QCD matter, just an analytic crossover.
Contrary to the isotropic EMD model at finite (T,μ_B) from Refs. <cit.>, whose phase diagram has been deeply investigated, the phase diagram of the anisotropic EMD model at finite (T,eB) from Refs. <cit.> still remains largely unexplored. One challenge, however, is that the anisotropic EMD model typically requires a much larger set of background black hole solutions than the isotropic model in order to allow for smooth interpolations of physical observables as functions of T and eB.
In Ref. <cit.>, the holographic anisotropic EMD model at finite (T,eB) was further employed to calculate the magnitude of the expectation value of the renormalized Polyakov loop operator <cit.>,[The holographic renormalization procedure for the calculation of the Polyakov loop involves only the on-shell Nambu-Goto (NG) action for a probe string extending from an isolated quark at the boundary up to the background black hole horizon deep into the bulk, and not the bulk action (which generates the black hole backgrounds, over which the probe string described by the NG action is defined) <cit.>.] P_r(T,eB)=|⟨L̂_P⟩_r|=e^-F_Q^r(T,eB)/T, where F_Q^r(T,eB) is the renormalized free energy of a single static heavy quark at the boundary.[The renormalization scheme at nonzero magnetic field employed in Ref. <cit.> was the same one used in the LQCD simulations of Refs. <cit.>.] In holography, this quantity depends on the `t Hooft coupling coming from the NG action, which in a bottom-up setup is taken as an extra free parameter. Since √(λ_t)=L^2/α'=(L/l_s)^2,[See the discussion in section <ref>.] where l_s is the fundamental string length and L is the asymptotic AdS radius (which is set here to unity), one expects that in the classical gauge-gravity regime of the holographic duality the `t Hooft coupling should be large, since in this limit, l_s≪ L. Indeed, by matching the overall magnitude of the holographic Polyakov loop, P_r(T,eB), with the corresponding LQCD results from <cit.>, as illustrated in Fig. <ref> (e), in Ref. <cit.> it was fixed the large value √(λ_t)=1450, which hints at a nontrivial consistency between top-down theoretical expectations and bottom-up phenomenological results within this holographic approach. Furthermore, one also notices that the anisotropic EMD model provides a reasonable description of the LQCD results for the Polyakov loop in the deconfined regime of QCD matter corresponding to the strongly coupled hot and magnetized QGP, for magnetic fields up to eB≲ 1 GeV^2 with T≳ 150 MeV.
Also in Ref. <cit.>, the holographic EMD prediction for the heavy quark entropy, S_Q(T,eB)=-∂ F_Q^r(T,eB)/∂ T was computed. The ratio between any two different values of S_Q is particularly interesting because it does not depend on the extra free parameter √(λ_t) present in the holographic calculation of the Polyakov loop. Consequently, once the background black hole solutions are obtained, there are no extra free parameters to fix in such a calculation. In Fig. <ref> (f), there are shown the EMD predictions for the ratio S_Q(T,eB)/S_Q(T=200MeV,eB=0), with the result at zero magnetic field being compared to the corresponding available LQCD result from <cit.>. Interestingly enough, the EMD prediction at B=0 is in perfect quantitative agreement with the LQCD result in the deconfined regime for T≳150 MeV, while completely missing the correct behavior for the heavy quark entropy in the confined hadronic regime.
The disagreement found with the lattice results for the Polyakov loop and the heavy quark entropy below the crossover temperature in the hadronic regime with the contrast of the quantitative agreement found above the crossover temperature in the partonic regime, is because the holographic EMD model is suited to describe the deconfined QGP phase of hot QCD matter but not the confined hadronic phase.
The anisotropic EMD model at finite (T,eB) results in Fig. <ref> and the isotropic EMD model at finite (T,μ_B) in Figs. <ref>, <ref>, <ref> (d) compared to first principles LQCD results on several thermodynamic observables and transport coefficients posteriors from Bayesian analyses using heavy-ion data, comprise the main argument for the actual phenomenological applicability of EMD holography in the description of many aspects of the hot QGP produced in heavy-ion collisions.
These results are interesting mainly due to the following reasons:
* The class of relatively simple bottom-up holographic EMD constructions reviewed here may be used to make physically reasonable predictions for the QGP, providing not only qualitative insight but also some quantitatively reliable results, which may extend beyond the current reach of first principles approaches in QCD.
* As a class of bottom-up holographic constructions, the phenomenological EMD models reviewed here provide further evidence that the holographic dictionary may be useful in practice even when the precise form of the holographic dual QFT at the boundary of the higher dimensional bulk spacetime is unknown.
* Even though the precise holographic dual is unknown, the results reviewed here show that this holographic dual must be some effective 4D strongly coupled QFT which very closely mimics several aspects of QCD. While EMD holography differs from QCD in several aspects e.g. the lack of asymptotic freedom and the thermodynamic behavior in the confining hadronic regime, it is still able to capture several other key features of QCD.
§.§.§ Anisotropic holographic transport coefficients
The presence of an external magnetic field (or, more generally, of any source of anisotropy) in the medium splits the transport coefficients into several anisotropic components, when compared to the more simple case of an isotropic medium.
Holographic analyses regarding the anisotropic heavy quark drag forces and the Langevin momentum diffusion coefficients, and also the anisotropic jet quenching parameters involving light partons, were done e.g. in Refs. <cit.>.
Additionally, anisotropic shear and bulk viscosities were analyzed in <cit.>.
At this time systematic checks of these transport coefficients have not yet been performed, since the field of relativistic magnetohydrodynamics is currently under intense development <cit.>.
The purpose of the present section is to briefly review some of the main results obtained in Refs. <cit.> regarding the anisotropic EMD predictions at finite (T,eB) for some transport coefficients of the strongly coupled hot and magnetized QGP.
The holographic formulas of the anisotropic heavy quark drag forces and Langevin momentum diffusion coefficients can be found in Appendix A of <cit.>, and then applied to the anisotropic EMD model at finite (T,eB) as done in sections III.B and III.C of the same reference. The general conclusion[Which was shown, in section II of <cit.>, to also hold for the top-down magnetic brane model of Ref. <cit.>.] is that energy loss and momentum diffusion for heavy quarks traversing a strongly coupled anisotropic plasma are enhanced by the presence of an external magnetic field, being larger in transverse directions than in the direction of the magnetic field. In Ref. <cit.> it was found that also the anisotropic jet quenching parameters for light partons display an overall enhancement with increasing values of the external magnetic field, with the phenomenon of transverse momentum broadening being larger in transverse directions than in the direction of the magnetic field.[These conclusions were shown in <cit.> to also hold for the top-down magnetic brane model of Ref. <cit.>.] Consequently, one generally predicts more energy loss for heavy and light partons traversing a strongly coupled quantum medium in the presence of an external magnetic field.
The holographic formulas for the anisotropic shear viscosities in the plane transverse to the magnetic field, η_⊥, and along the direction of the magnetic field, η_∥, were derived in <cit.> and reviewed in Appendix A of <cit.>. The anisotropic η/s ratios are then
η_⊥/s = 1/4π, η_∥/s = 1/4π g_zz(r_H)/g_xx(r_H),
from which one recovers the isotropic result η_⊥/s=η_∥/s≡η/s=1/4π when B=0, since in this case the background metric is isotropic g_zz=g_xx.
At nonzero magnetic fields, only η_∥/s varies with the value of the external magnetic B while η_⊥/s=1/4π is constant.
In Fig. <ref> the results for the ratio η_∥/η_⊥ in the anisotropic EMD model at finite (T,eB) are shown.
The anisotropic shear viscosity is lower in the direction parallel the magnetic field than in the transverse plane, with its magnitude being reduced as one increases the value of B.
Along the external magnetic field direction, a strongly coupled magnetized medium becomes progressively closer to the idealized perfect fluid limit field by enhancing the value of the magnetic field.[See also Ref. <cit.> for a discussion about the breaking of rotational invariance and its
effects in the calculation of the shear viscosity of a p-wave superfluid model. In the case considered in <cit.>, the rotational symmetry breaking does not lead to a value of η/s below 1/4π.]
§ SUMMARY AND OUTLOOK
In this work, we provided an up-to-date review of quantitative holographic EMD models for the hot and strongly coupled QGP produced in relativistic heavy-ion collisions.
We reviewed both isotropic EMD constructions at finite temperature and baryon chemical potential with vanishing electromagnetic fields and anisotropic EMD models at finite temperature and magnetic field with zero chemical potential.
Evidence that the holographic duality can quantitatively provide reliable predictions for the hot and deconfined QGP phase of QCD, depending on the class(es) of gauge-gravity models considered and how their free parameters are fixed by phenomenological inputs, was discussed. These key results highlight precisely this evidence for the reliability of the EMD predictions:
* Isotropic EMD model for the (T,μ_B)-plane of QCD: in Figs. <ref> and <ref> we displayed, respectively, the holographic predictions for the equation of state at finite temperature and baryon chemical potential, and for the 6th and 8th order baryon susceptibilities at μ_B=0, compared to state-of-the-art first principles LQCD results; and in Fig. <ref> (d), we have shown the EMD prediction for the bulk viscosity to entropy density ratio at vanishing baryon density compared to the profiles favored by the latest phenomenological multistage models that simultaneously describes several different experimental data from relativistic heavy-ion collisions. As an isotropic and translationally invariant holographic model with two derivatives of the metric field in the bulk gravity action, the model naturally encompasses a small shear viscosity, η/s=1/4π, compatible with the overall magnitude estimated for the strongly coupled QGP produced in heavy-ion collisions. A number of other holographic EMD models are currently available in the literature which have been also shown to successfully describe LQCD results at the quantitative level, such as the works presented in Refs. <cit.>.
* Anisotropic EMD model for the (T,eB)-plane of QCD: in Fig. <ref> we displayed the holographic predictions for the anisotropic equation of state, the crossover transition temperature, the renormalized Polyakov loop, and the heavy quark entropy at finite temperature and magnetic field compared to the available first principles LQCD results.
The holographic EMD model allows one to go beyond the current capabilities of LQCD simulations. For instance, one prediction of this model is the existence of a critical end point. While different competing EMD models do provide differences in the predicted location of this critical point after fitting to LQCD results for μ_B=0, they all lead to the existence of a critical point in approximately a similar region of the QCD phase diagram. Such a spread of critical points clearly motivates a more systematic investigation of different parametrizations of the free functions and parameters of the bottom-up class of holographic EMD models through Bayesian statistical inference.
A detailed Bayesian analysis of such models is currently underway, but preliminary results were discussed in section <ref>.
This Bayesian analysis considered uniform prior distributions of the free parameters.
Using the LQCD results for the entropy density and the baryon susceptibility at μ_B=0 as constraints, the posterior distributions for the free parameters of the holographic EMD setup become strongly constrained, as shown in Table <ref>.
Thousands of different EMD models were generated within the constrained posterior distributions that provided holographic predictions for the behavior of the QCD equation of state at finite temperature and baryon density.
The resulting equation of state has remarkably thin bands, as shown in Fig. <ref>, which are in quantitative agreement with state-of-the-art lattice results for the QCD equation of state also at finite baryon density.[Although some deviations exist for the baryon charge density at high temperatures and high baryon chemical potentials, as depicted in Fig. <ref>. However, that is also precisely in the regime where the expansion scheme may begin to break down from lattice QCD and/or weaker coupling may be relevant.] A complete analysis considering regions of the phase diagram beyond the reach of current lattice simulations and the distribution of critical points predicted by a broader class of holographic EMD models will be presented elsewhere.
A critical assessment of the most relevant limitations and the drawbacks of holographic approaches to the description of hot QCD phenomenology were also discussed in detail. First, classical holographic gauge-gravity models with two derivatives of the metric field in the bulk gravity action lack asymptotic freedom, with the dual effective QFT at the boundary of the higher dimensional bulk spacetime being strongly coupled at all energy scales. This is explicitly manifest in the temperature-independent value of η/s=1/4π found in these models, which is in contrast to the gas-like pQCD results at asymptotically high temperatures.
Instead of a trivial ultraviolet fixed point, classical holographic gauge-gravity models which are asymptotically AdS feature a strongly coupled ultraviolet fixed point, being asymptotically safe but not asymptotically free.
The lack of asymptotic freedom and η/s=const are presumably tied to the neglected contributions from massive string states and quantum string loops in the classical gravity bulk theory.
This can be possibly improved by considering higher derivative corrections associated with massive string states in the bulk action, which in the presence of a nontrivial dilaton background has already been shown in the literature <cit.> to produce temperature-dependent profiles for η/s in holographic models. However, the systematic construction of phenomenologically realistic and fully-backreacted dilatonic models with higher-order derivative corrections is a challenging task still not accomplished in the literature.
Another very general limitation of classical holographic gauge-gravity models regards the inability to describe the thermodynamic and transport properties of the confining hadron resonance gas phase of QCD.
This limitation is related to the large N_c character of classical gauge-gravity models, in which the pressure in the confining phase is largely suppressed by a multiplicative factor of ∼ N_c^-2 relatively to the deconfined QGP phase.[One very clear manifestation of such a limitation has been shown in Fig. <ref> (f), where the holographic prediction for the heavy quark entropy was found to be in perfect agreement with the corresponding LQCD results above the pseudocritical crossover temperature, while for temperatures below the crossover region the holographic heavy quark entropy suddenly completely misses the correct LQCD behavior.] In principle, this situation can be improved by considering quantum string loops contributions to the dilatonic bulk theory. However, this task is considerably more complicated than the one discussed in the previous paragraph.
Specific limitations and drawbacks of the holographic EMD models reviewed here have been also identified in the literature. For instance, the strangeness neutrality condition realized in heavy-ion collisions is not implemented in the EMD model, as it only features a single chemical potential (in the case considered here, the baryon chemical potential). Moreover, in the investigation of the phase diagram of the EMD model of Refs. <cit.> no regions were found where the square of the speed of sound exceeds its conformal limit (c_s^2|_CFT=1/3), strongly indicating that such models are inadequate to describe the dense QCD equation of state of the most massive neutron stars <cit.>. As mentioned in section <ref>, the anisotropic EMD model is not versatile enough to simultaneously describe the magnetic and the electric sectors of the QGP with a single Maxwell-dilaton coupling function f(ϕ).
For future work, it is important to extend dilatonic holographic approaches to simultaneously include fully backreacted effects from conserved baryon, electric, and strangeness charges. Such an endeavor would enable the implementation of strangeness neutrality, which is relevant for applications in heavy-ion collisions. In order to pursue this task within a consistent implementation of QCD flavor symmetry in the holographic setup, the EMD class of holographic models should be substituted by a more general class of (fully backreacted) Einstein-Yang-Mills-Dilaton (EYMD) models.
Still within the class of holographic EMD models, the more complicated anisotropic EMD setups at finite temperature and magnetic field remain largely unexplored. Most of its phase diagram has yet to be investigated. Additionally, a Bayesian analysis would be another important next step to understand properties at large B fields (as in the case of the Bayesian analysis currently under development for the isotropic setup at finite baryon density).
Other important developments to be pursued in the future include the consideration of rotation effects for the strongly coupled dual plasma by taking into account more general ansatze for the bulk fields allowing for rotating and charged asymptotically AdS black holes. Also numerical simulations of far-from-equilibrium holographic dynamics <cit.> should be further pursed, such as the consideration of holographic Bjorken flow and holographic collisions of shockwaves in the context of the phenomenologically realistic EMD models reviewed in this manuscript.
§ ACKNOWLEDGEMENTS
This material is based upon work supported in part by the
National Science Foundation under grants No. PHY-2208724 and No. PHY-2116686 and in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-SC0022023, DE-SC0021301, DE-SC0020633, DE-SC0023861. This work was supported in part by the National Science Foundation (NSF) within
the framework of the MUSES collaboration, under grant
number No. OAC-2103680. This research was supported in part by the National Science Foundation under Grant No. PHY-1748958.
|
http://arxiv.org/abs/2307.04029v1 | 20230708183856 | On "Indifference" and Backward Induction in Games with Perfect Information | [
"Nimrod Megiddo"
] | cs.AI | [
"cs.AI"
] |
=480pt
=0pt
6pt
corollaryCorollary
definitionDefinition
factFact
exampleExample
lemmaLemma
propositionProposition
remarkRemark
*remarknonumRemark
theoremTheorem
=0pt
#1 1## ###1
|
http://arxiv.org/abs/2307.04352v1 | 20230710052537 | Phase Diagram and Crossover Phases of Topologically Ordered Graphene Zigzag Nanoribbons: Role of Localization Effects | [
"Hoang Anh Le",
"In Hwan Lee",
"Young Heon Kim",
"S. -R. Eric Yang"
] | cond-mat.str-el | [
"cond-mat.str-el",
"cond-mat.stat-mech",
"quant-ph"
] |
On Sufficient Graphical Models
Bing Li [email protected]
Department of Statistics, Pennsylvania State University
326 Thomas Building, University Park, PA 16802
Kyongwon Kim [email protected]
Department of Statistics, Ewha Womans University
52 Ewhayeodae-gil, Seodaemun-gu, Seoul, Republic of Korea, 03760
August 12, 2023
=======================================================================================================================================================================================================================================================================================================================
empty
Is the formation of a fractional charge <cit.> a necessary and sufficient condition<cit.> for topologically ordered insulators <cit.> such as fractional quantum Hall systems <cit.> and interacting disordered zigzag graphene nanoribbons <cit.> (ZGNRs)
with anyonic fractional charges? This issue is related to whether the electron localization effects of doped systems destroy or enhance the topological order <cit.> and quantization of fractional charges. In a Laughlin state on a disordered sphere (no edges are present), the added electrons fractionalize and form a quasi-degenerate peak in the gap of the tunneling density of states (DOS).
Electron localization <cit.> is expected to suppress the quantum fluctuations of these fractional charges of the quasi-degenerate gap states because these localized quasi-degenerate energy states are spatially separated from each other, as explained in Ref. <cit.> (if fractional charges are delocalized they overlap and become ill-defined). However, excessive disorder is considered detrimental to topological order.
In this study, we investigate similar issues with ZGNRs <cit.>.
A recent study showed that weak randomness (disorder) in ZGNRs can generate e^-/2 fractional charges <cit.>, which is a disorder effect closely related to the change in the disorder-free symmetry-protected topological insulator of ZGNRs to a topologically ordered <cit.> Mott-Anderson insulator <cit.>. These systems have a universal value for topological entanglement entropy (TEE) <cit.> in the weak-disorder regime <cit.>. The shape of entanglement spectrum is also found <cit.> to be similar to the DOS of the edge states, as expected of topologically ordered systems <cit.>.
In interacting disordered ZGNRs, the gap is filled further by edge states <cit.> with an increasing strength of the disorder potential. (We call these states gap-edge states.)
The ground states have the opposite edge site spins in the absence of disorder <cit.>. In the presence of disorder
a spin reconstruction of the zigzag edges can take place <cit.>. Nonetheless, a topologically ordered ZGNR has two degenerate ground states, see Fig. <ref>(a).
Mixed chiral edge states play an important role in this effect.
A short-range disorder potential couples two nearly chiral gap-edge states residing on opposite zigzag edges <cit.>, and mixed chiral gap-edge states with split probability densities may form
to display e^-/2 fractional semion charges <cit.> (see Fig.<ref>(b))(these states with midgap energies are solitonic with the half of the spectral weight originating from the conduction band and the other half from the valence band <cit.>). Note that a mixed chiral gap-edge state has a nonzero fractional probability at the A- and B-carbon sites. In other words, it is split into two nonlocal parts, each residing on the edges of the A or B sublattice. The formation of mixed chiral gap-edge states is a nonperturbative instanton effect <cit.>. (They are similar to the bonding and antibonding states of a double quantum well.)
It should be noted that well-defined e^-/2 fractional charges in the weak-disorder regime are emergent particles, i.e., they have new qualitative features and appear only in sufficiently long ribbons.
In a weak-disorder regime, the number of fractional charges is proportional to the length of zigzag edges.
Although weak disorder leads to formation of fractional charges strong disorder may destroy them. Similar to fractional quantum Hall systems, the topological order of a ZGNR is not immediately destroyed upon doping because electron localization partially suppresses quantum fluctuations between quasi-degenerate mid-gap states. The system may still be an insulator with a fractional charge. However, in the presence of strong disorder or doping, zigzag edge antiferromagnetism is expected to diminish, and thereby, the topological order.
(Away from the low doping region, a disordered anyon phase with a distorted edge spin density wave was found <cit.>.) These results suggest that there may be several topological phase transitions in the zigzag ribbons. What is the nature of these topological phase transitions and the physical properties of the ground states? Does the presence of a fractional charge imply a universal value of the TEE? Does the TEE become nonuniversal and vary <cit.> with an increase in the disorder strength or doping level?
We explored the phase diagram of ZGNRs in the parameter space comprising on-site repulsion (U), disorder strength (Γ), and doping concentration (δ N/N_s) (δ N and N_s are, respectively, the number of doped electrons and the total number of sites in the ribbon). The competition between localization and electron interactions can have detrimental effects on the topological order and lead to several different phases, which includes crossover phases. We found a number of different phases with a topological order, quasi-topological order, and no order. Each of these phases is defined by the value of TEE β and its variance. These properties of β are related to the presence or absence
of charge fractionalization and charge transfer correlations between zigzag edges. When both of these properties are present, in addition to correlations leading to spin-charge separation, β is universal, with small variances. In low-doped ZGNRs the interplay between electron localization and on-site repulsion contributes to the spatial separation of quasi-degenerate gap-edge states and protects the charge fractionalization against quantum fluctuations.
There are two other types of phases with a quasi-topological order. We refer to these phases as crossover phases, in which the variance of β is significant. In one of these phases, both e^-/2 fractional charges and spin-charge separation are absent; however, the charge transfer (± e^-/2) correlations exist between the zigzag edges.
Another phase may contain stable e^-/2 fractional charges but no charge transfer correlations between the zigzag edges.
The ground state and zigzag edge properties of the various crossover and nontopological phases are explored.
§ MODEL HAMILTONIAN
The following mechanisms can all lead to fractional charges: the coupling between the valleys mediated by short-range scatterers <cit.> and the sublattice mixing facilitated by alternation of the nearest neighbor hopping parameters <cit.>. Here we will consider only the effect of short-range scatterers.
The self-consistent Hartree-Fock (HF) approximation works well for graphene systems <cit.>. The HF Hamiltonian of a ZGNR with length L and width W is
H_MF=-t∑_n.n.,σ c^†_i,σc_j,σ +∑_i,σ V_ic_i,σ^†c_i,σ
+U∑_i[ n_i,↑⟨ n_i,↓⟩ +n_i,↓⟨ n_i,↑⟩-⟨ n_i,↓⟩⟨ n_i,↑⟩]
+ ∑_i [s_ix⟨ h_ix⟩+s_iy⟨ h_iy⟩],
where the site index is given by i=(k,l) (k labels sites along the ribbon direction and l along the width), c^†_i, σ and n_i_,σ represent creation and occupation operators at site i with spin σ = {↑, ↓}, respectively (periodic boundary conditions are used along the ribbon direction). The site spin operators are given by s_i x (y) = 1/2( c^†_i, ↑, c^†_i, ↓) σ^x (y) ( c_i, ↑, c_i, ↓)^T, where σ^x (y) is the conventional Pauli matrix. The first term represents the kinetic energy with hopping parameter t, n.n implies the summation over the nearest-neighbor sites. The second term represents the short-range impurities parameterized by V_i, which is randomly chosen from the energy interval [ -Γ, Γ]. Throughout this study, the density of the impure sites is fixed at 10 %. U denotes the on-site repulsive strength. The last term in Eq. (<ref>) represents self-consistent “magnetic fields", where ⟨ h_i x⟩ = -2 U ⟨ s_i x⟩ and ⟨ h_i y⟩ = -2 U ⟨ s_i y⟩. (These fields are present only in doped ZGNRs. In the initial stage of the HF iteration, the values of ⟨ h_i x⟩ and ⟨ h_i y⟩ can be selected from small random numbers). In the presence of these fields, the HF eigenstates are mixed spin states. The HF single-particle states |k⟩ (k=1,2,…,2N_s) can be written as a linear combination of site states |i,σ⟩. In the language of second quantization this is equivalent to
a_k=∑_i,σ A_k,i,σc_i,σ.
These magnetic fields are rather small for the disorder strength and doping level considered in this study.
There may be several nearly degenerate HF ground states. We select the HF initial ground state such that ⟨ n_i,σ⟩ represents a paramagnetic state with a small spin splitting.
In addition, we choose small random numbers of ⟨ h_i x⟩ and ⟨ h_i y⟩ (they do not significantly affect the final results).
The HF matrix dimension scales with the number of carbon atoms, which is typically <50000. The HF eigenstates and eigenenergies are self-consistently computed
(this requires approximately 20 iterations). The TEE is computed using the disorder-averaging results of numerous disorder realizations. Here, we used gpu to speed up the solution of the HF matrix. The gpu calculations were intensive and performed on a supercomputer. In the presence of disorder and in the low-doping region, the obtained HF ground-state properties with solitons are in qualitative agreement with those of the density matrix renormalization group (DMRG) in the matrix product representation <cit.>. (In this work we do not investigate the high doping region. The DMRG result is difficult to obtain in this region because the computation is rather time consuming, and therefore, it is not possible to determine which nearly degenerate HF ground state is the true ground state.)
Note that the Mott gap Δ is well-developed only when L≫ W (the excitation spectrum of a ribbon with L∼ W is similar to that of a gapless two-dimensional graphene sheet <cit.>). The localization properties of ZGNRs are unusual because both localized and delocalized states can exist <cit.>. Gap-edge states with energy |E|∼Δ/2 (Δ represents the Mott gap in the absence of disorder) can have localization lengths ∼ W and overlap significantly with each other.
§ EFFECTS OF ANDERSON LOCALIZATION
Anderson localization plays a crucial role in the quantization of fractional charges <cit.>. The effects of Anderson localization can be described using self-consistent Hartree–Fock approximation (HFA) <cit.>. The first important effect is as follows:
Anderson localization reduces the correlation length. Therefore, in comparison to that of the nondisordered case, we can use a smaller Wilson loop <cit.> to calculate the TEE of disordered interacting ZGNRs .
The correlation length may be computed from the entanglement entropy of an area A. It is computed from the correlation function, which is also known as the reduced density matrix of region A.
The HF correlation function <cit.> between i∈ A and j∈ A decays exponentially as a function of distance x between i and j
C(x) = C_i↑, j↑ = ⟨Ψ| c^†_i↑ c_j↑|Ψ⟩∼exp(- | x|/ξ),
where Ψ represents the HF ground state of the ZGNR and ξ represents the correlation length.
By inverting the relation given in Eq.(<ref>), we can write c_iσ as a linear combination of a_k. This makes it straightforward to compute C(x). To compute it accurately, the area must be larger than the correlation length <cit.>.
We compute the correlation function and determine the correlation length ξ, as shown in Fig. <ref>.
For the disordered case Γ≠ 0, the correlation length is obtained by averaging over several disorder realizations. The Anderson localization reduces the correlation length compared to that of disorder-free ribbons, as shown in Fig. <ref>(b). (Disorder-free ZGNRs have a large correlation length for small U.) In contrast, doping increases the correlation length, as shown in Fig. <ref>(c).
Another important effect of Anderson localization in the presence of on-site repulsion is that quasi-degenerate localized states
are spatially separated <cit.>, leading to well-defined fractional charges <cit.>. (In low-doped ZGNRs added electrons fractionalize and form a narrow peak in the DOS near E=0 consisting of quasi-degenerate localized states <cit.>.)
The probability densities of such two mid-gap states carrying fractional charges are shown in Fig. <ref>(d). These gap-edge states are mixed chiral states <cit.>, whose probability densities peak at the two edges and rapidly decays inside the ribbon.
Note that these states do not overlap with each other. Non-interacting electrons of disordered ZGNRs also display mixed chiral states near E=0. However, although the overlap between nearly degenerate states in the weak-disorder regime is small, it is not negligible. Thus well-defined fractional charges do not readily form in non-interacting disordered ZGNRs.
§ PHASE DIAGRAM
Topological order can be detected by investigating the TEE (β) <cit.> within the HFA <cit.>. We first select a set of values for
(L, W, w, l_zig,l_arm), as defined in Fig. <ref>(a), to compute β. Next, these quantities are increased by the same ratio and a new β is computed. This process is repeated several times (see Ref. <cit.> for details). We apply finite-size scaling analysis to extract the value of the TEE in the limit L→∞ (see Fig. <ref>(b)).
We divide the parameter space (Γ,U,δ N) into three-dimensional grid points, and at each grid point, we compute β (see Fig. <ref>(c)). The three-dimensional phase diagram obtained is shown in Fig. <ref>(d).
We find that β can have three types of values: (i) A universal value in the topologically ordered phase, (ii) nonuniversal values of β with large variances in the crossover phases, and (iii) a zero value of
β in the normal-disordered phase. Projections of the phase diagram, namely U-Γ,
Γ-δ N, and
U–δ N planes, are shown in Figs. <ref>(e)-(g).
In undoped ZGNRs a TO phase is found in regions Γ/U≲ 1 and U≲ t, see Fig. <ref>(c). The topological phase transition into the symmetric protected phase at Γ=0 is abrupt, consistent with the result of Ref. <cit.> (The TEE of the symmetric phase is zero). There are also other topological phase transitions but they are smooth transitions with crossover regions <cit.>.
Figs. <ref>(e)-(g) display the presence of crossover regimes lying beyond the TO phase with an increase in the disorder, doping, and interaction strength.
The phase boundaries between topologically ordered and normal phases are “blurred”, which indicate the presence of crossover phases (there are two types of crossover (CO) phases, labeled COI and COII). The numerical results of the TEE is shown in Fig. <ref>(c).
The error bars in this figure include, besides random fluctuations caused by disorder, the uncertainties that occur in the extrapolation process of the finite scaling analysis.
As Γ/U increases, β decreases (see the red line
in Fig. <ref>(c)). The value of the TEE thus changes across a crossover phase.
In such a phase, β has a large variance, but the average values are not zero, which implies that the topological order is not completely destroyed. In this regime, the TEE becomes nonuniversal and decays.
In crossover phases charge transfer correlations between the opposite zigzag edges are present but fractional charges are not well defined, or vice versa.
One can use a different but equivalent
procedure to determine the phase diagram. We verified that the same phase diagram can be obtained by analyzing the presence of fractional charges and nonlocal correlations between the opposite zigzag edges. At each grid point (Γ,U,δ N) in the parameter space, we find the ground state and investigate whether the gap-edge states display fractional charges and whether nonlocal correlations exist between the opposite zigzag edges. By ultilizing this method, we have successfully recovered the phase diagram shown in Fig. <ref>(d).
§ TOPOLOGICALLY ORDERED PHASE
The universal region was investigated in Ref. <cit.>, and therefore, we do not describe this phase in detail here. However, we would like to mention some new results.
We elucidate the nature of correlations in topologically ordered ZGNRs. A ZGNR is shown in Fig. <ref>(a). It consists of 8 carbon lines labeled l = 1, 2, 3, 4, 5, 6, 7, 8.
In each pair of carbons lines (1, 8), (2, 7), (3, 6), and (4, 5), an increase/decrease in the occupation number of one line is correlated with a decrease/increase in that of the other line (see, for example, lines 1 and 8 in Fig. <ref>(c)).
It is not only the zigzag edges that are correlated in this way, but also other carbon lines inside the ribbon that are away from the edges. The corresponding site spins of the ribbon are shown in Fig.<ref> (d).
Mixed chiral gap states contribute to this effect (these gap-edge states can decay slowly from the zigzag edges, unlike the fractional edge states. A schematic picture of a mixed chiral state is shown in Fig. <ref>(b)). Changes in the occupation numbers δ n_i,↑ and δ n_i,↓ of an edge often coincide at nearly the same values as k, which labels the site position along the ribbon direction. This effect can
lead to n_i,↑≈ n_i,↓ of the occupation numbers in the presence of disorder, resulting in s_i≈ 0, i.e., the appearance of spin-charge separation around a site on one of the edges <cit.>.
The following points should be also noted. The results in Fig.<ref>(c) show that
the variance of β decreases in the singular limit Γ/U→ 0. (Additional numerical results confirm this conclusion.) This result is consistent with the previous finding that
fractional charge of a midgap state becomes accurate in the weak disorder regime and in the thermodynamic limit (see Ref.<cit.>).
In the opposite limit Γ/U ≫ 1, the value of the TEE is non-universal and decreases with increasing U (see Fig. <ref>(c)). In addition, the functional dependence of DOS on E in the universal region is given by an exponentially suppressed function, a linear function <cit.>, or something in between. The actual shape of the DOS is determined by the competition between the strength of disorder and the on-site repulsion <cit.>; for example, the DOS is linear for (U,Γ, δ N) = (2t, 0.5t, 0), but it is exponentially suppressed in the weak disorder limit.
§ CROSSOVER PHASE I
We describe in detail the properties of undoped ZGNRs in the COI phase, where U ≳ t and U≳Γ (the on-site repulsion U is the dominant energy in this phase). The TO phase gradually changes
into the COI phase as U increases, as illustrated in Fig. <ref>(e). In this phase, β is nonzero, but its variance is significant, as shown in Fig <ref>(a). This phase has the following properties:
(i) The disorder-induced change in the edge occupation numbers δ n_i,↑= 1/2 for one type of spin σ is entirely transferred to the opposite edge, i.e., the zigzag edges are correlated in a nontrivial manner (see Fig. <ref>(c) for Γ/U = 0.17).
However, the site positions k on the opposite zigzag edges, where changes in δ n_i,↑ and δ n_i,↓ occur , do not coincide.
(In contrast, these positions are correlated at the nearly same values of k in the TO phase, as we mentioned before.) We believe that these edge transfer correlations between zigzag edges change the ground state entanglement pattern and yield a nonzero fluctuating TEE. The edge charge transfer correlations become weaker when the disorder is stronger (see Fig. <ref>(d) for Γ/t=4), leading to a smaller value of β (see Fig. <ref>(c)).
(ii) Although the zigzag edge changes are fractional, δ n_i,↑= 1/2 (Figs. <ref>(b) and (c)), the A- and B-probability densities of the mixed chiral states responsible for this feature overlap, see Fig. <ref>(c). Thus, fractional charges are ill-defined.
(iii) For spin-charge separation to be present, charge transfers for both spins must occur at the same values k. These effects are not observed in the COI phase. Note that, the condition S_z=1/2(n_i,↑-n_i,↓)=0 at site i is not sufficient for spin-charge separation. To fulfill the conditions, well-defined fractional charges must exist.
§ CROSSOVER PHASE II
For undoped ZGNRs, there is another CO phase for Γ≫ U but U/t≲ 1. We call this phase COII where the disorder strength Γ is the dominant energy. As Γ increases, the TO phase undergoes a gradual transition into the COII phase, as demonstrated in Figure <ref>(e). Concurrently, the gap is progressively filled with states, as depicted in the upper graph of Figure <ref>(a).
Similar to the COI phase (Fig. <ref>(a)) β is finite with a significant variance. But there are no charge-transfer correlations between the zigzag edges (see the lower graph of Figure <ref>(a)). However, some fractional charges may exist, see Fig. <ref>(b). This is consistent with the following obtained results: (i) Some changes in the edge occupation number are δ n_i,σ≈±1/2.
(ii) There are gap-edge states with q_A ≈ 1/2. (Here q_A=∑_i∈ A|ψ_iσ(E)|^2, where ψ_iσ(E) is the HF eigenstate with energy E, see Ref.<cit.>. The probability densities are summed over all sites of the A sublattice.) However, the variance of q_A in the energy interval [E-δ E,E+δ E] is large because q_A varies substantially in this interval, as shown in Fig. <ref>(a). (But the disorder averaged mean charge value of the states in this energy interval is e^-/2.) Despite this, a fractional charge of a state in the interval [E-δ E,E+δ E] near E=0 does not overlap significantly with the probability densities of other fractional and non-fractional states in the same energy interval, provided that δ E is small (for U∼ t and Γ∼ t this happens when δ E∼ 0.01t). We believe that the interplay between localization and on-site repulsion is responsible for this effect. However, since a gap is absent, the fractional charges are less stable compared to the TO phase.
We checked that several states in the same value of δ E overlap in the absence of on-site repulsion.
Thus far, we investigated the undoped case. Upon doping, the disorder-free ZGNRs exhibit edge spin density waves instead of edge ferromagnetism of undoped ribbons. If disorder is added to a doped ZGNR, the spin waves become distorted <cit.>: There is a topological phase transition from modulated ferromagnetic edges at zero doping to distorted spin-wave edges at finite doping. Our results indicated that, when doping is substantial, this phase is also a COII phase.
The dependence of the mean value of q_A of the states in the mid-gap peak on the number of doped electrons is shown in Fig. <ref>(d) (the DOS shows a sharp peak at the mid-gap energy, see Fig. <ref>(c)). At a low doping concentration, the disorder-averaged value of q_A is close to 0.5. The states in the mid-gap peak display well-defined fractional charges, as we discussed below Fig.<ref> (note that the width of the midgap peak is δ E∼ 0.005t). As the doping concentration increases further, q_A significantly deviates from 0.5, and simultaneously, the DOS mid-gap peak starts to decrease <cit.>. These findings imply that even though fractional charges can be found, their number decreases with an increase in doping. The gradual change in q_A as a function of δ N / N_sindicates that the transition from the phase of the distorted ferromagnetic edge to the phase of the distorted edge spin-wave is not sharp. Figure <ref>(e) shows how β decreased with an increase in δ N / N_s. For a large δ N / N_s, it is computationally demanding to calculate β because the correlation length is expected to be longer (see Fig. <ref>(c)) in comparison to that of undoped ZGNRs.
§ STRONGLY DISORDERED AND STRONGLY REPULSIVE PHASES
We discuss the strongly disordered phase in region Γ/U≫ 1 (see Fig. <ref>(e)). The topological order is destroyed once the disorder strength reaches a sufficiently large value (e.g., β = 0 at (U,Γ, δ N) = (t,15t, 0)). In this region the edge charge-transfer correlations and charge fractionalization are not well-defined, which implies that TEE is zero.
In Fig. <ref>(a), site occupation numbers in weak (Γ = 0.03 t) and strong (Γ = 15 t) disorder regimes are shown side by side to highlight the difference, where the ones in strong disorder regime highly fluctuate from site to site.
Edge magnetization is zero almost everywhere (see Fig. <ref>(b)). The occupation numbers display sharp values of n_i,σ=1 at some sites (see Fig. <ref>(b)) (they were also present in the DMRG calculations of disordered ribbons, see Ref.<cit.>).
Another phase, that is, the strongly repulsive phase (U≫ t) with no fractional charges and zigzag edge correlations, is shown in Fig. <ref>(e). In this case, β≈ 0.
The q_A–E diagram in Fig. <ref>(c)
displays the nonperturbative nature of disorder in this regime: the values q_A are scattered between 0 and 1 in the limit Γ→0, whereas they are restricted to
the four solid lines at Γ = 0.
Also, the (E, q_A) distribution indicates that in a strongly repulsive regime, even with the presence of disorder, there is still a large energy gap.
There are no states with q_A≈ 1/2 near the mid-gap energy, as shown in Fig. <ref>(c). The A and B components of the wave function of the states with q_A≈ 1/2 near the gap edges ±Δ/2 overlap (see Fig. <ref>(d)).
For a stronger disorder (larger values of Γ), the gap is filled with states such that the DOS is finite at E = 0, as shown in Fig. <ref>(e).
The main physics of this phase is illustrated by investigating the zigzag edge structure: the occupation numbers are n_i,σ=1 or 0 so charge transfers are one (δ n_i,σ=± 1) in the strongly repulsive phase (the total site occupation number of each site is n_i≈ 2, 1, or 0 despite a strong U). No transfer of fractional charges was observed between zigzag edges. This is because mixed chiral gap edge states are not present. Note that the edge magnetization displays sharp domain walls, as indicated in Figs. <ref>(f)-(g).
§ SUMMARY AND DISCUSSION
We computed the phase diagram of zigzag graphene nanoribbons as a function of the on-site repulsion U, doping δ N, and disorder strength Γ.
We identified the universal, crossover, strongly disordered, and strongly repulsive phases.
Each phase of the phase diagram was defined by the TEE value and its variance. We also investigated how the values of the TEE are related to the following physical properties: the presence of charge fractionalization and the edge charge transfer correlations between the opposite zigzag edges. When both properties are present, in addition to correlations leading to spin-charge separation,
β was universal. If only one of these properties is present, β was nonuniversal and its variance was significant. However, when both were absent, β was approximately zero. In addition, we found a strongly repulsive phase with zero TEE
in large on-site repulsion and weak disorder limits. Its ground state contains abrupt kinks in zigzag edge magnetizations without charge fractionalization, which is a consequence of the singular perturbative nature of the disorder potential.
There is another phase with zero TEE, i.e., the strongly disordered phase in regime Γ≫ U. In this phase, the edge site occupation numbers fluctuate highly from site to site, and antiferromagnetic coupling between the two edges are nearly destroyed.
Each phase of the phase diagram has a different zigzag-edge structure.
We also investigated the effect of the interplay between localization and on-site repulsion on the charge quantization. In low-doped and/or weakly disordered ZGNRs this interplay contributes to the spatial separation of quasi-degenerate gap-edge states and protects the charge fractionalization against quantum fluctuations. Even in the presence of moderately strong disorder charge fractionalization is not completely destroyed.
We briefly discuss some experimental implications. It would be interesting to observe the presence of nonlocal charge transfers between the zigzag edges of the COI phase. This can be investigated by measuring correlations between the edge site occupation numbers using a scanning tunneling microscope <cit.>.
In the COII phase, fractional e^-/2 edge charges are present; however, unusual transport and magnetic susceptibility
properties are not expected because spin-charge separation is not present. (In contrast, the TO phase is expected to display unusual transport and magnetic susceptibility because of spin-charge separation <cit.>.) Similarly, scanning tunneling microscopy can be used to verify the predicted edge occupation numbers
of the strongly repulsive and strongly disordered phases.
In addition, investigation of tunneling between zigzag edges, as in fractional quantum Hall bar systems <cit.>, may be fruitful.
10
rm
url<#>1urlprefixURL doiprefixDOI:
Lei13
authorLeinaas, J. M. & authorMyrheim, J.
journaltitleOn the theory of identical
particles.
Nuovo Cimento B
volume37, pages1–23,
<10.1007/BF02727953> (year1977).
Wilczek03
authorWilczek, F.
journaltitleQuantum mechanics of
fractional-spin particles.
Phys. Rev. Lett.
volume49, pages957–959,
<10.1103/PhysRevLett.49.957> (year1982).
Arovas
authorArovas, D., authorSchrieffer, J. R. &
authorWilczek, F.
journaltitleFractional statistics and the
quantum hall effect.
Phys. Rev. Lett.
volume53, pages722–723,
<10.1103/PhysRevLett.53.722> (year1984).
Nakamura01
authorNakamura, J., authorLiang, S.,
authorGardner, G. C. & authorManfra, M. J.
journaltitleDirect observation of anyonic
braiding statistics.
Nature Physics volume16,
pages931–936,
<https://doi.org/10.1038/s41567-020-1019-1>
(year2020).
Barto1
authorBartolomei, H. et al.
journaltitleFractional statistics in anyon
collisions.
Science volume368,
pages173–177, <10.1126/science.aaz5601>
(year2020).
GV2019
authorGirvin, S. M. & authorYang, K.
titleModern condensed matter physics
(publisherCambridge University Press,
addressCambridge, year2019).
Pach
authorPachos, J. K.
titleIntroduction to Topological Quantum
Computation (publisherCambridge University Press,
addressCambridge, year2012).
Wen11
authorWen, X.-G.
journaltitleColloquium: Zoo of
quantum-topological phases of matter.
Rev. Mod. Phys.
volume89, pages041004,
<10.1103/RevModPhys.89.041004> (year2017).
NPlaugh
authorLaughlin, R. B.
journaltitleAnomalous quantum hall effect: An
incompressible quantum fluid with fractionally charged excitations.
Phys. Rev. Lett.
volume50, pages1395–1398,
<10.1103/PhysRevLett.50.1395> (year1983).
Yang
authorS.-R. Eric Yang.
titleTopologically Ordered Zigzag Nanoribbon
(publisherWorld Scientific, Singapore, year2023).
Kitaev11
authorKitaev, A. & authorPreskill, J.
journaltitleTopological entanglement entropy.
Phys. Rev. Lett.
volume96, pages110404,
<10.1103/PhysRevLett.96.110404> (year2006).
Levin11
authorLevin, M. & authorWen, X.-G.
journaltitleDetecting topological order in a
ground state wave function.
Phys. Rev. Lett.
volume96, pages110405,
<10.1103/PhysRevLett.96.110405> (year2006).
Haldane191
authorLi, H. & authorHaldane, F. D. M.
journaltitleEntanglement spectrum as a
generalization of entanglement entropy: Identification of topological order
in non-abelian fractional quantum hall effect states.
Phys. Rev. Lett.
volume101, pages010504,
<10.1103/PhysRevLett.101.010504> (year2008).
Altshuler
authorAltshuler, B.
titleInductory anderson localization.
In booktitleAdvanced Workshop on Anderson
Localization, Nonlinearity and Turbulence: a Cross-Fertilization
(organizationInternational Centre for Theoretical Physics,
year2010).
GV2000
authorGirvin, S. M.
titleThe quantum hall effect: novel excitations and broken
symmetries.
In booktitleAspects topologiques de la physique en
basse dimension. Topological aspects of low dimensional systems,
pages53–175 (publisherSpringer,
year1999).
Fujita
authorFujita, M., authorWakabayashi, K.,
authorNakada, K. & authorKusakabe, K.
journaltitlePeculiar localized state at zigzag
graphite edge.
J. Phys. Soc. Jpn.
volume65, pages1920–1923,
<10.1143/JPSJ.65.1920> (year1996).
Brey2006
authorBrey, L. & authorFertig, H. A.
journaltitleElectronic states of graphene
nanoribbons studied with the dirac equation.
Phys. Rev. B volume73,
pages235411, <10.1103/PhysRevB.73.235411>
(year2006).
Lyang
authorYang, L., authorPark, C.-H.,
authorSon, Y.-W., authorCohen, M. L. &
authorLouie, S. G.
journaltitleQuasiparticle energies and band
gaps in graphene nanoribbons.
Phys. Rev. Lett.
volume99, pages186801,
<10.1103/PhysRevLett.99.186801> (year2007).
Pisa1
authorPisani, L., authorChan, J. A.,
authorMontanari, B. & authorHarrison, N. M.
journaltitleElectronic structure and magnetic
properties of graphitic ribbons.
Phys. Rev. B volume75,
pages064418, <10.1103/PhysRevB.75.064418>
(year2007).
Cai2
authorRuffieux, P. et al.
journaltitleOn-surface synthesis of graphene
nanoribbons with zigzag edge topology.
Nature volume531,
pages489–492, <10.1038/nature17151>
(year2016).
Kolmer
authorKolmer, M. et al.
journaltitleRational synthesis of atomically
precise graphene nanoribbons directly on metal oxide surfaces.
Science volume369,
pages571–575, <10.1126/science.abb8880>
(year2020).
Brey
editorBrey, L., editorSeneor, P. &
editorTejeda, A. (eds.) titleGraphene
Nanoribbons.
2053-2563 (publisherIOP Publishing,
year2019).
Yang2019
authorJeong, Y. H., authorS.-R. Eric Yang &
authorCha, M.-C.
journaltitleSoliton fractional charge of
disordered graphene nanoribbon.
Journal of Physics: Condensed Matter
volume31, pages265601,
<10.1088/1361-648X/ab146b> (year2019).
yang1
authorS.-R. Eric Yang.
journaltitleSoliton fractional charges in
graphene nanoribbon and polyacetylene: similarities and differences.
Nanomaterials volume9,
pages885, <10.3390/nano9060885>
(year2019).
Yang2020
authorS.-R. Eric Yang, Cha, M.-C., authorLee,
H. J. & authorKim, Y. H.
journaltitleTopologically ordered zigzag
nanoribbon: e/2 fractional edge charge, spin-charge separation, and
ground-state degeneracy.
Phys. Rev. Research
volume2, pages033109,
<10.1103/PhysRevResearch.2.033109> (year2020).
Dob
authorDobrosavljevic, V., authorTrivedi, N. &
authorValles, J. M., Jr.
titleConductor-Insulator Quantum Phase
Transitions (publisherOxford University Press,
year2012).
Belitz
authorBelitz, D. & authorKirkpatrick, T. R.
journaltitleThe anderson-mott transition.
Rev. Mod. Phys.
volume66, pages261–380,
<10.1103/RevModPhys.66.261> (year1994).
Byczuk
authorByczuk, K., authorHofstetter, W. &
authorVollhardt, D.
journaltitleCompetition between anderson
localization and antiferromagnetism in correlated lattice fermion systems
with disorder.
Phys. Rev. Lett.
volume102, pages146403,
<10.1103/PhysRevLett.102.146403> (year2009).
Yang2021
authorKim, Y. H., authorLee, H. J. &
authorS.-R. Eric Yang.
journaltitleTopological entanglement entropy of
interacting disordered zigzag graphene ribbons.
Phys. Rev. B volume103,
pages115151, <10.1103/PhysRevB.103.115151>
(year2021).
Yang2022
authorKim, Y. H., authorLee, H. J.,
authorLee, H.-Y. & authorS.-R. Eric Yang.
journaltitleNew disordered anyon phase of doped
graphene zigzag nanoribbon.
Scientific Reports
volume12, pages14551,
<10.1038/s41598-022-18731-6> (year2022).
Efros
authorÉfros, A. L. & authorShklovskii, B. I.
journaltitleCoulomb gap and low temperature
conductivity of disordered systems.
Journal of Physics C: Solid State Physics
volume8, pagesL49–L51,
<10.1088/0022-3719/8/4/003> (year1975).
Lima2012
authorLima, L. R. F., authorPinheiro, F. A.,
authorCapaz, R. B., authorLewenkopf, C. H. &
authorMucciolo, E. R.
journaltitleEffects of disorder range and
electronic energy on the perfect transmission in graphene nanoribbons.
Phys. Rev. B volume86,
pages205111, <10.1103/PhysRevB.86.205111>
(year2012).
Canri
authorCanright, G. S., authorGirvin, S. M. &
authorBrass, A.
journaltitleSuperconductive pairing of fermions
and semions in two dimensions.
Phys. Rev. Lett.
volume63, pages2295–2298,
<10.1103/PhysRevLett.63.2295> (year1989).
Heeger
authorHeeger, A. J., authorKivelson, S.,
authorSchrieffer, J. R. & authorSu, W. P.
journaltitleSolitons in conducting polymers.
Rev. Mod. Phys.
volume60, pages781–850,
<10.1103/RevModPhys.60.781> (year1988).
Zeng
authorZeng, B., authorChen, X., authorZhou,
D.-L. & authorWen, X.-G.
titleQuantum Information Meets Quantum Matter
(publisherSpringer New York, addressSpringer,
year2019).
Stau
authorStauber, T. et al.
journaltitleInteracting electrons in graphene:
Fermi velocity renormalization and optical response.
Phys. Rev. Lett.
volume118, pages266801,
<10.1103/PhysRevLett.118.266801> (year2017).
Neto
authorCastro Neto, A. H., authorGuinea, F.,
authorPeres, N. M. R., authorNovoselov, K. S. &
authorGeim, A. K.
journaltitleThe electronic properties of
graphene.
Rev. Mod. Phys.
volume81, pages109–162,
<10.1103/RevModPhys.81.109> (year2009).
Yang1995
authorS.-R. Eric Yang, authorMacDonald, A. H. &
authorHuckestein, B.
journaltitleInteractions, localization, and the
integer quantum hall effect.
Phys. Rev. Lett.
volume74, pages3229–3232,
<10.1103/PhysRevLett.74.3229> (year1995).
Mac1
authorS.-R. Eric Yang & authorMacDonald, A. H.
journaltitleCoulomb gaps in a strong magnetic
field.
Phys. Rev. Lett.
volume70, pages4110–4113,
<10.1103/PhysRevLett.70.4110> (year1993).
Peschel119
authorPeschel, I.
journaltitleCalculation of reduced density
matrices from correlation functions.
Journal of Physics A: Mathematical and
General volume36, pagesL205,
<10.1088/0305-4470/36/14/101> (year2003).
Bal
authorJiang, H.-C., authorWang, Z. &
authorBalents, L.
journaltitleIdentifying topological order by
entanglement entropy.
Nature Phys volume8,
pages902–905, <10.1038/nphys2465>
(year2012).
Andrei
authorAndrei, E. Y., authorLi, G. &
authorDu, X.
journaltitleElectronic properties of graphene:
a perspective from scanning tunneling microscopy and magnetotransport.
Reports on Progress in Physics
volume75, pages056501,
<10.1088/0034-4885/75/5/056501> (year2012).
Chung
authorChung, T.-C., authorMoraes, F.,
authorFlood, J. D. & authorHeeger, A. J.
journaltitleSolitons at high density in
trans-(CH)_x: Collective transport by mobile, spinless charged
solitons.
Phys. Rev. B volume29,
pages2341–2343, <10.1103/PhysRevB.29.2341>
(year1984).
Kang
authorKang, W., authorStormer, H.,
authorPfeiffer, L., authorBaldwin, K. &
authorWest, K.
journaltitleTunnelling between the edges of two
lateral quantum hall systems.
Nature volume403,
pages59–61, <https://doi.org/10.1038/47436>
(year2000).
Iyang
authorYang, I., authorKang, W.,
authorBaldwin, K. W., authorPfeiffer, L. N. &
authorWest, K. W.
journaltitleCascade of quantum phase
transitions in tunnel-coupled edge states.
Phys. Rev. Lett.
volume92, pages056802,
<10.1103/PhysRevLett.92.056802> (year2004).
§ ACKNOWLEDGEMENTS
S.R.E.Y. was supported by the Basic Science Research Program of the National Research Foundation of
Korea (NRF), funded by the Ministry of Science and ICT (MSIT) NRF-2021R1F1A1047759. Two other grants are also acknowledged: BK21 FOUR (Fostering Outstanding Universities for Research)
and KISTI Supercomputing Center with supercomputing resources including technical support KSC-2022-CRE-0345.
§ DATA AVAILABILITY
On reasonable request, the corresponding author will provide all relevant data in this paper.
§ CODE AVAILABILITY
On reasonable request, the corresponding author will provide all numerical codes in this paper.
§ AUTHOR CONTRIBUTIONS
L.H.A., Y.H.K. and I.H.L. performed the HF calculations. S.R.E.Y. conceived the project and supervised the study. All authors contributed to the writing of the manuscript.
§ COMPETING INTERESTS
The authors declare no competing interests.
|
http://arxiv.org/abs/2307.04269v1 | 20230709215514 | Big-Bang Nucleosynthesis within the Scale Invariant Vacuum Paradigm | [
"V. G. Gueorguiev",
"A. Maeder"
] | nucl-th | [
"nucl-th"
] |
Ronin Institute, Montclair, NJ, USA
Institute for Advanced Physical Studies, Sofia, Bulgaria
Geneva Observatory, University of Geneva, Switzerland
The Scale Invariant Vacuum (SIV) paradigm is applied to the Big-Bang Nucleosynthesis
using the known analytic expressions for the expansion factor a and
the plasma temperature T as functions of the SIV time τ since the Big-Bang
when a(τ=0)=0. The results are compared to the known standard
BBNS model as calculated with the PRIMAT code. Potential SIV-guided deviations from the
local statistical equilibrium are explored. Overall, we find that smaller than usual baryon
and non-zero dark matter content, by a factor of three to five times reduction,
result in compatible to the standard reproduction of the light elements abundances.
Keywords:
Cosmology: theory – primordial nucleosynthesis – dark matter
Big-Bang Nucleosynthesis within
the Scale Invariant Vacuum Paradigm
A. Maeder
August 12, 2023
====================================================================
§ INTRODUCTION
It is often ignored that when the cosmological constant Λ is assumed to be equal to zero,
the equations of General Relativity are scale invariant,
which is also a property present in Maxwell equations of electrodynamics.
However, for a non-zero Λ, the field equations of the gravitation no longer show the property of scale invariance.
A fact, which as discussed by <cit.>, was one of the reasons of Einstein's disenchantment with the cosmological constant.
It is thus of interest to examine at what conditions the scale invariant properties of General Relativity may be restored,
since current cosmological observations support a positive cosmological constant.
A theoretical framework has been developed by <cit.> and <cit.>,
in the so-called co-tensorial calculus based on the Weyl's Integrable Geometry (WIG) based on the original idea by <cit.>.
It offers a consistent basis to account for the properties of scale invariance of gravitation via a scale factor λ,
as also illustrated by several properties studied by <cit.>.
Scale invariant derivatives, modified affine connexions, modified Ricci tensor and curvatures can be obtained leading to a general scale
invariant field equation. Dirac and Canuto et al. have expressed an action principle in the scale invariant framework,
with a matter Lagrangian, as a coscalar of power n=-4 (varying like λ^-4).
By considering the variations of this action, they also obtain the generalization of the Einstein field equation.
This is Equation (7) in <cit.> from which the scale invariant cosmological equations are derived.
In the Weyl’s Integrable Geometry (WIG), the scale factor is undetermined without any other constraints
as shown by Dirac and Canuto et al. Thus, these authors were fixing the scale factor by some external
considerations based on the so-called Large Number Hypothesis; this hypothesis, however, is often disputed <cit.>.
Thus, it seems appropriate to explore other conditions for setting the gauge.
The proposition was made to fix the gauge factor by simply assuming the scale invariance of the empty space <cit.>.
This means that the properties of the empty space should not change under a contraction or dilatation of the space-time.
Indeed, we note, as shown by <cit.>, that the current equation of the vacuum
p_vac= - ϱ_vac c^2 already implies that
ϱ_vac should remain constant
“if a volume of vacuum is adiabatically compressed or expanded”.
On this basis, the cosmological equations derived by <cit.> were simplified <cit.>.
A number of cosmological tests were performed, with positive results. These equations were
then shown to have rather simple analytical solutions <cit.> for models of a matter dominated Universe with a zero curvature.
In order to express the motions of free particles, a geodesic equation was obtained <cit.>
from a minimum action in the Weyl's integrable geometry (WIG). In the weak field approximation,
the geodesic equation leads to a modification of the Newton equation <cit.> where it contains
a (currently very small) additional acceleration term proportional to the velocity of the particles.
This equation was applied to study the internal motions of clusters of galaxies,
the flat rotation curves of spiral galaxies and
the age increase of the “vertical” velocity dispersion of stars in galaxies <cit.>.
The interesting result was that the observational properties of these various systems
could be accounted without requiring to the current hypothesis of dark matter,
and the same for the radial acceleration relation (RAR) of galaxies <cit.>.
The growth of the density fluctuations in the early Universe was also studied by <cit.>
who showed that dark matter is not needed, within the Scale-Invariant Vacuum (SIV) paradigm,
to achieve the growth of the density fluctuations to the currently observed inhomogeneities <cit.>.
Such studies suggested a connection between the Scale-Invariant Vacuum (SIV) theory and
the origin of Dark Matter and Dark Energy <cit.>.
This was further reenforced by the study of scale-invariant dynamics of galaxies,
MOND, dark matter, and the dwarf spheroidals <cit.>.
Furthermore, it was shown that the SIV framework naturally relates the scale invariance, horizons, and inflation,
while providing also a graceful exit from inflation <cit.>.
Summary of the main results was compiled and presented at the conference on
Alternative Gravities and Fundamental Cosmology (AlteCosmoFun'21) and published
in the journal Universe <cit.>.
The above successes naturally inquire further studies as to the applicability of the SIV paradigm to other well-known phenomenon.
A study of the Cosmic Microwave Background <cit.> is one such phenomenon along with
the Big-Bang nucleosynthesis (BBNS), which could be understood without a complicated numerical simulations
<cit.>. The BBNS phenomenon is very relevant for us
since the SIV possesses analytic expressions suitable for first exploration of such a problem <cit.>.
This could be very useful as an approach to study BBNS within the SIV paradigm given
by the recent Mathematica code PRIMAT <cit.>.
The objective of the present work is to apply the analytic expressions derived within the SIV paradigm <cit.>
to the BBNS via the use of the PRIMAT code and to see how well the SIV will perform compared to the standard BBNS model.
For this purpose, in Section <ref>, we provide a summary of the background needed by the reader to understand
the framework to be utilized. In Section <ref> are discussed the main methods, similarities and difference of various relevant functions,
and the equations that need to be employed within the computational process. In Section <ref> we present our main results and
explain the various model choices described in the tables shown.
Finally, summary and conclusions are presented in the Section <ref>.
§ BACKGROUND FRAMEWORK
We start this section with a summary of the commonly used fundamental
physical constants and expressions relevant for the description of
the early Universe and their relation to the observations during the
current epoch:
H_0 = h H_100 , h=70/100 , H_100=100 km/s/Mpc=3.2408×10^-18 s^-1 ,
ρ_c0 = 3H_0^2/(8π G) , G=6.6743×10^-11 m^3/kg/s^2 , τ_0=4.355×10^17s ,
T_0 = 2.7255 K , a_BB=π^2/(15 ħ^3 c^5)=2.31674×10^59 s^2/m^5/J^3 ,
k_B = 1.3806×10^-23J/K , ρ_γ0=a_BB(k_B T_0)^4/c^2=4.6485×10^-34 g/cm^3 ,
N_ν = 3 , K_0=1+7/8(4/11)^4/3N_ν=1.6813 , ρ_γ0 h^2/ρ_c0=2.47476×10^-5.
Here, H_0 is the Hubble constant expressed via the reduced dimensionless
Hubble parameter h and the scale fixing formal constant H_100.
The usual current critical density based on H_0 is ρ_c0, G is Newton's gravitational constant,
and τ_0 is the current age of the Universe
(13.8 Gyr with 365.25 days in a year as in <cit.>).
Some minor differences from ref. <cit.> are to be noted here:
the choice h=0.7 is used in ref. <cit.> while PRIMAT uses Planck's
CMB value of h=0.677 <cit.> and
the pre-factor defining v_eq, in Eq. (27) of <cit.>, is 2.4741×10^-5
rather than the above value for ρ_γ0 h^2/ρ_c0,
furthermore, the current value of the CMB temperature is T_0=2.726 K in <cit.>,
and a=a_BBk_B^4 in Eq. (27) of ref. <cit.>.
PRIMAT <cit.> uses units such that c=1, k_B=1, and ħ=1 with Planck's
CMB value of h=67.66/100 along with the number of effective neutrino flavors as N_ν^eff=3.01,
while Ω_m=0.31 and Ω_b=0.05 correspondingly.
The relevant SIV analytic expressions based on <cit.> are summarized
in the next set of formulas where the prefix “A” is used to indicate
that the subsequent equation number refers to the corresponding original equation
in ref. <cit.>:
(A27) v_eq=K_0 ρ_γ0/(Ω_mρ_c0) , c_2=(v_eq^2+√(v_eq^4+C_rel))/t_eq^2 , (A21)
(A20) C_m=4Ω_m/(1-Ω_m)^2, C_rel=C_m v_eq ,
(A25) t_eq=2^-2/3(v_eq^3/2(1-Ω_m)+√(v_eq^3(1-Ω_m)^2+4Ω_m))^2/3,
(A29) t_ in=C_rel^1/4/c_2^1/2 , Δt=(t_0-t_ in) τ/τ_0 , (A30)
(A33) a(Δt)=√(2c_2t_ in^3 Δt) , τ(T)=T_0^2τ_0/2(t_0-t_ in)√(C_rel)1/T^2, (A39)
(A37) ρ_r(Δt)=ρ_γ0K_0/4C_relΔt^2 , ρ_m(Δt)=ρ_m0c_2^1/4/C_rel^7/8(2Δt)^3/2.
where in (39) of <cit.> one has 1.272×10^9 instead of T_0√(τ_0/2)=1.271×10^9 here.
The quantities v_eq and c_2 are integration constants for the SIV modified FLRW equation and are related to the
matter energy content C_m and the radiation energy content C_rel, while t_eq is the moment of matter–radiation
equality given in the SIV t–time such that the current time satisfies t_0=1. The moment of the Big-Bang (BB),
when a=0, is denoted by t_ in∈[0,1), while Δt is the time since the BB.
Δt is related via (A30) to the conventional time τ=Δτ since the BB in seconds
when τ_0 is the current age of the Universe in seconds and the BB is at τ=0.
The expansion factor a(τ), also known as RW spatial scale factor,
is given by substituting (A30) in (A33),
while (A39) gives the relationship between age τ and temperature T of the radiation.
The last two expressions are the energy-densities for radiation and matter within the SIV.
In PRIMAT, the thermonuclear reaction equations that describe the rate of change
of the abundances of the various nuclear species are defined via in-scalar variables
Y_i=n_i/n_b based on the number density of the nucleus i relative to
the total baryon number density n_b.
For PRIMAT based expressions, we will use prefix P followed by the corresponding equation number in ref. <cit.>.
These equation numbers may differ by ±1 between the arXiv version and the published version of the paper.
The usual reaction rates for the production and reduction of a specific nucleus
are re-expressed from the traditional form, i.e. a two body reaction i+j↔ k+l (P131),
into the new form (P136), and also into a more general case of more bodies (P138),
but the overall co-tensor structure stays the same since all Γ-s are now only in units of inverse seconds:
(P131) ṅ_i⊃ n_kn_lγ_kl→ij-n_in_jγ_ij→kl , γ_ij→kl=⟨σ v⟩_ij→kl , (P132)
(P136) Ẏ_i⊃ Y_kY_lΓ_kl→ij-Y_iY_jΓ_ij→kl , Γ_ij→kl=n_bγ_ij→kl . (P137)
Here, the reaction rate γ_j…→i… is in units cm^3/s but when multiplied by
the appropriate n_b factor it results in Γ_j…→i… being in inverse seconds.
The forward γ_j…→i… and the reverse reaction rates
γ̅_j…→i…=γ_i…→j…
are related due to the assumption of a local thermodynamic equilibrium;
thus, there is a simple three-parameter factor
containing the reaction constants α,β, γ and
expressed using temperature T_9 in GK units,
γ̅_j…→i…=γ_i…→j…
=γ_j…→i…×α T_9^β exp(γ/T_9)
(see (P141) and (P142) for details).
The constant α is an overall reaction constant related to the
stoichiometric coefficients of the reaction species and their spins,
while γ is a factor in a Boltzmann like exponent and
depends on the reaction Q-factor over a temperature constant of 1GK;
as such ∼ Q/T is an in-scalar quantity if mass and thermal energy
have the same λ-scaling.
Thus, the only co-scalar of non-zero power is related to the constant β
since it is coming from a factor of the type m× T (see (P141)).
This means that if energy is scaling as m→ m'=mλ^n_m
and k_BT→ k_BT'=k_BTλ^n_T,
then the effective T_9 re-scaling in the reverse reaction factor T_9^β
should be scaling as λ^(n_m+n_T).
For most of our study we would assume that the scaling power of the rest-mass energy
and thermal energy are the same[
It is possible to argue for different scaling powers of the
radiation and rest-mass energies based on the different conservation laws
for matter (w=0) and radiation (w=1/3) based on the SIV conserved quantity
ρ_w a^3(1+w)λ^1+3w=ρ_0 within SIV.
In doing so, one may induce a deviation from the usual energy conservation.
To avoid such deviation one will have to use the
appropriate equation of state w=p/ρ to determine the unique λ-scaling
for the energy.], that is, n_m=n_T, otherwise it may result in apparent deviations
from the law of energy conservation. Furthermore, we also adopt the
PRIMAT view that one can choose the system of units so that k_B=1
and therefore temperature is directly measuring thermal energy[
One can consider scaling for k_B, that is,
k_B→ k'_B=k_Bλ^n_k_B.
However, since k_B is a conversion constant from temperature to energy (erg/K),
as such it is related to the choice of units that once made should not be subject to change.
Thus, choosing k_B=1 fixes/eliminates this λ-scaling just like the the choice c=1
fixes the time and space units.
However, one has to keep in mind whether the energy is related to thermal energy or rest-mass energy.],
thus there is no question of how the constant k_B scales with
λ since it is just a conversion constant that an observer
can choose to be 1.
§ METHOD
In PRIMAT one finds first the expansion factor a(T), then
builds the energy density and its corrections as function of a
and/or T, finally the time variable τ is obtained
from the FLRW equation[
This is a first order ordinary differential equation that needs
proper initial conditions. In this case it is set to be
τ(a(T_i))=1/(2H(a(T_i)) when integrating
dτ/da=1/(a H(a)) for τ(a), where T_i≈10^12K is the
initial temperature to be considered for the BBNS processes.]
ȧ/a=H=√(8/3π Gρ). In
our approach to study BBNS within SIV, we bypass the numerical solution
of the FLRW equation in favor of using the analytic SIV
functions above. In particular, the functions used in the SIV-PRIMAT
are: a(T)=a(τ(T)) based on τ(T) and a(τ) above,
while the inverse function T(a) is computed within PRIMAT.
[
The PRIMAT (numerical) inverse function process is validated by
comparing the numerically inverted function to the known SIV analytic function T(a).
We use the same validation approach as done by the original PRIMAT code for other similar cases.]
This way densities are not needed for FLRW integration to obtain
τ(a) and a(τ).
As usual, the functional form of the expansion factor a is
inversely proportional to the temperature T. As can be seen from (A39),
(A30), and (A33) a(T)=a(τ(T))= const/T where the constant is
T_0 C_ rel^-1/4√(c_2t_ in^3) which by
(A29) becomes T_0√(t_ in)=T_0λ^-1/2. This
constant depends only on Ω_m and the CMB temperature T_0.
During the BBNS λ is practically constant since it is
very close to 1/t_ in. However, λ is generally
evolving during the evolution of the Universe towards the
value λ_0=1 at the current epoch. In the case of Ω_m→1
one also has λ→1. Either way, one obtains a→ T_0/T→1
towards the current epoch. In PRIMAT for neutrino decoupled scenario
the expansion factor a(T) is also of the form T_0/T but has a
distortion factor of 𝒮(T)^-1/3 due to
neutrino physics and entropy injection by the electron-positron annihilation
process around T∼1GK. The distortion factor 𝒮(T)
becomes 1 for T≪1GK and therefore recovers the usual a(T)=T_0/T
behavior in the low-temperature regime (see Fig. <ref> for details).
Note that if one interprets Ω_m and Ω_b as the current epoch (now)
values, then Ω_m=1 is not realistic limit here given the known current values;
however, since the BBNS is in the radiation epoch far from the matter-radiation equality
when radiation dominates then Ω_m=1 for the total matter and radiation seems reasonable;
thus, we consider Ω_m=1 along with Ω_m=0.3 and Ω_m=Ω_b=0.05
to illustrate the trend due to Ω_m in the graphs.
To define the time variable, PRIMAT solves the FLRW equation
using the relevant energy density. Within the SIV we have an analytic
expressions for the SIV time. In the standard units
(age of the Universe is 13.8 Gyr), the SIV analytic form is
τ(a)=const× a^2 and from (A30) and (A33) the constant is:
τ(a)/a^2 = τ_0/(2c_2 t_ in^3(t_0-t_ in))
=τ_0/(2C_rel^1/2 t_ in(t_0-t_ in))=
= τ_0(1-Ω_m)/(4√(Ω_m v_eq) t_ in(t_0-t_ in))
=τ_0 f(Ω_m).
This constant depends only on Ω_m and for
Ω_m→1 goes to 3τ_0/(4√( v_eq(Ω_m=1))).
As it is seen in Fig. <ref>, during the relevant interval of time,
there is a clear quadratic relation of τ(a) ∼ a^2
that becomes obvious on the τ(a)/a^2 plot. In the limit
Ω_m→1, one has C_m, C_rel, and c_2→∞
in this respect t_eq is sandwiched between t_ in and
t_0=1 and based on (A25) t_eq→1. Notice that the
PRIMAT τ(a)/a^2 is larger in the initial stages of the BBNS
and then becomes smaller then the SIV τ(a)/a^2/λ but
it is about the same order of magnitude.
The λ–scaling is due to the corresponding λ^2–scaling
of the 8π Gρ within the SIV and the fact that the PRIMAT
time is based on integrating the FLRW equation ȧ/a=√(8/3π Gρ).
The time keeping, between the PRIMAT and SIV, is non-uniform (as seen in Fig. <ref>);
this has an impact on the overall time related observables, such as life-time of processes and particles,
i.e. neutron life-time and nuclear reaction rates within the SIV framework.
The details of the PRIMAT τ(a)/a^2 variations
could be understood to be due to the high-temperature behavior of the relativistic density
containing correction terms δρ(T) and also via the dependence on a(T),
which in PRIMAT has the distortion factor 𝒮(T)
that is missing in the SIV model.
As it can bee seen τ(a)/a^2 decreases with Ω_m
but always stays above the PRIMAT value.
The gap between Ω_m=1 (λ=1)
and the PRIMAT τ(a) can be resolved by the
λ–scaling of 8π Gρ as seen in the bottom curves.
This means that if one uses the SIV a(T) instead of the default PRIMAT functions,
but utilizes the PRIMAT densities and usual FLRW to define τ(a)
then one would obtain a curve similar to the one displayed but pushed
by a factor λ to the corresponding SIV τ/a^2
line shown at Ω_m=0.3. The reason for this is the
factor λ in the SIV T∼λ^-1/2, which propagates
through the density ρ∼ T^4 to λ^-2 factor, that
becomes λ^-1 due to the square root in ȧ/a=√(8/3π Gρ),
and ultimately resulting in dτ'=λ dτ, which is consistent
with the SIV view about its effects on time and space intervals.
Below are the densities used in PRIMAT for the case of decoupled neutrinos:
ρ_γ = a_BB(k_BT)^4/c^2(1+δρ(T)+7/8N_ν(⟨ T_ν⟩/T)^4)
=T^4ρ_γ,
ρ_m = n_b0m_b0/c^2a^3(1+Ω_c0/Ω_b0+3/2k_BT/m_b0)
=a_0^3ρ_m0(T)/a^3.
In the high-temperature regime ρ_tot∼ a_BB(k_BT)^4/c^2.
Where the proportionality constant is (1+δρ_max+7N_ν/8)
since at high-temperature the neutrinos are in thermal equilibrium with
the radiation; that is ⟨ T_ν⟩ =T. In
the low-temperature regime this is K_0 since the plasma corrections
δρ are becoming negligible
and ⟨ T_ν⟩ /T→(4/11)^1/3
see PRIMAT (P32). This is demonstrated in Fig. <ref>.
For the rate of change of the nuclear species,
we have to consider that the right hand side of the equation (P136) is in SIV time τ
while the original PRIMAT reaction rates Γ'_j…→i…
are in the Einstein GR frame,
we will use a prime to indicate that, but these rates need to be expressed in the SIV frame.
Based on (P136) we have for the transition from EGR to WIG (SIV) frame:
dY_i/dτ'=Y_kY_lΓ'_kl→ij-Y_iY_jΓ'_ij→kl, ⇒1/λdY_i/dτ=Y_kY_lΓ'_kl→ij-Y_iY_jΓ'_ij→kl.
Thus, Γ_kl→ij in the SIV frame is related to the measured EGR laboratory rates
Γ'_kl→ij via a simple rescale factor λ; that is,
Γ_kl→ij=λΓ'_kl→ij, which is
based on the relationship[Even though
dτ' in EGR is in seconds and so is the WIG time interval dτ,
these two time units (seconds) are not necessarily the same. If they were the same,
then the EGR and WIG frames would coincide since this would imply λ=1.
Furthermore, the relation dτ'=λ dτ
could be viewed as a consequence of the λ scaling of ρ and the definition of
τ via the solution of the FLRW equation as it was discussed in connection to the ρ/T^4 relationship.]
dτ→ dτ'=λ dτ;
thus, the original PRIMAT rates need to be rescaled as
Γ'_j…→i…→ Γ_j…→i…
=λ×Γ'_j…→i….
This is accomplished by using Forwards Rescale Factor (FRF) to all the reaction rates.
That is, FRF=λ for SIV guided studies[
One can argue that FRF should be 1 because the reaction
cross-sections σ should not be modified since the sizes of the nuclei are not governed by gravitation.
Such argument ignores the possibility that ħ may not be an in-scalar object.
Nevertheless, we can carry on and consider
ṅ_i⊃ n_i n_j ⟨σ v⟩_ij→kl based on (P131) and (P132).
Thus, because v is inscalar, there is no change on the RHS of the equation.
However, our argument was about the change of the time parametrization
in the LHS that takes into account dτ→ dτ'=λ dτ.
So, the λ in the denominator of the LHS becomes a FRF scale-factor on the RHS
when one is using a different time parametrization in switching from EGR to WIG (SIV) formulation.].
Furthermore, the T_9 argument of the factor T_9^β in the reverse reaction rates
may have to be rescaled as well with the appropriate λ-factor
for mass and thermal energy scaling.
That is, for SIV guided studies,
mT_9→ mT_9×λ^n_m+n_T.
In our results section, we refer to the scale-factor λ^n_m+n_T
as mŤ scale-factor λ^2n when n_m=n_T=n.
However, the two scale-factors powers n_m and n_T may not be the same,
but we have deferred the discussion on this topic to the Appendix <ref>.
In this paragraph we drop the primes since the discussion is about the standard BBNS.
The validity of Ta=T_0a_0 is affected during the standard BBNS due to the e^+e^- annihilations,
as such it is related to the distortion factor 𝒮(T) (see P31);
that is, a_0T_0=aT𝒮^1/3(T). Furthermore,
since n_b∝ a^-3 and n_γ∝ T^3,
this also affects the baryon to photon in-scalar ratio
η=n_b/n_γ=η_0𝒮^1/3(T),
where η_0=6.0913×10^-10 is the current ratio of baryons to photons;
it is often written as η_10=η_0×10^10 which removes the factor 10^-10 in η_0.
Finally, we would like to point out that if the SIV paradigm is valid, and since during the BBNS λ
is practically constant, then one could include the effect of the e^+e^- annihilation via the distortion factor
𝒮(T) as an equivalent effect within the SIV background.
To do so, we recognize that for EGR ↔ SIV with a'=aλ and T'=Tλ^-1/2,
one has a_0T_0(𝒮'(T'))^-1/3=T'a'=Taλ^1/2 and therefore the new
ã(T)=a_0T_0/(Tλ^1/2)/𝒮^1/3(T)=a_SIV(T)/𝒮^1/3(T),
where 𝒮(T) is defined according to the discussion in the Appendix <ref>
via the known EGR laboratory function 𝒮'(T'), that is,
𝒮(T)=𝒮'(T'(T))=𝒮'(Tλ^-1/2).
Furthermore, the new ã(T) is also equivalent to a'(T')/λ,
as it should be based on a'=aλ.
As a measure of goodness of fit for the theory against the experimental data,
we use √(χ_ϵ^2), which is the 2-norm of
the deviation of theory (th) from observation (ob) with respect to
the experimental error ϵ.
√(χ_ϵ^2)=√(1/N∑_i^N(y_i^(ob)-y_i^(th)/ϵ_i)^2).
A number less than one indicates that all the theory
values are within the observational uncertainties.
§ RESULTS
In Table <ref> we have shown the values of a(T), τ(T), and
ρ(T) for PRIMAT when using standard cosmological parameters
for Ω_CDM=0.26 and Ω_b=0.05 along with
the corresponding values for the relevant SIV functions for the same
cosmological parameters[
We have used temperature T as the
control variable, which is customary;
however, an in-scalar quantity will be more appropriate
since there could be a λ-scaling for T within SIV.].
From Table <ref> and from Fig. <ref> we see that the two “clocks”
are irregular in the few first moments after the Big-Bang
with SIV time ticking about 1.008 times faster than the PRIMAT time;
however, at low temperatures they become synchronized
and shifted only by a few seconds.
This may be taken as a justification to use FRF=1 since
the 1.008 is practically 1.
The relevant element abundances are given in Table <ref>.
In the second column are shown the observational values,
while in the third column are the results of the PRIMAT code
when run with small reaction network and decoupled neutrinos,
without QED dipole corrections[When the partially decoupled neutrinos
and QED dipole corrections are turned on there are minor insignificant changes
to the results that are not relevant to the current discussion.],
with standard Ω _b and Ω _m values.
The forth column shows the SIV results for the same values of
Ω _b and Ω _m using the
analytic SIV functions a_SIV(T) and τ_SIV(T).
The results reveal under-production of
^4He, deuterium, and ^3He with significant
over-production of ^7Li. Abundances improve if we fit the
^4He and D/H by changing the values of
Ω _b and Ω _m as seen in the fifth column.
Now ^3He and ^7Li are compatible with the PRIMAT results,
but at much smaller values of the baryon and total matter.
In this case the dark matter (DM) is less than 3× of the baryon matter (BM),
unlike the usual case where the DM is more than 5× of the BM.
In order to study the contribution of 𝒮(T) within SIV
we consider the last two runs (columns six and seven in Table <ref>).
The sixth column is based on the parameters in the fifth column.
Not shown in the Table, but if we use the PRIMAT values for Ω _b and Ω _m (based on column three)
instead of column five, then we have over-production of ^4He and significant under-production of deuterium and
very high production of ^7Li - all in the further enhanced directions of the results in column four.
So, the last two columns are for a_SIV(T) distorted by 𝒮(T)
which is equivalent to PRIMAT a̅(T) distorted by λ.
Such choice of modification is relevant since
the SIV runs (column four and five) don't include the
electron–positron annihilation and neutrino effects
encoded in the function 𝒮(T).
Adding the distortion function 𝒮(T)
to the a_SIV(T) or equivalently modifying PRIMAT a̅(T) by λ,
results in light increase of ^4He and under production of deuterium and tritium
with an over production of ^7Li (comparing column six to five).
The next, seventh column, fit* is the best possible, but not perfect,
fit for ^4He and D/H and seems to require a significant mass content.
This is a simple (“naive”) SIV model without the utilization of the λ-modifications of the
various reaction parameters (FRF, mŤ, Q/Ť)
that have been discussed in the Appendix <ref>.
Thus, this fit* failure to achieve a perfect 2D fit (on ^4He and D/H) is likely reflecting the need of proper λ-scaling.
To check this, we have added the calculations discussed in the Appendix <ref>.
The last two columns use modified a(T) and therefore one has to rely on the numeric
integrations in PRIMAT for τ(a) and a(τ).
That is, we do not have, as far as we know,
analytic SIV solutions for τ(a) and a(τ) when a(T) is distorted;
thus, we do instead the PRIMAT numeric integration to get the relevant time variable τ(T).
The last column seems to be close to the default PRIMAT run.
That is because λ is close to 1, which is practically PRIMAT
since the last two columns are using the a_SIV(T)
augmented by the distortion factor of 𝒮(T)^-1/3
as indicated in column six, or equivalently this is PRIMAT
a(T) rescaled by λ since
a_SIV/𝒮^1/3=a̅(T)/λ,
where the a̅(T) is the PRIMAT a(T).
The need to have λ close to 1 is not an indicator of dark matter content but
indicates the goodness of the standard PRIMAT results that allows only for λ close to 1
as an augmentation, as such leads to a light but important improvement in D/H
as seen when comparing columns three and seven.
In order to study further the SIV guided modifications to the reverse reactions,
we study only the effects due to mŤ scaling
by utilizing the SIV analytic functions for a(τ) and τ(T)
as in the corresponding middle a_SIV columns in Table <ref>.
The relevant element abundances are given in Table <ref>.
The value of λ is set to 1/t_ in.
The last five columns were fitted on Ω _b and Ω _m
to reproduce ^4He and D/H since these are known
within 1% uncertainty. Given that we have chosen Q/Ť=1; then,
due to the traditionally preferred energy scaling by λ^n_m with n_m=1,
the SIV scalings of the thermal energy k_B T
with λ (k_B=1) should be λ^+1.
However, the BBNS is in the radiation dominated epoch where,
n_T=-1/2 is expected, as discussed in the Appendix <ref>;
therefore, we have also considered a few other scaling options
λ^n with 2n∈{±2,±1,0}.
We have also explored the case n=±2 but
it was not possible to find perfect reproduction for ^4He and D/H for the
mŤ=λ^2 scaling. That is, we had fit* problem. The minimum was at Ω_b=0.02075 and
Ω_m=0.0863 with λ=2.26 and √(χ_ϵ^2)=4.8.
On other hand, mŤ=λ^-2 resulted in reproduction of ^4He and D/H at
Ω_b=0.0149 and Ω_m=0.0809 with λ=2.31 and √(χ_ϵ^2)=7.5.
These cases are consistant with what is seen in Table <ref> but do not have any specific
justification for choosing such mŤ=λ^±2 scaling of the T^β-term in the revers reaction formulas.
From Table <ref> one can conclude that SIV-guided modifications to the local statistical equilibrium
implemented to the T^β-term in the reverse reactions, which is induced by the mŤ-scaling,
that are consistent with ^4He and D/H data, are actually mŤ-scaling independent.
The overall result is a reduced baryon and dark matter content in general but no significant
λ-scaling dependence.
§ SUMMARY AND CONCLUSIONS
The SIV analytic expressions for a(T) and τ(T) can be utilized to study the BBNS within the SIV paradigm.
The functional behavior is very similar to the standard models such as PRIMAT except during the very early universe
where electron-positron annihilation and neutrino processes affect the a(T) function
see Table <ref> and Fig. <ref>.
The distortion due to these effects encoded in the function 𝒮(T)
could be incorporated by considering the SIV paradigm
as a background state of the universe where the processes could take place.
It has been demonstrated that incorporation of the 𝒮(T)
within the SIV paradigm results in a compatible outcome with the standard BBNS
see Table <ref> and if one is to fit the observational data the result is λ≈1
for the SIV parameter λ (see last column of Table <ref>).
However, a pure SIV treatment results in Ω_b≈1% and less total matter,
either around Ω_m≈23%
when all the λ-scaling connections are utilized (see Table <ref>), or around
Ω_m≈6% without any λ-scaling factors
(see the fit column of Table <ref>).
The SIV paradigm suggests specific modifications to the reaction rates,
as well as the functional temperature dependences of these rates,
that need to be implemented to have consistence between the
Einstein GR frame and the WIG (SIV) frame.
In particular, the non-in-scalar factor T^β in the reverse reactions rates
may be affected the most due to the SIV effects.
As shown in Table <ref>, we have studied a specific case of dependences and have seen that
within the assumptions made the SIV model requires three times less baryon matter,
usually around Ω_b≈1.6% and less total matter - around Ω_m≈6%.
The lower baryon matter content leads to also a lower photon to baryon ratio η_10≈2
within the SIV, which is three tines less that the standard value of η_10=6.14.
The results in Table <ref> indicate insensitivity to the specific
λ-scaling dependence of the mŤ-factor in the reverse reaction expressions.
Thus, one may have to explore further the SIV-guided λ-scaling relations as done
for the last column in Table <ref>, however, this would require the
utilization of the numerical methods used by PRIMAT and as such will take us away from the
SIV-analytic expressions explored in this paper that provide simple model for understanding the BBNS within the SIV paradigm.
Furthermore, it will take us further away from the accepted local statistical equilibrium and
may require the application of the reparametrization paradigm that seems to result in SIV like
equations but does not impose a specific form for λ <cit.>.
Our main conclusion is that the SIV paradigm provides a concurrent model of the BBNS
that is compatible to the description of ^4He, D/H, T/H, and ^7Li/H achieved in the standard BBNS.
It suffers of the same ^7Li problem as in the standard BBNS but also suggests a possible
SIV-guided departure from local statistical equilibrium which could be a fruitful direction to be explored
towards the resolution of the ^7Li problem.
§ APPENDIX: EXPLORING THE SIV-GUIDED Λ-SCALING RELATIONS
As mentioned earlier the two scale-factors powers n_m and n_T may not be the same
since one can argue for different scaling powers of the
radiation and rest-mass energies based on the different conservation laws.
For example, in a matter dominated state with w=0 one has
ρ_m a^3λ=ρ_m0 with
m∝ρ_m a^3 R_0^3⇒ m→ m'_0=mλ,
while for radiation dominated epoch w=1/3 one has
ρ_r a^4λ^2=ρ_r0, then by using
ρ_r∝ T^4⇒ Taλ^1/2=T_0a_0 along with
a→ a'=aλ this gives T→ T'=Tλ^-1/2,
so that the usual T'a'=T_0a_0 holds. Thus, while mass scales as λ
when matter is dominating, then the thermal energy scales[
This scaling for radiation is consistent with the mass scaling by λ
since ρ_γ∝ T^4 then the total energy in a comoving 3D volume
will be E_γ=ρ_γ a^3R^3_0∝ T^4 a^3
=T'^4λ^4/2 a'^3λ^-3∝ E'_γ/λ;
that is, E_γλ=E'_γ just as mλ=m'.
This argument shows that there is no contradiction with the
law of energy conservation; that is, while the radiation (thermal energy)
has a different λ-scaling from the rest-mass energy,
when radiation is absorbed from a finite 3D-region of space the process
results in the correct energy scaling as for a system with a rest-mass energy,
which is also finite and localized. The key difference is the different λ-scaling
of a thermal radiation with a state label T in an infinite volume compared to
the λ-scaling of a 3D-localized rest-mass system with a state label m
but consistent upon absorption and emission of localized photons.]
as λ^-1/2 when radiation is dominating.
Thus, this is the correct thermal energy scaling during the BBNS!
Such λ-scalings to FRF, mŤ, and Q/Ť are easy to be implemented in our SIV study
during the BBNS where λ is practically constant.
In doing so, one has to pull-back the known functions; that is, for a function within the SIV frame
f(T) one has to define its value via the corresponding function f'(T') measured within the EGR laboratory frame.
This way one has f(T)=f'(T'(T))=f'(Tλ^n_T) when the two temperatures are related via T'=Tλ^n_T.
We will use this to define the λ-scalings for functions that depend on mŤ and Q/Ť.
In particular, since our control variable is T then we will adjust only the corresponding scale that comes along
with T but will not include any mass related scaling since the formulas for evaluating f'(T',m') are already using
EGR laboratory frame values for these quantities; that is, for functions containing m T
the scaling mŤ must be by λ^-1/2 and those that depend on Q/T should be scaled by λ^+1/2.
The SIV runs shown in Table <ref> could be viewed as a “naive” SIV model because
we have not utilized any λ-modifications of the
various reaction parameters (FRF, mŤ, Q/Ť).
The study of using such modifications are presented in column six and nine of Table <ref>,
while the other columns are the same as in Table <ref>.
The fit* here is the best possible, but not perfect, fit for ^4He and D/H and
seems to require a reduced total mass content relative to PRIMAT
but much more than in the previous case (column five).
The failure of this fit* (column six) to achieve perfect fit to the
^4He and D/H reflects the importance of the electron-positron annihilation process
accounted by 𝒮(T).
We have already discussed the contribution of 𝒮(T) within SIV
as shown in Table <ref> which uses a simple (“naive”) SIV model without the utilization of the
λ-modifications of the various reaction parameters (FRF, mŤ, Q/Ť).
We already suggested that, the second fit* failure to achieve a perfect 2D fit (on ^4He and D/H)
is likely reflecting the need for such λ-modification implementation.
To check this, we have performed the calculations shown in the last column in Table <ref>.
The closeness of Ω _m to 1 in the last two columns is actually reflecting the
need of having FRF close to 1.
This is because FRF is present in the thermonuclear reactions
as well as in the weak reactions where the time scale is set naturally by the neutron life-time.
Thus, in order to not change the weak reactions significantly,
which are related to the neutron life-time, one has to have FRF close to 1.
Since, FRF is expected to be equal to λ,
this leads to λ close to 1 as well. Thus, we kept FRF=1 in the runs for the two
columns shown that use modified a(T) and therefore one has to rely on the numeric
integrations in PRIMAT for τ(a) and a(τ). That is because we do not have
analytic SIV solutions for τ(a) and a(τ) when a(T) is distorted;
thus, we do the numeric integration instead.
The last column seems to be close to the default PRIMAT run.
That is because λ is close to 1, which is practically PRIMAT
since the last three columns are using the a_SIV(T)
augmented by the distortion factor of 𝒮(T)^-1/3
as indicated in column six, or equivalently this is PRIMAT
a(T) rescaled by λ since
a_SIV/𝒮^1/3=a̅(T)/λ,
where the a̅(T) is the PRIMAT a(T).
The need to have λ close to 1 is not an indicator of dark matter content.
These results indicate that unmodified FRF (=1) is preferred,
which pushes λ towards 1 when considered as a possible modification option
as the fit in the last column indicates. For this fit we allowed the
λ-scaling for mŤ and Q/Ť to depend on λ as
stated in the table caption and discussed above but we got back
to the PRIMAT case with a little smaller Ω _b and
Ω _m very close to 1 since this is an effective way of getting λ=1.
99
[Bondi(1990)]Bondi90 Bondi, H. 1990, in Modern Cosmology
in Retrospect, Eds. Bertotti, B., Balbinot, R., & Bergia, S.,Cambridge Univ. Press., 426 pp.
[Bouvier & Maeder(1978)]BouvierM78
Bouvier, P. & Maeder, A. 1978, , 54, 497
[Bronstein & Semendiaev(1974)]Bronstein74
Bronstein, L.N., Semendiaev, K.A. 1974, Aide-memoire de mathematiques,
Ed. Eyrolles, Paris, 935 p.
[Canuto et al.(1977)Canuto, Adams, Hsieh, & Tsiang]Canu77
Canuto, V., Adams, P. J., Hsieh, S.-H., & Tsiang, E. 1977, , 16, 1643
[Carroll et al.(1992)Carroll, Press, & Turner]Carr92
Carroll, S. M., Press, W. H., & Turner, E. L. 1992, , 30, 499
[Carter(1979)]Carter79
Carter, B. 1979, in ”Confrontation of cosmological theories with observational data”,
IAU Symp. 63, Reidel Publ. Co., Dordrecht, p. 291.
[Coles & Lucchin(2002)]Coles02
Coles, P., Lucchin, F. 2002, Cosmology. The Origin and Evolution of Cosmic Structure,
Wiley & Sons Ltd, 492 p.
[Dirac(1973)]Dirac73
Dirac, P. A. M. 1973, Proceedings of the Royal Society of London Series A,
333, 403
[Durrer(2008)]Durrer08
Durrer, R. 2008, The Cosmic Microwave Background, Cambridge Univ. Press, 401 p.
[Jesus(2018)]Jesus18 Jesus, J.F. 2018, arXiv:1712.00697
[Maeder(2017a)]Maeder17a Maeder, A. 2017a, , 834, 194
[Maeder(2017b)]Maeder17b Maeder, A. 2017b, , 847, 65
[Maeder(2017c)]Maeder17c Maeder, A. 2017c, , 849, 158
[Maeder(2018)]Maeder18 Maeder, A. 2018, A. arXiv:1804.04484
[Maeder & Bouvier (1979)]MBouvier79 Maeder, A., Bouvier, P. 1979, Astron. Astrophys., 73, 82
[Maeder & Gueorguiev(2019)]MaedGueor19
Maeder, A.; Gueorguiev, V.G. The growth of the density fluctuations in the scale-invariant vacuum theory.
Phys. Dark Univ. 2019, 25, 100315.
[Maeder & Gueorguiev(2020)]MaedGueor20a
Maeder, A.; Gueorguiev, V.G. The Scale-Invariant Vacuum (SIV) Theory: A Possible Origin of Dark Matter and Dark Energy.
Universe 2020, 6, 46.
[Maeder & Gueorguiev(2020)]MaedGueor20b
Maeder, A.; Gueorguiev, V.G. Scale-invariant dynamics of galaxies, MOND, dark matter, and the dwarf spheroidals.
MNRAS 2019, 492, 2698.
[Maeder & Gueorguiev(2021)]SIV-Inflation'21
Maeder, A.; Gueorguiev, V.G. Scale invariance, horizons, and inflation.
MNRAS 2021, 504, 4005.
[Gueorguiev & Maeder(2021)]univ8040213
Gueorguiev, V.G.; Maeder, The Scale Invariant Vacuum Paradigm: Main Results and Current Progress.
Universe 2022, 8, 213.
[Gueorguiev & Maeder(2021)]sym13030379
Gueorguiev, V.G.; Maeder, A. Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces.
Symmetry 2021, 13, 379.
[Mukhanov(2004)]Mukhanov04 Mukhanov, V. 2004,
Intnl. J. Theoretical Physics, 43, 669
[Steigman(2007)]Steigman07 Steigman, G. 2007, Ann. Rev.
Nuclear and Particle Sci. 57, 463
[Weinberg(2008)]Weinberg08 Weinberg, S. 2008, Cosmology,
Oxford Univ. press, 593 p.
[Weyl(1923)]Weyl23
Weyl, H. 1923, Raum, Zeit, Materie. Vorlesungen über allgemeine
Relativitätstheorie. Re-edited by Springer Verlag, Berlin, 1970
Maeder18
A. Maeder, “Evolution of the early Universe in the scale invariant theory," ArXiV: 1902.10115
PRIMAT
Pitrou, C., Coc, A., Uzan, J.-P., Vangioni, E.
“Precision big bang nucleosynthesis with improved Helium-4 predictions.”
Physics Reports 754, 1–66 (2018).
ArXiV: 1801.08023
|
http://arxiv.org/abs/2307.04546v1 | 20230710132437 | Safety Analysis of Parameterised Networks with Non-Blocking Rendez-Vous | [
"Lucie Guillou",
"Arnaud Sangnier",
"Nathalie Sznajder"
] | cs.LO | [
"cs.LO",
"cs.MA",
"C.2.4; F.4.3"
] |
[
Simon R. Eugster1
August 12, 2023
=====================
We consider networks of processes that all execute the same
finite-state protocol and communicate via a rendez-vous
mechanism. When a process requests a rendez-vous, another process can
respond to it and they both change their control states
accordingly. We focus here on a specific semantics, called
non-blocking, where the process requesting a rendez-vous can
change its state even if no process can respond to it. In this context, we study the
parameterised coverability problem of a configuration, which consists in determining whether there is an initial
number of processes and an execution allowing to reach a
configuration bigger than a given one. We show
that this problem is EXPSPACE-complete and can be solved in polynomial time if the protocol is
partitioned into two sets of states,
the states from which a process can request a rendez-vous and the
ones from which it can answer one. We also prove that the
problem of the existence of an execution bringing all the
processes in a final state is undecidable in our context. These
two problems can be solved in polynomial time with the classical rendez-vous semantics.
§ INTRODUCTION
Verification of distributed/concurrent systems. Because of their ubiquitous use in applications we rely on constantly, the development of formal methods to guarantee the correct behaviour of distributed/concurrent systems has become one of the most important research directions in the field of computer systems verification in the last two decades. Unfortunately, such systems are difficult to analyse for several reasons. Among others, we can highlight two aspects that make the verification process tedious. First, these systems often generate a large number of different executions due to the various interleavings generated by
the concurrent behaviours of the entities involved. Understanding how these interleavings interact is a complex task which can often lead to errors at the design-level or make the model of these systems very complex. Second, in some cases, the number of participants in a distributed system may be unbounded and not known a priori. To fully guarantee the correctness of such systems, the analysis would have to be performed for all possible instances of the system, i.e., an infinite number
of times.
As a consequence, classical
techniques to verify finite state systems, like testing or
model-checking, cannot be easily adapted to distributed systems and
it is often necessary to develop new techniques.
Parameterised verification. When designing systems with an
unbounded number of participants, one often provides a schematic program (or protocol)
intended to be implemented by multiple identical processes, parameterised by
the number of participants. In general, even if the verification problem is
decidable for a given instance of the parameter, verifying all possible instances
is undecidable (<cit.>).
However, several settings come into play that can be
adjusted to allow automatic verification.
One key aspect
to obtain decidability is to assume that the processes do not manipulate
identities in the protocolsand use simple communication mechanisms like
pairwise synchronisation (or rendez-vous) <cit.>, broadcast of
a message to all the entities <cit.> (which can as well be lossy in order to
simulate mobility <cit.>), shared register containing values of a finite
set <cit.>, and so on (see <cit.> for a survey).
In every aforementioned case, all the entities execute
the same protocol given by a finite state automaton.
Note that parameterised verification, when decidable like in the above models, is
also sometimes surprisingly easy, compared to the same problem with a fixed number of participants.
For
instance, liveness verification of parameterised systems with
shared memory is Pspace-complete for a fixed number of
processes and in NP when parameterised
<cit.>.
Considering rendez-vous communication. In one
of the seminal papers for
the verification of parameterised networks <cit.>, German and Sistla
(and since then <cit.>)
assume that the entities communicate by “rendez-vous”,
a synchronisation mechanism in which two processes (the sender and the receiver)
agree on a common action by which they jointly change their local state.
This mechanism is synchronous and symmetric, meaning that if no process is ready to receive a message, the sender cannot send it.
However, in some applications, such as Java Thread programming, this is not exactly the primitive that is implemented. When a Thread is suspended in a waiting state, it is woken up by the reception of a message sent by another Thread. However, the sender is not blocked if there is no suspended Thread waiting for
its message; in this case, the sender sends the anyway and the message is simply lost.
This is the reason why Delzanno et. al.
have introduced non-blocking
rendez-vous in <cit.>
a communication primitive in which the sender of a message is not blocked
if no process receives it. One of the problems of interest in
parameterised verification is the coverability problem: is it
possible that, starting from an initial configuration, (at least)
one process reaches a bad state?
In <cit.>, and later in <cit.>,
the authors introduce variants of Petri nets to handle this type
of communication.
In particular, the authors investigate in <cit.>
the coverability problem for an extended class of Petri nets
with
non-blocking arcs, and show that for this model the coverability
problem is decidable using the techniques of Well-Structured
Transitions Systems
<cit.>.
However, since
their model is an extension of Petri nets, the latter problem is
-hard <cit.> (no upper bound
is given). Relying on Petri nets to obtain algorithms for
parameterised networks is not always a good option.
In fact, the
coverability problem for parameterised networks with rendez-vous can
be solved in polynomial timeis in P<cit.>, while it is -complete for Petri
nets <cit.>. Hence, no upper
bound or lower bound can be directly deduced for the verification of
networks with non-blocking rendez-vous from <cit.>.
Our contributions. We show that the coverability
problem for parameterised networks with non-blocking
rendez-vous communication over a finite alphabet is
-complete. To obtain this result, we consider an
extension of counter machines (without zero test) where we add non-blocking decrement
actions and some restore mechanism, i.e.edges that can bring back
the machine to its initial location at any moment. We show that
the coverability problem for these extended counter machines is
-complete (<ref>) and that it is equivalent to our problem
over parameterised networks (<ref>).
We consider then a subclass of parameterised networks –
wait-only protocols – in which
no state can allow to both request a rendez-vous and wait for one. This
restriction is very natural to model concurrent programs since when a thread
is waiting, it cannot perform any other action. We show that coverability problem
can then be solved in polynomial time (<ref>).
Finally, we show that the synchronization problem, where we look for a reachable configuration with all the processes in a given state,
is undecidable in our framework, even for wait-only protocols (<ref>).
Due to lack of space, some proofs are only given in the appendix.
§ RENDEZ-VOUS NETWORKS WITH NON-BLOCKING SEMANTICS
For a finite alphabet Σ, we let Σ^* denote the set of finite sequences over Σ (or words).
Given w∈Σ^*, we let |w| denote its length: if w=w_0… w_n-1∈Σ^*, then |w|=n.
We write to denote the set of natural numbers and [i,j] to
represent the set k∈| i≤ k k ≤
j for i,j ∈. For a finite set E, the set ^E represents
the multisets over E. For two elements m,m' ∈^E, we denote
m+m' the multiset such that (m+m')(e) = m(e) +m'(e) for
all e ∈ E. We say that m ≤ m' if and only if m(e) ≤
m'(e) for all e ∈ E. If m ≤ m', then m'-m is the multiset
such that (m'-m)(e) = m'(e)-m(e) for
all e ∈ E. Given a subset E' ⊆ E and m ∈^E, we denote by ||m||_E' the sum Σ_e∈ E'm(e) of elements of E' present in m. The size of a multiset m is given by
||m|| =||m||_E. For e ∈ E, we use sometimes the
notation e for the multiset m verifying m(e)=1 and
m(e')=0 for all e' ∈ E∖e and, to represent for instance the multiset
with four elements a, b,b and c, we will also use
the notations a, b, b, c or a, 2· b, c.
§.§ Rendez-Vous Protocols
We can now define our model of networks. We assume that all processes in the network follow the same protocol. Communication in the network is pairwise and is performed by rendez-vous through a finite communication alphabet Σ. Each process can either perform an internal action using the primitive τ, or request a rendez-vous by sending the message m using the primitive !m or answer to a rendez-vous by receiving the message m using the primitive ?m (for m ∈Σ). Thus, the set of primitives used by our protocols is RV(Σ)=τ∪?m,!m | m ∈Σ.
A rendez-vous protocol (shortly protocol) is a tuple = (Q, Σ, , q_f, T) where Q is a finite set of states, Σ is a finite alphabet, ∈ Q is the initial state, q_f ∈ Q is the final state and T ⊆ Q × RV(Σ) × Q is the finite set of transitions.
For a message m ∈Σ, we denote by m the set of states q from which the message m can be received, i.e. states q such that there is a transition (q, ?m, q') ∈ T for some q' ∈ Q.
A configuration associated to the protocol is a non-empty multiset C over Q for which C(q)
denotes the number of processes in the state q and ||C|| denotes the total number of processes in the configuration C.
A configuration C is said to be initial if and only if C(q)=0 for all q ∈ Q∖. We denote by () the set of configurations and by () the set of initial configurations. Finally for n ∈∖0, we use the notation _n() to represent the set of configurations of size n, i.e. _n()=C ∈() | ||C||=n. When the protocol is made clear from the context, we shall write , and _n.
We explain now the semantics associated with a protocol. For this
matter we define the relation ⊆⋃_n≥
1_n ×(τ∪Σ∪𝐧𝐛(m)
| m ∈Σ) ×_n as follows (here · is a special symbol). Given n ∈∖0 and C,C' ∈_n and m ∈Σ, we have:
* C τ C' iff there exists (q, τ, q') ∈ T
such that C(q) > 0 and
C' = C - q + q' (internal);
* C m C' iff there exists (q_1, !m, q_1') ∈ T and
(q_2, ?m, q_2')∈ T such that C(q_1)>0 and C(q_2)>0 and C(q_1)+C(q_2)≥ 2 (needed when q_1 = q_2) and
C' = C - q_1, q_2 + q_1', q_2' (rendez-vous);
* C 𝐧𝐛(m) C' iff there exists (q_1, !m, q_1') ∈ T, such
that C(q_1)>0 and (C-q_1)(q_2)=0 for all (q_2, ?m, q_2') ∈ T and
C' = C - q_1 + q'_1 (non-blocking request).
Intuitively, from a configuration C, we allow the following
behaviours: either a process takes an internal transition
(labeled by τ), or two processes synchronize over a
rendez-vous m, or a process requests a rendez-vous to which no
process can answer (non-blocking sending).
This allows us to define S_ the transition system ((),
) associated to . We will write C C' when there exists a ∈τ∪Σ∪𝐧𝐛(m)
| m ∈Σ such that C a C' and denote by
^∗ the reflexive and transitive closure of . Furthermore, when made clear from the context, we might
simply write instead of .
An execution is a finite sequence of configurations
ρ = C_0C_1… such that, for all 0≤ i< |ρ|,
C_i C_i+1. The execution is said to be initial if C_0∈().
Figure <ref> provides an example of a rendez-vous protocol
where is the initial state and the final state. A
configuration associated to this protocol is for instance the multiset
2 · q_1, 1· q_4, 1 · q_5 and the following
sequence represents an initial execution: 2 ·𝐧𝐛(a), b, c 2 ·.
When we only allow behaviours of type (internal)
and (rendez-vous), this semantics corresponds to the classical rendez-vous semantics (<cit.>).
In opposition, we will refer to the semantics defined here as the
non-blocking semantics where a process is not
blocked if it requests a rendez-vous and no process can
answer to it.
Note that all behaviours possible in the classical rendez-vous semantics are as well possible in the non-blocking semantics but the converse is false.
§.§ Verification Problems
We now present the problems studied in this work. For this matter,
given a protocol = (Q, Σ, , q_f, T), we define two
sets of final configurations. The first one () = { C ∈() | C(q_f)> 0}
characterises the configurations where one
of the processes is in the final state. The second one () =
{ C ∈() | C(Q ∖{q_f})= 0} represents the
configurations where all the processes are in the final state. Here
again, when the protocol is clear from the context, we might use the
notations and .
We study three problems: the
coverability problem (), the synchronization problem
() and the termination problem () which all takes as input
a protocol and which can be stated as follows:
We study three problems: the
state coverability problem (), the configuration coverability problem () and
the synchronization problem
(), which all take as input
a protocol
and can be stated as follows:
Problem name Question
Are there C_0 ∈ and C_f ∈, such
that C_0 ^∗ C_f?
Given C ∈, are there C_0 ∈ and C' ≥ C, such that C_0 ^∗ C'?
Are there C_0 ∈ and C_f ∈, such
that C_0 ^∗ C_f?
Does _∞ (S_) = ∅?
expresses a safety property: if q_f is an error state and the answer is negative, then for any number of processes, no process will ever be in that error state. , in another hand, is a liveness property: if q_f is a deadlock state (a state in which no action is possible), and the answer is negative, then for any number of processes, all processes together are never blocked at the same time.
The difficulty in solving these problems lies in the fact that we
are seeking for an initial configuration allowing a specific execution
but the set of initial configurations is infinite. The difference
between and is that in the first one we ask for at least one
process to end up in the final state whereas the second one requires all the processes to end in this state.
Note that is an instance of but is not.
The rendez-vous protocol of Figure <ref> is a positive instance of , as shown in <ref>.
However, this is not the case for : if an execution
brings a process in , this process cannot be brought afterwards to . If is the final state, is now a positive instance of (see Example <ref>).
Note that if the final state is , is not a positive
instance of anymore. In fact, the only way to reach a
configuration with a process in is to put (at least) two
processes in state as this is the only state from which
one process can send the message b.
However, this cannot happen, since from an initial configuration,
the only available action consists in sending the message a as a non-blocking request. Once
there is one process in state q_5, any other attempt to put another process in this state will induce a reception of message a by the process already in q_5, which will hence leave q_5.
Finally, note that for any n ∈ℕ, the configuration n · is coverable, even if with as final state is not a positive instance of .
§ COVERABILITY FOR NON-BLOCKING COUNTER MACHINES
We first detour into new classes of counter machines, which we call non-blocking counter machines and non-blocking counter machines with restore, in which a new way of decrementing the counters is added to the classical one: a non-blocking decrement, which is an action that can always
be performed. If the counter is strictly positive, it is decremented; otherwise it is let to 0. We show that the coverability of a control
state in this model is -complete, and use this result to solve coverability problems in rendez-vous protocols.
To define counter machines, given a set of integer variables (also called counters) , we use the notation to represent the set of associated actions given by ,,|∈∪. Intuitively, increments the value of the counter , while decrements it and checks if it is equal to 0. We are now ready to state the syntax of this model.
A counter machine (shortly CM) is a tuple M = (, , Δ, ) such that is a finite set of locations, ∈ is an initial location, is a finite set of counters, and Δ⊆×× is finite set of transitions.
We will say that a CM is test-free (shortly ) whenever Δ∩×{|∈}× = ∅. A configuration of a CM M = (, , Δ, ) is a pair (ℓ, v) where ℓ∈ specifies the current location of the CM and v∈^ associates to each counter a natural value. The size of a CM M is given by |M|= || + || + |Δ|.
Given two configurations (ℓ, v) and (ℓ',v') and a transition δ∈Δ, we define (ℓ, v) δ_M (ℓ', v') if and only if δ = (ℓ, op, ℓ') and one of the following holds:
[t]7cm
* op = and v =v';
* op = and v'() = v() + 1 and
v'(') = v(') for all ' ∈∖;
[t]7cm
* op = and v'() = v() - 1 and v'(') = v(') for all ' ∈∖;
* op = and v() = 0 and v'= v.
In order to simulate the non-blocking semantics of our rendez-vous protocols with counter machines, we extend the class of test-free CM with non-blocking decrement actions.
A non-blocking test-free counter machine (shortly ) is a tuple M=(, , Δ_b, Δ_nb, ) such that (, , Δ_b, ) is a and Δ_nb⊆×{|∈}× is a finite set of non-blocking transitions.
Observe that in a , both blocking and non-blocking decrements are possible, according to the definition of the transition relation.
Again, a configuration is given by a pair (ℓ,v)∈×^.
Given two configurations (ℓ, v) and (ℓ, v') and δ∈Δ_b∪Δ_nb, we extend the transition relation (ℓ,v)δ_M (ℓ,v')
over the set Δ_nb in the following way: for δ= (ℓ, , ℓ') ∈Δ_nb, we have (ℓ,v) δ_M (ℓ',v') if and only if v'() = max(0, v() - 1), and v'(') = v(') for all ' ∈∖.
We say that M is an with restore (shortly ) when (ℓ, , ) ∈Δ for all ℓ∈, i.e. from each location, there is a transition leading to the initial location with no effect on the counters values.
For a CM M with set of transitions Δ (resp. an with sets of transitions Δ_b and Δ_nb),
we will write (ℓ, v) _M (ℓ', v') whenever there exists δ∈Δ (resp. δ∈Δ_b∪Δ_nb) such that (ℓ, v) δ_M (ℓ', v') and use ^∗_M to represent the reflexive and transitive closure of _M. When the context is clear we shall write instead of _M. We let 0_ be the valuation such that 0_()=0 for all ∈.
An execution is a finite sequence of configurations (ℓ_0, v_0) (ℓ_1, v_1) …(ℓ_k, v_k).
It is said to be initial if (ℓ_0,v_0)=(, 0_). A configuration (ℓ,v) is called reachable if (, 0_) ^∗ (ℓ,v).
We shall now define the coverability problem for (non-blocking test-free) counter machines, which asks whether a given location can be reached from the initial configuration. We denote this problem [ℳ], for ℳ∈{CM, , , }. It takes as input a machine M in ℳ (with initial location and working over a set of counters) and a location ℓ_f and it checks whether there is a valuation v ∈ℕ^ such that (, 0_) ^*(ℓ_f, v).
In the rest of this section, we will prove that [] is -complete. To this end, we first establish that [] is in , by an adaptation of Rackoff's proof which shows that coverability in Vector Addition
Systems is in Expspace <cit.>. This gives also the upper bound for , since any is a . This result is established by the following theorem, whose proof is omitted due to lack of space.
[] and [] are in .
To obtain the lower bound, inspired by Lipton's proof showing that coverability in Vector Addition Systems is -hard <cit.>, we rely on 2Exp-bounded . We say that a CM M = (,, Δ,) is 2Exp-bounded if there exists n ∈ O(|M|) such that any reachable configuration (ℓ, v) satisfies v() ≤ 2^2^n for all ∈.
We use then the following result.
[2Exp-bounded ] is -hard.
We now show how to simulate a 2Exp-bounded by a , by carefully handling restore transitions that may occur at any point in the
execution. We will ensure that each restore transition is followed by a reset of the counters, so that we can always extract from an execution of the
a correct initial execution of the original . The way we enforce resetting of the counters is inspired by the way Lipton simulates 0-tests of a CM in a .
As in <cit.>, we will describe the final by means of several submachines. To this end, we define procedural non-blocking counter machines that are with several identified output states: formally, a procedural- is a tuple N = (, , Δ_b, Δ_nb, ℓ_in, L_out) such that (, , Δ_b, Δ_nb, ℓ_in) is a , L_out⊆, and there is no outgoing transitions from states in L_out.
Now fix a 2Exp-bounded M = (,, Δ,), ℓ_f∈ the location to be covered. There is some c, such that, any reachable configuration (ℓ, v) satisfies v() < 2^2^c |M| for all ∈, fix n = c|M|.
We build a N as pictured in <ref>. The goal
of the procedural 𝚁𝚜𝚝𝙸𝚗𝚌 is to ensure that all counters in are reset. Hence, after each restore transition, we are sure that we start over a fresh execution of the M.
We will need the mechanism designed by Lipton to test whether a counter is equal to 0. For a counter bounded by some value K, this is done by duplicating into and ensure along any execution that the sum of and is equal to K.
So, we define two families of sets of counters (Y_i)_0≤ i ≤ n and (Y_i)_0≤ i≤ n as follows. Let Y_i = {_i, _i, _i } and
Y_i = {_i, _i, _i} for all 0≤ i < n and Y_n = and Y_n = ∅ and '=⋃_0≤ i≤ n Y_i∪Y_i. All the machines we will describe from now on will work over the set of counters '.
Procedural- 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i().
We use a family of procedural- defined in <cit.>: for all 0≤ i <n, for all ∈Y_i, 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i() is a procedural- with an initial location ^𝚃𝚂,i,, and two output locations ℓ^𝚃𝚂,i,_z and
ℓ^𝚃𝚂,i,_nz. It tests if the value of is equal to 0, using the fact that the sum of the values of and is equal to 2^2^i. If =0, it swaps the values of and , and the execution ends in the output location ℓ^𝚃𝚂,i,_z. Otherwise, counters values are left unchanged and the execution ends in ℓ^𝚃𝚂,i,_nz. In any case, other counters are not modified by the execution. Note that
𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i() makes use of variables in ⋃_1≤ j< i Y_i∪Y_i.
Formally, these machines have the following property:
We use this proposition.
Let 0≤ i < n, and ∈Y_i.
For all v,v'∈ℕ^X', for ℓ∈{ℓ^𝚃𝚂,i,_z,ℓ^𝚃𝚂,i,_nz},
we have (^𝚃𝚂,i,v)^*(ℓ,v') in 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i()
if and only if :
* (PreTest1): for all 0 ≤ j < i, for all _j ∈Y_j, v(_j) = 2^2^j and for all _j ∈ Y_j, v(_j) = 0;
* (PreTest2): v(_i) = 2^2^i and v( _i) = 0;
* (PreTest3): v() + v() = 2^2^i;
* (PostTest1): For all ∉{,}, v'() = v();
* (PostTest2): either (i) v() = v'() = 0, v() = v'() and ℓ = ℓ^i_z, or (ii) v'() = v() >0, v'() = v() and ℓ = ℓ^𝚃𝚂,i,_nz.
Moreover, if for all 0 ≤ j ≤ n, and any counter ∈ Y_j ∪Y_j, v()≤ 2^2^j, then for all 0 ≤ j ≤ n, and any counter ∈ Y_j ∪Y_j, the value of will never go above 2^2^j during the execution.
Note that for a valuation v∈ℕ^X' that meets the requirements (PreTest1), (PreTest2) and (PreTest3), there is only one configuration
(ℓ,v') with ℓ∈{ℓ^𝚃𝚂,i,_z,ℓ^𝚃𝚂,i,_nz} such that (ℓ_in,v) ^* (ℓ,v').
Procedural 𝚁𝚜𝚝_i.
We use these machines to define a family of procedural- (𝚁𝚜𝚝_i)_0≤ i≤ n that reset the counters in Y_i∪Y_i, assuming that
their values are less than or equal to 2^2^i.
Let 0≤ i≤ n, we let 𝚁𝚜𝚝_i=(^𝚁,i, ',Δ_b^𝚁,i,Δ^𝚁,i_nb, ℓ^𝚁,i_in, {ℓ_out^𝚁,i}). The machine 𝚁𝚜𝚝_0 is pictured Figure <ref>.
For all 0≤ i< n, the machine 𝚁𝚜𝚝_i+1 uses counters from Y_i∪Y_i and procedural- 𝚃𝚎𝚜𝚝𝚜𝚠𝚊𝚙_i(_i) and
𝚃𝚎𝚜𝚝𝚜𝚠𝚊𝚙_i(_i) to control the number of times variables from Y_i+1 and Y_i+1 are decremented. It is pictured Figure <ref>. Observe that since Y_n=, and Y_n=∅, the machine 𝚁𝚜𝚝_n will be a bit different from the picture: there will only be non-blocking decrements over counters from Y_n, that is over counters from the initial M. If _i, _i (and 𝚜_i) are set to 2^2^i and _i, _i (and 𝚜_i) are set to 0, then each time
this procedural- takes an outer loop, the variables of Y_i+1∪Y_i+1 are decremented (in a non-blocking fashion) 2^2^i times.
This is ensured by Proposition <ref>the properties of 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(). Moreover, the location ℓ^𝚃𝚂, i, _z will only be reached when the counter _i
is set to 0, and this will happen after 2^2^i iterations of the outer loop, again thanks to Proposition <ref>the properties of 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(). So, all in all, variables from Y_i
and Y_i+1 will take a non-blocking decrement 2^2^i.2^2^i times, that is 2^2^i+1.
These properties are formalized in the following proposition.
For all 0≤ i≤ n, for all v∈ℕ^' such that
* (PreRst1): for all 0 ≤ j < i, for all ∈Y_j, v() = 2^2^j and for all ∈ Y_j, v() = 0,
for all v' ∈ℕ^', if (^𝚁,i, v) ^* (ℓ^𝚁,i_out,v') in 𝚁𝚜𝚝_𝚒 then
* (PostRst1): for all ∈ Y_i ∪Y_i, v'() = max(0, v() - 2^2^i),
* (PostRst2): for all ∉Y_i ∪Y_i, v'() = v().
For all ∈', we say that
is initialized in a valuation v if ∈ Y_i for some 0≤ i≤ n and v()=0, or ∈Y_i for some
0≤ i≤ n and v()=2^2^i.
For 0≤ i≤ n, we say that a valuation v∈ℕ^' is i-bounded if for all ∈ Y_i ∪Y_i, v() ≤ 2^2^i.
The procedural- 𝚁𝚜𝚝_i is taking care of resetting counters in Y_i∪Y_i.
The following lemma states that no counter in Y_j∪Y_j,
for 1≤ j≤ n,
will be increased over 2^2^j during this process, and that it reset properly counters in Y_i ∪Y_i.
Let 0≤ i ≤ n, and let v∈ℕ^' satisfying (PreRst1) for 𝚁𝚜𝚝_𝚒. If for all 0≤ j ≤ n, v is j-bounded, then for all (ℓ,v')∈^𝚁,i×ℕ^' such that
(ℓ^𝚁,i_in,v) ^* (ℓ, v') in 𝚁𝚜𝚝_i, v' is j-bounded for all 0≤ j ≤ n. Furthermore, the unique configuration such that (ℓ^𝚁,i_in,v) ^* (ℓ^𝚁,i_out, v') in 𝚁𝚜𝚝_i is defined by v'() = 0 for all
∈ Y_i ∪Y_i and v'() = v() for all ∉ Y_i ∪Y_i.
The construction ensures that when one enters 𝚁𝚜𝚝_i with a valuation v that is i-bounded, and in which all variables in ⋃_0≤ j<i Y_j∪Y_j are initialized, the location ℓ^𝚁,i_out is reached with a valuation v' such that: v'() = 0 for all
∈ Y_i ∪Y_i and v'() = v() for all ∉ Y_i ∪Y_i. Moreover, if v is j-bounded for all 0≤ j≤ n, then
any valuation reached during the execution remains j-bounded for all 0≤ j≤ n.
Procedural 𝙸𝚗𝚌_i.
The properties we seek for 𝚁𝚜𝚝_i are ensured whenever the variables in ⋃_0≤ j<iY_j∪Y_j are initialized. This is taken care of by
a family of procedural- introduced in <cit.>. For all 0≤ i< n, 𝙸𝚗𝚌_i is a procedural- with initial location ^𝙸𝚗𝚌, i, and unique output location ℓ^𝙸𝚗𝚌, i_out.
They enjoy the following property: for 0≤ i<n, when one enters 𝙸𝚗𝚌_i with a valuation v in which all the variables in ⋃_0≤ j<i Y_j∪Y_j
are initialized and v()=0 for all ∈Y_i, then the location ℓ^𝙸𝚗𝚌_i_out is reached with a valuation v' such that v'()=2^2^i
for all ∈Y_i, and v'()=v() for all other ∈'. Moreover, if v is j-bounded for all 0≤ j≤ n, then
any valuation reached during the execution remains j-bounded for all 0≤ j≤ n.
For all 0≤ i< n, for all v,v'∈ℕ^', (^𝙸𝚗𝚌, i,v) ^* (ℓ_out^𝙸𝚗𝚌, i, v') in 𝙸𝚗𝚌_i if and only if:
* (PreInc1) for all 0 ≤ j < i, for all ∈Y_j, v() = 2^2^j and for all ∈ Y_j, v() = 0;
* (PreInc2) for all ∈Y_i, v( ) = 0,
* (PostInc1) for all ∈Y_i, v'() = 2^2^i;
* (PostInc2) for all ∉Y_i, v'() = v().
Moreover, if for all 0≤ j ≤ n, v is j-bounded, then for all (ℓ,v”) such that
(ℓ^𝙸𝚗𝚌,i_in,v) ^* (ℓ, v”) in 𝙸𝚗𝚌_i, then v” is j-bounded for all 0≤ j≤ n.
Procedural 𝚁𝚜𝚝𝙸𝚗𝚌.
Finally, let 𝚁𝚜𝚝𝙸𝚗𝚌 be a procedural- with initial location ℓ_a and output location ℓ_b, over the set of counters ' and built as an
alternation of 𝚁𝚜𝚝_i and 𝙸𝚗𝚌_i for 0≤ i<n, finished by 𝚁𝚜𝚝_n. It is depicted in <ref>. Thanks to the
properties of the machines 𝚁𝚜𝚝_i and 𝙸𝚗𝚌_i, in the output location of each 𝙸𝚗𝚌_i machine, the counters in Y_i are set
to 2^2^i, which allow counters in Y_i+1∪Y_i+1 to be set to 0 in the output location of 𝚁𝚜𝚝_i+1. Hence, in location ℓ^𝙸𝚗𝚌,n_out, counters in Y_n= are set to 0.
The reduction.
To build the final N, we compose the procedural 𝚁𝚜𝚝𝙸𝚗𝚌 with the M in the way described <ref>, and
we add to every location ℓ of 𝚁𝚜𝚝𝙸𝚗𝚌 and M a restore transition (ℓ, ∅,') which is represented in the figure in an abstract way with dashed arrows, for readability's sake.
From <cit.>, each procedural machine 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i() and 𝙸𝚗𝚌_i has size at most C × n^2 for some constant C. Hence, observe that N is of size at most B for some B∈ O(|M|^3).
One can show that (, 0_) ^*_M (ℓ_f, v) for some v∈ℕ^, if and only if (', 0_') ^*_N (ℓ_f, v') for some v'∈ℕ^'. Using <ref>, we obtain:
[] is -hard.
§ COVERABILITY FOR RENDEZ-VOUS PROTOCOLS
In this section we prove that and problems are both -complete for rendez-vous protocols. To this end, we present the following reductions: reduces to [] and [] reduces to .
This will prove that is in and is -hard (from <ref> and <ref>). As is an instance of , the two reductions suffice to prove -completeness for both problems.
§.§ From Rendez-vous Protocols to
Let = (Q, Σ, , q_f, T) a rendez-vous protocol and C_F a configuration of to be covered. We shall also decompose C_F as a sum of multisets 𝐪_1 + 𝐪_2 + … + 𝐪_s. Observe that there might be 𝐪_i=𝐪_j for i≠ j.
We build the M = (, , Δ_b, Δ_nb, ) described in <ref>. Here, with =Q. A configuration C of is meant to be represented in M
by (,v), with v(q)=C(q) for all q∈ Q. The only meaningful location of M is then . The other ones are here to ensure correct updates of the
counters when simulating a transition.
We let = {}∪{ℓ_(t,t')^1, ℓ_(t,t')^2,ℓ_(t,t')^3| t=(q,!a,q'), t'=(p,?a,p')∈ T}∪{ℓ_t, ℓ_t,p_1^a,⋯,ℓ_t,p_k^a|
t=(q,!a,q')∈ T, a={p_1,…, p_k}}∪{ℓ_q| t=(q,τ,q')∈ T}∪{ℓ_1 …ℓ_s}, with final location ℓ_f = ℓ_s, where m for a message m ∈Σ has been defined in <ref>. The sets Δ_b and Δ_nb are shown <ref>. Transitions pictured <ref> show how to
simulate a rendez-vous protocol with the classical rendez-vous mechanism.
The non-blocking
rendez-vous are handled by the transitions pictured <ref>(where the
only non-blocking transitions of the occur): to simulate the occurrence of (q,!a,q'), the M decrements the value of q by
a transition of the form (3). It then takes a sequence of non-blocking decrements for each state in a. The last transition of the simulation of a non-blocking rendez-vous is to increment the counter q' by a transition of the form (3).. If the M faithfully simulates , then this loop of non-blocking
decrements is taken when the values of the counters in a are equal to 0, and the configuration reached still corresponds to a configuration in .
However, it could be that this loop is taken in M while some counters in a are strictly positive. In this case, a blocking rendez-vous has to be taken in , e.g. (q,!a,q') and (p,?a,p') if the counter p in M is strictly positive. Therefore, the value of the reached configuration (, v) and the corresponding configuration C in
will be different: first, C(p')>v(q'), since the process in p has moved in the state p' in when there has been no increment of p' in M. Furthermore,
all other non-blocking decrements of counters in a in M may have effectively decremented the counters, when in no other process has left a state of a. However, this ensures that C≥ v. The reduction then ensures that if (, v) is reachable in M, then a configuration C≥ v is reachable in . Then, if it is possible to reach a configuration (, v) in M whose counters are high enough to cover ℓ_F, then
the corresponding initial execution in will reach a configuration C≥ v, which hence covers C_F.
over rendez-vous protocols is in .
§.§ From to Rendez-Vous Protocols
The reduction from [] to in rendez-vous protocols mainly relies on the mechanism that can ensure that at most one process evolves in some given set of states, as explained in <ref>. This will allow to somehow select a “leader” among the processes that will simulate the behaviour
of the whereas other processes will simulate the values of the counters.
Let M = (, , Δ_b, Δ_nb, ) a and ℓ_f ∈ a final target location.
We build the rendez-vous protocol pictured in <ref>, where (M) is the part that will simulate the M. The locations
{1_|∈} will allow to encode the values of the different counters during the execution: for a configuration C, C(1_) will represent
the value of the counter .
We give then (M)=(Q_M,Σ_M,,ℓ_f,T_M) with
Q_M = ∪{ℓ_δ|δ∈Δ_b}
Σ_M = {inc_,inc_, dec_, dec_, nbdec_|∈}
T_M ={(ℓ_i,!inc_, ℓ_δ), (ℓ_δ, ?inc_, ℓ_j)|δ=(ℓ_i, , ℓ_j)∈Δ_b}
∪{(ℓ_i, !dec_, ℓ_δ), (ℓ_δ, ?dec_, ℓ_j)|δ = (ℓ_i, , ℓ_j)∈Δ_b}
∪{(ℓ_i, !nbdec_, ℓ_j)| (ℓ_i, x,ℓ_j)∈Δ_nb}
∪{(ℓ_i, τ, ℓ_j)| (ℓ_i, ,ℓ_j)∈Δ_b, ℓ_j }
Q_M=∪{ℓ_δ|δ∈Δ_b}, Σ_M= {inc_,inc_, dec_, dec_, nbdec_|∈}, and T_M={(ℓ_i,!inc_, ℓ_δ),(ℓ_δ, ?inc_, ℓ_j)|δ=(ℓ_i, , ℓ_j)∈Δ_b}∪{(ℓ_i, !dec_, ℓ_δ), (ℓ_δ, ?dec_, ℓ_j)| δ = (ℓ_i, , ℓ_j) [0]∈Δ_b}∪{(ℓ_i, !nbdec_, ℓ_j)| (ℓ_i, ,ℓ_j)∈Δ_nb}∪{(ℓ_i, τ, ℓ_j)| (ℓ_i, ,ℓ_j)∈Δ_b}. Here, the reception of a message inc_
(respectively dec_) works as an acknowledgement, ensuring that a process has indeed received the message inc_ (respectively dec_), and that the corresponding counter has been incremented (resp. decremented). For
non-blocking decrement, obviously no acknowledgement is required.
We define =(Q,Σ,T,, ℓ_f) as follows.
Q = Q_M∪{1_, q_, q'_|∈}∪{, q, q_}
Σ = Σ_M∪{L, R}
T =T_M∪{(, !L, q), (q, !R, ), (q, ?L, q_)}∪{(ℓ, ?L, q_)|ℓ∈ Q_M}
∪{(, ?inc_, q_), (q_, !inc_, 1_), (1_, ?dec_, q'_), (q'_, !dec_, ), (1, ?nbdec_, )|∈}
{(q_, ?R, ), (q'_, ?R, )|∈}
The protocol =(Q,Σ,, ℓ_f,T) is then defined with Q= Q_M∪{1_, q_, q'_|∈}∪{, q, q_},
Σ=Σ_M∪{L, R} and T is the set of transitions T_M along with the transitions pictured in <ref>. Note
that there is a transition (ℓ,?L,q_) for all ℓ∈ Q_M.
With two non-blocking transitions on L and R at the beginning, protocol can faithfully simulate the M without further ado, provided that the initial configuration contains enough processes to simulate all the counters values during the execution: after having sent a process in state , any transition of M can be simulated in . Conversely, an initial execution of can send multiple processes into
the 𝒫(M) zone, which can mess up the simulation. However, each new process entering 𝒫(M) will send the message L, which will send the process already in
{q}∪ Q_M in the deadlock state q_, and send the message R, which will be received by any process in {q_,q'_|∈}. Moreover, the construction of the protocol ensures that there can only be one process in the set of states
{q_,q'_|∈}. Then, if we have reached a configuration simulating the configuration (ℓ, v) of M, sending a new process in the 𝒫(M) zone will lead to a configuration (, v), and hence simply mimicks a restore transition of M. So every initial execution of corresponds to an initial execution of M.
and over rendez-vous protocols are complete.
§ COVERABILITY FOR WAIT-ONLY PROTOCOLS
In this section, we study a restriction on rendez-vous protocols in which
we assume that a process waiting to answer a rendez-vous cannot perform another
action by itself. This allows for a polynomial
time algorithm for solving .
§.§ Wait–Only Protocols
We say that a protocol = (Q, Σ, ,
q_f, T) is wait-only if the set of states Q can be
partitioned into Q_A — the active states — and Q_W — the waiting states — with ∈ Q_A
and:
* for all q ∈ Q_A, for all (q',?m,q”)∈ T, we have q'≠ q;
* for all q∈ Q_W, for all (q', !m, q”) ∈ T, we have q' ≠ q and for all (q', τ, q”) ∈ T, we have q'≠ q.
From a waiting state, a process can only perform receptions (if it can perform anything), whereas in an active state, a process can only perform internal actions or send messages.
Examples of wait-only protocols are
given by Figures <ref> and <ref>.
In the sequel, we will often refer to the paths of the underlying
graph of the protocol. Formally, a
path in a protocol = (Q, Σ, ,
q_f, T) is either a control state q ∈ Q or a finite sequence of
transitions in T of the form
(q_0,a_0,q_1)(q_1,a_1,q_2)…(q_k,a_k,q_k+1), the first case
representing a path from q to q and the second one from q_0 to
q_k+1.
§.§ Abstract Sets of Configurations
To solve the coverability problem for wait-only protocols in polynomial time, we
rely on a sound and complete
abstraction of the set of reachable configurations. In the sequel, we consider a wait-only protocol = (Q, Σ, ,
q_f, T) whose set of states is partitioned into a set of active
states Q_A and a set of waiting states Q_W. An abstract set of
configurations γ is a pair (S,) such that:
* S ⊆ Q is a subset of states, and,
* ⊆ Q_W ×Σ is a subset of pairs
composed of a waiting state and a message, and,
* q ∉S for all (q,m) ∈.
We then abstract the set of reachable configurations as a set
of states of the underlying protocol. However,
as we have seen, some states, like states in Q_A, can host an unbounded number of processes together (this will
be the states in S), while some states can only host a bounded number (in fact, 1) of processes together (this will be the states stored in ).
This happens when a waiting state q answers a rendez-vous m, that has necessarily been requested for a process to be in q.
Hence, in , along with a state q, we remember the last message m having been sent in the path leading from to q, which is necessarily in Q_W. Observe that, since several paths can lead to q,
there can be (q,m_1),(q,m_2)∈ with m_1≠ m_2. We denote
by Γ the set of abstract sets of configurations.
Let γ=(S,) be an abstract set of
configurations. Before we go into the configurations represented by
γ, we need some preliminary definitions. We note (-1𝑝𝑡) the set q ∈ Q_W
|there exists m∈Σ such that (q,m) ∈ of control states
appearing in . Given a state q ∈ Q, we let
q be the set m ∈Σ|there exists q'∈ Q such that
(q,?m,
q') ∈ T of messages that can be received in state q (if q is
not a waiting state, this set is empty). Given two different waiting states q_1 and q_2 in
, we say q_1 and q_2 are conflict-free in
γ if there exist m_1,m_2 ∈Σ such that m_1 ≠ m_2,
(q_1,m_1),(q_2,m_2) ∈ and m_1 ∉q_2 and
m_2 ∉q_1. We now say that a configuration C∈() respects
γ if and only if for all q ∈ Q such that C(q)>0 one of the following two
conditions holds:
* q ∈ S, or,
* q ∈ and C(q)=1 and for all q' ∈∖q such that
C(q')=1, we have that q and q' are conflict-free.
Note that the condition is on states q such that C(q) > 0 and not all states q ∈ Q because it might be that some states don't appear in S∪ st(Toks) (non-reachable states for instance).
Let
γ be the set of configurations respecting γ. Note
that in γ, for q in S there is no restriction on
the number of processes that can be put in q and if q in
, it can host at most one process. Two
states from can both host
a process if they are conflict-free.
Finally, we will only consider abstract sets of configurations that
are consistent. This property aims to ensure that concrete configurations
that respect it are indeed reachable from states of S.
Formally, we
say that an abstract set of
configurations γ=(S,) is consistent if (i) for all (q,m) ∈, there exists a path
(q_0,a_0,q_1)(q_1,a_1,q_2)…(q_k,a_k,q) in
such that q_0
∈ S and a_0= !m
and for all 1≤ i ≤ k, we have that a_i= ?m_i and that there
exists (q'_i,!m_i,q”_i) ∈ T with q'_i ∈ S, and (ii) for two tokens (q,m), (q',m') ∈ either m∈q' and m'∈q, or, m∉q' and m'∉q.
Condition (i) ensures that processes in S can indeed lead to a process in the states from . Condition (ii) ensures that if in a configuration C,
some states in are pairwise conflict-free, then they can all host a process together.
Given γ∈Γ and a configuration C, there exists C' ∈γ such that C' ≥ C if and only if C ∈γ. Checking that C∈γ can be done in polynomial time.
§.§ Computing Abstract Sets of Configurations
Our polynomial time algorithm is based on the computation of a
polynomial length sequence of consistent abstract sets of
configurations leading to a final
abstract set characterising in a sound and complete manner (with
respect to the coverability problem), an abstraction for the set of
reachable configurations. This will be achieved by a function F:Γ→Γ,
that inductively computes this final abstract set
starting from γ_0=(, ∅).
Formal definition of the function F relies on intermediate sets
S”⊆ Q and ”⊆ Q_W ×Σ, which are the smallest sets satisfying the conditions described in <ref>.
From S and , rules described in <ref> add states and tokens to S” and ” from the outgoing transitions from states in S and ().
It must be that every state added to S” can host an unbounded number of processes, and every state added to ” can host at least one process, furthermore, two conflict-free states in ” should be able to host at least one process at the same time.
We now provide the formal definition of this function. For an
abstract set of configurations γ=(S,), we will have
γ'=F(γ) if and only if γ'=(S',') where S'
and ' are built as follows. First we use some intermediate sets
of states
S”⊆ Q and ”⊆ Q_W ×Σ which are
the smallest sets
satisfying the following conditions S ⊆ S” and ⊆” and:
* for all (p,τ,p') ∈ T with p ∈ S, we have p' ∈ S”;
* for all (p,!a,p') ∈ T with p ∈ S, we have: (a) p' ∈ S” if a ∉p' or if there exists
(q,?a,q') ∈ T with q ∈ S; (b) (p',a) ∈” otherwise (i.e. when a ∈p' and there
does not exists (q,?a,q') ∈ T with q ∈ S);
* for all (q,?a,q') ∈ T with q ∈ S or (q,a) ∈, we have q' ∈
S” if there exists (p,!a,p') ∈ T with p ∈ S;
* for all (q,?a,q') ∈ T with (q,m) ∈ with m
≠ a, we have: (a) q' ∈ S” if m ∉q' and there exists
(p,!a,p') ∈ T with p ∈
S; (b) (q',m) ∈” if m ∈q' and there exists
(p,!a,p') ∈ T with p ∈ S.
We have then that S' is the smallest set including S” and such
that:
* for all (q_1, m_1), (q_2, m_2) ∈” such that m_1
m_2 and m_2 ∉q_1 and m_1 ∈q_2, we
have q_1 ∈ S';
* for all (q_1, m_1), (q_2, m_2), (q_3,m_2) ∈” s.t m_1 m_2
and (q_2, ?m_1, q_3) ∈ T, we have q_1 ∈ S';
* for all (q_1, m_1), (q_2, m_2), (q_3, m_3) ∈” such
that m_1 m_2 and m_1 m_3 and m_2 m_3 and
m_1 ∉q_2, m_1 ∈q_3 and m_2∉q_1, m_2 ∈q_3, and m_3 ∈q_2 and m_3 ∈q_1,
we have q_1 ∈ S'.
And finally '=(q,m) ∈”| q ∉S'.
Consider the wait-only protocol _1 depicted on Figure
<ref>. From (q_in,∅), rules described in <ref> construct the following pair (S_1”, _1”) = (q_in,q_4,(q_1,a),[0](q_1,b),(q_5,c)).
In
_1, it is indeed possible to reach a configuration with as
many processes as one wishes in the state q_4
by repeating the transition (q_in,!d,q_4) (rule <ref>). On the other hand, it
is possible to put at most one process in the waiting state q_1
(rule <ref>), because any other attempt from a process in will yield a reception
of the message a (resp. b) by the process already in q_1.
Similarly, we can put at most
one process in q_5. Note that in
_1”, the states q_1
and q_5 are conflict-free and it is hence possible to have
simultaneously one process in both of them.
If we apply rules of <ref> one more time to (S”_1, ”_1), we get S_2”=, q_2, q_4, q_6,q_7 and
_2”=(q_1,a), (q_1,b) ,(q_3,a),(q_3,b),(q_5,c).
We can put at most one process in q_3: to add one, a process will take the transition
(q_1,?c,q_3). Since (q_1,a), (q_1,b)∈”_1, there can be at most one process in
state q_1, and this process arrived by a path in which
the last request of rendez-vous was !a or !b.
Since {a,b}⊆q_3, by rule <ref>, (q_3,a),(q_3,b) are added. On the other hand we can put as many processes as we want in the
state q_7 (rule <ref>): from a configuration with one process on state q_5, successive non-blocking request on letter c, and
rendez-vous
on letter d will allow to increase the number of processes in state q_7.
However, one can observe that q_5 can in fact host an unbounded number of processes:
once two processes have been put on states q_1 and q_5 respectively (remember that q_1 and q_5 are conflict-free in (S”_1, ”_1)), iterating rendez-vous on letter c (with transition (q_1, ?c, q_3)) and rendez-vous on letter a put as many processes as one wants on state q_5.
This is why we need another transformation from S_2”, _2” to F(S”_1, ”_1).
As we shall see, this transformation does not have any impact on S”_1 and ”_1 and so it holds that F((, ∅)) = (S”_1, ”_1).
Note F(γ) = (S', '), <ref> describes the construction of S' from (S”, ”), while ' = ”∖ (S ×Σ), i.e. all states added to S' are removed from ' so a state belongs either to S' or to '.
Now the case of state q_5 evoked in the previous example leads to application of rule <ref>, since (q_5,c), (q_1,a) ∈”_2, and (q_3,a) (q_1,?c,q_3)∈ T.
Finally, F(F(q_in,∅))=(q_in, q_2,q_4, q_5, q_6,q_7,[0](q_1,a), (q_1,b) ,(q_3,a),(q_3,b)). Since q_1 and q_3 are not
conflict-free, they won't be reachable together in a configuration.
We consider now the wait-only protocol _2 depicted on Figure
<ref>. In that case, to compute
F((q_in,∅)) we will first have S”=q_in
and ”=(q_1,a),(q_2,b),(p_1,m_1),(p_2,m_2),[0](p_3,m_3)
(using rule <ref>), to finally get F((q_in,∅))=(q_in,q_1,p_1,(q_2,b),(p_2,m_2),[0](p_3,m_3))).
Applying rule <ref> to tokens (q_1,
a) and (q_2, b) from ”, we obtain that q_1∈ S': whenever one manages
to obtain one process in state q_2, this process can answer the requests on message a instead of processes in state q_1, allowing one
to obtain as many processes as desired in state q_1.
Now since (p_1,m_1), (p_2, m_2) and (p_3, m_3) are in ”
and respect the conditions of rule <ref>, p_1 is added to the set S' of unbounded states.
This case is a generalisation of the previous one, with 3 processes. Once one process has
been put on state p_2 from , iterating the following actions: rendez-vous over m_3, rendez-vous over m_1, non-blocking request of m_2,
will ensure as many processes as one wants on state p_1.
Finally applying successively F, we get in this case
the abstract set (q_in,q_1,q_3,p_1,p_2,p_3,p_4,(q_2,b)).
We show that F satisfies the following properties.
* F(γ) is consistent and can be computed in polynomial time
for all consistent γ∈Γ.
* If (S',')=F(S,) then S ≠ S' (and S ⊆ S') or
⊆'.
* For all consistent γ∈Γ, if C ∈γ and C C' then C' ∈F(γ).
* For all consistent γ∈Γ, if C' ∈F(γ), then there exists C”∈ and C ∈γ such that C”≥ C' and C ^∗ C”.
Point 1. and 2, ensures us that if we apply successively the function
F to (q_in,∅) then the computation will reach a consistent
abstract set γ_f such that γ_f=F(γ_f) and it will
take a polynomial time. Points 3. ensures that the computed abstraction
is complete whereas Point 4. guarantees its soundness.
§.§ Polynomial Time Algorithm
We now present our polynomial time algorithm to solve for
wait-only protocols.
We
define the sequence (γ_n)_n ∈ as follows:
γ_0=(,∅) and γ_i+1=F(γ_i) for all
i ∈. First note that γ_0 is consistent and that γ_0= is the set of
initial configurations. Using Lemma <ref>, we deduce
that γ_i is consistent for all i ∈. Furthermore, each time we apply F to an
abstract set of configurations (S,) either S or
increases, or (S, ) stabilises. Hence for all n ≥ |Q|^2*|Σ|, we have
γ_n+1=F(γ_n)=γ_n. Let
γ_f=γ_|Q|^2*|Σ|. Using Lemma <ref>,
we get:
Given C ∈, there exists C_0 ∈ and C' ≥ C
such that C_0 ^∗ C' if and only if there exists C”∈γ_f such that C”≥ C.
We need to iterate |Q|^2*|Σ| times the function F to
compute γ_f and each computation of F can be done in
polynomial time. Furthermore checking whether there exists C”∈γ_f such that C”≥ C for a configuration C ∈ can be done in polynomial time by Lemma
<ref>, hence using the previous lemma we obtain the desired result.
and restricted to wait-only protocols are in .
§ UNDECIDABILITY OF
It is known that [CM] is undecidable in its full generality <cit.>. This result holds for a very restricted class of counter machines, namely Minsky machines (Minsky-CM for short), which are CM over 2 counters, _1 and _2.
Actually, it is already
undecidable whether there is an execution (,0_{_1,_2})^* (ℓ_f, 0_{_1,_2}).
Reduction from this last problem gives the following result.
is undecidable, even for wait-only protocols.
Fix M = (, ℓ_0, {_1, _2}, Δ ) with ℓ_f ∈ the final state. W.l.o.g., we assume that there is no outgoing transition from state ℓ_f in the machine.
The protocol is described in <ref>.
The states {0_i,p_i,1_i,p'_i| i=1,2} will be visited by processes simulating values of counters, while the states in will be visited by a process simulating
the different locations in the Minsky-CM. If at the end of the computation, the counters are equal to 0, it means that each counter has been incremented and
decremented the same number of times, so that all processes simulating the counters end up in the state ℓ_f. The first challenge is to appropriately check when
a counter equals 0. This is achieved thanks to the non-blocking semantics: the process sends a message !zero_i to check if the counter i equals 0. If it is does not, the message will be received by a process that will end up in the deadlock state . The second challenge is to ensure that only
one process simulates the Minsky-CM in the states in . This is ensured by the states {w, w'}. Each time a process arrives in the state, another must arrive in the w' state, as a witness that the simulation has begun. This witness must reach ℓ_f for the computation to be a testifier of a positive instance of , but
it should be the first to do so, otherwise a process already in ℓ_f will receive the message “w” and reach the deadlock state . Thus, if two processes simulate the Minsky-CM, there will be two witnesses, and they won't be able to reach ℓ_f together.
§ CONCLUSION
We have introduced the model of parameterised networks communicating by non-blocking rendez-vous, and showed that safety analysis of such networks becomes
much harder than in the framework of classical rendez-vous. Indeed, and become -complete and undecidable in our
framework, while these problems are solvable
in polynomial time in the framework of <cit.>. We have introduced a natural restriction of protocols, in which control states
are partitioned between active states (that allow requesting of rendez-vous) and waiting states (that can only answer to rendez-vous) and showed that can then be solved in polynomial time. Future work includes finding further restrictions that would yield decidability of
. A candidate would be protocols in which waiting states can only receive one message. Observe that in that case, the reduction of <ref> can be adapted to simulate a , hence for this subclass of protocols is as hard as reachability in Vector Addition Systems with States, i.e. non-primitive recursive <cit.>. Decidability remains open though.
§ PROOFS OF <REF>
We present here the omitted proofs of <ref>.
§.§ Proof of <ref>
We will in fact prove the upper bound for a more general model: Non-Blocking Vector Addition Systems ().
A is composed of a set of transitions over vectors of dimension d, sometimes called counters, and an initial vector of d non-negative integers, like in VAS. However, in a , a transition is a pair of vectors: one is a vector of d integers and is called the blocking part of the transition and the other one is a vector of d non-negative integers and is called the non-blocking part of the transition.
Let d ∈ℕ. A Non-blocking Vector Addition System () of dimension d is a tuple (T, v_0) such that T ⊆ℤ^d ×ℕ^d and v_init∈ℕ^d.
Formally, for two vectors v, v' ∈ℕ^d, and a transition t=(t_b, t_nb) ∈ T, we write v t v' if there exists v”∈ℕ^d such that v” = v + t_b and, for all i ∈ [1,d], v'(i) = max(0, v”(i) - t_nb(i)). We write for ⋃_t ∈ Tt. We define an execution as a sequence of vectors v_1 v_2 … v_k such that for all 1 ≤ i < k, v_i v_i+1.
Intuitively, the blocking part t_b of the transition has a strict semantics: to be taken, it needs to be applied to a vector large enough so no value goes below 0. The non-blocking part t_nb can be taken even if it decreases some component below 0: the corresponding component will simply be set to 0.
We can now define what is the problem on .
problem for a V = (T,v_init) of dimension d ∈ℕ and a target vector v_f, asks if there exists v∈ℕ^d, such that v ≥ v_f and v_init^∗ v.
Adapting the proof of <cit.> to the model of yields the following result.
The problem for is in .
Fix a (T,v_init) of dimension d, we will extend the semantics of to a slightly relaxed semantics: let v,v' ∈ℕ^d and t = (t_b, t_nb) ∈ T, we will write v t v' when for all 1≤ j ≤ d, v'(j) = max(0, (v+t_b -t_nb)(j)).
Note that v t v' implies that v t v' but the converse is false: consider an of dimension d = 2, with t = (t_b, t_nb) ∈ T
such that t_b =(-3, 0) and t_nb = (0, 1), and let v = (1, 2) and v' =(0, 1). One can easily see that there does not exist v”∈ℕ^2 such that v” = v + t_b, as 1 - 3<0. So, t cannot be taken from v and it is not the case that vt v', however, v t v'.
We use for ⋃_t ∈ Tt.
Let J ⊆ [1,d], a path v_0 v_1 … v_m is said to be J-correct if for all v_i such that i < m, there exists t = (t_b, t_nb) ∈ T, such that v_i t v_i+1 and for all j ∈ J, (v_i + t_b)(j) ≥ 0. We say that the path is correct if the path is [1,d]-correct.
It follows from the definitions that for all v,v'∈ℕ^d, v^* v' if and only if there exists a correct path between v and v'.
Fix a target vector v_f ∈ℕ^d, and define = |v_f| + max_(t_b, t_nb)∈ T(|t_b| + |t_nb|), where |·| is the norm 1 of vectors in ℤ^d. Let ρ = v_0 v_1 … v_m and J ⊆ [1,d]. We say the path ρ is J-covering if it is J-correct and for all j ∈ J, v_m(j) ≥ v_f(j). Let r ∈ℕ, we say that ρ is (J,r)-bounded
if for all v_i, for all j ∈ J, v_i(j) < r. Let v ∈ℕ^d, we define m(J,v) as the length of the shortest J-covering path starting with v, 0 if there is none.
Note 𝒥_i = {J⊆ [1,d]| |J| = i } and define the function f as follows: for 1 ≤ i ≤ d, f(i) = max{m(J_i, v) | J_i ∈𝒥_i, v∈ℕ^d}. We will see that f is always well defined, in .
f(0) = 1.
From any vector v ∈ℕ^d, the path with one element v is ∅-covering.
For all 0 ≤ i < d, f(i+1) ≤ (· f(i))^i+1 + f(i).
Let J ∈𝒥_i+1 and v∈ℕ^d such that there exists a J-covering path starting with v. Note ρ = v_0t^1…t^mv_m the shortest such path.
First case: ρ is (J, .f(i))-bounded.
Assume, for sake of contradiction, that for some k < ℓ, for all j∈ J, v_k(j)=v_ℓ(j). Then we show that v_0… v_kv_ℓ+1…v_m is also a J-correct path, with
the vectors (v_ℓ')_ℓ< ℓ'≤ m, defined as follows.
v_ℓ+1(j)=v_ℓ+1(j) for all j∈ J
max(0,(v_k(j)+t^ℓ+1_b(j)-t^ℓ+1_nb(j))) otherwise.
And for all ℓ + 1< ℓ'≤ m,
v_ℓ'(j)=v_ℓ'(j) for all j∈ J
max(0, (v_ℓ'-1(j)+t_b^ℓ'(j)-t_nb^ℓ'(j))) otherwise.
Then v_0… v_kv_ℓ+1…v_m is also a J-correct path. Indeed, since v_k(j)=v_ℓ(j)
for all j∈ J, we have that v_ℓ+1(j)=v_ℓ+1(j)=max(0,(v_ℓ(j) + t^ℓ+1_b(j) - t^ℓ+1_nb(j)))=max(0,(v_k(j) + t^ℓ+1_b(j) - t^ℓ+1_nb(j))).
Moreover, for j∈ J, since v_ℓ(j)+t^ℓ+1_b(j)≥ 0, we get that v_k(j)+ t^ℓ+1_b(j)≥ 0.
By definition, for j∉ J, v_ℓ+1(j)=max(0,(v_k(j) + t^ℓ+1_b(j) - t^ℓ+1_nb(j))). Hence, v_k^t^ℓ+1v_ℓ+1, and
v_0^t^1… v_k^t^ℓ+1v_ℓ+1 is J-correct. Now let ℓ<ℓ'<m.
By definition, for j∈ J, v_ℓ'+1(j)=v_ℓ'+1(j). Then, v_ℓ'+1(j)=max(0,(v_ℓ'(j)+t^ℓ'+1_b(j) - t^ℓ'+1_nb(j))) = max(0,(v_ℓ'(j)+t^ℓ'+1_b(j) - t^ℓ'+1_nb(j))). Again, since ρ is J-correct, we deduce that for j∈ J, v_ℓ'(j)+t^ℓ'+1_b(j)≥ 0, hence v_ℓ'(j)+t^ℓ'+1_b(j)≥ 0. For j∉ J, v_ℓ'+1(j)=max(0, (v_ℓ'(j)+t_b^ℓ'+1(j)-t_nb^ℓ'+1(j))). So v_ℓ'^t^ℓ'+1v_ℓ'+1, and v_0^t^1… v_k^t^ℓ'+1v_ℓ'+1 is J-correct.
Then, ρ'=v_0… v_kv_ℓ+1…v_m is a J-correct path, and since v_m(j)=v_m(j) for all
j∈ J, it is also J-covering, contradicting the fact that ρ is minimal.
Hence, for all k < ℓ, there exists j ∈ J such that v_k(j) ≠ v_ℓ(j). The length of such a path is at most (.f(i))^i+1, so m(J,v)≤ (.f(i))^i+1≤ (.f(i))^i+1+f(i).
Second case: ρ is not (J, .f(i))-bounded. We can then split ρ into two paths ρ_1 ρ_2 such that ρ_1 is (J,.f(i))-bounded and ρ_2 = v'_0 … v'_n is such that v'_0(j) ≥.f(i) for some j ∈ J.
As we have just seen, |ρ_1|≤ (.f(i))^i+1.
Note J' = J ∖{j} with j such that v'_0(j) ≥.f(i). Note that ρ_2 is J'-covering, therefore, by definition of f, there exists
a J'-covering execution ρ = w_0 … w_k with w_0=v'_0, and such that |ρ|≤ f(i). Also, by definition of , for all 1≤ j' ≤ d, for all (t_b,t_nb)∈ T, ≥ |t_b(j')|+|t_nb(j')|, then t_b(j')≥ -, and t_b(j')-t_nb(j')≥ -. Hence, for all v∈^d,
1≤ j'≤ d, and c∈ such that v(j')≥ + c, for all (t_b,t_nb)∈ T, (v+t_b)(j') ≥ c and (v+t_b-t_nb)(j') ≥ c.
Now, since w_0 = v'_0, we get w_0(j)≥.f(i).
We deduce two things: first, for all 0 ≤ℓ < k, if t=(t_b,t_nb)∈ T is such that w_ℓ^t w_ℓ+1,
it holds that (w_ℓ + t_b)(j)≥.(f(i)- ℓ - 1). Since k = f(i) - 1, it yields that ρ is J-correct.
Second, for all 0 ≤ℓ≤ k, w_ℓ(j)≥(f(i) - ℓ). Again, k = f(i) - 1, so w_k(j) ≥≥ v_f(j). Hence
ρ is also J-covering.
Since ρ is the shortest J-covering path, we conclude that |ρ|≤ (.f(i))^i+1 + f(i), and so m(J,v)≤ (.f(i))^i+1 + f(i).
We define a function g such that g(0) = 1 and g(i+1) = (+1)^d(g(i))^d for 0 ≤ i < d; then f(i)≤ g(i) for all 1 ≤ i ≤ d. Hence, f(d) ≤ g(d) ≤ (+1)^d^d+1≤ 2^2^cnlog n for some n ≥max( d, , |v_init|) and a constant c which does not depend on d, v_0, nor v_f or the . Hence, we can cover vector v_f from v_init if and only if there exists a path (from v_init) of length ≤ 2^2^cn log n which covers v_f. Hence, there is a non-deterministic procedure that guesses a path of length ≤ 2^2^cn log n, checks if it is a valid path and accepts it if and only if it covers v_f. As |v_init|≤ n, |v_f| ≤ n and for all (t_b, t_nb) ∈ T, |t_b| + |t_nb| ≤ n, this procedure takes an exponential space in the size of the protocol. By Savitch theorem, there exists a deterministic procedure in exponential space for the same problem.
We are now ready to prove that the problem for is as hard as the problem for .
[] reduces to in .
Let a M = (, , Δ_b,Δ_nb, ), for which we assume wlog that it does not contain any self-loop
(replace a self loop on a location by a cycle
using an additional internal transition and an additional location).
We note = {_1, …, _m}, and = {ℓ_1…ℓ_k}, with ℓ_1= and ℓ_k=ℓ_f, and let d = k+m.
We define the V = (T, v_init) of dimension d as follows: it has one counter by location of the , and one counter by
counter of the . The transitions will ensure that the sum of the values of the counters representing the locations of M will always be equal to 1, hence
a vector during an execution of V will always represent a configuration of M. First, for a transition δ = (ℓ_i, op, ℓ_i')∈Δ, we define (t_δ, t'_δ)∈ℤ^d×ℕ^d by t_δ(i) = -1,
t_δ(i')= 1 and,
* if op=, then
t_δ(y)= 0 for all other 1≤ y≤ d, and t'_δ=0_d (where 0_d is the null vector of dimension d), i.e. no other modification is made on the counters.
* if op=_j, then t_δ(k+j)=1, and
t_δ(y)= 0 for all other 1≤ y≤ d, and t'_δ=0_d, i.e. the blocking part of the transition ensures the increment of the corresponding counter, while the non-blocking part does nothing.
* if op=_j, then t_δ(k+j)=-1, and
t_δ(y)= 0 for all other 1≤ y≤ d, and t'_δ=0_d, i.e. the blocking part of the transition ensures the decrement of the corresponding counter, while the non-blocking part does nothing.
.
* if op=_j, then
t_δ(y)= 0 for all other 1≤ y≤ d, and t'_δ(k+j)=1 and t'_δ(y)=0 for all other 1≤ y≤ d,
i.e. the blocking part of the transition only ensures the change in the location, and the non-blocking decrement of the
counter is ensured by the non-blocking part of the transition.
We then let T={t_δ|δ∈Δ},
and v_0 is defined by v_init(1)=1 and v_init(y)=0 for all 2≤ y≤ d.
We also fix v_f by v_f(k)=1, and v_f(y)=0 for all other 1≤ y≤ d.
One can prove that v_f is covered in V if and only if ℓ_f is covered in M.
If there exists w ∈ℕ^ such that (, 0_) ^* (ℓ_f, w), then there exists v ∈ℕ^d such that v_0 ^* v and v ≽ v_f.
Any configuration (ℓ,w) of M can be turned into a valuation v(ℓ_i,w) of T such that v(ℓ_i,w)(i)=1, for all 1≤ i≤ m, v(ℓ_i,w)(k+i)=w(_i)
and for all other 1≤ y≤ k, v(ℓ_i,w)(y)=0. Observe that v(,0_)=v_0. It follows from the definitions that (ℓ_i,w)(ℓ_i',w') if and only if v(ℓ_i,w) v(ℓ_i',w'). Hence, v_0^*v(ℓ_f,w)≥ v_f.
If there exists v ∈ℕ^d such that v_0 ^* v and v ≽ v_f, then there exists w ∈ℕ^ such that (, 0_) ^* (ℓ_f, w).
One can prove by induction that every vector v reachable from v_0 is such that there exists only one 1 ≤ i ≤ k such that v(i) = 1 and for all 1 ≤ i' ≤ k such that i ≠ i', v(i') = 0. Hence, given a reachable vector v, one can define γ_v a machine configuration as (ℓ_i, w) where i is the unique index 1≤ i≤ k such that v(i) = 1 and, for all 1 ≤ j ≤ m, w(_j) = v(k+j). Note v_0 v_1 … v_n = v, and observe that γ_v_n = (ℓ_f, w) for some w ∈ℕ^. Again, by a simple induction, one can prove that γ_v_0γ_v_1…γ_v_n, which concludes the proof.
Putting together Lemma <ref> and Lemma <ref>, we obtain the proof of <ref>.
§.§ Proof of <ref>
In this subsection, we prove <ref> by proving that the [] problem is hard. Put together with <ref>, it will prove the -completeness of [].
§.§.§ Proofs on the Pocedural Defined in <ref>
We formalize some properties on the procedural presented in <ref> used in the proof.
As for the procedural 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i, we use this proposition from <cit.>.
Let 0≤ i < n, and ∈Y_i.
For all v,v'∈ℕ^X', for ℓ∈{ℓ^𝚃𝚂,i,_z,ℓ^𝚃𝚂,i,_nz},
we have (^𝚃𝚂,i,v)^*(ℓ,v') in 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i()
if and only if:
* (PreTest1): for all 0 ≤ j < i, for all _j ∈Y_j, v(_j) = 2^2^j and for all _j ∈ Y_j, v(_j) = 0;
* (PreTest2): v(_i) = 2^2^i and v( _i) = 0;
* (PreTest3): v() + v() = 2^2^i;
* (PostTest1): For all ∉{,}, v'() = v();
* (PostTest2): either (i) v() = v'() = 0, v() = v'() and ℓ = ℓ^i_z, or (ii) v'() = v() >0, v'() = v() and ℓ = ℓ^𝚃𝚂,i,_nz.
Moreover, if for all 0 ≤ j ≤ n, and any counter ∈ Y_j ∪Y_j, v()≤ 2^2^j, then for all 0 ≤ j ≤ n, and any counter ∈ Y_j ∪Y_j, the value of will never go above 2^2^j during the execution.
Note that for a valuation v∈ℕ^X' that meets the requirements (PreTest1), (PreTest2) and (PreTest3), there is only one configuration
(ℓ,v') with ℓ∈{ℓ^𝚃𝚂,i,_z,ℓ^𝚃𝚂,i,_nz} such that (ℓ_in,v) ^* (ℓ,v').
*Procedural 𝚁𝚜𝚝_i.
We shall now prove that the procedural s we defined and displayed in <ref> meet the desired requirements.
For all 0≤ i≤ n, any procedural 𝚁𝚜𝚝_i has the following property:
For all 0≤ i≤ n, for all v∈ℕ^' such that
* (PreRst1): for all 0 ≤ j < i, for all ∈Y_j, v() = 2^2^j and for all ∈ Y_j, v() = 0,
for all v' ∈ℕ^', if (^𝚁,i, v) ^* (ℓ^𝚁,i_out,v') in 𝚁𝚜𝚝_𝚒 then
* (PostRst1): for all ∈ Y_i ∪Y_i, v'() = max(0, v() - 2^2^i),
* (PostRst2): for all ∉Y_i ∪Y_i, v'() = v().
For 𝚁𝚜𝚝_0, (PreRst1) trivially holds, and it is easy to see that (PostRst1) and (PostRst2) hold. Now fix 0 ≤ i < n, and consider the procedural- 𝚁𝚜𝚝_𝚒+1. Let v_0 ∈ℕ^' such that for all 0 ≤ j < i+1, for all ∈Y_j, v_0() = 2^2^j and for all ∈ Y_j, v_0( ) = 0, and let v_f such that (^𝚁,i, v_0) ^+ (ℓ^𝚁,i_out,v_f) in 𝚁𝚜𝚝_i.
First, we show the following property.
Property (∗): if there exist v,v'∈ℕ^' such that v(_i)=k,
(^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_z,v') with no other visit of
ℓ^𝚃𝚂,i,_z in between, then v'(_i)=2^2^i, v'(_i)=0, for all ∈ Y_i+1∪Y_i+1, v'()=max(0, v()-k), and v'()=v() for all other ∈'.
If k=0, then Proposition <ref> ensures that v'(_i)=2^2^i, v'(_i)=0, and for all other ∈',
v'()=v(). Otherwise, assume that the property holds for some k≥ 0 and consider (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_z,v') with no other visit of
ℓ^𝚃𝚂,i,_z in between, and v(_i)=k+1. Here, since v(_i)=k+1, Proposition <ref> and the construction
of the procedural- ensure that (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_nz,v)(ℓ^𝚁,i+1_2,v)^*(^𝚃𝚂,i,,v_1) with v_1(_i)=k, v_1(_i)=v(_i)+1, for all ∈ Y_i+1∪Y_i+1,
v_1()=max(0, v()-1), and for all other ∈', v_1()=v(). Induction hypothesis tells us that (^𝚃𝚂,i,,v_1)^* (ℓ^𝚃𝚂,i,_z,v') with v'(_i)=2^2^i, v'(_i)=0, for all ∈ Y_i+1∪Y_i+1, v'()=max(0, v()-k-1), and v'()=v() for all other ∈'.
Next, we show the following.
Property (∗∗): if there exist v,v'∈ℕ^' such that v(_i)=k, v(_i)=2^2^i, v(_i)=0,
and (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_z,v') with no other visit of
ℓ^𝚃𝚂,i,_z in between, then v'(_i)=2^2^i, v'(y_i)=0, for all ∈ Y_i+1∪Y_i+1, v'()=max(0, v()- k.2^2^i), and v'()=v() for all other ∈'.
If k=0, then Proposition <ref> ensures that
v'(_i)=2^2^i, v'(_i)=0, and v'()=v() for all other ∈'. Otherwise, assume that the property holds for some k≥ 0 and consider (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_z,v') with no other visit of
ℓ^𝚃𝚂,i,_z in between, and v(_i)=k+1. Again, since v(_i)=k+1, Proposition <ref> and the construction
of the procedural- ensure that (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_nz,v)(^𝚁,i+1,v)^*(^𝚃𝚂,i,,v_1)^* (ℓ^𝚃𝚂,i,_z,v'_1) (^𝚃𝚂,i,,v'_1), with v_1(_i)=v(_i)-1=k, v_1(_i)=v(_i)+1, v_1(_i)=v(_i)-1=2^2^i-1, v_1(_i)=v(_i)+1=1, for all ∈ Y_i+1∪Y_i+1, v_1()=max(0,v()-1), and for all other ∈', v_1()=v(). By Property (∗), v'_1(_i)=2^2^i, v'_1(_i)=0, for all ∈ Y_i+1∪Y_i+1, v'_1()=max(0, v()-2^2^i), and v'_1()=v_1() for all other ∈'. Induction hypothesis allows to conclude that
since (^𝚃𝚂,i,,v'_1)^* (ℓ^𝚃𝚂,i,_z,v'), v'(_i)=2^2^i, v'(_i)=0, for all ∈ Y_i+1∪Y_i+1, v'()=max(0, v'_1()- k.2^2^i) = max(0, v() - (k+1).2^2^i), and v'()=v'_1()=v() for all other ∈'.
Since (^𝚁,i, v_0) ^+ (ℓ^𝚁,i_out,v_f), we know that (^𝚁,i, v_0) ^* (^𝚃𝚂,i,,v)^*(ℓ^𝚃𝚂,i,_z,v')(^𝚃𝚂,i,,v')^*(ℓ^𝚃𝚂,i,_z,v”) (ℓ^𝚁,i_out,v_f).
By construction, v(_i)=2^2^i-1, v(_i)=2^2^i-1, v(_i)=1, v(_i)=1, for all ∈ Y_i+1∪Y_i+1,
v()=max(0,v_0()-1), and for all other counter , v()=v_0(). By Property (∗), v'(_i)=2^2^i=v_0(_i), v'(_i)=0=v_0(_i),
for all ∈ Y_i∪Y_i+1, v'()=max(0, v_0()-2^2^i) and for all other ∈', v'()=v(). By Property (∗∗),
v”(_i)=2^2^i=v_0(_i), v”(_i)=0=v_0(_i), for all ∈ Y_i∪Y_i+1,
v”()=max(0, v_0()-2^2^i - (2^2^i-1).2^2^i)=max(0, v_0()-2^2^i.2^2^i)=max(0, v_0()-2^2^i+1), and for all other ∈', v”()=v'()=v_0().
We get the immediate corollary:
Let 0≤ i≤ n, and v∈ℕ^' satisfying (PreRst1) for 𝚁𝚜𝚝_i. If v is i-bounded, then
the unique configuration such that (ℓ^𝚁,i_in,v) ^+ (ℓ^𝚁,i_out, v') in 𝚁𝚜𝚝_i is defined v'() = 0 for all
∈ Y_i ∪Y_i and v'() = v() for all ∉ Y_i ∪Y_i.
Let 0≤ i ≤ n, and let v∈ℕ^' satisfying (PreRst1)
for 𝚁𝚜𝚝_𝚒. If for all 0≤ j ≤ n, v is j-bounded, then for all (ℓ,v')∈^𝚁,i×ℕ^' such that
(ℓ^𝚁,i_in,v) ^* (ℓ, v') in 𝚁𝚜𝚝_i, v' is j-bounded for all 0≤ j ≤ n.
We will prove the statement of the property along with some other properties: (1) if ℓ is not a state of 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i) or 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i), then for all 0 ≤ j < i, for all ∈Y_j, v'() = 2^2^j and for all ∈ Y_j, v'() = 0, and v'(_i) =2^2^i and v'(_i) = 0.
(2) if ℓ is not a state of 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i) or 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i) and if ℓℓ_1^𝚁, i+1, then v'(_i) + v'(_i) = 2^2^i, and if ℓℓ_3^𝚁, i+1, then v'(_i) + v'(_i) = 2^2^i.
For 𝚁𝚜𝚝_0, the property is trivial. Let 0≤ i <n, and a valuation v∈ℕ^'
such that for all 0 ≤ j ≤ i, for all ∈Y_j, v() = 2^2^j and for all ∈ Y_j, v() = 0, and such that, for all 0≤ j≤ n, v is j-bounded. Let now (ℓ,v') such that (ℓ^𝚁,i+1_in,v) ^* (ℓ, v') in 𝚁𝚜𝚝_i+1.
We prove the property by induction on the number of occurences of ^𝚃𝚂,i,z and ^𝚃𝚂,i,y. If there is no occurence of such state between in (ℓ^𝚁,i+1_in,v) ^* (ℓ, v'), then, for all ∈ Y_j ∪Y_j∪{_i, _i} and j i, j i+1, then v'() = v() and so v' is j-bounded. Furthermore, for ∈ Y_i ∪ Y_i+1∪Y_i+1, v'() ≤ v(), and for all ∈Y_i, v'() ≤ v() + 1 = 1. The property (2) is easily verified.
Hence the properties hold.
Assume now we proved the properties for k occurrences of ^𝚃𝚂,i,z and ^𝚃𝚂,i,y, and let us prove the clam for k+1 such occurrences. Note ℓ_k+1∈{^𝚃𝚂,i,z,^𝚃𝚂,i,y} the last occurence such that:
(ℓ^𝚁,i+1_in,v) ^+ (ℓ_k, v_k) (ℓ_k+1, v_k+1) ^* (ℓ, v'). By induction hypothesis, v_k is j-bounded for all 0 ≤ j ≤ n and it respects (1) and (2), and by construction, (ℓ_k, , ℓ_k+1) and ℓ_k ℓ_1^𝚁,i+1, ℓ_k ℓ_3^𝚁, i+1, hence v_k+1 is j-bounded for all 0 ≤ j ≤ n and respects (PreTest1), (PreTest2), and (PreTest3) for 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i) and 𝚃𝚎𝚜𝚝𝚂𝚠𝚊𝚙_i(_i). As a consequence, if ℓ is a state of one of this machine such that (ℓ_k+1, v_k+1)^* (ℓ, v'), then by <ref>, for all 0 ≤ j ≤ n, as v_k+1 is j-bounded, so is v'.
Assume now ℓ to not be a state of one of the two machines. And keep in mind that v_k+1 respects (1) and (2). Then, either ℓ = ℓ_out^𝚁, i+1 and so v'() = v_k+1() for all ∈ Y_j ∪Y_j for all j i, and v'(_i) = 2^2^i and v'(_i) = 0 and so the claim holds, either ℓ∈{ℓ_in^𝚁,𝚒+1, ℓ_j'^𝚁, i+1}_j' = 1, 2, 3, 4, 5, 6, …, r. In this case, the execution is such that: (ℓ_k+1, v_k+1) ^+ (ℓ_nz, k+1, v_k+1) ^* (ℓ, v'), where if ℓ_k+1 =^𝚃𝚂,i,z, ℓ_nz, k+1 = ℓ^𝚃𝚂, i ,z_nz and otherwise ℓ_nz, k+1 = ℓ^𝚃𝚂, i ,y_nz.
In any cases, for all j i, j i+1, ∈ Y_j∪Y̅_j ∪{_i, _i}, v'() = v_k+1(), hence (1) holds and v' is j-bounded for all j < i and j > i+1.
Observe as well that for all ∈ Y_i+1∪Y_i+1, v'() ≤ v_k+1(), and so v' is i+1-bounded. The last thing to prove is that (2) holds. This is direct from the fact that v_k+1 respects (2).
About the procedural 𝙸𝚗𝚌_i, we use this proposition from <cit.>.
For all 0≤ i< n, for all v,v'∈ℕ^', (^𝙸𝚗𝚌, i,v) ^* (ℓ_out^𝙸𝚗𝚌, i, v') in 𝙸𝚗𝚌_i if and only if:
* (PreInc1) for all 0 ≤ j < i, for all ∈Y_j, v() = 2^2^j and for all ∈ Y_j, v() = 0;
* (PreInc2) for all ∈Y_i, v( ) = 0,
* (PostInc1) for all ∈Y_i, v'() = 2^2^i;
* (PostInc2) for all ∉Y_i, v'() = v().
Moreover, if for all 0≤ j ≤ n, v is j-bounded, then for all (ℓ,v”) such that
(ℓ^𝙸𝚗𝚌,i_in,v) ^* (ℓ, v”) in 𝙸𝚗𝚌_i, then v” is j-bounded for all 0≤ j≤ n.
*Procedural 𝚁𝚜𝚝𝙸𝚗𝚌.
We shall now prove the properties in the procedural 𝚁𝚜𝚝𝙸𝚗𝚌 defined in <ref>.
The next proposition establishes the correctness of the construction 𝚁𝚜𝚝𝙸𝚗𝚌.
Let v ∈ℕ^' be a valuation such that for all 0≤ i ≤ n and for all ∈ Y_i ∪Y_i, v() ≤ 2^2^i. Then the
unique valuation v' ∈ℕ^' such that (ℓ_a, v) ^* (ℓ_b, v') in 𝚁𝚜𝚝𝙸𝚗𝚌 satisfies the following: for all 0≤ i ≤ n,
for all ∈Y_i, v'() = 2^2^i and for all ∈ Y_i, v'() = 0.
Moreover, for all (ℓ,v”) such that (ℓ_a, v) ^* (ℓ, v”) in 𝚁𝚜𝚝𝙸𝚗𝚌, for all 0≤ i≤ n, v” is i-bounded.
We can split the execution in (ℓ_a,v) (^𝚁,0,v)^*(ℓ^𝚁,0_out, v_0) (^𝙸𝚗𝚌,0,v_0)^*
(ℓ_out^𝙸𝚗𝚌,0,v'_0)
(^𝚁,1,v'_0)^*(ℓ^𝚁,1_out,v_1)^*(^𝙸𝚗𝚌,n-1, v_n-1)^*(ℓ^𝙸𝚗𝚌,n-1_out, v'_n-1)
(^𝚁,n, v'_n-1)^*(ℓ_out^𝚁,n,v_n)(ℓ_b,v'), with v'=v_n and v=v'_-1. We show that for all 0≤ i≤ n:
* P_1(i): For all ∈ Y_i∪Y_i, v_i()=0, and for all ∉ (Y_i∪Y_i), v_i()=v'_i-1().
* P_2(i): For all 0≤ j <i, for all ∈ Y_j, v'_i-1()=0 and for all ∈Y_j, v'_i-1()=2^2^j, and
for all other ∈', v'_i()=v_i().
* P_3(i): For all v” such that (ℓ_a, v) ^* (ℓ, v”)^* (ℓ^𝚁,i_out, v_i), v” is i-bounded, for all 0≤ i≤ n.
For k=0, <ref> implies that for all ∈ Y_0∪Y_0, v_0()=0, and that for all other ∈',
v_0()=v(). Moreover, for all v” such that (^𝚁,0,v)^*(ℓ, v”)^*(ℓ_out^𝚁,0,v_0), <ref>
ensures that v” is i-bounded, for all 0≤ i≤ n. P_2(0) is trivially true.
Let 0≤ k< n, and assume that P_1(k), P_2(k) and P_3(k) hold. P_1(k) and P_2(k) and <ref> imply that for all ∈Y_k, v'_k()= 2^2^k, and that for all other counter ∈', v'_k()=v_k(). Thanks to P_1(k), P_2(k+1) holds. Moreover, we also know by <ref> that for all v” such that (ℓ_out^𝚁,k,v_k) (^𝙸𝚗𝚌,k, v_k)^*(ℓ, v”)^*(ℓ_out^𝙸𝚗𝚌,k, v'_k), v” is i-bounded for all 0≤ i≤ n. Since v'_k is then i-bounded for all 0≤ i≤ n, and since P_2(k) holds, <ref>
implies that v_k+1()=0 for all ∈ Y_k+1∪Y_k+1, and that, for all other ∈', v_k+1()=v'_k). So P_1(k+1) holds.
Moreover, by <ref>, for all v” such that (ℓ_out^𝙸𝚗𝚌,k, v'_k)(^𝚁,k+1,v'_k)^*(ℓ,v”)^*
(ℓ_out^𝚁,k+1,v_k+1), v” is i-bounded for all 0≤ i≤ n. Hence P_3(k+1) holds.
By P_1(n), v'()=0 for all ∈ Y_n, and since Y_n=∅, v'()=2^2^n for all ∈Y_n. Let ∉ (Y_n∪Y_n). Then v'()=v'_n-1(), and by P_2(n), for all 0≤ i <n, for all ∈Y_i, v'()=2^2^i, and for all ∈ Y_i,
v'()=0. By P_3(n), for all (ℓ,v”) such that (ℓ_a, v) ^* (ℓ, v”) in 𝚁𝚜𝚝𝙸𝚗𝚌, for all 0≤ i≤ n, v” is i-bounded.
§.§.§ Proofs of the Reduction
We are now ready to prove <ref>, i.e. that the reduction is sound and complete.
For some subset of counters Y, we will note v_| Y for the valuation v on counters Y, formally, v_| Y : Y →ℕ and is equal to v on its domain.
If there exists v ∈ℕ^ such that (, 0_) ^*_M (ℓ_f, v), then there exists v' ∈ℕ^' such that (', 0_') ^*_N (ℓ_f, v').
From <ref>, we have that (', 0_') ^*_N (, v_0) where v_0 is such that, for all 0 ≤ j ≤ n, for all ∈Y_j, v_0() = 2^2^j and for all ∈ Y_j, v_0( ) = 0. By construction of N, (, v_0)^*_N (ℓ_f,v') with v' defined by: for all 0≤ i <n, for all ∈Y_j, v'() = 2^2^j, for all ∈ Y_j, v'() = 0, and, for all ∈, v'() = v(). Note that in this path, there is no restore step.
If there exists v' ∈ℕ^' such that (', 0_') ^*_N (ℓ_f, v'), then there exists v ∈ℕ^ such that (, 0_) ^*_M (ℓ_f, v).
We will note v_0 the function such that for all 0≤ i ≤ n, and for all ∈Y_i, v_0() = 2^2^i and for all ∈ Y_i, v_0() = 0.
Observe that there might be multiple visits of location in the execution of N, because of the restore transitions. The construction of 𝚁𝚜𝚝𝙸𝚗𝚌
ensures that, every time a configuration (,v) is visited, v=v_0.
Formally, we show that for all (, v) such that (',0_')^*_N(,v), we have that v=v_0. First let (',w)^*_N(',w'), with w()≤ 2^2^i, and ', not visited in between. Then for all 0≤ i≤ n, for all ∈ Y_i∪Y_i, w'()≤ 2^2^i. Indeed, let (ℓ,w) be such that (',w)^*_N(ℓ, w)_N(',w'). By <ref>, we know that, for
all 0≤ i≤ n, for all ∈ Y_i∪Y_i, w()≤2^2^i. Since the last transition is a restore transition, we deduce that,
for all 0≤ i≤ n, for all ∈ Y_i∪Y_i, w'()=w()≤ 2^2^i.
* Let v∈ℕ^'
be such that (',0_')^*_N(,v), and (,v) is the first configuration where is visited. The
execution is thus of the form (',0_')^*_N(',w)^*_N(,v), with (',w) the last time ' is
visited. We have stated above that w()≤ 2^2^i. Then, we have that (',0_')
^*_N(',w)_N(ℓ_a,w)^*_N(ℓ_b,v)_N(,v), and by <ref>, v=v_0.
* Let now v_k,v_k+1∈ℕ^' be such that (',0_')^*_N(,v_k)^*_N(,v_k+1),
and v_k and v_k+1 are respectively the k^th and the (k+1)^th time that is visited, for some k≥ 0. Assume that
v_k=v_0. We have (, v_k)^*_N(ℓ,v)_N(',v)^*_N(',v)_N(ℓ_a,v)_N^*(ℓ_b,v_k+1)
_N(, v_k+1). Since the M is 2EXP-bounded, and v_k=v_0, we obtain that for all ∈=Y_n,
v()≤ 2^2^n. For all 0≤ i<n, for all ∈ Y_i∪Y_i, v()=v_0(), then for all 0≤ i≤ n, for all ∈ Y_i∪Y_i,
v()≤ 2^2^i. Then, as proved above, v()≤ 2^2^i for all 0≤ i≤ n, for all ∈ Y_i∪Y_i. By <ref>, v'=v_0.
Consider now the execution (',0_')^*_N(,v)^*_N(ℓ_f,v'), where (,v) is the last time the location
is visited. Then, as proved above, v=v_0. From the execution (,v)^*_N(ℓ_f,v'), we can deduce an execution
(, v_|)^*_M (ℓ_f, v'_|). Since v=v_0 and for all ∈=Y_n, v()=0, we can conclude the proof.
The two previous lemmas prove that the reduction is sound and complete. By <ref>, we proved the -hardness of the problem, and so <ref>.
§ PROOFS OF <REF>
In this section, we present proofs omitted in <ref>.
§.§ Proof of <ref>
We present here the proof of <ref>. The two lemmas of this subsection prove the soundness and completeness of the reduction presented in <ref>. Put together with <ref>, we prove <ref>.
Let C_0 ∈, C_f ≥ C_F. If C_0 ^* C_f, then there exists v∈ℕ^Q such that (, 0_)^*(ℓ_f, v).
For all q∈ Q, we let v_q(q)=1 and v_q(q')=0 for all q'∈ such that q'≠ q. Let n=||C_0||=C_0(), and let C_0C_1⋯ C_mC_f be the configurations visited in . Then, applying
the transition (, , ), we get (, 0_) (, v^1) … (, v^n) with v_0 = v^n and v_0()=n and v_0()=0 for all ≠. Let i≥ 0 and assume that (,0_)^*(, C_i). We show that
(, C_i)^*(, C_i+1).
* If C_im C_i+1, let t=(q_1,!m,q'_1), t'=(q_2, ?m, q'_2)∈ T such that C_i(q_1)>0, C_i(q_2)>0, C_i(q_1)+C_i(q_2)≥ 2, and C_i+1=
C_i - q_1,q_2+q'_1,q'_2. Then (, C_i)
(ℓ_(t,t')^1, v_i^1)(ℓ_(t,t')^2, v_i^2)(ℓ_(t,t')^3, v_i^3)(, v_i^4), with v_i^1= C_i - v_q_1, v_i^2=v_i ^1 - v_q_2,
v_i^3 = v_i^2 + v_q'_1, v_i^4 = v_i^3+v_q'_2. Observe that v_i^4=C_i+1 and then (, C_i)^*(, C_i+1).
* If C_iτ C_i+1, let t=(q,τ,q') such that C_i(q)>0 and C_i+1=C_i-q+q'. Then, (, C_i) (ℓ_q, v_i^1) (, v_i^2) with v_i^1=C_i- v_q and v_i^2 = v_i^1+ v_q'. Observe that v_i^2 = C_i+1, then (, C_i)^*(, C_i+1).
* If C_im C_i+1, let t=(q,!m,q') such that C_i+1=C_i-q+q', and m = {q_1,…, q_k}. Then C_i(p_j)=0 for all 1≤ j≤ k. We then have that (, C_i) (ℓ_t, v_i^1) (ℓ_t,q_1^m, v_i^1)⋯(ℓ_t,q_k^m, v_i^1) (, v_i^2) with v_i^1= C_i - v_q and v_i^2= v_i^1 + v_q'.
Indeed, v_i^1(q_j)=0 for all q_j∈m, so the transitions
(ℓ^m_t,q_j,
q_j+1), ℓ^m_t,q_j+1) do not change the value of the counters. Hence, v_i^2= C_i+1 and (, C_i)^* (, C_i+1).
So we know that (, 0_)^* (, C_f).
Moreover, since C_f ≥ C_F, it holds that C_f ≥ v_𝐪_1 + v_𝐪_2 + … + v_𝐪_s. Then (, C_f)^s (ℓ_f, v) with v=C_f-(v_𝐪_1 + v_𝐪_2 + … + v_𝐪_s).
Let v∈^Q. If (, 0_)^*(ℓ_f, v), then there exists C_0 ∈, C_f ≥ C_F such that C_0 ^* C_f.
Let (, v_0), (, v_1) … (, v_n) be the projection of the execution of M on {}×ℕ^.
We prove that, for all
0≤ i≤ n, there exists C_0 ∈, and C≥ v_i
such that C_0 ^* C. For i = 0, we let C_0 be the empty multiset, and the property is trivially true. Let 0≤ i < n, and assume that there exists C_0 ∈, C≥ v_i such that C_0 ^* C.
* If (, v_i)δ(, v_i+1) with δ=(, , ), then v_i+1 = v_i +v_. The
execution C_0^* C built so far cannot be extended as it is, since it might not include enough processes. Let
N be such that C_0 C_1… C_N = C, and let C'_0∈ with C'_0()=C_0()+N+1. We build, for all 0≤ j ≤ N, a configuration
C'_j such that C'_0^j C'_j,
C'_j≥ C_j and C'_j()>C_j()+N-j. For j=0 it is trivial. Assume now that, for 0≤ j < N, C'_j≥ C_j and that C'_j() > C_j()+N-j.
If C_jm C_j+1 for m∈Σ, with t_1=(q_1,!m, q'_1) and t_2=(q_2,?m,q'_2). Then, C_j+1=C_j - q_1,q_2 + q'_1,q'_2. Moreover, C'_j(q_1) ≥ C_j(q_1)>0 and C'_j(q_2) ≥ C_j(q_2) >0 and C'_j(q_1) + C'_j(q_2)≥ C_j(q_1) + C_j(q_2) ≥ 2. We let
C'_j+1 = C'_j - q_1,q_2 + q'_1,q'_2, and C'_jm C'_j+1. It is easy to see that C'_j+1≥ C_j+1. Moreover,
C'_j+1() > C_j+1() +N -j > C_j+1 + N -j-1.
If C_jm C_j+1 and
for all q∈m, C'_j-q_1(q)=0, with t=(q_1,!m,q_2), (respectively C_jτ C_j+1 with t=(q_1,τ,q_2)), we let
C'_j+1=C'_j - q_1+q_2, and C'_jmC'_j+1 (respectively C'_jτ C'_j+1).
Again, thanks to the induction hypothesis, we get that C'_j+1≥ C_j+1, and C'_j+1 ()> C_j+1() + N - j > C_j+1() + N - j-1.
If now C_jm C_j+1, with t_1=(q_1,!m,q_2) and there exists q'_1∈m such that C'_j - q_1(q'_1) >0.
Let (q'_1,?m,q'_2)∈ T, and then C'_j+1=C'_j - q_1,q'_1 + q_2, q'_2. Since C'_j≥ C_j, C'_j(q_1)≥ 1, and since
C'_j-q_1(q'_1) >0, C'_j(q'_1)≥ 1 and C'_j(q_1) + C'_j(q'_1) ≥ 2. Hence, C'_jm C'_j+1. We have that C'_j(q'_1) > C_j(q'_1), so C'_j+1(q'_1) ≥ C_j+1(q'_1) and C'_j+1(q)≥ C_j+1(q) for all other q∈ Q. Hence C'_j+1 > C_j+1. Also, C_j+1() = C_j() + x, with x∈{0,1}. If q'_1≠, then C'_j+1() = C'_j() + y, with y≥ x. Hence, since C'_j() > C_j() + N - j,
we get C'_j+1() > C_j+1() + N - j > C_j+1() + N -j - 1. If q'_1 =, then we can see that C'_j+1() = C'_j() +y, with
x-1≤ y ≤ x. In that case, C'_j+1() > C_j() + N-j+y≥ C_j() + N- j + x-1 ≥ C_j+1() + N-j-1.
So we have built an execution C'_0 ^* C'_N such that C'_N≥ C_N and C'_N() > C_N(). Hence, C'_N≥ v_i+1.
* If (,v_i) (ℓ_(t,t')^1, v_i^1) (ℓ_(t,t')^2, v_i^2) (ℓ_(t,t')^3, v_i^3) (, v_i+1), with
t= (q_1,!m,q_2) and t'=(q'_1, ?m, q'_2), then
v_i^1 = v_i - v_q_1, v_i^2= v_i^1 - v_q'_1, v_i^3 = v_i^2 + v_q_2, and v_i+1 = v_i^3+ v_q'_2. Then by induction hypothesis,
C(q_1)≥ 1, C(q'_1)≥ 1, and C(q_1) + C(q'_1) ≥ 2. We let C' = C - q_1, q'_1 + q_2, q'_2. We have
Cm C' and C' ≥ v_i+1.
* If (, v_i) (ℓ_q, v_i^1) (, v_i+1) with (q,τ, q')∈ T and v_i^1 = v_i - v_q and v_i+1 =
v_i^1 + v_q', then by induction hypothesis, C≥ 1,
and if we let C'=C- q+q', then CτC', and C'≥ v_i+1.
* If (, v_i) (ℓ_t, v_i^1) (ℓ_t,p_1^m, v_i^2)… (ℓ_t,p_k^m, v_i^k+1)
(, v_i+1) with t=(q,!m,q') and m = {p_1,…,p_k}, and (C-q)(p)=0 for all p∈m. We let
C' = C- q+q', hence Cm C'. Moreover, v_i^1 = v_i - v_q,
and, for all 1≤ j <k, it holds that v_i^j+1(p_j) = max(0, v_i^j(p_j) - 1) and v_i^j+1(p)=v_i^j(p) for all p≠ p_j. By induction hypothesis, C≥ v_i,
hence v_i^j(p)=0 for all p∈m, for all 1≤ j≤ k+1. Hence, v_i+1 = v_i^k+1 + v_q' = v_i^1 + v_q', and C' ≥ v_i+1.
* If (, v_i) (ℓ_t, v_i^1) (ℓ_t,p_1^m, v_i^2)… (ℓ_t,p_k^m, v_i^k+1)
(, v_i+1) with t=(q,!m,q') and m = {p_1,…,p_k}, and (C-q)(p_j)>0 for some p_j∈m. Let
(p_j,?m,p'_j)∈ T and
C' = C - q,p_j+q',p'_j. Obviously, Cm C'. It remains to show that C'≥ v_i+1. This is due to the fact that in
the
M, the counter p'_j will not be incremented, unlike C(p'_j). Moreover, in the protocol , only p_j will lose a process, whereas in M,
other counters corresponding to processes in m may be decremented. Formally, by definition and by induction hypothesis, C-q≥ v_i^1. Also, for all p∈m, either v_i^1(p)=v_i^k+1(p) = 0, or v_i^k+1(p) = v_i^1(p)-1. Remark
that since C≥ v_i, then C-q≥ v_i-v_q = v_i^1, hence (C-q,p_j)(p_j) = (C-q)(p_j) - 1 ≥ v_i^1(p_j)-1. Also, (C-q)(p_j) - 1≥ 0, hence (C-q)(p_j) - 1≥max(0,v_i^1(p_j)-1)=v_i^k+1(p_j). Observe also that, for all p≠ p_j∈m, if v_i^1(p)>0, then (C-q,p_j)(p)=
(C-q)(p) ≥ v_i^1(p) > v_i^k+1(p). If v_i^1(p) = 0, then (C-q,p_j)(p)≥ v_i^1(p)= v_i^k+1(p). For all other p∈ Q,
(C-q,p_j)(p) = (C-q)(p) ≥ v_i^1(p)= v_i^k+1(p). Hence, C-q,p_j≥ v_i^k+1. By definition, v_i+1 = v_i^k+1 + v_q'.
Hence, (C-q,p_j+q',p'_j)(p)≥ v_i+1(p), for all p≠ p'_j, and (C-q,p_j+q',p'_j)(p'_j)> v_i+1(p'_j).
So, C'> v_i+1.
Now we know that the initial execution of M is: (, 0_)^∗(, v_n)^∗ (ℓ_f, v_f) with v_f = v_n - (v_𝐪_1 + v_𝐪_2 + … + v_𝐪_s). Thus
v_n>v_𝐪_1 + v_𝐪_2 + … + v_𝐪_s. We have proved that we can build an initial execution of P: C_0^*C_n and that C_n≥ v_𝐪_1 + v_𝐪_2 + … + v_𝐪_s. Hence C_n ≥ C_F.
§.§ Proofs of <ref>
To prove <ref>, we shall use <ref> along with the reduction presented in <ref>. If the reduction is sound and complete, it will prove that is -hard. As is a particular instance of the problem, this is sufficient to prove <ref>.
The two lemmas of this subsection prove the soundness and completeness of the reduction presented in <ref>, put together with <ref>, it proves that is -hard.
For all v∈ℕ^d, if (, 0_)_M^*(ℓ_f, v), then there exists C_0 ∈, C_f ∈ such that C_0 ^* C_f.
For all ∈, we let N_ be the maximal value taken by in the initial execution (, 0_)^*(ℓ_f, v), and N=Σ_∈ N_. Now, we let C_0∈∩ C_N+1 be the initial configuration with N+1 processes. In the initial execution of that we will build, one of the processes will evolve in the (M) part of the protocol, simulating the execution of the , the others will simulate the values of the counters in the execution.
Now, we show by induction on k that, for all k≥ 0, if (, 0_)^k (ℓ, w), then C_0^* C, with C(1_)=w()
for all ∈, C(ℓ)=1, C()=N-Σ_∈ w(), and C(s)=0 for all other s∈ Q.
C_0L C_0^1R C_0^2, and C_0^2()=N, C_0^2()=1, and C_0^2(s)=0 for all other s∈ Q. So the property
holds for k=0. Suppose now that the property holds for k≥ 0 and consider (, 0_)^k (ℓ,w)δ (ℓ',w').
* if δ=(ℓ,,ℓ'), then Cinc_C_1 with C_1=C-ℓ, +ℓ_δ,q_. Indeed, by induction
hypothesis, C(ℓ)=1> 0, and C()>0, otherwise Σ_∈ w()=N and w() is already the maximal value taken by so
no increment of could have happened at that point of the execution of M. We also have C_1inc_C', since
C_1(ℓ_δ)>0 and C_1(q_)>0 by construction, and C'=C_1-ℓ_δ,q_+ℓ', 1_. So C'(ℓ')=1, for all ∈,
C'(1_)=w'(), and C'()=N-Σ_∈ w'().
* if δ=(ℓ,,ℓ'), then C(ℓ)=1>0 and C(1_)>0 since w()>0. Then Cdec_C_1 with
C_1=C-ℓ,1_+ℓ_δ,q'_. Then C_1dec_C', with C'=C_1-q'_, ℓ_δ+, ℓ'. So C'(ℓ')=1, C'(1_)=C(1_)-1, C'()=C()+1.
* if δ=(ℓ,,ℓ') and w()>0 then Cnbdec_C', and C'=C-ℓ, 1_+ℓ', and
the case is proved.
* if δ=(ℓ,,ℓ') and w()=0 then by induction hypothesis, C(1_)=0 and Cnbdec_C',
with C'=C-ℓ+ℓ'. Then, C'(1_)=0=w'(), and C'(ℓ')=1.
* if δ=(ℓ,,ℓ'), then CτC',
avec C'=C-ℓ+ℓ'. This includes the restore transitions.
Then C_0^* C with C(ℓ_f)=1 and C∈.
Let C_0 ∈, C_f ∈ such that C_0 ^* C_f, then (ℓ_0, 0_)^*_M(ℓ_f, v) for some v∈ℕ^.
Before proving this lemma we establish the following useful result.
Let C_0 ∈. For all C∈ such that C_0^+ C, we have Σ_p∈{q}∪ Q_M C(p)= 1.
Note C_0C_1…C_n = C_f.
Now, thanks to <ref>, for all 1≤ i≤ n, we can note 𝗅𝖾𝖺𝖽𝖾𝗋(C_i) the unique state s in {q}∪ Q_M such that C_i(s) = 1. In particular, note that 𝗅𝖾𝖺𝖽𝖾𝗋(C_n) = ℓ_f. We say that a configuration C is
M-compatible if 𝗅𝖾𝖺𝖽𝖾𝗋(C)∈.
For any M-compatible configuration C∈, we define the configuration of the π(C_i)=(𝗅𝖾𝖺𝖽𝖾𝗋(C), v) with v=C(1_) for all ∈.
We let C_i_1⋯ C_i_k be the projection of C_0C_1… C_n onto the M-compatible configurations.
We show by induction on j that:
P(j): For all 1≤ j≤ k, (,0_)^*_M π(C_i_j), and Σ_∈C_i_j(q_)+C_i_j(q'_)=0. Moreover, for all C such that C_0^*CC_i_j, Σ_∈C(q_)+C(q'_)≤ 1.
By construction of the protocol, C_0L C_1(L)^k C_2R C_i_1 for some k ∈ℕ. So π(C_i_1)=(, 0_), and for all C such that C_0^*CC_i_1, Σ_∈C(q_)+C(q'_)=0, so P(0) holds true.
Let now 1≤ j <k, and suppose that (,0_)^*_M π(C_i_j), and Σ_∈C_i_j(q_)+C_i_j(q'_)=0. We know that C_i_j^+C_i_j+1.
* If there is no C∈ such that C(q)=1 and C_i_j^+C^*C_i_j+1, the only possible transitions from C_i_j are in T_M. Let π(C_i_j)=(ℓ,v).
* if C_i_jinc_C then C=C_i_j-ℓ,+ℓ_δ,q_ for δ=(ℓ,,ℓ')∈Δ_b. Σ_∈C(q_)+C(q'_)=1. Note that the
message inc_ is necessarily received by some process, otherwise C(q_)=0 and C has no successor, which is in contradiction with the fact the the execution reaches C_f. Moreover, the only
possible successor configuration is Cinc_ C_i_j+1, with C_i_j+1=C-q_, ℓ_δ+1_, ℓ'. Hence, obviously, π(C_i_j)π(C_i_j+1).
* if C_i_jdec_C then C=C_i_j-ℓ,1_+ℓ_δ,q'_ for δ=(ℓ,,ℓ')∈Δ_b. Σ_∈C(q_)+C(q'_)=1. Note that the
message dec_ is necessarily received by some process, otherwise C(q'_)=0 and C has no successor, which is in contradiction with the fact the the execution reaches C_f. Besides, C_i_j(1_)>0 hence v()>0. Moreover, the only
possible successor configuration is Cdec_ C_i_j+1, with C_i_j+1=C-q'_, ℓ_δ+, ℓ'. Hence, obviously, π(C_i_j)π(C_i_j+1).
* if C_i_jnbdec_C_i_j+1 then C_i_j+1=C_i_j-ℓ,1_+ℓ', for δ=(ℓ,,ℓ')∈Δ_nb. Σ_∈C(q_)+C(q'_)=0. Besides, C_i_j(1_)>0 hence v()>0. Hence, obviously, π(C_i_j)π(C_i_j+1).
* if C_i_j𝐧𝐛(nbdec_)C_i_j+1 then C_i_j+1=C_i_j-ℓ+ℓ' for δ=(ℓ,,ℓ')∈Δ_nb. Σ_∈C(q_)+C(q'_)=0. Besides, C_i_j(1_)=0 hence v()=0. Hence, obviously, π(C_i_j) π(C_i_j+1).
* if C_i_jτC_i_j+1 then C_i_j+1=C_i_j-ℓ+ℓ' for δ=(ℓ,,ℓ')∈Δ_nb. Σ_∈C(q_)+C(q'_)=0. Besides, C_i_j(1_)=C'_i_j+1(1_) for
all ∈. Hence, obviously, π(C_i_j)π(C_i_j+1).
* Otherwise, let C be the first configuration such that C(q)=1 and C_i_j^+C^*C_i_j+1. The transition leading to C is necessarily
a transition where the message L has been sent. Remember also that by induction hypothesis,
Σ_∈C_i_j(q_)+C_i_j(q'_)=0.
* if C_i_jLC, then C(q)=1, and by induction hypothesis, Σ_∈C(q_)+C(q'_)=0. Then the only possible successor configuration is CRC_i_j+1, with Σ_∈C_i_j+1(q_)+C_i_j+1(q'_)=0, and π(C_i_j+1)=(, v), so π(C_i_j)π(C_i_j+1), by a restore transition.
* if C_i_jinc_C_1LC then C_1=C_i_j-ℓ,+ℓ_δ,q_ for δ=(ℓ,,ℓ')∈Δ_b and Σ_∈C_1(q_)+C_1(q'_)=1. Now, C=C_1 - ℓ_δ, +
q_, q, so C(q)=1=C(q_), and Σ_∈C(q_)+C(q'_)=1.
* If CRC_i_j+1, then C_i_j+1 =
C - q,q_+,, then Σ_∈C_i_j+1(q_)+C_i_j+1(q'_)=0 and π(C_i_j+1)=(, v), hence
π(C_i_j)π(C_i_j+1) by a restore transition.
* Now C(q_)=1 so it might be that Cinc_ C', with C'=C - q_+1_. Here, Σ_∈C'(q_)+C'(q'_)=0. However, 𝚕𝚎𝚊𝚍𝚎𝚛(C')={q} so C' is not M-compatible. The only
possible transition from C' is now C'R C_i_j+1 with C_i_j+1=
C'-q+. Hence, C_i_j+1(1_)= C'(1_)=C_i_j(1_)+1=v()+1, and
C_i_j+1(1_)=C'(1_)=C_i_j(1_)=v() for all ≠.
So π(C_i_j)=(ℓ,v)δ (ℓ',v+v_)(, v+v_)=π(C_i_j+1), the last step being a restore transition. Finally, Σ_∈C_i_j+1(q_)+C_i_j+1(q'_)=0.
* if C_i_jdec_C_1L C, then C_1=C_i_j-ℓ,1_+ℓ_δ,q'_ for δ=(ℓ,,ℓ')∈Δ_b and Σ_∈C_1(q_)+C_1(q'_)=1. Now, C=C_1 - ℓ_δ, +
q_, q, so C(q)=1=C(q'_), and Σ_∈C(q_)+C(q'_)=1. Again, two transitions are available:
* If CRC_i_j+1, then C_i_j+1 =
C - q,q'_+,, then Σ_∈C_i_j+1(q_)+C_i_j+1(q'_)=0 and π(C_i_j+1)=(, v), hence
π(C_i_j)π(C_i_j+1) by a restore transition.
* Now C(q'_)=1 so it might be that Cdec_ C', with C'=C - q'_+. Here, Σ_∈C'(q_)+C'(q'_)=0. However, 𝚕𝚎𝚊𝚍𝚎𝚛(C')={q} so C' is not M-compatible. The only
possible transition from C' is now C'R C_i_j+1 with C_i_j+1=
C'-q+. Hence, C_i_j+1(1_)= C'(1_)=C_i_j(1_)-1=v()-1, and
C_i_j+1(1_)=C'(1_)=C_i_j(1_)=v() for all ≠.
So π(C_i_j)=(ℓ,v)δ (ℓ',v-v_)(, v+v_)=π(C_i_j+1), the last step being a restore transition. Finally, Σ_∈C_i_j+1(q_)+C_i_j+1(q'_)=0.
* If C_i_jinc_ C_1 then, it means that C_i_j()=0. In
that case, let δ=(ℓ,,ℓ')∈Δ_b, and C_1=C_i_j -ℓ+ℓ_δ. Since, by induction hypothesis, C_1(q_)=C_i_j()=0,
the only possible transition from C_1 would be C_1LC_i_j+1. However,
C_i_j()=C_1()=0, so this transition is not possible, and C_1 is a deadlock
configuration, a contradiction with the hypothesis that C_i_jC_i_j+1.
* If C_i_jdec_ C_1 then it means that C_i_j(1_)=0. In
that case, let δ=(ℓ,,ℓ')∈Δ_b, and C_1=C_i_j -ℓ+ℓ_δ. Since, by induction hypothesis, Σ_∈C_1(q_)+C_1(q'_) = Σ_∈C_i_j(q_)+C_i_j(q'_) = 0, the only
possible transition from C_1 is C_1LC, with C=C_1 - ,ℓ_δ + q, q_. Again, Σ_∈C(q_)+C(q'_) = 0,
and C(ℓ)= for all ℓ∈ Q_M, so the only possible transition is
CR C_i_j+1. Observe that C_i_j+1 is M-compatible,
with C_i_j+1()=1, and C_i_j+1(1_)=C_i_j(1_) for all ∈. Hence π(C_i_j+1)=(, v), and π(C_i_j)π(C_i_j+1), thanks to a restore transition of M.
We then have, by P(k), that (,0_)^*_M π(C_i_k),
with C_i_k M-compatible and such that C_i_k^* C_f, and C_i_k is the
last M-compatible configuration.
Then, by definition of an M-compatible configuration, C_i_k=C_f, and π(C_i_k)=(ℓ_f,v)
for some v∈ℕ^.
§ PROOF OF SECTION <REF>
We present here omitted proofs of <ref>.
§.§ Technical Lemma
We provide here a lemma which will be useful in different parts of this section.
Let be rendez-vous protocol and C,C' ∈ such that C=C_0 C_1 ⋯
C_ℓ=C'. Then we have the two following properties.
* For all q ∈ Q verifying
C(q)=2.ℓ+a for some a ∈, we have C'(q)≥ a.
* For all D_0 ∈ such that D_0 ≥ C_0, there exist D_1,…,D_ℓ such that D_0 D_1 ⋯
D_ℓ and D_i ≥ C_i for all 1 ≤ i ≤ℓ.
According to the semantics associated to (non-blocking) rendez-vous
protocols, each step in the execution from C to C' consumes at
most two processes in each control state q, hence the result of the
first item.
Let C,C' ∈ such that C C'. Let D ∈ such
that D ≥ C. We reason by a case analysis on the operation
performed to move from C to C' and show that there exists D'
such that D D' and D'≥ C'. (To obtain the final
result, we repeat k times this reasoning).
* Assume C m
C' then there exists (q_1, !m, q_1') ∈ T and
(q_2, ?m, q_2')∈ T such that C(q_1)>0 and C(q_2)>0 and C(q_1)+C(q_2)≥ 2 and
C' = C - q_1, q_2 + q_1', q_2'. But since D
≥ C, we have as
well D(q_1)>0 and D(q_2)>0 and D(q_1)+D(q_2)≥ 2 and as
a matter of fact D m
D' for D' = D - q_1, q_2 + q_1', q_2'. Since D≥ C, we
have D' ≥ C'.
* The case C τ C' can be treated in a
similar way.
* Assume C 𝐧𝐛(m) C', then there exists (q_1, !m, q_1') ∈ T, such
that C(q_1)>0 and (C-q_1)(q_2)=0 for all (q_2, ?m, q_2') ∈ T and
C' = C - q_1 + q'_1. We have as well that
D(q_1)>0. But we need to deal with two cases:
* If (D-q_1)(q_2)=0 for all (q_2, ?m, q_2') ∈
T. In that case we have D 𝐧𝐛(m) D'
for D' = D - q_1 + q'_1 and D' ≥ C'.
* If there exists (q_2, ?m, q_2') ∈
T such that (D-q_1)(q_2)>0. Then we have that D m
D' for D' = D - q_1, q_2 + q_1', q_2'. Note that
since (C-q_1)(q_2)=0 and D ≥ C, we have here again
D' ≥ C'.
§.§ Properties of Consistent Abstract Sets of Configurations
§.§.§ Proof of Lemma <ref>
Let C' ∈γ such that C' ≥ C. Let q ∈ Q
such that C(q)>0. Then we have C'(q)>0. If q ∉ S, then q
∈ and C'(q)=1 and C(q)=1 too. Furthermore for all q' ∈∖q such
C(q')=1, we have that C'(q')=1 and q and q' are
conflict-free. This allows us to conclude that C ∈γ. Checking whether C belongs to γ can be done in
polynomial time applying the definition of ·.
§.§.§ Building Configurations from a Consistent Abstract Set
Let γ be a consistent abstract set of configurations. Given a
subset of states U ⊆ Q, if for all N ∈ and for all
q ∈ U there exists C_q ∈γ and C'_q ∈ such
that C_q ^∗ C'_q and C'_q(q)≥ N, then for all N
∈, there exists C ∈γ and C' ∈ such that C ^∗
C' and C'(q) ≥ N for all q ∈ U.
We suppose γ=(S,) and reason by induction on the number of
elements in U∖ S. The base case is obvious. Indeed assume U
∖ S=∅ and let N∈. We define the configuration C
such that C(q)=N for all q ∈ S and C(q)=0 for all q ∈
Q∖ S. It is clear that C ∈γ and that C(q)
≥ N for all q ∈ U (since U
∖ S=∅, we have in fact U ⊆ S).
We now assume that the property holds for a set U and we shall see
it holds for U ∪p, p∉ S. We assume hence that for all N ∈ and for all
q ∈ U ∪p there exists C_q ∈γ and C'_q ∈ such
that C_q ^∗ C'_q and C'_q(q)≥ N. Let N ∈. By induction
hypothesis, there exists C_U ∈γ and C'_U ∈ such that C_U ^∗ C'_U and C_U'(q) ≥ N for all q
∈ U. We denote by ℓ_U the minimal number of steps in an execution from
C_U to C'_U. We will see that that we can build a configuration C
∈γ such that C ^∗ C”_U with C”_U
≥ C_U and C”_U(p) ≥ N+2*ℓ_U. Using Lemma <ref>, we will
then have that C”_U ^∗ C' with C' ≥ C'_U and C'(p)
≥ N. This will allow us to conclude.
We as well know that there exist C_p ∈γ and C'_p ∈ such
that C_p ^∗ C'_p and C'_p(p)≥
N+2*ℓ_U+(k*ℓ). We denote by ℓ_p the minimum number of steps in an execution from
C_p to C'_p. We build the configuration C as follows:
we have C(q)=C_U(q)+2*ℓ_p+(k*ℓ)+C_p(q) for all q ∈ S,
and we have C(q)=C_p(q) for all q ∈. Note that
since C_p ∈γ, we have that C ∈γ. Furthermore, we have C ≥ C_p, hence using
again Lemma <ref>, we know that there exists a
configuration C”_p such that C ^∗ C”_p and C”_p
≥ C'_p (i.e. C”_p(p) ≥ N+2*ℓ_U+(k*ℓ) and
C”_p(q) ≥ C_U(q)+(k*ℓ) + C_p(q) for all q ∈ S by <ref>,<ref>)
Having C_U ∈γ, we name (q_1, m_1) … (q_k,
m_k) the
tokens in such that C_U(q_j) = 1 for all 1 ≤ j ≤
k, and for all q ∈∖{q_j}_1 ≤ j ≤
k, C_U(q) =0. Since γ is consistent, for each (q_j,
m_j) there exists a path
(q_0,j,!m_j,q_1,j)(q_1,j,?m_1,j,q_2,j)…(q_ℓ_j,j,?m_ℓ_j,j,q_j)
in such that q_0,j∈ S and such that there
exists (q'_i,j,!m_i,j,q”_i,j) ∈ T with q'_i,j∈ S for
all 1 ≤ i ≤ℓ_j. We denote by ℓ = max_1 ≤ j≤
k(ℓ_j)+1.
Assume there exists 1≤ i≤ j≤ k such that (q_i,m_i),(q_j,m_j)∈ and C_U(q_i)=C_U(q_j)=1, and m_i∈q_j and m_j∈q_i.
Since C_U respects γ, q_i and q_j are conflict-free: there exist (q_i,m), (q_j,m')∈ such that m∉q_j and
m'∉q_i. Hence, (q_i,m_i), (q_i, m), (q_j,m_j), (q_j,m')∈, and m∉q_j and m_j∈q_i. Therefore, we have
(q_i,m), (q_j,m_j)∈ and m∉q_j and m_j∈q_i, which is in contradiction with the fact that γ is consistent. Hence, for all
1≤ i≤ j≤ k, for all (q_i,m_i), (q_j,m_j)∈, m_i∉q_j and m_j∉q_i.
We shall now explain how from C”_p we reach C”_U in k*ℓ
steps, i.e. how we put (at least) one token in
each state q_j such that q_j ∈ and C_U(q_j)=1
in order to obtain a configuration C”_U ≥ C_U. We begin by
q_1. Let a process on q_0,1 send the message m_1 (remember
that q_0,1 belongs to S) and let ℓ_1 other processes on
states of S send the messages needed for the process to reach
q_1 following the path
(q_0,1,!m_1,q_1,1)(q_1,1,?m_1,1,q_2,1)…(q_ℓ_1,1,?m_ℓ_1,1,q_1). At this stage, we have that the number of processes in each state q in S
is bigger than C_U(q)+((k-1)*ℓ) and we have (at least) one process in
q_1. We proceed similarly to put a process in q_2, note that the
message m_2 sent at the beginning of the path cannot be received by the
process in q_1 since, as explained above, m_2 ∉q_1.
We proceed again to put a process in the
states q_1 to q_K and at the end we obtain the configuration
C”_U with the desired properties.
§.§ Proof of Lemma <ref>
In this subsection, the different items of Lemma <ref> have been separated in distinct lemmas.
F(γ) is consistent and can be computed in polynomial time
for all consistent γ∈Γ.
The fact that F(γ) can be computed in polynomial time is a direct consequence of the definition of F (see <ref>).
Assume γ = (S,) ∈Γ to be consistent. Note (S”, ”) the intermediate sets computed during the computation of F(γ), and note F(γ) = (S', ').
To prove that F(γ) is consistent, we need to argue that (1) for all (q, m) ∈”∖, there exists a finite sequence of transitions (q_0, a_0, q_1) … (q_k, a_k, q) such that q_0 ∈ S, and a_0 = !m and for all 1 ≤ i≤ k, we have that a_i = ?m_i and that there exists (q'_i, !m_i, q'_i+1) ∈ T with q'_i ∈ S, and (2) for all (q,m), (q',m') ∈' either m∈q' and m'∈q or m∉q' and m'∉q.
We start by proving property (1).
If (q, m) has been added to ” with rule <ref>, then by construction, there exists p ∈ S such that (p, !a, p') ∈ T, and (q, m) = (p', a). The sequence of transition is the single transition is (p, !a, q).
If (q, m) has been added to ” with rule <ref>, then there exists (q',m) ∈, and (q', ?a, q) with m a. Furthermore, m ∈q and there exists (p, !a,p') ∈ T with p ∈ S. By hypothesis, γ is consistent, hence there exists a finite sequence of transitions (q_0, q_0, q_1) … (q_k, a_k, q') such that q_0 ∈ S, and a_0 = !m and for all 1 ≤ i≤ k, we have that a_i = ?m_i and that there exists (q'_i, !m_i, q'_i+1) ∈ T with q'_i ∈ S. By completing this sequence with transition (q', ?a, q) we get an appropriate finite sequence of transitions.
It remains to prove property (2). Assume there exists (q, m), (q',m') ∈' such that m ∈q' and m' ∉q, then as ' ⊆”, (q, m), (q',m') ∈”. By condition <ref>, q ∈ S', therefore, as ' = {(p, a) ∈”| p ∉ S'}, we have that (q, m) ∉', and we reached a contradiction.
If (S',')=F(S,) then S ⊊ S' or
⊆'.
From the construction of F (see <ref>), we have S ⊆ S”⊆ S'.
Assume now that S=S'. First note that ⊆” (see Table <ref>) and that ∩ S=∅. But '=(q,m) ∈”| q ∉S'=(q,m) ∈”| q ∉S. Hence the elements that are removed from ” to obtain ' are not elements of . Consequently ⊆'.
For all consistent γ∈Γ, if C ∈γ and C C' then C' ∈F(γ).
Let γ = (S,)∈Γ be a consistent abstract set of configurations, and C ∈ such that C ∈γ and C C'. Note F(γ) = (S', ') and γ' = (S”, ”) the intermediate sets used to compute F(γ).
We will first prove that for all state q such that C'(q) > 0, q ∈ S' or q ∈('), and then we will prove that for all states q such that q ∈(') and C'(q)>0, C'(q) = 1 and for all other state p∈(') such that C'(p) >0, p and q are conflict-free.
Observe that S ⊆ S”⊆ S', ⊆”, and (”) ⊆(') ∪ S'.
First, let us prove that for every state q such that C'(q)>0, it holds that q ∈ S' ∪('). Note that for all q such that C(q) > 0, because C respects γ, q ∈() ∪ S. As () ∪ S ⊆(') ∪ S', the property holds for q.
Hence, we only need to consider states q such that C(q) = 0 and C'(q) > 0. If C τ C' then q is such that there exists (q', τ, q) ∈ T, q' is therefore an active state and so q' ∈ S, (recall that ⊆ Q_W ×Σ). Hence, q should be added to (”) ∪ S” by condition <ref>. As (”) ∪ S”⊆(') ∪ S', it concludes this case. If C a C' then q is such that there exists (q', !a, q) ∈ T, with q' an active state. With the same argument, q' ∈ S and so q should be added to (”) ∪ S” by condition <ref> or <ref>. If C a C', then q is either a state such that (q', !a, q) ∈ T and the argument is the same as in the previous case, or it is a state such that (q', ?a, q) ∈ T, and it should be added to (”)∪ S” by condition <ref>, <ref>, or <ref>. Therefore, we proved that for all state q such that C'(q) >0, it holds that q ∈(') ∪ S'.
It remains to prove that if q ∈(), then C'(q) = 1 and for all q' ∈(') ∖{q} such that C'(q') = 1, we have that q and q' are conflict-free. Note that if q ∈() and C(q) = C'(q) = 1, then for every state p such that p ∈() and C(p) = C'(p) = 1, it holds that q and p are conflict-free.
Observe that if C τ C', then note q the state such that (q', τ ,q), it holds that {p | p ∈(') and C'(p) > 0}⊆{p | p ∈() and C(p) = 1}: q' is an active state, q might be in () but it is added to S”⊆ S' with rule <ref>, and for all other states, C'(p) = C(p). If p ∈(') and C(p) > 0, it implies that C'(p)= C(p) = 1 and p∈() (otherwise p is in S ⊆ S'). Hence, there is nothing to do as C respects γ.
Take now q ∈(') ∖() with C'(q) > 0, we shall prove that C'(q) =1 and for all p ∈(') and C'(p) > 0, q and p are conflict-free. If q ∈(') ∖(), it implies that C(q) = 0 because C respects γ. Hence: either (1) C a C' with transition (q', !a, q) ∈ T, either (2) C a C' with transitions (q_1, !a, q'_1) ∈ T and (q_2, ?a, q'_2) ∈ T and q = q'_1 or q=q'_2. In the latter case, we should be careful as we need to prove that q'_2 q'_1, otherwise, C'(q) = 2.
Case (1): Note that as only one process moves between C and C' and C(q)= 0, it is trivial that C'(q) = 1. In this first case, as it is a non-blocking request on a between C and C', it holds that: for all p ∈() such that C(p) = 1, a ∉p. Take p ∈('), such that p q and C'(p) = 1, then C'(p) = C(p) = 1 and so p ∈(), and a ∉p. Suppose (p, m) ∈' such that m ∈q, then we found two tokens in ' such that m ∈q and a ∉p which contradicts F(γ)'s consistency. Hence, p and q are conflict-free.
Case (2): Note that if q'_2 ∈('), then q_2 ∈() (otherwise, q'_2 should be in S' by condition <ref>), and note (q_2, m) ∈, with (q'_2, m) ∈'. Note as well that if q'_1 ∈('), then a ∈q'_1 (otherwise, q'_1 should be in S' by condition <ref>) and (q'_1 ,a) ∈' by condition <ref>. Furthermore, if q'_1 ∈('), q_2 ∈() as well as otherwise q'_1 should be added to S' by condition <ref>.
We first prove that either q'_1 ∈ S', or q'_2 ∈ S'. For the sake of contradiction, assume this is not the case, then there are three tokens (q'_1, a), (q_2, m), (q'_2, m) ∈' ⊆”, such that (q_2, ?a, q'_2) ∈ T. From condition <ref>, q'_1 should be added to S' and so (q'_1, a) ∉'. Note that, as a consequence q'_1 q'_2 or q'_1 = q'_2 ∈ S'. Take q ∈(') ∖() such that C'(q) >0, if such a q exists, then q = q'_1 or q = q'_2 and q'_1 q'_2. As a consequence, C'(q) = 1 (note that if q'_1 = q_2, C(q_2) = 1).
Take p ∈(') ∖{q} such that C'(p) > 0, it is left to prove that q and p are conflict-free. If p q and p ∈('), then C'(p) = C(p) (because q'_1 ∈ S' or q'_2 ∈ S'). Hence, p ∈() and C'(p) = 1.
Assume q = q'_1 and assume q and p are not conflict-free. Remember that we justified that q_2 ∈(), and therefore, C(q_2) = 1. Hence, either C'(q_2) = 0, or q_2 = q'_2 and in that case q_2,q_2' ∈ S' or q_2' = q_1' and then q_2=q. In any case, p q_2. As C respects γ, there exists (p, m_p) and (q_2, m) ∈ such that m_p ∉q_2 and m ∉p (q_2 and p are conflict-free). As p ∈('), (p,m_p) ∈' and so m_p∈q or a ∈p (q and p are not conflict-free).
As F(γ) is consistent, m_p∈q and a ∈p.
Note that a m_p because a ∈q_2, a m because m ∉p, and obviously m m_p. Note also that if m ∉q, then we found two tokens (q,a) and (q_2,m) in ' such that a ∈q_2 and m ∉q, which contradicts the fact that F(γ) is consistent (Lemma <ref>). Hence, m∈q.
Note that even if q_2 is added to S”, it still is in ”. As ' ⊆” we found three tokens (p, m_p), (q_2,m), (q, a) in ”, satisfying condition <ref>, and so p should be added to S', which is absurd as p ∈('). We reach a contradiction and so q and p should be conflict-free.
Finally assume q = q_2'. If q = q_2, then, because C respects γ, q and p are conflict-free. Otherwise, as q_2 is conflict-free with p, there exists (q_2, m ) and (p, m_p) in such that m ∉p and m_p ∉q_2. Note that (q,m) ∈” from condition <ref> (otherwise, q ∈ S” which is absurd). Hence, (q, m) ∈' and, as p ∈('), (p,m_p) is conserved from to '. It remains to show that m_p ∉q. Assume this is not the case, then there exists (p,m_p) and (q,m) ∈' such that m∉p and m_p∈q which is absurd given F(γ)'s consistency.
As a consequence, q and p are conflict-free.
We managed to prove that for all q such that C'(q) >0, q ∈ S' ∪('), and if q ∈('), then C'(q) = 1 and for all others p∈(') such that C'(p) = 1, p and q are conflict-free.
For all consistent γ∈Γ, if C' ∈F(γ), then there exists C”∈ and C ∈γ such that C”≥ C' and C ^∗ C”.
Let γ be a consistent abstract set of configurations and C'∈F(γ). We suppose that γ=(S,) and F(γ)=γ'=(S','). We will first show that for all N ∈, for all q ∈ S' there exists a configuration C_q ∈γ and a configuration C_q' ∈ such that C_q ^∗ C_q' and C'_q(q) ≥ N. This will allow us to rely then on Lemma <ref> to conclude.
Take N ∈ and q ∈ S', if q ∈ S, then take C_q ∈γ to be N · q. Clearly C_q ∈F(γ), C_q(q) ≥ N and C_q ^∗ C_q. Now let q ∈ S' ∖ S. Note (”, S”) the intermediate sets of F(γ)'s computation.
Case 1: q ∈ S”. As a consequence q was added to S” either by one of the conditions <ref>, <ref>, <ref> or <ref>. In cases <ref> and <ref> when a ∉q, note q' the state such that (q', τ, q) or (q', !a, q), and consider the configuration C_q = N · q'. By doing N internal transitions or non-blocking requests, we reach C'_q= N · q. Note that the requests on a are non-blocking as q' ∈ Q_A and a ∉q. C'_q ∈F(γ).
In cases <ref> with a∈q and in case <ref>, note (q_1, !a, q_1') and (q_2, ?a, q_2') the two transitions realizing the conditions. As a consequence q_1, q_2 ∈ S. Take the configuration C_q =N · q_1, N · q_2. C_q ∈γ and by doing N successive rendez-vous on the letter a, we reach configuration C'_q = N· q'_1 + N · q'_2. C'_q ∈F(γ), and as q ∈{q'_1, q'_2}, C'_q(q) ≥ N.
In case <ref>, there exists (q', m) ∈ such that (q', ?a, q) ∈ T, m ∉q, and there exists p ∈ S such that (p, !a,p') ∈ T. Remember that γ is consistent, and so there exists a finite sequence of transitions (q_0, !m, q_1) (q_1, a_1, q_2) … (q_k, a_k, q') such that q_0 ∈ S and for all 1 ≤ i ≤ k, a_i = ?m_i and there exists (q'_i , !m_i, q”_i) ∈ T with q'_i ∈ S.
Take C_q = (N-1) · q_0 + (N-1) · q'_1 + … + (N-1) · q'_k + N · p + q'. Clearly C_q ∈γ as all states except q' are in S and q' ∈(), C_q(q') = 1. We shall show how to put 2 processes on q from C_q and then explain how to repeat the steps in order to put N. Consider the following execution: C_q a C_1 x_m C_2 m_1…m_k C_k+2a C_k+3. The first rendez-vous on a is made with transitions (p, !a, p') and (q', ?a, q). Then either m ∉p' and x_m = m, otherwise, x_m = m, in any case, the rendez-vous or non-blocking sending is made with transition (q_0, !m, q_1) and the message is not received by the process on q (because m ∉q) and so C_2 ≥q + q_1. Then, each rendez-vous on m_i is made with transitions (q'_i, !m_i,q”_i) and (q_i, ?m_i, q_i+1) (q_k+1 = q'), . Hence C_k+3≥(N-2)· q_0+ (N-2) · q'_1 + … + (N-2) · q'_k + (N-2) · p + 2 · q. We can reiterate this execution (without the first rendez-vous on a) N-2 times to reach a configuration C'_q such that C'_q ≥N · q.
Case 2: q ∉ S”. Hence, q should be added to S' by one of the conditions <ref>, <ref>, and <ref>.
If it was added with condition <ref>, let (q_1, m_1), (q_2, m_2) ∈” such that q =q_1, m_1 m_2, m_2 ∉q_1 and m_1 ∈q_2.
From the proof of Lemma <ref>, one can actually observe that all tokens in ” correspond to "feasible" paths regarding states in S, i.e there exists a finite sequence of transitions (p_0, !m_1, p_1) (p_1, a_1, p_2) … (p_k, a_k, q_1) such that p_0 ∈ S and for all 1 ≤ i ≤ k, a_i = ?b_i and there exists (p'_i , !b_i, p”_i) ∈ T with p'_i ∈ S. The same such sequence exists for the token (q_2, m_2), we note the sequence (s_0, !m_2, s_1)… (s_ℓ, a_ℓ, q_2) such that s_0 ∈ S and for all 1 ≤ i ≤ℓ, a_i = ?c_i and there exists (s'_i , !c_i, s”_i) ∈ T with s'_i ∈ S. Take C_q = N · p_0 + N · s_0 + N p'_1 + … + N p'_k + N · s'_1 + … + N · s'_ℓ. Clearly, C_q ∈γ, as all states are in S. Consider the following execution: C_q m_1 C_1 b_1…b_k C_k+1, the non-blocking sending of m_1 is made with transition (p_0, !m_1, p_1) and each rendez-vous on letter b_i is made with transitions (p'_i, !b_i, p_i”) and (p_i, ?b_i, p_i+1) (p_k+1 = q_1). Hence, C_k+1 is such that C_k+1≥q_1. From C_k+1, consider the following execution: C_k+1x_m_2 C_k+2c_1…c_ℓ C_k+ℓ +2m_1C_k+ℓ +3, where x_m_2 = m_2 if no process is on a state in R(m_2), or x_m_2 = m_2 otherwise. In any case, as m_2 ∉q_1, C_k+2≥q_1. And each rendez-vous on letter c_i is made with transitions (s'_i, !c_i, s_i”) and (s_i, ?c_i, s_i+1) (s_k+1 = q_2), the last rendez-vous on m_1 is made with transitions (p_0, !m_1, p_1) and (q_2, ?m_1, q_2') (such a q_2' exists as m_1 ∈q_2). Hence, C_k+ℓ +3≥p_1 + q_1. By repeating the two sequences of steps (without the first non-blocking sending of m_1) N-1 times (except for the last time where we don't need to repeat the second execution), we reach a configuration C'_q such that C'_q≥N · q_1.
If it was added with condition <ref>, then let (q_1, m_1), (q_2,m_2), (q_3,m_2) ∈” such that m_1 m_2 and (q_2, ?m_1, q_3) ∈ T with q =q_1. From the proof of Lemma <ref>, ” is made of "feasible" paths regarding S and so there exists a finite sequence of transitions (p_0, !m_2, p_1) (p_1, a_1, p_2) … (p_k, a_k, q_2) such that p_0 ∈ S and for all 1 ≤ i ≤ k, a_i = ?b_i and there exists (p'_i , !b_i, p”_i) ∈ T with p'_i ∈ S. The same sequence exists for the token (q_1, m_1), we note the sequence (s_0, !m_1, s_1)… (s_ℓ, a_ℓ, q_1) such that s_0 ∈ S and for all 1 ≤ i ≤ℓ, a_i = ?c_i and there exists (s'_i , !c_i, s”_i) ∈ T with s'_i ∈ S. Take C_q = N · p_0 + N · s_0 + N p'_1 + … + N p'_k + N · s'_1 + … + N · s'_ℓ. Clearly, C_q ∈γ, as all states are in S. We do the same execution from C_q to C_k+1 as in the previous case: C_q m_2 C_1 a_1…a_k C_k+1. Here C_k+1 is then such that C_k+1≥q_2. Then, from C_k+1 we do the following: C_k+1m_1 C_k+2c_1…c_ℓ C_k+ℓ+2m_2 C_k+ℓ+3: the rendez-vous on letter m_1 is made with transitons (s_0, !m_1, s_1) and (q_2, ?m_1, q_3). Then, each rendez-vous on letter c_i is made with transitions (s'_i, !c_i, s_i”) and (s_i, ?c_i, s_i+1) (s_k+1 = q_1), and the last rendez-vous on letter m_2 is made with transitions (p_0, !m_2, p_1) and (q_3, ?m_2,q_3') (such a state q_3' exists as (q_3, m_2) ∈” and so m_2∈q_3). Hence, C_k+ℓ+3 is such that C_k+ℓ +3≥q_1 + p_1. We can repeat the steps from C_1 N-1 times (except for the last time where we don't need to repeat the second execution), to reach a configuration C'_q such that C'_q≥N · q_1.
pas encore relu condition 8If it was added with condition <ref>, then let (q_1, m_1), (q_2, m_2), (q_3, m_3) ∈”, such that m_1 m_2, m_2 m_3, m_1 m_3, and m_1 ∉q_2, m_1 ∈q_3, and m_2 ∉q_1, m_2 ∈q_3 and m_3 ∈q_2 and m_3 ∈q_1, and q_1 = q. Then there exists three finite sequences of transitions (p_0, !m_1, p_1) (p_1, ?b_1, p_2) … (p_k, ?b_k, p_k+1), and (s_0, !m_2, s_1) (s_1, ?c_1, s_2) … (s_ℓ, ?c_k, s_ℓ +1), and (r_0, !m_3, r_1) (r_1, ?d_1, r_2) … (r_j, ?d_j, r_j+1) such that p_k+1 = q_1, s_ℓ +1 = q_2 and r_j+1 = q_3, and for all messages a ∈{ b_i_1, c_i_2, d_i_3}_1 ≤ i_1 ≤ k, 1 ≤ i_2 ≤ℓ, 1 ≤ i_3 ≤ j = M, there exists q_a∈ S such that (q_a, !a, q'_a). Take C_q = Np_0 + Ns_0 + Nr_0 + ∑_a ∈ MNq_a. From C_q there exists the following execution: C_q m_1 C_1 b_1…b_k C_k +1 where the non-blocking sending is made with the transition (p_0, !m_1, p_1) and each rendez-vous with letter b_i is made with transitions (q_b_i, !b_i, q'_b_i) and (p_i, ?b_i, p_i+1). Hence, C_k+1≥q_1.
Then, we continue the execution in the following way:
C_k+1x_m_2 C_k+2c_1…c_ℓ C_k+ ℓ +2 where
x_m_2 = m_2 if there is no process on R(m_2), and x_m_2 = m_2 otherwise. In any case, the rendez-vous is not answered by a process on state q_1 because m_2 ∉q_1.
Furthermore, each rendez-vous with letter c_i is made with transitions (q_c_i, !c_i, q'_c_i) and (s_i, ?c_i, s_i+1). Hence, C_k +ℓ+2≥q_2 + q_1. From C_k+ℓ +2 we do the following execution: C_k+ℓ +2m_3 C_k+ℓ +3d_1…d_j C_k +ℓ + j +3 where the rendez-vous on letter m_3 is made with transitions (r_0, !m_3, r_1) and (q_2, ?m_3, q_2') (this transition exists as m_3 ∈q_2). Each rendez-vous on d_i is made with transitions (q_d_i, !d_i, q'_d_i) and (r_i, ?d_i, r_i+1). Hence, the configuration C_k+ ℓ +j+3 is such that C_k+ℓ +j +3≥q_3 + q_1. Then from C_k+ℓ +j +3: C_k+ℓ + j +3m_1 C_k+ℓ + j +4 where the rendez-vous is made with transitions (p_0, !m_1, p_1) and (q_3, ?m_1, q'_3) (this transition exists as m_1 ∈q_3). By repeating N-1 times the execution from configuration C_1, we reach a configuration C'_q such that C'_q(q_1) ≥ N.
Hence, for all N ∈ℕ, for all q ∈ S', there exists C_q ∈γ, such that C_qC'_q and C'_q(q) ≥ N. From Lemma <ref>, there exists C'_N and C_N ∈γ such that C_N ^∗ C'_N and for all q ∈ S', C_N(q) ≥ N.
Take C' ∈F(γ), we know how to build for any N ∈, a configuration C'_N such that C'_N(q) ≥ N for all states q ∈ S' and there exists C_N ∈γ, such that C_N ^∗ C'_N, in particular for N bigger than the maximal value C'(q) for q ∈ S', C'_N is greater than C'_N on all the states in S'.
To conclude the proof, we need to prove that from a configuration C'_N' for a particular N', we can reach a configuration C” such that C”(q) ≥ C'(q) for q ∈ S' ∪('). As C' respects F(γ), remember that for all q ∈('), C'(q) = 1. The execution is actually built in the manner of the end of the proof of Lemma <ref>.
Note N_max the maximum value for any C'(q).
We enumerate states q_1, …, q_m in (') such that C'(q_i) = 1. As C' respects F(γ), for i j, q_i and q_j are conflict free.
From Lemma <ref>, F(γ) is consistent, and so we note (p^j_0, !m^j, p^j_1) (p^j_1, ?m^j_1, p^j_2) … (p^j_k_j, ?m^j_k_j, p^j_k_j+1) the sequence of transitions associated to state q_j such that: p^j_k_j+1 = q_j, (q_j, m^j) ∈ and for all m^j_i, there exists (q_m^j_i, !m_i^j, q'_m^j_i) with q_m^j_i∈ S'.
Note that for all i j, q_i and q_j are conflict-free and so there exists (q_i, m), (q_j,m') ∈' such that m ∉q_j and m' ∉q_i. As F(γ) is consistent, it should be the case for all pairs of tokens (q_i, a), (q_j, a'). Hence m^j ∉q_i and m^i ∉q_j.
Note ℓ_j = k_j + 1.
For N' = N_max + ∑_1≤ j ≤ mℓ_j, there exists a configuration C'_N' such that there exists C_N'∈γ, C_N'^*C'_N', and C'_N'(q) ≥ N' for all q ∈ S'. In particular, for all q ∈ S', C'_N'(q) ≥ C'(q) + ∑_1≤ j ≤ mℓ_j.
Then, we still have to build an execution leading to a configuration C” such that for all q ∈('), C”(q) ≥ C'(q). We then use the defined sequences of transitions for each state q_j. With ℓ_1 processes we can reach a configuration C_1 such that C_1(q_1) ≥ 1: C_1 x_m^1 C_2 m_1^1…m_k_1^1 C_ℓ_1+ 1. x_m^1 = m^1 if there is no process on R(m^1), and x_m^1 = m^1 otherwise. Each rendez-vous on m_i^1 is made with transitions (p_i^1, ?m_i^1, p_i+1^1) and (q_m_i^1, ! m_i^1, q'm_i^1). As a result, for all q ∈ S', C_ℓ_1+1(q) ≥ C'(q) +∑_2≤ j ≤ mℓ_j and C_ℓ_1 +1(q_1) ≥ 1. We then do the following execution form C_ℓ_1 + 1: C_ℓ_1 +1x_m^2 C_ℓ_1+2m_1^2…m_k_2^2 C_ℓ_1+ ℓ_2+ 2. x_m^2 = m^2 if there is no process on R(m^2), and x_m^2 = m^2 otherwise. Remember that we argued that m^2 ∉q_1, and therefore C_ℓ_1 + 2(q_1) ≥ C_ℓ_1 +1(q_1) ≥ 1.
Each rendez-vous on m_i^2 is made with transitions (p_i^2, ?m_i^2, p_i+1^2) and (q_m_i^2, ! m_i^2, q'm_i^2). As a result, C_ℓ_1+ℓ_2 +2(q) ≥ C'(q) +∑_3≤ j ≤ mℓ_j for all q ∈ S' and C_ℓ_1+ ℓ_2 + 2≥q_1 + q_2. We can then repeat the reasoning for each state q_i and so reach a configuration C” such that C”(q) ≥ C'(q) for all q ∈ S' and, C”≥q_1 + q_2 + …q_m.
We built the following execution: C_N'^∗ C'_N'^∗ C”, such that C”≥ C', and C'_N'∈γ.
§.§ Proof of Lemma <ref>
Assume that there exists C_0 ∈ and C' ≥ C
such that C_0 C_1 … C_ℓ =C'. Then
using the Lemma <ref> iteratively, we get that C'
∈γ_ℓ. From the definition of F and ·,
one can furthermore easily check that γ⊆F(γ) for all γ∈Γ. Hence we have
γ_ℓ⊆γ_f and C' ∈γ_f.
Before proving the other direction, we first prove by induction
that for all i ∈ and for all D ∈γ_i, there
exists C_0 ∈ and D' ≥ D such that C_0 ^∗
D'. The base case for i=0 is obvious. Assume the property holds
for γ_i and let us show it is true for γ_i+1. Let E ∈γ_i+1. Since γ_i+1=F(γ_i), using Lemma
<ref>, we get that there exists E' ∈ and D ∈γ_i such that E' ≥ E and D ^∗
E'. By the induction hypothesis, there exist C_0 ∈ and D'
≥ D such that C_0 ^∗ D'. Using the monotonicity
property stated in Lemma <ref>, we deduce that there
exists E”∈ such that E”≥ E' ≥ E and C_0
^∗ D' ^∗ E”.
Suppose now that there exists C”∈γ_f such that C”≥ C. By the previous
reasoning, we get that there exist C_0 ∈ and C' ≥ C”≥ C
such that C_0 ^∗ C'.
|
http://arxiv.org/abs/2307.07527v1 | 20230712102019 | Machine Learning for Autonomous Vehicle's Trajectory Prediction: A comprehensive survey, Challenges, and Future Research Directions | [
"Vibha Bharilya",
"Neetesh Kumar"
] | cs.LG | [
"cs.LG",
"cs.AI"
] |
Machine Learning for Autonomous Vehicle's Trajectory Prediction: A comprehensive survey, Challenges, and Future Research Directions
Vibha Bharilya, and Neetesh Kumar, Member, IEEE
Vibha Bharilya is with Department of Computer Science and Engineering, Indian Institute of Technology, Roorkee, Uttarakhand- 247667, India (e-mail: [email protected]).
Neetesh Kumar is with Department of Computer Science and Engineering, Indian Institute of Technology, Roorkee, Uttarakhand- 247667, India (e-mail: [email protected]).
===============================================================================================================================================================================================================================================================================================================================================================================================================================
The significant contribution of human errors, accounting for approximately 94% (with a margin of ±2.2%), to road crashes leading to casualties, vehicle damages, and safety concerns necessitates the exploration of alternative approaches. Autonomous Vehicles (AVs) have emerged as a promising solution by replacing human drivers with advanced computer-aided decision-making systems. However, for AVs to effectively navigate the road, they must possess the capability to predict the future behavior of nearby traffic participants, similar to the predictive driving abilities of human drivers. Building upon existing literature is crucial to advance the field and develop a comprehensive understanding of trajectory prediction methods in the context of automated driving. To address this need, we have undertaken a comprehensive review that focuses on trajectory prediction methods for AVs, with a particular emphasis on machine learning techniques including deep learning and reinforcement learning-based approaches. We have extensively examined over two hundred studies related to trajectory prediction in the context of AVs. The paper begins with an introduction to the general problem of predicting vehicle trajectories and provides an overview of the key concepts and terminology used throughout. After providing a brief overview of conventional methods, this review conducts a comprehensive evaluation of several deep learning-based techniques. Each method is summarized briefly, accompanied by a detailed analysis of its strengths and weaknesses. The discussion further extends to reinforcement learning-based methods. This article also examines the various datasets and evaluation metrics that are commonly used in trajectory prediction tasks. Encouraging an unbiased and objective discussion, we compare two major learning processes, considering specific functional features. By identifying challenges in the existing literature and outlining potential research directions, this review significantly contributes to the advancement of knowledge in the domain of AV trajectory prediction. Its primary objective is to streamline current research efforts and offer a futuristic perspective, ultimately benefiting future developments in the field.
Autonomous Vehicle, trajectory prediction, machine learning, deep learning, reinforcement learning
§ INTRODUCTION
[findent=2pt]Annually, approximately 1.35 million deaths occur due to road crashes, with 1,140 reported deaths in 2018 according to the Australian Automobile Association (AAA) <cit.>. There were 1,194 fatal car accidents in 2022 in Australia. This is an increase of 5.8% from 2021. National fatalities have stayed basically flat during the past ten years <cit.>. In the United States, the NHTSA's investigation reveals that around 94% of severe road crashes can be attributed to driver errors [4]. Further, human error is consistently identified as a major factor in road crashes, emphasizing the need to address this preventable distress. To assist human drivers in avoiding critical situations, advanced motorized vehicles employ Advanced Driver Assistance Systems (ADAS), which have rapidly evolved since their inception in the 1950s. Researchers are actively exploring the efficiency of ADAS in warning drivers and preventing crashes. The rapid technological progress, including the use of high-end sensors, powerful machine learning techniques, and innovations from companies like Google and Tesla Motors, has significantly impacted the automation industry. Automotive and tech companies have demonstrated the feasibility of Automated Driving Systems (ADS) through successful test fleet operations. The Society of Automotive Engineers (SAE) classifies ADS into six levels of vehicle automation <cit.>, with a focus on full automated operation.
Autonomous Vehicles (AVs) are expected to play a significant role in reducing crashes and enhancing road safety in the foreseeable future.
The rapid development of perception, planning, and control systems for AVs in recent years is noteworthy. However, the production of AVs in large quantities will not be feasible until their safety is fully established. One of the critical technologies in AVs is the ability to forecast the future states of the surrounding environment in real time, as human drivers can. This capability will further enhance safety measures.
Before beginning a new driving operation, such as acceleration or a lane change, a human driver typically scans the surrounding traffic to predict how it will behave in the future. Future trajectories can be used to model future traffic participant states, which can then be used to construct decision-making or planning algorithms as well as to foresee potential dangers. However, accurately predicting future traffic participant trajectories is attracting a lot of attention and is quickly becoming one of the key points to improving the safety of autonomous driving. This is because of the variety of maneuvers that traffic participants make, the complex interactions between traffic participants and environments, the uncertainty of sensory information, the computation burdens, and the computing time requirements of AVs.
§.§ Motivation
Numerous techniques have been developed for the trajectory prediction, and several scholars have pursued this area of research. Some of the review papers have covered various trajectory prediction techniques, in the same line, Lefèvre et al. <cit.> provided an analysis of motion prediction and risk assessment techniques used for AVs before 2014. Mohammad et al. <cit.> discussed strategies for behavior prediction at crossings based on drivers' actions. Further, Mozaffari et al. <cit.> offered a review of deep learning-based approaches focused on vehicle behavior analysis. Leon et al. <cit.> and Liu et al. <cit.> wrote reviews on trajectory prediction for AVs, where, Leon et al. covered deep learning and stochastic methods, and Liu et al. focused solely on deep learning methods. Karle et al. <cit.> offered three distinct prediction models as a classification of these models and compared them based on the underlying study methodology. Gomes et al. <cit.> reviewed the literature on Intention-Aware and Interaction-Aware trajectory prediction for autonomous vehicles and examined how maneuver goals and their interaction with other maneuvers affect the performance of trajectory prediction techniques. In Ghorai et al. <cit.>, a survey covered the identification and monitoring of dynamic agents and objects encountered by an autonomous ego vehicle. The main topics of the review were delved into 2D and 3D dynamic object identification techniques based on DL employed in AV research. Huang et al. <cit.> thoroughly examined trajectory prediction techniques for AVs put forth over the last two decades, excluding vision-based techniques. Recently, Benrachou et al. <cit.> encompassed research on both data-driven and model-based algorithms, which aim to forecast the movement of surrounding traffic. Table I provides a summary of the related state-art-of surveys, along with the different categorizations approaches, and contributions.
Motion prediction involves anticipating the behavior, maneuvers, or trajectory of an object, depending on the desired level of abstraction. The term "behavior" encompasses general actions and their execution style, such as "following the road and maintaining a safe distance." On the other hand, "maneuvers" refer to discrete actions that an object can perform without requiring a detailed specification, such as "turning right." Trajectories, on the other hand, provide the most detailed type of prediction by describing an object's position over discrete time steps <cit.>. Previous surveys have predominantly emphasized motion prediction and behavior prediction in the realm of Autonomous Vehicles (AVs). Further, other state-of-the-art surveys are mixed of trajectory prediction for vehicles and pedestrians. A comprehensive and dedicated review on autonomous vehicles trajectory prediction accounting machine learning methods is remained relatively unexplored. Furthermore, several advancements in the domain of trajectory prediction, in recent years, including computer vision-based methods, reinforcement learning etc., have not been addressed in the existing surveys which are also needed to be explored. Consequently, there exists substantial potential for further exploration and investigation within this domain. Thus, the motivation behind writing this survey paper is to actively contribute to the research in the trajectory prediction field specifically for AVs.
§.§ Google Trends
In recent years, Autonomous Driving (AD) has become increasingly popular in the automotive industry. Prominent automobile manufacturers, including Tesla, General Motors, and BMW, have made significant investments and focused on trajectory prediction and related technologies for Autonomous Vehicles (AVs) for the development of AD technology. The worldwide search trends for keywords related to AD, such as trajectory prediction, driving assistance, behavior prediction, trajectory planning, and obstacle avoidance, are illustrated in Fig. 1 and Fig. 2. In Fig. 1, the comparison of related keywords demonstrates changes over the same time period. It shows that driving assistance and obstacle avoidance generate similar levels of excitement worldwide. However, within the field of AD, there have been recent advancements in keywords such as trajectory prediction, trajectory planning, and behavior prediction. Notably, trajectory prediction focuses on a more specific domain within autonomous driving. In Fig. 2, the average number of searches related to the keyword worldwide is depicted. Notably, driving assistance and obstacle avoidance keywords receive a higher number of searches compared to other terms like trajectory prediction, trajectory planning, and behavior prediction. Trajectory prediction is currently evolving in the field of AD, indicating increasing interest and development in this area.
§.§ Key Contributions
This comprehensive survey on the state-of-the-art machine learning-based trajectory prediction methods for Autonomous Vehicles (AVs) provides a taxonomy of the different approaches, as shown in Fig. 3, including conventional methods, deep learning-based methods, and reinforcement learning-based methods, and discusses the advantages and limitations of each method. This study focuses on vehicle trajectory prediction algorithms, as other traffic participants, such as adjacent vehicles, directly affect the ego vehicle.
In the end, the paper highlights the challenges and future research directions in this field.
The significant contributions of this survey are enlisted as follows:
* This survey offers an empirical study on autonomous vehicles trajectory prediction methods and extensively focuses on machine learning-based methods. For the better understanding, an overview on AV's trajectory prediction problem, related terminology, and conventional methods are also briefly provided.
* A concise assessment of conventional methods such as Physics-based methods, Sampling methods, and Probabilistic models in trajectory prediction is presented, along with a discussion of their advantages and disadvantages.
* A comprehensive evaluation is provided for the prevalent deep learning and reinforcement learning based-methods used in trajectory prediction for autonomous vehicles.
* An analytical summary is provided for the metrics and datasets used to evaluate the performance of trajectory prediction methods.
* A comparison of the methods is conducted, analyzing the strengths and weaknesses of each approach. Furthermore, challenges and potential research avenues are identified.
§.§ Paper Organization
The road map of the survey has been presented in pictorial form in Fig. 4. There are nine sections in this paper. Section II presents a generic problem formulation, provides definitions of the terminologies used, and the methods are categorized based on various criteria. In Sections III, IV, and V of the paper, comprehensive reviews on conventional-based methods, deep learning-based methods, and reinforcement learning-based methods are conducted respectively. Section VI discusses the commonly used evaluation metrics and datasets. Section VII discusses the performance of different methods, and Section VIII highlights the current challenges in
the literature and potential new research directions. The key concluding remarks are given in section IX.
§ PROBLEM DESCRIPTION
In the context of Automated Driving (AD), accurately predicting the trajectories of other road users poses a significant challenge for AV's software. It requires a comprehensive understanding of the spatio-temporal dynamics of the environment, including the past states of observable road users and their interaction patterns, irrespective of their quantity and types.
Trajectory prediction involves two main steps. First, it is essential to track and gather relevant information about neighboring road users to obtain precise and reliable trajectories. Second, based on the acquired knowledge, future trajectories of these neighboring road users need to be predicted. To accomplish these tasks, the AV's software must have access to mapping data encompassing the road scene and the surrounding area (referred to as road context). This includes information such as road and crosswalk locations, lane directions, and other relevant map-related details. Additionally, the software needs to identify and monitor Surrounding Vehicles (SVs) as well as Target Vehicles (TVs) for accurate trajectory prediction.
To tackle the inherent ambiguity of the problem, we approach vehicle trajectory prediction as a probabilistic task. We define the future trajectories of TVs as the sequence of their future states, denoted as Y_TVs:
Y_TVs = {e_j^t, e_j^t+1, e_j^t+2, ..., e_j^t+f}_j=1^N
Here, N represents the number of TVs, f is the size of the prediction window, and e_j^t denotes the state of vehicle j at time step t. The problem is formulated by computing the posterior distribution P(Y_TVs|C), where C = X ∪ I represents the available information to the ego vehicle.
The historical states, captured in X, encompass the observations of N traffic participants up to time step t-1:
X = {e_j^0, e_j^1, e_j^2, ..., e_j^t-1}_j=1^N
These historical states typically include attributes such as position, velocity, acceleration, orientation, etc. Additionally, I denote optional environmental information that can be considered or omitted based on availability.
In this formulation, the goal is to estimate the future trajectories Y_TVs of the traffic participants given the available information C. The posterior distribution P(Y_TVs|C) represents the probability distribution of the future trajectories conditioned on the available information.
To manage computational complexity, the prediction of each TV can be performed independently. At each stage, one vehicle is selected as the target TV, and its trajectory distribution, P(Y_TV|C), is computed:
Y_TVs = {e_j^t,e_j^t+1,e_j^t+2,......,e_j^t+f}_j=1^N
Here, T represents the chosen TV, and the trajectory prediction for that specific vehicle is determined.
§.§ Terminolgy
In the field of trajectory prediction, several terminologies are commonly used to describe different aspects of the prediction process. Here are some key terminologies.
* Trajectory: A trajectory refers to the path or motion of an object or entity over time. It represents the series of positions or states that an object traverses.
* Manoeuvre: The term "maneuver" refers to the specific actions or movements performed by a vehicle or object as it navigates through its environment. Maneuvers can include various actions, such as lane changes, turns, merges, accelerations, decelerations, and stops.
* The vehicles whose trajectory we are interested in forecasting are called Target Vehicles (TVs).
* Ego Vehicle (EV) is an autonomous vehicle that monitors its surroundings to forecast TV trajectory.
* The prediction model examines Surrounding Vehicles (SVs) since they may have an impact on how TV will behave in the future. SVs may be chosen using a variety of criteria depending on the modelling assumptions used in the study.
* The remaining vehicles in the driving environment that are deemed to have no bearing on the behaviour of the TV are known as Non-Effective Vehicles (NVs).
* Unimodal Trajectory - Generate the single trajectory of single or multiple traffic participants in the given scenes.
* Multimodal Trajectory - Generate the multiple trajectories of single or multiple traffic participants in the given scenes.
The proposed terminology is illustrated in Fig. 5, through a driving scenario. The vehicles in the scenario are divided into SVs and NVs using a distance-based criterion as an example.
§.§ Techniques applied for Trajectory-Prediction Methods
Trajectory prediction methods in autonomous driving can be broadly classified into the following categories:
§.§.§ Conventional Methods
Conventional methods for trajectory prediction refer to traditional approaches that have been commonly used to forecast the future trajectories of road users in Autonomous Driving (AD). These methods typically rely on well-established mathematical and statistical techniques to make predictions based on historical data and predefined models.
Some of the commonly used conventional methods are:
* Physics-based Models: These methods rely on the laws of physics and kinematics principles to predict the future trajectory of a vehicle. They consider factors such as current position, velocity, acceleration, and road constraints to estimate the future path <cit.>.
* Kinematic models: These models assume that the motion of objects can be described by simple mathematical equations, such as constant velocity or constant acceleration models. They estimate future positions based on the object's current state and its assumed motion dynamics <cit.>.
* Kalman filters: Kalman filters are widely used for tracking and prediction tasks. They combine measurements from sensors with predictions from a mathematical model to estimate the current state of an object and make predictions about its future trajectory <cit.>.
* Markov models: Markov models capture the probabilistic dependencies between successive states of an object. They use historical data to estimate transition probabilities and make predictions based on the most likely sequence of states <cit.>.
* Probabilistic Models: Probabilistic approaches consider uncertainty in trajectory prediction by representing the future trajectories as probability distributions. These models leverage statistical techniques to estimate the most likely trajectory and provide a measure of confidence <cit.>.
* Bayesian Filters: Bayesian filters, such as Kalman filters and particle filters, are widely used for trajectory prediction. These filters combine measurements from sensors with a dynamic model to estimate the future trajectory of a vehicle. They can handle noisy sensor data and provide real-time predictions <cit.>.
Conventional methods for trajectory prediction are often computationally efficient and relatively easy to implement. However, they may have limitations in handling complex scenarios with intricate interactions and uncertainties. As a result, there has been a growing interest in exploring more advanced machine learning-based approaches, such as deep learning and reinforcement learning, to improve the accuracy and robustness of trajectory predictions.
§.§.§ Deep learning-based methods
Deep learning-based methods have gained significant attention in recent years for trajectory prediction in Autonomous Vehicles (AVs). These methods leverage the power of artificial neural networks to learn complex patterns and relationships from large amounts of data. Here are some common deep learning-based approaches for trajectory prediction:
* Recurrent Neural Networks (RNNs): RNNs are widely used in trajectory prediction due to their ability to model sequential data. Models such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) can capture temporal dependencies and predict future trajectories based on past observations <cit.>.
* Convolutional Neural Networks (CNNs): CNNs are primarily used for image processing tasks, but they can also be applied to trajectory prediction by treating trajectory data as image-like representations. CNNs can extract spatial features from trajectory data and learn to predict future trajectories based on these features <cit.>.
* Generative Adversarial Networks (GANs): GANs consist of a generator network and a discriminator network. They can be employed for trajectory prediction by training the generator to generate realistic future trajectories and the discriminator to differentiate between real and generated trajectories. GANs can capture the distribution of training data and generate diverse and plausible trajectory predictions <cit.>.
* Variational Autoencoders (VAEs): VAEs are generative models that learn a latent representation of the input data. They can be used for trajectory prediction by learning the latent space representation of past trajectories and generating future trajectories conditioned on this latent representation. VAEs enable the generation of diverse and probabilistic trajectory predictions <cit.>.
* Transformer Models: Transformer models, originally introduced for natural language processing tasks, have also shown promise in trajectory prediction. These models can capture long-range dependencies and interactions between different agents in the scene. By attending to relevant spatial and temporal information, transformer models can generate accurate trajectory predictions <cit.>.
Deep learning-based methods have demonstrated improved performance in capturing complex patterns, handling diverse scenarios, and generating more accurate trajectory predictions compared to conventional approaches. However, they require large amounts of labeled training data and computational resources for training and inference. Additionally, the interpretability of the learned models can be a challenge, making it important to validate the predictions and understand the model's limitations in real-world scenarios.
§.§.§ Reinforcement learning-based methods
Reinforcement Learning (RL) methods have been explored for trajectory prediction in Autonomous Driving (AD), offering a unique approach to learn optimal policies for predicting future trajectories. While RL is traditionally associated with decision-making and control, it can also be utilized in the context of trajectory prediction. Here are some RL methods used for trajectory prediction:
* Inverse Reinforcement learning (IRL): The key idea behind IRL is to observe and analyze expert demonstrations, typically provided by human drivers, and then infer the underlying reward function that motivates their actions. This inferred reward function can be used to predict future trajectories that align with the observed expert behavior <cit.>.
* Deep Inverse Reinforcement Learning (Deep IRL): Deep IRL is an extension of Inverse Reinforcement Learning (IRL) that combines deep neural networks with the IRL framework to predict trajectories in AD. Deep IRL aims to infer the underlying reward function from expert demonstrations using deep learning techniques, allowing for more complex and high-dimensional representations of the reward function <cit.>.
* Imitation learning (IL): IL for trajectory prediction enables autonomous systems to mimic the behavior of human drivers and generate trajectories that align with expert demonstrations. It leverages the knowledge and expertise of human drivers to make more human-like predictions and navigate the environment in a manner that is similar to how humans would drive <cit.>.
By applying RL methods to trajectory prediction, models can learn from data and interactions with the environment to make accurate predictions about future trajectories. However, it is important to consider the trade-off between the complexity of RL algorithms and the availability of training data, as well as the challenges of generalization to various driving scenarios and uncertainties in the real-world environment.
§ CONVENTIONAL TRAJECTORY PREDICTION METHOD
This section classifies prediction methods into three dominant classes, Physics-based Models, Sampling Methods, and Probabilistic models, and Table II presents a brief overview of the Conventional methods for trajectory prediction with their limitations and advantages. In Fig. 6, several conventional methods and their involvement in addressing the trajectory prediction task in Autonomous Vehicles (AVs) are depicted. The analysis of the papers reveals that 56% of the papers focus on probabilistic methods, 35% of the papers focus on sampling methods, and the remaining 9% of the papers are dedicated to sampling methods in this survey.
§.§ Physics based Models
The first class of suggested physics-based prediction models uses classical mechanics' motion equations as a foundation for modeling the target object's future motion. Either dynamic or kinematic models can be used to describe the physical behavior. A dynamics model considers the lateral and longitudinal tire forces causing the motion, but a basic dynamics model is typically chosen to balance predictive accuracy and computational effort. In contrast, kinematics models are more commonly used due to their simple form, and the Kalman Filtering (KF) techniques can handle disturbances, such as uncertainty or noise, in the vehicle's current condition. For instance, Zhang et al. <cit.> proposed a vehicle-to-vehicle communication and KF-based approach to enable a host vehicle to predict the trajectories of remote vehicles and avoid obstacles. Lefkopoulos et al. <cit.> also introduced the Interacting Multiple Model Kalman Filter (IMM-KF), a new technique that incorporates interaction-related parameters for more accurate trajectory prediction using a physics-based model over a few seconds.
The Monte Carlo approach can be used to roughly simulate the state distribution by applying a physics model to a sample of input variables at random, generating potential future trajectories. This method can be used to predict traffic participant trajectories from either a fully known or an unknown state evaluated by a filtering mechanism. Okamoto et al. <cit.> use the Monte Carlo approach in their maneuver-based model to predict future trajectories based on the recognized maneuver. Similarly, Wang et al. <cit.> use the Monte Carlo approach to predict trajectories and then use Model Predictive Control (MPC) to refine the reference trajectories.
In summary, Physics-based Models are characterized by their excellent explainability, robust performance, and high accuracy, particularly for short-term prediction tasks. These models are well-suited for safety assessment purposes, typically focusing on predictions within a time horizon of no more than 1 second.
However, they may have limitations in capturing complex maneuvers, often relying on simplified assumptions and limited adaptability in unknown or dynamic environments.
§.§ Sampling Methods
These techniques involve sampling the future states of traffic participants. Instead of predicting a single trajectory, these approaches generate a distribution of possible vehicle states, which makes them more robust to noise and uncertainty. There are two main types of sampling: generating multiple trajectory segments or particle states. In their study, Houenou et al. <cit.> combined a maneuver-based approach with a model-based approach assuming Constant Yaw Rate and Acceleration (CYRA) to develop a trajectory prediction method. They identified the maneuver and selected the best trajectory from a set generated by minimizing a cost function.
Meanwhile, Tran and Firl <cit.> utilized Monte Carlo Simulation (MCS) to predict multimodal trajectories and a normalized three-dimensional Gaussian Process (GP) regression model to learn vehicle behavior at a junction. Similarly, Wissing et al. <cit.> proposed an interaction-aware trajectory prediction method that used MCS to simulate interactions and forecast the distribution of potential future positions for the target vehicle. This approach leveraged the Intelligent Driver Model (IDM) <cit.> to account for the interacting behaviors of traffic participants, and the particles were disseminated using a lane-change driving model that considered three different lateral moves and aspects of the driving scenario with each run of the MCS. To summarize, sampling methods are essential tools for trajectory prediction, and the choice of method depends on the specific problem and the properties of the distribution of trajectories. However, these methods face challenges such as computational complexity, the requirement for efficient sampling strategies, and the possibility of overlooking important trajectory regions.
§.§ Probabilistic Models
A probabilistic framework in trajectory prediction refers to the use of probability theory to model and estimate the likelihood of future trajectories of objects or entities, such as vehicles, pedestrians, or other moving objects. It involves representing uncertainty and variability in the prediction process and providing probabilistic distributions or confidence measures for the predicted trajectories.
In a probabilistic framework, trajectory prediction is typically formulated as a conditional probability problem, where the goal is to estimate the probability distribution of future trajectories given the observed past trajectories, sensor measurements, and other relevant information. This involves incorporating probabilistic models, statistical techniques, and machine learning algorithms to capture the uncertainties and dependencies in the data.
§.§.§ Gaussian Mixture Model
A Gaussian Mixture Model (GMM) is a probabilistic model that is often used in trajectory prediction to capture the uncertainty and complexity of the data. It represents the distribution of the trajectories as a combination of multiple Gaussian distributions, each representing a possible mode or cluster of trajectories.
A Semantic-based Intention and Motion Prediction (SIMP) was proposed by Hu et al. <cit.>. It uses multiple 2D GMM to model the probability distribution of movement patterns in driving scenarios and Deep Neural Networks (DNNs) to calculate the likelihood of entering the intersection area. GMM was also employed in other methods to model specific motion patterns <cit.>,<cit.>. Although classic Hidden Markov Model (HMM) approaches have been quite successful at predicting drivers' moves, they do not take the impact of interaction-related aspects into account during the prediction process, therefore the results of their predictions are insufficiently accurate in real-world traffic situations. An interaction-related vehicle trajectory prediction model based on HMM and Variational GMM is proposed by Deo et al. <cit.>.
The knowledge about vehicle interactions is discovered by locating the energy function's ideal solution. A GMM-HMM manoeuvre prediction model that takes interaction-aware elements into account is proposed by Zhang et al. <cit.> based on game theory. Jiang et al. <cit.> developed a GMM-HMM recognizer based on joint mutual information maximization to estimate the driver's lane-changing intention. This recognizer was incorporated as a node in the Dynamic Bayesian Network (DBN) framework. To summarize, GMMs provide a versatile and robust method for trajectory prediction by capturing complex patterns and variations in the data. They are capable of handling multimodal distributions, which allows for representing different maneuver types or behavior patterns exhibited by vehicles. However, it's important to note that training and inference with GMMs can be computationally demanding. Additionally, determining the optimal number of Gaussian components or modes in the model can be a challenging task.
§.§.§ Gaussian Process
When utilizing Gaussian Process (GP) for trajectory forecasting, trajectories are considered as samples taken from a GP along the time axis. These samples are represented by N discrete points, which are mapped to an N-dimensional space. In this N-dimensional space, the samples adhere to a Gaussian distribution. During the modeling step, the GP model's main objective is to estimate the GP parameters based on these samples. By fitting the GP to the observed trajectory samples, the model captures the underlying patterns and dynamics of the data. The GP parameters, such as the mean and covariance, define the characteristics of the GP and determine the shape and uncertainty of the predicted trajectories <cit.>. In the study by Laugier et al. <cit.>, GPs were employed to predict trajectories following the evaluation of likely behaviors using Hidden Markov Models (HMMs). Trautman et al. <cit.> addressed the frozen robot problem by utilizing GP for joint collision avoidance. Additionally, GP can be utilized to simulate interaction-related aspects in trajectory prediction tasks.
Guo et al. <cit.> employed GPs and the Dirichlet process (DP) to construct motion processes and utilized a non-parametric Bayesian network to extract potential motion patterns. The prototype set was trained to represent each trajectory using methods based on prototype trajectories. The primary differentiation among these approaches lies in the technique employed to generate the prototype trajectory. Govea et al. <cit.>'s statistical analysis of the mean and variance of each sample of a trajectory yields the prototype trajectories. In their study, Hermes et al. <cit.> focused on capturing variations in vehicle movement through training. They divided the sample trajectories into multiple subsets and generated several prototype trajectories as an outcome of their research. In summary, GP is a valuable tool in trajectory prediction for AVs, providing several advantages such as flexibility, probabilistic forecasts, adaptability, and potential integration with other techniques. However, one limitation of approaches based on trajectory samples is their limited applicability to new contexts, which hinders their adaptability to diverse scenarios and environments.
§.§.§ Hidden Markov Model
In trajectory prediction using the Hidden Markov Model (HMM), the observation sequences are comprised of the previous states of the traffic participants. The HMM algorithm is applied to estimate the most likely future observation sequence based on these past observations. Qiao et al. <cit.> offers a technique called HMTP* based on HMM that adaptively selects parameters to replicate real situations at a pace that changes over time. In <cit.>, HMM and fuzzy logic are utilised to anticipate driver maneuvers. HMM can also be included in planning and decision-making processes. HMM is employed in <cit.> for risk assessment and trajectory prediction, with the outcomes being supplied into the system for making decisions. A behavior prediction method based on the HMM is proposed by Li et al. <cit.>, considering the direction of incoming cars. To ensure the reliability of the prediction results, multiple sets of initial values are generated. Additionally, this approach aims to reduce the model's dependency on data, resulting in improved prediction performance. Ren et al. <cit.> propose the lane-changing behavior recognition model based on the Continuous Hidden Markov Model (CHMM) is developed to identify the lane-changing behavior of nearby vehicles. In summary, The HMM is highly beneficial for trajectory prediction due to its ability to capture temporal dependencies, handle missing or noisy data, and account for the uncertainty involved in predicting future trajectories. However, an assumption of HMM is that the hidden states are Markovian, implying that the probability of transitioning to a future state depends solely on the current state.
§.§.§ Dynamic Bayesian Network
By incorporating time sequence and leveraging the Bayesian Network framework, the Dynamic Bayesian Network (DBN) offers a maneuver-based approach for trajectory prediction. DBN and Bayesian networks share fundamental concepts and methodologies for conducting probabilistic inferences. However, one distinction is that Kevin et al. <cit.> introduced the concept of time templates to address timing considerations in probabilistic models, while Bayesian Networks typically represent static systems. In the context of DBN, a time segment refers to a time template that discretizes continuous time into discrete points with a predetermined time granularity.
Multiple vehicles driving manoeuvres are modelled by Gindele et al. <cit.>. All vehicle states, interaction relationships, observation statuses, road structures, etc. are included in the input data. DBN is used by Schreieret al. <cit.> to evaluate driving manoeuvres, and they use the kinematics model associated with each maneuver to forecast the trajectory. Game theory is used in <cit.> to anticipate the vehicle movement, and DBN, which takes interaction-related elements into account, then judges the vehicle motion. He et al. <cit.> employ DBN to recognise lane-change and vehicle-following motions and to forecast the trajectory. In <cit.>, DBN is designed to consider the kinematic factors of vehicles, Vehicle manoeuvres, and their interdependence, and road-related information.
To summarize, when utilized for trajectory prediction, DBN takes into account the interactions between traffic participants, leading to improved performance in conventional machine learning-based methods. However, DBN still encounters challenges in accurately recognizing maneuvers and generating trajectories. Many existing methods are limited to distinguishing only two or three maneuvers, such as lane-keeping and lane-changing, which restricts the model's ability to generalize to a wide range of scenarios. Consequently, there is a need for further advancements in DBN-based approaches to enhance their maneuver recognition capabilities and improve the model's generalization ability in trajectory prediction tasks.
§ DEEP LEARNING-BASED PREDICTION METHODS
Conventional prediction techniques are only effective in basic prediction scenarios and short-term prediction assignments. Deep learning-based trajectory prediction models have gained popularity due to their ability to consider various factors that contribute to accurate predictions. These models take into account physical factors, such as the position, velocity, acceleration, size, and shape of vehicles. They also consider road-related factors like traffic signs, traffic lights, road geometry, and road obstacles. Additionally, interaction-related factors, including the distance between vehicles, relative speeds, and the presence of communication systems, are considered. Fig. 7 provides a general overview of these methods. The following sections outline the most prevalent deep learning-based methods used for trajectory prediction in Autonomous Vehicles (AVs).
§.§ Sequential Modelling
Deep learning-based trajectory prediction methods often involve using a sequential network to extract features from historical trajectories and can serve as the output layer. These networks typically include Recurrent Neural Networks (RNNs), Temporal Convolutional Neural Networks (TCNs), Attention Mechanisms (AMs), and Transformers. Fig. 8 provides a visual representation, in percentages, of the distribution of research papers utilizing different algorithms in sequential modeling for trajectory prediction. It can be observed that TCNs are less commonly used in the AVs trajectory prediction task compared to other algorithms such as RNNs, AMs, and Transformers.
§.§.§ Recurrent Neural Network
The Recurrent Neural Network (RNN) was designed to handle temporal information, as opposed to conventional machine learning methods and Convolution Neural Networks (CNNs), which excel at processing spatial information <cit.>, <cit.>. It maintains a record of past-time step data and combines input and hidden states to generate the desired output. However, when dealing with a large number of time steps, the RNN's gradient can either weaken or explode, causing issues. To tackle this problem, gated RNNs like the Long Short-Term Memory Network (LSTM) and Gated Recurrent Unit (GRU) have been developed. RNN-based trajectory prediction models are categorized as either single RNN models or multiple RNN models.
1) Single RNN: To predict trajectories based on maneuvers or single-modal trajectory prediction, a single RNN model is employed. Additionally, it can be incorporated into auxiliary models to facilitate more complex capabilities, such as interaction-aware forecasting. In various studies <cit.>, LSTM has been utilized as a sequence classifier for vehicle maneuver prediction. In these studies, LSTM cells extract vehicle attributes, and the output layer predicts movements using the final cell's hidden states. Fully connected layers are used to extract features and input them into three-layer LSTMs in <cit.>, while two LSTM layers without embedding are employed in <cit.>. Altché et al. <cit.> used a single-layer LSTM to estimate the target vehicle's trajectory. Ding et al. <cit.> used an LSTM encoder to predict the target vehicle's maneuver by encoding its states.
The LSTM encoder-decoder incorporating social and geographic information was compared to the Nearest Neighbor (NN) regression method in <cit.>. In <cit.>, Zyner et al. utilized a weighted Gaussian Mixture Model (GMM) to forecast multi-modal trajectories. The GMM's parameters were obtained using an encoder-decoder three-layer LSTM model, and the predicted trajectories were clustered using the modal with the highest probability. Xing et al. <cit.> used GMM to identify driving styles, LSTM and fully connected regression layers to assess sequence data and driving styles to predict vehicle trajectory, with the first vehicle in the fleet following its predicted trajectory. Kawasaki et al. <cit.> integrated LSTM and KF for multi-modal trajectory prediction while considering lane information.
2) Multiple RNNs: The development of neural networks has resulted in the widespread usage of various types of RNN architectures. In Xin et al. <cit.>, two separate LSTMs are employed to predict the target lane and trajectory of a vehicle based on its current state and expected lane. Deo et al. <cit.> suggest six LSTM decoders, each connected to a different maneuver, to forecast multi-modal trajectories. Dai et al. <cit.> utilizes two groups of LSTM networks to simulate the motion and interaction of nearby vehicles. Ding et al. <cit.> present a group of GRU encoders to characterize paired interactions between vehicles. Min et al. <cit.> use multiple RNNs and fully connected layers to generate the cubic polynomial coefficients that describe the target vehicle's future trajectory. Tang et al. <cit.> employ an attention mechanism to create a dynamic state encoder consisting of multiple RNNs sharing parameters to predict the multi-modal trajectory. Multi-modal trajectories are generated using an LSTM encoder-decoder and a multi-head attention layer in <cit.>. A paradigm with several LSTMs is proposed by Zhang et al. in <cit.> for both trajectory and intention prediction. Xu et al. <cit.> introduce a student-teacher network for trajectory prediction, where the student algorithm is based on an LSTM Encoder-Decoder model, and the instructor algorithm is based on a Convolutional Graph Network. Although RNNs are widely used for analyzing and predicting data series, such as trajectory prediction, they have limitations in simulating spatial relationships, such as vehicle interaction, and processing image-like data, such as the context of a driving scene. This is why complex RNN-based solutions often require multiple techniques to overcome the limitations of a single RNN. In summary, RNNs offer a powerful approach for trajectory prediction by effectively modeling temporal dependencies. They can handle variable-length sequences and provide interpretability. However, they can suffer from vanishing or exploding gradients and sequential computation limitations. Understanding these factors is crucial when applying RNNs to trajectory prediction tasks.
Table III summarizes the RNN-based approaches for trajectory prediction, providing information on the Prediction Horizon (PH) in seconds (s) and the number of predicted trajectories. The table also includes the Evaluation Metrics (EM) used for training and testing, along with highlighting the strengths and weaknesses of each study.
§.§.§ Temporal convolutional networks
Temporal Convolutional Networks (TCNs) are a popular type of deep neural network architecture used in trajectory prediction tasks. In trajectory prediction tasks, TCNs are trained on historical trajectory data and are used to predict the future trajectory of a vehicle or pedestrian. Compared to recurrent networks, TCNs have been shown to outperform them in tasks such as handwritten recognition <cit.>, audio synthesis <cit.>, and time-series data <cit.>. One advantage of TCNs is their ability to handle variable-length sequences without information leakage. Bai et al. <cit.> employed causal convolution, dilated convolution, residual connection, and a completely connected network to create TCN. Zhang et al. <cit.> utilized TCN to predict lane-change maneuvers and trajectories. In <cit.>, CNN processes the rasterized image while TCN collects features from historical trajectory data that are combined with the raster feature and the present state. DeepTrack <cit.> is a lightweight deep learning algorithm with accuracy comparable to top trajectory prediction algorithms, but with a much smaller model size and reduced computational complexity. DeepTrack encodes the vehicle dynamic using TCN and reduces model complexity by using depthwise convolution as the fundamental building block. In <cit.>, a TCN encoder and a Multi-Layer Perceptron (MLP) decoder are used, where the position and speed of the vehicles are sequentially entered and encoded as a context vector during the encoding procedure. To improve prediction accuracy, an intention recognition module is included with a TCN encoder. In <cit.>, Mozhgan et al. integrate a dilated convolutional network-based encoder-decoder with a mixture density network to predict potential multimodal pathways taken by target vehicles. It is evident that TCN possesses benefits when it comes to handling time-series data. In summary, TCNs offer a powerful approach for modeling temporal dependencies in trajectory prediction tasks. They excel at capturing short-term and long-term dynamics, perform efficient parallel computation, and have interpretable receptive fields. However, spatial relationships and long-term memory might require additional considerations.
Table IV presents a summary of TCN-based approaches for trajectory prediction, including the prediction horizon (in second (s)), the number of trajectories predicted, and the evaluation metrics used for training and testing. The table also highlights the strengths and weaknesses of each study.
§.§.§ Attention Mechanism
The Attention Mechanism (AM) is a cognitive model that approximates human thought processes by allowing for the efficient extraction of high-value information from a large volume of data using limited attentional resources. It is frequently used in deep learning tasks such as speech recognition <cit.>, image classification <cit.>, and natural language processing <cit.>, with self-attention <cit.> being a popular method for identifying the weights and new context vectors based on the input sequence.
Several recent studies have employed the attention mechanism for trajectory prediction and intention estimation. Hao et al. <cit.> proposed an encoder-decoder architecture combining GRU and self-attention, while Yan et al. <cit.> investigated a self-attention architecture with two types of self-attention mechanisms for the driving lane and driving context. Kim et al. <cit.> used self-attention to concentrate on features from the target vehicle's preferred lane, and Fu et al. <cit.>and Yu et al. <cit.> employed attention between the encoder and decoder components to selectively draw attention to particular context vector properties.
According to Wu et al. <cit.> and Meng et al. <cit.>, the model can learn important spatial and temporal components for predicting and anticipating the movements of nearby vehicles. These models use a spatial attention layer to combine data from surrounding vehicles and a temporal attention layer to account for the temporal relationships between object agents.
Lin et al. <cit.> proposed the STA-LSTM, which combines spatial and temporal information with an attention mechanism to explain how past trajectories and nearby vehicles affect the ego vehicle. Additionally, Kim et al. <cit.> proposed a model with a Baseline Network and Trajectory Proposal Attention, which is designed to simulate interaction-aware prediction. More recent work includes TP2Net <cit.>, a trajectory prediction network that uses temporal pattern attention to extract latent multimodal driving information, and yang et al. <cit.> investigating the spatiotemporal dynamics between the ego vehicle and nearby cars, utilized spatiotemporal attention mechanisms in LSTM networks to perform lane change prediction and the trajectories of the vehicles.
Several studies have employed multi-head attention and AM to extract information from lanes and vehicles, and model traffic interactions by analyzing attentions extracted from LSTM encoders. For instance, Kim et al. <cit.> utilizes multi-head attention to extract lane and vehicle information to predict future trajectory distributions. Messaoud et al.
<cit.> also employs attention extracted from LSTM encoders to model traffic interactions. In Messaoud et al.'s <cit.> model, each attention head simulates a possible interaction between the target and context features.
Hasan et al. <cit.> involves two Multi-Head Attention layers to capture the social and temporal interactions among vehicles. Additionally, incorporated a Multi-Head Attention-based decoder that includes an LSTM layer to decode the social and temporal interactions of the vehicles in a step-by-step manner. In summary, the attention mechanism in trajectory prediction improves the model's ability to focus on relevant information, handle variable-length sequences, provide interpretability, and enhance robustness to noise. However, it comes with potential drawbacks related to computational overhead, model complexity, attention bias, and data dependency.
Table V presents a comprehensive summary of Attention-based approaches for trajectory prediction. It includes important information such as the prediction horizon (measured in seconds (s)), the number of trajectories predicted, and the evaluation metrics used for training and testing. Additionally, the table provides insights into the strengths and weaknesses of each study.
§.§.§ Transformer
Transformer is a neural network design that utilizes an attention mechanism concept and has been employed in various projects such as object detection <cit.>, image segmentation <cit.>, posture estimation <cit.>, tracking, and trajectory prediction <cit.>. It was initially utilized for machine translation in Natural Language Processing (NLP) <cit.> and outperformed recurrent neural networks. Researchers have found the Transformer model to be effective for trajectory prediction, with Quintanar et al. <cit.> modifying a standard transformer to incorporate past trajectories as an input feature extracted from aerial view photo datasets. Another approach suggested by Liu et al. <cit.> involves a multi-modal prediction architecture consisting of stacked transformers that gather features from historical trajectories, road data, and social interaction. Meanwhile, Zhao et al. <cit.> utilized a transformer network with residual layers to predict trajectories that account for interaction, using fully linked feed-forward networks and pooling operations to integrate geographical data and enable the transformer to learn interaction aspects.
The Spatio-Temporal Transformer Networks (S2TNet) <cit.> uses spatio-temporal Transformer to represent spatio-temporal interactions and temporal Transformer to handle temporal sequences. Chen et al. <cit.> propose a novel non-autoregressive model for predicting vehicle trajectories based on transformers, utilizing a self-attention module to define the dynamic variation in social behavior and a graph attention module to represent the interactions between vehicles. The Structural Transformer suggested by Hou et al. <cit.> is a recurrence-free multi-sequence learning network that grasps interactions between surrounding vehicles along both temporal and geographical dimensions simultaneously. Huang et al. <cit.> propose a Transformer-based multi-modal trajectory prediction model using a multi-head attention Transformer layer to model the relationship between interacting agents. The SceneTransformer is a transformer-based model introduced by Ngiam et al. <cit.> that uses attention to mix features from agent interactions and road graphs in both space and time. The LaneTransformer proposed by Wang et al. <cit.> combines the characteristics of the features between the roadways and the agents using a stack of transformer blocks, and high-order interactions are aggregated using an attention-based block. Wang et al. <cit.> propose a mixture-of-experts approach utilizing a transformer to model the interactions between vehicles explicitly considering their driving styles for building a multimodal motion planner. The study in <cit.> proposes a dual Transformer model to demonstrate the relationship between intentions and trajectories for the target vehicle. As demonstrated by these studies, the use of transformers provides several advantages in handling time-series data in trajectory predictions. To summarize, transformers have shown their potential in trajectory prediction by capturing complex dependencies and interactions. They offer scalability, transfer learning capabilities, and the ability to handle multiple agents. However, they require substantial computational resources and may have challenges in interpretability and data efficiency.
Table VI summarizes the Transformer-based approaches for trajectory prediction, presenting key details such as the prediction horizon (measured in seconds (s)), the number of trajectories predicted, and the evaluation metrics employed for training and testing. Furthermore, the table highlights the strengths and weaknesses of each study.
§.§ Vision Based Modelling
There are two types of prediction methods, which differ in how they formulate predictions. The first is the Bird-Eye-View (BEV) approach, which uses an algorithm to process data from a top-down, map-like view. The second is ego-camera prediction, which involves viewing the world through the perspective of the ego-vehicle. However, the ego-camera approach is generally more challenging than the BEV approach due to various factors <cit.>. Firstly, the BEV approach offers a broader field of view and more accurate predictions, whereas the ego-camera approach has a narrower field of vision and a limited prediction horizon. Additionally, the ego-camera approach is more prone to obstructions than the BEV approach. Despite these difficulties, the ego-camera approach is still more beneficial than the BEV approach because most vehicles do not have access to cameras that can locate target agents and BEVs on the road.
Therefore, a prediction system should be able to view the world from the perspective of the ego vehicle, as demonstrated in Fig. 9. The illustrations of various vision-based techniques and their contribution to solving the trajectory prediction task in Autonomous Vehicles (AVs) are depicted in Fig. 10. Each technique makes a roughly equal contribution to the trajectory prediction in AVs research paper. This section highlights the inclusion of Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs) in addressing this domain.
§.§.§ Convolutional Neural Network
Convolutional Neural Networks (CNNs) have been successfully applied to various computer vision tasks, including trajectory prediction. Although CNNs are primarily designed for image data, they can be adapted for trajectory prediction by treating the trajectory sequence as a structured grid-like input. Recently, CNN has shown success in various tasks, including machine translation <cit.> and computer vision <cit.>. In the context of trajectory prediction for autonomous vehicles, CNN is commonly used for vision-based prediction, where features are extracted from images captured by frontal cameras. Nikhil et al. <cit.> found that using CNN for trajectory prediction is superior to RNN, as trajectory has significant spatio-temporal continuity. They stacked the convolutional layer after a fully connected layer to create time continuity and used a fully connected layer to output the future trajectory, taking the past trajectory as input. This CNN-based network operates faster, according to experiments.
However, most techniques that use the CNN framework take a Bird's-Eye View (BEV) as their input, displaying a top-down view of the traffic situation. BEV images can be created using multiple data sources, including LiDAR point clouds, Occupancy Grids (OG), and High-Definition Maps (HD-Maps). Some recent studies utilized CNN to extract features from sophisticated BEV representations. For example, MobileNetV2 <cit.>, a memory-effective CNN designed for mobile apps, was used in <cit.>,<cit.> to output potential trajectories and their likelihoods. The trajectory prediction of Vulnerable Road Users (VRUs) is addressed in <cit.> through a new rapid CNN architecture that utilizes context rasterization techniques <cit.>. In <cit.>, the vehicle state and the raster image were used to build a set of potential future trajectories, and the trajectory with the highest probability was selected as the future trajectory by examining semantic properties. A novel rapid CNN architecture was proposed in <cit.> for trajectory prediction of vulnerable road users, where a Memory Augmented Neural Network (MANN) was used to produce multimodal trajectories.
Recent studies have also proposed new techniques that forecast trajectory using CNN and produce cutting-edge results. For instance, Gilles et al. <cit.> generates a heatmap of the agent's potential future, while Ye et al. <cit.> uses the point cloud learning method to incorporate both spatial and temporal data into trajectory prediction. Zhuoren et al. <cit.> used ResNet-50 <cit.> to anticipate the trajectories of AVs such as vehicles and pedestrians. ResNet-50 <cit.> can effectively collect information from multiple dimensions to produce superior forecasts with the three trajectories and their confidence levels. While processing raster maps with CNN involves significant computational costs and information loss, vector maps can be used as nodes in Graph Neural Networks (GNN) for trajectory prediction.
To summarize, CNNs offer advantages in capturing spatial patterns and recognizing spatial relationships in trajectory data. They are efficient in terms of parameter sharing and can handle larger datasets. However, they may struggle with modeling temporal dependencies and handling variable-length sequences. Table VII presents a summary of CNN-based approaches for trajectory prediction, including the prediction horizon measured in seconds (s) and the number of trajectories predicted. The table also provides an overview of the evaluation metrics used for training and testing, as well as highlighting the strengths and weaknesses of each study.
The approaches for predicting vehicle trajectory based on GNN will be covered in the following sections.
§.§.§ Graph Neural Network
When considering prediction techniques that take interaction-related factors into account, each element of the environment can be viewed as a node in a graph. However, many real-world applications generate data from non-Euclidean spaces, and traditional deep learning-based methods that analyze Euclidean spatial data perform poorly in such cases. Each scene can be represented as an irregular graph with variable-sized unordered nodes, and some crucial operations, such as convolution, are not directly applicable to the graph due to variations in the number of nearby nodes. Nevertheless, every node in the graph is connected to other nodes by edges, which can be used to determine the interdependence of various objects. Graph Neural Networks (GNNs) are highly suited for vehicle trajectory prediction challenges based on interaction-related elements <cit.>. The methodology is described in Fig. 11. This idea is supported by Diehl et al. <cit.>, who demonstrate the effectiveness of trajectory prediction using two well-known graph networks: the Graph Convolutional Network (GCN) and the Graph Attention Network (GAT).
Graph Convolutional Network
The Graph Convolutional Network (GCN) is a popular technique in the field of graph neural networks. It extends the convolution operation from traditional image data processing to graph data processing. The key idea is to create a mapping function that can extract interaction-aware features from the node features in the network and their neighboring nodes.
Li et al. <cit.> proposed GRIP, a graph convolutional network-based trajectory prediction model that considers the interaction-related factors by treating vehicles as nodes within the network at each sampling time. GRIP utilizes a fixed graph network to describe the interaction-related characteristics between traffic participants and employs an LSTM encoder-decoder to forecast the trajectory of nearby vehicles using the output of GCN. To improve the accuracy of GRIP, Li et al. <cit.> proposed GRIP++, which uses both fixed and dynamic graph networks and achieved top ranking in the Baidu Apolloscape dataset <cit.> at the end of 2019. Jeon et al. <cit.> proposed SCALE-Net, which can predict the trajectories of any number of nearby vehicles while maintaining performance by using an Edge-Enhance Graph Convolutional Network (EGCN) <cit.> to learn edge features in the traffic flow. Chandra et al. <cit.> proposed a two-layer GNN-LSTM structure to resolve the trajectory prediction issue by using an LSTM encoder-decoder in the first layer to predict the future trajectories of traffic participants and a weighted dynamic geometric graph network in the second layer to represent the interaction-related characteristics of traffic participants. Zhao et al. <cit.> proposed a spectrum-based GCN network that allows all vehicles in the scene to communicate information to take into account how the surrounding vehicles are changing and adapting to the environment. Sheng et al. <cit.> proposed the GSTCN network, which uses a GCN to address spatial interactions, a CNN to capture temporal data, and a gated recurrent unit network to encrypt and decrypt the spatiotemporal properties to produce future trajectory distributions. Xu et al. <cit.> proposed a group vehicle trajectory prediction model with a global spatiotemporal graph that can thoroughly analyze the temporal and geographical association between previous vehicle trajectories. Dongwei et al. <cit.> suggested the MVHGN forecast, a graph neural network-based model for predicting the future paths of heterogeneous traffic-agents that employs a multi-view logical network by fusing various logical correlations and the multi-view logical characteristics derived by the graph convolution module. In summary, GCNs offer a promising approach for trajectory prediction by explicitly modeling the spatial dependencies and relationships among objects. They can effectively capture contextual information and handle irregular graph structures. However, scalability, graph construction, and temporal dependency modeling should be carefully considered when applying GCNs to trajectory prediction tasks. Table VIII provides a summary of GCN-based approaches for trajectory prediction including the prediction horizon measured in seconds (s) and the number of trajectories predicted, along with the evaluation metrics used for training and testing. The table also highlights the strengths and weaknesses of each study.
Graph Attention Network
The method for collecting data from the one-hop neighborhood varies greatly between Graph Attention Network (GAT) and GCN, with GAT employing the attention mechanism in place of the statically normalized convolution process. Velickovi'c et al. <cit.> proposed the GAT. In <cit.>, an encoder-decoder design was used along with GAT to extract spatial interaction information from a heterogeneous digraph, consisting of automobile and local road map vertices. The Repulsion and Attraction Graph Attention (RAGAT) model was introduced in <cit.>, which uses two stacked GATs to predict trajectories based on free space and vehicle condition information. In <cit.>, a three-channel system with a heterogeneous edge-enhanced graph attention network was developed to address the heterogeneity of vehicles in a scene. A directed edge-featured heterogeneous graph was used to represent inter-agent interactions in traffic, and a gate mechanism was added for selective map sharing among target agents. Liu et al. <cit.> proposed a multi-agent, multi-modal trajectory prediction framework using Graph Attention Isomorphism Networks (GAIN), which consisted of three attention blocks. AI-TP was introduced in <cit.> to forecast multiple SV trajectories using GAT for interaction information, followed by two convolutional Gated Recurrent Units (GRU) networks. Zhang et al. suggested the Gatformer model in <cit.> for predicting future movements of nearby traffic agents while considering spatial-temporal connections, using graphs and GAT to capture environmental interactions, and integrating the Transformer encoder-decoder. In <cit.>, a two-layer GAT was used for information aggregation and node correlation explanation, with a multi-head attention mechanism to project surrounding states to the graph and explain interactions between vehicles and the traffic flow state.
In summary, GATs enable the model to attend to relevant nodes (e.g., vehicles, pedestrians) in the graph, assigning different weights to capture the importance of each node's features for predicting the trajectory of a specific object. However, The performance of GATs heavily depends on the quality and representation of the graph structure. Designing an appropriate graph representation and considering the selection of nodes and edges is crucial for achieving optimal results.
Table IX summarizes the GAN-based approaches for trajectory prediction, highlighting the number of trajectories predicted and the prediction horizon measured in seconds (s). The table also provides insights into the strengths and weaknesses of each study, along with the evaluation metrics used for training and testing.
Other Graph Neural Network
High Definition (HD) maps play a crucial role in trajectory prediction for autonomous vehicles. HD maps provide detailed information about the road network, including lane markings, traffic signals, and road boundaries, which can help predict the future trajectory of a vehicle or pedestrian more accurately. Initially, Benz et al. <cit.> utilized HD maps for predicting trajectories, followed by determining the vehicle's future trajectory along the lane based on map topology using related lane information. However, this method did not consider interaction-related factors. To improve trajectory prediction accuracy, researchers have incorporated GNN to capture interaction features between vehicles and maps as well as between vehicles, following the introduction of the Argoverse dataset <cit.> with vector maps.
Gao et al. <cit.> proposed VectorNet, a GNN-based system that employs nodes to represent both the vector maps and vehicles in the scene for trajectory prediction. Liang et al. <cit.> integrated CNN-extracted vehicle features and GCN-extracted lane features from vector maps for trajectory prediction. Zhao et al. <cit.> presented a target-driven method called target-driven trajectory prediction (TNT) that selects sparse goal anchors and the optimal route to the target using VectorNet-extracted map features. DenseTNT
<cit.> outperforms TNT in performance by evaluating dense goal candidates. Zeng et al. <cit.> utilized LaneRCNN to represent local lane maps and interaction modules to account for interaction factors between participants' historical trajectories and local map topology. Researchers are exploring ways to integrate multiple sources of information, including HD maps, sensor data, and machine learning algorithms, to improve the accuracy and robustness of trajectory prediction for autonomous vehicles.
Table X provides a summary of other graph neural network-based approaches for trajectory prediction, focusing on the number of trajectories predicted and the prediction horizon measured in seconds (s). The table also provides the strengths and weaknesses of each study, along with the evaluation metrics used for training and testing.
§.§ Combination of CNNs and RNNs
Several researchers have proposed models that use a combination of RNN and CNN to handle temporal and spatial information for trajectory prediction. For instance, Deo et al. <cit.> use an LSTM encoder to extract temporal data from nearby vehicles, which is then fed into a social pooling layer that collects interaction-related parameters between vehicles. A social tensor is created and fed into a collection of CNNs to determine the spatial correlation of vehicles. MATF <cit.> introduces a fully convolutional network that resembles a U-net <cit.> for Multi-Agent Tensor Fusion (MATF) encoding and decoding. The fused vectors of each vehicle are taken from the output layer of the U-net <cit.>-like network, added to the LSTM-encoded vectors of the vehicles' dynamics, and then supplied to LSTM decoders. Schreiber et al. <cit.> use a CNN on condensed BEV images and an Encoder-Decoder LSTM to learn the temporal dynamics of the input data. TraPHic <cit.> uses a CNN-LSTM hybrid network to derive features from the state and nearby objects of the primary vehicle. Xie et al. <cit.> use a "box" to find and remove outliers in the vehicle's trajectory and extract interaction-aware features by feeding them into a convolutional layer and a maximum pooling layer. Xu et al. <cit.> propose a model that uses a convolutional network and a graph operation layer to capture spatiotemporal features and an LSTM encoder-decoder to forecast the traffic-related future trajectories of multiple vehicles. Table XI presents a summary of CNN-based approaches for trajectory prediction, emphasizing the number of trajectories predicted and the prediction horizon measured in seconds (s). The table also highlights the strengths and weaknesses of each study including evaluation metrics used for training and testing.
§.§ Generative Model
Predicting multi-modal trajectories presents challenges and uncertainties due to the potential diversity of outcomes. To address this issue, some researchers have turned to generative models to create multi-modal trajectories that can capture the underlying diversity. However, in order for a multi-modal trajectory prediction model to be effective, its output distribution must meet certain requirements, including diversity, social acceptability, and controllability. Achieving an optimal distribution using only one ground truth can be difficult and may lead to less diverse and unacceptable predictions. To overcome this challenge, Generative Adversarial Networks (GANs) and Variational Auto Encoders (VAEs) have been proposed as solutions. Fig. 12 illustrates the involvement of research papers, depict in percentages, of both generative models in assisting Autonomous Vehicles (AVs) with the task of trajectory prediction. Both models contribute approximately equally to the prediction process, showcasing their shared responsibility in generating accurate trajectory predictions.
Generative Adversarial Network
In trajectory prediction tasks, Generative Adversarial Networks (GANs) are used to generate realistic trajectories based on the input data. The generator takes in the historical trajectory data as input and generates a future trajectory, while the discriminator evaluates the generated trajectory for realism. The generator is trained to improve the realism of the generated trajectories by fooling the discriminator into believing they are real. This methodology is shown in Fig. 13. GAN was introduced by Ian Goodfellow in 2014 <cit.>. When GANs are used for trajectory prediction, the discriminator assesses the accuracy of the predicted trajectory while the generator constructs it. In <cit.>, Hegde et al. forecast vehicle trajectories using the vehicle's coordinate information. The TS-GAN model presented by wang et al. <cit.> utilizes a self-developed convolutional social mechanism and a recurrent social mechanism to extract vehicle spatial and temporal information from the GAN network. To create model-based multi-modal trajectories, Song et al. <cit.> employ vector maps and vehicle status information and apply a learning-based discriminator to extract information about vehicle interactions for providing the best trajectories. In <cit.>, the GAN-VEEP model is proposed for short-term vehicle trajectory prediction, utilizing a vehicle coordinate normalization model to convert position coordinates into normalized coordinates. In <cit.>, two strategies for incorporating Signal Temporal Logic (STL) rules into a GAN-style trajectory predictor are presented. In <cit.>, the STSF-Net framework is proposed, which utilizes a GAN for multi-modal trajectory distribution, with a generator that has an LSTM encode-decoder framework with a 3D CNN network for temporal correlations modeling and a discriminator that uses a Multi-Layer Perceptron (MLP) to identify the true trajectory. Additionally, Guo et al. <cit.> suggests using a map-enhanced GAN for trajectory prediction by fusing vehicle motion with high-definition (HD) maps to create contextual features. Table XII presents A summary of GAN-based approaches for trajectory prediction, focusing on the number of trajectories predicted and the prediction horizon measured in seconds (s). The table also provides the strengths and weaknesses of each study and highlights the evaluation metrics used for training and testing.
Variational Auto Encoder
The Auto Encoder (AE) compresses data using an Encoder and decodes it with a Decoder to produce a reconstructed output with minimal reconstruction errors. However, AE has been criticized for merely "memorizing" data and having limited data generation capacity. In contrast, the Variational Autoencoder (VAE) has a generative capability that spans the entire space, and it addresses the issue of non-regularized latent space in autoencoders. VAE aims to minimize both reconstruction loss and similarity loss. Bhattacharyya et al. <cit.> proposes the use of a Conditional Variational Autoencoder (CVAE) for structured prediction tasks.
Cho et al. <cit.> suggested using CVAE and LSTM to estimate possible future positions of vehicles. To ensure compliance with traffic laws and social navigation principles, they also utilized Signal Temporal Logic (STL) to eliminate irrational scenarios. Hu et al. <cit.> proposed a multi-modal trajectory prediction framework based on CVAE, but it only considered situations where two vehicles were involved. Zhang et al. <cit.> proposed using Stacked Sparse AutoEncoders (SSAE) to handle a high-dimensional input vector with motion and interaction data in a multi-modal scenario. Sriram et al. <cit.> presented an architecture that predicts the multi-modal trajectory of all traffic participants simultaneously using Convolutional LSTM and CVAE for scene context feature extraction and trajectory prediction, respectively. Dulian and Murray <cit.> utilized CNN networks to extract spatial information from Bird's Eye View (BEV) images of an HD-Map and used a CVAE to predict future trajectories, sampling the conditional variable from a prior distribution during the testing phase. liu et al. <cit.> developed a CVAE-based model to generate potential trajectories while considering motion uncertainty and then created a driving risk map. They also developed a probability model based on the trajectory risk value and used a random selection method to produce a unifying rendering of the scene's traffic agents' interactions. Based on the findings of these studies, CVAEs can take into account various conditions such as the current state of the vehicle, surrounding traffic, road conditions, or any other relevant contextual information. These conditions can be encoded as additional inputs to the CVAE model, which then learns to generate future trajectories conditioned on these inputs. Additionally, the performance of CVAEs heavily relies on the effectiveness of the chosen conditioning inputs.
Table XIII summarizes the Variational Autoencoder-based approaches for trajectory prediction, highlighting the number of trajectories predicted and the prediction horizon measured in seconds (s). The table also provides insights into the strengths and weaknesses of each study, along with the evaluation metrics used for training and testing.
§ REINFORCEMENT LEARNING-BASED METHODS
Recent years have seen the rapid growth of Reinforcement Learning (RL), which offers a new method for comprehending high-dimensional complex policies <cit.>. It offers innovative solutions for Autonomous Vehicles (AVs)' challenges involving trajectory prediction <cit.>. The Markov Decision Process (MDP) is typically utilised when RL is used to AVs trajectory prediction to maximise the projected cumulative reward.
RL techniques are utilized to estimate the underlying cost function or directly identify the optimal policy for trajectory prediction. In either approach, it is assumed that the observed agent always seeks to reach its objective by utilizing the optimal policy based on a specific cost function. Fig. 15 illustrates the application of RL methods in AVs. Within the framework of MDP, RL-based methods can be categorized into Inverse Reinforcement Learning (IRL) methods, Imitation Learning (IL) methods, and Deep IRL methods, as explained in the following sections. Fig. 14 illustrates the distribution of research articles, expressed as percentages, for different variants of RL and their involvement in addressing the trajectory prediction task in AVs.
§.§ Inverse Reinforcement Learning
The main idea behind Inverse Reinforcement Learning (IRL) is to learn the reward function that explains the observed behavior of the agents. Instead of directly imitating the observed trajectories, IRL aims to understand the underlying motivations or objectives that drive those trajectories. By inferring the reward function, the algorithm can generalize beyond the observed trajectories and make predictions about future trajectories.
Manually specifying the weight of the reward function is inappropriate due to the complex nature of driver behavior, according to Wang et al. <cit.> and Guan et al. <cit.>. To address this issue, IRL learns the optimal driving policy by inferring the reward function based on expert demonstrations (trajectories), as depicted in Fig. 16.
Liting et al. <cit.> utilize a spatiotemporal state lattice to describe driver behavior based on expert demonstrations. The driving maneuvers create a distribution for upcoming trajectories <cit.>. Interaction-related elements are considered to achieve probabilistic prediction for AVs. DriveIRL, presented by Tung et al. <cit.>, is the first learning-based planner that uses IRL to control a vehicle in congested urban traffic. They build an architecture divided into ego trajectory generation, checking, and scoring, using simple and reliable techniques to solve the very complex problem of ego trajectory generation.
A significant challenge in IRL is that an optimal policy may be ambiguous, since it can result from multiple
reward functions <cit.>. Because of this, a modified algorithm
called Maximum Entropy IRL (MaxEntIRL) was developed
by Ziebart et al. <cit.>. The MaxEntIRL algorithm aims to resolve the ambiguity in IRL by maximizing the entropy across the distributions of potential state-action pairs for a learned policy.
Some MaxEnt-IRL techniques use sampled trajectories to carry out prediction tasks. Xu et al. <cit.> sample candidate trajectories with the lowest cost that will be selected as the anticipated trajectory. Wu et al. <cit.> propose a method for learning reward functions in the continuous domain by estimating the partition function using the speed profile sampler. State sequences from the MaxEnt policy are sampled in <cit.> and provided to an attention-based trajectory generator to produce valuable future trajectories. To estimate the best policy while reducing computing costs, Xin et al. <cit.> utilize randomly pre-sampled policies in sub-spaces. Yifei et al. <cit.> propose an Inverse Optimum Control (IOC) method utilizing Langevin Sampling to determine the cost function of other vehicles in an energy-based generative model.
In summary, while IRL has the potential to provide deeper insights and more flexible trajectory predictions, the requirement for expert demonstrations and the challenges associated with their quality and computational complexity should be carefully considered in practical applications.
§.§ Deep Inverse Reinforcement Learning
Deep Inverse Reinforcement Learning (Deep IRL) is an extension of Inverse Reinforcement Learning (IRL) that incorporates Deep Neural Networks (DNNs) to learn the reward function from expert demonstrations. The deep IRL framework is introduced in <cit.> to approximate complex and nonlinear reward functions. To approximate rewards, this article uses a fully Convolutional Neural Network (CCN). You et al. <cit.> consider driving behavior and road geometry, constructing the MDP first using RL, learning the best driving strategy using IRL, and approximating the reward function using DNN. In <cit.>, driving behavior is represented by Deep IRL utilizing camera images, while CNN extracts the corresponding state information. Zhu et al. <cit.> encode the vehicle's kinematics using RL ConvNet and State Visiting Frequency (SVF) ConvNet by back-propagating the loss gradient <cit.> between expert SVF from expert demonstration and policy SVF from LiDAR data. Jung et al. <cit.> using neural LSTM to extract the feature map from the LiDAR and trajectory data, which will then be merged into the output reward map to forecast the traversability map. In <cit.>, a fused dilated convolution module is proposed to improve the extraction of raster features. Subsequently, a reward update policy with inferred goals is enhanced by learning the state rewards of goals and pathways individually instead of the original complex rewards, which can reduce the need for preset goal states. In summary, Deep IRL offers the potential for more powerful and adaptive trajectory prediction models by leveraging deep neural networks. However, challenges related to data requirements, computational complexity, interpretability, and overfitting need to be carefully addressed for successful application in trajectory prediction for autonomous driving.
§.§ Imitation Learning
One disadvantage of Inverse Reinforcement Learning (IRL) algorithms is their difficulty in training with scenarios where there are few rewards or no direct reward function. To address this issue, Imitational Learning (IL) has been suggested as a solution. IL aims to quickly determine a policy based on an expert's observation without requiring a cost function. One of the pioneering methods in imitation learning for autonomous driving is ALVINN, developed by Pomerleau <cit.>. Another notable approach by Anthony et al. <cit.> introduces a novel model-based architecture that leverages 3D geometry as an inductive bias. This method is trained solely on an offline dataset of expert driving data, eliminating the need for reward signals or online interaction. This approach shows great promise for real-world applications.
The author utilizes behavior cloning (BC) <cit.>, a technique that focuses solely on imitating the expert's policy. BC is straightforward and effective, but it struggles with unknown states, requiring a substantial amount of data. To address this limitation and produce a policy instead of a cost function,
Generative Adversarial Imitation Learning (GAIL), proposed by Ho et al. <cit.>, uses the Generative Adversarial Network (GAN) approach for imitation learning in RL. GAIL extracts policies directly from data rather than relying on expert demonstrations. GAIL, similar to GAN, is based on the fundamental concept of a generator and discriminator. The generator in GAIL produces trajectories that resemble those of an expert as closely as possible, while the discriminator determines whether the generated trajectories are from the expert or not, as shown in Fig. 17. To address GAIL's limitations in only using the current state to model the subsequent state, Choi et al. in <cit.> propose a method that incorporates a Partially Observable Markov Decision Process (POMDP) within the GAIL framework and uses the reward function from the discriminator to train the model. The high-dimensional solution space of a POMDP makes complex scenario modelling computationally expensive. Additionally, the ambiguity of state observations makes it difficult to differentiate state-action pairs. However, notable advancements have been made in online POMDPs, as demonstrated in <cit.> and <cit.>.
Bronstein et al. <cit.> modify the default model-based GAIL with a hierarchical model to enable generalization to any goal pathways and evaluate performance with simulated interactive agents in a closed-loop evaluation framework. Kuefler et al. <cit.> employ GAIL to model human driving behavior on highways and propose an RNN integrated into the GAIL architecture. Bansal et al. <cit.>'s ChaffeurNet utilizes IL to train a robust policy while penalizing implausible events and introduces an explicit loss to prevent the algorithm from solely imitating such undesirable behavior.
In summary, IL and GAIL are promising approaches to address the challenges of training RL algorithms in scenarios with limited rewards or no direct reward function. Their success in modeling human driving behavior and generating realistic predictions opens up possibilities for their application in other real-world scenarios.
Table XIV provides a summary of Reinforcement learning-based approaches for trajectory prediction and also highlights the strengths and weaknesses of each study.
§ TRAINING AND EVALUATION
Various standard datasets are used to test prediction algorithms, and appropriate metrics are used to assess their performance.
Datasets
To evaluate the accuracy of a trajectory prediction model, the projected trajectory and ground truth trajectory are usually compared. These trajectories are obtained from multiple datasets that are collected using sensors such as LiDAR, cameras, radar etc. The vehicle movements in these datasets are either automatically generated or manually annotated.
Modern benchmarks have made significant progress in the AVs prediction field, overcoming the limitations of older datasets which were constrained in terms of environments and agent categories. The NGSIM-180 <cit.> and highD <cit.> dataset are examples of such benchmarks that utilized drones and surveillance cameras to capture cars on highways. These datasets focused on a single type of agent with a limited set of possible actions, which included moving left or right and maintaining a straight path.
The KITTI <cit.> dataset, introduced by Geiger et al. in 2013, was among the earliest multimodality datasets that included LiDAR point clouds in addition to camera frames for input scenes. This development has generated a recent interest in object detection using 3D bounding boxes <cit.>. Moreover, KITTI <cit.> offers annotations for both cars and pedestrians.
As the depth of AI models increases, more images are required for efficient generalization. Recent datasets such as Lyft <cit.>, Waymo <cit.>, nuScenes <cit.>, and Argoverse <cit.> have significantly increased the number of annotated frames, thereby facilitating the training of deep models. These datasets not only include camera and LiDAR data but also provide High Definition (HD) maps that capture the road's topology <cit.>. The addition of HD maps has made it possible to investigate global navigation abilities, thus enabling the training of models for longer prediction horizons. Unlike previous datasets, the aforementioned datasets cover more classes, record ego-vehicle odometry data, encompass various cities, different weather and lighting conditions (including rain and night), and provide labels for other agents such as traffic lights and road rules. However, they still lack labels related to intention prediction.
To summarize, modern datasets have effectively addressed many of the challenges associated with prediction by providing a vast amount of diverse, multi-agent, multi-modal data. This data can be used to train models capable of predicting the behavior of various interacting agents in diverse weather conditions. Furthermore, these datasets offer annotations that are useful for high-level comprehension of the driving scene, including information on location, action, and events. Table XV presents an overview of the popular datasets commonly utilized in trajectory prediction tasks. The table includes information about the sensors used, scene descriptions, applications, and the research articles that have utilized these datasets.
The majority of the techniques described in this paper employ trajectories as input, however, some also make use of vehicle states or map data.
Evaluation Metrics
Evaluation Metrics (EMs) are crucial for assessing the effectiveness of vehicle trajectory prediction models. One common metric used for evaluating model output is the Average Displacement Error (ADE), which measures the mean l_2 distance between the predicted trajectory's locations and the corresponding ground truth. Another metric, the Final Displacement Error (FDE), calculates the same distance but only for the final predicted location and its ground truth at the prediction horizon.
Probabilistic generative models that produce multi-modal predictions require additional metrics. The Best of N metric calculates ADE and FDE for the best N samples out of all generated trajectories. When N equals 1, the method is called minADE and minFDE, respectively, and only the generated trajectory which is closest to the ground truth is selected. Other metrics for the multi-modal distribution include various versions of Negative Log Likelihood (NLL), which compares the distribution of generated trajectories against the ground truth. To evaluate the performance of the model on the ApolloScape <cit.> trajectory dataset in the literature, two metrics, the Weighted Sum of Average Displacement Errors (WSADE) and the Weighted Sum of Final Displacement Errors (WSFDE), are frequently used. Table XVI highlights the commonly used Evaluation metrics for trajectory prediction tasks with their formula and description.
§ DISCUSSION
In this section, a fair evaluation of the proposed models is presented through a comparison of representative models. The selected criteria encompass different factors that pertain to the task of trajectory prediction, as well as the overall prerequisites for utilizing the models in the field. Nonetheless, the comparison reveals prevailing patterns and provides an understanding of particular characteristics and scenarios of use. Deep Learning-based models and Reinforcement learning-based approaches shall be compared. The comparison results are summarized in Table XVII.
Deep learning-based models have demonstrated their ability to produce accurate predictions over an extended period, as they can conduct long-term predictions of up to 8 seconds. However, these models are typically comprised of neural networks and are therefore considered black-box models, which reduces their explainability and could pose challenges in terms of validation and approval. Despite this, these models have the advantage of being holistic since they can integrate various features from multiple sources, including object interaction and semantic data, into the neural network. However, to achieve good prediction performance, it is crucial to carefully select valid features. The use of spatial features and corresponding representation enables the consideration of the interaction between agents, which makes interaction awareness possible.
Deep learning-based models have the capability to describe complex processes at varying levels of abstraction, with the ability to output trajectories as prediction results. However, these models require valid training data that reflects the specific field of application to enable comprehensive and robust predictions. As a result, these models are highly data-dependent. Additionally, the adaptivity of these models is limited to scenarios that fall within the data the model has been trained on. Due to their holistic approach, Deep learning-based models are typically associated with high computational costs, which are strongly influenced by the size of the neural networks used. Nevertheless, in the current state of the art, Deep learning-based models offer the highest prediction accuracy.
Reinforcement learning-based methods are also capable of conducting long-term predictions. However, the degree of explainability varies depending on the specific approach used. Indirect models generate a cost function that is mapped to state-action tuples, which can be used to interpret the proposed output of a policy. Nevertheless, it is challenging to explain how the cost function is determined from an expert's demonstration. Direct models that output a policy do not explicitly derive a cost function from demonstration, making them less explainable. These models can directly consider the interaction between multiple objects as an input feature. Additionally, a wide range of features, including semantical information from road maps, can be used as input, making these models holistic.
Reinforcement learning-based models have the ability to describe complex maneuvers by utilizing the underlying policy. However, the model's output typically consists of discrete maneuvers because policies comprise state-action tuples that objects may execute. Although explicit trajectories can be derived from subsequent modules, such as a Recurrent Neural Network (RNN) demonstrated in <cit.>, these models heavily rely on diverse data, including demonstrations, for training. Extracting comprehensive cost functions or robust policies is particularly challenging as it strongly relies on expert behavior observations, making it difficult to train correctly. Reinforcement learning-based models are designed to reason about an object's motion, allowing them to adapt well to unknown scenarios. However, similar to Deep learning-based models, holistic models based on the reinforcement learning approach have high computational costs. Moreover, the complexity of learning a robust policy negatively affects prediction accuracy.
§ CHALLENGES AND FUTURE RESEARCH DIRECTIONS
Based on the above survey, this section highlights the research challenges and future research directions in the domain of Autonomous Vehicles (AVs) trajectory predictions.
§.§ Challenges
Trajectory prediction is a critical component of AV systems, as it enables them to anticipate the future motion of traffic agents such as vehicles, pedestrians, and cyclists in their environment. However, there are several challenges specific to the domain of AVs that make trajectory prediction exceptionally challenging:
* Uncertainty: The future trajectory of traffic agents is inherently uncertain, and it is impossible to predict it with 100 percent accuracy. Various factors such as noise in sensor measurements, unpredictable environmental changes, and unknown intentions of other traffic agents can contribute to this uncertainty.
* Complex dynamics: The motion of traffic agents can be affected by various physical laws, including gravity, friction, and aerodynamic forces. These dynamics can be highly complex and nonlinear, making it difficult to model accurately.
* Limited sensor coverage: Autonomous vehicles rely on a suite of sensors, including cameras, LiDAR, and radar, to perceive their environment. However, the coverage of these sensors is limited, as depicted in Fig. 18, and can be affected by occlusions, weather conditions, and other factors that can make it difficult to accurately track the motion of other traffic agents.
* Limited data: In some cases, there may be limited or incomplete data available for trajectory prediction. This can occur when sensors are malfunctioning, or when the historical data is missing or corrupted.
* Long-term prediction: Predicting trajectories over a long time horizon (no less than 3 seconds) can be challenging, as small errors in the initial prediction can compound and result in significant deviations from the true trajectory.
* Complex road environments: Autonomous vehicles operate in complex and dynamic road environments, which can include intersections, roundabouts, and crowded urban areas. Predicting trajectories in these environments requires models that can handle complex interactions between multiple agents, including other vehicles, pedestrians, and cyclists.
* Multimodal Outputs: In autonomous driving, agents' behaviors exhibit multimodality, where a single past trajectory can have multiple potential future trajectories, as depicted in Fig. 19.
* Sparse and noisy data: The data from sensors can be sparse and noisy, particularly in urban areas where buildings and other structures can obstruct the line of sight between the sensors and the objects being tracked. This can make it difficult to accurately model the motion of other traffic agents over time.
* Multi-agent interactions: In many real-world scenarios, multiple agents interact with each other, and their trajectories are interdependent. Predicting the trajectory of one agent may depend on the actions of other agents, as depicted in Fig. 20, making the problem even more challenging.
* Heterogeneous environment: A Heterogeneous environment refers to an environment that contains a diverse range of elements, such as various types of vehicles, pedestrians, cyclists, different road types, and complex interactions among them. In order to effectively predict trajectories in such environments, prediction models need to account for the different types of agents, incorporate contextual information, fuse sensor data, model interactions among multiple agents, estimate uncertainty, and enable adaptability.
* Safety-critical applications: Autonomous vehicles are safety-critical systems, and errors in trajectory prediction can have serious consequences, including accidents and injuries. As a result, trajectory prediction algorithms need to be highly accurate and reliable, with well-defined safety margins.
* Real-time constraints: Autonomous vehicles operate in real-time environments, and trajectory prediction algorithms need to be able to process data and generate predictions in real-time. This requires efficient algorithms and hardware architectures that can handle the large amounts of data generated by the sensors.
§.§ Future Research Directions
The field of trajectory prediction is undergoing rapid evolution in research, offering numerous opportunities for future investigations, particularly within the realm of autonomous vehicles. Several potential research directions can shape the trajectory prediction landscape. Here, we outline the identified futuristic research directions as follows:
* Incorporating context and intention: One limitation of current trajectory prediction methods is that they often focus solely on the motion of other vehicles, without considering the context or intention behind that motion. Future research could explore how to incorporate contextual information, such as road layout and traffic rules, as well as the intention of other drivers, to improve trajectory prediction accuracy.
* Integration of multiple sensors:
Autonomous vehicles rely on a suite of sensors to perceive their environment, and future research could explore how to integrate data from multiple sensors to improve trajectory prediction accuracy. This could involve developing new algorithms for fusing data from cameras, LiDAR, radar, and other sensors, as well as exploring new sensor modalities such as acoustic or thermal sensors.
* Uncertainty modeling: Trajectory prediction is inherently uncertain, and future research could explore how to model and propagate uncertainty through the prediction pipeline. This could involve developing new probabilistic models, such as Bayesian neural networks, or exploring new techniques for uncertainty quantification and propagation.
* Human-aware trajectory prediction: Autonomous vehicles operate in environments that include not only other vehicles but also pedestrians and cyclists. Future research could explore how to develop trajectory prediction methods that are aware of human behavior and can accurately predict the motion of pedestrians and cyclists in crowded urban environments.
* Real-time implementation & Hardware acceleration: Autonomous vehicles operate in real-time environments, and trajectory prediction algorithms need to be able to process data and generate predictions in real-time. Future research could explore how to optimize trajectory prediction algorithms for real-time performance, as well as developing new hardware architectures for efficient computation.
* Ensuring safety and robustness: Safety is of paramount importance in autonomous driving systems. Future research should aim to develop trajectory prediction methods that prioritize safety and robustness. This includes investigating techniques for handling rare or anomalous events, improving prediction accuracy in challenging weather conditions, and considering ethical aspects in trajectory prediction algorithms.
* Relative trajectory prediction: Relative trajectory prediction refers to the task of predicting the future motion or path of surrounding objects or agents relative to the ego vehicle or coordinate system. Future research should focus on estimating the relative displacement, velocities, and trajectories of other vehicles, pedestrians, and cyclists with respect to the ego vehicle.
* Random obstacle aware trajectory prediction: This approach refers to predicting the future trajectories of a vehicle while considering the presence of unexpected or random obstacles in the surrounding environment. These obstacles can be animals or objects in between roads, the sudden arrival of pedestrians, and road accidents that lead to an uncertain obstacle in between roads. Future research should focus on incorporating rare events into the prediction models and collecting and analyzing data related to these rare events to develop more comprehensive and robust prediction models.
* Challenging Weather condition: Adverse weather conditions, such as heavy rain, snow, fog, or low visibility, can affect the performance of sensors and limit the availability of critical data for trajectory prediction. Future research should focus on involves incorporating techniques such as sensor fusion, adaptive filtering, probabilistic modeling, and machine learning to improve the reliability and accuracy of trajectory predictions under adverse weather conditions.
* Vehicle-to-Vehicle (V2V) communication and Vehicle-to-Everything (V2X) communication
strategies:
V2V communication refers to the exchange of information directly between vehicles. V2X communication expands beyond V2V and includes communication with other entities such as infrastructure, pedestrians, cyclists, and traffic management systems. By sharing real-time data such as position, speed, acceleration, and intentions, vehicles can collaborate and cooperate to enhance trajectory prediction.
* Hybridization of several approaches:
Multiple strategies are suggested in Sections 3, 4, and 5 for solving the task of trajectory prediction. Hybridization can take different forms depending on the specific context and requirements. This can lead to more accurate and robust trajectory predictions
§ CONCLUSION
This paper provides an extensive survey of the current state-of-the-art Machine Learning (ML)-based trajectory prediction methods for Autonomous Vehicles (AVs). These ML-based approaches have demonstrated significant promise in accurately predicting trajectories, employing several techniques like deep learning-based methods and reinforcement learning-based methods. Deep learning-based methods including sequential models, vision-based models, and generative models are thoroughly explored, highlighting their respective strengths and weaknesses in trajectory prediction tasks. Furthermore, the review focuses on the discussion of reinforcement learning methods, including Inverse reinforcement learning, deep inverse reinforcement learning, and imitation learning techniques. Multiple informative tables and figures are provided to facilitate a comprehensive comparative study of various approaches used to address trajectory prediction tasks. The review paper includes an analysis of multiple datasets and evaluation metrics used to assess the accuracy of trajectory prediction tasks. This conducts a comparative analysis between deep learning-based methods and reinforcement learning methods across various characteristics. Recent advances in trajectory prediction for AVs show promise, but there are still several challenges that need to be addressed. The paper outlines potential research directions, emphasizing the need for more robust and interpretable models and the exploration of new sensor modalities. The survey aims to provide a valuable reference for researchers and practitioners in this field and guide future advancements in the trajectory prediction domain.
IEEEtran
[
< g r a p h i c s >
]Vibha Bharilya received his M.Sc degree in Mathematics and Scientific Computing from the National Institute of Technology, Warangal in 2019, and an M.Tech. degree in Computer Science & Technology from Jawaharlal Nehru University, New Delhi. Currently, she is pursuing Ph.D. in the Department of Computer Science and Engineering from Indian Institute of Technology, Roorkee, India. His research interests include Autonomous Vehicles motion, Autonomous Vehicle Driving, and Self Driving Cars.
[
< g r a p h i c s >
]Dr. N. Kumar received his M.tech and Ph.D. degrees from the School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi. He has received Special Mention Awarded Xerox Research Centre Bangalore and several best paper awrads. Currently, he is working as a faculty member at IIT-Roorkee, India. He has published sevral research publications in world’s top tier publishers like IEEE journals and transactions, Elsevier journals (including Future Generation Computer System (FGCS), Information Sciences etc.) and Springer Journals. One of his articles published in IEEE sensors journal has been notified by IEEE council in the list of world’s top 15 most downloaded articles in the month of Oct-Nov 2018. He is also acting as a lead PI for four sponsored projects from IIT Roorkee/DST/CSIR agencies, Government of India, and few proposals are in under process by several agencies. He has also filed four patents individual and in collaboration as well. He has been a technical program committee member in several conferences. He has also been invited as a keynote speaker in the conference held at Amity Gwalior, Conference Chair at IEEE CICT-2019 at IIIT Allahabad, IIT Indore etc. Broadly, his research interests include Algorithm Design, IoT, High Performance Computing (Cloud, Fog and Parallel computing), Applied Evolutionary Computing, Software Defined Networking, WSN and Intelligent Transportation Systems.
|
http://arxiv.org/abs/2307.04412v1 | 20230710083245 | Enhancing Biomedical Text Summarization and Question-Answering: On the Utility of Domain-Specific Pre-Training | [
"Dima Galat",
"Marian-Andrei Rizoiu"
] | cs.CL | [
"cs.CL"
] |
2023
Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
CLEF 2023: Conference and Labs of the Evaluation Forum, September 18–21, 2023, Thessaloniki, Greece
mode=sub]University of Technology Sydney participation in BioASQ Task 11b Phase B
]Dima Galat[
orcid=0000-0003-3825-2142,
email=dima.galat [@] student.uts.edu.au,
url=https://github.com/dimagalat/
]
]Marian-Andrei Rizoiu[
orcid=0000-0003-0381-669X,
email=Marian-Andrei.Rizoiu [@] uts.edu.au,
url=https://www.rizoiu.eu/,
]
[]University of Technology Sydney (UTS), Australia
Biomedical summarization requires large datasets to train for text generation. We show that while transfer learning offers a viable option for addressing this challenge, an in-domain pre-training does not always offer advantages in a BioASQ summarization task. We identify a suitable model architecture and use it to show a benefit of a general-domain pre-training followed by a task-specific fine-tuning in the context of a BioASQ summarization task, leading to a novel three-step fine-tuning approach that works with only a thousand in-domain examples. Our results indicate that a Large Language Model without domain-specific pre-training can have a significant edge in some domain-specific biomedical text generation tasks.
natural language processing biomedical summarization biomedical question answering transfer learning language modeling domain-specific pre-training BioASQ CEUR-WS
[
[
Received / Accepted
========================
§ INTRODUCTION
The fields of question-answering and summarization have witnessed significant advancements in recent years, with a shift from classification-based extractive approaches to the emergence of abstractive summarization models.
This transition has been driven by the superior performance and enhanced generalization capabilities exhibited by abstractive models, effectively blurring the boundary between long-form question answering and summarization.
This paper addresses the summarization challenge presented by BioASQ Task B Phase B in the biomedical domain, for which we propose a novel approach.
The healthcare sector holds immense potential for leveraging health research data sharing to enhance clinical care, informed decision-making, and scientific discovery <cit.>.
Sharing biomedical and healthcare studies and research data with the wider public requires robust and efficient methods.
Large pre-trained language models (LLMs) have emerged as promising candidates for this purpose.
LLMs have the potential to store medical knowledge while accommodating variations in data and application tasks <cit.>.
This paper aims to analyze the impact of the training process on LLMs' ability to store biomedical knowledge, explicitly focusing on their utilization for a question-answering and summarization task.
Traditionally, achieving state-of-the-art performance on natural language processing tasks involves a two-phase approach <cit.> that is shown in blue in the top row of <ref>:
pre-training the models on an extensive range of texts and topics, followed by task-specific fine-tuning <cit.>.
This approach has revolutionized various areas of natural language processing <cit.>, with LLMs such as BERT, GPT, and BART demonstrating remarkable capabilities.
However, pre-training models is a time-consuming and a resource-intensive process, and the literature lacks comprehensive insights into the performance of these models for domain-specific applications with limited data availability.
Therefore, this study aims to address this gap by examining the performance of LLMs in the context of the BioASQ summarization task.
This paper investigates two open questions concerning biomedical domain question-answering and text summarization tasks.
Over the past five years, the biomedical domain has increasingly relied on in-domain pre-training and fine-tuning of BERT <cit.> for a wide range of datasets and benchmarks <cit.>.
In-domain pre-training has proven effective in enhancing performance for discriminatory biomedical tasks.
However, BERT's architecture is not optimized for text generation tasks <cit.>, lacking an autoregressive decoder to generate tokens based on previously generated ones.
Consequently, BERT is suboptimal for generation tasks, necessitating exploring alternative approaches.
Previous studies evaluating biomedical models across diverse tasks have not reported results on generation problems due to using non-autoregressive models <cit.>.
The first question is is there a better-suited architecture for biomedical text generation tasks?
A significant amount of research suggests that domain-specific pre-training significantly outperforms mixed-domain pre-training.
However, we could not find any convincing evidence for supporting this belief when it comes to text generation problems <cit.>.
The second question is do LLMs need to be pre-trained in domain to achieve optimal performance?
We answer the above two questions.
To investigate the efficacy of domain-specific pre-training and fine-tuning for biomedical text generation, we propose an alternative three-step approach (shown in the bottom row of <ref>).
In this approach, we initially train a general-domain LLM, followed by fine-tuning for a specific task in the general domain (text summarization) and subsequent fine-tuning for the target biomedical domain task.
Contrary to established theories in the biomedical domain <cit.>, our findings suggest that having a large task-specific dataset can be more valuable than domain-specific pre-training for biomedical text generation tasks.
This approach aligns with studies indicating that diverse pre-training objectives, larger and more diverse datasets, and tasks contribute to the robustness of the fine-tuning process even without domain adaptation <cit.>.
We explore alternative architectures for biomedical text generation.
In this study, we focus on BART <cit.>, a comprehensive architecture that incorporates pre-training objectives from both BERT <cit.> and GPT <cit.> models.
BART has demonstrated state-of-the-art performance in abstractive dialogue, question-answering, and summarization tasks, making it particularly effective for text generation and comprehension.
Our experimental results showcase the benefits and effectiveness of utilizing the BART architecture for transfer learning techniques in a context of a biomedical summarization task.
The main contributions of this work can be summarized as follows:
* Evaluating the advantages of domain-specific pre-training in the context of text generation tasks.
* Evaluating the impact of task-specific training on improving text generation tasks.
* Assessing the performance of BART, an encoder with an auto-regressive decoder architecture, in the biomedical question answering task B of BioASQ 11.
§ RELATED WORK
We are looking for a LLM which has an architecture suitable for long-form question answering and has been trained on relevant in-domain data. There are several important model architectures and pre-training objectives used to optimize the models worth considering <cit.>.
First, lets briefly mention BERT <cit.> in the context of text generation, since most biomedical Transformer-based <cit.> models still rely on this architecture. BERT does not have an autoregressive decoder, preventing it from generating text. Despite this fact, a well-known summarisation approach called PreSumm <cit.> uses this architecture by inserting additional tokens for teaching models which sentences should be included in the summary. We followed the process proposed by the authors while using a BioBERT <cit.> model; we first trained an extractive summariser, which did perform a little better on BioASQ data than a regular BERT trained the same way. Unfortunately, when training an abstractive summarization architecture, PreSumm <cit.> process uses a randomly initialised Transformer <cit.> for a decoder. It appears that there is a significant mismatch between this decoder and a BioBERT <cit.> encoder leading to unstable abstractive fine-tuning process and poor generation outputs in our experiments. Based on these findings, we have concluded that BERT is a wrong architecture to be using for text generation tasks.
BART <cit.> is an architecture that uses an encoder with an auto-regressive decoder, similarly to the original Transformer <cit.>. BART relies on an architecture which can be seen as generalising BERT (because it also uses a bi-directional encoder) and GPT <cit.> (because it also uses the left-to-right decoder). This model is using a masked language modeling objective (also known as denoising) introduced by BERT <cit.> and adds two additional denoising objectives (token deletion and sentence permutation). Authors conduct experiments that are focused on text generation, and show that denoising objectives are particularly well-suited for summarization tasks. Because it can be easily fine-tuned directly for generation tasks, authors achieved a remarkable success on a wide range of abstractive summarization and long-form question answering problems <cit.>.
BioBART <cit.> is a BART model pre-trained on PubMed <cit.> abstracts. Authors have reported that they have trained without one of the objectives proposed by BART, namely the sentence permutation, showing that models trained without this objective have a better performance. Overall, this is the only study that we are aware of that applies a LLM to a range of generation tasks and reports the results (another BioGPT <cit.> study we found has not reported any numeric results on text generation problems). We are also not completely convinced that some of the results, like those reported for a BioASQ task could not be a result of a random chance, since the differences in the scores are very small and there are a few possible sources of non-determinism in training and generation procedures we discuss later in this paper.
§ OUR CONTRIBUTION
In the biomedical domain, the majority of models we have reviewed are focused on the pre-training process, perhaps because pre-training data is readily available <cit.>. However, question answering and summarization are plagued by a lack of a large domain specific dataset for fine-tuning LLMs directly for text generation problems. More specifically, when we are looking at the biomedical text generation tasks, it's hard to find a large (and clean) sequence-to-sequence dataset for fine-tuning for a long-form question answering and summarization. BioASQ is the closest dataset currently available, however it is still a few orders of magnitude away from what we would require to fine-tune a LLM for a previously unseen generation task. Therefore, we conclude that this two-step fine-tuning process offers a limited utility for this problem.
Following a conventional transfer learning definition we use a task to refer to training on labeled data, seeking to transfer the knowledge from a source task and a source domain (𝒯_S and 𝒟_S) to a target task and a target domain (𝒯_T and 𝒟_T) <cit.>. One of the common transfer learning scenarios involves learning the tasks sequentially, one after another; and we could also have an intermediate fine-tuning task making it a three-step fine tuning process, where a second step is only used to get a representation that is more suitable for the task of summarization in a biomedical domain. This means that an intermediate 𝒯_inter (which could be both in/out domain) should lead to a performance improvement in 𝒯_T. This could be potentially useful, since task-domain specific data is hard to come by.
Since we need to perform text generation, a reasonable option is to train for an 𝒯_inter which teaches the model to perform this task. Unfortunately, large question answering and summarization datasets like SQUAD <cit.> and CNN/DM <cit.> have nothing to do with biomedical domain, but because we need 10-100 times more biomedical summarization data than what we have available, we believe that task-specific datasets could offer just as much value as a domain-specific pre-training. We believe that CNN/DM is the most suitable (clean, large, easily available) task-specific dataset; especially because summaries there are typically closely related to source sentences, which is also the case with the BioASQ data. Moreover, lengths of summaries are similar to those in BioASQ. Therefore, we are interested in this task, even though a newsmedia domain would likely have completely different marginal probability distributions of generated text. This approach means that in addition to sequential transfer learning (two and three step fine-tuning processes described above), models competing with a two-step fine-tuning strategy would have to also adapt for the domain difference (i.e. differences in prior and conditional distributions). Second 𝒯_inter we have considered for training is Pubmed <cit.> article-abstracts combinations. While these are not summaries in the stricter sense of the word, this is the closest domain-specific dataset that we could find, and we would like to understand if it adds useful information to a LLM.
§ MODELS COMPARED
We select LLMs that reduce the amount of overall training required.
We select a mix of domain-specific pre-training and general pre-training datasets, and we attempt different 𝒯_inters to see how well the resulting models generalize to 𝒯_T, namely BioASQ Task 11b Phase B. Hence, the final list of LLMs we are considering are:
* BART - a baseline two-step LLM (without additional fine-tuning) used to establish a baseline for a general domain model without specialized domain knowledge or 𝒯_inter fine-tuning
* BioBART - a biomedical two-step LLM (without fine-tuning 𝒯_inter), used to establish a baseline for an in-domain model
* BART CNN - a baseline LLM three-step LLM with task-specific fine-tuning 𝒯_inter but without any deep domain knowledge
* BioBART CNN - a biomedical three-step LLM with task-specific fine-tuning 𝒯_inter
* BART CNN Pubmed - a general domain three-step LLM fine-tuned for 𝒯_inter summarisation task, and then further fine-tuned on a domain-specific 𝒯_inter dataset containing Pubmed articles
Based on the data available, we believe that these tasks and LLMs offer the greatest benefit for biomedical summarization, and we limit our selection to 5 models that will participate in the BioASQ competition.
We are only considering large models because we want the model to analyze as much context as possible, and therefore having a large model helps to double the context length (1024 tokens vs. 512 tokens).
We are using pre-trained BART, BioBART, and BART CNN models available via Huggingface[<https://huggingface.co/models>]; and we are fine-tuning BART CNN on Pubmed data and BioBART on CNN data for one epoch each (our 𝒯_inter). Subsequently, all models are fine-tuned on the latest complete BioASQ 11 dataset (𝒯_T) for five epochs using a 10-fold cross-validation process. We empirically chose the number of training epochs to maximize the final model scores.
We've tried training on Pubmed (with and without training on CNN), and found it beneficial when using a general-domain model. Despite this, CNN dataset is a much better 𝒯_T for BioASQ. Using Pubmed (summarisation) data for fine-tuning BioBART before or after CNN training didn't offer advantages (<ref>, <ref>) and was excluded from the top five models under consideration.
§ RESULTS
Our experiments have revealed a substantial (over 10%) variation in ROUGE <cit.> score results based on a simple choice of a seed parameter for cross-validation.
This indicates that the fine-tuning process is susceptible to changes in data.
Future studies should consider which BioASQ information snippets are passed to the model as the input for summarization training. Working with small healthcare question-answering datasets can require a more careful knowledge extraction process <cit.>.
We have experimented with fine-tuning for up to 10 epochs on the 𝒯_T, and found that this problem consistently persists across a range of training scenarios. In-domain studies we have reviewed show that the generation results can often differ by a minimal margin, significantly lower than the variation in scores we have observed in cross-validation.
To our knowledge, this research is the first to draw attention to this specific problem, and we decided to overcome this by repeating the 10-fold cross-validation training process 𝒯_T four times using a different seed value.
Therefore, we effectively report the average of 400 runs for each model (95% t-test confidence interval is given in parentheses), with 100 runs for each seed choice (ten for each fold).
We are primarily focused on SU4-F1 scores (<ref>) since they have been shown to correlate with human scores the best <cit.>.
However, ROUGE is recall-oriented; therefore, we also look at Recall results separately (<ref>).
Our experiments (<ref>) suggest that LLMs without domain-specific pre-training show a better capacity for domain-specific text generation. This becomes particularly clear when comparing BART and BioBART results before any additional task-specific fine-tuning, suggesting that BioASQ data is not as similar to Pubmed pre-training data as we would expect based on other results reported on discriminatory tasks.
Moreover, we believe that currently a non-domain specific CNN summarization task 𝒯_inter is required to accomplish the best results on a BioASQ task.
Adding in-domain Pubmed data improves Recall; however, Pubmed data is unsuitable for training for a summarization task from scratch. ROUGE Recall scores (<ref>) show one notable difference, BART CNN has a higher recall, whereas BART CNN Pubmed has a higher precision, likely because the Pubmed training after the task-specific training introduces a task-specific vocabulary to the model.
Overall, LLMs have established some remarkable results in various practical applications.
However, since LLMs require task-specific datasets to train to generate text, and such domain-specific datasets are scarce, we need to find ways to overcome these challenges.
We have presented an approach that focuses on applications of transfer learning to a domain with limited task-specific training data.
§ CONCLUSION AND FUTURE WORK
In this work, we have observed that task-specific data is critical for generating text in a biomedical domain.
Based on our experiments, models without in-domain pre-training are better at summarizing BioASQ data.
Unfortunately, our models have achieved fairly modest automated ROUGE scores during BioASQ 11 runs, and we are waiting for the final results to determine how the models have performed overall. The generation process is non-deterministic, and while the answers generated by the models appear sensible, we need better ways to evaluate the candidates.
We have discussed how transfer learning can overcome challenges with data availability.
We see a lot of exciting possibilities for using generator models (more specifically paraphrasing, simplification, and rewriting models <cit.>) for creating synthetic training data, as well as for providing a differentiable loss function which allows sampling a wider space of possible answers without over-penalizing exploration.
Abstractive summarization models are trained to generate specific gold sequences, even when they start making errors in the first steps (a problem known as exposure bias <cit.>).
One recent improvement over BART proposes generating multiple candidates and comparing them, showing a new SOTA on several popular summarization datasets <cit.>.
This could address a common shortcoming of autoregressive models, leading to further performance improvements.
Another possibility that shows a significant promise would be generating synthetic data to augment BioASQ.
This approach has recently shown good results in machine translation <cit.>, and we believe it can be used for other text-generation problems.
§ ADDITIONAL RESULTS
|
http://arxiv.org/abs/2307.04438v1 | 20230710092926 | Reconfigurable Intelligent Surface Assisted Railway Communications: A survey | [
"Aline Habib",
"Ammar El Falou",
"Charlotte Langlais",
"Marion Berbineau"
] | eess.SP | [
"eess.SP"
] |
Reconfigurable Intelligent Surface Assisted Railway Communications: A survey
Aline Habib1, Ammar El Falou2, Charlotte Langlais1, Marion Berbineau4
1 Mathematical and electrical engineering department, CNRS UMR 6285 Lab-STICC, IMT Atlantique, Brest, France
2 CEMSE Division, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
4 COSYS-LEOST, Université Gustave Eiffel, Villeneuve d'Ascq, France
Email: {aline.habib, charlotte.langlais}@imt-atlantique.fr, [email protected], [email protected]
========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
The number of train passengers and the demand for high data rates to handle new technologies such as video streaming and IoT technologies are continuously increasing. Therefore the exploration of millimeter waves (mmWave) band is a key technology to meet this demand. However, the high penetration loss makes mmWave very sensitive to blocking, limiting its coverage area. One promising, efficient, and low-cost solution is the reconfigurable intelligent surface (RIS). This paper reviews the state of the art of RIS for railway communications in the mmWave context. First, we present the different types of RIS and review some optimization algorithms used in the literature to find the RIS phase shift. Then, we review recent works on RIS in the railway domain and provide future directions.
RIS, Railway communications, mmWave.
§ INTRODUCTION
The need to double the capacity of the existing rail networks and, at the same time to increase the overall quality of service is leading to a drastic increase in the need for high data rates and robust and low latency data exchange between the different actors in the rail system. This multiplication of transmission needs ultimately leads to problems of spectrum scarcity. In this context, using mmWave bands opens up new opportunities. However, mmWaves suffer from very high attenuation and high sensitivity to various masking effects. In this context, Reconfigurable Intelligent Surfaces offers promising application use cases.
Reconfigurable Intelligent Surface, known in the literature by several nomenclatures as Software-Controlled Metasurface <cit.>, Intelligent Reflecting Surface (IRS) <cit.>, Large Intelligent Surface (LIS) <cit.>, and Reconfigurable Smart Surface (RSS) <cit.>, is an electromagnetic-based reconfigurable structure
that turns the random nature of the propagation channel into a controllable and programmable radio environment. RIS is a thin planar meta-surface made of several low-cost reflective elements <cit.>. Each RIS element adjusts the phase and amplitude of the incident wave to reflect it into a beam toward the target direction. This improves the signal quality and extends the coverage area especially when the direct link is blocked. The paper's main objective is to provide the reader with the basic elements to understand RIS and its interest in a railway communication environment. To do so, we review the literature in the domain and propose some future research directions.
The rest of the paper is organized as follows. Section <ref> provides a literature overview related to RIS, such as the different RIS structures and types, and their opportunity in the context of mmWave communications. We stress the need for realistic channel models in order to properly evaluate the performance of RIS-assisted systems. Section <ref> focuses on very recent works investigating RIS-assisted systems for railway communications. Finally, in Section <ref>, some future directions are drawn, and Section <ref> concludes the paper.
§ RECONFIGURABLE INTELLIGENT SURFACE
§.§ RIS General Overview
The main objective of a RIS is to provide a programmable radio environment between a transmitter (Tx), typically a base station (BS) in the downlink case, and a receiver (Rx), typically a remote user equipment (UE), by changing the phase shifts and amplitude of the RIS incident wave as follows <cit.>
z_n=β_ne^jθ_n,
where z_n is the reflection coefficient of the n^th element, β_n and θ_n are the adjustments in amplitude and phase due to the n^th element.
As the RIS should not encompass too many RF and signal processing resources to maintain a low level of energy consumption and complexity, the BS computes the needed tunable parameters and transfers commands to each RIS element thanks to a smart controller <cit.> as seen in Fig.<ref>.
To adjust phase shifts and amplitude of the incident wave, RIS consists of adjustable components, such as diodes and liquid crystals. The diodes adjust the signal by changing the bias voltage, while the liquid crystals adjust the electromagnetic signal by changing material parameters such as conductivity and permeability <cit.>. Indeed, the PIN diode-based RIS consists of three layers: 1) The outer layer with printed metal patches on a dielectric substrate. This layer directly processes the incident signals. 2) The intermediate layer
composed of a copper panel to avoid signal energy loss.
3) The inner layer is a control board activated by a programmable digital electronic circuit (FPGA), allowing the real-time adjustment of the RIS elements' reflection coefficients <cit.>.
Two reflexion paradigms govern propagation in the context of RIS-assisted communication systems, namely, the specular reflection paradigm and the scattering reflection paradigm, <cit.>. The differences are mainly related to the relation between the size of RIS A_t and the distance D between BS-RIS or RIS-UE, as follows:
* The specular reflection paradigm: the transmission occurs in the near-field, i.e., D<d_lim = 2A_t/λ[d_lim denotes the Rayleigh distance and is defined by d_lim=2A_t/λ with A_t the RIS area and λ the wavelength <cit.>.]. The path loss, in this case, depends on the summation of the distances between BS-RIS and RIS-UE.
* The scattering reflection paradigm: the transmission occurs in the far-field, i.e., D>d_lim. In this case, the path loss depends on the product of the BS-RIS and RIS-UE separation distances.
In the case of a passive RIS (β_n ≤ 1), the RIS elements reflect the signal without amplification. Thus, in the context of scattering reflection communications (far-field), and by assuming the optimal phase shifts, the received power at the UE for the indirect link via passive RIS is expressed as <cit.>
P_r^UE=P_tG_tG_r(λ/4π) ^4(d_0)^2μ-4/(d_1d_2)^μN^2
where P_t is the transmitted power at the BS, G_t and G_r are the transmit and receive antenna gains at the BS and the UE, respectively, d_0 the reference distance in the free space, d_1 and d_2 are BS-RIS distance and RIS-UE distance, μ is the path loss exponent depending on the environment type (e.g., μ≥ 3 for urban environments), and N is the number of RIS elements. Thus, the passive RIS gives a gain proportional to N^2. However, the passive RIS has limitations due to the double-path loss effect. Indeed, the signal traverses two cascaded channels, the Tx-RIS link and the RIS-Rx link <cit.>. Thus, the received power via the indirect link could be greater
than the power of the direct link, if N is large, or/and if
the direct link is weak or blocked. To illustrate this concept, we plot in Fig. <ref> the power received at the UE via an
attenuated direct link, a direct link without attenuation, and the indirect
link via the RIS, versus the number of RIS elements. The mmWave channel links are generated using an extended version of the New York University simulator NYUSIM <cit.>. Note that to verify the RIS scattering reflection paradigm, the distances d_1, and d_2 must be in the far-field region. As RIS size increases, the distance where the RIS is in the near-field also increases. Thus, for the distances d_1, and d_2 to be in the far-field and equation (<ref>) to be valid, N must not exceed a certain N_max, computed from d_lim, the Rayleigh distance, and represented by a square in Fig. <ref> <cit.>. The behavior of the RIS in the near-field is an interesting research topic.
§.§ RIS types
To overcome this limitation and obtain an efficient RIS when the direct link exists or the number of RIS elements is low, the authors of <cit.> propose an active RIS that can amplify the reflected signals through amplifiers embedded in the RIS elements. The simulation results in a direct link scenario without attenuation for 256 RIS elements reveal a negligible sum-rate gain
of 3 % using the passive RIS, while their proposed active RIS offers a significant sum-rate gain of 67 % compared to the case without RIS.
Nevertheless, a RIS with a large number of active elements consumes more energy. Thus, the authors in <cit.> propose a novel type of RIS composed of active and passive reflective elements, called hybrid RIS, to deal with the limited power budget of the RIS.
RIS based on continuous phase shifts is considered an ideal system that is difficult to implement in practice. Therefore, RIS based on finite discrete phase shifts is the alternative solution to cope with this hardware constraint. To this end, the authors in <cit.> compare the performance of RIS systems with continuous and discrete phase shifts and they find that 3 levels of quantization are sufficient to obtain full diversity.
§.§ RIS optimization
The efficient functioning of the RIS is strongly affected by the adapted phase shifts θ_n. For instance, in Single Input Single Output (SISO) systems, the optimal phase shift of a RIS is easily determined analytically as follows <cit.>
θ_n=θ_tn+θ_nr.
where θ_tn and
θ_nr are the phase of the LoS path in the BS-RIS
and RIS-UE channels, respectively.
However, it is hard to find the optimal phase shifts analytically in the case of Multiple Input Multiple Output (MIMO) systems. To this end, an optimization algorithm is needed. <cit.> studied multi-user Multi Input Single Output (MISO) downlink communications assisted by RIS, where the objective is to maximize the weighted sum rate to find the optimized passive beamforming θ_n and the optimized precoding at the BS.
To solve this non-convex problem, they used the Lagrangian Dual Transform which transforms the sum-of-logarithms-of-ratio to an alternative form.
The authors in <cit.> discussed an indoor MISO multi-user system with a channel model based on the Rician K-factor. The RIS phase shift were configured as follows
θ_n^*=(H_d^H) - (H_l^H)-(H),
where H_d is the direct channel between the Tx and the Rx, H is the channel between Tx and RIS, and H_l is the channel between the RIS and the lth user.
In <cit.>, the authors adopted a low-complex algorithm called the cosine similarity algorithm. The latter aims to find the sub-optimal phase shifts of the RIS that maximize the channel gain. Moreover, to minimize the transmitted power given the bit error rate for a RIS-assisted single-user multipath uplink system,
the authors of <cit.> propose an iterative algorithm to jointly optimize precoding and passive beamforming. In addition, a deep learning algorithm is applied in <cit.> to maximize the received signal-to-noise ratio and find the optimal phase shifts of RIS.
§.§ RIS versus Relay
Both RIS and relay aim to improve signal quality and coverage. However, there are two main differences.
* In the case of RIS, a power supply is only needed to configure the RIS components based on low-cost materials (diodes, switches...). Once the configuration is done, the RIS becomes passive, and no power supply is needed <cit.>. However, relays are generally considered active devices connected to active electronics such as analog-to-digital converters, digital-to-analog converters, amplifiers, etc., which require a power supply for operation. As a result, relays are more complex to implement and consume more energy than RIS <cit.>.
* A RIS operates in a full duplex mode while relays generally work in a half-duplex mode. Relays can still operate in full duplex mode, but this increases their cost, since appropriate antennas and analog and/or digital signal processing, to eliminate loop-back self-interference, are required <cit.>.
§.§ RIS is an opportunity for mmWave communications
The mmWave band, ranging from 30 to 300 GHz, offers enormous free bandwidth and high data rate possibilities <cit.>, unlike the overloaded low-frequency spectrum. However, it is very vulnerable to oxygen absorption and rain attenuation, and also suffers from penetration loss that makes mmWave signals easily blocked. Therefore, the coverage of mmWave communications is limited<cit.>. On the other hand, when the direct link is blocked or largely attenuated, a RIS is a competitive solution to extend coverage area and connectivity <cit.>. The location of the RIS should be optimized to obtain two efficient connections: the BS-RIS link and the RIS-UE link.
The authors in <cit.> discuss the size limitation of the RIS in low frequencies below 6 GHz, which makes their deployment in this band inefficient. A study of the specific propagation characteristics of the terahertz band is needed to use RIS in these frequencies, and the most important implementation of RIS today is in the mmWave band.
In the literature, the most used channels in RIS-assisted systems are the theoretical channels such as Rice for Line-of-Sight (LOS) environments, and Rayleigh for non-LOS (NLOS) <cit.>, <cit.>. To fill the gap towards realistic channel modeling and simulator, the authors in <cit.> propose a novel geometrical channel simulator, called SimRIS. This simulator is based on statistical modeling and can be used in indoor and outdoor environments at 28 and 73 GHz frequencies. Moreover, in <cit.>
the authors extend QuaDRiGa, a simulator used to model MIMO radio channels at sub-6GHz and mmWave frequencies, to handle RIS. This simulator is convenient for RIS-assisted MIMO systems with a mobile Rx or mobile RIS. In addition, <cit.> discusses the extension of NYUSIM, a mmWave channel simulator based on extensive measurements and well-used to assess MIMO systems <cit.>, to generate realistic channels for RIS-assisted systems.
§ RIS-ASSISTED RAILWAY COMMUNICATIONS
§.§ Railway environments characteristics
Railway environments are known to be very complex and harsh from a radio point of view. Various obstacles such as pylons supporting the catenary and rapid transitions between different scenarios (cutting/tunnel, cutting/viaduct) can create severe radio impairments. Railway tunnel size and shape are very specific, depending on the category of the train. Radio propagation inside tunnels is often modeled using Ray tracing tools <cit.>, <cit.>. It is also important to mention that MIMO system performance in tunnels is subject to possible impairments depending on spatial correlation in the tunnel and also Key holes phenomenon <cit.>. Due to high speed, the train can rapidly go through diverse scenarios. In addition, Doppler effects and possible interference due to the proximity of high voltage (catenary) in the vicinity of the antennas render the railway environments very specific compared to the indoor, urban, or suburban environments generally considered today for the use of mmWave communication systems. A detailed description of railway-specific environments can be found in <cit.>.
Considering the capability of RIS to solve the blockage problems in mmWave wireless communications, the use of RIS for railway communications has recently been considered as a promising candidate.
§.§ RIS-assisted railway communications
§.§.§ RIS for high-speed trains
<cit.> discusses the need for RIS in High-Speed Railway (HSR) environment for mmWave communications to improve the signal quality, which suffers from frequent blockages due to high-speed trains. The authors apply Deep reinforcement Learning (DRL) based approach to jointly optimize the RIS phase shifts and the BS beamforming for spectral efficiency maximization. The results show a significant improvement in spectral efficiency performance using DRL compared to the traditional approach.
In <cit.>, the authors describe how to use RIS on high-speed trains to improve communication performance by providing beamforming, interference mitigation, and reducing signal attenuation. They present a detailed discussion of the challenges associated with the RIS deployment on these trains, such as the need for tracking of the train, low latency, and high-speed RIS control, and the impact of train vibration on the RIS performance. They also propose the DRL approach to solve the sum rate maximization problem.
<cit.> deals with interference suppression in an HSR network, composed of a BS, a mobile relay (MR) located on the train, a RIS located near the MR, and an interference
source. The authors maximize the channel capacity
using a DRL solution and they consider outdated channel state information (CSI) to take into account the motion of the train. The authors found that deploying a RIS in close proximity to the embedded MR improves interference suppression and that their algorithm is more effective in suppressing interference than other optimization algorithms
based on mathematical formulations.
<cit.> proposes a new interrupt flow scheduling approach for RIS-assisted downlink mmWave HSR communications where multiple mobile relays exist. Given the existence of eavesdroppers, the BS schedules a number of flows for each MR when the MR flow quality of service (QoS) exceeds the QoS requirement. The authors seek to maximize the scheduled flow number, find the optimal beamforming, the optimal RIS phase shifts, and the scheduling or not of the RIS discrete phase shift, and they find that RIS can intend communication security by reducing eavesdropping capacity and extending coverage area in the HSR environments.
§.§.§ RIS in railway tunnels
In <cit.>, the authors have considered a simple two dimensions empty tunnel. Using the image theory approach and a vertical blocking element between a Tx and an Rx inside the tunnel, they have shown that the use of RIS located on the ceiling of the tunnel can reduce the Blocking Probability (BP) of the signal between Tx
and Rx.
An increase in the number of RIS
and optimization of the Tx position conduct to an additional decrease in BP. The increase in
distance between RIS and Tx can extend the effective range of RIS for a given BP. This study could be extended by considering a train inside a 3D tunnel.
§.§.§ RIS for passengers inside trains
Recently RIS technology has been studied to extend the coverage area in the mmWave band inside an airplane cabin <cit.>. The authors aim to minimize the number of RIS deployed in this system while ensuring the user data rate remains above a threshold. Besides, they compare the performance of this system for two RIS positions in the cabin corridor near the seat and above the center seat. This study could be easily transposed to the case of the inside of a high-speed train or inside a metro to guarantee a given throughput for the passengers.
§ FUTURE DIRECTIONS
As discussed in the previous sections, RIS offers a promising low-cost solution to solve the blocking problems in railway networks since it improves the efficiency and reliability of high-speed trains, solves the interference problem, and extends the coverage area through controlled signal reflection. In the case of high-speed trains, the channel estimation for RIS-assisted communications is a crucial challenge due to the unexpected rapid change of environments. Future research directions could explore the case of RIS-assisted wireless communications in tunnels, especially when the vertical cross-section of the train is large compared to the tunnel cross-section, which increases the probability of signal blockage. In addition, the case where the train moves in the tunnel from the inside to the outside is particularly difficult due to the development of urban transport and in particular driverless metro systems which require high data rate transmissions. The optimization of RIS-assisted communications in this case will require the development of realistic channel models. It would also be interesting to study the optimal location of the RIS, the number of RIS elements, or the number of RIS itself, needed in these systems to maximize the coverage inside the tunnel and also maximize the ever-increasing passenger throughput demand onboard the trains.
§ CONCLUSION
This paper presents a survey on RIS-assisted communications for railway applications, particularly in the mmWave band. First, we have defined the RIS concept, explaining its structure, and different types of RISs. A review of the various optimization algorithms used in the literature for RIS-assisted systems is proposed, and we highlight the ability of RIS to solve the blocking problem of mmWave. In the last section, the paper outlines
the characteristics of the railway environments and details some recent works concerning the use of RIS in high-speed trains. This topic is a very active field of research and we have proposed some future directions for RIS-assisted railway communications.
§ ACKNOWLEDGMENT
This work was funded by the council of the Region Bretagne, under the grant MILLIRIS.
IEEEtran
|
http://arxiv.org/abs/2307.04935v1 | 20230710230522 | Seeing quantum effects in experiments | [
"Victoria Borish",
"H. J. Lewandowski"
] | physics.ed-ph | [
"physics.ed-ph"
] |
[][email protected]
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 80309, USA
Quantum mechanics is a field often considered very mathematical, abstract, and unintuitive. One way some instructors are hoping to help familiarize their students with these complex topics is to have the students see quantum effects in experiments in undergraduate instructional labs. Here, we present results from an interview study about what it means to both instructors and students to see quantum effects in experiments. We focus on a popular set of quantum optics experiments, and find that students believe they are observing quantum effects and achieving related learning goals by working with these experiments. Although it is not possible to see the quantum phenomena directly with their eyes, students point out different aspects of the experiments that contribute to them observing quantum effects. This often includes seeing the experimental results, sometimes in conjunction with interacting with or understanding part of the experiment. There is additional variation across student achievement of the various related learning goals, ranging from many of the students being excited about these experiments and making a connection between the mathematical theory and the experiments to only some of the students seeing a connection between these experiments and quantum technologies. This work can help instructors consider the importance and framing of quantum experiments and raises questions about when and how in the curriculum quantum experiments can be best utilized and how to make related learning goals available to all students.
Seeing quantum effects in experiments
H. J. Lewandowski
August 12, 2023
=====================================
§ INTRODUCTION
Quantum mechanics, one of the pillars of modern physics, has long been seen as particularly difficult for students to learn <cit.>. This is due, in part, to it being very mathematical and abstract <cit.>, counter- or un-intuitive <cit.>, not seen in the real world <cit.>, and difficult to visualize <cit.>. These factors can also lead to some students losing interest in the subject <cit.>. Nonetheless, demand for a quantum workforce is increasing around the world <cit.>, and many new educational programs are being designed <cit.>. Although current work in quantum technologies involves both theoretical and experimental components <cit.>, much of the education research so far has focused on the theoretical side <cit.>.
There are many open questions about how to best utilize experiments to improve students' quantum education and preparation for the quantum workforce and if experiments can provide students a concrete, non-mathematical approach to the field.
One benefit of incorporating quantum experiments in undergraduate courses is that students have the chance to observe quantum effects with actual experimental equipment rather than just from a textbook. Many instructors have implemented some variation of a sequence of quantum optics experiments, which we refer to as the “single-photon experiments,” into their undergraduate courses <cit.>. This allows students to work with experiments that demonstrate fundamental quantum phenomena, including ones similar to recent Nobel-prize-winning experiments that laid the foundation for quantum information science <cit.>. In previous work, we found that one of the most important learning goals for instructors using the single-photon experiments was for students to “see” quantum mechanics in real life. In fact, all of the surveyed instructors ranked this goal as somewhat or very important <cit.>. Many instructors believe there is a large distinction between students performing quantum experiments and watching videos, demonstrations, or simulations (vide infra), yet there is no concrete evidence demonstrating exactly what students uniquely learn from working with quantum experiments.
In this work, we perform a phenomenographic study that investigates how students observe quantum effects in experiments and why it is important for them. We are interested in identifying the variation of possible ways students can experience quantum lab experiments, and whether this differs from the experiences of their instructors. We interview both students and instructors who work with the single-photon experiments in undergraduate courses to answer the following research questions:
0em
* How do students think about seeing quantum effects in experiments, and how does that compare with instructors’ ideas?
* Do students see quantum effects while working with the single-photon experiments, and what contributes to that?
* Do students achieve learning goals related to seeing quantum effects while working with the single-photon experiments?
Here, we present answers to these research questions, by first providing a framework for understanding what it means to students and instructors to see quantum effects and then using those ideas to see how effective the single-photon experiments are at helping students observe quantum effects and achieve related learning goals. We begin with a brief description of prior work studying quantum education in Sec. <ref>. This is followed with details about the interviews and analysis methods in Sec. <ref>. We then present the results of our study in Sec. <ref>, starting with results for the three research questions and ending with a discussion about different ways experiments can be considered quantum, which spans the research questions. We then present implications for both instructors and researchers in Sec. <ref> and summarize this work in Sec. <ref>.
§ PRIOR RESEARCH IN QUANTUM MECHANICS: FROM STUDENT DIFFICULTIES TO LAB WORK
Physics education researchers have studied quantum education for decades, with much of the work focusing on students' conceptual understanding of quantum theory. In this section, we briefly summarize some of this work, beginning with the way quantum mechanics is often perceived as being particularly mathematical and abstract <cit.>. This has led to many student difficulties in learning quantum concepts and prompted the creation of new curricula focused on simpler mathematical systems (e.g., two-level quantum systems). To help students better visualize these complex topics, a variety of simulations of quantum phenomena have been created, which have been shown to help students learn concepts <cit.> and improve student interest in the field <cit.>. However, they do not afford students the opportunity to see quantum mechanics in physical experiments, an aspect that may help students build a quantum intuition <cit.>. This body of literature motivates our study where we investigate possible learning gains from students seeing quantum experiments themselves in the context of the single-photon experiments. We end this section with a description of the single-photon experiments, their utilization in courses <cit.>, and the prior results of their efficacy in specific implementations <cit.>.
§.§ Studies on conceptual learning in quantum mechanics
Quantum mechanics has long been considered an abstract and mathematical subject <cit.> that is challenging to visualize <cit.>. Some students perceive being good at quantum mechanics as being good at performing calculations, instead of modeling or understanding the world <cit.>. The way many courses do not explicitly bring up interpretations of quantum mechanics can make it difficult for students to connect the abstract math with their conceptual understanding <cit.>. Some research in introductory courses has suggested that instructors can help students think about physics concepts in terms of their everyday lives <cit.>; however, students do not have everyday experience with quantum systems. Students see little relation between quantum mechanics and the real world <cit.>, but they may be able to make sense of the new ideas by learning that quantum mechanics is not about memorizing how to perform calculations <cit.>. Partly because students rarely encounter quantum mechanics in their everyday lives or see it with their own eyes, many find quantum mechanics to be counter- or un-intuitive <cit.>.
The abstraction inherent in quantum mechanics and the required mathematical sophistication have contributed to student difficulties in learning quantum mechanics <cit.>. The sophisticated mathematics needed to describe quantum systems can increase students' cognitive load <cit.>, and students have trouble building mental models of quantum mechanics since they cannot support the models with their own experiences <cit.>. Some students report being discomforted by the concepts and the math-physics connection in quantum courses <cit.> and feel like the physics is harder when it is less intuitive <cit.>. The common emphasis on the math at the expense of the concepts has additionally reduced some students' excitement about quantum physics and caused them to switch into other, clearer areas of physics <cit.>.
In part due to the challenging nature of the subject, there has been a long history of understanding student reasoning and difficulties within quantum courses, ranging from high school through graduate education <cit.>. Much of this work has investigated student understanding of specific concepts, such as tunneling <cit.>, conductivity <cit.>, quantum measurement <cit.>, and particle-wave duality <cit.>. Other work has investigated how student reasoning is connected to various aspects of the instruction, such as terminology <cit.>, notation <cit.>, epistemological framing <cit.>, and visualization <cit.>. This body of work has shown that students often struggle to come up with good mental models and therefore solve problems by applying known methods of calculation without having a good conceptual understanding <cit.>.
To help improve students' conceptual understanding, there has been a push towards new curricula in quantum courses, where some are incorporating earlier discussions about two-level systems <cit.>. These systems can be used to describe single-photon interference experiments and entanglement of spin-1/2 particles. They are mathematically simpler than continuous systems, allow students to directly think about quantum systems with no classical analogue, and can lead to discussions about interpretations of quantum mechanics and related quantum information applications <cit.>. Discussions of quantum optics experiments can additionally provide students the opportunity to learn about photons and their properties through the use of experimental evidence <cit.>. Incorporating discussions about interpretations and quantum optics experiments has been shown to increase student interest and improve their quantum reasoning <cit.>.
§.§ Visualizing quantum mechanics
A complementary approach to improving student understanding of quantum mechanics is through the use of visualizations <cit.>. Good visual representations can help students construct mental models and therefore learn abstract physics concepts, but some visualizations can also lead to student misconceptions <cit.>.
Even when instructors do not explicitly discuss ways to interpret or visualize quantum phenomena, students can develop their own mental images that are different than those intended by their instructors <cit.>. It is therefore important to consider productive ways students can visualize quantum mechanics, especially since some students find visualizations necessary to understand quantum theory beyond the mathematical formalism <cit.>. Different visual representations have been shown to influence student learning in the context of the single-photon experiments <cit.>.
One way visualizations of quantum mechanics are incorporated into classrooms is with research-based simulations developed to improve student understanding <cit.>. Simulations allow students to visualize parts of physics they cannot directly observe <cit.> in addition to helping students relate quantum mechanics to reality <cit.>, engage in inquiry driven learning <cit.>, and build intuition <cit.>.
The interactive component of the simulations more than the visual representation (e.g., screenshots) has been shown to lead to student enjoyment of the activity <cit.>.
Compared with actual experiments, simulations can reduce the cognitive load for students and allow them to explore a system in a situation where they do not need to worry about breaking equipment <cit.>. Simulations of the single-photon experiments have been shown to improve student understanding <cit.>.
Simulations, however, often do not help students understand how knowledge about quantum mechanics was gained from observations, something that can most easily be done with real experiments. A step in that direction is the use of interactive screen experiments, which are multimedia representations of experiments that allow students to interact with certain experimental settings without needing access to the actual experimental set-up <cit.>. Interactive screen experiments of quantum optics experiments have been shown to help students learn quantum concepts that are less influenced by classical physics <cit.>. Compared with real experiments, simulations and interactive screen experiments are cheaper, less complicated, and do not require access to an experimental apparatus. There are additionally analogy experiments (experiments demonstrating the idea of quantum phenomena without utilizing actual single-photon states) that require fewer resources and expertise than the full quantum optics experiments <cit.>. We are not aware of prior work investigating if there is any learning about quantum mechanics that students can only achieve by working with a physical experiment utilizing quantum states.
§.§ The single-photon experiments
Some institutions with enough resources to support it, have started incorporating quantum optics experiments into their curricula to teach about fundamental quantum effects such as particle-wave duality and entanglement. These experiments often involve a laser passing through a non-linear crystal in which spontaneous parametric down-conversion takes place. During that process, some of the photons in the laser beam are converted into pairs of lower-energy photons that are entangled in energy and momentum. The resulting photon pairs may then be measured simultaneously either to demonstrate their entanglement or to use as a heralded single photon source. Although there is not an exact set of experiments that falls into this category, there are many related experiments that can be done with similar apparatus, which have been incorporated into undergraduate courses over the past 20 years (see, for example, Refs. <cit.>). We refer to any of these similar experiments as the single-photon experiments.
The single-photon experiments have become popular in the advanced lab community. They are often taught at the Immersion workshops hosted by the Advanced Laboratory Physics Association [The Advanced Laboratory Physics Association (ALPhA) is an organization aimed at fostering communication and interaction among advanced laboratory physics instructors at colleges and universities in the United States and the rest of the world. More information can be found at <https://advlab.org/>.] where new instructors can learn how to implement these experiments in courses. Instructors are continuing to publish papers about new ways to incorporate extensions or similar experiments in undergraduate instructional labs. In our recent work studying the implementation and goals of these experiments across undergraduate courses, we found that they are primarily used in upper-level quantum or beyond-first-year lab courses, although some instructors are beginning to use them in introductory courses as well <cit.>. Instructors have a variety of goals for using these experiments including helping students learn about quantum concepts, improve lab skills such as aligning optics, gain interest and motivation, and see quantum mechanics, which is the motivation behind this work <cit.>.
Some of the instructors who have published new ways to use the single-photon experiments in their courses have additionally studied the effect these experiments had on their students. These experiments have been shown to improve students' conceptual understanding both self-reported <cit.> and through an assessment <cit.>. They additionally can motivate students to want to better understand the theory <cit.> and to pursue a career in quantum optics and quantum information <cit.>. A larger scale study across different implementations has not yet been implemented, so we begin that process here by investigating if and how students meet instructor goals related to seeing quantum effects in experiments.
§ METHODOLOGY
This study follows a previous survey about instructor usage and goals of the single-photon experiments <cit.>, and allows us to understand in-depth instructors' and students' ideas related to seeing quantum effects and how the single-photon experiments are an example of that. To investigate these ideas across course contexts, we interviewed instructors and students at many different institutions. We first performed semi-structured interviews of 14 instructors who had set up or utilized the single-photon experiments in their courses, and followed this with interviews of 14 students who had each performed a subset of the experiments in at least one physics course within the previous year. To analyze the interviews, we performed a thematic coding analysis, using the emergent results from the instructor interviews as a basis for our analysis of the student interviews. In this section, we present the details of our methodology including limitations of this kind of study.
§.§ Participants and courses
This work studies instructors and students from a range of courses and institutions. Instructors were recruited to participate in the previous survey through the Advanced Labs Physics Association, and all U.S. instructors who completed the survey and agreed to be contacted for future research opportunities were invited to participate in follow-up interviews. At the end of the instructor interviews, we asked if the instructors would be willing to forward a recruitment email from us to the students in their course(s) after the students had finished working with the single-photon experiments. Because many of the courses that use the single-photon experiments are offered only once every year or two, we asked instructors to reach out to students who were currently enrolled in their course(s) or who had taken them earlier that academic year. Presumably due to the small class sizes, we were not able to recruit a sufficient number of students initially, so we carried out a second round of recruitment the following semester, during which we additionally reached out to instructors who had completed our survey but had not participated in a interview and instructors we had recently met in the context of the single-photon experiments (e.g., at conferences, through campus visits, etc.).
Table <ref> shows the self-reported demographics of the interviewed instructors and students. Each participant was given the option to self-report their gender, race, and ethnicity in a free response format at the end of the interviews. We combined the responses at a level that balanced honoring the participants' responses with keeping them from being potentially identified. All students were assigned pseudonyms without any intentional racial or ethnic significance, and we use these pseudonyms along with the pronouns the students requested. The students were in their second year of study or above.
Information about the courses in which the participants utilized the single-photon experiments, as well as their institutions, are also included in Table <ref>. The instructors worked at 14 unique institutions, and some discussed multiple courses at their institution. The students were enrolled at seven distinct institutions with between one and three students from each institution, including both the same and different courses. The majority of the students were enrolled in lab courses for students beyond the first year in their major.
The students worked with between one and four of the single photon experiments in their courses, with the number of students working on the most common experiments reported in Table <ref>. These experiments include setting up the spontaneous parametric down-conversion (SPDC) source, measuring the anti-correlation of single photons sent through a beam splitter (“existence of a photon”), a single-photon interferometer (possibly with a quantum eraser), and violating Bell's inequality. Further descriptions of these experiments can be found in Ref. <cit.>. The amount of time spent working on these experiments ranged from a single three-hour lab to labs for the entire semester plus a prior course with the same experiments. Some of the students set up the experiments themselves, whereas others slightly manipulated optics and then took data on experiments set-up by their instructors.
§.§ Interviews
The primary goal of the instructor interviews was to understand what instructors meant by the phrase seeing quantum mechanics and why it was important to them. This was a follow-up to a previous survey in which all of the instructors ranked the goal “Seeing” quantum mechanics in real life as somewhat or very important, yet it was not clear how the instructors interpreted this goal <cit.>.
The interviews included questions about the courses the instructors taught with the single-photon experiments, the idea of seeing quantum mechanics, and other learning goals for using these experiments. Only the section about seeing quantum mechanics is analyzed in this work since the other parts have been analyzed in prior work <cit.>. Example questions about seeing quantum mechanics include:
0em
* What does the term seeing quantum mechanics mean to you?
* Do you think students in your course see quantum mechanics while working with the single-photon experiments?
Additional questions can be found in the Supplemental Material <cit.>.
The student interview protocol consisted of similar questions to those in the instructor interviews followed by additional, specific questions based off of ideas arising from the instructor interviews. The instructor interviews had been completed, but not yet analyzed, by the start of the student interviews. We additionally changed the wording of the phrase “seeing quantum mechanics in real life” to be “seeing/observing quantum effects in experiments.” We believed this change better encompassed the way instructors were talking about this idea while eliminating some possible confusion for the students. The student interviews included sections about the idea of seeing quantum effects generally, whether the students observed quantum effects while working the single-photon experiments, and pointed questions related to what instructors had told us they hoped students would achieve while working with the experiments. Example questions include:
0em
* Do you think it's important to see quantum effects in experiments?
* What specific parts of the experiment caused you to observe these quantum effects?
* Did working with these experiments help you build intuition about quantum effects?
The relevant part of the student interview protocol is provided in the Supplemental Material <cit.>.
Both sets of interviews occurred over Zoom, and all participants were compensated for their time. The instructor interviews ranged from 49 to 69 minutes, and the student interviews ranged from 33 to 59 minutes. The instructors were interviewed in the spring of 2022 and the students were interviewed between the spring and fall of 2022 about their experiences with the single-photon experiments in courses between fall 2021 and fall 2022.
§.§ Analysis
To analyze the data, we performed thematic coding analyses of the interview transcripts. Our initial coding of the instructor interviews is described in Ref. <cit.>, where we found 14 emergent themes related to the idea of seeing quantum mechanics. Many of these themes are interconnected, so we assigned all the codes throughout the entire section of the interview related to seeing quantum mechanics. We chose to focus on existence of these themes instead of the number of instructors assigned each code since we wanted to understand the range of possible ideas.
We used the resulting codes from the instructor interviews as an initial codebook for the student interviews and also allowed for new emergent codes. For our first research question about how students think about seeing quantum effects, we coded for existence in the same way as with the instructor interviews because we again wanted to emphasize the extent of possible student views. For our other two research questions, we chose to present the number of students assigned each code in order to investigate the efficacy of the single-photon experiments. Each research question was focused on a specific part of the interview.
The student codebook was created iteratively, with the codes being discussed by the research team throughout the coding process. After completion of the codebook, we recruited a colleague unfamiliar with this project to perform an interrater reliability (IRR) check on a subset of the student quotes, which achieved 94% agreement. We chose to present the percent agreement instead of Cohen's kappa because of the low prevalence of some codes across the dataset, which can make Cohen's kappa unreliable <cit.>. The IRR process led to the clarification of one code name and a few code definitions.
§.§ Limitations
There are three primary limitations to this study. First, as with any study at this level of detail, we had a relatively small sample size. To mitigate this, we tried to reach as wide a range of students as possible. The students in our dataset were enrolled in many different courses and used the experiments in different ways, so we effectively averaged over their experiences. This allowed us to better understand the idea of seeing quantum effects more broadly, but we did not have enough data from each individual course to make claims based on the specific experimental implementations.
Second, our sample may be biased towards students from more well-resourced institutions who had positive experiences with the single-photon experiments. Since students who had a bad experience or felt like they did not learn much from working with the experiments may have chosen not to participate in the interviews, we may be presenting a more positive outlook on the single-photon experiments than the average student experienced. Additionally, we interviewed only instructors and students who had access to working with these quantum optics experiments, so we do not know how instructors and students who have not worked with these experiments think about these topics. Since the single-photon experiments are expensive, the population of students working with them in courses may not be representative of the overall population of students in upper-level physics courses.
Finally, we are relying entirely on student self-assessment for all the learning goals discussed in this work, including conceptual learning. It has been shown that students are not always reliable at assessing their own learning <cit.>. Nonetheless, the focus of this work is on non-conceptual learning gains, for which there are no existing assessments and we must rely on students' self-reporting. One of the goals of this work is to identify the non-conceptual learning goals in the hope that better ways to assess them can be established in the future.
Another potential concern arising from student self-assessment is the delay between students performing the experiments and the interviews. Due to the timing of this study, some of the students were interviewed months after working with the single-photon experiments and some discussed how they did not remember well all of the experiments they had performed. Also, although this may be true no matter how student learning is assessed, it was difficult to distinguish what students learned from working with the experiments compared with other parts of the course in which the experiments were integrated.
§ RESULTS
In this section, we present the results of our thematic coding analysis, focusing on the student interviews. This is divided into four sub-sections, the first three of which answer our three research questions: what seeing quantum effects means to students, what contributed to students seeing quantum effects with the single-photon experiments, and whether or not students achieved other learning goals related to seeing quantum effects. The final sub-section discusses an idea that runs throughout these three questions related to how quantum an experiment needs to be.
§.§ How do students think about seeing quantum effects?
To answer our first research question, we analyzed the section of the interview where the students discussed seeing quantum effects in experiments generally. Some of the students explained their ideas broadly, while others used concrete examples of experiments, including the single-photon experiments, they had performed in various physics courses. This, along with our prior analysis of the instructor interviews in Ref. <cit.>, has led to a set of 14 codes that together describe the range of both instructor and student ideas surrounding seeing quantum effects in experiments. We divided these codes into two categories during the coding process. Our emergent codes are:
0em
0em
* Seeing quantum effects may include...
0em
* experiments described by quantum physics.
* seeing experimental results.
* clear results that require little interpretation.
* interactions with the experiment.
* seeing and understanding the experimental apparatus.
* understanding the theory behind the experiment.
* not literally seeing quantum objects.
* Seeing quantum effects can help students...
0em
* believe quantum mechanics describes the physical world.
* gain familiarity with quantum mechanics.
* improve conceptual understanding.
* think about philosophy of quantum mechanics.
* learn about topics of technological and societal importance.
* generate excitement and motivation.
* make learning quantum seem attainable.
Definitions, explanations, and examples of these codes can be found in the Appendix.
Overall, the students and instructors in our data set talked about seeing quantum effects in similar ways. The emergent codes from the instructor interviews worked well to categorize student ideas, and we only made a few wording changes to the codes presented in Ref. <cit.> to better align with the student data. The only new emergent themes appearing in the student interviews were about ways in which experiments could be considered quantum. During the iterative coding process, these were combined with the slightly re-worded code Experiments described by quantum physics and are further discussed in Sec. <ref> since they relate to a broader theme that has shown up in other aspects of this work as well.
However, instructors did discuss some of the ideas related to seeing quantum effects with more nuance than the students. This is not surprising since instructors are more knowledgeable about quantum physics, lab work, and the role physics can play in society. Although some students brought up the technological implications of quantum mechanics in the context of seeing quantum effects, some instructors went further and discussed the broader societal implications (e.g., how the students could “be stewards” and explain quantum physics to non-specialists or combat misinformation). Some instructors also considered additional kinds of interactions, such as by discussing decision-making instead of just physical interactions. The only theme that did not appear in the student data related to seeing quantum effects is the code Think about philosophy of quantum mechanics. This may be because many instructors, both of the set we interviewed and quantum instructors more broadly, often do not focus on interpretations of quantum mechanics <cit.>.
There is no single definition to students or instructors about what seeing quantum effects means, as evidenced by the many emergent codes. All of the students and instructors were each assigned multiple codes, sometimes even for the same quote, demonstrating how these codes represent many interconnected ideas. Although we asked separate questions to try to distinguish the meaning of seeing quantum effects from its importance, these ideas were mixed together in responses. We suspect this is because these are complicated ideas that instructors and students may not have ever had to clearly define for themselves before. The range of ideas provided us a starting point to look at the prevalence of these ideas occurring in the context of the single-photon experiments.
§.§ What contributed to students seeing quantum effects with the single-photon experiments?
Overall, students do think they are observing quantum effects while working with the single-photon experiments. When directly asked about this, 13 out of 14 students responded affirmatively, although the certainty in their responses varied from “I guess” to “Yes, 100%.” In order to understand what contributes to students feeling like they have observed quantum effects, we assigned codes similar to the first set in Sec. <ref> to parts of the interview where the students were talking about seeing quantum effects with the single-photon experiments. Table <ref> shows these codes along with example quotes and the number of students assigned each code. All of the students were assigned at least one of the codes.
§.§.§ Varied combinations of codes contributed to students observing quantum effects
Multiple aspects of working with the single-photon experiments contributed to students feeling like they observed quantum effects. On average, each student was assigned more than three of the codes in Table <ref>, although the exact combination varied by student. The most prevalent code was Seeing results contributed to students observing quantum effects, which was often assigned at the same time as other codes. For example, students discussed how the results matched quantum and not classical predictions, how seeing the results let them know if the results were particularly clear, how seeing the results with the theory in their head helped them understand what was happening, how they interacted with the experiment and then looked at the results, and how they understood the results though understanding the different parts of the apparatus. Many of the quotes in Table <ref> include both the listed code, as well as a part about the students looking at the resulting data or graphs. Possibly because of the different ways students needed to engage with the experiments in order to observe quantum effects, many students discussed how they observed quantum effects more in some experiments than others within the same course.
Not all aspects were useful to all students. For example, Jaime explained how he understood each component of the apparatus, yet that “is not what makes my mind click OK, this is quantum or classical. This is just an instrument for me to get data.” Additionally, not being able to physically see the photons mattered more to some students than others. Hayden, the only student for whom it was not clear that he thought he had observed quantum effects while working with the single-photon experiments, explained:
“I know that we did. Because obviously I saw the Bell inequality being violated. I saw the coincidence counts of the entangled photons and everything. But I guess it was just a little harder to... actually know that it was quantum effects. Because with normal physics experiments, you can see everything that's happening... But with quantum, like everything's so small... you can't see anything. And so I saw all these numbers and everything, but... they could have been... random computer generated for all I knew... So I did witness it, but I didn't like really witness it.”
On the other hand, the other three students who acknowledged they could not see the down-converted photons felt that they were observing quantum effects through seeing the results and understanding how each part of the apparatus affected the photons.
§.§.§ Students focused on different aspects of the results when determining what was clear
Some of the students talked about not only seeing the results, but having some aspect of the results be particularly clear as an important part of seeing quantum effects. There were two distinct ways students talked about the clarity of experimental results. First, similar to the quote in Table <ref>, some students discussed how it was obvious that the results they were seeing lined up with the experiment. This was often brought up when talking about rotating a polarizer and seeing the counts of photons increase and decrease. Other students instead discussed how the prediction for whether the results were quantum or classical was clear. For example, when comparing different experiments he performed in his course, Casey said:
“I think it's because those experiments were far more formulated in terms of a classical versus quantum prediction... They both were sort of similarly formatted in the sense that... here's this quantity that effectively, that allows us to test, directly to test a classical versus quantum prediction, and now let's measure it. And I think that's sort of a straightforward formulation. I think being able to formulate it in a straightforward manner, made it sort of easier to see.”
These two kinds of clarity may not always be compatible, as was evident with the different ways students discussed the Bell's inequality experiment. Students who fell into the first category (those who wanted it to be clear how the results lined up with the experiment) expressed that simpler experiments were clearer to them than Bell's inequality. When asked to compare different quantum effects he saw in his course, Briar said: “I mean, obviously the Bell inequality was really important because it sort of proved what we were doing, but the Malus' law things, to me, sort of gave a more tangible example of what was happening.” By Malus' law, he is referring to rotating a polarizer and detecting how the change in counts varied based on the polarizer angle. On the other hand, for students focused on a prediction that clearly distinguishes between quantum and classical outcomes, Bell's inequality may be very clear. When discussing how he observed quantum effects while working with this experiment, Nicky said: “And the number that pops up on the screen after we take the coincidence data is 2.8 or whatever, or 2.3... that clearly goes against the assumptions of local realism.”
§.§.§ The depth and timing of the requisite theory was varied
Another common code was related to students understanding the theory underlying the experiments. Students had varied responses about how much of the theory they needed to know in order to observe quantum effects, ranging from all of the details to just parts of it. For example, Jaime discussed how in experiments that were not obvious, it was necessary to “fully understand the quantum theory in order to think yeah this is quantum not classical.” On the other hand, Casey said that he “didn't quite fully get the derivations” for the two experiments that he felt were “satisfying in a sense that I felt like I observed a truly quantum mechanical effect.” These were also the two experiments he remembered best after his course was over. Nonetheless, of those two experiments, the one that was more satisfying for Casey was the one that was easier to understand.
The timing of when the students needed to understand the theory also varied by student. For example, Frankie, when asked if it mattered to her whether she understood the concepts before or after taking the measurement, said:
“Oh definitely before... You understand the concepts, you set up your equations, and then after, you compare the results to them. So, it's good to know what you're doing or why you're looking for what you're looking for.”
For other students, it was sufficient to understand the theory after performing the experiment, such as while writing up the accompanying lab report. When asked if he needed to fully understand the concepts in order to observe quantum effects, Hayden said:
“I don't think you need to understand all the concepts to feel like you're witnessing quantum effects, because, I remember, while I was doing the experiment I didn't fully understand the equations that went into it. I didn't really understand what a Bell inequality was. I just followed the directions, and I just kind of like saw everything happening as it was supposed to... Later on, when I actually sat down to write the paper and actually learned all the concepts that went into it, I was like oh wait, this is why this happened.”
Nicky, who also talked about only feeling like he had observed quantum effects when he had gone through the derivations after performing the Bell's inequality experiment, acknowledged that he would have found it less frustrating and “appreciated it more” if he understood the reasoning behind the procedure while working with the experiment.
§.§ Did students achieve learning goals related to seeing quantum effects while working with the single-photon experiments?
Independent of whether or not students are observing quantum effects by their own definition, they may still accomplish other learning goals related to seeing quantum effects set by their instructors. To answer our third research question, whether students report achieving some of these related learning goals by working with the single-photon experiments, we looked at student responses to questions based on the second set of codes related to seeing quantum effects that were discussed in Sec. <ref>.
§.§.§ Students achieved many learning goals
Table <ref> shows the emergent codes from this analysis, grouped into categories, and the number of students assigned each code. Almost all of these codes came from specific questions about those topics (instead of about seeing quantum effects more generally), so most students were assigned at least one of the codes for each question. However, for each set of codes, the numbers do not add up to 14 students for several reasons. First, there were some instances where students ended up talking about related topics without directly answering the interview questions, so we could not classify their responses. Second, some students discussed seemingly incompatible ideas at different parts in the interview, so they may have been assigned two ostensibly contradictory codes. This often occurred when students discussed either two different experiments or a specific part of one experiment compared with their experience overall. Note also that there are varying levels of agreement for the students assigned to each code; we were not able to capture all of the nuances.
Overall, students achieved many of the learning goals instructors had related to the idea of seeing quantum mechanics, although there was variation by student of exactly which goals and the degree to which they were accomplished. All of the students were assigned at least one code related to a positive outcome, and the majority of the interviewed students reported achieving many of these positive outcomes. These include making a math–experiment connection, gaining intuition or familiarity with quantum mechanics, confirming a belief about the validity of quantum mechanics, being excited by at least part of the experiment, improving their understanding about quantum concepts and the experimental apparatus, and seeing at least some connection between the experiments and quantum technologies. On the other hand, there are some negated versions of these codes as well, showing that not all students accomplished the different goals.
§.§.§ Improved conceptual learning and math-experiment connection
Although the majority of beyond-first-year lab courses, where the single-photon experiments are most commonly implemented, focus on developing lab skills over reinforcing concepts <cit.>, instructors using the single-photon experiments often have learning goals related to student conceptual learning about topics such as particle-wave duality, entanglement, and quantum states <cit.>. Many of the students in our dataset believed that working with the single-photon experiments helped them learn about quantum concepts and make a connection between the math and the physics.
At least 10 of the students in our dataset reported learning about quantum concepts. This is a lower bound because other students discussed improving their learning in some capacity, without specifying whether it was about quantum concepts or some part of the apparatus. The only students who mentioned not learning about quantum concepts while working with the single-photon experiments did so when discussing how they understood the concepts well already. The students in our study had a large range of levels of conceptual knowledge about quantum mechanics before working with the single-photon experiments; some students had not previously taken a quantum class, while others had taken several or had studied the subject on their own. Some of the students coming in with a solid grasp on the concepts still thought they improved their understanding while working with the single-photon experiments. For example, when asked if these experiments helped him better understand quantum concepts, Indigo said:
“Yeah, I think it really did. I actually think that even though before the experiment, I would have thought, yeah I understand this concept, I feel like I had moments in the experiment where I had somewhat unexpected results, and there was some nuance to the experiment which I didn't know, which led to results that we didn't expect and didn't get us the right results. And because of those nuances, I had to better understand what was happening conceptually on the quantum level to fix those results... That's a conceptual understanding that I don't think I could have gotten by just learning about it. I feel like I had to go through that process of having to solve the problem in the experiment to understand those kind of things.”
In addition to improving their conceptual understanding, many students felt that they were able to make the connection between the math they had studied in prior courses or read about in relation to the experiment and the experimental results they found. Eleven of the students made a math–physics connection for at least one part of the experiment. For example, Dana talked about how he had learned about qubits mathematically prior to working with the single-photon experiments, yet while manipulating the optics of the experiment, he was “able to put an experimental setup to the theory.” This taught him how he could experimentally test out the math. Nicky discussed how understanding the connection between the procedure and the theory helped him better understand the concept of entanglement:
“I think everyone has this kind of sense of what we mean when we say quantum entanglement. It's kind of like, it's just fuzzy. But I think this, and like the process, the experiment kind of, just kind of fixes it and connects it to... the actual physics.”
Just as with seeing quantum effects generally, various aspects of the experiment helped students make this connection, including performing the data analysis, manipulating the experiment and seeing the result, and evaluating if the experimental result matched the student's expectation.
There were only three students for whom working with the single-photon experiments did not help them make this connection, with one additional student who was not able to make the connection for one specific part of the experiment. These students cited several reasons for not making the connection including that “there's always a little bit of a disconnect,” the experiment “didn't really require that much mathematics at all,” or the student did not “have the math to back it up.”
§.§.§ Quantum was not surprising for students, but experiments still provided confirmation
Quantum mechanics did not seem to be particularly jarring or surprising to the students, but they still obtained benefits from working with the experiments. For example, the majority of the students had never doubted the validity of quantum mechanics, yet many of them discussed how seeing the experiment still confirmed their belief in the field. When asked if his views of the validity of quantum mechanics changed after working with these experiments, Casey said “I didn't disbelieve it before, but I definitely believe it more now.” Frankie talked about how her excitement for working with the single-photon experiment came about from wanting to be the one that proved that photons existed:
“...it was the first one I chose, because I saw it said proving photons exist, and I was like I want to do that, I need to do that. I need to know, and I need to do it myself, just to like make sure.”
Confirming previously known results was important for the students when working with concepts that were not intuitive or easy to understand.
The single-photon experiments helped students become more familiar with quantum mechanics in other ways as well. Prior to working with the experiments, most of the students thought quantum mechanics was weird or mysterious, and working with the experiments helped five of the students think of it as less weird or mysterious than before. For Logan, this was because working with the experiment made it so quantum “seem[ed] like a more everyday thing.” For other students, working with the experiment did not change their views on the weirdness of quantum mechanics, or even made them think of the subject as more weird or mysterious due to learning about concepts they had not previously known existed. Morgan, when discussing how she did not have the math to fully understand how the down-conversion crystal worked said “And I'm worried that's why it's mysterious. Just like any unexplained effect is automatically magic.” When asked if working with these experiments had helped them gain intuition about quantum effects, four students said they were already familiar with quantum ideas through self-study or popular media, yet two of those students still talked about gaining additional intuition by working with these experiments.
One student also discussed how seeing an experimental realization made him realize that quantum experiments were not as difficult as he expected based on the theoretical thought experiments discussed in his course. When discussing if his views about whether or not quantum mechanics is weird or mysterious changed after working with the experiment, Greer said:
“Having worked with experimental setup and stuff, it's easier to work with on an experimental level than I kind of would have expected doing the math... when we go back to our quantum mechanics class, we're always talking about... you have an oven that's shooting electrons out. And you're like what's an electron oven? I don't know what that means... Our experimental setup just kind of taught me that it's easier to work with than I kind of initially expected it to be in class... There's easier ways to find the effects and measure the effects than the theoretical perfect setups we talked about in class.”
§.§.§ Students were generally excited
Almost all of the students were excited to work with the single-photon experiments during at least part of the process. When asked if he found these experiments to be exciting or motivating, Casey discussed how these experiments were “far more interesting than the labs that I've done previously” because of both cool equipment and the way the experiments measure “more sophisticated predictions.” He continued to explain, “I mean showing that photons have to exist is a pretty big deal as opposed to just finding the gain of a particular filter.” Other students also talked about their excitement about the concepts covered in the single-photon experiments. When asked what he found exciting about the Bell's inequality experiment, Briar talked about how cool the experiment's goal was:
“I mean it's sort of those things where... it starts off as super buzzwordy, where you're like oh, you know this experiment breaks reality and defies our assumptions about the way the world works. And you know it sounds super cool to start with, and then you find out that that's true. That's not just like sort of a sensationalized advertisement. That's actually what we're doing with the experiment is challenging our assumptions about reality.”
Whether or not this excitement led to students being motivated to pursue physics more broadly is still an open question.
Although most students were excited about these experiments, one student was never excited about them and three students mentioned how the large amount of time spent on optical alignment detracted from their excitement. Some students liked the alignment aspect of the single-photon experiments, but for others it caused frustration, even if they may have been excited about the results. Nicky discussed how he had been excited going in to the experiment, but how “start[ing] from scratch after like not having the lasers and stuff aligned... kind of diminished my interest in it.” This emphasizes the concern many instructors had about the challenge of figuring out the optimal amount of alignment students should perform <cit.>.
§.§.§ Connection to quantum technologies
Some of the students identified a connection between the single-photon experiments and quantum technologies, such as quantum computing and quantum cryptography, while others did not. This mix of student responses is not too surprising since this was important to some instructors, but not others. Although eight of the students claimed they saw at least some connection between the single-photon experiments and quantum technologies, not all of the connections the students discussed were necessarily accurate. Nonetheless, we assigned codes based on the students' assessments of whether or not a connection existed because that could motivate the students to pursue a career in quantum information science and technology.
The connections the students saw between the single-photon experiments and quantum technologies ranged from seeing basic quantum principles in action to ideas of how these could be applied. Some students discussed thinking about light as photons or seeing a superposition or entanglement in practice. Other students were more explicit about the perceived connections, such as how entangled photons can “`transfer' information across time and space” as a way that quantum computers work or how “erasing data, but then kind of bringing it back” could be useful for quantum cryptography. Several of the students acknowledged that photons are not the dominate platform currently being used for quantum computing, but still saw the experiments as at least somewhat related in that they allowed experimenters to “test ideas and to test better solutions,” or to “measure these effects at this small time scale.”
There were a few reasons students did not see a connection between the single-photon experiments and quantum technologies. This occurred because the course focused on fundamental tests of quantum mechanics instead of applications, because the students did not know enough about quantum technologies, or because the students just did not see a connection. Five of the students reported not knowing enough about quantum technologies to say whether or not there was a connection.
Even if students did not see a connection to quantum technologies, they may have learned about experimental processes that could help them pursue a related job. All of the students reported learning at least one method of experimentally creating quantum superposition or quantum entanglement, a key resource for the current generation of quantum technologies. The students had a range of confidence in their answers. Some students could not explain anything beyond shining a laser through a crystal whereas other students generalized slightly more. Nicky discussed how he did not know any other methods to generate entanglement, but that he would need “some physical process that kind of imbues chance into the experiment.” Even the students who had already learned in prior courses how to theoretically generate entangled particles learned something additional from working the experiment. When discussing how his explanation about experimentally creating quantum entanglement would have changed from before working with the single-photon experiments to after, Kai said: “I don't think I would have been necessarily wrong, but I think my answer... might be longer now.”
§.§ What is quantum about experiments?
One of the codes in Sec. <ref>, Experiments described by quantum physics, is seemly obvious, but also very important for whether or not students and instructors feel like they are seeing quantum effects. Although we did not explicitly ask about this in the context of observing quantum effects, the idea of how quantum mechanics manifests in an experiment or how “quantum” an experiment needs to be came up in several ways throughout these interviews. Here we discuss three aspects of this idea: (1) a comparison of ways quantum effects can be exhibited in different types of experiments, (2) a comparison of some students' experiences with “single-photon” sources of various degrees of authenticity (e.g., heralded single photons versus an attenuated laser), and (3) a discussion of student responses when asked what is quantum about the single-photon experiments.
§.§.§ Ways quantum mechanics manifests in experiments
There are a variety of quantum effects one can observe in an experiment. Some students discussed having performed prior labs with equipment that was based on quantum mechanics (e.g., lasers and detectors), but explained how those were different than the single-photon experiments. Indigo pointed out how when the equipment was based on quantum mechanics, “the quantum effects were kind of like side parts of the whole experiment; they weren't too important.” That was in contrast to the single-photon experiment where “the entire part of the experiment was about these quantum effects.” This also came about when Dana discussed the results from experiments in an electricity and magnetism lab course he had previously taken; instead of “measuring the quantum effects,” he measured quantities such as current and magnetism so “it doesn't really count.” Alex discussed how quantum effects usually showed up as “errors because we aren't accounting for this quantum mechanical effect.” Although Indigo and Dana did not think experiments with quantum effects integrated into them were as useful for seeing quantum effects as experiments where it was the focus, Alex discussed how these were “two sides of the same coin,” and went on to explain why she thinks both are important:
“I think it's good to see examples of both. To see if we're sort of isolating a quantum effect, how do we examine it and how do we study it and what can we learn from it. But at the same time, seeing that it is everywhere and it's integrated into a lot of the other things that we do.”
Students and instructors also compared the way quantum mechanics showed up in the single-photon experiments with other experiments that also explicitly utilized quantum phenomena. When comparing the single-photon experiments with a lab using a scanning tunneling microscope, Alex said:
“The entirety of the experiment was more built around sort of coming to a conclusion about quantum mechanics as opposed to we're using a quantum phenomenon to do something else, where you still see it at play, but it's not necessarily like explicitly focused on the quantum stuff that we're talking about.”
One of the instructors compared the single-photon experiments with experiments demonstrating nuclear decay, by saying that the students “don't really get into the nuclear decay models, right, it's really phenomenological”; however, with the single-photon experiments,
“...you can actually do the analysis. You can write out and solve for the interference pattern. And so I think that's one of the values of it is it's a very tangible way to sort of use quantum mechanics... [In] these experiments, you can write down the wave function essentially and follow it through. So I think that's maybe what's a little bit different than a lot of the other experiments... you have to use the quantum formalism, the math, a little bit to solve, to analyze the data. Or to explain the data... It really feels like they're having to use the quantum mechanics, not just understand the phenomenon.”
There are some ways quantum mechanics exhibits differently in the single-photon experiments than other quantum experiments, and this made some, but not all, students feel like they could more easily see quantum effects with these experiments.
§.§.§ Different kinds of “single-photon” sources
Although the single-photon experiments we have been focusing on use light in a single-photon state (when the other photon in the pair is detected), some similar experiments instead use attenuated lasers. These put neutral density (ND) filters in front of a laser until on average there is less than one photon in the experiment at a time; however, the state of the light is a coherent state and not a Fock state. This distinction is more important to some of the instructors and students than others. One instructor explained how these are entirely different:
“...there are experiments out there purporting to be single-photon experiments that are just a HeNe laser with an ND filter. And I don't think that's right. I think you can say that on average you've got less than one photon in there, but if it's a coherent state with n̅ less than one, it's still a coherent state... But I think that's too much for me to get at my [students] with.”
On the other hand, another instructor discussed how an interferometer with an attenuated laser was “a good example of seeing quantum mechanics,” but still went on to claim that the single-photon experiments “are even better” because they are “clearer” and “more irrefutable.” This instructor also acknowledged that the attenuated laser not being an actual single photon source “doesn't occur to [the students].”
Some of the students in this dataset worked with single-photon interferometers where the “single photons” came from highly attenuated lasers instead of heralded single-photon sources. We did not ask about this distinction since our focus was on the experiments using heralded single-photon sources, so we cannot make any definitive claims about the efficacy of the two options. However, even the students who used attenuated lasers talked about observing quantum effects with them. One student who had the opportunity to work with both of these light sources discussed the different benefits from working with each.
Setting up an interferometer with an attenuated laser is easier and cheaper than doing so with heralded photons, so future work could investigate which learning goals can be achieved with each option.
§.§.§ What students think is quantum about the single-photon experiments
When directly asked what is quantum about the single-photon experiments, students gave a range of responses, some more and some less expert-like. Many of the students contrasted different behaviors they would expect for phenomena explained by quantum versus classical models, in a similar manner to how several of the instructors discussed seeing quantum mechanics. A few students, however, mentioned that these experiments were quantum because they dealt with small particles, without any further explanation. When asked if he understood what was quantum about the Bell's inequality experiment and how that differs from classical ideas, Briar said:
“I think so. I mean from from what I understand, the word quantum sort of means like the smallest possible unit of something. And so it's like if you're working with a photon, that's like the smallest possible unit of light.”
When later asked if he had thought about how Bell's inequality is quantum aside from a violation of it being demonstrated with small particles, Briar said:
“I don't know. I haven't really thought about the inequality itself, other than just how we got it and what it means about sort of the nature of reality... How can an equation itself be like quantum?”
Not all students may be thinking about why these experiments are quantum in the same way as instructors.
§ IMPLICATIONS AND FUTURE WORK
In this section we synthesize the results and present a few take-away points for both instructors utilizing these experiments in courses and researchers interested in continuing to study the ideas presented in this work. Instructors can use these results when thinking about how to help their students feel that they are observing quantum effects, ways to frame their experiments to help students achieve related learning goals, and how best to assess student learning with these experiments. Researchers can follow-up on this work to further investigate best practices for using these experiments in different contexts across the physics curriculum, the uniqueness or lack-there-of about quantum for student outcomes related to observing experimental effects, and strategies for achieving learning goals related to quantum experiments with limited resources.
§.§ Implications for instruction
The single-photon experiments were effective at causing the students in our study to feel like they observed quantum effects. Many students felt like they did so by seeing experimental results in combination with interacting with the experiment, understanding the experimental apparatus, or understanding the theory behind the experiment. Whether or not students thought they were observing quantum effects was additionally affected by how the experiment was framed in terms of quantum versus classical models and how clear the results were without additional interpretation. Because not all students feel that they observe quantum effects by taking the same actions, instructors may consider providing opportunities for all of these actions within their courses. Instructors should acknowledge that what is needed for students to feel like they are seeing quantum effects may be somewhat different than for the instructors themselves.
There are some goals related to seeing quantum effects that students are likely to achieve while working with the single-photon experiments no matter the differences in instruction, while other goals may depend more on how instructors frame the experiments. For example, if instructors' goals are to help students believe that quantum mechanics describes the physical world or have some excitement about the experiments, instructors may not need to explicitly focus on those goals. Almost all of the students in our dataset achieved some positive outcome related to both of those, although the students we report on did self-select into this study and thus are likely to have had positive experiences. However, if instructors goals are more specific, such as thinking about the philosophy and interpretations of quantum mechanics or learning about the current technological and societal importance of quantum experiments, instructors may need to intentionally build these ideas into their courses. Students are less likely to consider these more nuanced ideas on their own.
Throughout these interviews, we heard some student comments about quantum mechanics in general or these experiments in particular that were not always completely correct, although the students were not aware of it. Since we were not focusing on students' understanding of quantum concepts, we did not probe this deeply. We do not know whether this came about from imprecise use of terminology; students' experiences with the experiments; or outside sources, such as popular media, where quantum ideas to various degrees of accuracy are becoming more prevalent <cit.>. If instructors want to ensure their students have successfully grasped all the complex concepts exhibited in the single-photon experiments, they could consider assigning an assessment specifically geared towards those ideas or investigate student learning through some format other than student self-assessment.
§.§ Implications for future research
There are many remaining open questions about best practices for various methods of incorporating the single-photon experiments into courses. In this study, we were able to show that certain ways of working with the experiments led to students being more likely to observe quantum effects, but we were not able to distinguish student responses by course type or the way the experiments were implemented. These experiments are incorporated into quantum courses with large lecture components, beyond-first-year lab courses, and even some introductory courses, and the way they are integrated into the courses can depend on the context and the course itself <cit.>. Future work could investigate which of the learning goals related to seeing quantum effects students learn from different implementations of these experiments, what specifically contributes to those learning goals in each context, and how to most effectively use the experiments in each. A large-scale survey could be implemented to obtain these data from a wider range of students.
Instructors and students both discussed the importance of seeing quantum effects in a lab, with some thinking it is more important than seeing other areas of physics and others thinking it is equally as important as seeing other areas of physics. Most of the codes in Sec. <ref> related to how seeing quantum effects can help students could be seen as examples of the purpose of physics experiments more generally <cit.>. However, some of our codes may be more relevant for quantum mechanics than other areas of physics due to the way the field is often perceived and the public awareness of quantum information science and technology. Further work could examine the degree to which the importance of seeing experimental effects is unique to quantum or whether it is similarly important for other areas of upper-level physics.
This work implies that students do think they are learning and obtaining benefits from seeing quantum effects in experiments; however, many research-level and instructional quantum experiments are more expensive than many institutions can afford. The single-photon experiments are a way to bring quantum experiments to students at undergraduate focused institutions without expensive research labs, yet they still cost tens of thousands of dollars. Future work should investigate cheaper ways to achieve some of the same learning goals including addressing questions such as: What other kinds of quantum experiments not considered here could be cheaper and still lead to students feeling like they observe quantum effects? Which learning goals can be accomplished with an attenuated laser instead of a heralded single-photon source? Which learning goals can be accomplished with cloud based quantum experiments <cit.>?
These questions are especially important as a few students in our study discussed unprompted how these experiments made them feel more capable of understanding quantum mechanics instead of being intimidated by it. As physics educators are trying to make the field more approachable to a wide variety of students, more work needs to be done to investigate if seeing quantum effects in experiments can help with that, and if so how to make these or comparable experiments available to students at institutions with all levels of resources.
In this discussion, we have been focusing on student learning that can be achieved by working with experiments; however some possible learning goals (such as improving conceptual understanding) have already been demonstrated in free platforms such as simulations <cit.> and interactive screen experiments <cit.>. More investigation is necessary to understand if there are certain concepts that are easier learned about with one platform or another, and to directly compare the benefits to students of the different platforms for the overlapping learning goals.
§ CONCLUSION
The idea of students observing quantum effects in real experiments was important to at least some degree to all interviewed students and instructors. For some, it was important because quantum mechanics is a pillar of modern physics, while for others, it is particularly important due to the way quantum is thought of as abstract, mathematical, and hard to see. The single-photon experiments, in part due to the way they demonstrate basic quantum phenomena, overall helped students feel that they were observing quantum effects and achieve other related learning goals, although there was variation across students. This work provides suggestions for instructors about different parts of the experimental process that may help students realize they are observing quantum effects. It additionally raises new research questions about what concepts students are successfully learning with these experiments, the degree to which this is unique to quantum, and how learning goals related to seeing quantum effects can be achieved with fewer resources.
We thank the interviewed students and instructors for participating in this study, Kristin Oliver for performing the IRR check, and the rest of the CU PER group for useful conversations and feedback. This work is supported by NSF Grant PHY 1734006 and NSF QLCI Award OMA 2016244.
*
§ DEFINITIONS OF CODES ABOUT WHAT IT MEANS TO SEE QUANTUM EFFECTS
In this appendix, we present more details about what seeing quantum mechanics or quantum effects means to both instructors and students. All of the codes discussed in Sec. <ref> are presented with their definitions in Table <ref> with the following sections discussing each one in detail including example quotes. We focus on student quotes when the student and instructor ideas lined up, and additionally include instructor quotes when they provided more nuance. Other instructor quotes can be found in Ref. <cit.> and some of the student quotes in Sec. <ref> also help explain the ideas presented here.
We divided the codes related to seeing quantum effects into two main categories: one about the aspect of the experience of working with the experiment that led to students or instructors feeling like they observed quantum effects (labeled as “seeing quantum effects may include...” in Sec. <ref>) and the other about potential benefits for students from working with the experiment (labeled as “seeing quantum effects can help students...”). Although we made this distinction, instructors and students did not always separate these ideas. Both sets of codes came up when asked what seeing quantum effects means, as well as why it is important.
§.§ Components that contribute to seeing quantum effects
One component of seeing quantum effects for some students and instructors was observing some form of the results (the code Seeing experimental results). This could be the raw data (e.g., counts from a detector recorded as either numbers or a graph, or a build up of statistics) or the data after it had been analyzed. When asked what seeing quantum effects in experiments meant to him, Casey said, “And so I guess in a literal sense, it would just be seeing a number on a screen that shows that classical physics can't work.” Frankie, when asked what specific parts of the experiment caused her to observe quantum effects, discussed both the “raw sort of numbers coming out of [the detectors]” as well as “the values com[ing] out” of the Mathematica and Python programs (see the first quote in Table <ref>).
Some students and instructors took it a step further and claimed it was not just seeing the experimental results, but seeing results that were particularly clear and did not require additional interpretation. The code Clear results that require little interpretation was often assigned when students and instructors compared their experiences of seeing quantum effects in different experiments within the set of single-photon experiments. Because of this, some of the student quotes were double coded with the codes in Table <ref>. In particular, Sec. <ref> contains a discussion of the different ways students perceived the experimental results as clear. Instructors also discussed this idea, often with the example of how the Bell's inequality experiment involves “many steps” and students
“have to be willing to accept many things that [they’re] maybe not quite as comfortable with. Whereas if [they’ve] only got to make one leap, okay, [they] can get that, but if [they] need to make three leaps, it gets that much harder to sort of see.”
This experiment was sometimes compared with the existence of a photon experiment, which one instructor described as “that's really clear that they see the result and it's really clear what it means.”
For some students and instructors, seeing not just the results but also the experimental apparatus was important to feeling like they saw quantum effects (the code Seeing and understanding experimental apparatus). When asked to compare the importance of seeing quantum effects with seeing other areas of physics, Morgan explained: “quantum effects are very hard and difficult to visualize. And so seeing the setup and seeing what needs to happen is more important.” One instructor gave the example of the physical layout being particularly important to understand the concept of superposition, because a superposition of two positions is clearer to students than a superposition of polarization states. Another instructor talked about how the students were “closer to the quantum aspect of what was happening, because it was spread out on the table.”
For others, not just seeing, but also understanding how the apparatus worked was important. However, students differed in the amount of importance they attributed to it. When explicitly asked how much of the experimental apparatus they need to understand to see quantum effects, the students' responses ranged from none (the equipment was not the important part) to pointing out which parts they thought were important. For example, Alex said:
“...the parts of equipment that are really integral to the operation of the system, you should probably know how they work and how they're affecting the system. Some of those more peripheral elements, I don't think are quite as important.”
She gave the example of the beamsplitter and polarizing films as being integral to the single-photon experiments in contrast with the power supply for the laser, which was not the focus of the lab.
In addition to understanding the apparatus, understanding the theory behind the experimental results was also part of seeing quantum effects for many students and instructors (the code Understanding theory behind the experiment). For example, when asked if she had experimentally observed quantum effects in her course, Alex discussed a lab she had recently performed and explained how it was “a really good example of like yes I've seen this and I understand how this is operating on a quantum level.” Some students and instructors instead focused on having a solid understanding of classical physics. Indigo said:
“But I think most of all, what I need to know is not necessarily the quantum concepts, but the classical concepts. Because if I have the classical concepts, then I know what to expect. And then when those expectations aren't what I see, then I know that it's something else, and obviously it'll be quantum. So I think having a classical understanding of concepts is maybe even more important than the quantum understanding in doing this kind of experiment.”
When students were asked if they needed to fully understand the concepts of the experiment to feel like they observed quantum effects, responses ranged from “no” to “a hundred percent” with many responses in between. Frankie, the student who thought it was a hundred percent necessary, further explained: “Because otherwise it's just me looking at numbers and saying yeah they line up with some math I did that didn't make sense to me.” Some students talked about how they think fully understanding the concepts is not necessary to observe quantum effects, but having a solid background in the theoretical side of the experiment makes it more valuable.
The code Interactions with the experiment was assigned to students and instructors who thought that different kinds of interactions with the experiment were necessary to observe quantum effects. Students rarely mentioned interaction being important for seeing quantum effects more generally, but it came up many times when students discussed seeing quantum effects in the context of the single-photon experiments (see Sec. <ref>). The students discussed interaction in the form of adjusting polarization optics. Some instructors defined interactions more broadly, using phrases such as “doing an experiment” or “direct experimental interaction with quantum effects.”
The instructors were additionally asked if students could see quantum effects while watching a video of the experiments or a demonstration instead of interacting with the experiments themselves. Most of the instructors discussed how directly working with the experiment was necessary. One explained: “But the farther you are from the experiment, the less seriously you take it as being an actual experiment versus a dog and pony show.” They then discussed how watching a video is “special effects” and watching a demonstration is watching “a magician” before talking about how “it's not that the students deliberately don't believe it, it's just that it's separated from the experience of I set this up myself, and I verified these are the paths that the light is taking...”
Another instructor explained how “it's much better if they do the experiment” because “they have to spend a lot more time thinking about it... And thinking about all the details, so that they understand their results.”
A few other instructors were less convinced that physical interaction was completely necessary; they were open to the possibility that properly designed simulations, remote labs, or demonstrations where the students could direct the instructor might be able to improve students' understanding. For them, students being able to make decisions or change parts of the experiment was key, as evidenced when one instructor said
“I don't think the physical interaction is as necessary as just having a large enough kind of parameter space to be able to change in the experiment.”
The fact that the experiments themselves needed to be quantum is seemingly trivial, but is also a part of many students' and instructors' ideas about seeing quantum effects. There are a variety of ways experiments could be considered quantum, and not all of them may lead to students feeling like they observe quantum effects, as is discussed in Sec. <ref>. The code Experiments described by quantum physics was assigned to students and instructors who talked about how the experiments needed to be quantum.
In addition to the ideas in discussed in Sec. <ref>, one of the main ways students and instructors brought this up was by explaining that they see quantum effects when an experiment can be explained by a quantum, not a classical, model. One instructor said:
“I try to emphasize not so much seeing quantum mechanics... I really try to emphasize what's the difference between classical physics and quantum physics... a big thing is trying to draw that line between what can we explain classically using Newtonian physics or whatever, and then... when is it absolutely necessary to use quantum mechanics.”
Casey discussed the experiments in a similar way. When asked what it means to him to see quantum effects in experiments, he said:
“... it was mainly showing that there were certain quantities that had I guess different predictions. If you use classical versus non classical models... and so the entire experiment was usually... about showing that those quantities would lie in a non classical regime. And so I guess in a literal sense, it would just be seeing a number on a screen that shows that classical physics can't work... you formulate some test to show that... quantum mechanics provides a more accurate prediction, and then you explicitly show that.”
Some students, however, were less explicit and instead discussed experiments in relation to what they expected based on their classical experience with the world. For example, Nicky talked about “when stuff doesn't accord with what we think would happen, or what like classical mechanics tells us would happen,” and Hayden talked about “things not reacting the way that they should.”
Lastly, some students and instructors brought up the fact that even though they were talking about seeing quantum effects, it is not possible to actually see the photons themselves (the code Not literally seeing quantum objects). This was always discussed in the context of the single-photon experiments, so student ideas about this code are briefly discussed in Sec. <ref>. One instructor, for whom this led to skepticism about the idea of seeing quantum effects, said: “yeah I mean I am also kind of skeptical about this idea of seeing quantum mechanics because you don't see the beams, right. They're single photons, you don't see them.” Another instructor discussed how they wished students could see more: “Yeah, it would be... a much nicer lab in my mind if you could see, if the down converted beams were bright enough to see.” Another instructor instead talked about ways students could understand they were working with photons even if they could not physically see them. They described the photons as “stuff you can't see with your eyes... And yet, you can move knobs and get signals from it.”
§.§ Learning goals achievable by seeing quantum effects
One of the frequent codes for both students and instructors was Believe quantum mechanics describes the physical world. This code encompasses primarily two intertwined ideas. The first is that quantum mechanics is not just math or theory; it is what happens in the physical world. When asked what it meant to her to see quantum effects in experiments, Frankie discussed this idea:
“Quantum is like pretty abstract, for the most part, when you're learning about it. It's just like this thing that happens. And classical is much easier to get behind because it's something you can see. You see it every day. But this is like a really good opportunity to actually see quantum physics in play. And so it just sort of like grounds it a little more in reality, instead of just being this vague like, I don't know, sort of thought experiment, for the most part. So yeah I like just being able to see the physics happening. Or just having some like actual visual proof.”
Others talked about this as a math–physics connection. Although many students thought quantum mechanics was taught in a mathematical way, Kai brought up how at his institution, quantum is framed as being experimental, which makes it particularly important to see in an experimental setting:
“And the way [quantum is] taught is that it should be confusing in that it doesn't have answers, and that it's very experimental, and that everything we're going off of is experimental evidence. But it's hard to connect with that and `understand' that if you don't get to see how it's `just experimental evidence.' ”
The second main idea encompassed by the code Believe quantum mechanics describes the physical world is that seeing something helps students believe it more than just being told about it. For example, when asked if it is important to see quantum effects, Greer said:
“I think it was... very important to see that the stuff that we're talking about that sounds unrealistic as you're first learning it, like things can exist in two states, but being able to see that the effects in the lab really just say no, this is really what's going on. And just kind of... adds confidence to what you're learning. And makes it a little bit easier to learn, because you're like okay... I've seen it work now. I can more easily accept that this is the way it works.”
As discussed in Sec. <ref>, the students already believed in quantum mechanics, but working with experiments still helped confirm their beliefs. Indigo explained: “When I get to college, I'm already not in the level that I need to see things in order to believe them, but it's nice to get a confirmation...”
Another common code for both students and instructors was Gain familiarity with quantum mechanics. Just as with the previous code, this one encompasses several related ideas and could be considered a bridge between the codes about believing that quantum mechanics describes the physical world and conceptual understanding. One of the ideas contained within this code is that seeing quantum effects can help students build intuition. For example, when asked to compare seeing quantum effects with seeing experimental effects from other areas of physics, Dana said:
“In almost all other areas of physics, we do see it in our everyday life as well. We have an intuition for like kinematics and motion and somewhat of an intuition for light and how it'll react. But we don't see quantum effects at all. So, it's like I feel like that's the best thing to do in a lab is look at things that you don't normally get to see.”
Although the concept of intuition was brought up often by both students and instructors, intuition has different meanings for different students and can also be related to the math–physics connection <cit.>.
Another part of the code Gain familiarity with quantum mechanics is that seeing quantum effects can make the field seem more concrete since it is often perceived as abstract, intangible, and inaccessible. When asked if it was important to see quantum effects in experiments, Logan said:
“I think it makes... a field that oftentimes seems intangible, because it's... generally such a small scale that you need very high end equipment to see it. That it's inaccessible to a lot of people, so it makes it feel more real and less like an esoteric concept that's in a class.”
Others instead focused on how weird or mysterious quantum mechanics is. When asked what seeing quantum effects means to him, Briar talked about quantum as being “super weird... because it sort of goes against my intuition and everything that I think about the way the world works.” Seeing quantum effects in an experiment can help students become more familiar with the abstract and seemingly weird concepts.
Another reason instructors and students want students to see quantum effects is to help them learn concepts (the code Improve conceptual understanding). Concepts is a broad phrase that can include many different ideas, so for our coding scheme in Sec. <ref>, we created sub-codes for improving understanding of quantum concepts (e.g., particle-wave duality or entanglement), the apparatus, and uncertainty and statistics. The third category was included since quantum mechanics is not deterministic, so students need a solid basis in probability, statistics, and the way uncertainty is different in quantum mechanics than classical mechanics <cit.>. This code, however, focuses primarily on the first of these three categories, since students often can learn how parts of the apparatus work in non-quantum experiments and discussions of statistics unique to quantum mechanics were less frequent.
Instructors discussed learning about quantum concepts in the context of seeing quantum effects in more specific ways than the students did. Students talked about conceptual learning with only vague words, such as “better understanding.” Instructors went into more details, including how there are some concepts that are difficult to understand entirely theoretically. For example, one instructor
said:
“...there are some things like entanglement which I feel like they're really difficult to understand just on the basis of math, manipulating mathematical formula. And so I think it's helpful to see a lab where you're seeing a result which you can only understand on the basis of entanglement.”
Other instructors talked about about how seeing quantum mechanics could help students get rid of misconceptions: “And so seeing quantum mechanics could be a way to to defeat some of the wrong things that one might think about what an entangled state is.”
Some of this learning can have broader implications outside of the physics classroom as well, as is evidenced by the code Learn about topics of technological and societal importance. Some students discussed the technological side of this. They knew that quantum experiments could be related to quantum technologies, even if they did not fully understand how quantum technologies worked. For example, when describing what seeing quantum effects in experiments meant to him, Greer said:
“Just seeing [effects] in the experiment shows me that they could have and will have potential beyond just the niche physics experiments in the lab. And that there definitely are ways to, or there's probably ways to utilize it in the future in a more macro, daily scale. And whether that's through quantum computers or something else, I don't know yet.”
Greer is referring to an example of a quantum 2.0 technology, the more recent quantum technologies, such as quantum computing, that utilize the manipulation of quantum entanglement <cit.>. Some instructors discussed seeing quantum mechanics as helping students learn about quantum 2.0 technologies, whereas others focused on students seeing quantum experiments from the previous generation of quantum technologies. This second category, which includes semiconductor physics, may be more relevant for their engineering students to see.
The second component of the code Learn about topics of technological and societal importance is related to the way quantum mechanics shows up in society at large. This idea was not mentioned by the students. Instructors talked about how students may “appreciate being able to say something about [quantum ideas].” They could even use this knowledge to teach others. One instructor
talked about how students can “be stewards” and answer questions people in the public have about quantum physics. They went on to explain how after working with quantum experiments, students will
“have authority to say I've gone through that from theory and shown it in experiment and I've seen, it's not just something that somebody is saying and writing down on a blackboard or whiteboard, it's something that I've seen in the lab.”
The knowledge students gain from quantum experiments can also be used to combat misinformation. When talking about ways entanglement is often misused, one instructor discussed performing Bell's inequality, which is a proof of entanglement:
“We need to go out of our way, we go into the lab, we use lasers, we use spontaneous parametric down-conversion. Like you can't, there's no way that I know of to entangle a vaccine with the Google credit score... if you realize that you have to spend hours and hours for three weeks to achieve this entanglement... they can use our words, like we don't have to be possessive of our words, but when they say quantum entanglement it's not what we mean.”
Although less commonly discussed, one theme that appeared in some instructor interviews is the code Think about philosophy of quantum mechanics. This often related to the various interpretations of quantum mechanics, and how seeing an experiment can help students think about them. When asked what seeing quantum mechanics means to them, one instructor
said:
“For something like the entangled photons, I think it has a lot to do with interpretations of quantum mechanics, because if we're visualizing, we're sort of imposing on the photons some kind of state prior to measurement.”
Doing an experiment can lead to thinking about the state of the photons throughout the entire experimental sequence, which can differ based on the favored interpretation. Other instructors also mentioned how there are “different ways to conceptualize [what's happening]” or how an important idea in these experiments is “whether or not you view photons as real physical objects or as artifacts of measurement.” None of the students brought up philosophy with respect to seeing quantum effects.
Seeing quantum effects can have an affective element as well (the code Generate excitement and motivation). This could come about through student excitement about the experiments themselves or the experiments helping motivate students to complete their coursework and pursue physics in the future. When asked what it meant to students to see quantum effects in experiments, one of the first things some of the students said was “it's really cool.” There is a large range of reasons these experiments may be exciting for students, including the concepts <cit.>. In relation to this idea of seeing quantum mechanics, some instructors wanted their students to have the “feeling of wow I just saw magic” or “view it as this hidden knowledge.” Excitement caused by these experiments could also help students be motivated to spend time understanding their coursework or study more physics in the future. For example, Jaime discussed how it's “very important to see quantum effects because they are what motivate new physics... So, I think those are important to really motivate people to study more physics, more than the basic level.”
In addition to motivation, some instructors and students think these experiments can improve students self-confidence in their ability to understand quantum mechanics (the code Make learning quantum seem more attainable). Students talk about how “you hear from sources that like quantum physics is so hard” or how they “have classmates that basically are like I get it, but it's too hard, it intimidates me, I don't want to do it.” Instructors discuss how seeing quantum mechanics can be an entry point for some students to feel that they are capable of understanding it. For students who are intimidated by the math, experiments can be a way for them to still experience quantum mechanics without all of the associated math. One instructor said: “And I think this system... opened up quantum mechanics for folks who were more interested in the applications and who liked more experimental physics.” Additionally, having experience working with a quantum experiment that is similar to some kinds of current research can make future research opportunities “more accessible” to the students because “[they]'ve talked about quantum in modern physics or [they]'ve taken the quantum course and this is something [they] can do.”
[pages=,1,,2]Supplement.pdf
|
http://arxiv.org/abs/2307.05645v1 | 20230711114040 | Description length of canonical and microcanonical models | [
"Francesca Giuffrida",
"Tiziano Squartini",
"Peter Grünwald",
"Diego Garlaschelli"
] | cond-mat.stat-mech | [
"cond-mat.stat-mech",
"physics.data-an"
] |
[email protected]
IMT School for Advanced Studies, Lucca (Italy)
Lorentz Institute for Theoretical Physics (LION), University of Leiden, Leiden (The Netherlands)
IMT School for Advanced Studies, Lucca (Italy)
Institute for Advanced Study (IAS), Amsterdam (The Netherlands)
Centrum Wiskunde & Informatica, Amsterdam (The Netherlands)
Mathematical Institute, University of Leiden, Leiden (The Netherlands)
IMT School for Advanced Studies, Lucca (Italy)
Lorentz Institute for Theoretical Physics (LION), University of Leiden, Leiden (The Netherlands)
Non-equivalence between the canonical and the microcanonical ensemble has been shown to arise for models defined by an extensive number of constraints (e.g. the Configuration Model). Here, we focus on the framework induced by entropy maximization and study the extent to which ensemble non-equivalence affects the description length of binary, canonical, and microcanonical models. Specifically, we consider the description length induced by the Normalized Maximum Likelihood (NML), which consists of two terms, i.e. a model log-likelihood and its complexity: while the effects of ensemble non-equivalence on the log-likelihood term are well understood, its effects on the complexity term have not been systematically studied yet. Here, we find that i) microcanonical models are always more complex than their canonical counterparts and ii) the difference between the canonical and the microcanonical description length is strongly influenced by the degree of non-equivalence, a result suggesting that non-equivalence should be taken into account when selecting models. Finally, we compare the NML-based approach to model selection with the Bayesian one induced by Jeffreys' prior, showing that the two cannot be reconciled when non-equivalence holds.
Description length of canonical and microcanonical models
Diego Garlaschelli
August 12, 2023
=========================================================
§ INTRODUCTION
Entropy maximization represents a principled approach to inference, allowing the functional form of a constrained probability distribution to be determined without resorting to unjustified assumptions <cit.>. The models that have been derived within such a framework have found application across a wide range of scientific disciplines, such as network science <cit.>, biology <cit.> and neuroscience <cit.>.
Depending on the way constraints are enforced, two distinct formulations of maximum-entropy models are possible: while canonical models are defined by requiring the value of the constraints to be satisfied on average, microcanonical models are defined by requiring the value of the constraints to be satisfied exactly. These two formulations do not necessarily lead to the same description of a given system, i.e. are, in general, non-equivalent.
Non-equivalence between canonical and microcanonical models can be formally defined in a number of ways. The measure level definition is rooted in information theory and involves the Kullback-Leibler (KL) divergence between the related probability distributions: according to this definition, (non-)equivalence is characterized by a (non-)vanishing relative entropy density <cit.>, as the system size approaches infinity. Interestingly, the asymptotic behavior of the KL divergence can be inspected upon considering the difference between the canonical and the microcanonical log-likelihood functions in the same regime <cit.>.
Whenever the two approaches are equivalent, they identify the same set of typical configurations. In these cases, preferring one formulation over the other becomes a matter of convenience (e.g. computational efficiency); whenever the two approaches are non-equivalent, instead, the choice of formulation depends on the specific characteristics of the system under study as well as the nature of the constraints involved. Non-equivalence has been recently shown to arise in models that involve an extensive number of constraints (e.g. the Configuration Model <cit.>) and have practical implications on the prediction of the macroscopic behavior of the system under study (see <cit.> for an example concerning bipartite networks).
Information theory also provides a powerful framework for model selection, by offering criteria that capture the trade-off between a model `goodness of fit', quantified by its log-likelihood, and a model `complexity'. The two most popular model selection criteria are the Akaike information criterion (AIC) <cit.> and the Bayesian information criterion (BIC) <cit.>, according to which one should pick the model minimizing
AIC=-lnℒ+k
and
BIC=-lnℒ+k/2ln V,
respectively, where ℒ is the model likelihood, k is the number of its parameters and V is the sample size. Unfortunately, neither AIC nor BIC can be computed for microcanonical models since they are derived under regularity assumptions that are not met by the latter ones <cit.>. Moreover, canonical and microcanonical models defined by the same set of constraints share the same number of parameters: therefore, according to both AIC and BIC, their complexity is exactly the same and a comparison based on these criteria would merely reduce to a comparison between their log-likelihood functions. While the impact of non-equivalence on the difference between log-likelihoods is quite well understood, its effects on the difference between complexities remain a topic still to be explored in further detail.
To overcome these issues, here we consider the Minimum Description Length (MDL) principle <cit.>, a family of information theoretical approaches to model selection seeking the model that provides the shortest description of data. Specifically, we focus on the approach based upon the Normalized Maximum Likelihood (NML) <cit.>, our aim being that of investigating the interplay between the difference of log-likelihood functions and that of complexities in light of ensemble non-equivalence. More precisely, we aim at determining i) whether the influence of non-equivalence persists even when the complexity term is accounted for and ii) if non-equivalent canonical and microcanonical models should, in fact, be considered as different even from a model selection perspective. To encompass a broad spectrum of applications, our study focuses on maximum-entropy models that are suitable for studying systems representable as binary, rectangular matrices (e.g. bipartite networks, time series).
The rest of the paper is organized as follows. In section <ref>, we introduce MDL for discrete data; specifically, we define two alternative approaches to MDL, i.e. the NML-induced one and the Bayesian one, that are compared in the last section. In section <ref> we provide a general expression for the difference between the description length of canonical and microcanonical models as a function of the Kullback-Leibler divergence between the two distributions. In section <ref>, we explicitly derive the description length of both the canonical and the microcanonical variant of models defined by one global constraint (i.e. the sum of the entries of a matrix) as well as one-sided local constraints (i.e. the row-specific sums of a matrix). While the first constraint leads to ensemble equivalence, the second set of constraints leads to ensemble non-equivalence. Finally, the findings of section <ref> are used in section <ref> to review the relationship between the NML-based approach and the Bayesian approach when non-equivalence holds.
§ THE MINIMUM DESCRIPTION LENGTH PRINCIPLE
The MDL principle embodies the idea that the best model to describe a given dataset is the one providing the shortest description of it.
We now introduce the basic formalism by focusing on the discrete data x∈𝒳, where 𝒳 represents the space of all possible samples of fixed size V. A parametric statistical model ℳ consists of a family of probability distributions sharing the same functional form, i.e.
ℳ={P_ℳ(x;θ)}_θ∈Θ
where θ is the vector of parameters whose values lie in the set Θ.
In order to apply the MDL principle, one needs to compute the description length of the data x provided by a model ℳ: the optimal number of natural digits (nats) needed to describe the outcome x by a probability distribution P(x) is given by -ln P(x). Since, however, ℳ encompasses many distributions corresponding to different parameter values, a natural choice is that of considering the description length provided by the distribution minimizing the quantity -ln P_ℳ(x;θ), reading
ℒ_ℳ(x)≡ P_ℳ(x;θ̂(x))
with θ̂(x) being the maximum-of-the-likelihood (ML) estimator of θ, i.e.
θ̂(x)=max_θ∈ΘP_ℳ(x;θ).
However, the distribution ℒ_ℳ(x) is affected by (at least) two problems having the same origin: the parameters appearing in its definition are evaluated for each possible data in 𝒳. As a consequence, i) ℒ_ℳ(x) cannot be a probability distribution with support in 𝒳 as it is not properly normalized, and ii) adopting ℒ_ℳ(x) may lead to severe overfitting as it is defined a posteriori with respect to the observations.
To solve these problems, one can build a so-called universal distribution relative to ℳ, i.e. a unique, representative distribution P̅_ℳ(x) of the parametric, statistical model ℳ. Such a distribution (which doesn't necessarily belong to ℳ) can be defined in several ways. In what follows we will consider the recipe based upon the NML distribution (for a more formal introduction to universal coding, we redirect the interested reader to <cit.>).
§.§ The Normalized Maximum Likelihood distribution
The NML, or Shtarkov, distribution corresponds to the distribution attaining the minimum point-wise regret in the worst-case scenario <cit.>, i.e.
P̅^NML_ℳ≡min_P̅_ℳmax_x∈𝒳{-lnP̅_ℳ(x)-[-ln P_ℳ(x; θ̂(x))]}.
In words, it is the distribution that a priori requires the minimum number of nats when compared to the shortest description length a posteriori, that is -ln P_ℳ(x;θ̂(x)); the minimum is found in the worst-case scenario, i.e. for the data x for which this difference is maximal. The unique solution to this minimax problem reads
P̅^NML_ℳ(x)=P_ℳ(x;θ̂(x))/∑_y∈𝒳P_ℳ(y;θ̂(y)),
which is a normalized maximum likelihood. Following the universal coding approach, the description length of x through model ℳ is defined as
DL^NML_ℳ(x)≡ -lnP̅^NML_ℳ (x)
= -ln P_ℳ(x;θ̂(x))+ln∑_y∈𝒳P_ℳ(y;θ̂(y));
while the first term is (minus) a model maximum log-likelihood, the second term
COMP_ℳ≡ln∑_y∈𝒳P_ℳ(y;θ̂(y))=ln∑_y∈𝒳ℒ_ℳ(y)
is named parametric complexity of ℳ. Notice that 𝒳 represents the set of all possible data sets of fixed size V, i.e. all the observable samples of size V: thus, COMP_ℳ depends on both the model and 𝒳. Finally, we can rewrite equation (<ref>) as
DL^NML_ℳ(x)=-lnℒ_ℳ(x)+COMP_ℳ.
Once we have an evaluation method to compute the description length of data given a model, the MDL principle prescribes selecting the one whose description length is minimum. As AIC and BIC and many other criteria based on information theory, the description length (<ref>) is made of two terms: the likelihood term, evaluated on the data, which favors models with a high goodness-of-fit, and the complexity term, which depends only on the chosen model and penalizes models that are too complex. Minimizing the description length results in a trade-off between goodness-of-fit and complexity, thus preventing over-fitting. Notice that this trade-off is not imposed ad-hoc but derives naturally from implementing the principle.
The complexity term COMP can be quite hard to evaluate, and it is ill-defined for many models. Nevertheless, if the model is a k-dimensional i.i.d. exponential family, and under some weak additional regularity conditions on ℳ, the following asymptotic result, first introduced by Rissanen <cit.>, is available:
COMP_ℳ = k/2lnV/2π + ln∫_Θ√(det𝐈(θ)) dθ + o(1),
and then
DL^nml_ℳ (x) = - lnℒ_ℳ(x) + k/2lnV/2π
+ ln∫_Θ√(det𝐈(θ)) dθ + o(1),
where det𝐈(θ) is the determinant of the k × k normalized Fisher information matrix
𝐈(θ) = lim_V→∞1/Vℐ(θ)
and ℐ(θ) is the Fisher information matrix of the model. Notice that when the model is a k-dimensional i.i.d. exponential family, detI(θ) does not depend on the sample size V and the correspondent term in the asymptotic formula is finite. Then, the NML description length (<ref>) provides a finite correction to the BIC criterion (<ref>). Nevertheless, this correction can have an important role in finite sample sizes. Moreover, the very definition of DL^nml is way more general than BIC and does not rely on any asymptotic assumption.
§.§ The Bayesian universal distribution
NML holds a notable position among universal distributions as it arises from solving the minimax problem (<ref>). A further reason to justify its special role will be clear at the end of this section, where we introduce another important universal distribution, the Bayesian universal distribution
P̅^B, w_ℳ(x) = ∫_Θ P_ℳ(x; θ) w(θ)dθ
defined for some prior on the parameters, w(θ), which must be supplied by the user. This implementation is formally equivalent to the Bayesian approach to model selection, the Bayes factor method <cit.>. Once equipped with P̅^B, w_ℳ, the description length of x through ℳ is defined as
DL_ℳ^B, w(x) - lnP̅^B, w_ℳ(x).
Similarly to the NML case, if the model is a k-dimensional i.i.d. exponential family and under some weak additional regularity conditions, the following asymptotic result holds:
DL_ℳ^B, w(x) = - lnℒ_ℳ(x) +k/2lnV/2π
- ln w(θ̂(x)) + ln√(det𝐈(θ̂(x))) +o(1).
By comparing this expression with its NML-analogous (<ref>), one notices that if both asymptotic expressions hold and w(θ) does not depend on the sample size, the difference in the NML and the Bayesian description lengths is at most finite. Moreover, the two asymptotic expressions coincide if we equip the Bayesian universal distribution with the Jeffreys prior <cit.>
J(θ) =
√(det𝐈(θ))/∫_Θ√(det𝐈(θ)) dθ.
Thus, in these cases, the NML approach asymptotically coincides with the Bayesian one equipped with the Jeffreys prior. This prior has a special role in Bayesian inference: it was introduced by Jeffreys <cit.> as the least-informative prior, to be used when there is no useful prior information available, and it has the important feature of being invariant under smooth monotone transformations of the parameter vector θ.
§ DESCRIPTION LENGTH OF CANONICAL AND MICROCANONICAL MODELS
Although computing the complexity term of the NML description length can be challenging, for models admitting a sufficient statistics the expression for COMP can be
simplified. In this section, we will introduce an important class of models that admit a sufficient statistic, the Maximum Entropy Models (MEM), both in their canonical and microcanonical definitions. We will apply the NML formalism outlined in the previous section to determine their description lengths, and we will emphasize the strong correlation between the description length of a model and the equivalence (or lack thereof) between microcanonical and canonical MEM.
We focus on discrete data that can be represented as a binary n× m matrix 𝐆:
𝐆= {g_ij}_i = {1, ..., n}
j = {1, ..., m}
and denote by 𝒢 the set of all such matrices, which, when endowed with a probability distribution P(𝐆), constitutes an ensemble of matrices. The matrix representation is often used to describe systems with n elements being modeled, corresponding to the number of rows, each with m state variables or degrees of freedom, corresponding to the number of columns. The size of the data set is fixed and taken as the number of independent entries of the matrix. In the simplest case, this will be equal to n· m.
In the most typical scenario, we have access to a single data set, that is to a single matrix 𝐆^*. In the context of the problem under consideration, it is sufficient that only a vector of quantities 𝐜^* = 𝐜(𝐆^*) evaluated on 𝐆^* is available, rather than the entire matrix. This vector of quantities will represent the sufficient statistic of the model. Maximum Entropy Models are then obtained by looking for the probability distribution maximizing the entropy
𝒮[P] = - ∑_𝐆∈𝒢 P(𝐆)ln P(𝐆),
while constraining the sufficient statistic. If the constraints are reproduced exactly by the model, i.e., the hard constraints 𝐜(𝐆) = 𝐜^* are enforced, one obtains a microcanonical model. Otherwise, if the constraints are reproduced on average, i.e., the soft constraints ⟨𝐜(𝐆)⟩_P = 𝐜^* are enforced, one obtains the corresponding canonical model. This approach, first introduced by Jaynes <cit.>, ensures us that the available information, represented by the sufficient statistic, is used while assuming full ignorance of everything else. Indeed, these models lead to ensembles of matrices that are maximally random upon reproducing the constrained quantities.
Different choices of the constraints result in different models, and for each choice, it is possible to define a microcanonical and a canonical model that shares the same sufficient statistic. In both cases, by definition of sufficient statistic, every quantity defined by the model on a realization 𝐆 will depend only on the values of the sufficient statistic 𝐜(𝐆). For this reason, with a little abuse of notation, when needed we will denote by θ̂(𝐜) and ℒ(𝐜) respectively the ML estimator and the maximized likelihood when the sufficient statistic value is equal to 𝐜. Moreover, we will denote by Ω(𝐜) the number of configurations verifying a certain value 𝐜 of the sufficient statistic:
Ω(𝐜) = ∑_𝐆∈𝒢δ_𝐜(𝐆),𝐜,
where δ_i,j is the Kronecker delta. This set may be empty for some values of 𝐜. We will consider the set 𝒞 of values, known as graphical values, for which this does not happen, that is, the set of values verified by at least one configuration:
𝒞 = {𝐜: Ω(𝐜) ≠∅}.
Finally, we denote by 𝒩_𝒞 = |𝒞| the number of these values.
We will assume no prior knowledge on the parameters, making the NML approach a suitable choice for computing description lengths. Thus, in what follows, the description length DL is implicitly taken as the NML description length defined in expression (<ref>).
For models that admit a sufficient statistic, the complexity COMP takes the convenient form
COMP = ln∑_𝐜∈𝒞Ω(𝐜)ℒ(𝐜),
where the sum is computed running over the graphical values of the sufficient statistic rather than over all possible configurations. In what follows, this expression will be used to compute the complexity of microcanonical and canonical MEM.
§.§ Description length of microcanonical models
In the microcanonical case, the exact values of the sufficient statistic are enforced. The entropy maximization yields the following functional form for microcanonical MEM:
P_mic(𝐆; 𝐜) =
1/Ω(𝐜) if 𝐜(𝐆)= 𝐜
0 else,
where 𝐜 is the parameter vector with values in the discrete set 𝒞. In fact, the parameter vector of the microcanonical model corresponds to the sufficient statistic itself, and
P_mic(𝐆; 𝐜) is a uniform probability that puts mass only on those configurations verifying the condition 𝐜(𝐆)= 𝐜.
By inserting (<ref>) back in (<ref>) we obtain the microcanonical entropy
S_mic(𝐜) 𝒮[P_mic(𝐆; 𝐜)] = lnΩ(𝐜).
which is also known as Boltzmann entropy.
To evaluate the description length of the model, one needs to compute its maximum log likelihood and complexity. The likelihood of the model given a matrix 𝐆^* is always null apart from 𝐜= 𝐜(𝐆^*). Thus, the likelihood is maximized by 𝐜̂(𝐆) = 𝐜(𝐆) and the maximum log-likelihood reads
lnℒ_mic(𝐜) = ln1/Ω(𝐜) = -lnΩ(𝐜).
Using expression (<ref>) and the definition of microcanonical probability, one gets the following expression for the microcanonical complexity
COMP_mic = ln∑_𝐜∈𝒞Ω(𝐜) ·1/Ω(𝐜) = ln∑_𝐜∈𝒞 1 = ln𝒩_𝒞.
Finally, the description length of a microcanonical maximum entropy model is
DL_mic(𝐆^*) = lnΩ(𝐜^*)+ ln𝒩_𝒞.
This expression can be easily interpreted in a Bayesian context. If we apply definition (<ref>) to the microcanonical model (<ref>) we obtain
DL_mic^B, w(𝐆^*) = -lnℒ_mic(𝐆^*) -ln w(𝐜^*)
= lnΩ(𝐜^*) -ln w(𝐜^*).
This expression coincides with the NML description length (<ref>) if w(𝐜) is the uniform prior over 𝒞:
w^unif(𝐜) = 1/𝒩_𝒞 ∀𝐜∈𝒞.
Thus, for microcanonical MEM, the NML approach is always equivalent to a Bayesian approach with a uniform prior on the sufficient statistics, which plays the role of the less informative prior in the discrete parameters case.
§.§ Description length of canonical models
In the canonical case, where the average values of the sufficient statistics are enforced, the entropy maximization leads to the exponential expression
P_can(𝐆; θ) = e^-θ·𝐜(𝐆)/Z(θ),
where Z(θ) is known as partition function in the jargon of statistical physics and has the role of a normalization constant:
Z(θ) = ∑_𝐆∈𝒢 e^-θ·𝐜(𝐆).
If the system under study is a network, the maximum entropy canonical models are known as Exponential Random Graph Models (ERGM).
The parameter vector θ takes, in this case, continuous values and corresponds to the vector of Lagrange multipliers resulting from the constrained maximization of the entropy. The maximum likelihood estimators θ̂(𝐆) are found by solving, often numerically, the constraint equations
⟨𝐜⟩_θ̂(𝐆) = 𝐜(𝐆)
where ⟨·⟩_θ stands for the ensemble average with respect to P_can(𝐆; θ).
The maximum log-likelihood of the canonical model, irrespectively of the specific choice of the constraint, is linked to the microcanonical one through the relative entropy or Kullback-Leibler divergence
. S(P_mic || P_can) |_𝐜,θ = ∑_𝐆∈𝒢 P_mic(𝐆; 𝐜) lnP_mic(𝐆; 𝐜)/P_can(𝐆; θ).
Indeed, it can be shown <cit.> that if we consider the ML parameters evaluated on the same value of the sufficient statistic 𝐜^* for both models, then
D_KL(𝐜^*) S(P_mic || P_can)|_𝐜^*, θ^*
= lnP_mic(𝐆^*; 𝐜^*)/P_can(𝐆^*; θ^*)
= lnℒ_mic(𝐜^*) - lnℒ_can(𝐜^*),
where θ^* = θ̂(𝐜^*). It follows that, for any value 𝐜 of the sufficient statistic
ℒ_can(𝐜) = ℒ_mic(𝐜)· e^-D_KL(𝐜)
= 1/Ω(𝐜)· e^-D_KL(𝐜)
and then by (<ref>)
COMP_can = ln∑_𝐜∈𝒞Ω(𝐜) ·1/Ω(𝐜)· e^-D_KL(𝐜)
= ln∑_𝐜∈𝒞 e^-D_KL(𝐜).
Finally, the description length of a canonical model can be expressed as
DL_can(𝐆^*) = -lnℒ_can(𝐜^*) + ln∑_𝐜∈𝒞 e^-D_KL(𝐜),
or equivalently, by using equation (<ref>):
DL_can(𝐆^*) = lnΩ(𝐜^*) + D_KL(𝐜^*)+ ln∑_𝐜∈𝒞 e^-D_KL(𝐜).
§.§ Comparing microcanonical and canonical models
We can now compare microcanonical and canonical log-likelihoods, complexities, and description lengths. Given that D_KL is guaranteed by definition to be non-negative, we have
Δlnℒ(𝐆^*) lnℒ_can (𝐆^*) -lnℒ_mic(𝐆^*)
= -D_KL(𝐜^*) ≤ 0.
i.e., the canonical likelihood is always smaller than (or equal to) the microcanonical one.
For the same reason, it is straightforward to see that the canonical complexity is always smaller than or equal to the microcanonical one:
ΔCOMP COMP_can - COMP_mic
=ln∑_𝐜∈𝒞 e^-D_KL(𝐜)/∑_𝐜∈𝒞1≤ 0.
Thus, microcanonical models reach a higher goodness-of-fit, encompassed in a higher likelihood, and at the same time, they are more complex. The interplay between the two differences determines the difference in description length between the canonical and microcanonical models:
ΔDL (𝐆^*) DL_can(𝐆^*) - DL_mic(𝐆^*)
= -Δlnℒ(𝐆^*) + ΔCOMP
= D_KL(𝐜^*) + ln∑_𝐜∈𝒞 e^-D_KL(𝐜)/∑_𝐜∈𝒞1.
This expression shows that the entity of the difference between the description lengths depends crucially on the Kullback-Leibler divergence, which is also a fundamental quantity in describing the equivalence or non-equivalence between the canonical and the microcanonical ensemble. Ensemble (non-)equivalence can be defined in multiple ways, all of which have been shown to be equivalent <cit.>. In this paper, we adopt the definition of non-equivalence in the measure sense, which involves the Kullback-Leibler divergence D_KL between the two ensembles. According to this definition, the two ensembles are equivalent if D_KL, when scaled by the system size, approaches zero in the thermodynamic limit, i.e., the limit of infinite system size. In cases where the data is structured, such as with matrices, choosing the system size as the number of entries n× m may not be appropriate. For example, it is not possible to add only one entry to a matrix without altering its structure. In such cases, it is more reasonable to consider the number n of items being modeled as the system size. Increasing the system size from n to n+1 thus corresponds to the addition of a node to a network or a time-point to a time series. The asymptotic behavior of the number of state variables m with respect to n determines different regimes. Consistent with <cit.>, thus, we define the thermodynamic limit as n→∞ and we further consider two scenarios:
* m stays finite as n →∞: m= O(1);
* m diverges as n →∞: we assume that the rate at which m diverges cannot exceed that of n. In other words, we consider that m is at most of the order O(n). All in all, it seems reasonable to assume that the number of state variables or degrees of freedom grows at most as the number of items.
Finally, we define the two ensembles to be equivalent if the specific relative entropy tends to zero in the thermodynamic limit:
lim_n→∞D_KL(𝐜^*) /n = 0
or, equivalently, if
Δlnℒ(𝐆^*) = o(n).
It has been shown <cit.> that the equivalence between the canonical and microcanonical ensembles is disrupted by an extensive number of local constraints, while global constraints preserve ensemble equivalence. Notably, <cit.> thoroughly investigates the order of D_KL(𝐜^*), which corresponds to the likelihood contribution to the description length difference (<ref>), in the context of ensembles of n × m binary matrices.
We incorporate these findings into the MDL framework defined here and seek to compare the resulting likelihood difference to the complexity difference through the formula (<ref>). This comparison is important because, as we have shown, the two terms have opposite signs and compete, potentially rendering the distinction between equivalent and non-equivalent models meaningless from a model selection perspective, even when non-equivalence between the two holds. In the next section, we will test this hypothesis for two examples of Maximum Entropy Models.
Before moving to it, let us consider another way of looking at the comparison between canonical and microcanonical description lengths. The canonical model {P_can(𝐆; θ)}_θ∈Θ induces a model on the sufficient statistic 𝐜, that we call {Q_can(𝐜; θ)}_θ∈Θ, whose functional form is
Q_can(𝐜; θ) = Ω(𝐜) P_can(𝐜; θ),
where P_can(𝐜; θ) is the canonical probability of a matrix 𝐆 such that 𝐜(𝐆) = 𝐜 for a given value of the parameters vector θ. The maximum likelihood assigned to 𝐜 by the model Q_can(𝐜, θ) is
Q_can(𝐜; θ̂(𝐜)) = Ω(𝐜) P_can(𝐜; θ̂(c))
= ℒ_can(𝐜)/ℒ_mic(𝐜) = e^-D_KL(𝐜)
It follows that the difference in description length (<ref>) can be expressed in terms of Q_can:
ΔDL (𝐆 ^*) = DL_can(𝐆^*) - DL_mic (𝐆^*) =
-ln Q_can(𝐜^*; θ^*) + ln∑_𝐜∈𝒞Q_can(𝐜; θ̂(𝐜))/∑_𝐜∈𝒞1
If we further define the average of Q_can(𝐜; θ̂(𝐜)) over all the possible values of 𝐜∈𝒞:
Q̅_can = ∑_𝐜∈𝒞Q_can(𝐜, θ̂(𝐜))/∑_𝐜∈𝒞 1 = ∑_𝐜∈𝒞Q_can(𝐜, θ̂(𝐜))/𝒩_𝒞
we find that the description length difference can be expressed as
ΔDL (𝐆^*) = lnQ̅_can/Q_can^*,
where Q_can^* = Q_can(𝐜^*, θ^*).
This expression, which in fact depends only on the canonical model and not on the microcanonical one, provides a new interpretation of the comparison between canonical and microcanonical description lengths. If we consider only the sign of ΔDL(𝐆^*) when making a choice, the canonical model is the preferred option whenever ΔDL(𝐆^*)<0, i.e. if Q_can^* > Q̅_can. In other words, the canonical model is preferred over the microcanonical one whenever it assigns a higher maximum likelihood to the observed value 𝐜^*, compared with the average maximum likelihood assigned to all possible values of the sufficient statistic; or, more intuitively, if it performs better on the observed data than it does on average.
§ APPLICATION TO GLOBAL AND LOCAL CONSTRAINTS
This section aims to explicitly compare the NML description lengths of canonical and microcanonical ensembles of matrices subject to global and local constraints. Initially, we consider a single global constraint, namely the sum of the elements of the matrix, which in the case of binary matrices corresponds to the number of ones in the matrix:
l(𝐆) ∑_i=1^n ∑_j=1^m g_ij.
In the context of bipartite networks, this quantity is equivalent to the total number of links, and the resulting canonical Maximum Entropy Model is the bipartite version of the Erdös-Rényi model, or random graph model. While this model is a prominent approach in network science, it is widely recognized that it falls short in capturing and accurately describing the topological heterogeneity that characterizes most real-world systems. In the case of n× m matrices, one possible approach to introduce heterogeneity is to constraint the sum of elements in each row:
r_i(𝐆) ∑_j=1^m g_ij for i=1,...,n.
This sequence of quantities is enclosed in the vector of sufficient statistics 𝐫(𝐆). The resulting model is later considered an example of local constraints and non-equivalence between ensembles.
Notably, for bipartite networks, the sequence r_i(𝐆) represents the degree sequence of one layer, and the correspondent MEM is known as Bipartite Partial Configuration Model <cit.>.
In both cases, we will employ asymptotic results to compare the microcanonical and canonical description lengths. This entails considering the scaling of the constraints 𝐜^* in the thermodynamic limit, in line with <cit.>. In the present paper, we focus on dense matrices: We assume that the system is such that the elements g_ij^* of the observed matrix 𝐆^* are, on average, of order O(1) with respect to n. As a consequence, the sum of each row r_i^* is of order O(m), and the total sum l^* is of order O(nm). The actual order of these quantities in the thermodynamic limit will depend on the growth rate of m, according to the different scenarios previously described. Finally, regardless of the behavior of m, in the dense regime the matrix density
p^* = l^*/nm
and row densities
p_i^* = r_i^*/m for i = 1,..., n
will be of order O(1). This is in contrast with the sparse case, where p^*=O(1/n), and the matrix is dominated by null entries in the limit.
§.§ One global constraint
As one global constraint, we consider the total sum of entries of the matrix, l(𝐆), as defined in (<ref>). Since we are considering binary matrices, this corresponds to constraining the number of 1s in the matrix.
The number of configurations for a given value l of the sufficient statistic is Ω(l) = nml. Thus, the microcanonical probability distribution reads
P_mic(𝐆; l) =
1/nm l if l(𝐆)= l
0 else,
and the maximum log-likelihood
lnℒ_mic(l) = - lnnm l.
The complexity is equivalent to the logarithm of the number of graphical values:
COMP_mic = ∑_l=0^nm 1 = ln(nm +1).
Finally, the microcanonical description length reads
DL_mic(𝐆^*) = lnnml^* +ln(nm +1).
The canonical probability can be computed explicitly, as this model represents one of the few canonical MEM whose partition function Z(θ) can be computed analytically:
Z(θ) = (1+e^θ)^nm,
so that
P_can(𝐆; θ) = e^-θ l(𝐆)/(1+e^-θ)^n m .
If we further define
p = e^-θ/1+e^-θ,
we can re-write the canonical probability in the so-called mean-value parametrization, that leads to the expression
P_can(𝐆; p) = p^l(𝐆) (1-p)^n m -l(𝐆).
It is straightforward to recognize this expression as the distribution of a collection of n· m i.i.d. Bernoulli variables, meaning that in the canonical model, each matrix element is either 1 with probability p or 0 with probability 1-p. The ML estimator of p is given by the matrix density
p̂(𝐆) = l(𝐆)/nm
and the maximum log-likelihood, obtained by putting (<ref>) into (<ref>), reads:
lnℒ_can(l) = - n m· h(l/nm),
where h(p) = -pln(p) -(1-p)ln(1-p) is the entropy of a Bernoulli distribution of parameter p. An exact expression for the canonical complexity has been derived in <cit.>:
COMP_can= ln( e^nmΓ(nm, nm)/nm^nm-1 +1),
where Γ(s, x) is the upper incomplete gamma function, which if s is a positive integer, as it happens to be our case, can be expressed as
Γ(s, x) = (s-1)! e^-x∑_k=0^s-1x^k/k!.
For practical implementations, <cit.> provides an approximation of (<ref>):
COMP_can≃ln[√(π nm/2)+2/3 +√(2π)/24√(nm) -4/135nm.
.+ √(2π)/576√((nm)^3) + 8/2835(nm)^2],
and shows that it is already a very good approximation for very small values of the number of entries nm. Finally, the canonical description length reads
DL_can(𝐆^*) = n m · h(l^*/nm)+ ln(e^nmΓ(nm, nm)/nm^nm-1+1).
Notably, the matrix structure has no impact on these results, which solely depend on global quantities, namely the number of 1s and the sample size (n· m in our case).
This implies that the same conclusion would hold true for any data structure, such as a triangular matrix or a strip with random entries.
This observation will prove useful later when dealing with local constraints.
Thus far, all the reported results are exact. Nevertheless, in order to compare the microcanonical and canonical description lengths, it is useful to move to asymptotic results in the thermodynamic limit of n→∞. Specifically, we are interested in comparing the asymptotic behaviors of the log-likelihood difference Δlnℒ(𝐆^*) and the complexity difference ΔCOMP. By expanding the exact results above (see <ref>), we obtain the following expressions for the log-likelihood and complexity differences
Δlnℒ(𝐆^*) = -1/2ln(nm) - 1/2ln (2π p^*(1-p^*)) +o(1),
ΔCOMP = -1/2ln(nm) + 1/2ln(π/2) +o(1).
Combining the asymptotic results above, we obtain the asymptotic description length difference:
ΔDL(𝐆^*) =1/2ln(π^2 p^*(1-p^*)) + o(1).
These expressions provide two significant insights. Firstly, the order of the log-likelihood difference is o(n) regardless of the growth rate of m. This implies that condition (<ref>) is met, and ensemble equivalence holds. Secondly, the leading terms of the log-likelihood and complexity differences are identical, and they cancel out when the two terms are compared. The resulting asymptotic difference in description length depends on the size of the system only through the matrix density p^*, which is assumed to be finite in the dense regime. Thus, in this regime, the description length difference between the canonical and microcanonical description lengths is finite in the limit.
In what follows, we will replicate these results for local constraints and highlight the differences and similarities between the two scenarios.
§.§ Local constraints
We define the vector of constraints 𝐫(𝐆) as the sum of entries in each row of the matrix 𝐆, as expressed in (<ref>), and we refer to these constraints as one-sided because they restrict the individual sums along a single dimension of the matrix. This choice yields microcanonical and canonical models with an extensive, i.e., scaling with the size of the system, number of parameters, specifically n parameters, where each parameter corresponds to a different row. Notably, with this choice, the matrix can be viewed as a collection of n independent strips of size 1 × m. For each strip, we impose a sum constraint, leading to a model that is analogous to the one discussed in the previous section but applied to individual strips. Therefore, all the prior results apply to each row, with m replacing nm and r_i replacing l.
As a result of the independence between the individual rows, the microcanonical probability distribution is the product of the individual microcanonical probabilities:
P_mic(𝐆; 𝐫) =
∏_i=1^n 1/mr_i if 𝐫(𝐆)= 𝐫
0 else.
The microcanonical maximum log-likelihood is the sum of the individual microcanonical maximum log-likelihoods:
lnℒ_mic(𝐆^*) = - ∑_i=1^n lnm r_i^*;
and analogously for the microcanonical complexity:
COMP_mic = n ln[m+1].
Finally
DL_mic(𝐆^*) = ∑_i=1^n lnmr_i^* + n ln (m+1).
Similarly, the canonical probability is the product of the independent row-individual canonical probabilities
P_can(𝐆; θ) = ∏_i=1^ne^-θ_i r_i(𝐆)/(1+e^-θ_i)^m.
If we consider the mean-value parameter vector 𝐩, with components
p_i = e^-θ_i/(1+e^-θ_i)^m for i = 1, ..., n,
the canonical probability (<ref>) becomes
P_can(𝐆; 𝐩) = ∏_i=1^n p_i^r_i(𝐆) (1-p_i)^m -r_i(𝐆).
Thus, each row is a collection of m independent Bernoulli variables with probability p_i that depends on the row. The ML estimator of 𝐩 is given by the row densities
p̂_i(𝐆) = r_i(𝐆)/m for i=1,...,n.
As for the microcanonical case, we apply the results found in the global case to each row, yielding the maximum log-likelihood
lnℒ_can(𝐫) = - m∑_i=1^n h (r_i/m)
and the complexity
COMP_can= nln( e^mΓ(m, m)/m^m-1 +1).
Finally
DL_can(𝐆^*) = m ∑_i=1^n h(r_i/m) + nln( e^mΓ(m, m)/m^m-1 +1).
In the case of one global constraint, every quantity depends on the matrix dimensions n or m through their product nm. For this reason, we did not need to distinguish among different regimes of m in the thermodynamic limit. This is needed, instead, when local constraints are enforced and n and m play different roles.
If m=O(1), and consequently r_i^* = O(1), we do not need any asymptotic expansion because, by comparing the exact results above, it is straightforward to notice that all differences, and in particular the description length difference, are of order O(n).
If m grows linearly with n, i.e., m=O(n), we need to turn to asymptotic results to evaluate the order of the log-likelihood and complexity differences. We consider the following asymptotic expressions (see <ref>):
Δlnℒ(𝐆^*) = -n/2ln m - 1/2∑_i=1^n [ln (2π p_i^*(1-p_i^*))]
+n/12m - 1/12m∑_i=1^n[1/p_i^*(1-p_i^*)] +o(1)
and
ΔCOMP = -n/2ln m + n/2lnπ/2
+ a·n/√(m) + b ·n/m+o(1),
where
a = 2/3√(2/π)≃ 0.53,
b = -11/12 - 4/9π≃ -1.06.
By combining these expressions we obtain the asymptotic ΔDL when m = O(n)
ΔDL (𝐆^*) =
1/2∑_i=1^n [ln (π^2 p_i^*(1-p_i^*))] +a ·n/√(m)
+ c ·n/m + 1/12m∑_i=1^n [1/p_i^*(1-p_i^*)] +o(1),
where a is the same as in (<ref>) and c = -1 - 4/9π≃ -1.14.
As in the global case, we focus on two aspects of these results. First of all, the order of the log-likelihood difference is O(n) or O(nln n) respectively if m=O(1) and m=O(n). According to (<ref>), this implies that the ensemble equivalence is broken. If m=O(n), the leading terms of the log-likelihood and complexity differences, both of order O(nln n), cancel out exactly as in the equivalent case. Nonetheless, the remaining terms in the description length difference are still of order O(n). Thus, the description length difference between the microcanonical and canonical models becomes bigger and bigger when considering increasing system size, regardless of the growth rate of m in the limit.
These results show that, in the examples we considered, the difference in description length remains finite when ensemble equivalence holds, while it grows with the size of the system when the equivalence is broken. To conclude, in the non-equivalent case, the two models can be considered 'far apart' in the thermodynamic limit, even in the context of model selection, when the complexity term is also taken into account. This result can serve as an incentive to consider the microcanonical and canonical models with the same constraints as two distinct models in the non-equivalent case.
§ NML AND BAYES
In section <ref>, we introduced two universal distributions that can be chosen to build the description length of data given a model: the NML and the Bayesian universal distributions, the latter properly equipped with a prior. There may be different reasons to choose one over the other, including practical reasons, according to the specific problem of interest. In the previous sections, we focused on the case when there was no prior knowledge about the parameters. Under this assumption, the NML distribution, when computable, is the "best choice" in a precise minimax sense, as shown in section <ref>. Nevertheless, in some cases, the NML approach can be retrieved in a Bayesian context if the Bayesian distribution is equipped with a prior chosen such that the two description lengths are, at least asymptotically, the same. We will refer to these priors as NML-optimal priors. For all microcanonical models, the uniform prior is an NML-optimal prior, as shown in section <ref>. The situation is more nuanced for canonical models: for those cases when the asymptotic formulas (<ref>) and (<ref>) hold, NML and Bayesian-Jeffreys description lengths are asymptotically equivalent, and the Jeffreys prior is NML-optimal.
This holds true for the one global constraint case that we considered. Indeed, the resulting canonical model is an exponential i.i.d. model for which the asymptotic formulas (<ref>) and (<ref>) are verified. The rate of convergence between the two description lengths is not provided by the asymptotic formulas but can be derived by directly comparing the exact expressions of the NML description length DL_can^nml, Eq. (<ref>), and the Bayesian-Jeffreys description length, DL_can^B, Jeff:
DL_can^B, Jeff(𝐆^*)= -ln(Γ(nm-l^*+1/2)Γ(l^*+1/2)/π (nm)!)
(see <ref>).
By expanding the two exact expressions of DL_can^nml and DL_can^B, Jeff in the limit of n→∞, one finds a first approximation for their difference in the case of one global constraint
DL_can^nml(𝐆^*) - DL_can^B, Jeff(𝐆^*) = 2/3√(2/π nm) + O(1/nm).
We can make the same comparison in the case of one-sided local constraints in order to verify whether or not Jeffreys prior is still NML-optimal when non-equivalence holds. The previous results, obtained in the case of one global constraint, are used to infer the analogous expressions when one-sided local constraints are enforced. Indeed, it is enough to apply the results above to the individual n matrix rows, each made of m entries. Thus, in this case, the Bayesian-Jeffreys description length, to be compared with the NML one of Eq. <ref>, reads:
DL_can^B, Jeff(𝐆^*)= nln[π (m)!]+ ∑_i=1^nln( Γ(m-k_i^*+1/2)/Γ(r_i^*+1/2)),
and the asymptotic difference
DL_can^nml(𝐆^*) - DL_can^B, Jeff(𝐆^*) = 2/3√(2)n/√(π m) + O(n/m).
To sum up:
(A) DL^nml_mic =DL^B, unif_mic always;
(B) DL^nml_can -DL^B, Jeff_can = o(1) in the case of one global constraint;
(C) DL^nml_can -DL^B, Jeff_can = O(n/√(m)) in the case of one-sided local constraints.
The last difference diverges as n→∞ in both regimes considered for the second dimension m. Consequently, in this case, it is no longer true that the NML approach can be retrieved from the Bayesian one by putting Jeffreys prior on the parameters. This result suggests that the equivalence between the NML approach and Bayes-Jeffreys one for canonical models should be reconsidered when an extensive number of constraints are taken into account. We can hypothesize that there exists a prior that recovers this equivalence and extends it to the case of local constraints. This topic may be addressed in future works. For the time being, we merely suggest not assuming the correspondence between NML and Bayes-Jeffreys in those cases where ensembles non-equivalence holds.
In the remainder of this section, we examine further the relationship between NML and Bayes description lengths to show that, where local constraints are present, the Bayesian description length becomes more sensitive to the prior choice. We start by considering the following identities, proven in <ref>, which hold true both in the case of global and local constraints:
(a)
DL_can^nml = DL_mic^B, Ŵ, that is the NML description length of the canonical model coincides exactly with the Bayesian description length of the microcanonical one, equipped with the canonical prior, first introduced by Rissanen in <cit.>, which is defined in all cases as
Ŵ(𝐜) = ∑_𝐆: 𝐜(𝐆) = 𝐜 P_can(𝐆;θ̂(𝐆) /∑_𝐆P_can(𝐆;θ̂(𝐆))
= Q_can(𝐜; θ̂(𝐜))/∑_𝐜∈𝒞 Q_can(𝐜; θ̂(𝐜)),
for 𝐜∈𝒞. Notice that this prior for the microcanonical model is nothing but the NML of the distribution Q_can defined in (<ref>).
Moreover
(b)
DL_mic^nml = DL_can^B, unif, that is the NML description length of the microcanonical model coincides exactly with the Bayesian description length of the canonical one, equipped with a uniform prior on the mean-value parameters, which is
w^unif(p) = 1 for p∈[0,1]
in the case of one global constraint, and
w^unif(𝐩) = 1 for 𝐩∈[0,1]^n
in the case of one-sided local constraints.
Simply put, these two identities show that in both the equivalent and the non-equivalent cases that we consider, we can find a prior such that the Bayesian description length of the canonical (microcanonical) model coincides exactly with the NML description length of the microcanonical (canonical) model. This might lead to the conclusion that, after all, the microcanonical and canonical models coincide in the framework of model selection, even when non-equivalence holds, if the right priors are chosen. Nevertheless, these priors are not NML-optimal, as defined at the beginning of this section. This non-optimality, while negligible in the case of one global constraint, could play a crucial role when local constraints are enforced. This is confirmed if we compare the microcanonical Bayesian description length equipped with the NML-optimal prior, DL_mic^B, unif, to the microcanonical Bayesian description length equipped with the canonical prior, DL_mic^B, Ŵ. Combining (A) and (a) and the findings of the previous sections, one finds that the two description lengths differ by O(1) if one global constraint is considered, while the same difference is of order O(n) when one-sided local constraints are enforced. This simple example shows that when non-equivalence holds, prior non-optimality matters, as the selection of different priors can result in increasingly larger differences in DL when the system size grows.
Similar reasoning can be applied to the canonical description length. As shown at the beginning of this section, Jeffreys prior is NML-optimal in the case of one global constraint (B) but not in the case of local constraints (C). If an NML-optimal prior existed in the latter case, we would reach the same conclusion just reached for microcanonical description lengths. With the current knowledge, though, we can still compare the canonical Bayesian description lengths with the uniform prior, DL_can^B, unif, to the canonical Bayesian description length with Jeffreys prior, DL_can^B, Jeff. Based on (B), (C), (b) and the previous findings, the difference between the two Bayesian DL is finite if one global constraint is enforced, and it is of order O(√(n)) if one-sided local constraints are enforced and m=O(n); nothing can be concluded, instead, in the regime where m=O(1). These observations still serve as a further example of how different choices of prior can result in very different Bayesian description lengths for microcanonical as for canonical models.
§ CONCLUSION
In this study, we examined the implications of ensemble non-equivalence within the framework of MDL-based model selection. Our focus was on comparing the NML description lengths of canonical and microcanonical maximum entropy models. Through our analysis, we discovered that this comparison can be rephrased in terms of the Kullback-Leibler divergence between the canonical and microcanonical distributions. We found that while microcanonical models fit the data better, they also possess higher complexity compared to their canonical counterparts.
To delve deeper into this topic, we explicitly compared the description lengths of canonical and microcanonical models by imposing constraints on the sum over the entire matrix (one global constraint) and on the individual rows (one-sided global constraints) of dense n× m binary matrices. The former case represents a homogeneous model, where the probability of an entry being 1 is the same for all entries. The latter case introduces some degree of locality and heterogeneity as the probabilities vary depending on the specific row. In this scenario, the microcanonical and canonical ensembles are no longer equivalent. By comparing the description lengths of the microcanonical and canonical versions of these two models, we observed that the difference in description lengths remains finite when a single global constraint is imposed but grows linearly with the system size n when one-sided local constraints are enforced. Hence, non-equivalence persists even when considering the complexity as well as the log-likelihood. These findings emphasize the significant consequences that arise from choosing between hard or soft constraints in terms of data compression. Additionally, when confronted with the task of model selection, it is crucial to recognize microcanonical and canonical models as distinct models in the presence of non-equivalence.
Furthermore, we used these results to explore the relationship between NML and Bayesian description lengths. Our investigation revealed that while the Jeffreys prior is NML-optimal for canonical models under a single global constraint, implying asymptotic equivalence between NML and Bayes-Jeffreys canonical description lengths, this is no longer valid when local constraints are enforced. In fact, the difference between these two description lengths diverges as the system size increases. This observation cautions against regarding the two approaches as interchangeable when non-equivalence is present and raises the question of whether an NML-optimal prior exists when local constraints are enforced. Furthermore, we demonstrated through an example that when local constraints are imposed, the difference between Bayesian description lengths with different priors can grow with the system size. This outcome underscores the importance of carefully selecting priors when dealing with an extensive number of constraints.
unsrt
§ ASYMPTOTIC EXPANSIONS
Here we derive the asymptotic expansions used to obtain the asymptotic results in section <ref>.
§.§ One global constraint
We start by deriving an asymptotic expansion for the microcanonical log-likelihood:
lnℒ_mic(l) = - lnnml = - ln (nm)! + ln(nm-l)! +ln l!.
In the dense regime, l is of order O(n) and Stirling's approximation
ln x! = xln x -x +1/2ln (2π x) + O(1/x)
can be used to evaluate all the factorials above, yielding
lnℒ_mic(l) = - nm ln (nm) + (nm-l)ln(nm -l) + l ln l
-1/2ln(nm/2π l (nm-l)) + o(1),
which can be further expressed as a function of the density p= l/nm:
lnℒ_mic(p) = - nm · h(p) + 1/2ln (nm)
+ 1/2ln (2π p(1-p)) + o(1).
By combining this expression with the canonical log-likelihood
lnℒ_can(p) = -nm· h(p) ,
we obtain the asymptotic log-likelihood difference of equation (<ref>).
Similarly, we combine the asymptotic expansion of the microcanonical complexity
COMP_mic = ln(nm+1) =ln (nm) + ln(1 + 1/nm)
= ln (nm) + o(1),
which follows from ln(1+1/x) = O(1/x), and the following asymptotic expansion of the canonical complexity, which can be easily derived from <ref> in the limit of infinite n
COMP_can = 1/2ln (nm) + 1/2ln(π/2) +o(1).
Finally, equation <ref> is obtained by comparing the two asymptotic complexities. Notice that the asymptotic formula <ref> for the canonical complexity represents a very well-known result, as it can be derived directly by applying the asymptotic formula <ref> to the case of i.i.d. Bernoulli random variables.
§.§ One-sided local constraints
Here we consider the regime in which m =O(n). Similarly to the previous case, we need to expand the microcanonical likelihood
lnℒ_mic(𝐫) = - ∑_i=1^n lnm r_i
= -nln m! + ∑_i=1^n ln(m - r_i)! + ln r_i!.
In the dense regime, all r_i's are of order O(n), and we could again apply Stirling's formula to the factorials above. Nevertheless, because of the factor n ahead of everything, Stirling's formula is not enough, and we turn to Stirling's series <cit.>
ln x! = xln x -x +1/2ln (2π x) + 1/12x + O(1/x^3).
The resulting asymptotic expression can be expressed as a function of the row densities p_i = r_i/m:
lnℒ_mic(𝐫) = -m ∑_i=1^n h(p_i) +n/2ln m + 1/2∑_i=1^n [ln (2π p_i(1-p_i))]
-n/12m + 1/12m∑_i=1^n[1/p_i(1-p_i)] +o(1)
By combining this expression with the canonical log-likelihood
lnℒ_can(𝐩) = -m∑_i=1^n h(p_i)
one the asymptotic log-likelihood difference of equation (<ref>).
Similarly, the asymptotic expansion of the microcanonical complexity
COMP_mic = nln(m+1) =ln m + nln(1 + 1/m)
= nln m + n/m + o(1),
which follows from ln(1+1/x) = 1/x + O(1/x^2), is compared to the following asymptotic expansion of the canonical complexity, derived by expanding <ref> to express <ref>:
COMP_can = n/2ln m + n/2lnπ/2
+ 2/3√(2/π)·n/√(m) + (1/12-4/9π) ·n/m+o(1).
The asymptotic expression <ref> is obtained as the difference between the two asymptotic complexities.
§ NML AND BAYES
§.§ Bayesian-Jeffreys description lengths
In what follows, we compute the Bayesian description length DL^B, Jeffreys_can of the canonical model obtained by constraining the sum l over the matrix. As already stated, this model is equivalent to modeling nm i.i.d. Bernoulli variable, and this result can be found in Example 8.3 of <cit.>. Later we extend this result to the case of one-sided local constraint.
First of all, we compute Jeffreys prior according to (<ref>). For a Bernoulli variable, the Fisher information for the parameter p is 𝐈(p)= p^-1(1-p)^-1 and ∫_0^1 𝐈(p) = π. Thus, the Jeffreys prior reads
J(p) = 1/π√(p(1-p))
and, according to definition (<ref>), the Bayesian-Jeffreys DL is
DL^B, Jeff_can(𝐆^*) = -ln∫_0^1 P_can(𝐆^*; p)J(p)dp
= -ln∫_0^1 p^l^*(1-p)^nm - l^*1/π√(p(1-p))
= -ln1/π∫_0^1 p^l^* -1/2 (1-p)^nm - l^* -1/2 dp.
The integral above is the Beta function
B(x, y) = ∫_0^1 t^x-1(1-t)^y-1dt = Γ(x)Γ(y)/Γ(x+y),
computed for x = l^*+ 1/2 and y=nm -l^* + 1/2, where Γ(x) is the Gamma function
Γ(x) = ∫_0^1 t^x-1 e^-tdt.
Thus
DL^B, Jeff_can(𝐆^*) = -lnΓ(l^* + 1/2)Γ(nm - l^* + 1/2)/π (nm)!
which corresponds to Equation (<ref>). By employing the Stirling series, this expression can be expanded asymptotically as
DL^B, Jeff_can(𝐆^*) = nm · h(l^*/nm) + 1/2lnπ nm/2 +O(1/nm).
The last expression is compared with the asymptotic expansion of the canonical NML description length DL^nml_can, obtained as the difference between the asymptotic expansion of (<ref>) and the canonical log-likelihood:
DL^nml_can(𝐆^*) = nm · h(l^*/nm) + 1/2lnπ nm/2 + 2/3√(2/π n m) +o(1).
Finally, Equation <ref> is obtained as the difference between (<ref>) and (<ref>)
§.§ Identities
In what follows, we prove identities (a) and (b) of section <ref>.
§.§.§ Identity (a)
We begin with identity (a), namely:
DL_can^nml = DL_mic^B, Ŵ,
with Ŵ being the canonical prior, which can be expressed as
Ŵ(𝐜) = ∑_𝐆: 𝐜(𝐆) = 𝐜 P_can(𝐆;θ̂(𝐆) /∑_𝐆P_can(𝐆;θ̂(𝐆))
= Ω(𝐜)ℒ_can(𝐜)/∑_𝐆ℒ_can(𝐆).
We recognize the canonical complexity in the denominator of Ŵ. Thus, putting this expression in the Bayesian microcanonical description length we get
DL_mic^B, Ŵ(𝐆^*) =lnΩ(𝐜^*) - lnŴ(𝐜^*) =
= - lnℒ_can(𝐜) + COMP_can
= DL_can^nml(𝐆^*),
which proves identity (a).
§.§.§ Identity (b)
We will now prove identity (b),
DL_mic^nml = DL_can^B, unif,
starting from the case of one global constraint. The uniform prior on the mean-value parameter p reads
w^unif(p) = 1 for p∈[0,1].
Putting this prior in the Bayesian description length yields:
DL_can^B, unif(𝐆^*) = -ln∫_0^1 P_can(𝐆^*; p) dp
= -ln∫_0^1 p^l^* (1-p)^nm - l^* dp
= lnnml^* +ln (nm+1)
=DL_mic^nml(𝐆^*),
where the integral is computed by integrating by parts l^* times. This proves identity (b) for the case of one global constraint.
The identity holds as well for the case of one-sided local constraint, whit the uniform prior reading
w^unif(𝐩) = 1 for 𝐩∈[0,1]^n
Indeed we have that
DL_can^B, unif(𝐆^*) = -ln∫_0^1 P_can(𝐆^*; p) dp
= -∑_i=1^nln∫_0^1 p_i^r_i^* (1-p)^m - r_i^* dp_i
= ∑_i=1^nlnmr_i^* +nln (m+1)
=DL_mic^nml(𝐆^*),
which proves identity (b).
|
http://arxiv.org/abs/2307.04783v1 | 20230710180000 | Confinement in $(1+1)$ dimensions: a holographic perspective from I-branes | [
"Carlos Nunez",
"Marcelo Oyarzo",
"Ricardo Stuardo"
] | hep-th | [
"hep-th"
] |
.
Confinement in (1+1) dimensions: a holographic perspective from I-branes
Carlos Nunez^a[[email protected]],
Marcelo Oyarzo^b[[email protected]] and Ricardo Stuardo^a[[email protected]]
^aDepartment of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
^bDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Abstract
In this paper we holographically study the strongly coupled dynamics of the field theory on I-branes (D5 branes intersecting on a line). In this regime, the field theory becomes (2 + 1) dimensional with 16 supercharges. The holographic dual background has an IR singularity. We solve this singularity by compactifiying the theory on a circle, preserving 4 supercharges. We study various aspects of the QFT: confinement, symmetry breaking, Entanglement Entropy, etc. We also discuss a black hole solution and make some comments on the string σ-model on our backgrounds.
Dedicated to the memory of Roman Jackiw.
tocempty
empty
§ INTRODUCTION
Shortly after the conjectured duality between super conformal field theories and string/M theory on spaces with an AdS factor <cit.> -<cit.>, these ideas were extended to non-conformal situations <cit.>-<cit.>.
In this work, we holographically study one particular two dimensional field theory that at weak coupling is defined on the intersection of two stacks of D5 branes (these are called I-branes). Dynamical features of these theories imply that, as the coupling is increased the field theory turns (2+1)-dimensional and doubles the amount of SUSY preserved. This dynamics is very well explained in <cit.>.
The string background, holographic dual to this strongly coupled QFT is well understood at large values of the radial coordinate, where it can be written as an intersection of NS five branes. Good control over the string σ-model in such background has been developed <cit.>, <cit.>. Nevertheless, this background is singular in the IR (at small values of a suitable radial coordinate). In this paper we propose a 'completion' of this dual background, making it trustable at low energies. The new solution is very explicit and simple. It preserves four supercharges and can be thought of as the dual to a (2+1)-dimensional field theory that is compactified to (1+1)-dimensions. The QFT is a two nodes quiver with Chern-Simons terms, connected by bifundamental matter.
In the bulk of this paper, we study holographically various aspects of the strongly coupled dynamics of this QFT. We define a suitable gauge coupling, that suggest a low energy confining behaviour. Theta angles and the breaking of U(1)-R symmetries are discussed, together with an estimate of a density of degrees of freedom as a function of the energy. Maldacena-Wilson loops are calculated (again indicating confinement). Also, 't Hooft loops and Entanglement Entropy on a strip are calculated, discussing how the non-local UV dynamics of the system impacts on these observables.
We also briefly touch upon two aspects that will be further developed in future publications: we present a black hole solution, obtained as analytic continuation of our new background and calculate some characteristic quantities.
Also, we shortly discuss some aspects of the string σ-model on our backgrounds.
The paper is organised as follows. In Section <ref> we present the supergravity backgrounds studied in the rest of the paper. In Section <ref> we propose the QFT dual to these backgrounds, with various characteristic observables calculated. This proposal is sharpened in Section <ref>, where the IR confining behaviour is determined and the influence of the high energy LST dynamics on observables like 't Hooft loops and Entanglement Entropy is discussed. Some aspects of the string σ-model are discussed in Section <ref>. The conclusions and future lines of research suggested by this paper are written in Section <ref>. Various appendices are included, these discuss in great detail the very many interesting technical aspects needed in the main body of this work.
§ THE SUPERGRAVITY BACKGROUNDS
In this section we write the supergravity backgrounds studied. The first background already appears in the bibliography <cit.>, the second background is new. We refer the reader to Appendices <ref> and <ref> for detailed derivations.
The associated charges are studied. A black hole solution is obtained as a bonus, by performing analytic continuations. Some characteristic observables of the black hole are calculated.
§.§ Background I
To describe the backgrounds we
use the coordinates ( t,x, φ, r ,θ_A,ϕ_A,ψ_A
,θ_B,ϕ_B,ψ_B). We set α'=g_s=1 and define two sets of left-invariant forms of SU(2),
ω̂_1= cosψ_A dθ_A +sinψ_Asinθ_A dϕ_A, ω̃_1= cosψ_B d θ_B +sinψ_Bsinθ_B dϕ_B,
ω̂_2= -sinψ_A dθ_A +cosψ_Asinθ_A dϕ_A, ω̃_2= -sinψ_B dθ_B +cosψ_Bsinθ_B d ϕ_B,
ω̂_3= dψ_A +cosθ_Adϕ_A, ω̃_3= dψ_B +cosθ_Bd ϕ_B.
In terms of these we present the first background. The string frame metric, the Ramond three form F_3, the potential C_2 and the dilaton Φ read,
ds^2_st = r{ -dt^2+dx^2+ (e_A^2+e_B^2) dφ^2/2+ 8 dr^2/r^2 (e_A^2+e_B^2)
+2/e_A^2[ ω̂_1^2+ω̂_2^2+ ω̂_3^2]
+2/e_B^2[ ω̃_1^2+ω̃
_2^2+ ω̃_3^2]} ,
F_3 = dC_2= -2/e_A^2ω̂_1∧ω̂
_2∧ω̂_3 -2/e_B^2ω̃_1∧ω̃_2∧ω̃_3 ,
C_2 =
-2/e_A^2ψ_A sinθ_Adθ_A
∧ dϕ_A - 2/e_B^2ψ_B sinθ_Bdθ_B
∧ dϕ_B .
Φ = log r .
Here (e_A,e_B) are parameters which are fixed when imposing charge quantisation. The coordinate φ is non-compact and could be rescaled to absorb the prefactor (e_A^2+e_B^2) /2.
There are two three-spheres labelled by Ŝ^3 and S̃^3 respectively parametrised by the Euler angles [θ_A, ϕ_A, ψ_A] and [θ_B,ϕ_B,ψ_B]. The range for these angles are θ
_A,B∈ 0,π ψ _A,B∈[0,4π], and ϕ _A,B∈[
0,2π].
The Ricci scalar for the metric in eq.(<ref>) is
R= -3( e_A^2 +e_B^2)/2r,
indicating a singularity at r=0. This is the singular behaviour found close to a stack of D5 branes–see equation (3.38) in the paper <cit.>. In fact, the background can be understood as the backreaction of two stacks of D5 branes that intersect along the non-compact coordinates (t, x) and extend respectively over (y_1,y_2,y_3,y_4) and (w_1,w_2,w_3,w_4), which can be written as radial coordinates and three spheres. After backreaction (at strong coupling), the two stacks share the directions [t,x,φ] wrap the spheres Ŝ^3[θ_A,ϕ_A,ψ_A] and S̃^3[θ_B,ϕ_B,ψ_B], as we find in the background (<ref>). See details in Appendix <ref>.
The singular behaviour at r∼ 0 indicates the need of a description in terms of other variables. On the other hand for large values of the radial coordinate r, the growth of the dilaton and the string coupling (g_s∼ e^Φ) requires an S-duality and the description of the system is in terms of an intersection of NS five branes. The system is then dual to two Little String Theories (LST) that intersect along (t,x,φ) each one wrapping the spheres Ŝ^3 and S̃^3.
We further elaborate on this background in Section <ref> and Appendix <ref>.
We are interested in resolving the singular behaviour at r=0 of the background in eq.(<ref>), making the solution in terms of D5 branes trustable in the ”IR-regime”. The goal is to write a trustable dual description for a strongly coupled QFT (that is UV-completed by a LST). We are also interested in preserving some amount of SUSY for stability purposes.
§.§ Background II
We write below a solution to the equations of motion of Type IIB supergravity that resolves the singular behaviour by compactifying the coordinate φ, with a precise period. A fibration between the spheres and the coordinate φ is also needed. This solution reads,
ds^2_st = r{ -dt^2+dx^2+f_s( r) dφ ^2+4/r^2f_s( r)
dr^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r)
dφ) ^2] } ,
F_3 = dC_2= 2 ζ'(r)dr∧ dφ∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ (r)dφ -ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ (r)dφ -ω̃_3) ,
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
Φ = log r .
Here (e_A, Q_A, e_B, Q_B) are parameters. The functions f_s(r), ζ(r) are given by
f_s(r) = e_A^2+e_B^2/2-m/r^2-2(
Q_A^2+Q_B^2) /r^4≡e_A^2+e_B^2/2r^4
(r^2-r_+^2)(r^2-r_-^2) ,
ζ( r) = 1/r^2-1/r_+^2 ,
r_±^2=m±√(
m^2+4(Q_A^2+Q_B^2)(e_A^2+e_B^2))/e_A^2+e_B^2.
If the parameter m=0 and e_A Q_B=± e_B Q_A, the background preserves four supercharges. For the SUSY study and the details of the construction of the background in eqs.(<ref>)-(<ref>), see Appendices <ref> and <ref>.
Note that the circle parametrised by the angle φ shrinks smoothly at r=r_+ if we choose its periodicity to be
φ∼φ +L_φ,
L_φ =8 π/r_+f^'_s(r_+) =4 π/e_A^2+e_B^2( 1+m/√(m^2+4 (e_A^2+e_B^2)(Q_A^2+Q_B^2) )) .
In the BPS limit, the Ricci scalar associated with the geometry in eq.(<ref>) is
R=-(e_A^2+e_B^2)/2 e_A^2 r^5(4 Q_A^2 + 3 e_A^2 r^4 ).
That is bounded for all the range of the radial coordinate [r_+,∞).
Notice that in the case Q_A=Q_B=m=0 (conversely, in the strict limit r→∞), the background in eq.(<ref>) becomes that in eq.(<ref>). In some of the observables we discuss below, we can perform a regularisation that takes away the effects on the observable that come from the background (<ref>) from the same observable computed in the background (<ref>). Also, in what follows, for any object, ξ _A,B, (like a D-brane) extended along the spheres we use the notation ξ_A≡ξ̂ and ξ _B≡ξ̃.
§.§ Conserved charges
To calculate the D5 brane charges associated with the Ramond Field F_3 in eq.(<ref>), we define the three-cycles,
ℳ_A = (ψ_A,θ_A,ϕ_A) , ℳ_B = (ψ_B,θ_B,ϕ_B).
Let us call the A-stack of branes to be the one extended along the coordinates [t,x,φ, θ_A,ϕ_A,ψ_A] and analogously for the B-stack.
To calculate the number of branes in the A-stack, we need to integrate F_3 over the three cycle ℳ_B–as this is orthogonal to the A-brane stack. Analogously, the number of branes in the B-stack will be obtained by integrating F_3 over ℳ_A.
Setting α'=g_s=1, the quantisation condition for Dp-branes is
(2π)^7-p g_s α'^7-p/2 N_Dp= ∫_Σ_8-p F_8-p, leads to N^i_D5 = 1/(2π)^2∫_ℳ_i F_3.
After choosing a convenient orientation for the three-cycles (equivalently, changing the sign of F_3) we find for the D5 charges,
N_A= 8/e^2_B ,
N_B = 8/e^2_A .
This implies a quantisation condition for the parameters (e_A, e_B). We could have chosen different three cycles, leading to the same conditions. The result is the same for the either of the backgrounds in eqs.(<ref>), (<ref>).
§.§ Bonus: a black hole solution
Let us consider our new background in eq.(<ref>)[The material in this section arose in discussion with Juan Maldacena, whom we gratefully acknowledge.]. We perform a double Wick rotation
φ→ it , Q_A,B→ -iQ_A,B , t → iy,
we find the black hole configuration, which we present in Einstein frame
ds_E^2 = √(r){ dy^2+dx^2-f_bh( r) dt^2+4/
r^2f_bh( r) dr^2+2/e_A^2[ ω̂
_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ(
r) dt) ^2] .
+. 2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r) dt)
^2] } ,
F_3 = dC_2=2ζ ^'( r) dr∧ dt∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ( r) dt-ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ( r) dt-ω̃_3) ,
Φ = log( r) .
where
f_bh( r) = e_A^2+e_B^2/2-m/r^2+
2( Q_A^2+Q_B^2) /r^4≡e_A^2+e_B^2/
2r^4( r^2-r_+^2) ( r^2-r_-^2) ,
ζ( r) = 1/r^2-1/r_+^2 , r_±^2=m±√(m^2-4( Q_A^2+Q_B^2) (
e_A^2+e_B^2) )/e_A^2+e_B^2 .
This configuration F_3 has both "electric" and magnetic
parts. Nevertheless, the integrating of the magnetic part of F_7=⋆ F_3 does not lead to charge of D1 brane since the 7-cycle in which one integrates is not closed, thus is not possible to apply Gauss's law.
In general f_bh( r) has two real roots r_±.
The extremal black hole is obtained when r_+=r_- that is,
m^2=4( Q_A^2+Q_B^2) ( e_A^2+e_B^2), r_+^2=r_-^2=2√(Q_A^2+Q_B^2/e_A^2+e_B^2),
f_bh( r) =e_A^2+e_B^2/2r^4(
r^2-r_+^2) ^2=
e_A^2+e_B^2/2r^4( r^2-2√((
Q_A^2+Q_B^2) /( e_A^2+e_B^2) ))
^2 ,
The preservation of SUSY imposes extremality and
e_AQ_A± e_BQ_B=0 .
The BPS background with Q_B=e_A/e_BQ_A reads,
ds_E^2 =√(r){ dy^2+dx^2+4/r^2e_A^2+e_B^2
/2r_+^4ζ( r) ^2dr^2+2/e_A^2[
ω̂_1^2+ω̂_2^2+ω̂_3^2-2ω̂
_3e_AQ_Aζ dt] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+ω̃_3^2-2ω̃_3e_AQ_Aζ dt]
} .
F_3 =2d[ ζ( r) dt∧( Q_A/e_A
ω̂_3+Q_B/e_Bω̃_3) ] -2/e_A^2ω̂_1∧ω̂_2∧ω̂_3-2/e_B^2ω̃_1∧ω̃
_2∧ω̃_3,
Φ= log(r).
Note that g_tt=0, thus the vector ∂ _t is null.
In what follows we consider the non-BPS black hole background (<ref>). This configuration is rotating along the directions ∂_ψ_A and ∂_ψ_B. Since the fibrations do not decay at infinity, the coordinate system used in (<ref>) correspond to a rotating frame, i.e. an observer at infinity is rotating with the system. We can move to a non-rotating frame via large gauge transformation which cancels the constant term of ζ(r) in (<ref>). Also for simplicity in the computation of the charges we shift the Dilaton by a constant and the F_3 by a factor
ζ(r)=1/r^2 , Φ→Φ-2log(e_A^2+e_B^2/2) , F_3 →e_A^2+e_B^2/2F_3.
The geometry is asymptotically locally flat and has the same causal structure as the Reissner-Nordström spacetime. Nevertheless,
the spacetime is asymptotically conformal to ℝ^1,3× S^3 × S^3.
We compute the conserved charges associated to the spacetime by using the Noether-Wald method <cit.>. The expression for the charges are derived in detail in Appendix <ref>. The energy (mass), angular momentum associated to ∂_ψ_A and ∂_ψ_B, the temperature and the entropy for this configuration are given by [The asymptotic form of the metric leaves an ambiguity in the normalisation of the time-like killing vector at infinity that appears in the computation of the energy and the temperature. Therefore, the temperature in ten dimensions is the same as in four dimensions up to a numerical factor.]
E = 2m/e_A^3e_B^3r_+^2κ ^2( 16π
^2) ^2L_xL_y ,
J_A = 8Q_A/e_A^4e_B^3κ ^2( 16π
^2) ^2L_xL_y ,
J_B =8Q_B/e_B^4e_A^3κ ^2( 16π
^2) ^2L_xL_y ,
T = e_A^2+e_B^2/16π-4(
Q_A^2+Q_B^2) /16π r_+^4 ,
S =2r_+^2/e_A^3e_B^3G_10( 16π)
^2L_xL_y .
According to our normalisation κ^2=8π G_10 where G_10 is the Newton constant in ten dimensions. These quantities satisfy the first law of thermodynamics as expected
dE=TdS+Ω _AdJ_A+Ω _BdJ_B ,
where the angular velocities are
Ω _A= e_A Q_A/r_+^2 , Ω _B= e_B Q_B/r_+^2 .
This background can be understood as the lift to Type IIB of the four dimensional planar black hole with electric charges, found in <cit.>. The presence of the electric charges in four dimensions corresponds to rotations of the branes in ten dimensions.
We leave this black hole background here, as it is not the focus of the rest of this work. We move into the study of the dual Field Theories to the backgrounds I and II in eqs.(<ref>) and (<ref>)
§ A PROPOSAL FOR THE DUAL FIELD THEORY AND ITS OBSERVABLES
Here we present a proposal for the field theory dual to our new background in eq.(<ref>). It is convenient to first discuss the field theory dual to
the D5-D5 intersection and the background in eq.(<ref>).
§.§ The holographic dual to the Background I
We start discussing the field theory on I-branes.
The result in eq.(<ref>) indicates the presence of two stacks of D5 branes, with N_A and N_B being the number of branes on each stack.
When taken at weak coupling these stacks intersect over two dimensions. In <cit.> it was shown that when two stacks of branes intersect along (4k+2), being the transverse dimensions a multiple of four (in our case k=0, D5 stacks intersect in two dimensions and have eight transverse directions) the massless spectrum contains chiral fermions, arising from the open strings connecting the branes. These fermions give rise to gauge (and gravitational) anomalies on the intersection. The anomalies are cancelled by anomaly inflow from the 'bulk of the brane'. This implies that the D-branes world-volume action must contain a Chern-Simons term.
When studied at weak coupling the D5-D5 system preserve chiral supercharges. We have two gauge groups SU(N_A)× SU(N_B) with chiral fermions transforming in the ( N_A, N̅_B) representation, the system has SO(1,1) Poincare symmetry. The anomaly is cured by inflow from the bulk of the D5 branes. In other words, the dynamics of the intersection is not decoupled from the brane dynamics. The system preserves eight SUSYs <cit.>. The weakly coupled field theory is summarised by the quiver in Figure <ref>.
As the couplings grow large, the above description breaks down and is replaced by a description in terms of the background in eq.(<ref>). This is carefully described in <cit.>. In the strong coupling regime, the system preserves SO(2,1) with SO(4)× SO(4) R-symmetry and sixteen SUSYs. The three dimensional field theory has gauge groups with Chern Simons terms SU(N_A)_N_B× SU(N_B)_N_A <cit.>, <cit.>.
At strong coupling, these stacks intersect in the coordinates (t,x,φ). One of the stacks extends along (θ_A,ϕ_A,ψ_A) whilst the other does it over (θ_B,ϕ_B,ψ_B).
The papers <cit.>, <cit.>, argue that the field theory is gapped.
Notice that the background in eq. (<ref>) is not trustable for all the range of the radial coordinate. In fact, for large values of r, the dilaton becomes large the type IIB system is better described by performing an S-duality and describing the dynamics in terms of the Little String Theory on the two stacks of NS five branes.
That is, the field theory above described has a non-field theoretical UV completion.
On the other hand, for r→ 0 the Ricci scalar in eq.(<ref>) diverges and the background is not trustable. We then need to replace the description by the one given by the configuration in eq.(<ref>).
§.§ The field theory dual to the Background II.
Here, we analyse the background in eq.(<ref>). The supergravity solution is smooth, hence the dual QFT is strongly coupled.
In fact, the Ricci scalar in eq.(<ref>) is finite for r>r_+ and the string coupling (proportional to e^Φ) is bounded below some value r< r_*. The value of r_* is determined by observing that the dilaton in eq.(<ref>) can be changed by Φ=Φ_0+ log r at the cost of rescaling the Ramond form F_3→ e^-Φ_0 F_3. These scalings make the string coupling g_s=e^Φ= r e^Φ_0. The value r_*∼ e^-Φ_0 (for which g_s∼ 1), can be made arbitrarily large by suitably choosing the parameter Φ_0. Notice that a chosen large and negative Φ_0, makes the charges of the D5 branes larger.
We then conclude that the background is trustable in a large region of the radial coordinate [r_+,r_*). Consequently, in a large regime of energies the dual field theory is strongly coupled. At very high energies, when the string coupling becomes large we should S-dualise arriving to a configuration of intersecting and wrapped NS-five branes. The high energy behaviour of the field theory is UV-completed in terms of a Little String Theory.
We now discuss the flowing to lower energies. In this case, the coordinate φ is compactified, shrinks to zero size and one ends with a QFT with less SUSY, smaller R-symmetry and effectively in (1+1) dimensions.
Whilst for large values of the radial coordinate, the backgrounds I and II coincide at leading order, a relevant operator is deforming the field theory dual to background II. This deformation is associated with the subleading terms, proportional to the parameters Q_A, Q_B. As we lower the energy (still at strong coupling)
a Kaluza-Klein spectrum of massive modes arises due to the compactification of the branes on φ.
At energies around the scale set by r_+, the QFT dual to the background in eq.(<ref>) should be a (1+1) dimensional QFT. This should be the reduction of the Yang-Mills-Chern-Simons SU(N_A)_N_B× SU(N_B)_N_A to (1+1), preserving four supercharges. The QFT is expected to be gapped, confine and break part of the R-symmetry.
In what follows, we start the study of this interesting field theory. We do so by calculating observables of the two-dimensional QFT using probes of the background in eqs. (<ref>)-(<ref>).
These probes inform us about gauge couplings, theta-angles, symmetry breaking, confinement, etc. We present a quantity that indicates the number of degrees of freedom (density of states in terms of the energy).
§.§ Gauge coupling
To study the background using D-branes probes, it is first useful to set our conventions for the Dirac-Born-Infeld-Wess-Zumino (DBIWZ) action, describing the dynamics of branes in our background.
Consider a ten dimensional manifold M_10 equipped with a metric tensor G_μν, Neveu-Schwarz two form B_μν, dilaton Φ and Ramond potentials encoded in the poly-form C. In this space, there is an embedded manifold Σ of dimension (p+1) with (9-p)
space-like normal vectors. This embedded manifold hosts a Dp-brane. We denote the coordinates on the
Dp-brane as X^M with M=0,1,… ,p. and the induced metric g_MN.
The action of a single Dp-brane is the Dirac-Born-Infeld-Wess-Zumino action
given by
S_Dp, DBI[ g_MN, F_MN] =T_p∫ d^p+1xe^-Φ√(-( g_MN+ F_MN) ) .
S_Dp, WZ[ C, F_MN]= - T_p∫_Σ C∧ e^- F_MN
Here F_MN= B_MN +2πα' F_MN, where B_MN is the pull-back of the background Neveu-Schwarz two-form on Σ, and F_MN is an Abelian gauge field strength defined on the brane. The tension of the Dp brane is T_p=1/(2π)^7-p, in our chosen units. For the case of our backgrounds, we have B_MN=0 and the poly-form C= C_2 given in eq.(<ref>), and/or its electric dual C_6, given in Appendix <ref>.
With our choice of units (g_s=α'=1) we can perform a small field F_MN expansion of the action in eq.(<ref>), equivalent to a small-α' expansion. We obtain an effective action for
the Dp-brane,
S_Dp, DBI=T_p∫ d^p+1xe^-Φ√(- g_MN)[ 1-1/4
( 2π) ^2F^BCF_BC+𝒪(F^3) ]
where F^BC=g^BMg^CNF_MN. The Wess-Zumino part of the action in eq.(<ref>) contains a finite number of terms.
Specialising for our backgrounds with B_MN=0 we have for any D_p brane probe,
S_Dp,WZ=
- T_p∫_Σ_p+1 C_p+1 - 2π C_p-1∧ F_2 +(2π )^2/2 C_p-3∧ F_2^2 -(2π)^3/6 C_p-5∧ F_2^3+(2π)^4/24C_p-7∧ F_2^4.
In what follows we study the backgrounds in eqs. (<ref>), (<ref>)-(<ref>) with various probe branes in Type IIB.
The first probe is a D5 brane that extends on the directions [t,x,φ, θ_A,ϕ_A,ψ_A]. This is like a probe that extends where the A-stack originally was. We will switch on an electric field on its worldvolume.
Calculation in the Background I
Let us start by performing the probe calculation in the background of eq.(<ref>). The dual QFT is (2+1)-dimensional, as the coordinate φ is not compact. By expanding the Born-Infeld action, we find a Maxwell term, with coupling (the details of this calculation are spelled out below)
1/g_YM,A^2= 16π^4/e_A^3T_5 √(e_A^2+ 2 e_B^2).
Similarly, the Wess-Zumino term gives,
S_D5,WZ= -2π^2 T_5∫ C_2∧ F_2∧ F_2=- ∫_S^3_A F_3 ∫_t,x,φ A_1∧ F_2= -N_B ∫_t,x,φ A_1∧ F_2.
We have performed an integration by parts, used the quantisation condition in eq.(<ref>) and set T_5=1/(2π)^2 (in our units).
There is a similarly symmetric calculation for a D5 probe along [t,x,φ, θ_B,ϕ_B,ψ_B]. In agreement with the field theory picture discussed above, we find two gauge groups with Yang-Mills Chern-Simons dynamics, SU(N_A)_N_B× SU(N_B)_N_A, with fixed gauge couplings. This is exactly in agreement with the field theory expectations <cit.>, that we summarised in the previous section.
Calculation in the Background II
Let us now study the case for which the (2+1) QFT has been compactified along the φ-direction and we are dealing with a (1+1) dimensional QFT.
We work with the background in eqs.(<ref>)-(<ref>), and follow the calculation above, by first writing the induced metric on the probe D5 brane,
ds_D̂_5^2=r{ -dt^2+dx^2+[ f_s( r)
+2Q_B^2ζ( r) ^2] dφ ^2+2/e_A^2
[ ω̂_1^2+ω̂_2^2+( ω̂
_3-e_AQ_Aζ (r)dφ) ^2] } .
From here we calculate
e^-Φ√(- g_MN^( D̂_5) )=r^2( 2/
e_A^2) ^3/2sinθ_A√(f( r)
+2Q_B^2ζ( r) ^2).
The effective action
for the brane in eq. (<ref>) reads,
S_D̂_5, BI = -T_D_5∫ dt dx dφ dθ_Adϕ_Adψ_A( 2/e_A^2) ^3/2r^2sinθ_A√(f( r) +2Q_B^2ζ( r) ^2)(1-(2π)^2/4 F^BCF_BC) ,
=
T_D_5L_φ(
4π) ^2( 2/e_A^2) ^3/2r^2√(f(
r) +2Q_B^2ζ( r) ^2)∫ dtdx(1 -(2π)^2/4
F^BCF_BC) .
We turn on F_tx so that
F^BCF_BC=2F_txF_txg^ttg^xx=21/r^2F_txF_txη
^ttη ^xx=1/r^2F_μν^2 .
From here we identify the Yang-Mills coupling for this probe D5,
1/g_YM,A^2=( 2π)^4T_D̂_5 L_φ( 2/e_A^2) ^3/2√(f( r) +2Q_B^2ζ( r) ^2).
If the D5 brane probes the SUSY preserving background, we impose Q_B=e_B/e_AQ_A and m =0 on the parameters appearing in eqs.(<ref>)-(<ref>). This implies r_±^2=±2Q_A/e_A and L_φ=4π
/(e_A^2+e_B^2). Together with eq.(<ref>) and the quantisation condition (<ref>), the gauge coupling reduces to[For large values of the radial coordinate r and decompactifying φ, this result reduces to that in eq.(<ref>). ]
1/ĝ_YM,A^2= 16π^4/e_A^3 T_D̂_5L_φ√(( 1-2Q_A/
e_Ar^2) ( 2e_AQ_A/r^2+e_A^2+2e_B^2) ) ,
with limiting values
1/ĝ_YM,A^2={[ 16 π^4/e_A^3T_D̂_5L_φ√(e_A^2+2e_B^2) , r→∞; 0 , r→ r_+ ]. .
In other words, the gauge coupling grows very large at low energies and asymptotes to a constant value for high energies. As discussed, at very high energies the field theory is best described in terms of a Little String Theory (LST). This calculation above refers to the gauge coupling g_YM,A. Other interactions in the QFT may become large at high energies, in such a way that the field theory is strongly coupled in the UV. This is in agreement with the background in eq.(<ref>) being weakly curved for all values of the radial coordinate [r_+,∞).
Had we studied a D5 probe extended along [t,x,φ, θ_B,ϕ_B,ψ_B], with an electric field F_tx switched on the brane, the result would be,
1/g_YM,B^2=( 2π)^4T_D_5 L_φ( 2/e_B^2) ^3/2√(f( r) +2Q_A^2ζ( r) ^2).
§.§ Theta angle
From the viewpoint of the (2+1) dimensional QFT, represented by Background I in eq.(<ref>), we can consider dimensionally reducing the Chern-Simons term obtained in eq.(<ref>). We then obtain a theta-term proportional to N_B ∮_φ A_φ for the QFT on the A-stack.
For the (1+1) viewpoint, additional probes calculate the Θ-angle of each gauge group. Let us use a D3 probe, extended along
[t,x,θ_A,ϕ_A] with an electric field F_tx switched on. We study the Wess-Zumino term following eq.(<ref>) and using the two-form potential pulled-back on this D3 probe
C_2|_D_3=-2/e_A^2ψ_Asinθ_Adθ_A∧ dϕ_A .
Replacing this C_2 in eq.(<ref>) and using that F_2= F_tx dt∧ dx we find the Wess-Zumino term for this probe is (note that C_4=0 in the background),
S_WZ,D3=-T_3 2π∫ C_2∧ F_2 = T_3 16 π^2 /e_A^2ψ_A∫ F_txdt ∧ dx .
The Θ-angle associated with the gauge group should be identified according to ,
S_WZ,D3= Θ_A/4π^2∫ dt dx F_tx, ⟶ Θ_A =T_D_364π^4 /e_A^2ψ_A= ψ_A N_B/2 .
We have used 16 π^4 T_D3=1 (in our units) and the quantisation condition in eq.(<ref>). Notice that the periodic identification Θ_A∼Θ_A + 2 k π implies that the angle ψ_A gets quantised to the values
Δψ_A= 4 k π/N_B, with k=0,1,2...., N_B-1.
For k=N_B we have Δψ_A=4π, covering the full circle.
Had we considered the D3 probe extended along [t,x, θ_B,ϕ_B] with all other coordinates fixed, we would have found (the calculation is identical),
Θ_B = ψ_B N_A/2.
Another way of understanding the R-symmetry breaking would be to consider euclidean D1 branes wrapping [θ_A,ϕ_A].
The Wess-Zumino action contributes to the partition function via
Z_D1∼ e^i/2π∫ C_2.
This contribution should not depend on a (large) gauge transformation parameter ϵ_A, that appears as we change ψ_A→ψ_A+ϵ_A. We enforce
(i/2π) (- 2ϵ_A/e_A^2)∫sinθ_A dθ_A dϕ_A= 2 i k π , leads to ϵ_A= 4 k π/N_B.
These results indicate that the background's continuous isometries transforming ψ_A,B→ψ_A,B + ϵ_A,B, with (ϵ_A,B being constants), are actually broken. In fact, the allowed changes are Δψ_A,B= 4 k π/N_B,A, with k=0,1,2...., N_B,A-1, which should be interpreted as the breaking of the two field theory global symmetries U(1)_A,B into discrete subgroups.
The argument used to derive eqs.(<ref>)-(<ref>) is not airtight. It uses a two manifold that has a boundary and at the same time a gauge choice is made for the potential C_2 in eq.(<ref>). It would be better to have an argument for R-symmetry breaking that is explicitly gauge invariant. In the coming section we present a different holographic perspective on the U(1)_A,B breaking with this property.
§.§ U(1)-A,B symmetry breaking pattern
Some supersymmetric field theories exhibit a classical U(1) R-symmetry that is quantum mechanically broken to a discrete subgroup. The symmetry breaking can be understood diagrammatically or in terms of instantons. The supergravity dual to the given field theory should encode this, but the mechanism should not involve instantons (as that are very suppressed in supergravity). The fact that the Ramond potentials are not gauge invariant under the U(1) R-symmetry is key. The breaking of the global R-symmetry in the field theory manifest as spontaneous breaking in supergravity. The vector field in the bulk, dual of the R-symmetry current acquires a mass. We find this below for our background of eq.(<ref>). The argument that follows is gauge invariant at all steps. All along this section, we set m=0 and focus only in the BPS case.
The U(1) symmetry of the metric is perturbed, the Lagrangian for this fluctuation is described by the usual F_μν^2-term. For the perturbation to be consistent, the Ramond fields must be also perturbed, this contributes to the mass term for the fluctuation. The massive gauge field is understood as symmetry breaking in supergravity.
In the previous section, we hinted at a breaking of the isometries represented by the Killing vectors ∂_ψ_A,B. To better understand the breaking of the field theory global U(1) symmetries associated with the translations in ψ_A,B we proceed as explained in <cit.>, <cit.>. We give full details in Appendix <ref>. In the holographic background, we gauge the isometry by replacing, both in the metric and in the Ramond potential of eq.(<ref>)
dψ_A,B→ dψ_A,B + A_A,B, ψ_A,B→ψ_A,B + ϵ_A,B.
Such that a change ψ_j →ψ_j + ϵ_j is compensated by A_j→ A_j + d ϵ_j.
We then study the Lagrangian for the gauge fields A_A,B, by replacing these changes in the string frame Lagrangian.
The metric perturbation changes the Ricci scalar to
e^-2Φ R → e^-2Φ(R -1/42r/e^2_AF^2_A
-1/42r/e^2_BF^2_B).
The kinetic term for F_3 changes as,
1/12F^μνλF_μνλ→ 1/12F^μνλF_μνλ
+ 1/24Q^2_A +e^2_A r^4/e^2_Ar^6(A_A-dϵ_A)^2
+ 1/24Q^2_B +e^2_B r^4/e^2_Br^6(A_B-dϵ_B)^2
+4Q_AQ_B/e_Ae_Br^6(A_A-dϵ_A)·(A_B-dϵ_B).
These perturbations around an isometry direction are usually known to be consistent. It is interesting to note that the kinetic term for the gauge field A_(A,B) comes only from the Ricci scalar. The information of ϵ_A,B on the other hand, comes only from F^2_3. Defining
W_(A,B) = A_A,B - dϵ_A,B,
and imposing the BPS condition e_AQ_B= e_BQ_A, the Lagrangian for the perturbation reads,
ℒ = -1/42/r e^2_AF^2_A
-1/42/r e^2_BF^2_B
+1/2r^2 (W^(A)_μW^(A)μ+W^(B)_μW^(B)μ)
+1/24Q^2_A/e^2_Ar^6(W^(A)_μ+ W^(B)_μ)^2.
We interpret this result as follows. There are two U(1) global symmetries in the QFT, holographically they are represented by the invariance of the metric under changes in ψ_A and ψ_B. These global symmetries are broken to discrete groups ℤ_N_B and ℤ_N_A, as indicated by eq.(<ref>). This breaking is an effect of the lack of invariance of the gauge potential C_2. The breaking of the global symmetries is addressed in this section without appeal to the Ramond potentials, by observing that gauging the metric isometries ∂_ψ_A,B leads to a breaking of the gauge symmetry, by a mass term. These mass terms are dependent on the radial coordinate.
Contrary to what happens for the duals to N=1 SYM, the metric in eq.(<ref>) does not break these discrete isometries to ℤ_2. In other words, there is not a radial-regime in the metric that explicitly breaks the isometry ∂_ψ_A,B. We interpret this result as the (presumably anomalous) breaking of the two U(1)_A× U(1)_B→ℤ_N_B×ℤ_N_A in the QFT not being followed by a further spontaneous breaking. One might argue that VEVs are not allowed in a two dimensional QFT, hence no further breaking can take place by the formation of a condensate. This argument is not completely rigorous, as our QFT is two dimensional in the far IR, but get UV completed around the confining scale to a higher dimensional QFT.
We study now a different observable that gives an approximate idea of the number of degrees of freedom as a function of the energy (a density of states).
§.§ Holographic central charge
Consider a generic holographic background dual to a QFT in (d + 1) spacetime dimensions, with metric and dilaton given by
ds^2 = a(r,y^i) [ dx^2_1,d+b(r) dr^2]+ g_ij(r,y^i)dy^i dy^j, Φ(r,y^i).
Following <cit.>, we define quantities V_int, H according to
V_int= ∫ dy^i √(e^-4 Φ a(r, y^i)^d [g_ij]), H= V_int^2.
From these we define the holographic central charge (or free energy),
c_hol = d^db(r)^d/2 H^2d+1/2/G^(10)_N (H')^d,
where G^(10)_N = 8π^6 is (in our conventions), the ten-dimensional Newton constant.
The holographic central charge of eq.(<ref>) makes perfect sense for backgrounds with an AdS-factor. In those cases the quantity in eq.(<ref>) is a number depending on the parameters of the background and it was successfully matched with the free energy of the CFT. In contrast for our case, without an AdS-factor we use eq.(<ref>) to give an indication of the number of degrees of freedom of the QFT.
Let us first compute the quantities in eqs.(<ref>)-(<ref>)
for the solution in eq.(<ref>), dual to a (2+1) dimensional QFT. We find,
d=2, a(r, y^i)=r, b(r)= 8/(e_A^2+e_B^2) r^2, V_int= N r^3, N= 8 (4π)^4/e_A^3e_B^3,
H= N^2 r^6, c_hol= 8 N/9 G_N (e_A^2+e_B^2) r^3.
This quantity diverges at large energies, hinting at a UV completion in terms of a system in higher dimensions (a LST). Also, it vanishes for r=0, indicating a gapped system.
Notice nevertheless that the calculation should not be trusted close to r=0, as the background (<ref>) is singular there.
Let us now calculate for the background in eq.(<ref>), as a dual to a (1+1)-dim QFT. We find
d=1, a(r,y^i)= r, b(r)= 4/r^2 f_s(r),
V_int= N r^2√(f_s(r)), Ĥ= N^2 r^4 f_s(r), N=(4π)^48/e_A^3 e_B^3 L_φ,
c_hol= N/2 G_Nf_s(r) r^2/( f_s(r) +r/4 f'_s(r) ).
At high energies
the number of degrees of freedom grows unbounded (as r^2), signalling the UV completion in terms of a decompatified QFT in higher dimensions. At very low energies the number of degrees of freedom vanish, as f_s(r_+)=0. In this case, the calculation is trustable, hence the gapped character of the system is clear.
A related calculation can be done that encodes the fact that we can think our field theory as a three dimensional QFT with anisotropies (or a QFT with a flow across dimensions). We follow the treatment described in section 8.2 of the paper <cit.>, see also <cit.>. Using the notation of <cit.> we have,
d=2, α_0=α_1=r, α_2=r f_s(r), β= 4/r^2 f_s^3/2, H= N^2 r^4 f_s(r), N=(2/e_A e_B)^3 (4π)^4,
c_flow= N/G_Nr^2 f_s/(f_s(r)+ r/4 f_s(r)')^2
.
The result in eq.(<ref>) has a similar interpretation. At low energies we have no degrees of freedom, at high energies an unbounded growth in the degrees of freedom indicates the UV completion. Note that the growth at high energies for the flow-central-charge–eq.(<ref>), is slower than it is for the case in which we consider the QFT to be two dimensional, see eq. (<ref>). This same feature occurs when considering flows between conformal points in different dimensions.
These results suggest that our QFT generates a mass gap at low energies. The behaviour of the gauge couplings at low energies, see eq.(<ref>) also suggest that the QFT is confining. To ascertain the confining behaviour we calculate Maldacena-Wilson and 't Hooft loops, that provide order parameters for confinement. We also investigate the Entanglement Entropy (EE), which gives information about the interplay between a confining IR and the non-local UV dynamics of the QFT.
§ MALDACENA-WILSON, 'T HOOFT LOOPS AND ENTANGLEMENT ENTROPY
In this section we calculate different observables to learn more about the proposed field theory.
We start calculating the Maldacena-Wilson loops <cit.>, in order to test the above proposal that at low energies the QFT presents a mass gap and confines. We start with a summary of the formalism to compute Maldacena-Wilson loops. This same formalism is then adapted for the study of 't Hooft loops and Entanglement Entropy.
§.§ General comments on Maldacena-Wilson loops and similar probes
We start summarising general results pertaining holographic Wilson loops. We follow the treatment of <cit.>, <cit.>. This generic treatment is also useful for the study of other probes that reduce to an 'effective string' in the background. Hence it will apply to 't Hooft loops, Entanglement Entropy, as we discuss below.
Consider a generic holographic background of the form
ds^2=-g_ttdt^2+g_xxdx⃗^2+g_rr
dr^2+g_ijdθ ^idθ ^j .
We assume that g_tt, g_xx, g_rr depend only on the radial coordinate r. As usual, we propose a string embedding (parametrised in terms of (τ,σ) the worldsheet coordinates), which leads to a Nambu-Goto action for the F1-string of the form,
t=τ, x=x(σ), r=r(σ).
S_NG= T_F1∫ dτ dσ√(g_tt(r) g_xx(r)x'^2 + g_tt(r)g_rr(r) r'^2).
From this action, the equations for the string moving in the generic background reduce to (see <cit.> for a detailed derivation)
dr /dσ=±dx/dσV_eff( r ) .
We defined the effective potential
V_eff( r ) =F( r ) /CG( r
) √(F^2( r ) -C^2) , F^2( r ) =g_ttg_xx, G^2( r )=g_ttg_rr.
The constant C=F^2x'/√(F^2 x'^2+ G^2 r'^2) is obtained from one of the equations of motion. In the simpler case in which we take x(σ)=σ we find eq.(<ref>) from the conserved Hamiltonian. In that case C= F(r_0), being r_0 the turning point of the string satisfying r'(σ)=0 (these are called U-shaped embeddings). We set C=F(r_0) in what follows. We enumerate below a set of properties of the U-shaped embeddings.
* This formalism applies to an open string whose end points are at r →∞, where we add a D-brane. Dirichlet boundary conditions for the string at r →∞ require that V_eff|_r →∞∼∞.
* We compute the separation between the two ends of the string on the D-brane, which can be thought as the separation between a quark-antiquark pair. The energy of the pair of quarks calculated from the Nambu-Goto action needs regularisation, implemented by subtracting the mass of two non-dynamical strings extended along the whole range of the radial coordinate [r_+, ∞).
The separation and energy are given as functions of r_0 (the distance from the origin of the radial coordinate, r_+, to the position of the turning point of the string). The expressions for these quantities are,
L_QQ( r_0) =2∫_r_0^+∞dz/V_eff(z) ,
E_QQ( r_0) =F( r_0) L_QQ( r_0)
+2∫_r_0^+∞dzG( z) /F( z) √(F( z) ^2-F( r_0) ^2)-2∫_r_+^+∞dz G( z) .
*
To obtain a finite contribution coming from the upper limit of the QQ̅ pair separation in (<ref>), a further restriction on the behaviour of the effective potential at infinity is needed. See <cit.> for a derivation,
V_eff|_r → +∞∼ r^β , with β >1 .
*
When expanded close to the end of the space, which is at r=r_+ in the case of the background in eq.(<ref>), we find V_eff∼ (r-r_+)^γ. If 1 ≤γ, the separation between the pair becomes infinite–see <cit.>. Otherwise (if γ<1) we have screening behaviour.
*
There is an analytic relation between E_QQ and L_QQ which is
dE_QQ/dr_0=F(r_0) dL_QQ/d r_0 ⟶ dE_QQ/dL_QQ=F(r_0).
We have inverted the relation (<ref>) as r_0=r_0(L_QQ).
*
For generic backgrounds, the evaluation of the integral in eq.(<ref>) need not be simple nor have an expression in terms of elementary functions. Nevertheless, the quantity
L̂_QQ(r_0)= πG/F'|_r_0,
provides a reasonable approximation to eq.(<ref>).
We check this approximate expression for different observables studied below.
*
Following <cit.> we define
Z(r_0)= d/dr_0L̂_QQ(r_0)= πd/dr_0( G(r_0)/F'(r_0)).
The stability of the U-shaped string embedding in eq.(<ref>) is guaranteed if Z(r_0)<0 <cit.>. This is valid for any observable that can be reduced to an effective string action of the form (<ref>)[
It was shown in <cit.>,<cit.> that embeddings for which Z(r_0)≥ 0 do not satisfy two physically well motivated criteria <cit.>:
The force between the quark and the antiquark is always attractive and positive dE_QQ/dL_QQ>0. It is also a non-increasing function of the separation d^2 E_QQ/d L_QQ^2≤ 0.
The proposal in <cit.> is that the two criteria above are equivalent to the stability of the U-shaped embedding or conversely Z(r_0)<0.]
*
In calculations like those in eqs.(<ref>)-(<ref>) we can introduce a cutoff r_UV to regulate divergences coming from the upper limit in the integrals. We
define a quantity analog to (<ref>),
L_QQ(r_0,r_UV)= 2∫_r_0^r_UVdz/V_eff(z).
Following <cit.> we calculate
L_a= lim_r_0→∞lim_r_UV→∞ L_QQ(r_0, r_UV) and L_b= lim_r_UV→∞lim_r_0→ r_UV L_QQ(r_0, r_UV).
Whilst L_b=0 by definition, it is sometimes the case that L_a is nonzero.
If in this case the U-shaped configuration is unstable (Z(r_0)>0), there exist 'short configurations' (that appear very close to the cutoff). These short configurations are energetically favoured and induce a phase transition in the observable calculated <cit.>, <cit.>.
The existence of these short configurations that appear when introducing the UV-cutoff is not an artefact of the cutoff. They indicate that the string embedding we proposed in eq.(<ref>) is not capturing the dynamically favoured configuration.
In the study of the 't Hooft loop and the Entanglement Entropy we encounter these short configurations, that cure the problem of the instability of the embedding and introduce a phase transition in the observable. We take the instability of the configuration (cured by the introduction of a cutoff) as an indication that the dynamics of the LST is driving the observable.
Let us now apply this general treatment to our background in eq.(<ref>).
§.§ Maldacena-Wilson loops in our background
Let us apply the expressions in eqs.(<ref>)-(<ref>) to our background in eq.(<ref>).
The relevant functions are
F(r)=r, G(r)=2/√(f_s(r)), V_eff(r)= √(e_A^2+e_B^2/8)( 1/r_0 r) √((r^2-r_0^2)(r^2-r_+^2)(r^2-r_-^2)).
The condition in eq.(<ref>) to have Dirichlet boundary conditions for the string at r→∞, is satisfied. When expanded close to r_0=r_+, V_eff∼ (r-r_+), indicating that L_QQ diverges when r_0 approaches the end of the space r_+. This points to confining behaviour.
Another quick way of determining a confining behaviour is to check the function F(r_+), that intuitively represents the tension of a QCD-string at low energies in the QFT (close to the end of the space of the geometry). In this case we find F(r_+)=r_+, pointing to confining behaviour. In fact, a finite QCD-string tension leads to an energy growing with the separation of the quark pair.
Note also that the approximate formula for the separation of the quark pair eq.(<ref>), gives for the functions in eq.(<ref>)
L̂_QQ(r_0)=2 π/√(f_s(r_0)).
For r_0→∞ this gives L̂_QQ= π√(8/e_A^2+e_B^2), which refers to a characteristic length of the UV completion (the Little String Theory scale). On the other hand, for r_0∼ r_+, the approximate length diverges and 1/√(r_0-r_+). This indicates that the quark-anti-quark pair can be infinitely separated. According to eq.(<ref>), this gives an energy that scales linearly with the separation, another signal of a confining behaviour.
The quantity Z(r_0) in eq.(<ref>) reads,
Z(r_0)= 2√(2)π r_0/√(e_A^2+e_B^2)( -r_0^2(r_+^2+ r_-^2) +2 r_+^2r_-^2)/[ (r_0^2-r_+^2)(r_0^2-r_-^2)]^3/2.
By inspection, one find that Z(r_0)<0 in all the range. Hence the U-shaped embeddings are stable and no phase transition is expected for the Maldacena-Wilson loops.
The rigorous way of determining the low energy behaviour of the QFT (either confining or screening) is to analyse the expressions for the distance and the energy of the quarks pair.
The distance between the quark-antiquark pair and its energy are written from eqs.(<ref>)-(<ref>)
L_QQ( r_0) = √(32 /e_A^2+ e_B^2 ) r_0 ∫_r_0^∞z
√(1/( z^2-r_+^2) ( z^2-r_-^2)
( z^2-r_0^2) )dz ,
E_QQ( r_0) =r_0 L_QQ( r_0) +√(32/e_A^2+ e_B^2 )[
∫_r_0^∞ dz √(z^2( z^2-r_0^2) /( z^2-r_+^2) ( z^2-r_-^2) ) -
∫_r_+^∞ dz z^2/√(( z^2-r_+^2) ( z^2-r_-^2) )] .
The integrals can be performed analytically and expressed in terms of in terms of Elliptic integrals of the first kind. It is interesting to study this in the BPS limit, when m=0 and r_+^2=- r_-^2. See Appendix <ref> for a detailed study. Let us quote the explicit results for the separation and energy of the quark pair in eqs.(<ref>)-(<ref>). Defining
the elliptic integrals,
K(x)=∫_0^π/2dθ/√(1-x sin^2θ), E(x)= ∫_0^π/2√(1 -x sin^2θ) dθ,
we can write the explicit expressions for the separation and energy of the quark-antiquark pair, which read
L_QQ( r_0) =2√(8/e_A^2+e_B^2)
r_0/r_+/√(r_0^2/r_+^2-1)𝐊(
1-r_-^2/r_+^2/1-r_0^2/r_+^2) ,
E_QQ( r_0) = 2r_+√(8/e_A^2+e_B^2)
[ r_0^2/r_+^2/√(r_0^2/r_+^2-1)𝐊
( 1-r_-^2/r_+^2/1-r_0^2/r_+^2) .
. -𝐄( 1-r_-^2/r_+^2/
1-r_0^2/r_+^2) √(r_0^2/r_+^2-1)+C(
r_-/r_+) ] ,
where
C( r_-/r_+) =𝐄( r_-^2/r_+^2)
+λ _-𝐊( 1-r_-^2/r_+^2) +iλ _-
𝐊( r_-^2/r_+^2) -( 1-r_-^2/r_+^2) 𝐊(
r_-^2/r_+^2) .
λ_-^2=-r_-^2/r_+^2.
See Appendix <ref> for a careful derivation of these expressions. We plot these results in Figure <ref>. The various panels of Figure <ref> show some conventional and other less conventional behaviours. First, note that the expression for L̂_QQ in eq.(<ref>) very well approximates the exact expression in eq.(<ref>). As is usual, the concavity of the curve E_QQ(L_QQ) is 'downwards', indicating that the Nambu-Goto string configuration of eq.(<ref>) is stable, as confirmed by the Z(r_0) in eq.(<ref>). Note also that for large separations between the quark pair L_QQ, the energy grows linearly (signalling confinement). What is less conventional is that there is a minimal separation, given by the Little String Theory scale. This indicates that the far UV of the QFT dynamics is not field theoretical (but has the dynamics of the LST). The plot of the strings profiles confirms this. Indeed, strings that barely explore the bulk (with r_0/r_+ large) show a minimal fixed separation between the quark pair. On the other hand, the strings that explore deeper into the bulk display a bigger quark separation and carry higher energy.
Let us now focus on a second interesting observable, the 't Hooft loop. In the next section we propose a string-like object (for the gauge theory observer) with magnetic charge. This characteristically is represented by a Dp brane that wraps a (p-1) cycle in the internal space. Once the Born-Infeld action for this Dp brane is written and integrals over the internal space are performed, we arrive at an action for the 'effective string'. This action is studied with the same formalism as that used for Wilson loops, described in Section <ref>.
§.§ 't Hooft loops
The 't Hooft loop can be calculated by proposing an object with magnetic charge that effectively appears like a string for the (1+1) dimensional QFT. The ends of this magnetic string appear as a pair of monopoles of oposite charge. To study this object, we adapt the formulas
summarised in eqs.(<ref>)-(<ref>) for the effective magnetic string.
We propose to calculate the 't Hooft loop by studying the effective magnetic string obtained when extending a D5 brane along the directions [t,x,φ,θ_A,ϕ_A,ψ_A], with r(x). There is an analog magnetic string for the second gauge group, for the D5 in the configuration [t,x,φ,θ_B,ϕ_B,ψ_B], with r(x). We do not discuss the latter object, as its result is analog to the one obtained below. Note that this object becomes string-like when we consider the size of the S^1_φ to be small enough.
The induced metric for the above D5 is,
ds^2 = r{ -dt^2+( 1+4/r^2f_s( r)
r^' 2) dx^2+( f_s( r) +2Q_B^2ζ( r) ^2) dφ ^2.
. +2/e_A^2[ ω̂_1^2+ω̂
_2^2+( ω̂_3-e_AQ_Aζ (r)dφ) ^2
] } .
Then we calculate,
e^-Φ√(- g_MN)=( 2/e_A^2) ^3/2r^2sinθ_A√(( 1+4r^' 2/r^2f( r) )
( f( r) +2Q_B^2ζ( r) ^2) ) .
The action for this D5 is,
S_D̂_5[ r] =T_D_5L_φ( 4π) ^2( 2/e_A^2) ^3/2∫ dtdx√((
f_s( r) +2Q_B^2ζ( r) ^2)
r^4+4r^2( 1+2Q_B^2ζ( r) ^2/f_s(
r) ) r^' 2) .
Comparing with (<ref>)-(<ref>) we identify
F^2 =( f_s(r) +2 Q_B^2ζ(r)^2) r^4 , G^2=4/
r^2f_s( r) F^2 , T_eff =T_D_5L_φL_t( 4π) ^2(
2/e_A^2) ^3/2 .
Where T_eff is the effective tension of the magnetic string.
Following eq.(<ref>), the effective potential
V_eff( r) =r^3/2C√(f_s( r) (
f_s( r) +2Q_B^2ζ( r) ^2-C^2/r^4
) ).
The constant C= F(r_0)= √(( f_s(r_0) +2 Q_B^2ζ^2(r_0)) )r_0^2. In the asymptotic region the effective potential in eq.(<ref>) scales as V_eff( r→∞) ∼ r^3, satisfying the condition in eq.(<ref>).
The intuitive criteria discussed in the previous section applied for this case indicate a screening behaviour. In fact F(r_+)=0 and V_eff∼(r-r_+)^1/2, signalling a vanishing tension of the magnetic QCD-string in the IR, and a finite maximal separation, after which the monopoles are screened (more about this is discussed below).
We perform a similar analysis using the approximate expression for the separation between the monopole-anti-monopole pair L̂_MM. Replacing in eq.(<ref>) the functions in eq.(<ref>), we find an involved expression that asymptotes as,
L̂_MM(r_0→∞)∼π√(2/e_A^2+ e_B^2), L̂_MM(r_0→ r_+ )∼ 8π√(r_+(r_0-r_+)/(e_A^2+e_B^2)(r_+^2-r_-^2)).
These asymptotic behaviours indicate that at high energies in the field theory, the pair is separated by a maximum distance characteristic of the UV completion (Little String Theory). In this sense, the magnetic string behaves oppositely to the electric one used to compute the Wilson loop (which shows a minimum separation). On the other hand, at low energies the separation decreases to zero (again, oppositely to the electric string case). We calculate Z(r_0) in eq.(<ref>) for this configuration and find that is positive in all the range. This indicates the instability of the U-shaped embeddings. A more general embedding that the one proposed here should drive the dynamics. Instead of finding this more complicated embedding, below we introduce a UV-cutoff. New short configurations appear close to a cutoff, that dominate the dynamics and produce a phase transition (to a deconfining behaviour).
More formally, we write expressions for the separation between the monopole-anti-monopole pair as those in eqs.(<ref>)-(<ref>). A careful analysis of these integrals is performed in Appendix <ref>. Let us quote the exact expression for the separation for the monopole-anti-monopole pair. We leave the study of the expression for the energy between the pair of monopoles for Appendix <ref>. In terms of the elliptic integral of the first kind
F(y| x)= ∫_0^y dθ/√(1-x sin^2θ),
and the definition for 𝐊 (x) in eq.(<ref>), working in the BPS limit and defining η =e_A/e_B, we find
L_MM^BPS( r_0) = 2/e_B√(2(
r_0^2/r_+^2-1) ( ( η ^2+2)
r_0^2/r_+^2+η ^2) /( η ^2+1) (
r_0^2/r_+^2( η ^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) r_0^2/r_+^2) /(
2r_0^2/r_+^2+η ^2( 1+r_0^2/r_+^2) ) )
|( 1+r_0^2/r_+^2) (
2r_0^2/r_+^2+η ^2( 1+r_0^2/r_+^2) ) /
-4+4( 2+η ^2) r_0^2/r_+^2) .
. +i𝐊( 1-(
1+r_0^2/r_+^2) ( 2r_0^2/r_+^2+η ^2(
1+r_0^2/r_+^2) ) /-4+4( 2+η ^2)
r_0^2/r_+^2) ] .
In Figure <ref> we compare this exact expression with the approximating function whose asymptotics we write in eq.(<ref>). As stated above, opposite to the Wilson loop case, we encounter a maximal separation for the monopole pair. This maximal separation is associated with the Little String Theory scale. We also plot the string profiles as they enter the bulk. We observe a different behaviour to that found in the case of the Wilson loop–compare with the lower right panel of Figure <ref>.
For magnetic strings that barely explore the bulk, the separation between the monopole pair is large (equal to the LST scale). As we decrease the separation between the monopole pair, the magnetic string dives into r_0→ r_+. These unconventional behaviours, together with the instability of the string embedding–note that E_MM(L_MM) has upwards concavity (and Z(r_0) is positive), indicate the presence of a second 'disconnected' configuration for which the pair of monopoles separate without energy expense, which is indicative of screening. The transition to a disconnected configuration is dynamically favoured.
To avoid the 't Hooft loop to be driven by the UV (LST) dynamics and to realise explicitly these short configurations, we introduce a hard cutoff in the radial direction and recalculate things. Doing so, the behaviour changes qualitatively.
In fact, after introducing the UV cut-off, the separation between the pair of monopoles does not show a maximum value, instead we find a 'double valued' behaviour as displayed in the left panel of Figure <ref>.
This leads to a phase transition in the curve E_MM(L_MM). Note that this curve now has the correct 'downwards' concavity, indicating that the configuration is stable. This phase transition is the physical manifestation of the magnetic string suddenly changing into two disconnected magnetic strings that move without energy expense. This is deconfinement for the pair monopole anti-monopole. We are finding confinement for the quark-anti-quark pair and screening for a pair of monopoles. These behaviours are consistent with the (electric) confining behaviour of the dual QFT.
The introduction of the UV cutoff might seem unsatisfactory. Here we use it as a device to show that the correct five brane embedding must be more elaborated than the one we proposed above. It is also used to avoid the LST overtaking the dynamics. The effect of the cutoff is clear considering the integral needed to calculate L(r_0). This integral vanishes for r_0→ r_MAX. This produces a double-valued L(r_0) and a consequent phase transition.
For the Entanglement Entropy a very similar behaviour occurs. We study this next.
§.§ Entanglement Entropy
The Entanglement Entropy (EE) between two regions for field theories with a string dual can be calculated as shown in <cit.>. The method is to find a minimal area eight-surface (Σ_8, a codimension-two surface to which we refer below as RT surface) such that the boundary of the surface coincides with the two entangled regions. We focus on the case in which one of the regions is a strip of size L_EE and the other region is the complement.
The EE between these regions is given in <cit.>, <cit.>, minimising the quantity
S_EE=1/4 G_N∫_Σ_8 d^8σ√(e^-4Φ[g_Σ_8]).
There are various eight-surfaces that minimise S_EE in eq.(<ref>). Due to this, in some cases there is a phase transition between different extremal surfaces.
It was suggested in <cit.> that a criterium for confinement is the presence of a phase transition in the EE. This proposal was critically analysed in <cit.>. It was found that for the case of field theories that confine, but have a non-local high energy behaviour, the phase transition in the EE is absent. The point is subtle, as introducing a UV cutoff or UV-completing the QFT to avoid the non-locality, recovers the phase transition.
It is in this way that the EE can serve as an order parameter for confinement, but also as a tool to diagnose non-locality in the UV-behaviour of the QFT (when used together with a confining Wilson loop).
In <cit.> it was found that introducing a UV-cutoff implies the existence of new configurations realising the phase transition (and resolving a stability issue with the original eight-surface). Below, we perform an analysis of these features in our background of eq.(<ref>).
We follow the approach of <cit.>, in particular the treatment for non-AdS backgrounds developed in <cit.>, <cit.>.
We calculate the Entanglement Entropy on a strip by computing the area of an eight-surface [x,φ, θ_A,ϕ_A,ψ_A,θ_B,ϕ_B,ψ_B] with r=r(x) in the background of eq.(<ref>).
The induced metric on the RT eight-surface, its determinant and the Entanglement Entropy are,
ds^2_st = r{ dx^2(1+4 r'^2/r^2 f_s(r)) +f_s( r) dφ ^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3^2-e_BQ_Bζ( r)
dφ) ^2] } ,
√(e^-4Φ[g_8]) = ( 8/e_A^3e_B^3)√(r^4 f_s(r) + 4 r^2 r'^2)sinθ_Asinθ_B.
S_EE = 1/4 G_N∫ d^8x √(e^-4Φ[g_8])=( 2 (4π)^4 L_φ/e_A^3e_B^3 G_N)∫_-L/2^L/2 dx √(r^4 f_s(r) + 4 r^2 r'^2).
From eqs.(<ref>)-(<ref>), this implies
F(r)= r^2√(f_s(r)), G(r)= 2 r.
To minimise the S_EE above, we follow the usual conserved Hamiltonian treatment. The Entanglement Entropy needs to be regularised by the area of two eight-surfaces that hang straight from infinity. Then, computing the regulated area for a surface that turns around at r_0, we find for the length of the interval and the Entanglement Entropy,
L= 4 r_0^2 √(f_s(r_0))∫_r_0^∞dr/√(r^2 f_s(r) ( r^4 f_s(r) - r_0^4 f_s(r_0) )),
S_EE= N/G_N[∫_r_0^∞√(r^6 f_s(r)/r^4 f_s(r) -r_0^4 f_s(r_0) ) dr -∫_r_+^∞ r dr ].
As in eq.(<ref>), we can write a simple expression that approximates L_EE in eq.(<ref>)—see <cit.>,
L̂_EE= π G(r_0)/F'(r_0)=2πH(r)√(β(r))/H'(r)|_r_0, with H(r)= N^2 r^4 f_s(r), β(r)=4/r^2 f_s(r).
Using eq.(<ref>) we find
L̂_EE= (π√(8/e_A^2+e_B^2))
√((r_0^2-r_+^2)(r_0^2-r_-^2))/(2r_0^2-r_+^2-r_-^2) .
This function is monotonous, going from a vanishing value at r_0=r_+ to a constant value at r_0→∞. This behaviour prevents the possibility of phase transitions, which require that for a given L_EE there are two possible values of r_0.
In fact, the conditions for the presence of a phase transition (see section 2.4 of the work <cit.>) are not satisfied, in particular equations (2.26)-(2.29) of <cit.> imply j=2 preventing a phase transition. The absence of a phase transition in a confining model was interpreted in <cit.> as an effect of the non-locality of the completion of the QFT, in this case, by a LST.
Since Z(r_0) defined in eq.(<ref>) gives
Z(r_0)= √(8)π (r_+^2-r_-^2)^2/r_0 f_s(r_0) (r_+^2+r_-^2-2r_0^2)^2>0,
the proposed embedding is unstable. Upon the introduction of a cutoff, new surfaces appear as found in <cit.>, <cit.>. These cure the instability problem of the embedding and give place to the phase transition, in agreement with confinement.
The treatment in the papers <cit.>, <cit.> applies to our background, even when the IR dynamics is different, the UV dynamics is similarly driven by a Little String Theory.
Before discussing the presence (or absence) of phase transitions we write the analytic expressions for the values of the separation between the two entangled regions L_EE(r_0) and the Entanglement Entropy S_EE(r_0). These expressions are explicitly derived in Appendix <ref>. Using the definitions in eqs.(<ref>),(<ref>) and recalling that λ_-^2=-r_-^2/r_+^2, we find
L_EE( r_0) = 2√(8/e_A^2+e_B^2)√(
r_0^2/r_+^2-1/r_0^2/r_+^2+λ _-^2)[ i
𝐊( ( r_0^2/r_+^2-1) ^2/(
r_0^2/r_+^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(
r_0^2/r_+^2-r_-^2/r_+^2/1-r_-^2/r_+^2)|( 1-r_-^2/r_+^2) (
-r_-^2/r_+^2+2r_0^2/r_+^2-1) /(
r_0^2/r_+^2-r_-^2/r_+^2) ^2) ] ,
and
S_EE^BPS( r_0) =𝒩/G_Nr_+^2[
r_+^2/2r_0^2( -r_0^4/r_+^4𝐄(
r_+^4/r_0^4) -𝐊( r_+^4/r_0^4)
+r_0^4/r_+^4𝐊( r_+^4/r_0^4) ) +
1/2] .
To analyse these expressions, it is useful to show some plots. First, we check that the approximate L̂_EE(r_0) in eq.(<ref>) approximates well the analytic expression in eq.(<ref>), see the left panel of Figure <ref>.
We also plot S_EE(r_0), see the right panel of Figure <ref>. The plot of S_EE in terms of L_EE in the left panel of Figure <ref>, shows an upwards concavity indicating that the configuration is unstable. This follows the prediction of <cit.> that indicates that new configurations should appear as we introduce a UV-cutoff in the geometry. The profiles of the effective strings shown in the right panel of Figure <ref>, display a behaviour similar to the one we encountered in the study of 't Hooft loops (and opposite to that of the Wilson loop), again suggesting the need for a phase transition.
In analogy with the case of the 't Hooft loop, if we introduce a UV-cutoff, the separation between the two entangled regions becomes multiple-valued, as shown in Figure <ref> (right panel). This is at the root of the phase transition. The plot of S_EE (L_EE) shows the correct concavity and the presence of a transition to the disconnected configuration is clearly displayed. See Figure <ref>
Following the findings of <cit.>, we state that if a field theoretical UV completion to our system (that is completed by a LST) were found, the phase transitions for the 't Hooft loops and the EE would become apparent. In this sense the UV-cutoff captures the correct dynamics.
§ SIGMA MODEL
In this section we review some results <cit.>, <cit.>, concerning the string σ-model on the background in eq.(<ref>)[We gratefully acknowledge conversations with Lewis Cole and Gastón Giribet on the topics discussed here.]. To properly study the string action we move to the S-dual frame, and work in terms of NS5 branes. After suitable coordinate changes detailed in Appendix <ref>, the background reads,
ds^2_st = -dt^2+ dx^2+dφ^2+ dρ^2
+ N_B ds^2(S^3_A)
+ N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -√(1/N_A+1/N_B) ρ.
For a careful derivation we refer the reader to Appendices <ref> and <ref>. The background is a product space of the form
ℝ^2,1×ℝ_ρ× S^3_N_A× S^3_N_B,
where ℝ_ρ denotes the direction (with the linear dilaton), and the subscript on the S^3 denotes the square of their radius. The metric on the spheres together with the H_3 flux on each of them allows us to write the σ-model on them as a WZW model on SU(2). This is due to the fact the the S^3 is a group manifold. Naively, the contribution of the ρ coordinate to the string action is (for clarity we reinstate the α'-factor)
S_ρ = 1/4πα'∫ d^2σ√(-h)( h^ab∂_aρ∂_bρ - α' √(1/N_A + 1/N_B)R^(2)ρ),
where R^(2) is the world-sheet Ricci scalar. However when ρ→ -∞ the theory becomes non-perturbative, since g_s∼ e^-ρ. In order to avoid the strong coupling region, it is necessary to add the tachyon operator e^2bρ to the action, so that
S_ρ = 1/4πα'∫ d^2σ√(-h)( h^ab∂_aρ∂_bρ - α' Q R^(2)ρ + Λ e^2bρ).
Here, b is related to the background charge Q = √(1/N_A + 1/N_B) as
Q = b+1/b,
such that it does not have a strong coupling region. Thus the contribution of the ρ direction to the σ-model corresponds to a Liouville field. The complete σ-model on this geometry is
U(1)^3×Liouville× SU(2)_N_AWZW× SU(2)_N_BWZW
where the subscripts denote the WZW level. One can check that this is indeed a good σ-model by computing the central charge. Here we use
c(U(1))=1, c(Liouville) = 1+ 6Q^2, c(SU(2)_k) = 3(k-2)/k.
In total we have
c_total=3 + 1 + 6 ( 1/N_A + 1/N_B)
+ 3(N_A-2)/N_A+ 3(N_B-2)/N_B = 10.
To this we should add the ten free fermions, that contribute to the central charge c_ferm=5. The central charge of the SUSY system c_SUSY=15, is then cancelled by the b-c and β-γ ghosts.
We now study the interesting case Q_A=Q_B=0, but with m>0. The configuration of interest is obtained by S-dualising
eq.(<ref>),
ds^2_st = -dt^2+ dx^2+ f_s(r)dφ^2+ 4 dr^2/r^2 f_s(r)
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - log(r).
Recall that here φ∼φ + π/Q^2, see eq.(<ref>), with Q^2=1/N_A+1/N_B, as above. Before proceeding to the σ-model analysis, it is convenient to perform some changes of variables. First, we want to rewrite the background (<ref>) in such a way that it reduces to (<ref>) when m=0. For this use
r = e^Q ρ, m̃ = m/4Q^2,
which leads to
ds^2_st = -dt^2+ dx^2+ 4Q^2(1-m̃ e^-2Qρ)dφ^2
+ dρ^2/1-m̃ e^-2Qρ
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - Q ρ .
We change coordinates as,
tanh^2(λ) = 1- m̃ e^-2Qρ, φ = ϕ/2Q^2,
which puts the geometry in the usual cigar form (note that ϕ has period 2π)
ds^2_st = -dt^2+ dx^2+ 1/Q^2( tanh^2(λ)dϕ^2 + dλ^2)
+ N_B ds^2(S^3_A) + N_A ds^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = - log( cosh(λ) ) - 1/2log(m̃) .
As explained in <cit.>, for more details see Section 2 of <cit.>, the above backgrounds leads to an exact σ-model
U(1)^2×SL(2,ℝ)_k/U(1)× SU(2)_N_B WZW× SU(2)_N_A WZW,
where k^-1 = Q^2=1/N_A+1/N_B. To check that this is also a good string σ-model we use
c( SL(2,ℝ)_k) = 3(k+2)/k, c( G/H ) = c(G)-c(H),
so that
c( SL(2,ℝ)_k/U(1)) = 2 + 6/k=2+6/N_A+6/N_B ,
from where is easy to see that c_total=10. We leave for future research the study of the σ-model in the background with m, Q_A,Q_B arbitrary.
§ CONCLUSIONS AND FUTURE RESEARCH
The I-brane QFT, defined as the (1+1) field theory on the intersection of two stacks of D5 branes, was studied in <cit.>. The field theory has the remarkable behaviour that as the coupling is increased, the system gains one more dimension and enhances its SUSY (with a peculiar SUSY algebra in (2+1) dimensions <cit.>). The background dual to this strongly coupled QFT was written in <cit.>, see our eq.(<ref>). This presents a singular behaviour for large values of the radial coordinate, where the dilaton diverges and string coupling effects cannot be neglected. This is solved by performing an S-duality and working with the NS branes system. The background of eq.(<ref>) is also singular for small values of the radial coordinate, r→ 0, as indicated by eq.(<ref>). This ill-defined IR behaviour is amended by our background in eq.(<ref>). Our simple and explicit solution describes the holographic dual to a (2+1) QFT that gets compactified to (1+1) dimensions, preserving four supercharges and ending the flow with a confining and gapped behaviour.
We holographically studied different aspects of this peculiar QFT. Maldacena-Wilson loops, 't Hooft loops, Entanglement entropy were discussed in dedicated sections, with emphasis on the effects of the UV-completion in terms of LST. R-symmetry and its breaking, a suitably defined gauge coupling and a quantity measuring the number of degrees of freedom as a function of the radial coordinate (the energy) are presented and discussed.
By a double Wick rotation, a black hole solution is found. Also, some of the NS string σ-model aspects are briefly mentioned.
It would be interesting to dedicate future efforts to
* The careful study of the black hole solution in Section <ref>. In particular if its entropy can be computed in terms of a (2+1) field theory compactified on a torus.
* The study of the string σ-model for the full solution in eq.(<ref>), in the NS5 branes frame.
* To achieve a cleaner understanding of the R-symmetry breaking in terms of anomalies in two dimensional QFT. To relate this to the Chern Simons coefficients discussed in eq.(<ref>). Note that while we find an anomalous breaking U(1)_A× U(1)_B→ℤ_N_B×ℤ_N_A, we do not find a further spontaneous breaking to ℤ_2×ℤ_2, as it normally occurs in holographic models to four dimensional N=1 dynamics. Understanding the reason of the absence of the spontaneous breaking would be of interest.
* Geometrically, it would be interesting to generalise the metric and fluxes in eq.(<ref>) adding warp factors in front of the ω_i's and more general fibrations. Finding a more general classification is of interest. It may be possible to relate this to the material in <cit.>.
* A fair amount of papers have been written studying the background in eq.(<ref>). See for example <cit.>-<cit.>. It would be interesting to understand the effects of the resolution provided in eq.(<ref>) on some of these observables.
We hope to report on some of these problems in the near future.
§ ACKNOWLEDGEMENTS:
The contents and presentation of this work much benefitted from extensive discussion with various colleagues. We are very happy to thank: Andres Anabalon, Adi Armoni, Fabrizio Canfora, Lewis Cole, Gaston Giribet, Nicolas Grandi, Nabil Iqbal, Prem Kumar, Juan Maldacena, Anibal Neira, Leo Pando Zayas, Julio Oliva, Niels Obers, Dibakar Roychowdhury, Kostas Skenderis, Christoph Uhlemann who shared their knowledge with us.
We are supported by STFC grant ST/T000813/1. The work of M.O. is partially funded by Beca ANID de
Doctorado 21222264. The work of R.S. is supported by STFC grant ST/W507878/1. The authors have applied to a Creative Commons Attribution (CC BY) licence.
§ DETAILS OF THE SUPERGRAVITY BACKGROUNDS
In this appendix we set some of the conventions used in this paper and study the SUSY preserved by the background in eq.(<ref>).
§.§ Type IIB Supergravity
We start this appendix by explicitly writing the Type IIB Supergravity action
and its SUSY variations. The field content of Type IIB is split into two sector. In the NS-NS sector we have: the metric g_μν, 2-form potential B_2 with field strength H_3, and the Dilaton Φ. In the R-R sector we have a set of Abelian p-form gauge fields: C_0, C_2, C_4. By defining
F_1 = dC_0, F_3 = dC_2 - C_0∧ H_3,
F_5 = dC_4 - C_2∧ H_3.
The bosonic part of the Type IIB action in String frame is
S_IIB = 1/2κ^2∫ d^10x√(-g)[e^-2Φ( R + 4∂_μΦ∂^μΦ - 1/2 |H_3|^2) - 1/2|F_1|^2 - 1/2|F_3|^2
- 1/4|F_5|^2]
= - 1/4κ^2∫ C_4∧ H_3∧ F_3
where |F_p|^2 = F_μ_1...μ_pF^μ_1...μ_p/p! and analogously for H_3. On the solutions of this theory we need to impose [To be precise, there is no covariant action for the effective theory of the Type IIB Superstring, but the presented here is close enough. The issue is that is not possible to implement the self-duality condition for F_5 at the level of the action.] self-duality of the F_5=⋆ F_5. The equations of motion of this theory are
∇^2Φ - ∂_μΦ∂^μΦ + 1/4R - 1/8|H_3|^2=0,
d( e^-2Φ⋆ H_3) = - F_5∧ F_3 - F_1∧ F_7,
d F_5- H_3∧ F_3 = 0,
d F_7 - H_3∧ F_5 = 0,
d F_9 - H_3∧ F_7 = 0,
R_μν + 2∂_μ∂_νΦ
- 1/2|H_3|^2_ μν
- e^2Φ/2( 1/2|F_1|^2_ μν
+ 1/2|F_3|^2_ μν
+1/2|F_5|^2_ μν -1/2g_μν( |F_1|^2+ |F_3|^2)) = 0
where |F_p|^2_μν = F_μν_1...ν_p-1F_ν^μν_1...ν_p-1/(p-1)!, similarly for H_3, and
F_7 = - ⋆ F_3, F_9 = ⋆ F_1.
The equations of motion are complemented by the Bianchi Identities
dF_1 = 0, dF_3 - H_3∧ F_1 = 0.
Due to the self-duality of F_5, its equation of motion and its the Bianchi identity are the same.
Solutions of the purely bosonic part of Type IIB has all the fermionic partners, the dilatino λ and gravitino Ψ_μ, set to zero. If we are interested in finding SUSY solutions, we need to be consistent with the fact that we turn off the fermions by asking for the SUSY variations of these fields to vanish. In string frame, the SUSY variations of the fermionic fields are <cit.>,
δλ = 1/2( Γ^μ∂_μΦ
+ 1/2· 3!H_μνλΓ^μνλσ^3 -e^Φ(F_μΓ^μ(iσ_2) + 1/2· 3!F_μνλΓ^μνλσ^1))ϵ,
δΨ_μ = ∂_μϵ
+ 1/4ω^μab_μΓ_abϵ + 1/4· 2!H_μνλΓ^νλσ^3ϵ
= +e^Φ/8(
F_νΓ^ν(iσ_2)
+ 1/3!F_νλρΓ^νλρσ^1
+ 1/2·5!F_νλρστΓ^νλρστ(iσ_2))Γ_μϵ.
Here ω^μab_μ is the spin connection of the 10D background, where the a,b indexes are flat space ones, and σ^1, σ^2 and σ^3 are Pauli matrices. Also
Γ^μ_1...μ_p = Γ^[μ_1...Γ^μ_p]
Here ϵ is a 64 component spinor,
ϵ = [ ϵ_1; ϵ_2 ]
where both 32-component parts are left-handed.
§.§ Checking SUSY for the Fibered Background
Here we aim to compute how many supercharges are preserved by the the backgrounds presented in this paper. For the un-fibered background in eq.(<ref>) we refer the reader to <cit.>, <cit.>, where it is shown that this solution preserves 16 Supercharges in an interesting way: the anti-commutator of two supercharges includes the R-Symmetry generators. Now we present the analysis for the fibered background in eq.(<ref>). We perform all the analysis in the S-dual system, in terms of NS5 branes, where we only have H_3 flux.
First, note that the dilatino variation is a matrix equation of the form Mϵ=0. In order to have non-trivial solutions to this equation, we require M to be non-invertible, for which we need to impose det(M)=0. It is also possible to obtain a matrix equation from the gravitino variation. Noting that we can write the gravitino variation as a covariant derivative, for which we define the connection
W_μ = 1/4ω^μab_μΓ_ab + 1/4· 2!H_μνλΓ^νλσ^3 +e^Φ/8( F_μΓ^μ(iσ_2)
+ 1/3!F_μνλΓ^μνλσ^1
+ 1/2·5!F_μνλρσΓ^μνλρσ(iσ_2))Γ_μ,
then we can write the gravitino variation as
δψ_μdx^μ = ( ∂_μϵ
+ W_μϵ)dx^μ≡𝒟ϵ.
We can get rid of the partial derivative of the spinor by acting with 𝒟 a second time
𝒟∧𝒟ϵ = ( dW + W∧ W )ϵ = 1/2Θ_μνdx^μ∧ dx^νϵ.
Each of the components of Θ_μν defines a matrix equation, giving a total of 45 independent equations. We need to make sure that det(Θ_μν)=0 for each of the components. The equations
M ϵ = 0 , Θ_μνϵ =0,
constrain the number of independent components of the spinor. After this procedure we use the gravitino variation to solve the dependence of the spinor on the spacetime coordinates.
Specialising to our background, the determinant of the Dilatino variation for the background in eq.(<ref>) reads
det(M) ∼( 4(e_BQ_A-e_AQ_B)^2+m^2)^8(4(e_BQ_A+e_AQ_B)^2+m^2)^8.
In order to have non-trivial solutions we need to impose the following BPS conditions on the parameters of the background
e_AQ_B = ± e_BQ_A, m = 0.
With this conditions it is possible to check that det(Θ_μν)=0 is also satisfied. Solving these matrix equations shows that the spinor has 8 independent components. Then, solving for the gravitino variation shows that these components are not independents, and in fact, the total number of independent components its reduced to 4. The solution for the spinor is
ϵ_1 = 0⃗
and
ϵ_2 = [ c_1 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; 0; c_1 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; i c_2 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; i c_2 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; 0; 0; c_1 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A-i e_B); c_1 (e_A+i e_B) e^-1/4 i φ(e_A^2+e_B^2)(2 Q_A-e_A r^2)/r √((e_A^2+e_B^2) (e_A r^2-2 Q_A)); 0; i c_2 (e_A+i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0; 0; c_2 (e_B-i e_A) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0; c_3 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A)/r; 0; i c_4 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; 0; 0; -i c_4 e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2-2 Q_A)/r; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A+i e_B); 0; 0; -c_3 e^-1/4 i φ(e_A^2+e_B^2)√((e_A^2+e_B^2) (e_A r^2-2 Q_A))/r (e_A+i e_B); 0; -i c_4 (e_A-i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; -i c_4 (e_A-i e_B) e^1/4 i φ(e_A^2+e_B^2)√(e_A r^2+2 Q_A/e_A^2+e_B^2)/r; 0 ]
We have found a spinor with four arbitrary constants (c_1,c_2,c_3,c_4). Being the spinor complex, we count four preserved supercharges.
§ HOW ARE THE BACKGROUNDS OBTAINED
In this appendix we describe the procedure followed to obtain the backgrounds in eqs.(<ref>) and (<ref>). These solutions are originally obtained in gauged supergravity, together with a lift procedure. Below, we review these steps.
§.§ 4D N=4 SU(2)xSU(2) Gauged Supergravity
The action of the bosonic part of the Freedman-Schwarz (FS) gauged Supergravity is
S_FS = ∫ d^4x√(-g_(4))(-R^(4)/4 + 1/2(∂ϕ)^2
+ 1/2e^4ϕ(∂a)^2 - V(ϕ)
.
=.
-e^-2ϕ/4Tr( F_(A)mnF_(A)^mn
+ F_(B)mnF_(B)^mn)
- a/2 Tr( F̃_(A)mnF_(A)^mn
+ F̃_(B)mnF_(B)^mn) ).
Here g_(4) and R^(4) are the determinant of the 4D metric and the 4D Ricci scalar, ϕ is the 4D dilaton, a is a pseudo-scalar called axion and F_(A)mn and F_(B)mn are the field strengths of two SU(2) gauge fields A_m and B_m,
F^i_(A)mn = ∂_mA^i_n - ∂_nA^i_m + e_Aϵ_ijkA^j_mA^k_n
F^i_(B)mn = ∂_mB^i_n - ∂_nB^i_m + e_Bϵ_ijkB^j_mB^k_n
where e_A and e_B are the gauge couplings of A_m and B_m respectively. Here the index i=1,2,3 transform in the adjoint of each of the copies of SU(2). Also, the duals of the field strengths are
F̃_(A)mn = 1/2√(-g)ϵ_mnλρ F_(A)mn, F̃_(B)mn = 1/2√(-g)ϵ_mnλρ F_(B)mn,
§.§.§ A BPS Solution
This theory admits a series of BPS solution preserving some amount of supersymmetry. Our main focus is the 1/4 BPS soliton presented in <cit.>. As we will review, this solution is particularly interesting because it manages to resolve a singularity by introducing a thermal cycle that preserves some SUSY (in the usual case, the non-extremal factor completely breaks SUSY). The field configuration of the Soliton is[In <cit.> the solution was presented in the mostly-minus signature. We write the solution in the mostly-plus one.]
ds^2_4D = -ρ dt^2 + dρ^2/g(ρ) + g(ρ)dφ^2 +ρ dx^2,
ϕ(ρ) = -1/2log(ρ),
A^1 =0, A^2=0, A^3= Q_Aζ(ρ)dφ,
B^1 =0, B^2=0, B^3= Q_Bζ(ρ)dφ,
where
g(ρ) = e^2_A+e^2_B/2ρ - m -2Q^2_A+Q^2_B/ρ
which has a zeros
ρ_± = m±√(4(e^2_A+e^2_B)(Q^2_A+Q^2_B)+m^2)/(e^2_A+e^2_B).
Also
ζ(ρ) = 1/ρ - 1/ρ_+,
where the last term ensures that both of the gauge fields vanish at ρ=ρ_+.
In order for the cycle φ to close smoothly at ρ=ρ_+, the period of φ needs to be
β_φ = 4π/g'(ρ_+)
= 8πρ^2_+/(e^2_A+e^2_B)ρ^2_++
4(Q^2_A+Q^2_B).
This solution was shown to preserve 4 supercharges <cit.>, when the parameters (e_A,e_B, Q_A, Q_B,m) satisfy
e_AQ_B = ± e_BQ_A, m=0.
In what follows it is convenient to perform the change of coordinates ρ = r^2. The background configuration now reads
ds^2_4D = -r^2 dt^2 + 4dr^2/f_s(r) + r^2f_s(r)dφ^2 +r^2 dx^2,
ϕ(r) = -log(r),
A^1 =0, A^2=0, A^3= Q_Aζ(r)dφ,
B^1 =0 B^2=0, B^3= Q_Bζ(r)dφ,
with
f_s(r) = e^2_A+e^2_B/2 - m/r^2 -2Q^2_A+Q^2_B/r^4, ζ(r) = 1/r^2-1/r^2_+.
§.§ Lift to 10D Supergravity
It was shown in <cit.> that the FS Supergravity has
a Kaluza-Klein interpretation as a compactification of 𝒩=1 Supergravity in 10D on the group manifold S^3× S^3. The action of the 10D theory is
S_10D = -1/4∫ d^10x√(-g)( R
-2∂_μΦ̃∂^μΦ̃
-e^2Φ̃/3H̃_μνλH̃^μνλ),
where R is the 10D Ricci scalar, Φ̃ the 10D Dilaton and H̃_μνλ is a 3-form field strength H̃_3=dB̃_2. We split the indexes as
x^μ ={ x^m=t,r,φ,x ;
z_A^i = ψ_A,θ_A,ϕ_A ;
z_B^i = ψ_B,θ_B,ϕ_B} .
The of the lift to 10D is
ds^2_10D=e^3ϕ /2ds^2_4D+2e^-ϕ /2(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i)
where the 1-forms
Θ^i_A = A^i+1/e_Aω_A^i,
Θ^i_B = B^i+1/e_Bω_B^i,
with ω_A^i and ω_B^i are the Maurer-Cartan forms of the two different SU(2), is so that Θ^i_A and Θ^i_B realise a fibration of the two 3-spheres.
The 10D Dilaton Φ̃ is written in terms of the four dimensional one ϕ
Φ̃=-ϕ/2,
while the 3-form field strength is written in term of the non-Abelian gauge fields and the SU(2) Maurer-Cartan forms as
H̃_3= -∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.
§.§.§ A Note on Conventions
We are interested in lifting the theory to Type II Supergravity, when the field content is purely of the NS-NS sector. The action in eq.(<ref>) can be mapped to Type II, after the field redefinitions
Φ̃→Φ =- 2Φ̃, H̃_3→ H_3= 2 H̃_3.
The action corresponds to the Type II in Einstein frame
S_Type II, E= -1/4∫ d^10x√(-g)(R
-1/2∂_μΦ∂^μΦ
- e^-Φ/12 H_μνρ H^μνρ).
We move to String frame by g^(S)_μν = e^1/2Φg^(E)_μν, then the action reads
S_Type II, S=-1/4∫ d^10x√(-g) e^-2Φ( R
+ 4∂_μΦ∂^μΦ
-1/2· 3!H_μνρ H^μνρ).
In this frame, the lift of the 4D FS Supergravity reads
ds^2_st = e^2ϕds^2_4D+2(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i),
H_3 = 2(-∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.)
,
Φ = ϕ(x^m).
It is convenient to write the lift in the S-dual frame, where instead of H_3 flux, we have a F_3 flux, the Dilaton is Φ'=-Φ and the metric now is g_μν' = e^-Φg_μν, explicitly this is
ds^2_st = e^ϕds^2_4D+2e^-ϕ(
Θ_A^iΘ_A^i
+Θ_B^iΘ_B^i) ,
F_3 = 2(-∑^3_i=1F_A^i∧Θ_A^i
-∑^3_i=1F_B^i∧Θ_B^i
+ e_AΘ_A^1∧Θ_A^2∧Θ_A^3
+e _BΘ_B^1∧Θ_B^2∧Θ_B^3.)
,
Φ = -ϕ(x^m).
§.§.§ Lift of the BPS Solution
Following the explicit construction of the lift, we now read the lift of the 4D solution that preserves 4 Supercharges. In the S-dual frame of eqs. (<ref>), we have
ds^2_st = r{ -dt^2+dx^2+f_s( r) dφ ^2+4/r^2f_s( r)
dr^2+2/e_A^2[ ω̂_1^2+ω̂_2^2+( ω̂_3-e_AQ_Aζ
(r)dφ) ^2] .
. +2/e_B^2[ ω̃_1^2+ω̃
_2^2+( ω̃_3-e_BQ_Bζ( r)
dφ) ^2] } ,
F_3 = dC_2= 2 ζ'(r)dr∧ dφ∧(
Q_A/e_Aω̂_3+Q_B/e_Bω̃
_3) +2/e_A^2ω̂_1∧ω̂
_2∧( e_AQ_Aζ (r)dφ -ω̂_3)
+2/e_B^2ω̃_1∧ω̃_2∧( e_BQ_Bζ (r)dφ -ω̃_3) ,
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
C_6 = -2e_Ar^2/e^3_Bdt∧ dx∧ dφ∧(S^3_B)
+2e_Br^2/e^3_Adt∧ dx∧ dφ∧(S^3_A) ,
= -8Q_B/e^3_Ae^2_Bcos(θ_B)dt∧ dx ∧(S^3_A)∧ dϕ_B
+8Q_A/e^3_Be^2_Acos(θ_A)dt∧ dx ∧(S^3_B)∧ dϕ_A,
Φ = log r .
This is the background in eq.(<ref>).
In the case for which Q_A=Q_B=m=0, the background fields read (note that we S-dualise moving to the NS5 brane frame),
ds^2_st = -dt^2+dx^2+ e^2_A+e^2_B/2dφ^2 + 8/e^2_A+e^2_Bdr^2/r^2
+ 8/e^2_Ads^2(S^3_A)
+ 8/e^2_Bds^2(S^3_B),
H_3 = -8/e^2_A(S^3_A)-8/e^2_B(S^3_B),
Φ = -log(r).
This is the background in eq.(<ref>).
We can rescale φ to absorb the prefactor. Also, it is convenient to set as in eq.(<ref>)
N_A = 8/e^2_B, N_B = 8/e^2_A,
and perform the change of coordinates
r = e^√(1/N_A+1/N_B) ρ,
after changing H_3→ -H_3, the background reads
ds^2_10D = - dt^2 + dx^2 + dφ^2 + dρ^2
+ N_B ds^2(S^3_A)
+ N_Ads^2(S^3_B),
H_3 = N_AS^3_A + N_BS^3_B,
Φ = -√(1/N_A+1/N_B) ρ.
This is the background written in Section <ref> to study the string σ-model on this field configuration.
§ ON THE UNFIBERED GEOMETRY
Here, we review a different derivation of the background (<ref>). The S-dual of this background (<ref>) was first introduced in <cit.>, <cit.>, here we review the derivation of the pure NS-NS frame for simplicity. Let us consider two stacks of NS5-branes, the first extended in (t,x,y_1,y_2,y_3,y_4) and while the second one spans (t,x,w_1,w_2,w_3,w_4). These stacks intersect in the (t,x) directions, thus in the weak coupling regime, the effective theory on the intersection is 1+1 dimensional and preserves 8 Supercharges.
We now move to the strong coupling regime. For this, we write the space ℝ^4_y = (y_1,y_2,y_3,y_4) in spherical coordinates (r_A, S^3_A), and similarly for ℝ^4_w we use (r_B, S^3_B). In terms of the harmonic functions
H_A(r_A) = 1 + N_B/r^2_A,
H_B(r_B) = 1 + N_A/r^2_B,
the backreacted fields are given by
ds^2_st = dx^2_1,1
+ H_A(r_A)( dr^2_A + r^2_Ads^2(S^3_A))
+ H_B(r_B)( dr^2_B + r^2_Bds^2(S^3_B)),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = 1/2log(H_A(r_A)H_B(r_B)).
By taking the near-horizon geometry we are led to
ds^2_st = dx^2_1,1
+ N_Bdr^2_A/r^2_A + N_Adr^2_B/r^2_B
+ N_Bds^2(S^3_A) + N_Ads^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -log(r_A) -log(r_B) + 1/2log(N_AN_B).
Under the change of coordinates r_A = e^ρ_A/√(N_B) and r_B = e^ρ_B/√(N_A) we obtain
ds^2_st = dx^2_1,1
+ dρ^2_A + dρ^2_B
+ N_Bds^2(S^3_A) + N_Ads^2(S^3_B),
H_3 = N_B(S^3_A) + N_A(S^3_B),
Φ = -√(1/N_B)ρ_A -√(1/N_A)ρ_B + 1/2log(N_AN_B).
Finally, taking the linear combination
√(1/N_A+1/N_B)ρ_A = √(1/N_B)ρ - √(1/N_A)φ,
√(1/N_A+1/N_B)ρ_B = √(1/N_A)ρ + √(1/N_B)φ,
we reach the background fields in (<ref>)
ds^2_st = -dt^2+ dx^2+dφ^2+ dρ^2
+ N_B ds^2(S^3_A)
+ N_A ds^2(S^3_B),
H_3 = N_B (S^3_A) + N_A (S^3_B),
Φ = -√(1/N_A+1/N_B) ρ,
where we cancelled the constant term of the Dilaton by a suitable shift of its zero mode. Note that the Dilaton in (<ref>) has functional dependence in two of the coordinates of the background, while the one in (<ref>) only depends only on one coordinate. This allows us to interpret the extra flat direction of (<ref>) as being part of the Field Theory ones. In this way, we see that in the strong coupling regime, the theory on the intersection acquires an extra dimension, becoming (2+1) dimensional. There is also a SUSY enhancement from 8 to 16 supercharges. We refer the reader to <cit.> to cover this matter.
§ CHARGES OF THE BLACK HOLE BACKGROUND
We compute the charges of the configuration considering the Noether-Wald method
<cit.>. The bulk action principle in string frame of IIB in the metric-dilaton-F_3 sector is
S_IIB,bulk=1/2κ ^2∫ d^10x√(-g)( R-
1/2( ∂Φ) ^2-1/12e^ΦF_μνρF^μνρ) =∫ d^10x√(-g)ℒ ,
A general variation of the action gives
δ S_IIB,bulk=∫ d^10x√(-g)[ δ g^μν
ℰ_μν^( g) +δΦℰ^(
Φ) +δ C_νρℰ_( F_3) ^νρ+∇ _μΘ ^μ( 𝐟,δ𝐟
) ] .
where 𝐟 denotes the fields collectively and
Θ ^μ( 𝐟,δ𝐟) = 1/
2κ ^2( g^δηδΓ _ ηδ^μ-g^δμδΓ _ λδ^λ-δΦ∂ ^μΦ -e^Φδ C_νρF^μνρ)
,
ℰ_μν^( g) = 1/2κ ^2[
R_μν-1/2g_μνR.
-1/2( ∂ _μΦ∂ _νΦ -1/2
g_μν∂ _ρΦ∂ ^ρΦ)
. -1/2e^Φ( 1/2F_μδρF_ν^ δρ-1/12g_μνF_δρσF^δρσ) ] ,
= 1/2κ ^2( G_μν-1/2T_μν^(
Φ) -1/2T_μν^( F_3) ) ,
ℰ^( Φ) = 1/2κ ^2( ∇
^ρ∇ _ρΦ -1/12e^ΦF_μνρF^μνρ) ,
ℰ_( F_3) ^νρ = 1/2κ ^2
∇ _μ( e^ΦF^μνρ) .
The Noether current is defined by
J^μ=Θ ^μ( 𝐟,ℒ_ξ𝐟)
-ξ ^μℒ ,
where ℒ is the Lagrangian scalar under diffeomorphisms in (<ref>) and ℒ_ξ is the Lie derivative along the
vector ξ. The Noether current is conserved on-shell, thus it can be
written locally as J^μ=∇ _νq^μν. The Noether
current (<ref>) for our system gives
J^μ =-1/κ ^2∇ _ν( ∇ ^μξ ^ν ]+1/2e^Φ2C_λρξ ^λF^μνρ) +2ξ ^λℰ_( g) λ^μ-2ξ
^λC_λρℰ_( F_3) ^μρ ,
on-shell it defines the Noether pre-potential:
q^μν( ξ) =-1/κ ^2( ∇
^μξ ^ν ]+1/2e^Φ2C_λρξ
^λF^μνρ) .
The Hodge dual of the Noether pre-potential gives the 8-form
Q[ ξ] =1/21/8!√(-g)ϵ
_μνρ _1…ρ _8q^μνdx^ρ _1∧…∧ dx^ρ _8 ,
that in differential forms is
Q[ ξ] =-1/κ ^2( ⋆ dξ
+e^Φξ¬ C_2∧⋆ F_3) .
¬ stands for the contraction operator.
The boundary term that allow us to have a well posed action principle and
finite mass is
S_full=S_IIB,bulk+∫_∂ Md^9x√(-h)1/κ ^2(
𝒦-e^-1/4Φ) ,
where the first term in the integral is the Gibbons-Hawking-York term and
the last term corresponds to a counter term, that depends only on intrinsic
quantities, that allow us to renormalize the mass term. The extrinsic
curvature is defined in terms of the normal unit outwards vector n^μ
to the boundary of the spacetime by
𝒦_μν=h_ μ^ρh_ ν^σ∇ _ρn_σ ,
and the induced metric is h_μν=g_μν-n_μn_ν for
our case.
Following <cit.> the energy, angular momentum and entropy are defined by
ℰ[ t] = ∫_∞( Q[
t] -ξ¬B) ,
𝒥[ ψ] = -∫_∞Q[
ψ] ,
S[ ξ] = 1/T∫_ℋ
Q[ ξ] .
The boundary terms are in the 9-form
B=-1/κ ^2( 𝒦-e^-1/4Φ) ⋆ n .
t is the time-like killing vector at infinity properly
normalized, ψ is the rotation generator and ξ is the horizon generator
ξ=t+Ωψ .
ξ is null at the Horizon which defines the angular
velocity Ω and satisfies the geodesic equation at the horizon
ξ ^μ∇ _μξ ^ν=κ _sξ ^ν ,
defining the surface gravity κ _s that is related to the
temperature as T=κ _s/2π. In this case we are in general
relativity, therefore the entropy give one-quarter of the horizon area.
Let us consider the black hole configuration in Einstein frame (<ref>), with
ζ(r)=1/r^2 , Φ→Φ-2log(e_A^2+e_B^2/2) , F_3 →e_A^2+e_B^2/2F_3 .
The in-going Eddington-Finkelstein coordinates are well-defined
at the horizon, which are defined by
dt = dv-2dr/rf_bh( r) ,
dψ _A = dψ _A^'-2e_AQ_Aζ( r) /
rf_bh( r) dr ,
dψ _B = dψ _B^'-2e_BQ_Bζ( r) /
rf_bh( r) dr .
Then, the metric becomes
ds_E^2 = √(r){ dy^2+dx^2-f_bh( r) dv^2+
4/rdrdv.
+2/e_A^2[ dθ _A^2+sin ^2θ _Adϕ
_A^2+( dψ _A^'+cosθ _Adϕ
_A-e_AQ_Aζ( r) dv) ^2]
. +2/e_B^2[ dθ _B^2+sin ^2θ
_Bdϕ _B^2+( dψ _B^'+cosθ _Bdϕ
_B-e_BQ_Bζ( r) dv) ^2] } ,
In these coordinates we consider the vector
ξ=t+Ω _Aψ_A+Ω
_Bψ_B
where
t=1/2∂/∂ v ,
ψ_A=∂/∂ψ _A , ψ
_B=∂/∂ψ _B .
The vector ξ is null at the horizon located at r_+
when
Ω _A=e_AQ_A/r_+^2 , Q_B=e_BQ_B/
r_+^2 .
Due to the fact that the spacetime that we are considering is not asymptotically Minkowski times S^3× S^3, instead is conformal to Minkowski times S^3× S^3, it is not clear how we should normalize the vector t time-like at infinity. This ambiguity propagates to the energy and the temperature. Therefore we expect to obtain the temperature in 4D up to a factor.
The energy (<ref>), angular momentum (<ref>), temperature
defined through (<ref>) and the entropy give
E = ℰ[t]=2m/e_A^3e_B^3r_+^2κ ^2( 16π
^2) ^2L_xL_y ,
J_A = 𝒥[ψ_A]=8Q_A/e_A^4e_B^3κ ^2( 16π
^2) ^2L_xL_y ,
J_B = 𝒥[ψ_B]=8Q_B/e_B^4e_A^3κ ^2( 16π
^2) ^2L_xL_y ,
T = e_A^2+e_B^2/16π-4(
Q_A^2+Q_B^2) /16π r_+^4 ,
S =2r_+^2/e_A^3e_B^3G_10( 16π)
^2L_xL_y .
They satisfy the first law of thermodynamics
dE=TdS+Ω _AdJ_A+Ω _BdJ_B .
§ R-SYMMETRY BREAKING
In this appendix we give a detailed derivation of the symmetry breaking pattern of three U(1) directions present in our background. In order to do this, we gauge this symmetries by introducing a gauge field A and a scalar ϵ. The presence of a mass term in the effective action of the gauge field, i.e. an explicit symmetry breaking of the gauge symmetry, signals the breaking of this U(1) symmetry on the dual field theory. In the QFT, the breaking of the global symmetry can be either spontaneous or anomalous.
§.§ U(1) R-Symmetry of psi A and psi B
Let us start by recalling the R-R C_2 potential
C_2 = ψ_A( 2Q_A/e_Aζ ^'(
r) dr∧ dφ -2/e_A^2sinθ_Adθ_A
∧ dϕ_A) +2/e_Acosθ_AQ_Aζ( r) dφ∧ dϕ_A
+ψ_B( 2Q_B/e_Bζ ^'( r)
dr∧ dφ -2/e_B^2sinθ_Bdθ_B
∧ dϕ_B) +2/e_Bcosθ_BQ_Bζ( r) dφ∧ dϕ_B .
Since this potential is not invariant under ψ_A,B→ψ_A,B+4π, we expect this symmetry to be broken in the dual field theory. We gauge these isometries by doing the following replacements in the R-R potential and the metric
dψ_A,B→ dψ_A,B + A_A,B, ψ_A,B→ψ_A,B + ϵ_A,B.
Where A_A,B is a U(1) gauge field and ϵ_A,B is a scalar charged under the gauged U(1) symmetry, which makes the combination D_A,Bϵ = ∂_A,Bϵ_A,B - A_A,B is gauge invariant (here ∂_A,B = ∂/∂ψ_A,B). These fields only depend on the coordinates of the field theory directions, i.e. (t,x). After these replacements the metric and the R-R 3-form read
ds^2 = ds^2_(0) +4r/e^2_A( ω̂_3-e_AQ_Aξ(r)dφ) A_A μdx^μ + 2r/e^2_AA_A μ A_A ν dx^μdx^ν
+4r/e^2_B( ω̃_3-e_BQ_Bξ(r)dφ) A_B μdx^μ + 2r/e^2_BA_B μ A_B ν dx^μdx^ν
F_3 = F^(0)_3 - 2 dϵ_A∧( -Q_A/e_Aξ'(r)dr∧ dφ + 1/e^2_A(S^2_A) )
- 2 dϵ_B∧( -Q_B/e_Bξ'(r)dr∧ dφ + 1/e^2_B(S^2_B) )
where ds^2_(0) and F^(0)_3 denotes the metric and the 3-form of the configuration before gauging the U(1) symmetries and (S^2_A,B) = sin(θ_A,B)dθ_A,B∧ dϕ_A,B. Now we want to obtain an effective lagrangian for A_A,B and ϵ_A,B. In order to do this, we consider how the Ricci scalar and the kinetic term of F_3 change under the gauging of the symmetry. Explicitly, the Ricci scalar transforms as
R = R^(0) - 1/42r/e^2_A F^2_A - 1/42r/e^2_B F^2_B,
where F^2_A,B = F_A,B μνF^μν_A,B and F_A,B μν is the field strengh of A_A,B μ, while the kinetic term of the R-R potential reads
1/12 F_μνλF^μνλ
= 1/12 F^(0)_μνλF_(0)^μνλ
+ 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)( A_A-dϵ_A)^2
+ 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)( A_B-dϵ_B)^2 + Q_AQ_B/e_Ae_B( A_A-dϵ_A)·( A_B-dϵ_B).
Finally, replacing this expression in the Type IIB action (in string frame), leads to the following effective lagrangian
ℒ =
- 1/42/e^2_Ar F^2_A - 1/42/e^2_BrF^2_B
- 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)( A_A-dϵ_A)^2
= - 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)( A_B-dϵ_B)^2
- Q_AQ_B/e_Ae_B( A_A-dϵ_A)·( A_B-dϵ_B)
Due to the coupling between A_A,B and ϵ_A,B the gauge field obtains a mass. This is the same as the Stueckelberg mechanism. Defining W_A,B = A_A,B - dϵ_A,B, we obtain an action for the massive gauge field.
ℒ =
- 1/42/e^2_Ar F^2_A - 1/42/e^2_BrF^2_B
- 1/2r^2( Q^2_Ar^2/e^2_Aξ'(r)^2+1)
W_A μW^μ_A
= - 1/2r^2( Q^2_Br^2/e^2_Bξ'(r)^2+1)W_B μW^μ_B
- Q_AQ_B/e_Ae_BW_A μW^μ_B
§.§ U(1)) R-Symmetry of varphi
Now we repeat the same procedure as above for the φ direction. The only difference is at the starting point. The potential (<ref>) does not depend on φ. We need to perform a gauge transformation to give it φ dependance, after which
C_2 = 2Q_A/e_Aφ[ ξ'(r)( dψ_A + cos(θ_A)dϕ_A)∧ dr + ξ(r) (S^2_A)] - 2/e^2_Aψ_A(S^2_A)
= + 2Q_B/e_Bφ[ ξ'(r)( dψ_B + cos(θ_B)dϕ_b)∧ dr + ξ(r) (S^2_b)] - 2/e^2_Aψ_B(S^2_B).
As before, we gauge the symmetry along φ by shifting the metric and the R-R potential as follows
dφ→ dφ + A_φ, φ→φ + ϵ_φ.
Repeating the procedure of the previous section lead to the following shifts for Ricci scalar
R = R^(0) - 1/4( r f_s(r) + 2(Q^2_A+Q^2_B)rξ(r)^2)F^2_φ
where F^2_φ = F_φ μνF^μν_φ, with F_φ μν the field strenght of A_φ, and the kinetic term of the R-R potential
1/12 F_μνλF^μνλ = 1/12 F^(0)_μνλF_(0)^μνλ
= +1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) ( A_φ - dϵ_φ)^2
which leads to the effective lagrangian
ℒ = - 1/4( f_s(r) + 2(Q^2_A+Q^2_B)ξ(r)^2)F^2_φ
= -1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) ( A_φ - dϵ_φ)^2
As before, we see from the action that after a gauge transformation the gauge field obtains a mass via Stueckelberg mechanism. Explicitly by defining W_φ = A_φ - dϵ_φ we obtain
ℒ = - 1/4( f_s(r) + 2(Q^2_A+Q^2_B)ξ(r)^2)F^2_φ
= -1/4( 2/r^2ξ(r)^2( e^2_AQ^2_A + e^2_BQ^2_B
+ (Q^2_A + Q^2_B)^2r^2ξ'(r)^2) + (Q^2_A + Q^2_B)f_s(r)ξ'(r)^2) W_φ μW^μ_φ
§ MALDACENA-WILSON, 'T HOOFT LOOPS AND EE. DETAILED CALCULATIONS
In this appendix, we study the integrals needed to compute the Wilson loops, 't Hooft loops and Entanglement Entropy. We express the analytic results in terms of r_0.
Let us define the following quantities that allow us to write the integrals
in a simpler way
λ _0 = r_0/r_+ , ξ =r/r_+ , η =e_A/e_B ,
λ _-^2 = -r_-^2/r_+^2≡ 1-m/
r_+^2( e_A^2+e_B^2) ,
where λ _-∈[ 0,1], λ _0>1 and η >0.
All the problems we will address here can be reduced to a one-dimensional
problem for the function r=r( x) which minimises the functional
in eq.(<ref>) once we impose that the parameter x(σ)=σ. Then, the equation for the function r(σ)=r(x) reduces to
dr/dx=± V_eff( r) ,
for a suitable effective potential which is case-dependent. In most
cases the function r( x) can be interpreted as a string (or a
section of a higher dimensional surface) with end points at r→∞. The profile of the string subject to the initial condition x( r_0) =0, can be obtained by performing the integral
x( r) =±∫_r_0^rdr/V_eff( r) .
From here we compute the end points separation as
L( r_0) ≡lim_r→∞2x( r) .
The definition in eq.(<ref>) coincides with the
quark-anti-quark separation, monopole-anti-monopole separation and the
interval length for the Maldacena-Wilson loop, t' Hooft loop and
entanglement entropy, respectively.
The results of the integrals that we compute analytically are given
in terms of elliptic integrals. The elliptic integral of first
kind 𝐅( ϕ |m) and the complete elliptic integral
of first kind 𝐊( m) are defined as
𝐅( ϕ |m) = ∫_0^ϕdθ1/√(
1-msin ^2θ) ,
𝐊( m) = 𝐅( . π/2
| m) ,
respectively for -π/2<ϕ <π/2. The elliptic
integral of second kind 𝐄( ϕ |m) and the complete
elliptic integral 𝐄( m) are defined respectively as
𝐄( ϕ |m) = ∫_0^ϕ√(1-msin ^2θ)dθ ,
𝐄( m) = 𝐄( . π/2
| m) ,
where -π/2<ϕ <π/2.
§.§ Wilson loop
The effective potential in terms of the variable ξ defined in (<ref>) reads
V_eff( ξ) =√(e_A^2+e_B^2/8)r_+/
λ _0ξ√(( ξ ^2-λ _0^2) ( ξ
^2+λ _-^2) ( ξ ^2-1) ) .
The string profile, considering the change of variables in eq.(<ref>)
x( ξ) = ±√(8/e_A^2+e_B^2)∫_λ
_0^ξλ _0ξ dξ/√(( ξ ^2-λ
_0^2) ( ξ ^2+λ _-^2) ( ξ
^2-1) ) ,
= ±√(8/e_A^2+e_B^2)λ _0/√(λ
_0^2-1)
×[ -𝐅( . arcsin√(λ
_0^2-1/ξ ^2-1)|1+λ _-^2/1-λ
_0^2) +𝐊( 1+λ _-^2/1-λ
_0^2) ] .
The definition of the quark-antiquark separation given in eq.(<ref>) can
be expressed in terms of the limit (<ref>) of
the string profile. Replacing in eq.(<ref>) we find
L_QQ( λ _0) =2√(8/e_A^2+e_B^2)
λ _0/√(λ _0^2-1)𝐊( 1+λ
_-^2/1-λ _0^2) .
This is our result in eq.(<ref>).
In order to get an analytic expression for the energy in eq.(<ref>) we compute the integrals
E_QQ=F( r_0) L_QQ( r_0) +I_2+I_3
where
I_2 = √(32/e_A^2+e_B^2)∫_r_0^+∞dz z
√(z^2-r_0^2/( z^2-r_-^2) (
z^2-r_+^2) ) ,
I_3 = -√(32/e_A^2+e_B^2)∫_r_+^∞dz
z^2/√(( z^2-r_-^2) ( z^2-r_+^2) ).
Considering the change of variable ξ =z/r_+ and the definitions (<ref>) the integrals become
I_2 = 4√(2)r_+/√(e_A^2+e_B^2)∫_λ
_0^+∞dξ ξ√(ξ ^2-λ _0^2/( ξ
^2+λ _-^2) ( ξ ^2-1) ) ,
I_3 = -4√(2)r_+/√(e_A^2+e_B^2)∫_1^∞dξξ ^2/√(( ξ ^2+λ _-^2) (
ξ ^2-1) ) .
We perform the indefinite integral of I_2 giving
∫^ξdξ ξ√(ξ ^2-λ _0^2/( ξ
^2+λ _-^2) ( ξ ^2-1) ) = √(
( ξ ^2-λ _0^2) ( λ _-^2+ξ
^2) )/√(ξ ^2-1)
+√(( λ _0^2-1) )𝐄𝐄( . arcsin√(λ _0^2-1/ξ ^2-1)|1+λ
_-^2/1-λ _0^2) .
Taking the limits
lim_ξ→∞∫^ξdξ ξ√(ξ
^2-λ _0^2/( ξ ^2+λ _-^2) ( ξ
^2-1) ) = lim_ξ→∞ξ +𝒪(
1/ξ) ,
lim_ξ→λ _0∫^ξdξ ξ√(ξ
^2-λ _0^2/( ξ ^2+λ _-^2) ( ξ
^2-1) ) = √(( λ _0^2-1) )𝐄𝐄
( 1+λ _-^2/1-λ _0^2) .
Hence,
I_2=4√(2)r_+/√(e_A^2+e_B^2)[ lim_ξ→∞ξ -√(( λ _0^2-1) )𝐄𝐄
( 1+λ _-^2/1-λ _0^2) ].
The indefinite integral of I_3 gives
∫^ξdξξ ^2/√(( ξ ^2+λ
_-^2) ( ξ ^2-1) )=iλ _m[ 𝐄
( arcsinξ| -1/λ _-^2. ) -
𝐅( arcsinξ| -1/λ _-^2.
) ] .
Computing the limits we get
lim_ξ→∞∫^ξdξξ ^2/√((
ξ ^2+λ _-^2) ( ξ ^2-1) ) = lim_ξ→∞ξ +ℐ_λ _-+O( ξ ^-1)
,
lim_ξ→ 1∫^ξdξξ ^2/√(( ξ
^2+λ _-^2) ( ξ ^2-1) ) = iλ _-
[ 𝐄( -λ _-^-2) -𝐊( -λ
_-^-2) ] ,
where
ℐ_λ _- = [ iλ _-𝐄( -λ
_-^-2) -𝐄( -λ _-^2) -λ _-
𝐊( 1+λ _-^-2) .
. -2iλ _-𝐊( -λ _-^-2) +𝐊
( -λ _-^2) +λ _-^2𝐊( -λ
_-^2) ].
Therefore, the integral becomes
I_3 = -4√(2)r_+/√(e_A^2+e_B^2)[lim_ξ→∞ξ -𝐄( -λ _-^2) -λ
_-𝐊( 1+λ _-^-2)
. -iλ _-𝐊( -λ _-^-2) +
𝐊( -λ _-^2) +λ _-^2𝐊(
-λ _-^2) ] .
Replacing into the energy in eq. (<ref>), we find the result in eq.(<ref>)
E_QQ( λ _0) = 2r_+√(8/e_A^2+e_B^2
)[ λ _0^2/√(λ _0^2-1)𝐊(
1+λ _-^2/1-λ _0^2) .
+𝐄( -λ _-^2) +λ _-𝐊(
1+λ _-^-2) +iλ _-𝐊( -λ
_-^-2)
. -√(( λ _0^2-1) )𝐄(
1+λ _-^2/1-λ _0^2) -( 1+λ
_-^2) 𝐊( -λ _-^2) ]
§.§ t' Hooft loop
The effective potential V_eff is given in eq.
(<ref>).
Replacing explicitly the functions and using the definition in eq. (<ref>) leads to
V_eff = r_+e_B/2√(η ^2+1)1/ξ√((
ξ ^2-λ _0^2) ( ξ ^2-1) ( ξ
^2+λ _-^2) )
×√(( 4Q_B^2r_+^-4e_B^-2( ξ
^2+λ _0^2-2) +( η ^2+1) ( ξ
^2+λ _0^2-1+λ _-^2) ) /2( λ
_0^2-1) [ 4Q_B^2r_+^-4e_B^-2( λ
_0^2-1) +( η ^2+1) ( λ _0^2+λ
_-^2) ] ) .
We compute analytically the integrals in the BPS bound in
which λ _-=1 and Q_B=±e_B/e_AQ_A implying Q_A=e_A/2r_+^2. In this limit the effective potential
simplifies to
V_eff^BPS = r_+1/2√(e_A^2+e_B^2/2(
λ _0^2-1) ( ( e_A^2+2e_B^2) λ
_0^2+e_A^2) )
×√(1/ξ ^2( ξ ^2-λ _0^2)
( ξ ^4-1) ( e_A^2( ξ ^2+λ
_0^2) +2e_B^2( ξ ^2+λ _0^2-1) ) .
)
The indefinite integral (<ref>) gives
∫^ξdξ r_+/V_eff^BPS = 2/e_B√(
2( λ _0^2-1) ( ( η ^2+2) λ
_0^2+η ^2) /η ^2+1)
×∫ξ dξ/√(( ξ ^2-λ _0^2)
( ξ ^4-1) ( η ^2( ξ ^2+λ
_0^2) +2( ξ ^2+λ _0^2-1) ) ) ,
= 1/e_B√(2( λ _0^2-1) (
( η ^2+2) λ _0^2+η ^2) /( η
^2+1) ( λ _0^2( η ^2+2) -1) )
×𝐅( . arcsin√(2( -1+(
2+η ^2) λ _0^2) ( ξ ^2-1) /(
2λ _0^2+η ^2( 1+λ _0^2) ) (
ξ ^2-λ _0^2) )|( 1+λ
_0^2) ( 2λ _0^2+η ^2( 1+λ
_0^2) ) /-4+4( 2+η ^2) λ _0^2
).
Taking the limit ξ→ r_0 we find
lim_ξ→λ _0r_+∫^ξdξ/V_eff
= 1/e_B√(2( λ _0^2-1) (
( η ^2+2) λ _0^2+η ^2) /( η
^2+1) ( λ _0^2( η ^2+2) -1) )
( -i) 𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) .
Therefore, the profile of the string is
± x( ξ) = 1/e_B√(2( λ
_0^2-1) ( ( η ^2+2) λ _0^2+η
^2) /( η ^2+1) ( λ _0^2( η
^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) λ _0^2) ( ξ
^2-1) /( 2λ _0^2+η ^2( 1+λ
_0^2) ) ( ξ ^2-λ _0^2) )
|( 1+λ _0^2) ( 2λ
_0^2+η ^2( 1+λ _0^2) ) /-4+4(
2+η ^2) λ _0^2) .
. +i𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) ] .
The monopole-anti-monopole separation can be deduced easily from the above
expression by taking the limit (<ref>). We obtain,
L_MM^BPS( λ _0) = 2/e_B√(
2( λ _0^2-1) ( ( η ^2+2) λ
_0^2+η ^2) /( η ^2+1) ( λ
_0^2( η ^2+2) -1) )
×[ 𝐅( . arcsin√(2(
-1+( 2+η ^2) λ _0^2) /( 2λ
_0^2+η ^2( 1+λ _0^2) ) )|( 1+λ _0^2) ( 2λ _0^2+η
^2( 1+λ _0^2) ) /-4+4( 2+η ^2)
λ _0^2) .
. +i𝐊( 1-( 1+λ _0^2)
( 2λ _0^2+η ^2( 1+λ _0^2) ) /
-4+4( 2+η ^2) λ _0^2) ] .
We compare it with the approximate function
(<ref>)
for the separation, replacing the functions explicitly we find
L̂_MM( r_0) =π√(2)/√(
e_A^2+e_B^2)( e_A^2+2e_B^2)
r_0^2/r_+^2+e_A^2/( e_A^2+2e_B^2)
r_0^2/r_+^2-e_B^2√(r_0^2/r_+^2-1/
r_0^2/r_+^2+1) .
For the energy of the t' Hooft loop we have a similar expression to the one obtained when computing the
energy of the Wilson loop,
E_MM( r_0) =F( r_0) L_MM( r_0)
+I_2+I_3 ,
where
I_2 = 4r_+^2/√(η ^2+1)∫_λ _0^+∞dξ√(ξ ^2( ξ ^2-λ _0^2) (
η ^2( ξ ^2+λ _0^2) +2( ξ ^2+λ
_0^2-1) ) /ξ ^4-1) ,
I_3 = -2∫_r_+^+∞dzG( z) =-4r_+^2/
√(( 1+η ^2) )∫_1^∞dξ ξ√(
2ξ ^2+η ^2( ξ ^2+1) /( 1+ξ ^2) ) .
These integrals are quite involved and present technical
difficulties to be performed in terms of known functions. Therefore we
compute them numerically up to a large value of the upper limit ξ̃
_max. Since ξ̃_max is finite the integrals are
convergents and we can write them together in terms of a single integral which depends
on ξ̃_max
E_MM( r_0) . =. F( r_0)
L_MM( r_0) -[ 4r_+^2/√(1+η ^2)
∫_1^λ _0 ξ√(2ξ ^2+η ^2( ξ
^2+1) /( 1+ξ ^2) ).
-lim_ξ̃_max→∞∫_λ _0^ξ̃_maxdξ( √(ξ ^2( ξ ^2-λ
_0^2) ( η ^2( ξ ^2+λ _0^2)
+2( ξ ^2+λ _0^2-1) ) /ξ ^4-1).
. . -ξ√(2ξ
^2+η ^2( ξ ^2+1) /( 1+ξ ^2) ))
] .
We verify that the integral converges to a limiting value for ξ̃
_max large enough and much bigger than λ _0=r_0/r_+.
The limiting value of the function L_MM( r_0) when r_0→∞ is non-zero and is given in terms of a
characteristic length of the Little String Theory (LST). This asymptotic
behavior matches with the asymptotic behavior of the background
(<ref>)
, to see this fact explicitly we compute the separation length of the 't
Hooft loop of the background
(<ref>)
by taking the limit of the end-points separations
(<ref>)
. The relevant functions for the t' Hooft loop of the background
(<ref>)
are the metric function
f_s( r) =1/2( e_A^2+e_B^2) ,
and the effective potential
V_eff( r,r_0) =√(2( e_A^2+e_B^2)
)/4r_0^2√(r^2( r^4-r_0^4) ) .
Then, the profile of the string in the bulk is
± x( r) =√(2/e_A^2+e_B^2)arctan(
√(r^4-r_0^4)/r_0^2) .
The end-point separation is given by the limit
(<ref>)
and leads to the following constant
L_MM( r_0) =π√(2/e_A^2+e_B^2) .
Thus, all the strings in the background
(<ref>)
that explores the bulk has the same end points separation. This value
coincides with the limiting value of the separation length of the background
(<ref>)
and with a LST characteristic length. Therefore the UV behavior of the dual
theory is driving by the LST. To capture the field theory behavior we
introduce a cut-off to rule out the non-local effects of the LST. In that
case the system present a phase transition between the unstable
configurations to the short strings configuration.
The energy of the t' Hooft loop of the background
(<ref>)
is given by
(<ref>)
with
F( r) =√(e_A^2+e_B^2/2) r^2, G(
r) =2√(e_A^2+2e_B^2/e_A^2+e_B^2)r .
We find that the energy is zero. This implies that energy of the t' Hooft
loop of the disconnected solution is the same to the connected one.
§.§ Entanglement entropy
The profile of the 8-dimensional surface is governed by the function r=r( x) with equations coming from the minimisation of eq.
(<ref>) which gives an equation like (<ref>) for the effective potential
V_eff=1/2r_0^2√(f_s( r_0) )√(
r^2f_s( r) ( r^4f_s( r)
-r_0^4f_s( r_0) ) ) .
The integration of (<ref>) for this potential subject to the initial
condition x( r_0) =0 gives
± x( ξ) = ∫_r_0^r2r_0^2√(f_s(
r_0) )dr/√(r^2f_s( r) ( r^4f_s(
r) -r_0^4f_s( r_0) ) )
= √(8/e_A^2+e_B^2)√(( λ
_0^2-1) ( λ _0^2+λ _-^2) )
×∫_λ _0^ξξ dξ/√(( ξ
^2-λ _0^2) ( ξ ^2-1) ( ξ
^2+λ _-^2) ( ξ ^2+λ _0^2+λ
_-^2-1) ).
Performing the indefinite integral
ℐ_L( ξ) ≡ ∫^ξξ dξ/√(
( ξ ^2-λ _0^2) ( ξ ^2+λ
_-^2) ( ξ ^2-1) ( ξ ^2+λ
_0^2+λ _-^2-1) )
= 1/λ _0^2+λ _-^2𝐅( .
arcsin√(( λ _0^2+λ _-^2) (
ξ ^2-1) /( λ _-^2+1) ( ξ ^2-λ
_0^2) )|( 1+λ _-^2) (
λ _-^2+2λ _0^2-1) /( λ
_0^2+λ _-^2) ^2) .
The limit ξ→λ _0 gives
lim_ξ→λ _0ℐ_L( ξ) =-
i/λ _0^2+λ _-^2𝐊( ( λ
_0^2-1) ^2/( λ _0^2+λ _-^2) ^2
) .
Thus, the profile of the surface in the bulk is
± x( ξ) = √(8/e_A^2+e_B^2)√(
λ _0^2-1/λ _0^2+λ _-^2)[ i𝐊
( ( λ _0^2-1) ^2/( λ
_0^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(( λ
_0^2+λ _-^2) ( ξ ^2-1) /( λ
_-^2+1) ( ξ ^2-λ _0^2) )|( 1+λ _-^2) ( λ _-^2+2λ
_0^2-1) /( λ _0^2+λ _-^2) ^2
) ].
The length of the interval is given by the limit in eq.(<ref>). In term of the variables in eq.(<ref>) gives
L_EE( λ _0) = 2√(8/e_A^2+e_B^2)
√(λ _0^2-1/λ _0^2+λ _-^2)[ i
𝐊( ( λ _0^2-1) ^2/( λ
_0^2+λ _-^2) ^2) .
. +𝐅( . arcsin√(( λ
_0^2+λ _-^2) /( λ _-^2+1) )
|( 1+λ _-^2) ( λ
_-^2+2λ _0^2-1) /( λ _0^2+λ
_-^2) ^2) ] .
This is our expression in eq.(<ref>).
We cannot see a phase transition in this background. However, if we put a
cutoff at ξ _cutoff=r_cutoff/r_+ the
coordinate λ _0<ξ <ξ _cutoff, the double-valued character of L_EE shows, as in Figure <ref>.
The renormalized EE in eq.(<ref>) written in the variables of eq.(<ref>) reads
S_EE( λ _0) =𝒩/G_Nr_+^2[
∫_λ _0^∞dξ√(ξ ^2( ξ
^2+λ _-^2) ( ξ ^2-1) /( ξ
^2-λ _0^2) ( ξ ^2+λ _0^2+λ
_-^2-1) )-∫_1^∞ξ dξ] .
This integral can be done analytically. In the BPS limit it becomes
particularly simple,
S_EE^BPS( λ _0) =𝒩/G_Nr_+^2
[ ∫_λ _0^∞dξ√(ξ ^2( ξ
^4-1) /( ξ ^2-λ _0^2) ( ξ
^2+λ _0^2) )-∫_1^∞ξ dξ] .
The indefine integral reads
ℐ_S^BPS( ξ) = ∫^ξdξ√(ξ
^2( ξ ^4-1) /( ξ ^2-λ _0^2)
( ξ ^2+λ _0^2) )
= 1/2𝐄( . arcsinξ ^2/λ
_0^2|λ _0^4) .
Expanding for large ξ and ξ→λ _0 we find
lim_ξ→∞ℐ_S( ξ)
= lim_ξ→∞1/2ξ ^2+1/2λ
_0^2[ -λ _0^4𝐄( λ _0^-4)
+λ _0^2𝐄( λ _0^4) -𝐊(
λ _0^-4) +λ _0^4𝐊( λ
_0^-4) ]
+𝒪( ξ ^-2) ,
lim_ξ→λ _0ℐ_S( ξ) =
1/2𝐄( λ _0^4) .
Replacing in these expressions in the entanglement entropy of the BPS
configuration (<ref>) we obtain the expression in eq.(<ref>),
S_EE^BPS( λ _0) =𝒩/G_Nr_+^2
[ 1/2λ _0^2( -λ _0^4𝐄(
λ _0^-4) -𝐊( λ _0^-4) +λ
_0^4𝐊( λ _0^-4) ) +1/2]
.
The limiting of the interval length
(<ref>)
value when r_0→∞ is non-zero and coincides with the
characteristic length of the Little String Theory. The background
(<ref>)
and the fibred one coincides in the UV leading to a regime in which the LST
dominates the behavior of dual theory. To verify this point we compute
profiles of the strings in the bulk
± x( r) =√(2/e_A^2+e_B^2)arctan(
√(r^4-r_0^4)/r_0^2) ,
which end points separation at r→∞ gives interval length
L_EE( r_0) =π√(2/e_A^2+e_B^2) .
One again, in order to capture the field theoretical behaviour of the dual
theory we add a cut-off to the observables which allow us to recover
expected behavior of a confining field theory.
99
Maldacena:1997re
J. M. Maldacena,
“The Large N limit of superconformal field theories and supergravity,”
Int. J. Theor. Phys. 38, 1113 (1999)
[Adv. Theor. Math. Phys. 2, 231 (1998)]
[hep-th/9711200].
Gubser:1998bc
S. S. Gubser, I. R. Klebanov and A. M. Polyakov,
“Gauge theory correlators from noncritical string theory,”
Phys. Lett. B 428, 105-114 (1998)
[arXiv:hep-th/9802109 [hep-th]].
Witten:1998qj
E. Witten,
“Anti-de Sitter space and holography,”
Adv. Theor. Math. Phys. 2, 253-291 (1998)
[arXiv:hep-th/9802150 [hep-th]].
Itzhaki:1998dd
N. Itzhaki, J. M. Maldacena, J. Sonnenschein and S. Yankielowicz,
“Supergravity and the large N limit of theories with sixteen supercharges,”
Phys. Rev. D 58, 046004 (1998)
[arXiv:hep-th/9802042 [hep-th]].
Witten:1998zw
E. Witten,
“Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,”
Adv. Theor. Math. Phys. 2, 505-532 (1998)
[arXiv:hep-th/9803131 [hep-th]].
Boonstra:1998mp
H. J. Boonstra, K. Skenderis and P. K. Townsend,
“The domain wall / QFT correspondence,”
JHEP 01, 003 (1999)
[arXiv:hep-th/9807137 [hep-th]].
I. Kanitscheider, K. Skenderis and M. Taylor,
“Precision holography for non-conformal branes,”
JHEP 09, 094 (2008)
[arXiv:0807.3324 [hep-th]].
Polchinski:2000uf
J. Polchinski and M. J. Strassler,
“The String dual of a confining four-dimensional gauge theory,”
[arXiv:hep-th/0003136 [hep-th]].
Klebanov:2000hb
I. R. Klebanov and M. J. Strassler,
“Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities,”
JHEP 08, 052 (2000)
[arXiv:hep-th/0007191 [hep-th]].
Maldacena:2000yy
J. M. Maldacena and C. Nunez,
“Towards the large N limit of pure N=1 superYang-Mills,”
Phys. Rev. Lett. 86, 588-591 (2001)
[arXiv:hep-th/0008001 [hep-th]].
Petrini:2018pjk
M. Petrini, H. Samtleben, S. Schmidt and K. Skenderis,
“The 10d Uplift of the GPPZ Solution,”
JHEP 07, 026 (2018)
[arXiv:1805.01919 [hep-th]].
Bobev:2018eer
N. Bobev, F. F. Gautason, B. E. Niehoff and J. van Muiden,
“Uplifting GPPZ: a ten-dimensional dual of 𝒩=1^∗,”
JHEP 10, 058 (2018)
[arXiv:1805.03623 [hep-th]].
Itzhaki:2005tu
N. Itzhaki, D. Kutasov and N. Seiberg,
“I-brane dynamics,”
JHEP 01, 119 (2006)
[arXiv:hep-th/0508025 [hep-th]].
Giveon:2019twx
A. Giveon and N. Itzhaki,
“Stringy Information and Black Holes,”
JHEP 06, 117 (2020)
[arXiv:1912.06538 [hep-th]].
Khuri:1993ii
R. R. Khuri,
“Remark on string solitons,”
Phys. Rev. D 48, 2947-2948 (1993)
[arXiv:hep-th/9305143 [hep-th]].
P. M. Cowdall and P. K. Townsend,
“Gauged supergravity vacua from intersecting branes,”
Phys. Lett. B 429, 281-288 (1998)
[erratum: Phys. Lett. B 434, 458-458 (1998)]
[arXiv:hep-th/9801165 [hep-th]].
Lozano:2019emq
Y. Lozano, N. T. Macpherson, C. Nunez and A. Ramirez,
“AdS_3 solutions in Massive IIA with small 𝒩=(4,0) supersymmetry,”
JHEP 01, 129 (2020)
[arXiv:1908.09851 [hep-th]].
Wald:1993nt
R. M. Wald,
“Black hole entropy is the Noether charge,”
Phys. Rev. D 48 (1993) no.8, R3427-R3431
[arXiv:gr-qc/9307038 [gr-qc]].
Klemm:1998in
D. Klemm,
“BPS black holes in gauged N=4, D = 4 supergravity,”
Nucl. Phys. B 545 (1999), 461-478
[arXiv:hep-th/9810090 [hep-th]].
Green:1996dd
M. B. Green, J. A. Harvey and G. W. Moore,
“I-brane inflow and anomalous couplings on d-branes,”
Class. Quant. Grav. 14, 47-52 (1997)
[arXiv:hep-th/9605033 [hep-th]].
Lin:2005nh
H. Lin and J. M. Maldacena,
“Fivebranes from gauge theory,”
Phys. Rev. D 74, 084014 (2006)
[arXiv:hep-th/0509235 [hep-th]].
Klebanov:2002gr
I. R. Klebanov, P. Ouyang and E. Witten,
“A Gravity dual of the chiral anomaly,”
Phys. Rev. D 65, 105007 (2002)
[arXiv:hep-th/0202056 [hep-th]].
Gursoy:2003hf
U. Gursoy, S. A. Hartnoll and R. Portugues,
“The Chiral anomaly from M theory,”
Phys. Rev. D 69, 086003 (2004)
[arXiv:hep-th/0311088 [hep-th]].
Macpherson:2014eza
N. T. Macpherson, C. Núñez, L. A. Pando Zayas, V. G. J. Rodgers and C. A. Whiting,
“Type IIB supergravity solutions with AdS_5 from Abelian and non-Abelian T dualities,”
JHEP 02, 040 (2015)
[arXiv:1410.2650 [hep-th]].
Bea:2015fja
Y. Bea, J. D. Edelstein, G. Itsios, K. S. Kooner, C. Nunez, D. Schofield and J. A. Sierra-Garcia,
“Compactifications of the Klebanov-Witten CFT and new AdS_3 backgrounds,”
JHEP 05, 062 (2015)
[arXiv:1503.07527 [hep-th]].
Merrikin:2022yho
P. Merrikin, C. Nunez and R. Stuardo,
“Compactification of 6d N=(1,0) quivers, 4d SCFTs and their holographic dual Massive IIA backgrounds,”
[arXiv:2210.02458 [hep-th]].
Maldacena:1998im
J. M. Maldacena,
“Wilson loops in large N field theories,”
Phys. Rev. Lett. 80, 4859-4862 (1998)
[arXiv:hep-th/9803002 [hep-th]].
S. J. Rey and J. T. Yee,
“Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,”
Eur. Phys. J. C 22, 379-394 (2001)
[arXiv:hep-th/9803001 [hep-th]].
Sonnenschein:1999if
J. Sonnenschein,
“What does the string / gauge correspondence teach us about Wilson loops?,”
[arXiv:hep-th/0003032 [hep-th]].
Nunez:2009da
C. Nunez, M. Piai and A. Rago,
“Wilson Loops in string duals of Walking and Flavored Systems,”
Phys. Rev. D 81, 086001 (2010)
[arXiv:0909.0748 [hep-th]].
Faedo:2014naa
A. F. Faedo, M. Piai and D. Schofield,
“Gauge/gravity dualities and bulk phase transitions,”
Phys. Rev. D 89, no.10, 106001 (2014)
[arXiv:1402.4141 [hep-th]].
A. F. Faedo, M. Piai and D. Schofield,
“On the stability of multiscale models of dynamical symmetry breaking from holography,”
Nucl. Phys. B 880, 504-527 (2014)
[arXiv:1312.2793 [hep-th]].
Brandhuber:1999jr
A. Brandhuber and K. Sfetsos,
“Wilson loops from multicenter and rotating branes, mass gaps and phase structure in gauge theories,”
Adv. Theor. Math. Phys. 3, 851-887 (1999)
[arXiv:hep-th/9906201 [hep-th]].
S. D. Avramis, K. Sfetsos and K. Siampos,
“Stability of strings dual to flux tubes between static quarks in N = 4 SYM,”
Nucl. Phys. B 769, 44-78 (2007)
[arXiv:hep-th/0612139 [hep-th]].
Bachas:1985xs
C. Bachas,
“Convexity of the Quarkonium Potential,”
Phys. Rev. D 33, 2723 (1986)
Kol:2014nqa
U. Kol, C. Nunez, D. Schofield, J. Sonnenschein and M. Warschawski,
“Confinement, Phase Transitions and non-Locality in the Entanglement Entropy,”
JHEP 06, 005 (2014)
[arXiv:1403.2721 [hep-th]].
Ryu:2006bv
S. Ryu and T. Takayanagi,
“Holographic derivation of entanglement entropy from AdS/CFT,”
Phys. Rev. Lett. 96, 181602 (2006)
[arXiv:hep-th/0603001 [hep-th]].
S. Ryu and T. Takayanagi,
“Aspects of Holographic Entanglement Entropy,”
JHEP 08, 045 (2006)
[arXiv:hep-th/0605073 [hep-th]].
Klebanov:2007ws
I. R. Klebanov, D. Kutasov and A. Murugan,
“Entanglement as a probe of confinement,”
Nucl. Phys. B 796, 274-293 (2008)
[arXiv:0709.2140 [hep-th]].
Barbon:2008ut
J. L. F. Barbon and C. A. Fuertes,
“Holographic entanglement entropy probes (non)locality,”
JHEP 04, 096 (2008)
[arXiv:0803.1928 [hep-th]].
J. L. F. Barbon and C. A. Fuertes,
“A Note on the extensivity of the holographic entanglement entropy,”
JHEP 05, 053 (2008)
[arXiv:0801.2153 [hep-th]].
Witten:1991yr
E. Witten,
“On string theory and black holes,”
Phys. Rev. D 44, 314-324 (1991)
Kluson:2005eb
J. Kluson,
“Dynamics of D1-brane in I-brane background,”
JHEP 12, 016 (2005)
[arXiv:hep-th/0510243 [hep-th]].
J. Kluson,
“Dynamics of probe brane in the background of intersecting fivebranes,”
Phys. Rev. D 73, 106008 (2006)
[arXiv:hep-th/0601229 [hep-th]].
L. Y. Hung,
“Comments on I1-branes,”
JHEP 05, 076 (2007)
[arXiv:hep-th/0612207 [hep-th]].
J. Kluson, B. H. Lee, K. L. Panigrahi and C. Park,
“Magnon like solutions for strings in I-brane background,”
JHEP 08, 032 (2008)
[arXiv:0806.3879 [hep-th]].
Dijkgraaf:2007sw
R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa,
“Supersymmetric gauge theories, intersecting branes and free fermions,”
JHEP 02, 106 (2008)
[arXiv:0709.4446 [hep-th]].
R. R. Nayak and P. K. Swain,
“Note on I-brane Near Horizon PP-wave Background,”
Nucl. Phys. B 848, 490-497 (2011)
[arXiv:1012.2772 [hep-th]].
S. Biswas and K. L. Panigrahi,
“Spiky Strings on I-brane,”
JHEP 08, 044 (2012)
[arXiv:1206.2539 [hep-th]].
A. Chakraborty, N. Padhi, P. Pandit and K. L. Panigrahi,
“Neumann-Rosochatius system for strings on I-brane,”
JHEP 12, 022 (2022)
[arXiv:2209.09933 [hep-th]].
S. Biswas,
“On spinning strings in I-brane background,”
[arXiv:2305.05164 [hep-th]].
Martucci:2005rb
L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Proeyen,
“Dirac actions for D-branes on backgrounds with fluxes,”
Class. Quant. Grav. 22 (2005), 2745-2764
[arXiv:hep-th/0504041 [hep-th]].
Canfora:2021nca
F. Canfora, J. Oliva and M. Oyarzo,
“New BPS solitons in 𝒩 = 4 gauged supergravity and black holes in Einstein-Yang-Mills-dilaton theory,”
JHEP 02 (2022), 057
[arXiv:2111.11915 [hep-th]].
Chamseddine:1997mc
A. H. Chamseddine and M. S. Volkov,
“NonAbelian solitons in N=4 gauged supergravity and leading order string theory,”
Phys. Rev. D 57 (1998), 6242-6254
[arXiv:hep-th/9711181 [hep-th]].
|
http://arxiv.org/abs/2307.04950v1 | 20230711003634 | Solvable models of many-body chaos from dual-Koopman circuits | [
"Arul Lakshminarayan"
] | nlin.CD | [
"nlin.CD",
"cond-mat.stat-mech",
"quant-ph"
] |
1.05
thmTheorem
lemmaLemma
corollaryCorollary
definitionDefinition
propositionProposition
proofsketch[1][Proof sketch.]
#1
example[1][Example.]
#1
remark[1][Remark.]
#1
protocol[1][Protocol.]
#1
⟩⟨
patterns,decorations.pathreplacing
breakPropProperty[section]
LemmaLemma[section]
|
http://arxiv.org/abs/2307.07286v1 | 20230714115210 | One-Shot Action Recognition via Multi-Scale Spatial-Temporal Skeleton Matching | [
"Siyuan Yang",
"Jun Liu",
"Shijian Lu",
"Er Meng Hwa",
"Alex C. Kot"
] | cs.CV | [
"cs.CV",
"cs.AI"
] |
TPAMI - Short Paper Submission
Shell et al.: Bare Demo of IEEEtran.cls for Computer Society Journals
One-shot skeleton action recognition, which aims to learn a skeleton action recognition model with a single training sample, has attracted increasing interest due to the challenge of collecting and annotating large-scale skeleton action data. However, most existing studies match skeleton sequences by comparing their feature vectors directly which neglects spatial structures and temporal orders of skeleton data.
This paper presents a novel one-shot skeleton action recognition technique that handles skeleton action recognition via multi-scale spatial-temporal feature matching.
We represent skeleton data at multiple spatial and temporal scales and achieve optimal feature matching from two perspectives. The first is multi-scale matching which captures the scale-wise semantic relevance of skeleton data at multiple spatial and temporal scales simultaneously. The second is cross-scale matching which handles different motion magnitudes and speeds by capturing sample-wise relevance across multiple scales. Extensive experiments over three large-scale datasets (NTU RGB+D, NTU RGB+D 120, and PKU-MMD) show that our method achieves superior one-shot skeleton action recognition, and it outperforms the state-of-the-art consistently by large margins.
One-Shot Action Recognition via Multi-Scale Spatial-Temporal Skeleton Matching
Siyuan Yang,
Jun Liu,
Shijian Lu,
Er Meng Hwa, Life Fellow, IEEE,
and Alex C. Kot, Life Fellow, IEEE
Siyuan Yang is with the Rapid-Rich Object Search Lab, Interdisciplinary Graduate Programme, Nanyang Technological University, Singapore.
E-mail: [email protected]
Jun Liu is with the Information Systems Technology and Design Pillar, Singapore University of Technology and Design, Singapore.
E-mail: [email protected]
Shijian Lu is with the School of Computer Science & Engineering, Nanyang Technological University, Singapore.
E-mail: [email protected]
Er Meng Hwa and Alex C. Kot are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.
E-mail: {emher, eackot}@ntu.edu.sg
Corresponding author: Jun Liu
August 12, 2023
=========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
§ INTRODUCTION
Human action recognition is a fast-developing research area due to its wide applications in human-computer interaction, video surveillance, game control, etc.
In recent years, human action recognition with skeleton data has attracted increasing attention as skeleton data encodes high-level representations of human actions and is generally lightweight and robust to variations in appearances, surrounding distractions, viewpoint changes, etc.
As of today, most existing studies expect large-scale labeled training data for learning effective human action representations. While facing skeleton data of a new category, they require to collect hundreds of action samples of the new category for adapting or fine-tuning some existing models.
How to achieve single-shot recognition for new action categories becomes critically important for circumventing the tedious and laborious data collection and labeling procedure.
One-shot skeleton action recognition is a very challenging task. Beyond the data constraint with one-shot for unseen new classes, the major challenge comes from the very rich variations in human actions. Take the action “put on glasses” as an example. Different persons could perform it by using their left hand, right hand, or both hands. The same person could also perform it at different paces with different motion dynamics.
Different approaches have been explored to address this challenging task, and most existing works <cit.> represent the anchor and target samples with certain pooled feature vectors and compute the sample distance based on the similarity of their pooled feature vectors. However, the adoption of such global feature similarity discards the very useful spatial structures and temporal order of the skeleton sequences. In addition, most existing works learn skeleton action representations at a single scale of the original body joints which tends to lose useful action features under the one-shot scenario. Specifically, human actions are multi-scales in both spatial and temporal spaces, e.g., multiple joints on arms and legs collaborate in walking and consecutive frames of human actions contain strong temporal correlations. Skeleton action representations should therefore capture the rich semantic correlations at different spatial and temporal scales.
We propose to capture spatial-temporal features by leveraging spatial structures and temporal orders of skeleton sequences as illustrated in Fig. <ref>. Inspired by the theory of optimal transport <cit.>, we measure the semantic relevance of two skeleton sequences by computing an optimal matching flow between their feature maps. Specifically, we adopt Earth Mover's Distance (EMD) <cit.> as the optimal matching metric for acquiring the optimal matching flow, where EMD maximizes the impact of relevant joints and minimizes the effect of irrelevant joints between two skeleton sequences. In addition, we model skeleton sequences at multiple spatial scales (joint-scale, part-scale, and limb-scale) and temporal scales as illustrated in Fig. <ref>, and perform multi-scale matching to capture scale-wise skeleton semantic relevance by using EMD. Further, human action could be performed with different motion magnitudes and motion paces, e.g., `hand waving' may be performed by hand (joint-level), forearm (part-level), or the whole arm (limb-level) at different paces. We thus design cross-scale matching that learns semantic relevance by measuring feature consistency across spatial and temporal scales.
The contributions of this work are threefold.
First, we formulate one-shot skeleton action recognition as an optimal matching problem and design an effective network framework for one-shot skeleton action recognition.
Second,, we propose a multi-scale matching strategy that can capture scale-wise skeleton semantic relevance at multiple spatial and temporal scales. On top of that, we design a novel cross-scale matching scheme that can model the within-class variation of human actions in motion magnitudes and motion paces. To the best of our knowledge, this is the first work that exploits multi-scale representations and cross-scale matching to capture multi-scale skeleton semantic relevance and maintain consistency across motion scales in one-shot skeleton action recognition. Third, extensive experiments on three public datasets (NTU RGB+D, NTU RGB+D 120, and PKU-MMD) show that our method outperforms the state-of-the-art consistently by large margins.
§ RELATED WORKS
Skeleton-based Action Recognition
has attracted increasing attention in recent years.
Traditional methods design hand-craft features to represent skeleton sequences <cit.>.
In recent years, deep learning has been widely explored
by leveraging the advances in Recurrent Neural Networks (RNNs) <cit.>, Convolutional Neural Networks (CNNs) <cit.>, and Graph Convolution Networks (GCNs) <cit.>.
Inspired by the observation that the human skeleton is naturally a topological graph,
GCNs have attracted increasing attention in skeleton-based action recognition.
For example,
Yan et al. <cit.> proposed a spatial-temporal graph convolutional network to learn spatial-temporal patterns from skeleton data.
Shi et al. <cit.> designed an adaptive graph convolutional network, which utilized self-attention with the spatial-temporal graph convolutional network.
Peng et al. <cit.> recognized actions by searching for different graphs at different layers via neural architecture search.
Though the prior studies achieve very impressive performance, most of them are supervised and require large-scale training data which is often laborious to collect.
We focus on one-shot skeleton action recognition, aiming to address the scenarios where only a few labeled skeleton samples are available for an unseen new class.
Few-Shot Learning.
Motivated by the human capability in learning new concepts from just a few samples, few-shot learning,
which aims at recognizing unseen concepts with only a few labeled training samples, has received increasing attention and witnessed significant advances in recent years.
It has also been widely explored in the computer vision research community. For example,
Snell et al. <cit.> presented the Prototypical Networks that compute distances between a datapoint and class-wise prototypes.
Ye et al. <cit.> defined set-to-set transformations to learn a task-specific feature embedding for few-shot learning.
Simon et al. <cit.> presented the DSN that employs a few-shot learning model via affine subspaces.
Xu et al. <cit.> proposed to learn a novel dynamic meta-filter for few-shot learning.
One-shot Skeleton Action Recognition
has attracted increasing interest in
recent years.
Leveraging the NTU RGB+D 120 dataset, Liu et al. <cit.> first presented
an Action-Part Semantic-Relevance aware (APSR) approach for one-shot skeleton action recognition.
Sabater et al. <cit.> presented a one-shot action recognition approach based on a Temporal Convolutional Network (TCN).
Memmesheimer et al. <cit.> proposed to formulate the one-shot skeleton action learning problem as a deep metric learning problem.
More recently, Memmesheimer et al. <cit.> presented an image-based skeleton representation, which performs well in the deep metric learning manner.
Most existing methods take the skeleton representation as a whole and measure the skeleton similarity globally which often misses useful structure and temporal information. We propose to treat one-shot skeleton action recognition as an optimal matching problem and design multi-scale matching and cross-scale matching which capture the scale-wise semantic relevance and maintain the spatial and temporal consistency across different scales, respectively
§ METHOD
We aim to train a model that can recognize human skeleton data of novel classes with only a single labeled sample. It is a very challenging task due to the very rich intra-class spatial-temporal variations in human skeleton action. We address this challenge by proposing a one-shot skeleton action recognition framework as illustrated in Fig. <ref>. Specifically, we design a novel optimal matching technique to capture the useful spatial structure and temporal order information which is largely neglected in most existing one-shot skeleton action recognition studies.
In the following, we first present the problem formulation of the one-shot skeleton action recognition task. We then introduce the embedding network and elaborate on how to construct multi-spatial and multi-temporal scale skeletons. Finally, we describe the proposed optimal matching technique in detail.
§.§ Problem Formulation
Inspired by prior studies in few-shot learning <cit.> and few-shot video action recognition <cit.>, we formulate the one-shot skeleton action recognition task as a meta-learning problem <cit.> that consists of a meta-training phase and a meta-testing phase. In a n-way and 1-shot problem, each episode consists of a support set S and a query set Q, where S contains 1 labeled sample for each of n unseen classes and Q is employed to evaluate the generalization performance. The algorithm aims to determine which support classes each query sample belongs to. Specifically, multiple n-way and 1-shot tasks are randomly sampled from the meta-training set D_train (with seen classes), and employed to train a model in an episodic manner. In meta-testing phase, n-way and 1-shot tasks are sampled from the meta-testing set D_test (with unseen classes) for evaluations.
§.§ Skeleton Feature Embedding
Following prior studies on few-shot learning <cit.> and few-shot video action recognition <cit.>, we first pretrain an embedding network on the whole meta-training set D_train before episodic training. We adopt the GCN-based model <cit.> as the baseline network which has an adaptive spatial-temporal graph for extracting the relation among body joints.
The GCN-based model just processes the features of the original scale. However, such single-scale modeling often misses meaningful skeleton information especially when only a single labeled sample is available as described in Sec. <ref>. Inspired by <cit.> which handle multi-scale features, we represent the human skeleton data at multi-spatial and multi-temporal scales.
Multi-Spatial Scale Skeleton:
We model skeleton actions at multiple spatial scales. Specifically, we adopt 3 spatial scales including the body-joint scale (s_1), the part-level scale (s_2), and the limb-level scale (s_3) as illustrated in Fig. <ref> (a). We first build GCN blocks on the first scale to capture joint-wise feature representations and then perform the average pooling. For skeleton-based representations, the pooling requires meaningful neighborhoods and we simply put joints of the same spatial scale into one group.
Multi-Temporal Scale Skeleton:
We represent skeleton data at multiple temporal scales with a temporal pooling operator that averages multiple consecutive frames into a new frame in a coarse-to-fine manner. The intuition is that consecutive frames have continuous motion and poses capturing similar abstract states. After processing GCN blocks at the original spatial scale, we introduce two average pooling layers along the temporal dimension to perform temporal pooling as illustrated in Fig. <ref> (b).
We implement 3 spatial scales and 3 temporal scales for illustration, where each skeleton structure captures unique perspectives of skeleton representations. To extract the multi-scale skeleton representations, the multi-spatial (multi-temporal) scale skeleton structures are trained individually in parallel[The detailed network structures are available in appendix.].
§.§ Optimal Matching Strategy
Unlike <cit.> that compute distances over sequence-level embeddings, we capture discriminative local information of each body joint and design a skeleton optimal matching scheme to compute semantic relevance based on optimal transport theory as discussed in Sec. <ref>. Specifically, we adopt the Earth Mover’s Distance (EMD) <cit.> as the optimal transport matching metric, which searches for the minimal cost transport plan between two joints' feature distributions by maximizing the impact caused by relevant joints and minimizing the effect between irrelevant joints.
Here, we first use the single-scale model as an example to show how we formulate the one-shot skeleton action recognition as the optimal matching problem by using EMD.
The skeleton representation embedded by the single-scale model can be represented as X∈ R^C × N × T, where C is the number of output channels, N denotes the number of skeleton joints, and T denotes the number of frames.
For two feature maps X, Y∈ R^C × N × T, we first flatten them into two sets of joint's local representations 𝒳 = {x_i|i = 1, 2, ...NT } and 𝒴={y_j| j=1,2, … NT}, where x_i and y_j (x_i, y_j∈ R^C) denote the joint's local representation at the corresponding spatial and temporal positions.
Then we define the EMD between two sets of local representations as the minimum “transport cost” from 𝒳 (suppliers) to 𝒴 (demanders).
Suppose for each supplier x_i, it has r_i units to transport, and for each demander y_j, it requires c_j units.
The overall optimal transport matching problem can be formulated as:
OT(r, c) = {π∈ℛ^NT × NT | π1 = 𝐫, π^T1 = 𝐜},
where π is the optimal matching flow between these two distributions, which can also be viewed as the optimal matching plan of two skeleton sequences.
r_i and c_j are called the weights of nodes, which control the total matching flows generated by each node, and 𝐫 and 𝐜 are vectorized representations of { r_i} and {c_j}.
EMD seeks an optimal matching flow π between “suppliers” 𝒳 and “demanders” 𝒴, such that the overall matching cost can be minimized.
Additionally, the transporting cost per unit is defined by computing the pairwise distance between supplier node 𝐱_i and demander node 𝐲_j from two skeleton features:
d_i j=1-𝐱_i^T𝐲_j/𝐱_i𝐲_j,
where nodes with similar local representations tend to generate small transporting costs between each other.
Then we can define the EMD as the optimal transportation problem, which is represented as:
D_emd(X, Y) =min_π∈ OT(r, c)∑_i=1^NT∑_j=1^NT d_ijπ_ij.
The weight of each node (e.g., r_i and c_j) plays an important role in optimal matching problems. Intuitively, the node with a larger weight is more important while matching two sets.
Therefore, in order to assign the more important node a higher weight, we follow <cit.> to generate the weight r_i by a cross-reference mechanism that uses the dot product between a joint representation and the average joint representation in the other skeleton features:
r_i = max {x_i^T·∑_j=1^NTy_j/NT, 0 },
where x_i and y_j denote the feature vectors from two skeleton feature maps, and function max(·) ensures the weight is always non-negative.
Above we take r_i as an example, and c_j can be calculated in the same manner.
Once acquiring the optimal matching flow π, we can compute the semantic relevance score s between two skeleton representations as:
s(X, Y) = ∑_i=1^NT∑_j=1^NT(1-d_ij)π_ij.
These semantic relevance scores allow studying the composition of the overall relevance, enabling us to assign high relevance to semantically similar joints no matter whether they are in the same spatial order or temporal frame.
We can thus tackle the problems that the semantic relevance of two skeleton sequences can occur at different temporal positions or different spatial joints.
Multi-Scale Matching.
As mentioned in Sec. <ref>,
human skeleton data can be represented in multi-spatial scales and multi-temporal scales, and each scale's representation contains unique semantic information.
Thus, we propose to capture the pair-wise skeleton semantic relevance at multiple scales, including multi-spatial scale matching and multi-temporal scale matching, to acquire optimal matching flow from multiple spatial and multiple temporal scales.
For the multi-spatial scale scenario, there are three pairs of feature embeddings, which can be represented as X_s1∈ R^C × N × T, X_s2∈ R^C × N_2 × T, and X_s3∈ R^C × N_3 × T, respectively.
N_2 denotes the number of nodes for the second-scale spatial graph, and N_3 stands for the number of third-scale spatial graph nodes.
The semantic relevance score between two skeleton sequences thus becomes:
s_ms(X, Y) = s(X_s1, Y_s1) + s(X_s2, Y_s2) + s(X_s3, Y_s3).
This enables us to seek the optimal matching flow using EMD, and measure the semantic relevance at multiple spatial scales as shown in Fig. <ref> (a) (highlighted in blue color).
For the multi-temporal scale, there are also three pairs of feature embeddings, which can be represented as X_t1∈ R^C × N × T, X_t2∈ R^C × N × T/2, and X_t3∈ R^C × N × T/4, respectively.
The semantic relevance score between two skeleton sequences becomes:
s_mt(X, Y) = s(X_t1, Y_t1) + s(X_t2, Y_t2) + s(X_t3, Y_t3).
Using Eq. (<ref>), semantic relevance between two skeleton sequences is measured at multiple temporal scales, as shown in Fig. <ref> (b) (highlighted in blue color).
Cross-Scale Matching.
As discussed in Sec. <ref>, different instances of the same action class may be performed at different magnitudes (spatial scales).
For the class, such as `hand waving', people may perform it by moving their palm only (joint-level), or by moving the forearm (part-level), or even by moving the whole arm (limb-level).
Additionally, the same-category samples can also be performed at different speeds (temporal scale).
Thus, there is also semantic relevance between different scales' skeleton representations for matching.
To address the cross-scale matching problem mentioned above, we further investigate how to measure the semantic relevance between skeleton sequences across different scales, including cross-spatial scale and cross-temporal scale matching, considering the possibility of different spatial magnitudes and temporal speeds of the same action.
For cross-spatial scale matching, the three spatial scales' skeleton representations are X_s1∈ R^C × N × T, X_s2∈ R^C × N_2 × T, and X_s3∈ R^C × N_3 × T.
It can be seen that all these three scales' representations contain T frame features.
Thus, we first perform 1D average pooling (AvgPool) on the spatial dimension to generate these three scales' features in the same shape (R^C × T), and then formulate the semantic relevance score as the T frame features optimal matching problem to acquire the optimal matching flow between different spatial-scale representations.
The cross-spatial scale semantic relevance score can be represented as:
s_cs(X, Y) = ∑_i=1^3∑_j=1, j ≠ i^3 s(AvgPool(X_si), AvgPool(Y_sj)).
This process is shown in Fig. <ref> (a) (highlighted in orange color).
In this way, we solve the problem of matching skeleton sequences with different motion magnitudes.
Furthermore, we also address the problem of matching the skeleton sequences with different motion speeds and design the cross-temporal scale matching.
The three temporal scales' skeleton features are represented as X_t1∈ R^C × N × T, X_t2∈ R^C × N × T/2, and X_t3∈ R^C × N × T/4.
It can be seen that these three scales' representations all contain N joint features.
Similarly, the 1D average pooling (AvgPool) can be performed on the temporal dimension to pool X_t1, X_t2, and X_t3 into the same shape (R^C × N).
Thus, the Earth Mover's Distance is used to measure the semantic relevance score across different temporal scales as:
s_ct(X, Y) = ∑_i=1^3∑_j=1, j ≠ i^3 s(AvgPool(X_ti), AvgPool(Y_tj)),
This process is shown in Fig. <ref> (b) (highlighted in orange color).
Action sequences with different motion speeds can be matched well through Eq. (<ref>).
Summary.
As above mentioned, we first introduce the single-scale semantic relevance score s(X, Y) (Eq. (<ref>)) that considers the useful spatial structure and temporal order information during matching.
To address the problem that different instances of the same action class samples may be performed at different paces with different motion dynamics, we introduce 4 types of semantic relevance scores including multi-spatial scale (s_ms(X, Y), Eq. (<ref>)), multi-temporal scale (s_mt(X, Y), Eq. (<ref>)), cross-spatial scale (s_cs (X, Y), Eq. (<ref>)), and cross-temporal scale (s_ct(X, Y), Eq. (<ref>)).
The semantic relevance score for our proposed model is averaging from multi-scale and cross-scale relevance scores, which is then used to predict the action category.
§.§ Objective Loss, Model Training, and Inference
The proposed method is trained in two sequential stages: The First is the pre-training stage. The embedding network is trained on meta-training set D_train in a standard supervised learning way (Sec. <ref>).
The Second is the meta-training stage. The embedding network and our optimal matching method (Sec. <ref>) are further optimized in an end-to-end manner.
Both training stages leverage the cross-entropy loss as the classification loss L_cls for optimization.
Given an unseen query sequence q and its support set S (both sampled from meta-testing set D_test) at test, the goal is to determine which support set classes q belongs to.
§ EXPERIMENTS
§.§ Datasets
NTU RGB+D dataset <cit.> consists of 56880 skeleton action sequences, which is the most widely-used dataset in skeleton-based action recognition research. In this dataset, action samples are performed by 40 subjects, with three camera views, and categorized into 60 classes.
The NTU RGB+D dataset provides two standard evaluation protocols, namely cross-view (CV) and cross-subject (CS).
NTU RGB+D 120 dataset <cit.> is currently the largest dataset with 3D joints annotations for human action recognition. The dataset contains 114480 action samples in 120 action classes. Action samples are captured by 106 volunteers with three camera views. This dataset contains 32 setups, and each setup denotes a specific location and background.
The evaluation protocols of this dataset are (1) cross-subject evaluation and (2) cross-setup evaluation.
PKU-MMD dataset <cit.> is a large-scale benchmark for continuous multi-modality 3D skeleton action understanding.
It contains 21,545 action instances performed by 66 distinct subjects in 51 action categories.
The evaluation protocols of PKU-MMD are similar to those of NTU RGB+D, i.e., a cross-view protocol and a cross-subject protocol.
PKU-MMD consists of two subsets, part I and part II, we conduct experiments on the part I subset in this work.
§.§ Training and Evaluation Protocol
Training Protocol.
As described in Sec. <ref>, we formulate the one-shot skeleton action recognition problem as a meta-learning problem <cit.>. Therefore, we adopt the meta-training phase as our training protocol.
All experiments are trained under the 5-way, 1-shot setting, and we set the number of query samples for each class to 15.
Evaluation Protocol 1.
Similarly, we conduct the evaluation on 5-way, 1-shot setting, and adopt this procedure as Evaluation Protocol 1 in the following experiments.
Evaluation Protocol 2.
To compare with the existing one-shot skeleton action recognition techniques <cit.> in a fair way, we also follow the official one-shot protocol described in <cit.> for the dataset NTU RGB+D 120.
Specifically, the testing set consists of 20 novel classes, and we pick one sample from each novel class as the exemplar[More exemplars & dataset splitting details are provided in the appendix. ] and leave the rest (except for the 20 exemplars) to test the recognition performance.
For datasets NTU RGB+D and PKU-MMD<ref>, we adopt a similar protocol in experiments.
§.§ Implementation Details
Pre-training Stage.
We use the SGD optimizer with Nesterov momentum (0.9) as the optimizer.
The learning rate is set as 0.1 and is divided by 10 at 30_th epoch and 40_th epoch.
The training process is ended at the 50_th epoch.
For NTU RGB+D and NTU RGB+D 120, the batch size is set as 64. The batch size for the PKU-MMD dataset is 32.
Meta-learning Stage.
the learning rate starts at 0.001 and decays every 10 epochs by 0.5.
We train for 100 epochs using SGD optimizer, and each epoch consists of 100 episodes from the meta-training set D_train. For the meta-testing phase, we sample 1000 episodes from meta-testing set D_test.
(D_train and D_test are defined in Sec. <ref>).
Embedding Network.
We use Adaptive Graph Convolutional Network <cit.> (AGCN) as our single-scale embedding network which has 9 GCN blocks.
For multi-spatial and multi-temporal networks, the first 6 blocks are shared to capture the single-scale features. Then each scale feature is processed by the other 3 individually and in parallel[The detailed network structures can be found in appendix.].
§.§ Dataset Splitting
For dataset NTU RGB+D 120, we adopt the one-shot skeleton action setting as described in <cit.> which splits the full dataset into a training set and a testing set. The action classes of the two sets are distinct which include 100 classes for training and 20 for testing<ref>. For dataset NTU RGB+D, the training set and testing set are determined by the selection of 50 classes and 10 classes from the 100 training and 20 testing classes of the NTU RGB+D 120, respectively. Similarly, for dataset PKU-MMD, we divide the action categories into a training set and a testing set which include 41 classes for training and 10 classes for testing<ref>.
As no hold-out validation set is defined in the one-shot skeleton action setting and all these three datasets contain the cross-subject setting for supervised action recognition. Therefore, we divide the training class data into the training set and validation set based on the cross-subject principle for the one-shot skeleton action recognition task.
We maintain the testing set of these three datasets as the testing set for one-shot skeleton action recognition.
§.§ Evaluating on One-Shot Skeleton Action Recognition
We conduct extensive experiments with five optimal matching strategies that include single-scale (`S-scale'), multi-spatial scale (`M_s'), multi-temporal scale (`M_t'), cross-spatial scale (`C_s'), and cross-temporal scale (`C_t'). We combine
`M_s' and `M_t' to form a new multi-scale strategy `M-scale'. In addition, we combine `M_s', `M_t', `C_s', and `C_t' to form another new strategy `M&C-scale' that matches spatial and temporal features simultaneously at multiple scales and also cross scales.
We compare our method with two groups of state-of-the-art methods on one-shot skeleton action recognition. The first group consists of state-of-the-art few-shot image classification methods including Subspace <cit.>, ProtoNet <cit.>, Dynamic Filter <cit.>, FEAT <cit.>. All these methods use the same embedding network as our method for fair comparisons. We re-implement <cit.> based on publicly available codes and conduct experiments on NTU RGB+D, NTU RGB+D 120 and PKU-MMD datasets. The implementation details are available in the appendix.
The second group consists of state-of-the-art one-shot skeleton action recognition techniques including APSR <cit.>, TCN <cit.>, SL-DML <cit.>, Skeleton-DML <cit.>, uDTW <cit.>, and JEANIE <cit.>.
For those one-shot skeleton action recognition works <cit.>, the results in Tabs. <ref>,<ref>,<ref> are from the original papers.
We compare our method with the first group methods under both evaluation protocols, while the second group with Evaluation Protocol 2 only.
§.§.§ Evaluation Protocol 1:
We evaluate the 5-way 1-shot setting on all three datasets and Tab. <ref> shows experimental results. It can be seen that our proposed single-scale optimal matching outperforms state-of-the-art few-shot learning methods on all three datasets. In addition, our proposed multi-scale and cross-scale matching strategies further improve one-shot skeleton action recognition by large margins,
demonstrating the effectiveness of our proposed method on the one-shot skeleton action recognition task.
§.§.§ Evaluation Protocol 2:
Following the one-shot setting in <cit.>, we also conduct experiments on NTU RGB+D, NTU RGB+D 120, and PKU-MMD datasets under Evaluation Protocol 2. Similar to the experiments on Evaluation Protocol 1, our proposed method outperforms the state-of-the-art one-shot skeleton action recognition and few-shot learning methods by large margins (up to 8% on NTU120 and PKU-MMD). Tab. <ref> shows more details of the experiments.
§.§ Ablation Studies
Matching Strategies: We compare our proposed optimal matching with the global matching scheme that adopts global average pooling to generate feature vector, as well as a local matching scheme that computes joint-to-joint distances (either Euclidean or Cosine distance) with local-level representations. For fair comparisons, we adopt the same backbone and training scheme for all compared methods, and Tab. <ref> reports experiment results. We can observe that models with local-level representations
perform better than models that rely on the global-level representations (in globally pooled feature vectors). In addition, our method which
works with the optimal matching flow between all pairs of joints outperforms all compared methods. Note that all experiments here were conducted on the single-scale model under evaluation protocol 2.
Reducing Training Classes: While evaluating one-shot action recognition methods, one interesting question is how many training classes are required to achieve fair recognition performance. We examine this issue under Evaluation Protocol 2 by following prior studies on NTU RGB+D 120 dataset <cit.>. Tab. <ref> shows experimental results. It can be seen that our method outperforms the state-of-the-art by large margins under different numbers of training classes.
With a training set of 60 classes, our method is on par with state-of-the-art methods that are trained by using 100-class training set. This clearly shows the effectiveness of our proposed optimal matching.
Effect of Multi-Scale and Cross-Scale Matching Manners:
We conduct experiments on different combinations of our proposed optimal matching strategies. Tab. <ref> shows experimental results under evaluation protocol 2. We can see that including any of our proposed matching strategies (M_s, M_t, C_s, and C_t) improves the one-shot skeleton action recognition clearly. Including all four matching strategies perform simply the best over all three datasets, demonstrating the effectiveness of our proposed optimal matching technique.
Please refer to the appendix for more ablation studies and visualization of the proposed multi-scale and cross-scale optimal matching strategies.
Effect of Multiple Spatial Scales and Multiple Temporal Scales: We also study how models with different spatial and temporal scales (as the backbone) perform for the one-shot skeleton action recognition task. Tab. <ref> shows the one-shot learning performance with different combinations of scales. We can observe that the model performs the best when combining scales 1, 2, and 3. In addition, employing two scales (scales 1 and 2 or scales 1 and 3) also outperform the model using scale 1 only, showing the benefits of the proposed multi-scale representations.
§ CONCLUSION
In this paper, we address the one-shot skeleton action recognition as a matching problem.
We obtain the multi-spatial and multi-temporal scale features by designing a hierarchical pooling that represents the same skeleton sequence at various spatial and temporal scales.
Moreover, based on the multi-scale skeleton features, we propose a multi-scale skeleton matching strategy and a cross-scale skeleton matching manner to measure the semantic relevance between two skeleton sequences for one-shot skeleton action recognition.
The experiments demonstrate that our proposed method achieves superior one-shot skeleton action recognition performance.
IEEEtran
§ EMBEDDING NETWORK STRUCTURES
Single scale.
We adopt the adaptive graph convolutional network (AGCN) <cit.> as our single scale skeleton embedding network, which is the stack of 9 adaptive graph convolutional (AGC) blocks, global average pooling layer, and a softmax classifier, as shown in Fig. <ref>.
For the details of the AGC block and the setting of each block, please refer to <cit.>.
Multi-spatial scale.
To construct the multi-spatial scale skeleton, we first build six AGC blocks on the original spatial scale to capture the joint-wise feature representation and then perform the spatial pooling to generate the spatial scale 2 and spatial scale 3 features.
To extract the multi-spatial scale skeleton representations, the multi-spatial scale skeleton structures are trained individually and paralleled, as shown in Fig. <ref>.
The vertically parallel AGC blocks share the same setting, except for the skeleton graph structure.
Multi-temporal scale.
Similarly, after processing the first six AGC blocks in the original temporal scale, we perform the temporal pooling to generate the coarser scales (temporal scale 2 and temporal scale 3).
To extract the multi-temporal scale skeleton representations, the multi-temporal scale skeleton structures are also trained individually, as shown in Fig. <ref>.
The vertically parallel AGC blocks share the same setting.
For all types of embedding networks (single scale, multi-spatial scale, and multi-temporal scale), we leverage the output feature maps of 9^th block as the skeleton feature representations in the meta-training stage.
§ ADDITIONAL EXPERIMENTAL RESULTS
More experimental results on different combinations of optimal matching manners:
We conduct more experiments on different combinations of optimal matching manners to show the advantages of our proposed optimal matching strategies.
Tabs. <ref> and <ref> show the experimental results under the evaluation protocol 1 and the “Reducing training classes” experiments, respectively. Similar to the conclusion we got in the main manuscript, we can see that including any of our proposed matching strategies (M_s, M_t, C_s, and C_t) improves the one-shot skeleton action recognition performance clearly. Including all four matching strategies perform the best on all three datasets, demonstrating the effectiveness of our proposed optimal matching manners.
Visualization Results:
In order to further show the effectiveness of our proposed multi-scale and cross-scale matching strategies, we provide some visualization examples, as shown in Fig. <ref>.
For Fig. <ref> (a),
two skeleton samples of the same action "wipe face" were performed by different body parts (two hands vs one hand only)
Similarity-based method (proto) fails in recognition, while our matching-based methods (e.g., `S-scale', `M-scale', `M&C-scale') can recognize the correct category, showing the effectiveness of the matching-based method.
As shown in Fig. <ref> (b), two `use a fan' action samples were performed in different ways.
The S-scale model fails in recognition since the joint-level representations are different.
However, if we focus on the limb level, these two samples all can be seen as `frequent shaking of arms toward the torso'.
Therefore, our multi-scale matching-based methods (e.g., `M-scale', `M&C-scale') successfully recognize the action category, demonstrating the advantage of our proposed multi-scale matching manner.
Fig. <ref> (c) shows two skeleton action samples (belonging to `throw') were performed at different motion magnitudes,
and Fig. <ref> (d) shows two samples (belonging to `falling ') were performed at different speeds.
Our `M&C-scale' method can still succeed in recognition in the above challenging two situations, showing that our designed cross-scale matching strategy is able to handle the challenging scenarios where the samples of the same action category can be performed at different magnitudes and different motion paces.
§ IMPLEMENTATION DETAILS OF COMPARED FEW-SHOT LEARNING METHODS
The implementation for the pre-training stage is the same as our single-scale setting in the main manuscript.
In this section, we focus on the setting for the meta-training stage.
To achieve the best results for those few-shot learning methods <cit.>, we use different settings.
There are also slight differences in the settings of the same method on different datasets.
ProtoNet <cit.>.
For NTU RGB+D and NTU RGB+D 120 datasets, we set the learning rate to 0.001.
For the PKU-MMD dataset, the initial learning rate is set at 0.0005.
We cut the learning rate to half every 10 epochs for all these three datasets.
FEAT <cit.>.
For NTU RGB+D, NTU RGB+D 120, and PKU-MMD datasets, we set the learning rate to 0.0005 and cut the learning rate to half every 10 epochs.
There is a weight value to balance the contrastive term in the learning objective. Here we set the balance value to 0.1 for NTU RGB+D and NTU RGB+D 120 datasets and 0.01 for the PKU-MMD dataset.
Subspace <cit.>.
For NTU RGB+D and NTU RGB+D 120 datasets, we set the learning rate to 0.005.
For the PKU-MMD dataset, the initial learning rate is set at 0.0005.
We cut the learning rate to half every 5 epochs for all three datasets.
Dynamic Filter <cit.>.
For NTU RGB+D, NTU RGB+D 120, and PKU-MMD datasets, we set the learning rate to 0.05 and cut the rate by half every 10 epochs.
There is a weight value to balance the few-shot classification objective function and global classification objective function.
We set the balance value to 0.2 for NTU RGB+D and NTU RGB+D 120 datasets and 0.1 for the PKU-MMD dataset.
Experiments for all these methods <cit.> are optimized by SGD optimizer with Nesterov momentum (0.9).
§ SPATIAL POOLING
We adopt 3 spatial scales in our work: the joint-level scale, the part-level scale, and the limb(super-part)-level scale, as shown in Fig. <ref>.
As all three datasets (NTU RGB+D, NTU RGB+D 120, and PKU-MMD) collected skeleton data, which consists of 3D locations of 25 body joints, we consider those 25 skeleton joints for spatial scale 1.
Additionally, in spatial scale 2 and spatial scale 3, we consider 10 parts and 6 super-parts, respectively.
Pooling details from spatial scale 1 to spatial scale 2 and from spatial scale 2 to spatial scale 3 can be found in Tabs. <ref> and <ref>, respectively.
§ DATASET SPLITTING
NTU RGB+D 120 <cit.>.
We follow the official one-shot setting as described in the NTU RGB+D 120 paper <cit.>. The action classes of the two sets are distinct, which include 100 classes for training and 20 for testing.
The testing set consists of 20 novel classes (i.e. A1, A7, A13, A19, A25, A31, A37, A43, A49, A55, A61, A67, A73, A79, A85, A91, A97, A103, A109, A115), and one sample from each novel class is picked as the exemplar.
The following 20 categories are selected: A1 (drink water), A7 (throw), A13 (tear up paper), A19 (take off glasses), A25 (reach into pocket), A31 (pointing to something with finger), A37 (wipe face), A43 (falling), A49 (use a fan (with hand or paper)/feeling warm), A55 (hugging other person), A61 (put on headphone), A67 (hush (quite)), A73 (staple book), A79 (sniff (smell)), A85 (apply cream on face), A91 (open a box), A97 (arm circles), A103 (yawn), A109 (grab other person’s stuff), A115 (take a photo of other person).
As suggested by the original dataset paper <cit.>, the following 20 samples are selected as the exemplars:
`S001C003P008R001A001', `S001C003P008R001A007', `S001C003P008R001A013', `S001C003P008R001A019', `S001C003P008R001A025', `S001C003P008R001A031', `S001C003P008R001A037', `S001C003P008R001A043', `S001C003P008R001A049', `S001C003P008R001A055', `S018C003P008R001A061', `S018C003P008R001A067', `S018C003P008R001A073', `S018C003P008R001A079', `S018C003P008R001A085', `S018C003P008R001A091', `S018C003P008R001A097', `S018C003P008R001A103', `S018C003P008R001A109', `S018C003P008R001A115'.
NTU RGB+D <cit.>. We select 10 novel classes and 10 exemplars from the NTU RGB+D 120 one-shot setting, of which the action label's no. is smaller than 60, as the novel classes and exemplars for the NTU RGB+D dataset.
The following 10 categories are selected: A1 (drink water), A7 (throw), A13 (tear up paper), A19 (take off glasses), A25 (reach into pocket), A31 (pointing to something with finger), A37 (wipe face), A43 (falling), A49 (use a fan (with hand or paper)/feeling warm), A55 (hugging other person).
The following 10 samples are selected as the exemplars: `S001C003P008R001A001', `S001C003P008R001A007', `S001C003P008R001A013', `S001C003P008R001A019', `S001C003P008R001A025', `S001C003P008R001A031', `S001C003P008R001A037', `S001C003P008R001A043', `S001C003P008R001A049', `S001C003P008R001A055'.
PKU-MMD <cit.>.
Similarly, we split PKU-MMD dataset into two parts: the training set (41 classes) and the testing set (10 classes).
The testing set consists of 10 novel classes, and one sample from each novel class is picked as the exemplar.
The following 10 categories are the novel classes: A1 (bow), A6 (clapping), A11 (falling), A16 (hugging other person), A21 (pat on back of other person), A26 (punching/slapping other person), A31 (rub two hands together), A36 (take off glasses), A41 (throw), A46 (typing on a keyboard).
The following 10 samples are the exemplars:
`0003-L_A_1', `0003-L_A_6', `0002-L_A_11', `0005-L_A_16', `0005-L_A_21', `0005-L_A26', `0002-L_A_31', `0003-L_A_36' , `0002-L_A_41', `0003-L_A_46'.
The videos in the PKU-MMD dataset are untrimmed, so we need to trim videos to the one-action segment level based on the given starting time and ending time.
While the videos' filenames contain only the part before the first `_' of exemplars' filenames, take the `0003-L_A_1' as an example, the original filename is `0003-L'.
Since we trim the video, we add the action category number in the filename, here `A_1' in `0003-L_A_1' means the corresponding segment of action category 1 in the video `0003-L'.
|
http://arxiv.org/abs/2307.07422v1 | 20230708172155 | Can LLMs be Good Financial Advisors?: An Initial Study in Personal Decision Making for Optimized Outcomes | [
"Kausik Lakkaraju",
"Sai Krishna Revanth Vuruma",
"Vishal Pallagani",
"Bharath Muppasani",
"Biplav Srivastava"
] | cs.CL | [
"cs.CL"
] |
Explicit a posteriori error representation for variational problems and application to TV-minimization
[
August 12, 2023
========================================================================================================
Increasingly powerful Large Language Model (LLM) based chatbots, like ChatGPT and Bard, are becoming available to users that have the potential to revolutionize the quality of decision-making achieved by the public. In this context, we set out to investigate how such systems perform in the personal finance domain, where financial inclusion has been an overarching stated aim of banks for decades.
We asked 13 questions representing banking products in personal finance: bank account, credit card and certificate of deposits and their inter-product interactions, and decisions related to high-value purchases, payment of bank dues, and investment advice, and in different dialects and languages (English, African American Vernacular English, and Telugu). We find that although the outputs of the chatbots are fluent and plausible, there are still critical gaps in providing accurate and reliable financial information using LLM-based chatbots.
§ INTRODUCTION
Consider a freshman that has just started making personal financial decisions. They open a bank account to save up money and get their first credit card. They are given some seed money by their family and they also start earning by working on campus.
The student is encouraged by their support system to start thinking about saving into products like Certificate of Deposits (CDs) that earn higher interest. As the student makes a series of decisions in their academic and subsequent professional life, they need to make sound financial decisions and may look for resources online to assist them. An optimal decision needs to consider how the banking products interact with each other along with the changing needs of the student.
For users like this student, increasingly powerful LLM-based chatbots that have the potential to revolutionize the quality of
decision for personal finance are becoming available. LLMs have demonstrated tremendous potential across diverse domains <cit.>, such as natural language processing <cit.> and protein structure <cit.>, and have been claimed to show sparks of artificial general intelligence <cit.>. These models have been implemented in several applications, ranging from mental health assistants <cit.> to financial advisement <cit.>. In the finance domain, LLMs have been used to develop applications such as fraud detection, risk management, and financial forecasting <cit.>. They have been used to analyze financial data, predict stock prices, and generate automated reports. However, with the advent of recent models such as OpenAI's ChatGPT, Google's Bard, and BloombergGPT <cit.>, a comparative chatbot study is needed to evaluate their ability to be financial advisors. In this paper, we present an initial study of ChatGPT and Bard in providing personal decision-making for optimized outcomes.
It is widely known that LLMs based systems have unique limitations.
For example, they may struggle with common-sense reasoning tasks <cit.>, encounter challenges when handling symbols <cit.>, and are susceptible to hallucinations <cit.>.
With this work, we make the following contributions:
* identify a personal financial planning scenario involving a series of tasks (plans) and optimization of decisions.
* show how leading LLM-based chatbots perform in them and analyze their behavior.
* lay out challenges that future chatbots in this area should overcome to provide trusted financial recommendations.
We thus highlight the potential and limitations of current LLM-based systems - ChatGPT and Bard - in their role as financial advisors. We included all the queries posed and responses from both ChatGPT and Bard in our GitHub repository[https://github.com/ai4society/LLM-CaseStudies/tree/main/Finance] along with a few snapshots of the actual conversations.
§ PERSONAL FINANCE USE CASE
§.§ Setup: Tools and Procedure
§.§.§ Chatbots Tested
* ChatGPT: ChatGPT <cit.> is an LLM-based chatbot created by OpenAI that was trained on large amount of text data from the internet, including books and articles. ChatGPT is capable of answering questions, generating text and converse with users in a natural way. It can also learn from users and adapt to new information.
* Bard: Bard <cit.> is an LLM-based chatbot created by Google that was trained on large amount of text data and is capable of generating human-like text in response to user prompts and queries. Like ChatGPT, it is also capable of conversing with users about wide variety of topics in a natural way and adapt to new information.
§.§.§ Product Interaction Categories
Product interaction refers to interaction between different products like Credit Card (CC), Certificate of Deposit (CD) and Account Balance (AB).
Each product has different quantitative properties. For example, credit card due, limit line and billing cycle are some of the properties that would provide credit card information (not private information) of the user. Different properties pertaining to these products are:
* Purchase Amount (PA): It is the amount spent by the user on purchase of a product.
* Billing Cycle (BC): It is the billing cycle of user's credit card.
* Due Amount (DA): The amount that is due on the user's credit card for the specified billing cycle.
* Credit Line (CL): The maximum amount that user could spend using their credit card. If the amount spent exceeds this value, the credit card company could charge additional interest.
* Cashback Percentage (CP): The % of amount which will be returned to the user in the form of cashback on buying furniture using their credit card.
* Account Balance (AB): The amount of cash present in user's personal bank account.
* Annual Percentage Rate (APR): The APR is charged if there is due on the credit card after the due date. Some financial institutions choose to charge a late fee if the minimum due (MD) is not paid. It is calculated by the formula, Daily Period Rate (DPR) x Billing Cycle (in days) x Average Daily Balance (ADB).
* Certificate of Deposit Percentage (CDP): The % of interest accumulated on the cash deposited by the user in the form of CD.
Based on different combinations of these products, we classified the queries into 4 categories. These four categories along with the queries posed under each category, the variables used in each query and the constraints the chatbot has to take into consideration to make a sound recommendation are shown in Table <ref>. In the CC category, we considered a different dialect of English called African American Vernacular English (AAVE) and Telugu, one of the well-known languages from India, to observe how the chatbots handle queries in a different language or dialect.
§.§ Findings
In this subsection, we present the findings from the interesting (and sometimes insightful) conversations we had with Bard and ChatGPT.
§.§.§ Differences Between the Chatbots
Table <ref> shows the differences that were identified between Bard and ChatGPT when queries listed out in Table <ref> were asked. We compare these models on various criteria related to their performance in answering queries. The criteria include accuracy, utilization of user information, personalized suggestions, use of visual aids, bias in recommendations, provision of multiple response drafts, learning from mistakes, and understanding of different dialects and languages.
§.§.§ Error Categories
We identified some limitations / errors in the responses generated by both the chatbots and classified them into the following categories:
* Lack of Personalized Recommendations: When the agent makes a generalized recommendation without using all the information provided by the user, we consider this as lack of personalized recommendation.
* Mathematical Errors: We consider errors like rounding errors, calculation errors, etc. as mathematical errors.
* Perceptual Errors: When the agent misinterprets information given by the user or makes assumptions on unknown data, we consider these as perceptual errors.
* Grammatical Errors: We consider typos, grammatical errors, etc. as grammatical errors (we encountered these errors only in Telugu text generated by ChatGPT).
* Lack of Visual Aids: When the agent doesn't use visual aids like tables, graphs, etc. in its response, we consider these as lack of visual aids.
Table <ref> shows the percentage of queries for which the chatbots exhibited each of these errors. We also list out the individual query identifiers. Qi denotes the query identifier as previously defined (and also shown in Table <ref>). ABi and ACi refer to the corresponding Bard and ChatGPT responses respectively. 'i' denotes the identifier (number). Figures <ref> and <ref> show the response generated by Bard and ChatGPT chatbots respectively. For this one query, Bard made use of a table (though it misinterpreted user information) and ChatGPT did not.
§ DISCUSSION AND CONCLUSION
The application of language models in the finance industry has witnessed a surge in recent times due to their ability to process vast volumes of unstructured data and extract valuable insights. This paper delves into the performance of two prominent language models, Bard and ChatGPT, within the finance domain.
We also find the following challenges in evaluating LLM-based systems for finance domains:
* C1: Changing nature of answers for the same question. How does one create reference test cases since the answers change over time?
* C2: Inability of the chatbots to do numeric reasoning
* C3: Presenting results with easy to follow graphics.
* C4: Support for languages used by customers from different population groups. We considered AAVE - (African American Vernacular English) and Telugu, an Indian language spoken by nearly 100m people world-wide.
* C5: Evaluation the response of users from a diverse set of background. We only considered college students in this study.
C1 can be mitigated by carefully cataloging questions and system answers by identifiers that account for changing behavior over time. For C2, integration with numeric solvers like Wolfram may help <cit.> although this makes the systems non-learnable over time. For C3, different data presentation strategies need to be tried. For C4, the LLM models or the chatbots need to be enhanced. For C5, more experiments are needed with inputs carefully modeling the characteristics of the different user groups. These are just preliminary challenges and we expect them to grow as more researchers will try LLM-based systems in complex and diverse application scenarios.
While our study only comprised thirteen queries, we meticulously selected them to cover various categories of credit card finance. However, there exists ample scope for more extensive testing of these chatbots by expanding the number of queries under each category or including additional categories like student loans and stock purchases. By doing so, we can gain a better understanding of the efficacy of language models in different financial domains and improve their functionality in real-world scenarios.
|
http://arxiv.org/abs/2307.04813v2 | 20230710180256 | Cohomologies of tautological bundles of matroids | [
"Christopher Eur"
] | math.AG | [
"math.AG",
"math.CO"
] |
Functional PCA and Deep Neural Networks-based Bayesian Inverse Uncertainty Quantification with Transient Experimental Data
[
August 12, 2023
==========================================================================================================================
Tautological bundles of realizations of matroids were introduced in <cit.> as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they depend only on the matroid of the realization. As an application, we show that the log canonical bundle of a wonderful compactification of a hyperplane arrangement complement, in particular the moduli space of pointed rational curves ℳ_0,n, has vanishing higher cohomologies.
§ INTRODUCTION
Let E = {1, …, n} be a finite set, and an algebraically closed field.
Let L⊆^E be an r-dimensional linear subspace.
The matroid of L is the data of the set
{B ⊆ E : the composition L↪^E ↠^B is an isomorphism}
called the set of bases of . We say that L realizes the matroid .
We will explain notions from matroid theory as necessary, and refer to <cit.> for a general background.
In <cit.>, Berget, Spink, Tseng, and the author introduced tautological bundles of realizations of matroids as a new geometric model for studying matroids.
Let us recall the construction. Let T be the algebraic torus (^*)^E, which acts standardly on ^E via (t_1, …, t_n) · (x_1, …, x_n) = (t_1x_1, …, t_nx_n).
Let T = T/^* be its projectivization, i.e. its quotient by the diagonal. The image of t∈ T in T is denoted t.
The permutohedral variety X_E (Definition <ref>) is a smooth projective toric variety with the open dense torus T,
considered here as a T-variety.
Let 𝒪_X_E^⊕ E be the trivial vector bundle X_E ×^E whose T-equivariant structure is given by the inverse action of T on ^E, i.e. t ·_inv x = t^-1 x.
The tautological subbundle and quotient bundle of L⊆^E are the T-equivariant vector bundles _L and _L (respectively) on X_E defined by
_L = the T-equivariant subbundle of 𝒪_X_E^⊕ E whose fiber over t∈ T ⊂ X_E is t^-1L, and
_L = the T-equivariant quotient bundle of 𝒪_X_E^⊕ E whose fiber over t∈ T ⊂ X_E is ^E/t^-1L.
For well-definedness, see <cit.>. The authors of <cit.> showed that the K-classes [_L] and [_L] of these vector bundles depend only on the matroid . Moreover, by studying the Chern classes and sheaf Euler characteristics of the tautological bundles, both of which depend only on the K-class, they were able to unify, recover, and extend various recent developments in algebro-geometric studies of matroids.
Here, we ask: How do the sheaf cohomologies of _L and _L depend on the matroid ?
Our main results are as follows. We say that an element e∈ E is a coloop (resp. loop) of L if the decomposition ^E∖ e⊕^{e} of ^E decomposes L into L' ⊕ (resp. L' ⊕{0}) for some L' ⊆^E∖ n, or equivalently, if every basis of includes (resp. excludes) e.
Exterior powers of _L and _L have vanishing higher cohomologies, i.e.
H^i(^p _L) = 0 and H^i(^p _L) =0 for all i > 0 and p≥ 0,
and we have
∑_p ≥ 0 H^0(^p _L)u^p = (u+1)^|coloops()| and ∑_p≥ 0 H^0(^p _L) u^p = ∑_S⊆ E
S contains
a basis of u^|E|-|S|
where u is a formal variable.
The symmetric powers of _L have vanishing higher cohomologies, i.e.
H^i(Sym^p _L) = 0 for all i > 0 and p≥ 0,
and we have
∑_p ≥ 0 H^0(Sym^p_L) u^p = (1/1-u)^|E| - |coloops()|
where u is a formal variable.
In particular, the theorems imply that the cohomologies of exterior and symmetric powers of _L, and those of exterior powers of _L, depend only on the matroid that L realizes.
One may contrast this to the fact that exterior and symmetric powers of a realization L of are not in general determined by the matroid <cit.>.
Similar results for the dual vector bundles _L^∨ and _L^∨ can be obtained as follows.
Using the standard dot product on ^E, let us identify ^E ≃ (^E)^∨, so the trivial bundle (𝒪_X_E^⊕ E)^∨ is identified with X_E ×^E where T now acts standardly on ^E. Denoting L^⊥ for the space (^E/L)^∨ considered as a subspace of ^E ≃ (^E)^∨, we identify _L^∨ as the subbundle of (𝒪_X_E^⊕)^∨ whose fiber over t∈ T ⊂ X_E is t L^⊥.
The permutohedral variety X_E has the Cremona involution crem: X_E ∼→ X_E, induced by sending t∈ T to t^-1 (see for instance <cit.>).
Our description of _L^∨ above shows that _L^∨≃crem_L^⊥, and similarly one has _L^∨≃crem_L^⊥. In particular, symmetric and exterior powers of _L^∨ have vanishing higher cohomologies.
We prove Theorems <ref> and <ref> by establishing a “deletion-contraction” property for the tautological bundles, which we now describe.
For a subset S⊆ E, we denote
L\ S = the image of L under the projection ^E ↠^E∖ S, and
L/S = L ∩ (^E∖ S×{0}^S), considered as a subspace of ^E∖ S.
When S = {e} is a singleton we often omit the brackets to write L\ e and L/e, called the deletion and contraction of L by e, respectively.
For an element of E, say n∈ E for concreteness, there is a natural projection map f: X_E → X_E∖ n (Definition <ref>).
We show the following deletion-contraction property for the pushforward f_* of the tautological bundles of L.
For all p ≥ 0, we have
R^if_*(^p _L) = 0 for all i>0, and
f_*(^p _L) = ^p(_L/n⊕𝒪_X_E∖ n) if n is a coloop in
^p _L/n if n is not a coloop in .
Similarly, for all p ≥ 0, we have
R^if_*(^p _L) = 0 for all i>0, and
f_*(^p _L) = ^p(_L\ n⊕𝒪_X_E∖ n) if n is a loop in
^p_L/n if n is a coloop in
^p_L/n⊕^p-1_L∖ n if n is neither a loop nor a coloop in .
For all p≥ 0, we have
f_*Sym^p_L = Sym^p(_L/n⊕𝒪_X_E∖ n) if n a coloop
Sym^p_L/n if n not a coloop.
For all p ≥ 0, we have
R^if_*_L = 0 for all i > 0, and
f_*_L = Sym^p _L/n if n a coloop
Sym^p(_L/n⊕𝒪_X_E∖ n) if n not a coloop.
We induct on the cardinality of E, where the statements in the base case |E| = 1 are straightforward since X_E is a point in that case.
When |E|>1, for all p≥ 0, the Leray spectral sequence E_2^a,b = H^a(X_E∖ n, R^b f_* (^p_L)) satisfies E_2^a,b = 0 for all b >0 by Theorem <ref>, so that
H^i(X_E, ^p_L) ≃ H^i(X_E∖ n, f_* (^p_L)) for all i≥ 0.
Similar statements hold for ^p_L and Sym^p_L by the same argument.
From the formula for the pushforward f_* of these bundles in Theorems <ref> and <ref>, we conclude by induction hypothesis the vanishing of higher cohomologies.
Moreover, the formula for f_*(^p _L) implies that the polynomial g(L,u) = ∑_p ≥ 0 H^0(^p _L) u^p satisfies the relation
g(L,u) =
(u+1) · g(L/n, u) if n is a coloop in
g(L/n,u) if n is not a coloop in ,
hence g(L,u) = (u+1)^|coloops()|.
One similarly computes ∑_p ≥ 0 H^0(Sym^p _L) u^p.
Lastly, the formula for f_*(^p_L) implies that the polynomial h(L,u) = ∑_p≥ 0 H^0(^p _L) u^p satisfies
h(L,u) =
(u+1) · h(L\ n, u) if n is a loop in
h(L/n, u) if n is a coloop in
u · h(L\ n, u)+ h(L/n,u) if n is neither a loop nor a coloop in .
Feeding this into the recipe formula for deletion-contraction invariants <cit.> gives h(L,u) = u^|E| - rT_(1, 1+ u^-1) where T_ is the Tutte polynomial of , whose corank-nullity description [(2.13), loc. cit.] gives the desired formula for h(L,u).
Introduced in <cit.>, a wonderful compactification (Definition <ref>) of L⊆^E is a compactification W_L of L ∩ T that served as a key geometric model behind the Hodge theory of matroids <cit.>.
Its boundary ∂ W_L = W_L ∖ ( L ∩ T) is a simple normal crossings divisor.
We use Theorem <ref> to deduce the following.
The log canonical divisor K_W_L + ∂ W_L of a wonderful compactification W_L of L has vanishing higher cohomologies, i.e.
H^i(𝒪_W_L(K_W_L + ∂ W_L)) = 0 for all i > 0,
and we have
H^0(𝒪_W_L(K_W_L + ∂ W_L)) = ∑_S⊆ E
S contains
a basis of (-1)^|S|-r.
The moduli space ℳ_0,n of pointed rational curves arises as a wonderful compactification of a linear subspace whose matroid is the cyclic matroid of the complete graph on n-1 vertices <cit.> (see also <cit.>).
Hence, the corollary in particular implies that the log canonical divisor of ℳ_0,n has vanishing higher cohomologies, and recovers the classical result that H^0(𝒪_ℳ_0,n(K_ℳ_0,n + ∂ℳ_0,n)) = (n-2)!.
Corollary <ref> is the “dual version” of the following outstanding question in matroid theory due to Speyer about the anti log canonical divisor.
Speyer asked whether
(-1)^r-1χ( 𝒪_W_L(-K_W_L - ∂ W_L)) ≥ 0
for all L⊆^E such that its matroid is a connected matroid.[This is an equivalent formulation of the original question, which asked whether <cit.> holds over positive characteristic. We omit the details of the equivalence, which was communicated to the author by David Speyer.]
One can ask more strongly whether H^i( 𝒪_W_L(-K_W_L - ∂ W_L)) = 0 for all i<r-1, which implies the nonnegativity.
Speyer showed that the validity of this nonnegativity implies a bound on the f-vectors of matroidal subdivisions <cit.>.
Over characteristic zero, he proved the nonnegativity via Kawamata–Viehweg vanishing.
Corollary <ref> also implies that the cohomologies of the log canonical divisor on a wonderful compactification W_L depends only on the matroid of L.
In tropical geometry, for an arbitrary matroid possibly with no realization, instead of the wonderful compactifcation we have its tropical linear space <cit.>, which serves as building blocks of tropical manifolds.
With the theory of tropical vector bundles in its infancy, we ask:
Is there a theory of tropical line bundles and their sheaf cohomology on tropical manifolds such that it agrees with Corollary <ref>?
Related discussions and questions can be found in Section <ref>.
§.§ Previous works
When the characteristic of is zero, so that tools like resolution of singularities and Kawamata–Viehweg vanishing are available, parts of the results here have been established in previous literature <cit.>. For instance, <cit.> states that any Schur functor applied to _L^∨ has vanishing higher cohomologies. The vanishing higher cohomologies of the log canonical divisor of W_L (Corollary <ref>) is also immediate from Kawamata–Viehweg vanishing when one notes that ∂ W_L is big and nef.
The proofs of these previous results crucially depend on characteristic zero methods.
The vanishing statements here are established over fields of arbitrary characteristic by elementary methods.
§.§ Organization
In Section <ref>, we review permutohedral varieties, and detail the behavior of the projection map f: X_E → X_E\ n.
In Section <ref>, after some preparatory computations on ^1, we prove Theorems <ref> and <ref>.
In Section <ref>, we explain the application to wonderful compactifications.
In Section <ref>, we collect some questions.
§.§ Acknowledgements
The author thanks Andrew Berget, Alex Fink, Dhruv Ranganathan, and David Speyer for helpful conversations, and thanks Matt Larson for helpful conversations and comments on a preliminary draft of the paper.
The author is supported by US National Science Foundation (DMS-2001854 and DMS-2246518).
§ PERMUTOHEDRAL VARIETIES
For a subset S⊆ E, let _S be the sum of standard basis vectors ∑_i∈ S_i ∈^E, and let _S be its image in ^E/_E.
For background and conventions for polyhedral geometry and toric geometry, we refer to <cit.>.
An ordered set partition ℱ of E is a sequence (F_1, …, F_ℓ) of nonempty subsets of E that partition E.
The permutohedral fan Σ_E is the fan in ^E/_E consisting of cones
σ_ℱ = cone{_F_1, _F_1∪ F_1, …, _F_1 ∪⋯∪ F_ℓ}
for each ordered set partition ℱ = (F_1, …, F_ℓ) of E.
The permutohedral variety X_E is the (smooth projective) toric variety associated the fan Σ_E, considered as a rational fan over ^E/_E.
We identify the cocharacter lattice of T = (^*)^E with ^E, which identifies the cocharacter lattice of T with ^E/_E. This identifies the dense open torus of X_E with T, and so we treat X_E as a T-variety.
We refer to <cit.> for a background on torus-orbits of toric varieties, and fix the following notations.
For an ordered set partition ℱ = (F_1, ⋯, F_ℓ) of E, denote by
* p_ℱ = lim_t→ 0λ(t) the limit point in X_E where λ: ^* → T is the one-parameter map of any cocharacter λ∈^E / _E in the relative interior relint(σ_ℱ) of σ_ℱ,
* O_ℱ the T-orbit of corresponding to σ_ℱ, i.e. the orbit T· p_ℱ, and
* Z_ℱ the closure of the T-orbit O_ℱ.
We now describe the map f: X_E → X_E∖ n. First, note that the projection map f: ^E →^E∖ n induces a map of fans Σ_E →Σ_E∖ n.
We record the following observation, whose verification is straightforward and is omitted.
Let ℱ =(F_1, …, F_ℓ) be an ordered set partition of E∖ n.
The inverse image of the cone σ_ℱ∈Σ_E∖ n under the map Σ_E →Σ_E∖ n consists of cones in Σ_E corresponding to the following two kinds of ordered set partitions of E:
* For 1≤ i ≤ℓ+1, let ℱ^i = (F_1, …, F_i-1, n, F_i, …, F_ℓ).
* For 1≤ i ≤ℓ, let ℱ(i) = (F_1, …, F_i-1, F_i ∪ n, F_i+1, …, F_ℓ).
Note that σ_ℱ^i∩σ_ℱ^i+1 = σ_ℱ(i).
Let f: X_E → X_E∖ n be the toric map associated to the map of fans Σ_E →Σ_E∖ n induced by the projection map f: ^E →^E∖ n.
Translating the polyhedral statement in Proposition <ref> to toric geometry gives the following.
The map f: X_E → X_E∖ n is a flat and projective map whose fibers are chains of rational curves. More specifically, for any (t,1)∈ T where t∈ (^*)^E∖ n and an ordered set partition ℱ = (F_1,…, F_ℓ) of E∖ n, we have that the fiber
f^-1(t · p_ℱ) = ⋃_i=1^ℓ C(t,i)
where
C(t, i) = {(t,1)· p_ℱ^i}⊔{(t,1) · p_ℱ^i+1}⊔{(t,t_n)· p_ℱ(i) : t_n ∈^*}
= {0}⊔{∞}⊔^* ≃^1.
One may also deduce the first statement of the corollary by noting that X_E is the Losev-Manin space <cit.>, which is a particular Hassett space of rational curves with weighted markings, and that the map f is the universal curve map.
For proving Thereoms <ref> and <ref>, Corollary <ref> primes us to use Grauert's theorem, which we recall here for convenience <cit.>:
If φ: X → Y is projective and ℱ∈Coh(X) is flat over Y such that H^i(X_y, ℱ_y) is constant over the fibers X_y = φ^-1(y) of y∈ Y, then R^iφ_*ℱ is a vector bundle on Y whose fiber at y∈ Y is H^i(X_y, ℱ_y).
In particular, if φ is itself flat, then because the Euler characteristic χ(X_y, ℰ_y) is constant for a vector bundle ℰ on X, the pushforward φ_* ℰ is a vector bundle on Y with fibers H^0(X_y, ℰ_y) if H^i(X_y, ℰ_y) = 0 for all i>0 and y∈ Y.
We conclude this section by discussing the behavior of the tautological bundles of L on the fibers of the map f.
Let us write L|S = L\(E\ S) for a subset S ⊆ E. For an ordered set partition ℱ = (F_1, …, F_ℓ) of E, let L_ℱ be the linear subspace
L_ℱ= L | F_1 ⊕ L|(F_1∪ F_2)/F_1 ⊕⋯⊕ L|(F_1 ∪⋯∪ F_ℓ-1)/(F_1 ∪⋯∪ F_ℓ-2) ⊕ L/(F_1 ∪⋯∪ F_ℓ-1)
of ^F_1⊕^F_2⊕⋯⊕^F_ℓ = ^E. We will need the following fact, which follows from <cit.>.
The restriction _L|_Z_ℱ (resp. _L|_Z_ℱ) is the unique T-equivariant subbundle (resp. quotient bundle) of (𝒪_X_E^⊕ E)|_Z_ℱ = Z_ℱ×^E whose fiber over p_ℱ is L_ℱ (resp. ^E/L_ℱ).
The tautological bundles of L restricted to a fiber of f are simple in the following sense.
Let notations be as in Corollary <ref>.
As a subbundle (resp. quotient bundle) of the trivial bundle 𝒪^⊕ E, the fibers of the restricted bundle _L|_f^-1(t· p_ℱ) (resp. _L|_f^-1(t · p_ℱ)) are constant along all ^1-components C(t,i) if n is a loop or a coloop of L, and non-constant at exactly one component if n is neither a loop nor a coloop of L.
If n is a loop or a coloop of L, then (t,t_n) · L = (t,1) · L for all t_n ∈^*, and for any S⊆ E∖ n, the element n is again a loop or a coloop of L\ S and L/S.
Thus, in this case n is a loop or a coloop of L_ℱ(i) for all 1≤ i≤ℓ, and so Lemma <ref> and Corollary <ref> together imply that _L and _L are constant along each component C(t,i).
Suppose now that n is neither a loop nor a coloop of L. We need show that n is neither a loop nor a coloop in L_ℱ(k) for exactly one 1≤ k ≤ℓ.
For this end, we will use the following property of matroids that follows from its greedy algorithm structure (see <cit.>):
For an ordered partition ℱ' = (F'_1, …, F'_ℓ) of E, let w_ℱ': E → be any weighting such that w_ℱ' is constant on each F'_i and w(f'_i) > w(f'_j) whenever f'_i∈ F'_i and f'_j∈ F'_j with i<j. Then, the w_ℱ'-maximal bases of the matroid of L are the bases of the matroid of L_ℱ.
Now, note that if n is neither a loop nor a coloop in L_ℱ(k), then it is a coloop in L_ℱ^k and a loop in L_ℱ^k+1 by construction. In particular, every w_ℱ^k-maximal bases of includes n, and every w_ℱ^k+1-maximal bases of excludes n. Thus, every w_ℱ(i)-maximal bases of must include n if i<k, and must exclude n if i>k.
§ PROOF OF THEOREMS <REF> AND <REF>
Since _L or _L along a fiber of the map f is non-constant on at most one ^1-component of the fiber (Proposition <ref>), we begin with preparatory observations for vector bundles on ^1.
Consider ^1 = {0}⊔{∞}⊔^* as a ^*-toric variety, and let 𝒪_^1^⊕ E be the trivial vector bundle ^1 ×^E where ^* acts on ^E by t· (x_1, …, x_n-1, x_n) = (x_1, …, x_n-1, t^-1 x_n).
We write ^{n} for last coordinate of ^E with the inverse standard action of ^*.
For a subspace L⊆^E, let _L' and _L' be the ^*-equivariant sub and quotient bundles of 𝒪_^1^⊕ E, respectively, fitting into a short exact sequence 0→_L' →𝒪_^1^E →_L' → 0 such that its fiber over the identity of ^* is 0→ L →^E →^E/L → 0.
Note that if n is a loop or a coloop of L, so that L = L' ⊕ L|n ⊆^E∖ n⊕^{n}, we have that '_L ≃𝒪_^1⊗ (L' ⊕ L|n) is a trivial bundle, and similarly for '_L.
We have short exact sequences
0→'_L/n ⊕ 0→'_L →ℒ_→ 0 and 0→ℒ_→'_L →'_L\ n ⊕ L|n→ 0,
where ℒ_ and ℒ_ are ^*-equivariant line bundles (or zero) defined by
ℒ_ = the ^*-equivariant subbundle of
𝒪_^1⊗( (^E∖ n / (L/n)) ⊕^{n})
whose fiber at identity is L/ (L/n ⊕ 0)≃
0 if n a loop
𝒪_^1⊗^{n} if n a coloop
𝒪_^1(-1) if n neither
and
ℒ_ = the ^*-equivariant quotient bundle of
𝒪_^1⊗( (L\ n)/(L/n) ⊕ L|n )
whose fiber at identity is the quotient by L/(L/n ⊕ 0)≃
0 if n a loop
0 if n a coloop
𝒪_^1(1) if n neither.
If n is a loop or a coloop, so that '_L and '_L are trivial bundles, the statements of the lemma are immediate. Let us now assume that n is neither a loop nor a coloop.
For all t∈^* ⊂^1, we have L/n ⊕ 0 = (t· L) ∩ (^E∖ n⊕ 0) ↪ t· L, and at the boundaries we have lim_t→ 0 t· L = L/ n ⊕ 0 and lim_t→∞t· L = L\ n ⊕ 0.
Since L/n ⊆ L\ n, we thus have an injective map of vector bundles '_L⊕ 0↪'_L.
We hence obtain the following diagram of commuting short exact sequences
0 [d] 0 [d]
0 [r] '_L/n ⊕ 0[r,equal][d] 𝒪_^1⊗ (L/n ⊕ 0) [r] [d] 0 [d]
0 [r] '_L [r] [d] 𝒪_^1⊗^E [r][d] '_L [r][d,equal] 0
0 [r] ℒ_[r] [d] 𝒪_^1⊗(^E∖ n/(L/n) ⊕^{n}) [r][d] '_L [r][d] 0
0 0 0
by starting with the first two rows and then applying the snake lemma.
We have the desired short exact sequence for '_L.
Now, over the identity point of ^* ⊂^1, the ^*-equivariant embedding ℒ_↪𝒪_^1⊗( (^E∖ n / (L/n)) ⊕^{n}) is L/(L/n⊕ 0) ↪ (^E∖ n / (L/n)) ⊕^{n}.
Because (L\ n)/(L/n) ⊕ L|n is the direct sum of the projections of L/(L/n ⊕ 0) to the two direct summands of (^E∖ n / (L/n)) ⊕^{n}, we have that
ℒ_ in fact embeds in 𝒪_^1⊗( (L\ n)/(L/n) ⊕ L|n ).
In other words, ℒ_ is the pullback of the tautological subbundle of ^1 ≃((L\ n)/(L/n) ⊕ L|n) where the isomorphism ^1 ∼→((L\ n)/(L/n) ⊕ L|n) is defined by
^*∋ t ↦the image in (L\ n)/(L/n) ⊕ L|n of t· L / (L/n ⊕ 0).
The Euler sequence 0→𝒪_^1(-1) →𝒪_^1^2 →𝒪_^1(1) → 0 on ^1 then becomes
0 →ℒ_→𝒪_^1⊗( (L\ n)/(L/n) ⊕ L|n ) →ℒ_→ 0,
which defines the line bundle ℒ_, and proves the statements about the isomorphism types of ℒ_ and ℒ_.
Lastly, we have the following commuting diagram of short exact sequences
0 [d] 0 [d] 0[d]
0 [r] ℒ_[r][d,equal] 𝒪_^1⊗( (L\ n)/(L/n) ⊕ L|n) [r] [d] ℒ_[d][r] 0
0 [r] ℒ_[r] [d] 𝒪_^1⊗(^E∖ n/(L/n) ⊕^{n})[r][d] '_L [r][d] 0
0 [r] 𝒪_^1⊗(^E∖ n/(L\ n) ⊕^{n}/(L|n) ) [r,equal] [d] '_L\ n ⊕ L|n[r][d] 0
0 0
by starting with the first two columns and then applying the snake lemma. The desired short exact sequence for '_L follows.
The two short exact sequences in <Ref> split: For the first sequence, it follows from the possible isomorphism types of ℒ_ that Ext^1_^1(ℒ_, '_L/n ⊕ 0) ≃ H^1(^1, ℒ_^∨⊗'_L/n ⊕ 0) = 0, and similarly for the second sequence.
The lemma implies the following about the cohomologies of exterior powers of '_L and '_L.
For all p≥ 0, we have H^1(⋀^p '_L) = 0 and H^1(⋀^p '_L) = 0, and we have natural isomorphisms
H^0(^p '_L) ≃^p (L/n ⊕) if n a coloop,
^p(L/n) if n not a coloop,
and
H^0(^p '_L) ≃^p(^E/(L/n ⊕ 0)) if n a loop
^p ( ^E / (L/n ⊕)) if n a coloop
^p ( ^E∖ n/(L/n) ) ⊕^p-1(^E∖ n/(L\ n)) if n neither.
By standard multilinear algebra (e.g. <cit.>), applying exterior powers to the short exact sequences of <Ref> yields short exact sequences
†
0 →^p '_L/n ⊕ 0→ ^p '_L →^p-1'_L/n⊕ 0⊗ℒ_→ 0 and
0→ℒ_⊗^p-1'_L\ n ⊕ L|n→ ^p '_L →^p '_L\ n ⊕ L|n→ 0
for all p ≥ 0.
In the resulting long exact sequences of cohomologies, we have H^1(^p '_L)=H^1(^p '_L) = 0 because of the descriptions of ℒ_ and ℒ_ in <Ref> and H^1(𝒪_^1(-1)) = H^1(𝒪_^1) = H^1(𝒪_^1(1)) = 0, keeping in mind that '_L/n ⊕ 0 and '_L\ n ⊕ L|n are trivial bundles.
As H^1's vanish, note that applying H^0 yields the short exact sequences of vector spaces.
We now treat the statements about H^0. When n is a loop or a coloop, the desired follows since all vector bundles involved are trivial in such case. So, assume now that n is neither a loop nor a coloop.
The statement for H^0(^p '_L) follows since H^0(𝒪_^1(-1)) = 0.
For H^0(^p '_L), note first that H^0(ℒ) = V for any V ≃^2 and ℒ≃𝒪_^1(1) such that
0 →𝒪_^1(-1) →𝒪_^1⊗ V →ℒ→ 0.
Applying this with V = (L\ n)/(L/n) ⊕ L|n and ℒ = ℒ_, we obtain that the short exact sequence from applying H^0 to the second sequence in (<ref>) with p = 1 is natural isomorphic to
0→ (L\ n)/(L/n) ⊕ L|n →^E∖ n/(L/n) ⊕^{n}→^E∖ n/(L\ n) ⊕^{n}/(L|n) → 0,
(i.e. the middle column of the second diagram in the proof of <Ref>) which is the direct sum of two sequences
0→ (L\ n)/(L/n) →^E∖ n/(L/n) →^E∖ n/(L\ n) → 0 and 0→→→ 0 → 0.
In general, applying H^0 for p≥ 1 yields the short exact sequence which is the direct sum of
0 → (L\ n)/(L/n) ⊗^p-1^E∖ n/(L\ n) → ^p ^E∖ n/(L/n) →^p^E∖ n/(L\ n) → 0 and
0 →⊗^p-1^E∖ n/(L\ n) → ^p-1^E∖ n/(L\ n) → 0 → 0.
The desired statement for H^0(^p'_L) follows.
We also use <Ref> to deduce the following symmetric powers analogue of <Ref>. Note that for any V ≃^2 and ℒ≃𝒪_^1(1) fitting into 0 →𝒪_^1(-1) →𝒪_^1⊗ V →ℒ→ 0, we have natural isomorphisms
H^0(ℒ^⊗ p) ≃Sym^p V and H^1(𝒪_^1(-p-2)) ≃ V^∨⊗Sym^p V^∨ for all p≥ 0.
For all p ≥ 0, we have a natural isomorphism
H^0(Sym^p 𝒮'_L) ≃Sym^p(L/n ⊕) if n a coloop
Sym^p(L/n) if n not a coloop.
When n is a loop or a coloop, we have H^1(Sym^p '_L) = 0, and when n is neither, we have a filtration H^1(Sym^p '_L) = F_0 ⊇ F_1 ⊇⋯⊇ F_p-2⊇ F_p-1 = 0 such that
F_i/F_i+1≃(((L\ n)/(L/n) ⊕ L|n)^∨)^⊗ p-1-i⊗Sym^p-2-i((L\ n)/(L/n) ⊕ L|n) ⊗Sym^i(L/n ⊕ 0)
for all 0≤ i ≤ p-2. Similarly, for all p≥ 0, we have H^1(Sym^p '_L) = 0, and we have a natural isomorphism
H^0(Sym^p'_L) ≃Sym^p (^E∖ n/(L/n)) if n a coloop
Sym^p (^E∖ n/(L/n)⊕) if n not a coloop.
When n is a loop or a coloop, the bundles '_L and '_L are trivial, and the claimed statements follow easily. Suppose n is neither a loop or coloop now. For the statements about Sym^p '_L, we first note that the short exact sequence 0→'_L/n ⊕ 0→'_L →ℒ_→ 0 in <Ref>, along with some multilinear algebra (e.g. <cit.>), gives a filtration Sym^p'_L = ℱ_0 ⊇ℱ_1 ⊇⋯⊇ℱ_p ⊇ℱ_p+1 = 0 with ℱ_i / ℱ_i+1≃Sym^i '_L/n ⊕ 0⊗ℒ_^⊗ p-i. In the long exact sequences
0 → H^0(ℱ_i+1) → H^0(ℱ_i) → H^0(ℱ_i / ℱ_i+1) → H^1(ℱ_i+1) → H^1(ℱ_i) → H^1(ℱ_i / ℱ_i+1) → 0,
we have H^0(ℱ_i / ℱ_i+1) = 0 if i < p since ℒ_S ≃𝒪_^1(-1), and thus we have H^0(Sym^p '_L) ≃ H^0(ℱ_p / ℱ_p+1) = Sym^p (L/n). The filtration for H^1 also follows since the H^1's form a short exact sequence for each i, and
H^1(ℱ_i / ℱ_i+1)
≃((L\ n)/(L/n) ⊕ L|n)^∨⊗Sym^p-2-i((L\ n)/(L/n) ⊕ L|n)^∨⊗Sym^i(L/n ⊕ 0)
≃(((L\ n)/(L/n) ⊕ L|n)^∨)^⊗ p-1-i⊗Sym^p-2-i((L\ n)/(L/n) ⊕ L|n) ⊗Sym^i(L/n ⊕ 0).
For the statements about Sym^p'_L, we similarly have from 0→ℒ_→'_L →'_L\ n⊕ L|n→ 0 a filtration
Sym^p'_L = ℱ_0 ⊇ℱ_1 ⊇⋯⊇ℱ_p ⊇ℱ_p+1 = 0 with ℱ_i / ℱ_i+1≃ℒ_^⊗ i⊗Sym^p-i'_L/n ⊕ 0. Note that ℒ_≃𝒪_^1(1), so all H^1(ℱ_i / ℱ_i+1) vanish, and hence H^1(ℱ_i) = 0 for all i as well. Now, the resulting short exact sequences
0 → H^0(ℱ_i+1) → H^0(ℱ_i) → H^0(ℱ_i / ℱ_i+1) → 0
give a filtration of H^0(Sym^p'_L) whose successive quotients are
Sym^i((L\ n)/(L/n) ⊕ L|n) ⊗Sym^p-i(^E∖ n/(L\ n) ⊕/(L|n)).
This is exactly the filtration of Sym^p(^E∖ n/(L/n) ⊕) arising from the short exact sequence 0→ (L\ n)/(L/n) ⊕ L|n →^E∖ n/(L/n) ⊕→^E∖ n/(L\ n) ⊕/(L|n) → 0.
We are now ready to prove the main theorems.
We prove the statement for ^p_L. The statements for ^p_L and Sym^p_L are proved similarly.
We will compute the cohomologies of the restriction of ^p_L to a fiber of f: X_E → X_E∖ n, and then apply Grauert's theorem.
A point y∈ X_E∖ n is of the form t· p_ℱ for some t∈ (^*)^E∖ n and an ordered set partition ℱ = (F_1, …, F_ℓ) of E∖ n.
To reduce notational burden (such as (t· L)_ℱ(i)), we assume without loss of generality that t is the identity.
By Corollary <ref>, the restriction ^p _L|_f^-1(y) is constant on all ^1-components of f^-1(y) except possibly on C(t,k) for some 1≤ k ≤ℓ (if no such then fix an arbitrary 1≤ k ≤ℓ). By Lemma <ref>, the restriction ^p _L|_C(t,k) to C(t,k) ≃^1 is isomorphic to ^p'_L_ℱ(k), where '_L_ℱ(k) is as defined in Definition <ref>.
Hence we have H^i(^p _L|_f^-1(y)) ≃ H^i(^p'_L_ℱ(k)) for all i.
Now, Proposition <ref> implies that H^i(^p'_L_ℱ(k)) = 0, and moreover, when we note that L_ℱ(i)/n = (L/n)_ℱ for any 1≤ i ≤ℓ, the proposition gives
H^0(^p '_L_ℱ(k)) ≃^p ((L/n)_ℱ⊕) if n a coloop,
^p((L/n)_ℱ) if n not a coloop.
The desired statements for ^p _L now follow from Grauert's theorem.
Let ℰ be any globally generated vector bundle on X_E, such as ^2_L ⊗Sym^3_L.
Its restriction to a fiber of f has no higher cohomology since it is a globally generated bundle on a chain of ^1, and hence f_*ℰ is a vector bundle with R^i f_*ℰ = 0 for all i>0. However, without sufficiently explicit description of f_*ℰ like the one in Theorem <ref>, one cannot conclude much about H^i(ℰ). We currently do not have an analogue of Proposition <ref> for arbitrary Schur/Weyl functors applied to '_L, and thus our treatment is restricted to exterior and symmetric powers.
§ WONDERFUL COMPACTIFICATIONS
We begin with a review of wonderful compactifications introduced in <cit.>.
To avoid trivialities, throughout this section we assume L⊆^E to be loopless, so that the intersection L ∩ T is nonempty.
Let 𝒜 be the arrangement of hyperplanes { L ∩ H_e : e∈ E} where H_e is the e-th coordinate hyperplane of (^E). Notice that L ∩ T = L ∖ (⋃𝒜).
Let 𝒫 be the poset whose elements are the linear subvarieties P⊆ L that arise as intersections of hyperplanes in 𝒜, with partial ordering G≤ G' given by reverse inclusion P ⊇ P'. The poset 𝒫 has the top and bottom elements 1̂ = ∅ and 0̂ = L, respectively.
In matroid theory, this poset is known as the lattice of flats of the matroid of L.
A building set 𝒢 is a subset of 𝒫∖{0̂, 1̂} such that for every P ∈𝒫∖{0̂ , 1̂}, the set max𝒢_≤ P of maximal elements of 𝒢 in the interval [0̂, P] satisfies
[0̂, P] ≃∏_G∈max𝒢_≤ P [0̂, G].
The wonderful compactification of L with building set 𝒢 is the variety W_L^𝒢 obtained from L by sequentially blowing-up the linear subvarieties of L in 𝒢, starting with the smallest dimensional ones to the largest.
The boundary ∂ W_L^𝒢 = W_L ∖ ( L ∩ T) of W_L^𝒢 is a simple normal crossings divisor <cit.>.
A stratum in the boundary is the intersection of a subset of the irreducible components of the boundary divisor ∂ W_L^𝒢, which is necessarily smooth.
Note that 𝒫∖{0̂, 1̂} itself is a building set, in which case we abuse notation to denote W_L^𝒫=W_L^𝒫∖{0̂, 1̂}.
We now recall as a lemma two facts from the literature to prepare for the proof of Corollary <ref>, which stated that the log canonical divisor K_W_L^𝒢 + ∂ W_L^𝒢 has vanishing higher cohomology.
Let notations be as above.
* If 𝒢 and ℋ are building sets on 𝒫 such that 𝒢⊇ℋ, then there exists a sequence of building sets (𝒢 = 𝒢_1, 𝒢_2, ⋯, 𝒢_ℓ = ℋ) such that W_L^𝒢_i is the blow-up of a stratum in the boundary of W_L^𝒢_i+1 for each i = 1, …, ℓ-1.
* The variety W_L^𝒫 is isomorphic to the vanishing locus in X_E of a global section of _L. Under this isomorphism, we have 𝒪_W_L^𝒫 (K_W_L^𝒫 + ∂ W_L^𝒫) ≃_L|_W_L^𝒫.
(1) is a translation of <cit.> into geometric language under the dictionary provided in <cit.> between the boundary strata structure of W_L^𝒢 and the simplicial complex known as the nested complex of 𝒢.
The first statement of (2) is <cit.>.
The second statement follows from <cit.>, which implies that 𝒯_W_L^𝒫(-log∂ W_L^𝒫) ≃_L|_W_L^𝒫, and _L^∨≃_L from 0→_L →𝒪_X_E^⊕ E→_L → 0.
We first claim that H^i(𝒪_W_L^𝒢(K_W_L^𝒢 + ∂ W_L^𝒢)) ≃ H^i(𝒪_W_L^𝒫(K_W_L^𝒫 + ∂ W_L^𝒫)) for any building set 𝒢.
Let π: W_L^𝒫→ W_L^𝒢 be the composition of blow-down maps given by Lemma <ref>(1).
Recall that, for any blow-up φ: X→ X of a smooth subvariety Y in a smooth variety X, we have K_X = φ^* K_W + (codim_X(Y) -1) E where E is the exceptional divisor <cit.>.
Applying this to each of the blow-down maps making up π, we find π^*(K_W_L^𝒢 + ∂ W_L^𝒢) =
K_W_L^𝒫 + ∂ W_L^𝒫.
Moreover, Lemma <ref>(1) further implies that π_* 𝒪_W_L^𝒫 = 𝒪_W_L^𝒢 and R^iπ_* 𝒪_W_L^𝒫 = 0 for all i>0 <cit.>. Our claim now follows from the projection formula.
To finish, the first statement of Lemma <ref>(2) implies that we have the Koszul resolution
0→_L^∨→⋯→^2 _L^∨→_L^∨→𝒪_X_E→𝒪_W_L^𝒫→ 0.
Since ℰ⊗^i ℰ^∨≃^rank(ℰ) - iℰ for a vector bundle ℰ, twisting the above resolution by _L and noting the second statement of Lemma <ref>(2) gives the resolution
0→𝒪_X_E→_L →^2 _L →⋯→_L→𝒪_W_L^𝒫 (K_W_L^𝒫 + ∂ W_L^𝒫) → 0.
Applying Theorem <ref> now yields the desired corollary by standard homological algebra <cit.>.
§ QUESTIONS
A broader theme behind Question <ref> is to ask: Which sheaf theoretic properties of realizations of matroids extend to all matroids?
We collect some related observations and questions.
We will now assume familiarity with matroid theory.
As in the previous section, we suppose L⊆^E to be loopless to avoid trivialities.
For an arbitrary not necessarily realizable matroid , there are K-classes [_] and [_] in the Grothendieck K-ring of vector bundles on X_E such that [_] = [_L] and [_] = [_L] whenever has a realization L⊆^E <cit.>.
Let us denote by D_-P() = c_1(_), the first Chern class of [_]. See <cit.> for an explanation of the notation D_-P().
If has a realization L, Lemma <ref>(2) states that the log canonical divisor of W_L^𝒫 is D_-P()|_W_L^𝒫.
Even if is not realizable, we may consider the line bundle 𝒪_X_E(D_-P()).
§.§ Immaculate line bundles
We have the following variation of Corollary <ref>.
Let L'⊆^E be a subspace containing L such that L' = L+1, and let ' be the matroid of L'. Then, the line bundle 𝒪_W_L^𝒫(D_-P() - D_-P(')) on W_L^𝒫 satisfies
H^i(𝒪_W_L^𝒫(D_-P() - D_-P('))) = 0 for all i>0, and
H^0(𝒪_W_L^𝒫(D_-P() - D_-P('))) = 1 if ' has loops
0 if ' is loopless.
In particular, the line bundle 𝒪_W_L^𝒫(D_-P() - D_-P(')) on W_L^𝒫 is immaculate, i.e. has no nonzero cohomologies, if ' is loopless.
By construction, from L'⊂ L ⊆^E we have the surjective map _L'→_L.
Let ℒ_', be the kernel, so that we have the short exact sequence
0 →ℒ_',→_L'→_L → 0.
By taking of the sequence, we see that ℒ_',≃𝒪_X_E(D_-P(') - D_-P()). Applying duality and exterior power, we obtain for each p≥ 1 a short exact sequence
0 →^p _L^∨→^p _L'^∨→^p-1_L^∨⊗ℒ^∨_',→ 0.
Applying Theorem <ref> and Remark <ref> to the long exact sequence of cohomologies, we thus obtain H^i(^p-1_L^∨⊗ℒ^∨_',) = 0 for all i>0 and p ≥ 1. Moreover, we have
H^0(^p-1_L^∨⊗ℒ^∨_',) = H^0(^p _L'^∨) - H^0(^p _L^∨) = |loops(')|p - |loops()|p,
where for the last equality we used that crem_L^∨≃_L^⊥, and L^⊥ realizes the dual matroid ^⊥ whose coloops correspond to the loops of the original matroid . To finish, recall the Koszul resolution ^∙_L^∨→𝒪_W_L^𝒫→ 0 from the proof of Corollary <ref>. Twisting the resolution by ℒ^∨_', and taking cohomology, keeping in mind standard homological algebra <cit.>, one obtains the desired result.
An elementary matroid quotient ↠' consists of two matroids and ' whose ranks differ by 1 such that every flat of ' is a flat of . It is realizable if there is a flag of linear subspaces L' ⊆ L ⊆^E such that L' and L respectively realize ' and .
In light of the corollary above, we ask the following question.
For any elementary matroid quotient ↠', not necessarily realizable, is the line bundle 𝒪_W_L^𝒫(D_-P() - D_-P(')) on W_L^𝒫 immaculate, i.e. has no nonzero cohomologies, if ' is loopless?
Is there a theory of tropical line bundles and their sheaf cohomology on tropical manifolds such that it agrees with the above corollary?
§.§ Log canonical image
We conclude with a discussion of the log canonical image of a wonderful compactification of L.
The line bundle _L ≃𝒪_X_E(D_-P()) is globally generated, with torus-invariant sections in bijection with the bases of (see <cit.> and <cit.>).
We may thus consider the embedded projective variety
X_L = the closure of the image of L ∩ T under the map φ: X_E →(H^0(_L)).
This variety X_L is also known as Kapranov's visible contour <cit.>. When is connected, as we shall assume from now, the variety X_L is the log canonical model of L ∩ T with (étale locally) toric singularities <cit.>. For a building set 𝒢, the map W_L^𝒢→ X_L given by the log canonical bundle of W_L^𝒢 is a (étale locally) toric resolution of singularities,
and thus H^i(𝒪_X_L(ℓ)) ≃ H^i(_L|_W_L^𝒫^⊗ℓ) for all ℓ∈.
In particular, applying Theorem <ref> and Remark <ref> to the Koszul complexes in the proof of Corollary <ref> yields the following.
The ideal sheaf ℐ_X_L satisfies
H^i(ℐ_X_L) = 0 and H^i(ℐ_X_L(1)) = 0 for all i>0,
and hence H^i(𝒪_X_L) =0 and H^i(𝒪_X_L(1)) = 0 for all i>0.
Over characteristic zero, applying <cit.> further implies that H^i(ℐ_X_L(ℓ)) = 0 and H^i(𝒪_X_L(ℓ)) = 0 for all i>0 and ℓ≥ 0.
Moreover, over characteristic zero, by Kawamata–Viehweg vanishing we have H^i(𝒪_X_L(-ℓ)) = 0 for all i< X_L and ℓ >0.
We thus ask the following, part of which is a strengthening of Speyer's question in Remark <ref>.
Suppose has positive characteristic. Is H^i(ℐ_X_L(ℓ)) = 0 and H^i(𝒪_X_L(ℓ))=0 for all i>0 and ℓ≥ 0? Is H^i(𝒪_X_L(-ℓ)) = 0 for all i< X_L and ℓ >0? In particular, is the embedded variety X_L projectively normal and/or arithmetically Cohen-Macaulay?[Matt Larson also proposed this question during the Banff workshop “Algebraic Aspects of Matroid Theory.”]
Given a fixed total order of E, one can show that the restrictions to W_L^𝒫 of the torus-invariant sections of _L are spanned by those that correspond to the nbc-bases of the matroid .
One can moreover show that they not only span but also form a basis of H^0(𝒪_X_L(1)), by using Corollary <ref> and by noting that the quantity
H^0(𝒪_X_L(1)) = H^0 (𝒪_W_L^𝒫 (K_W_L^𝒫 + ∂ W_L^𝒫) ) = ∑_S⊆ E
S contains
a basis of (-1)^|S|-r
is the Möbius invariant T_(1,0) of , which equals the number of nbc-bases of <cit.>.
alpha
|
http://arxiv.org/abs/2307.05826v1 | 20230711221659 | Detection, Instance Segmentation, and Classification for Astronomical Surveys with Deep Learning (DeepDISC): Detectron2 Implementation and Demonstration with Hyper Suprime-Cam Data | [
"G. M. Merz",
"Y. Liu",
"C. J. Burke",
"P. D. Aleo",
"X. Liu",
"M. C. Kind",
"V. Kindratenko",
"Y. Liu"
] | astro-ph.IM | [
"astro-ph.IM"
] |
firstpage–lastpage
SnakeSynth: New Interactions for Generative Audio Synthesis
Eric Easthope
University of British Columbia
Vancouver, British Columbia, Canada
[email protected]
August 12, 2023
===============================================================================================================================
The next generation of wide-field deep astronomical surveys will deliver unprecedented amounts of images through the 2020s and beyond. As both the sensitivity and depth of observations increase, more blended sources will be detected. This reality can lead to measurement biases that contaminate key astronomical inferences. We implement new deep learning models available through Facebook AI Research's Detectron2 repository to perform the simultaneous tasks of object identification, deblending, and classification on large multi-band coadds from the Hyper Suprime-Cam (HSC). We use existing detection/deblending codes and classification methods to train a suite of deep neural networks, including state-of-the-art transformers. Once trained, we find that transformers outperform traditional convolutional neural networks and are more robust to different contrast scalings. Transformers are able to detect and deblend objects closely matching the ground truth, achieving a median bounding box Intersection over Union of 0.99. Using high quality class labels from the Hubble Space Telescope, we find that the best-performing networks can classify galaxies with near 100% completeness and purity across the whole test sample and classify stars above 60% completeness and 80% purity out to HSC i-band magnitudes of 25 mag. This framework can be extended to other upcoming deep surveys such as the Legacy Survey of Space and Time and those with the Roman Space Telescope to enable fast source detection and measurement. Our code, DeepDISC, is publicly available at <https://github.com/grantmerz/deepdisc>.
techniques: image processing – methods: data analysis – galaxies: general – Sky Surveys
§ INTRODUCTION
The rise of machine learning/artificial intelligence has allowed for rapid advancement in many image analysis tasks to the benefit of researchers who wish to work with large sets of imaging data. This active field of study, known as computer vision, has led to developments in many disciplines including medical imaging <cit.>, urban planning <cit.>, autonomous systems <cit.> and more.
Tasks such as image compression, inpainting, object classification and detection, and many others have been extensively studied. Astronomy is no exception, and many methods that utilize deep learning have been applied to simulations and real survey data for tasks such as object detection, star/galaxy classification, photometric redshift estimation, image generation, deblending and more (see for a comprehensive review). Machine learning methods are already becoming instrumental in handling the large volume of data processed every day in survey pipelines <cit.>
The next generation of astronomical surveys such as the upcoming Legacy Survey of Space and Time <cit.> at the Vera C. Rubin Observatory, the Wide-Field Imaging Survey at the Nancy Grace Roman Space Telescope <cit.>, and Euclid <cit.> will produce unprecedented amounts of imaging data throughout the 2020s and beyond. LSST will provide incredibly deep ground-based observations of the sky, revealing a map of the universe including objects as faint as ∼25-27 mag at a 5σ detection for 10 year observing runs. Ground-based surveys such as the Hyper Suprime-Cam Subaru Strategic Program <cit.> and the Dark Energy Survey <cit.> have already mapped large swaths of the sky and produced catalogs of tens of millions of objects, with HSC depths being comparable to LSST. The astronomical research community is now in an era that demands robust and efficient techniques to detect and analyze sources in images.
Current surveys such as HSC already report large fractions of blended (overlapping) objects. For instance, 58% of objects in the the shallowest field (Wide) of the HSC survey are blended, i.e., detected in a region of sky above the 5σ threshold (26.2 mag) containing multiple significant peaks in surface brightness. As depths increase, line-of-sight projections and physical mergers cause the overall number of blends to increase. This fraction rises to 66% for the Deep and 74% for the UltraDeep layers, which are comparable to LSST depths <cit.>. If blends are not identified, they will bias results from pipelines that assume object isolation. For example, <cit.> show that the traditional detection/deblending methods can lead to a photometric error of >0.75 mag for ∼12% of their sample of artificially blended galaxies from the Cosmic Assembly Near-infrared Deep Extragalactic
Legacy survey <cit.>. Unrecognized blends can cause an increase in the noise of galaxy shear measurements by ∼14% for deep observations <cit.>. Deblending, or source separation, has been recognized as a high priority in survey science, especially as LSST begins preparations for first light.
Despite rigorous efforts to deblend objects, the problem of deblending remains, and in some sense will always remain in astronomical studies. Deblending involves separating a mixture of signals in order to independently measure properties of each individual object. This an imaging problem analogous to the “cocktail party problem”, in which an attempt is made to isolate individual voices from a mixture of conversations. However, since it is impossible to trace a photon back to an individual source, astronomical deblending is characterized as an under-constrained inverse problem. Deblending methods must rely on assumptions about source properties and models of signal mixing <cit.>.
A first step in deblending is object detection. Many codes have been developed for source detection and classification, including FOCAS <cit.>, NEXT <cit.> and SExtractor <cit.>. SExtractor is widely used in survey pipelines including HSC <cit.> and DES <cit.>, but can be sensitive to configuration parameters. While SExtractor also deblends by segmenting, or identifying pixels belonging to unique sources, modern deblenders have been developed such as Morpheus <cit.> and Scarlet, <cit.> with the latter implemented in HSC and LSST pipelines. With hopes for real-time object detection and deblending algorithms in surveys such as LSST, machine learning applications to crowded fields offer a promising avenue. The use of deep neural networks, or deep learning has seen particular success in image processing. In addition to efficiency and flexibility, neural networks may be able to overcome limitations of traditional peak-finding algorithms due to their fundamentally different detection mechanism.
There is a growing body of deep learning deblending methods in astronomy. <cit.> use a Generative Adversarial Network (GAN) to deblend small cutouts of Sloan Digital Sky Survey <cit.> galaxies from Galaxy Zoo <cit.>. <cit.> use a variational autoencoder to deblend small cutouts of simulated LSST galaxies. <cit.> use GANs to deblend images with HSC resolution and recover Hubble Space Telescope resolution. On larger scales, <cit.> use a probabilistic U-net model to deblend large simulated scenes of galaxies.
In addition to blending, another pressing issue with increased depth is the presence of many unresolved galaxies in the deep samples of smaller and fainter objects. This will prove difficult for star-galaxy classification schemes that rely on morphological features to distinguish between a point source star or a point source galaxy, although machine learning methods have been employed to combat this problem <cit.>. <cit.> use a Gaussian process classifier to perform star/galaxy classification on HSC images. This is an important area of study, as misclassifications can introduce biases in studies that require careful measurement of galaxy properties. For instance, it has been shown that stellar contamination can be a significant source of bias in galaxy clustering measurements <cit.>. Precise constraints of cosmological models require a correction of this systematic bias in measurements of clustering at high photometric redshifts.
The broader field of computer vision has seen a large growth in object detection, classification, and semantic segmentation models. Object detection and classification consist of identifying the presence of an object in an image and categorizing it from a list of possible classes. Semantic segmentation involves identifying the portion of an image which belongs to a specific class, i.e. deblending. Put together, these tasks amount to instance segmentation. This pixel-level masking can be used to deblend objects by selecting the pixels associated with each individual object by class. The benchmark leader in deep learning instance segmentation models has been the Mask-RCNN framework <cit.>.
The Mask R-CNN architecture was implemented in <cit.> to detect, deblend, and classify large scenes of simulated stars and galaxies. Other architectures have been tested in astronomical contexts, including You Only Look Once <cit.>. <cit.> use a combination of the instance segmentation model YOLOv4 and a separate classification network to perform source detection and classification on SDSS images, and <cit.> use a YOLO model to detect and morphologically classify SDSS galaxies. However, these models do not perform segmentation.
The rapid pace of research has led to many new variations and methods that can outperform benchmark architectures. To the benefit of computer vision researchers, Facebook AI Research (FAIR) has compiled a library of next-gen object detection and segmentation models under the framework titled Detectron2 <cit.>. This modular, fast, and well-documented library makes a fertile testing ground for astronomical survey data. In addition to a variety of architectures, pre-trained models are also provided. By leveraging transfer learning, i.e., the transfer of a neural network's knowledge from one domain to another, we can cut back on training time and costs with these pre-trained models. It is also possible to interface new models with detectron2, e.g., <cit.>, taking advantage of its modular nature and flexibility[See https://github.com/facebookresearch/detectron2/tree/main/projectshttps://github.com/facebookresearch/detectron2/tree/main/projects for a comprehensive list of projects.].
In this work, we leverage the resources of the detectron2 library by testing state-of-the-art instance segmentation models on large scenes, each containing hundreds of objects. We perform object detection, segmentation, and classification simultaneously on large multi-band HSC coadds. Many deep learning applications have been tested on simulated images, but methods applied to real data are often limited by a lack of ground truth. Here, we construct a methodology for using instance segmentation models on real astronomical data, and demonstrate the potential and challenges of this framework when applied to deep images. The HSC data is ideal for testing this framework, as it represents the state-of-the-art among wide/deep surveys, and is closest in quality to upcoming LSST data. By interfacing with detectron2, we are able to test new models as the repository is updated. We compare models with different performance metrics, and test how robust they are to contrast scalings that alter the dynamic range of the data, which will be important to consider for application to other datasets.
The major contributions of this work can be summarized as 1) Using instance segmentation models to deblend and classify objects in real images from HSC. This demonstrates the feasibility for future integration with wide/deep survey pipelines. We will show that the models can learn inherent features in the data that lead to classification performance gains above traditional morphological methods. 2) Comparing the performances of different models when the input data undergoes different contrast scalings. There is no standard method for scaling image data in astronomical studies that use deep neural networks, so we apply a variety of pre-processing scalings to the data for each model. Dynamic ranges can vary significantly across datasets, and raw data may not be ideal for feature extraction. We test sensitivity to contrast scalings to identify models that will be more easily adapted to different datasets. 3) Interfacing our pipeline with the detectron2 framework to test state-of-the-art models. Of particular note are our tests using transformer-based architectures, an emerging framework in computer vision studies. We will show that these architectures are more robust and accurate than traditional convolutional neural networks in both deblending and classifying objects in large scenes.
This paper is organized as follows. In <ref>, we present an overview of detectron2 in which we highlight the flexibility of its modular nature and describe the portion of the available deep learning models we implemented. In <ref>, we describe the curation of our datasets, production of ground truth labels, data preparation and our training procedure. In <ref> we present the results of training our suite of models and assess performance with different metrics. <ref>, we discuss the differences in model capabilities, compare the performance of our pipeline to existing results, and discuss the benefits and drawbacks of our method. In <ref>, we contextualize our findings and conclude.
§ DETECTRON2 FRAMEWORK
We leverage the modular power of detectron2 by implementing models with varying architectures. The pre-trained models we test in Detectron2's Model Zoo have a structure that follows the GeneralizedRCNN meta-architecture provided by the codebase. This architecture is a flexible overarching structure that allows for a variety of changes, provided they support the following components: (1) a per-image feature extraction backbone, (2) region-proposal generation, (3) per-region feature extraction/prediction. The schematic of this meta-architecture is shown in Figure <ref>.
The feature extraction backbone takes an input image and outputs “feature maps” by running the input through a neural network, often composed of convolutional layers. In our tests, we use ResNet backbones and transformer-based backbones. ResNets are convolutional neural networks that utilize skip connections that allow for deep architectures with many layers without suffering from the degrading accuracy problem known to plague deep neural networks <cit.>. In this paper we explore a few different ResNet backbones: ResNet50, ResNet101 and ResNeXt. A ResNet50 network consists of 50 total layers, with two at the head or ”stem” of the network and then four stages consisting of 3, 4, 6 and 3 convolutional layers, respectively. Each stage includes a skip connection. A ResNet101 network is similar to a ResNet50 setup, but with each stage consisting of 3, 4, 23 and 3 convolutional layers, respectively. Subsequent layers undergo a pooling operation that reduces the input resolution. We refer the reader to <cit.> for details regarding these layers. ResNeXt layers work similar to ResNet layers, but include grouped convolutions which add an extra parallel set of transforms <cit.>. We also test a network with deformable convolutions, in which the regularly spaced convolutional kernel is deformed by a pixel-wise offset that is learned by the network <cit.>.
The stages of a ResNet backbone produce feature maps, representing higher level image aspects such as edges and corners. While one can simply take the feature map outputted by the last layer of the backbone, this can pose a challenge in detecting objects of different scales. This motivates the extraction of features at different backbones stages (and thus scale sizes). A hierarchical feature extractor known as a feature pyramid network <cit.> has seen great success in object detection benchmarks. The FPN allows each feature map extracted by a ResNet stage to share information with other feature maps of different scales before ultimately passing on to the Region Proposal Network (RPN).
After the image features have been extracted, the next stage of Generalized-RCNN networks involves region proposal. This stage involves placing bounding boxes at points in the feature maps and sampling from the proposed boxes to curate a selection of possible objects. After this sampling has been done, bounding boxes are once again proposed and sent to the Region of Interest (ROI) heads, where they are compared to the ground truth annotations. The annotations consist of bounding box coordinates, segmentation masks, and other information such as class labels. Ultimately, many tasks can be done on the objects inside these regions of interest, including classification, and with the advent of Mask-RCNN frameworks, semantic segmentation. We do not include the details of the RPN and ROI heads, as these structures largely remain the same in our tests. We do test architectures with a cascade structure <cit.> which involves iterating the RPN at successively higher detection thresholds to produce better guesses for object locations. For specifics, we refer the reader to <cit.>, <cit.> and the detectron2 codebase.
We train a suite of networks to allow for several comparisons. We use a shorthand to denote network configurations as follows.
* R101c4: A ResNet50 backbone that uses features from the last residual stage
* R101fpn: A ResNet101 backbone that uses a FPN
* R101dc5: A ResNet101 backbone that uses a FPN with the stride of the last block layer reduced by a factor of two and the dilation increased by a factor of two
* R50def: A ResNet50 backbone that uses a FPN and deformable convolutions
* R50cas: A ResNet50 backbone that uses a cascaded FPN
* X101fpn: A ResNeXt101 backbone that uses a FPN
In addition to these ResNet based models, we also test transformer based architectures. A transformer is a encoder-decoder model that employs self-attention. Briefly, self-attention consists of applying linear operations to an encoded sequence to produce intermediate “query, key and value” tensors. A further series of linear operations and scalings are done to these intermediate tensors to produce an output sequence, and then a final linear operation is performed on the entire output sequence. Transformer models have exploded in popularity in the domain of natural language processing due to their scalability and generalizability on sequences, which translates well to language structure. Recently, transformers have been used in computer vision tasks such as image classification and object detection. These models been shown to be competitive with the dominant convolutional neural networks, and are seeing rapid advances in performance measures <cit.>. For example, MViTv2 utilizes multi-head pooling attention <cit.> to apply self-attention at different image scales, allowing for the detection of features of varying sizes. To obtain the input encoded sequences, an image is first divided into patches which are flattened and sent through a linear layer. MHPA is applied to the sequences to produce the image features. In an object detection context, these features are input to an FPN in the same way as features obtained from a ResNet in RCNN models. Another modern transformer model, the Swin Transformer <cit.>, applies multi-head attention to image patches, but rather than a pooling operation, use patch merging to combine features of different image patches. Swin models also use shifted window attention to allow for efficient computation and information propagation across the image. We test both MViTv2 and Swin backbones in our implementation.
§ IMPLEMENTATION
§.§ HSC coadds
In this work, the data we use consist of multi-band image coadds of roughly 4000 pixels^2 from the Deep and Ultra-Deep fields of the Hyper Suprime Cam (HSC) Subaru Strategic Program <cit.> Data Release 3 <cit.>. The HSC SSP is a three-tiered imaging survey using the wide-field imaging camera HSC. The HSC instrument <cit.> consists of a 1.77 deg^2 camera with a pixel scale of 0.168”, attached to the prime focus of the Subaru 8.2 m telescope in Mauna Kea. The Deep+UltraDeep component of the HSC survey covers ∼36 deg^2 of the sky in five broad optical bands (grizy; <cit.>) up to a full 5σ depth of ∼27 mag (depending on the filter). Despite limitations (e.g., sky subtraction and crowded field issues), the HSC DR3 data provides the closest match among all currently available deep-wide surveys to the expected data quality of LSST wide fields. The Deep/Ultra-Deep field properties are listed in Table <ref>. We use the g, r and i bands.
Given the large depth of the survey, a significant portion of objects are blended in comparison to other ground-based surveys such as the Dark Energy Survey <cit.>.
For reference, 58% of objects in the the shallowest field (Wide) of the HSC survey are blended. While a significant challenge, this lends the HSC fields to be an excellent set of data for testing deblending algorithms, particularly those suited for crowded fields. The pipeline to produce the image coadds is described in detail in <cit.>.
There are two sets of sky-subtracted coadds. The first set consists of global sky-subtracted coadds. The second set also uses the global sky-subtracted images, but an additional local sky subtraction algorithm is applied. This is to remove the wings of bright objects, artifacts that can cause problems in object detection algorithms. However, this process creates a trade-off with removing flux from extended objects, and <cit.> empirically find a local sky subtraction scale of 21.5 arcseconds to be a good balance. Ultimately, we use these local sky-subtracted images, as bright wings and artifacts can introduce problems of over-deblending or “shredding” and we want our “ground truth” detections to be as clean and accurate as possible.
To further ensure a clean training set, we apply a few quality cuts to the sample. Some images suffer from missing data in one or more bands, especially at the edge of the imaging fields. We use the bitmasks provided in the coadd FITS files to exclude images with >30% of the pixels assigned a NO_DATA flag. Given that the neural network takes multi-band images, if one of the g, r or i band images is flagged in this way, we exclude the other bands as well. There remain some imaging artifacts and issues, such as saturated regions around bright stars, and we discuss how these affect network performance in Section <ref>.
§.§ Ground Truth Generation
We must provide ground-truth object locations and masks to the network to perform pixel-level segmentation. We utilize the multi-band deblending code scarlet <cit.> to produce a model for each individual source from which we create an object mask. scarlet utilizes constrained matrix factorization to produce a spectral decomposition of an object. It is a non-parametric model that has been demonstrated to work well on individual galaxies and blended scenes. Before we run scarlet, we extract an object catalog using sep, the python wrapper for SExtractor. Then, each identified source is modelled and the “blend” or composition of sources is fit to the coadd image data. Once the final blend model is computed, the mask is determined by running sep on each individual model source and setting a mask threshold of 5σ above the background. Both the scarlet modelling and mask thresholding are done on the detection image, i.e., the sum over all bands. The run time of this process increases with the number objects in an image. In order to reduce run-time, we divide the 4k stitched coadd images into 16 images of ∼1000×1000 pixels^2. While scarlet on its own is a powerful deblender, the fits can take up to ∼30 minutes depending on the number of objects in the image, which motivates the use of efficient neural networks. After this process is complete, we compile a training set of 1000 1k×1k pixels^2 images. The distribution of the number of sources per image is shown in Figure <ref>.
The trade-off in using real over simulated data is that in supervised tasks, there is a lack of predetermined labels. For the classification task, we produce object labels with a catalog match to the HSC DR3 catalogs. We convert each detected source center to RA and DEC coordinates and then run the match_to_catalog_sky algorithm in astropy to find objects in the HSC catalog within 1 arcsecond. Then, we compare the i-band magnitude of the deblended source to the “cmodel” magnitude of the catalog objects and pick the object with the smallest magnitude difference. If no objects are within 1 arcsecond or no objects have a magnitude difference smaller than 1, we discard the object from our labelled set. Once an object is matched, we use the HSC catalog “extendedness value” to determine classes, which is based on differences in PSF magnitudes and extended model magnitudes. While yielding high accuracy at bright magnitudes, this metric becomes unreliable for star classification around a limiting magnitude of 24 mag in the i band <cit.>. We additionally discard objects with NaN values in the DR3 catalog, as the class is indeterminate. We show an example image and the results of our labelling methodology in Figure <ref>, with color-coded classes.
§.§ Data Preparation
We employ three common methods for scaling the raw data from the coadd FITS files to RGB values. These are: a z-scale, a Lupton scale, and a high-contrast Lupton scale. The z-scale transformations are commonly employed in computer vision tasks and are given by
R=A(i-I̅)/σ_I
G=A(r-I̅)/σ_I
B=A(g-I̅)/σ_I
where I=(i+r+g)/3 with a mean I̅ and standard deviation σ_I, R is pixel values in the red channel (and similarly for the green G and blue B channels using the r and g -bands respectively). We set A=10^3 for the training and cast the images to 16-bit integers. In addition to z-scaling, we also apply a Lupton scaling from <cit.>. This is an asinh scaling with
R=i(asinh(Q (I - minimum)/stretch)/Q
G=r(asinh(Q (I - minimum)/stretch)/Q
B=g(asinh(Q (I - minimum)/stretch)/Q .
We use a stretch of 0.5 and Q=10 and set the minimum to zero and cast the images to unsigned 8-bit integers. Lupton scaling brings out the fainter extended parts of galaxies while avoiding saturation in the bright central regions. These augmentations preserve the color information of objects to aid in classification. Lastly, we also use a high-contrast Lupton scaling, in which image brightness and contrast is doubled after applying the Lupton scaling. We test all of these scalings for each network architecture. In Figure <ref>, we show an example image and a histogram of pixel values in i, r and g bands (corresponding to RGB colors)
We apply data augmentation to the training and test sets. Data augmentation has become a staple of many deep learning methods. It allows the network to “see” more information without needing to store extra images in memory. We employ spatial augmentations of random flips and 90^∘ rotations. We do not employ blurring or noise addition, as the real data we train on is already convolved with a PSF and contains noise. For future generalizations of this framework to different datasets, then blur/noise augmentations may be useful, but for inference purposes on test data taken under the same conditions as the training data, spatial augmentations are sufficient. We also employ a random 50% crop on each image during training so that the data can fit into GPU memory. We considered applying all contrast scalings as a data augmentation, but did not find a significant improvement in network performance. However, this could be used in future work to reduce the training costs, as results were on par with networks trained with only one contrast scaling.
§.§ Training
Training is done using stochastic gradient descent to update the network weights by minimizing a loss function. The loss functions of these Mask-RCNN models is
L = L_cls + L_box + L_mask
where the classification loss L_cls is -log p_u or the log of the estimated probability of an object belonging to its true class u. Discrete probability distributions are calculated per class (plus a background class) for each ROI. L_box is a smoothed L1 loss calculated over the predicted and true bounding box coordinates as given in <cit.>. Finally, the mask loss L_mask is the per-pixel average binary cross-entropy loss between the ground truth and predicted masks.
All networks are pre-trained on either the MS-COCO <cit.> or ImageNet-1k <cit.> datasets of terrestrial images, and so we use transfer learning to apply these models to the our astronomical datasets. Transfer learning is a technique in deep learning where networks can generalize knowledge of one task to complete a different but related task (See for an overview of deep transfer learning). It is often used when applying a pre-trained deep learning model to a different domain than the one seen during training. By using pre-trained weights as initial conditions, training is likely to converge faster and be less prone to over-fitting. We use weights provided by Detectron2 as the starting point for our training procedure. We then train the networks for 50 total epochs, i.e. the entire training set is seen 50 total times by the network. In order to facilitate the transfer of knowledge, we first freeze the feature extraction backbones of the models and only train the head layers in the ROI and RPN networks for 15 epochs. We use a learning rate of 0.001 for this step. Then, we unfreeze the feature extraction backbone and train the entire network for 35 epochs. We begin this step with a learning rate of 0.0001 and decrease by a factor of 10 every 10 epochs.
We use two NVIDIA Tesla V100 GPUs in HAL system <cit.> to train on 1,000 images of size 500 pixels^2 paired with object annotations. When trained in parallel on each GPU, our models take roughly ∼3 hours to complete. Transformer architectures tend to use more memory, and thus are trained on 4 GPUs for roughly 4 hours.
§ HSC RESULTS
After training, we evaluate network performance on the test set of HSC images. The test set is taken from the patches in the UltraDeep COSMOS <cit.> field and consists of 95 images of 1000 pixels^2. No test set images were seen during training. A benefit of the instance segmentation models used in this work is their ability to infer on images of variable size. Thus, despite the need to crop images during training, we are still able to utilize the full size of the images in the test set.
We evaluate classification performance with precision and recall, given by
p=TP/TP+FP,
r=TP/TP+FN.
True positives (TP) are counted as a detection that has a confidence score outputted by the network above a certain threshold and additionally can be matched to a ground truth object by having an Intersection over Union (IOU) above another threshold. False negatives (FN) are those ground truth objects that do not have a corresponding detection. False Positives (FP) are those detections with a high confidence score but do not have a matching ground truth. The IOU is defined as
IOU=area(box_predicted∩box_truth)/area(box_predicted∪box_truth).
or the area of the intersection over the area of the union of the predicted and ground truth bounding boxes.
Precision and recall are often broken down by class, or combined into one value, the AP score,
AP=1/51∑_r∈{0,0.02,...,1.0} p(r)
where p(r) is maximum the precision in a recall bin of width Δ r. AP scores are computed for IOU thresholds of {0.5,0.55...0.95} and averaged.
AP scores on the HSC COSMOS test set are reported for all network configurations in Table <ref>. We report the per-class AP score for stars and galaxies separately, as well as the Small, Medium, and Large AP scores, defined by the object bounding box size of 0-32 pixels^2, 32-96 pixels^2 and >96 pixels^2, respectively. For galaxies and stars, AP score can vary significantly across network configurations. For ResNet-based architectures, AP for galaxies is consistently higher than stars, which may be due to the higher sample size of galaxies and morphological features that make galaxies easier to distinguish than compact stars. Among ResNet-based networks, a Lupton high-contrast scaling generally gives the highest galaxy AP score, while a z-scaling always gives the highest star AP score. It appears that these networks are very sensitive to the contrast scaling used, which is not desirable for application to other datasets with different dynamic ranges. However, transformer-based architectures perform more robustly with varying contrast scalings, and outperform ResNet architectures in almost all cases. For these networks, galaxy AP scores all lie within ∼50-52, showing a gain of about 5 over the highest performing ResNet configuration. Stellar AP scores for Lupton and z-scalings lie within ∼33-35, with high-contrast Lupton scalings performing worse by an AP of ∼8.
Among the Small, Medium, and Large AP metrics, transformer-based networks also outperform ResNet-based networks, in some cases seeing massive gains in AP score. The networks generally perform better on Small and Large object categories over Medium objects, again likely due to sample size.
Many studies of instance segmentation models use the MS-COCO or ImageNet-1k datasets as a benchmark to judge performance through the AP score. These data consist of terrestrial images with many object classes, so it can not necessarily be used as a comparison for our AP scores calculated on astronomical survey images with only 2 classes. However, to give a reader a sense of the range of typical values, the AP scores for models trained on terrestrial data typically range from ∼35-45 for convolutional backbones and push to ∼55 for transformer backbones (see the detectron2 repo for results). For a more fair comparison, we look to <cit.> in which instance segmentation models were tested on the simulated observations from the Dark Energy Camera <cit.>. The authors report an AP score for galaxies of 49.6 and score of 48.6 for stars, averaged to a combined score of 49.0. We also train our suite of models on the DECam dataset and report the results in Appendix <ref>. More recently, <cit.> use a combination of the instance segmentation model YOLOv4 <cit.> and a separate classification network to perform source detection and classification on SDSS images. They report an AP score of 52.81 for their single-class detection network.
§.§ Incorrect Label Bias Mitigation
There is an inherent bias in our measure of AP scores due to incorrect object class labels. In measurements described above, we test the network abilities to infer classes based on labels generated from HSC catalogs. However, these labels are known to become unreliable, especially for stars, around i-band magnitudes of ∼24 mag <cit.>. We use HSC coadds in the COSMOS field for our test dataset, and attempt to mitigate this mislabelling bias by exploiting the overlap of this field with space-based observations using the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST). Because of the lack of atmospheric seeing, morphological classification of stars/galaxies using the HST COSMOS catalog data is much more precise for faint objects, and can be used as ground truth instead of HSC labels. This will test how much poor classification behaviour is due to label generation as opposed to limitations of the models. We generate HST labels by cross-matching detected sources to the catalog of <cit.> within 1 arcsecond. If there is no object within 1 arcsecond, we discard the object. There is not necessarily a one-to-one match of HSC versus HST labels, as we are cross-matching to different catalogs, but the number of objects per image remains roughly the same for either labelling scheme. We will refer to this as the HST COSMOS test set.
This small set is not sufficient to train a network, so instead of training on HST-labelled data, we take the models trained on HSC-labelled data and test their evaluation performance on the HST COSMOS test set. To highlight the differences in class label generation, in Figure <ref> we show the number of stars and galaxies as a function of HSC i-band magnitude for the COSMOS set for both HSC and HST class labels. The unreliable quality of HSC labels at faint magnitudes is reflected in the increased counts of stars, especially the bump in stellar counts beginning at i∼25 mag. Also of note is the fewer amounts of star counts in the HSC COSMOS set at bright magnitudes. This is likely due to our HSC label generating procedure of discarding objects with NaN values in the HSC catalog. Bright stars are likely to have saturated pixels in their centers, causing these error flags to appear. With HST labels. we can test with a more astrophysically accurate baseline.
Using this new test set, we present AP scores in Table <ref>.
The results for galaxy/star AP scores are in line with the previous results on the HSC COSMOS test set. In all cases, transformer architectures outperform ResNet architectures and are more robust to different contrast scalings. AP scores for Small bounding box objects improves for all network configurations, Medium bounding box AP score roughly remains the same, and Large bounding box AP score worsens. The decrease in Large bounding-box AP scores is likely due to the initial label generation step with sep that over-deblends or “shreds” large extended galaxies and saturated regions around stars. With our HSC label generation, we exclude many of the shredded regions by enforcing the i-band Δ1 mag criterion and discarding labels matched to saturated catalog objects with NaN values. However, our HST label generation is solely based on a distance matching criterion, and so some of these shredded regions are included in the ground truth labels in the HST COSMOS test set. These spurious extra labels can lead to lower AP scores if the networks avoid shredding these regions at inference. In the next section, we examine metrics other than AP score that are less susceptible to this effect.
§.§ Missing and Extra Label Bias Mitigation
Since we have done the labelling ourselves using sep, scarlet and catalog matching to produce ground truth detections, masks and classes, traditional metrics of network performance may not be the best choice in characterizing efficacy. Consider the precision/recall and AP metric. An implicit assumption in these metrics is the completeness and purity of the ground truth labels. This assumption holds for large annotated sets of terrestrial images such as the MS-COCO set <cit.> commonly used as a benchmark in object detection/segmentation studies. It also holds for simulated datasets of astronomical images <cit.> as the ground truth object locations, masks, and classes are all known a priori when constructing the training and test set labels. However, real data of large astronomical scenes presents a challenge. Given that we must generate labels without a known underlying truth, any comparisons to this “ground truth” are really comparisons to the methods used to generate these labels. Issues in the label generating procedures will propagate to the performance metrics.
First, the ground truth detections are produced from running sep using a detection threshold of 5σ above the background. This causes a lack of complete labels, as some objects are missed. We could lower this threshold, but then run the risk of further over-deblending extended/saturated objects. This leads to the second issue in that there will still remain some level of shredding that will cause spurious extra objects to appear in the ground truth set, i.e, a lack of pure labels. If the networks do not shred extended/saturated objects as much as sep, (which is a desirable feature of the networks) then the AP metric will be lower due to less spurious network detections than the ground truth. Finally, the object detection mechanisms of the neural networks used in this work are fundamentally different from the peak-finding detection used in sep.
These issues lead to cases in which the neural networks detect objects that are not labelled in our ground truth catalog, despite being actual objects, or cases in which the networks do not detect unphysical objects that are in the ground truth. Any metric that considers true/false detections is subject to this effect. We do not wish to count these cases of fake true/false positives, as this would lead to a reduction in performance metrics that does not reflect network classification/detection accuracy, but rather the limitations of our label generation. Therefore, we construct a set of metrics similar to the canonical precision and recall, but slightly alter our definitions of positive and negative detections. We use equations <ref> and <ref>, but we limit our metrics to the set of objects D that are matched to a ground truth detection. The set of matched detections D is determined by selecting the inferred bounding box with the highest IOU to a ground truth bounding box, above a threshold of 0.5. Then for a given class C, true positives are the objects in D that are correctly classified, false positives are objects that are incorrectly assigned class C, and false negatives are matched objects with a ground truth class C that the network assigns to a different class. With these metrics, precision and recall measure purely the classification power of the network, without bias from missing labels or extra false labels. If we assume that the network's ability to classify remains consistent for objects outside of the matched set, we can generalize these metrics to overall classification performance.
We combine precision and recall into one metric to judge classification power, the F1 score, which is given by the harmonic mean between precision and recall,
F1=2×p*r/p+r.
The F1 score balances the trade-off between precision and recall, with a value close to unity being desirable. We report the F1 scores for the networks on the HST COSMOS test set in Table <ref>. The best performing configuration among ResNet architectures is the R50cas network with a z-scale scaling. A Swin network with a Lupton scaling achieves the highest overall galaxy and star F1 scores, although the MViTv2 architecture remains competitive. Nearly all transformer networks configurations perform better on star/galaxy classification than ResNet-based networks. Classification power of transformer-based networks is again more robust to contrast scalings than ResNet-based networks.
To examine network performance on faint objects, we show precision and recall as a function of i band magnitude for the HST COSMOS test set in Figure <ref>. Galaxy recall maintains a value close to one for all objects regardless of magnitude, with some fluctuations of a few percent for some models. Galaxy precision dips for some models at bright magnitudes, which may be due to compact galaxies with bright cores resembling stars. However, these dips are more likely due to inherent limitations of the models rather than label generation, as transformer architectures produce high galaxy precision and recall across magnitude bins compared to ResNet architectures. Most ResNet architectures suffer with stellar recall, with many showing poor performance even at bright magnitudes. Stellar precision reaches near unity at bright magnitudes for all architectures, but many networks configurations begin to drop in performance around i band magnitudes of 21 mag. The best performing networks maintain a stellar precision above 0.8 out to ∼25 mag in the i band. The transformer models we trained are able to achieve a 99.6 percent galaxy recall, 99.2 percent galaxy precision, 85.4 percent stellar recall and 91.5 percent star precision on our HST COSMOS test set, averaged over the whole magnitude range. For comparison, <cit.> perform deep neural network object detection and classification of stars, galaxies, and quasars in large SDSS images. With their sample of objects that covers an r band magnitude range of 14-25 mag, they report a galaxy recall of 95.1 percent, galaxy precision of 95.8 percent, stellar recall of 84.6 percent and stellar precision of 94.5 percent.
§.§ Deblending
In order to quantify deblending performance of the networks, we compute IOU scores for matched objects. The process is similar to the matching done in computing classification precision/recall. We first set a detection confidence threshold of 0.5 and then compute the bounding box IOUs for all detected and ground truth objects. For each ground truth object, we take the corresponding detected object with the highest IOU above a threshold of 0.5. We employ this threshold to avoid the biases discussed in Section <ref>. An IOU of one indicates a perfect match between the ground truth box and the inferred box. In addition to bounding box IOU, we also compute the segmentation mask IOU, which follows from Equation <ref>, but uses the area of the true and predicted segmentation masks. We report the median IOU for all matched objects in Table <ref>, and show the distributions in Figure and <ref>. Transformer-based networks generally produce a higher bounding box IOU than ResNet-based networks, although the R50cas, R101fpn and X101fpn networks remain competitive. Segmentation mask IOUs are lower than bounding box IOUs in all cases. This indicates that while the networks are able to identify overall object sizes quite well, the finer details of object shapes within the bounding boxes are not as well inferred.
The median IOUs measure the ability of the network to detect and segment objects, but it does not fully capture the deblending power of the networks. We examine the cases of a few close blends to get a sense of the ability of the networks to distinguish large overlapping objects. We demonstrate the deblending capabilities of the different networks in Figure <ref>. In very crowded scenes, the networks are able to distinguish the individual sources, and even pick up objects that are not present in the labelled set, which may present an advantage for studies of low surface-brightness galaxies. As discussed in Section <ref>, this is likely due to the difference in object detection abilities of the Region Proposal Networks compared to peak-finding methods, and highlights that the models are not limited by the training data, but are able to extrapolate beyond it. It is also possible to alter inference hyperparameters such as IOU or detection confidence thresholds, which could allow for more or less detections or overlap between detections. In Figure <ref> we demonstrate the effect of lowering the confidence threshold hyperparameter, allowing for more low-confidence detections. While not equivalent, this is similar to lowering the detection threshold in peak-finding algorithms. There are cases in which deblending is poor, and these are typically very large galaxies with one or more very large and very close companions. In such instances, it may be better to use a different contrast scaling. In Figure <ref>, a Lupton contrast scaling prevents the network from deblending multiple large sources. With the same IOU/confidence score thresholds, a z-scaling works to better isolate the two sources. This is likely due to much larger dynamic range of our z-scaling, which allows for less smearing of the sources and more distinguishing power in this case. Overall, there does not seem to be a one-size-fits-all network configuration for the cases of very large and very close blends. Training on more data would likely improve the ability to detect and segment these objects.
§ DISCUSSION
The effectiveness of instance segmentation models has been proven in many domains, boosted by the ability of networks to work “out-of-the-box” and without much fine-tuning. It has been shown that an object detection model based on the Mask R-CNN framework performs well in the classification and detection/segmentation of simulated astronomical survey images <cit.>. In this work, we have trained and tested a broad range of state-of-the-art instance segmentation models on real data taken from the HSC SSP Data Release 3 to push the direction of deep learning based galaxy detection, classification, and deblending towards real applications. Network training and evaluation performance is limited by the efficacy of our label generation methodology, a task not easily formulated when the ground truth is not completely known. This limitation also affects the choices of metrics we use to measure network performance. Often, classification and detection power are combined into the AP score, used throughout instance segmentation literature. However, this may not the best choice of metric for comparisons, as it implicitly assumes the completeness and correctness of the ground truth labels. To attempt to mitigate the effects of incorrect labels on performance metrics, we construct a test set of objects with class labels determined from more accurate space-based HST observations. However, since the AP metric artificially suffers from the detections of “false positives” that are true objects simply missing from the labelled set and/or the presence of spurious ground truth detections, we further attempt to mitigate this bias by constraining performance metrics to detected objects that have a matched ground truth label.
We find that all networks perform well at classifying galaxies, even out to the faintest in the sample. Despite the wide variety of colors, sizes, and morphologies in the real imaging data, our models can identify these objects. Stellar classification is worse, likely due to the smaller sample size in the training and test set. Transformer based networks generally outperform ResNet based networks in classification power of both stars and galaxies. They also appear to be more robust classifiers as magnitudes become fainter. Transformer based models maintain near 100% completeness (recall) and purity (precision) of galaxy selection across the whole sample and above 60% completeness and 80% purity of stars out to i-band magnitudes of 25 mag. These models are able to outperform the extendedness classifier used in the HSC catalogs, which depending on cuts yields near 100% galaxy purity, roughly 90% galaxy completeness, stellar completeness slightly above 50% and stellar purity slightly above 40% at i-band magnitudes of 25 mag <cit.>. The performance increase of our models is especially noteworthy because they are able to surpass the HSC class labelling despite being trained with it. Transformer models are also more robust to different contrast scalings than traditional convolutional neural networks, indicating that they may be more applicable to a wide range of images across surveys with different dynamic ranges.
The detection/deblending capabilities are measured by the median bounding box IOUs of the networks. Again, transformer based networks generally outperform convolutional ResNet based networks. The improved performance of transformer networks over convolutional based ones may be attributable to the ability of different attention heads to encode information at different image scale sizes <cit.>, allowing for more overall global information propagation than CNNs. While a convolutional neural network is able to learn spatial features through sliding a kernel across an image, a transformer learns features over the entire input at once, removing any limitations due to kernel sizes. It is possible that the transformer backbones are implicitly utilizing large scale features in the images such as the spatial clustering of objects, background noise or seeing and using these bulk properties to inform the network.
We examine a few cases of close blends to qualitatively see how the networks distinguish objects. There are cases in which the networks do not detect close objects, but these can sometimes be mitigated by altering the confidence and NMS IOU threshold hyperparmeters (which can be done after training). In other cases, using a different contrast scaling helps to isolate closely blended objects.
There is room to improve both classification and segmentation of these models in future work. One possibility is constructing a larger training set with more accurate labels. With better and larger samples of stars/galaxies, networks may perform better on classification. The more close blends of large galaxies seen during training, the more likely the networks will be able to distinguish these scenes. There could be more fine-tuning of hyperparmeters done to the architectures before training, rather than running them out-of-the-box. Additionally, the use of more photometric information could help in all tasks. We use the i, r and g bands on the HSC instrument in this work, corresponding to RGB color images, but could further investigate the performance if we include the z and y bands.
It is possible that these networks need to be trained longer, or that the fundamentally different properties of astronomical images over terrestrial ones limits the abilities of these architectures in extracting useful features for classification. Despite our attempts to mitigate measurement biases arising from label generation, classification remains a challenge for these models at faint magnitudes. A machine learning model has already been used to classify HSC data using photometry information with better accuracy than morphological methods, but relies on the upstream task of detection <cit.>. The instance segmentation models presented in this work are able to identify and assign classes after training using only an image as input.
§ CONCLUSIONS
It is already a necessary consequence of the current epoch of astronomical research for machine learning algorithms to parse through massive sets of images. A first step in catalog construction is detecting these objects from imaging data. Advancements in the broader computer vision community have given rise to a large ecosystem of models that perform many necessary tasks at once, including detection, segmentation, and classification. While tried and tested on terrestrial data and shown to work on simulated astronomical data, the application on real survey images remains a work in progress. Many methods rely on the object detection stage to produce measurements of individual objects. In this work, we employ a variety of instance segmentation models available through Detectron2 to perform the detection task as well as deblending and object classification simultaneously on images taken from the HSC-SSP Data Release 3. We carefully construct ground truth labels with existing frameworks and catalog matching, and caution that real data gives no straightforward way of producing labels. We find that the best networks perform well at classifying the faintest galaxies in the sample, and perform better than traditional methods at classifying stars up to i-band magnitudes of ∼25 mag. We find that even if trained on less accurate class labels, the neural networks still pick up on useful features that allow inference of the true underlying class. We expect more data with accurate labels to improve performance. The best performing models are able to detect and deblend by matching ground truth object locations and bounding boxes. Transformer networks appear to be a promising avenue of exploration in further studies.
There are many other areas for future study. While we tested a variety of models, there are many within Detectron2 that we did not implement. Some architectures are quite large and require significant resources to train. For example, we attempted to implement ViT backbones <cit.> among our set of transformer-based architectures, but were limited by the available GPU memory. Many models, especially transformers, are trained with state-of-the-art computing resources at FAIR or other organizations, and subsequently retraining them demands significant resources. Tests could be done on other sets of real data, with other downstream tasks in mind. For example, <cit.> investigate the application of instance segmentation models on SDSS data to classify galaxy morphologies. It would be straightforward to add additional classes, or implement a redshift estimation network using the modular nature of detectron2. In future work we plan to add a photo-z estimator branch to the Mask R-CNN/transformer networks and interface with the LSST software RAIL (Redshift Assessment Infrastructure Layers)[https://github.com/LSSTDESC/RAILhttps://github.com/LSSTDESC/RAIL]. The availability of realistic LSST-like simulations <cit.> for training will allow us to avoid biases from label generation. The efficiency of neural networks and the ability to perform multiple tasks at once is now a necessity with the amount of survey data pouring into pipelines.
As surveys push deeper into the sky, they will produce unprecedented amounts of objects that will be necessary to process. LSST will provide the deepest ground-based observations ever, and survey terrabytes of data every night, highlighting a need for accurate and precise object detection and classification, potentially in real-time. Correctly classifying and and deblending sources will be necessary for a wide range of studies, and deep instance segmentation models will be a valuable tool in handling these tasks.
§ ACKNOWLEDGEMENTS
We thank Dr. S. Luo and Dr. D. Mu at the National Center for Supercomputing Applications (NCSA) for their assistance with the GPU cluster used in this work. We thank Y. Shen for helpful discussion on the HST observations of the COSMOS field.
G.M., Y.L., Y.L. and X.L. acknowledge support from the NCSA Faculty Fellowship and the NCSA SPIN programs.
This work utilizes resources supported by the National Science Foundation's Major Research Instrumentation program, grant #1725729, as well as the University of Illinois at Urbana-Champaign.
We acknowledge use of Matplotlib <cit.>, a community-developed Python library for plotting. This research made use of Astropy,[http://www.astropy.orghttp://www.astropy.org] a community-developed core Python package for Astronomy <cit.>. This research has made use of NASA's Astrophysics Data System.
The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), 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 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 (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University.
This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at http://dm.lsst.org
The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE) and the Los Alamos National Laboratory.
Based [in part] on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center at National Astronomical Observatory of Japan.
This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.
mnras
§ DECAM RESULTS
For a baseline comparison of network performances, we utilize the PhoSim dataset created and used by <cit.>. We refer to the earlier work for a full description, but provide a brief summary here. Crowded fields as taken with DECam are produced using the Photon Simulator code <cit.>. Simulations account for equipment optics <cit.>, telescope options <cit.> and atmospheric conditions. Spiral, elliptical and irregular galaxies are produced by sampling three-dimensional sersic profiles with additional parameters for extra morphological features. Stars are modeled as point sources and created following an initial mass function from <cit.>. For both stars and galaxies, SEDs and metallicities are assigned based on physical models. Cosmic star formation history <cit.> is used to assign galaxy number density and population, while the distribution of stars is based on galactic latitude. To simulated crowded fields, the galactic overdensity is boosted by a factor of 4. A 512x512 pixel^2 image is produced with g,r, and z DECam bands. Integration time and magnitude ranges are assigned to roughly correspond to DECaLs DR7 coadds <cit.>. In order to assign object masks, a g-band image without background is produced for every object in the field. The PSF is configured to ∼1 arcsec. In total, 1000 images are produced for our training set, while an additional 250 are used for validation and another 50 as our test set for evaluation. Each image contains roughly 150 objects.
Here we present the results of two runs on the simulated DECam data, using the R101fpn and MViTv2 backbones. These backbones are chosen to compare the performance of convolutional versus transformer-based architectures. We use the same contrast scalings that were applied to the HSC data, but change the stretch parameter to 100 and Q to 10 for the Lupton and Lupton high-contrast scalings. The dynamic range of the simulated data is different from the HSC data, so the adjustment is done to make galaxy features more distinguishable. AP scores for each configuration are shown in Table <ref>. We adapt the ranges for Small, Medium, and Large bounding box sizes to match those used in <cit.>. Overall, we find that a Lupton scaling with a ResNet backbone works best for this dataset, giving the highest AP scores for almost all categories. This is in contrast to the results on HSC data, however we note that a transformer backbone is again more robust to contrast scalings. Although <cit.> use a z-scale with a R101fpn backbone, our results are different as we use a slightly altered z-scale formula in that we rescale each band by a constant σ_I rather than a per-band scale factor. This alteration makes galaxy classification performance worse (AP=29.80 compared to AP=49.6) but star classification performance better (AP=54.32 compared to AP=48.6). The large drop in galaxy AP suggests that the R101fpn backbone is very sensitive to the contrast scaling. All other configurations result in better galaxy and star AP scores than the <cit.> results. Our AP scores for Small objects are lower, but Medium and Large are much higher. For size categories, we use the same size definitions as in <cit.>, but compute an average AP for all IOU thresholds, rather than the AP at only the lowest threshold IOU=0.5. Thus, our results can be thought of as a kind of lower bound, as AP score tends to increase with a lower IOU threshold.
|
http://arxiv.org/abs/2307.07248v1 | 20230714094329 | Rigorous Runtime Analysis of Diversity Optimization with GSEMO on OneMinMax | [
"Denis Antipov",
"Aneta Neumann",
"Frank Neumann"
] | cs.NE | [
"cs.NE",
"cs.AI"
] |
Rigorous Runtime Analysis of Diversity Optimization with GSEMO on OneMinMax
Denis Antipov
Optimisation and Logistics
School of Computer Science
The University of Adelaide
Adelaide, Australia
Aneta Neumann
Optimisation and Logistics
School of Computer Science
The University of Adelaide
Adelaide, Australia
Frank Neumann
Optimisation and Logistics
School of Computer Science
The University of Adelaide
Adelaide, Australia
August 12, 2023
=========================================================================================================================================================================================================================================================================================================================================================================
The evolutionary diversity optimization aims at finding a diverse set of solutions which satisfy some constraint on their fitness. In the context of multi-objective optimization this constraint can require solutions to be Pareto-optimal. In this paper we study how the algorithm with additional diversity-enhancing heuristic optimizes a diversity of its population on a bi-objective benchmark problem , for which all solutions are Pareto-optimal.
We provide a rigorous runtime analysis of the last step of the optimization, when the algorithm starts with a population with a second-best diversity, and prove that it finds a population with optimal diversity in expected time O(n^2), when the problem size n is odd. For reaching our goal, we analyse the random walk of the population, which reflects the frequency of changes in the population and their outcomes.
§ INTRODUCTION
Diversity optimization is an area of optimization, where we aim at finding a set of solutions which all have a good quality and where the set is diverse. It is widely used in practice, e.g., in the area of quality diversity (QD). QD is a new paradigm, which aims at finding a set of high-quality solutions that differ based on certain user-defined features <cit.>. QD algorithms have been successfully applied to the area of robotics <cit.>, design <cit.>, and games <cit.>.
Diversity optimization problems in which the goal is to find the most diverse (according to some measure) set of solutions, all of which satisfy some quality constraint, are usually much harder than the same problems in which the goal is to find a single best solution. The main reason for this is the high dimensionality of the search space: instead of the original search space we have to perform the optimization in the space of population.
Often such problems are solved with evolutionary algorithms (EAs), which are considered as a good general-purpose solvers for such high-dimensional search spaces. This approach is called the evolutionary diversity optimization (EDO) and has been shown to be effective
to evolve high quality diverse sets of solutions for the traveling salesperson problem
<cit.>, the knapsack problem <cit.>, and minimum spanning tree problems <cit.>. EDO approach for constrained monotone submodular functions has been introduced in <cit.> to improve the initial population diversity obtained by a diversifying greedy sampling technique. A co-evolutionary approach for evolving an optimized population and interacting with a diversifying population has been introduced in <cit.>.
Furthermore, EDO algorithms were also recently utilized for constructing wireless communication networks that minimize the area covered by the senders' transmissions while avoiding adversaries <cit.>.
There are different ways to measure diversity.
The early study of Ulrich and Thiele <cit.> used the Solow-Polasky measure to measure the diversity of a population. Later studies that evolved solutions according to a set of features used the discrepancy measure <cit.> as well as population indicators such as the hypervolume indicator and inverted generational distance indicator from the area of evolutionary multi-objective optimization <cit.>.
In recent years, the analysis of evolutionary algorithms for computing diverse sets of solutions has become one of the hot topics in the theory of evolutionary computation. In particular, results have been obtained for computing diverse solutions for the traveling salesperson problem and the quadratic assignment problem for the basic case where there is no quality constraint is imposed on the desired solutions <cit.>. The EDO approach for constrained monotone submodular functions <cit.> builds on sampling greedy approaches that provably come with good approximation guarantees which directly translate to the EDO algorithms. For the classical knapsack problem, QD approaches have been presented in <cit.> that resemble dynamic programming and also provide a fully randomized polynomial time approximation scheme (FPRAS).
In the context of the multi-objective optimization the diversity optimization is used to get a diverse set of Pareto-optimal solutions. Doerr et al. in <cit.> studied how the (μ + 1)-SIBEA with population size μ = n + 1 finds a diverse population covering the whole Pareto front of benchmark problem, when it uses a heuristics to support the diversity and showed a O(n^3log(n)) upper bound on the runtime. The heuristic used in that paper helps the algorithm decide which of the two individuals with the identical fitness to kick out from the population at all times when this conflict arises, and it always removes the individual with the smaller contribution to the diversity of the population. This heuristic is very similar to the one used in single-objective optimization in <cit.>, where the authors used the same tie-breaking rule in the (μ + 1) GA, which resulted in a significant improvement of the algorithm's ability to escape local optima of benchmark functions.
In this paper, we contribute to better understanding of the population's behavior when optimizing the diversity on multiobjective problems. We study , that is, the same problem as in <cit.>, but optimized by a slightly different algorithm, the . The behavior of these two algorithms after they cover the whole Pareto front of is identical (see Section <ref> for details), but the perspective of simplifies the description of the process. We focus on the last step of the optimization, that is, we assume that the starts with a population with the second-best diversity. We prove a O(n^2) bound on the runtime, which improves the results of <cit.>, which implies the O(n^3) bound on this stage of the optimization. The previous O(n^3) bound comes from an observation that there is always a two-bits flip present in the population which allows us to find the optimally diverse population and which can be made with probability Θ(1/n^3). We show that during this last stage, the population of the performs a random walk (in the space of populations) and it often gets to the state when we have Ω(n) of such good two-bits flips, and therefore we have a O(1/n^2) probability to make one of them.
As a diversity measure we consider the total Hamming distance, that is, the sum of Hamming distances between all pairs of individuals in the population. We note that our result also holds for any measure which can be computed based on the known number of ones and zeros in each position in the population.
The rest of the paper is organized as follows. In Section <ref>, we describe formally the problem we are studying and also present some preliminary results. In Section <ref>, we prove the bounds on the probability to change an individual in the population of . Then, in Section <ref>, we analyse the random walk of the population and prove our main result.
Finally, in Section <ref>, we discuss our results and the further direction of the research.
§ PRELIMINARIES
In this section, we describe the algorithm, the problem and the diversity measure that we aim to optimize. We also define the main problem studied in the paper and state some preliminary results that help explain our goals. Additionally, we provide several auxiliary results that help to reach those goals.
§.§ The
The simple evolutionary multiobjective optimizer () is a multi-objective optimization algorithm which aims at finding a Pareto-optimal population. It is based on the dominance relation defined on the elements of the search space, which we define as follows. For elements x, y of the search space and for a k-objective fitness function f = (f_1, …, f_k) defined on that space we say that x dominates y (and write x ≽ y), iff for all i ∈ [1..k] (where [1..k] stands for an integer interval from 1 to k) we have f_i(x) ≥ f_i(y) and there exists i ∈ [1..k] for which f_i(x) > f_i(y).
The starts with a population consisting of one individual which is chosen uniformly at random from the search space. In each iteration it creates a new individual y by choosing a parent uniformly at random from its population and applying the mutation operator to it. If there is no individual in the population which dominates y, then we add y into the population and remove all the individuals which are dominated by y. If there is an individual x with exactly the same fitness as y, then the standard algorithm removes x, giving the priority to the newer individual y, which supports the exploration of the search space. However, if we have any additional objective to optimize, we can use different rules to decide whether we should remove x or y. In this paper we aim at optimizing diversity of the population (we define the measure of diversity later in Subsection <ref>), hence if after adding y to the population we have two individuals with the same fitness, we remove the one with the lowest contribution to the diversity. If this contribution is the same, we remove x to enhance the exploration ability of the algorithm. The , which uses this additional mechanism with some diversity measure D is denoted as the .
We note that similar ideas of enhancing diversity have been used in the single-objective optimization, e.g., in the in <cit.>, where a tie-breaking rule which prioritizes some diversity measure of the population helped to escape local optima.
In this paper we focus on the bi-objective pseudo-Boolean optimization, that is, our search space is the space of bit strings of a fixed length n (which is called the problem size). As the mutation operator used to create new individuals we consider the standard bit mutation, which flips each bit independently from other bits with probability 1/n. We follow the common notation and call the with the standard bit mutation the Global (for brevity). We also denote the population of the in the beginning of iteration t by P_t.
The pseudocode of the is shown in Algorithm <ref>.
Although we consider a different algorithm than in <cit.>, we note that once the and the (μ + 1)-SIBEA_D cover the whole Pareto front of , they both always have a population of n + 1 individuals with different fitness values, and they improve the diversity via the tie-breaking rule. Hence, given the same initial population which covers the whole Pareto front, both algorithms are described by the same stochastic process. This first population which covers the Pareto front can have a different distribution for the two algorithms with a random initialization, but this is not important in this paper, since we only consider the last stage of the optimization. In this light we find it easier to use notation of to ease the reading, while our results can still be compared to the ones from <cit.>.
§.§ Problem
The problem is a benchmark bi-objective problem, which is defined on bit strings of length n as
(x) = (|x|, n - |x|),
where |x| stands for the number of one-bits in x. In other words, the first objective is the number of one-bits in x and the second objective is the number of zero-bits in x. With this fitness, none of bit strings dominates any other bit string, since if there are x and y such that |x| > |y|, then we have (n - |x|) < (n - |y|). Therefore, when we aim at maximizing both objectives, then the whole search space lies on the Pareto front. The size of Pareto front (that is, the number of different fitness values) is n + 1. These observations make this problem a good benchmark for studying the diversity optimization.
When we have a population P in which all individuals have different value, by x_i we denote an individual of this population with (x_i) = (i, n - i), that is, i is the number of one-bits in x_i.
§.§ Diversity Measure: Total Hamming Distance
In this paper we consider the total Hamming distance, which is a diversity measure for the populations consisting of bit strings, which is equal to the sum of Hamming distances between each pair of individuals in the population. More formally, for population P = {x_1, …, x_m} consisting of m bit strings the total Hamming distance is
D(P) = ∑_i = 1^m - 1∑_j = i + 1^m H(x_i, x_j),
where H(·,·) is the Hamming distance between two bit strings. In our analysis we use the following observation, which allows us to compute the diversity based only on the number of one-bits in each position.
Let P be a population of size m. For all k ∈ [1..n] let m_k be the number of individuals of population P which have a one-bit in position k. Then the total Hamming distance of the population is
∑_k = 1^n m_k (m - m_k),
and it is maximized when for all m_k, k = 1, …,n, are equal to m ± 1/2, when m is odd, and are equal to m/2, when m is even.
For any two individuals x and y and for all k ∈ [1..n] let h_k(x, y) be one, if x and y have different values in position k and zero otherwise. Then we have H(x, y) = ∑_k = 1^n h_k(x, y). The total Hamming distance is therefore
D(P) = ∑_i = 1^m - 1∑_j = i + 1^m ∑_k = 1^n h_k(x_i, x_j) = ∑_k = 1^n ∑_i = 1^m - 1∑_j = i + 1^m h_k(x_i, x_j).
Consider the two inner sums (over i and j) for some particular position k. A pair of individuals (x_i, x_j) contributes one to this sum if and only if these two individuals have different bits in position k, and there are exactly m_k (m - m_k) such pairs. Hence, we have
D(P) = ∑_k = 1^n m_k (m - m_k).
Each term of this sum is a quadratic function of m_k, which is maximized when m_k = m/2. If m is even, then m/2 is also an integer, and therefore we can have all m_k = m/2, which maximizes the diversity. Otherwise, each term reaches its largest value when m_k = m ± 1/2.
From this lemma it trivially follows that for even m the maximal diversity is m^2n/4 and for odd m it is (m - 1)(m + 1)n/4 = n(m^2 - 1)/4.
§.§ Problem Statement
In this paper we study, how the optimizes when it uses the total Hamming distance as a diversity measure to break the ties between the individuals with the same fitness. By the runtime we denote the number of iterations made by before it finds a population, which (i) covers the whole Pareto front and (ii) has the maximal diversity.
This problem has already been studied in <cit.>, where it was shown that the expected runtime is O(n^3log(n)).
In this paper we study the algorithm's behavior in the very last stage of the optimization, that is, when we start with the population covering the whole Pareto front which has the second-best diversity value. The results of <cit.> imply that the expected runtime with this initialization is O(n^3). With a rigorous analysis of the population's dynamics we improve this upper bound and show that the expected runtime is O(n^2).
Lemma <ref> suggest that the set of populations is very different for the even and odd values of n. When n is odd, the population size n + 1 is even, and to reach the optimal diversity we must have exactly n + 1/2 one-bits in each position. For even n (and thus, odd n + 1) we have two options for each position, since we can have n + 1/2±1/2 one-bits in each position. The latter case gives us more freedom, hence intuitively it should be easier for the . For this reason in this paper we consider only the case when n is odd, which is harder for the algorithm (however, we believe that it is easier for the analysis).
To show that for all odd n there exists at least one population with the optimal diversity, we build such population as follows. For each individual x_i with i < n - 1/2 we can take any bit strings with exactly i one-bits in it. For larger i we take x_i, which is a bit-wise inverse of x_n - i. This population covers the whole Pareto front and due to the inverse operation it has equal number of one-bits and zero-bits in each position, which yields the optimal diversity.
To show how the population with the second-best diversity looks like, we introduce the following notation for the populations which cover the whole Pareto front of . We call position k balanced, if m_k = n + 1/2, that is, it has the same number of one-bits and zero-bits. We call it almost balanced, if m_k = n + 1/2± 1, that is we have a minimal deviation from the balanced number of one-bits. In all other cases we call position unbalanced. If all positions are balanced, we have the best diversity, therefore, a population with the second-best diversity must have at least one almost balanced or unbalanced position.
The total number of one-bits in any population covering the whole Pareto front of is the same as the number of zero-bits, which follows from the symmetry of . Therefore, when we have a position with more than n + 1/2 one-bits, we also have a position with more than n + 1/2 zeros. This brings us to conclusion that a population with exactly two almost balanced positions has the second-best diversity: we cannot have only one unbalanced or almost balanced position, and adding new almost balanced positions or making the almost balanced positions unbalanced reduces the total hamming distance.
When the has a population covering the whole Pareto front and it generates an offspring with i one-bits, it decides if this offspring should replace x_i or not. Since it does not accept an individual to the population which reduces the diversity, during the whole run until we find a population with the optimal diversity we have exactly two almost balanced positions. For each iteration t we call the only position with n + 3/2 one-bits in population P_t hot and the only position with n - 1/2 one-bits cold. Note that this definition depends on the population P_t, which can change during the algorithm's run, hence the hot and the cold position can move. Similar to the notation of position, we call a population balanced, if all positions are balanced, we call it almost balanced, if there are exactly two almost balanced positions and other positions are balanced, and we call a population unbalanced otherwise.
§.§ Useful tools
In our proof we use the following auxiliary lemma to estimate the upper bound on the probability of generating some particular individuals.
Let x and y be bit strings of length n. If the Hamming distance between x and y is H(x, y) ≥ d, then the probability that the standard bit mutation applied to x generates y is at most 1/e(n-1)^d. If d is constant when n tends to positive infinity, then this probability is at most 1 + O(1/n)/en^d.
Let the Hamming distance between x and y be d' ≥ d. Then to generate y via the standard bit mutation applied to x we must flip the d' bits which are different and do not flip any of n - d' bits which are the same in x and y. The probability to do it is
1/n^d'(1 - 1/n)^n - d' = 1/(n - 1)^d'(1 - 1/n)^n ≤1/e(n - 1)^d'
≤1/e(n - 1)^d = 1/en^d(n - 1/n)^-d.
If d is Θ(1) when n → +∞, then by Bernoulli inequality we have
(1 - 1/n)^-d = 1/(1 - 1/n)^d≤1/1 - d/n = 1 + 2d/n/(1 - d/n)(1 + 2d/n)
= 1 + 2d/n/1 + d/n - 2d^2/n^2≤ 1 + 2d/n = 1 + O(1/n),
where the last inequality holds when d/n≤1/2, which is true when n is large enough, since d is a constant. Therefore, the probability that the standard bit mutation applied to x results in y is at most 1/en^d(1 + O(1/n)).
In our analysis we split the algorithm run into phases, which can result either in a success or in a failure. The following lemma helps us to estimate the expected length of each phase and also helps us to estimate the probability that it ends successfully.
Consider a sequence of random trials {X_t}_t ∈ (not necessarily independent), where each trial results in one of three outcomes {ω_1, ω_2, ω_3}. For all t ∈ let A_t and B_t be the events when X_t = ω_1 and X_t = ω_2 correspondingly. For all t ∈∖{1} let C_t be the event that for all τ∈ [1..t - 1] the trial X_τ resulted in ω_3. Let p_t = [A_t | C_t] for all t ∈∖{1} and p_1 = [A_1]. Let q_t = [B_t | C_t] for all t ∈∖{1} and q_1 = [B_1]. Let T be the minimum t such that X_t ω_3.
If there exist some p ∈ (0, 1) and α > 0 such for all t ∈ we have p_t + q_t ≥ p and also q_t/p_t < α, then we have E[T] ≤1/p and [X_T = ω_1] ≥1/1 + α.
We have [T ≥ 1] = 1 - (p_1 + q_1) ≤ 1 - p and for all t ≥ 2 we have [T ≥ t] = [C_t + 1] = [C_t](1 - [A_t ∪ B_t | C_t]) ≤[C_t](1 - p). By induction we have that [T ≥ t] ≤ (1 - p)^t, hence T is dominated by the geometric distribution (p), and thus E[T] ≤1/p.
To estimate the probability that X_T = ω_1 we consider some arbitrary t ∈ and condition on T = t, which is the same event as (A_t ∪ B_t) ∩ C_t. If t = 1, then we have
[X_1 = ω_1 | T = 1] = [A_1]/[A_1 ∪ B_1] = p_1/p_1 + q_1 = 1/1 + q_1/p_1≥1/1 + α
For all t ≥ 2, since A_t ∩ C_t and B_t ∩ C_t are disjoint events, we have
[X_T = ω_1 | T = t] = [X_T = ω_1 ∩ T = t]/[T = t]
= [A_t ∩ ((A_t ∪ B_t) ∩ C_t)]/[((A_t ∪ B_t) ∩ C_t)]
= [A_t ∩ C_t]/[(A_t ∩ C_t) ∪ (B_t ∩ C_t)]
= [A_t ∩ C_t]/[A_t ∩ C_t] + [B_t ∩ C_t]
= 1/1 + [B_t ∩ C_t]/[C_t]·[C_t]/[A_t ∩ C_t]
= 1/1 + q_t/p_t≥1/1 + α.
This bound is independent from t, hence [X_T = ω_1] ≥1/1 + α.
§ THE PROBABILITY TO MAKE CHANGES
In this section we consider some particular individual x_i which has i one-bits and estimate the probability that we replace it in one iteration with a different bit string. This probability depends on the values of bits in x_i in the two almost balanced positions. Note that x_i can be replaced only by another individual x_i', which also has i one-bits. We cannot accept an individual which reduces the diversity, hence we can only accept x_i', which makes the two almost balanced positions balanced and keeps other positions balanced as well (then we get the optimal diversity), or we can accept x_i', which makes at least one of the almost balanced positions balanced and makes at most two balanced positions almost balanced. In the latter case we move either the hot, the cold, or both these positions to another place.
The cold position can be balanced or moved to another place only when x_i has a zero-bit in the cold position, and x_i' has a one-bit there. To move it to another position j, we must decrease the number of one-bits in j, hence we can move it only to a balanced position in which x_i has a one-bit, and x_i' must have a zero-bit there. Similarly, the hot position can be balanced or moved only when x_i has a one-bit in it and x_i' has a zero bit there. It can be moved only to a position, where x_i has a zero-bit, and x_i' must have a one-bit there. These observations trivially imply the following lemma.
Consider an individual x_i with a one-bit in the cold position and a zero-bit in the hot position. If we replace it with any other individual with exactly i one-bits, it will reduce the diversity.
By the discussion before the lemma, we cannot balance neither the hot, nor the cold positions by replacing x_i, hence any replacement of x_i with a different bit string will either make one of these positions unbalanced or it will add new almost balanced positions. In both cases the diversity is reduced.
In the following lemma we show, how and with what probability we can replace x_i which has one-bits both in the cold and in the hot positions.
Consider individual x_i with one-bits in both the hot and the cold positions (thus, i ≥ 2). x_i can be replaced by n - i different bit strings without decreasing the diversity, and all these replacements can only move the hot position to another place, but they can neither move the cold position nor find the optimal population.
If x_i - 1 is different from x_i only in the hot position (and has a zero-bit in it), then the probability that we create an individual which can replace x_i without reducing the diversity is in
[n - i/en^2(1 - O(1/n)), n - i + 2/en^2(1 + O(1/n))].
Otherwise this probability is in
[n - i/en^3(1 - O(1/n)), 7/en^2(1 + O(1/n))].
For i = n the only string which has i one-bits is the all-ones bit string, which cannot be replaced by any other bit string. By the discussion in the beginning of this section, we cannot move the cold position by replacing x_i, but we can move the hot position to one of the n - i positions, where x_i has a zero. Hence, the bit string which can replace x_i (and move the hot position) must be different from x_i in exactly two positions: in the hot position and in one of zero-bits of x_i. Therefore, there are n - i bit strings different from x_i which can replace it without reducing the diversity. In the rest of the proof we call these bit strings valid.
We now estimate the probability to generate a valid bit string conditional on the number j of one-bits in the parent x_j that we choose. If we choose x_j as a parent with |i - j| ≥ 3, then the Hamming distance from x_j to any valid bit string is at least 3. Hence, by Lemma <ref> the probability to generate a valid bit string from x_j is at most 1/en^3(1 + O(1/n)). By the union bound over all n - i valid bit strings, the probability that we generate any of them is at most
n - i/en^3(1 + O(1/n) ).
If we choose x_j as a parent with |i - j| = 2, then the distance from it to any valid bit string is at least 2 and by Lemma <ref> and by the union bound over n - i valid bit strings, the probability to create a valid bit strings is at most
n - i/en^2(1 + O(1/n) ).
If we choose x_i + 1 as a parent, then we argue that it can have a distance one to at most two valid bit strings, and the distance to the rest of them is at least 3. Let S̃_0 and S̃_1 be the sets of positions, in which there is a zero-bit and a one-bit in x_i correspondingly. Then each valid bit string has exactly one one-bit in S̃_0, each in a unique position. It also has i - 1 one-bits in S̃_1 (in all positions, except the hot one). If x_i + 1 is in distance one from a valid bit string x_i', then it must have one-bits in all position which are ones in x_i'. If it in distance one from three or more valid bit strings, then it must have at least i - 1 + 3 ≥ i + 2 one-bits, but it has only i + 1. Hence, there are at most two valid bit strings in distance 1 from x_i + 1. The others must have an odd distance to x_i + 1, hence it is at least 3. Consequently, by Lemma <ref> and by the union bound over all n - i valid bit strings, we generate a valid bit strings with probability at most
2/n(1 + O(1/n) ) + n - i - 2/n^3(1 + O(1/n) ) = 2/en(1 + O(1/n) ).
We now consider the event when we choose x_i - 1 as a parent and distinguish two cases.
If x_i - 1 is different from x_i only in the hot position (and thus has a zero-bit in it), then the Hamming distance from it to any valid bit string is one (they are different in that one-bit of the valid bit string which is in one of the zero-bits positions of x_i). By Lemma <ref> and by the union bound the probability that we replace x_i is therefore at most
n - i/en(1 + O(1/n) ).
To compute the lower bound on generating a valid bit string from x_i - 1 in this case we note that for this we need to flip any of the n - i zero bits which are not in the hot position and not to flip any other bit, the probability of which is at least
n - i/n(1 - 1/n)^n - 1≥n - i/en.
If x_i - 1 is different from x_i ether in a non-hot position or in more than one position. Assume that it is in distance one to a valid bit string x_i'. Then it must have zero-bits in all positions which are zero in x_i' and one additional zero-bit in a position which is one in x_i'. Consequently, it is different from any other valid bit string x_i” in at least three bits: this additional zero-bit, the additional one of x_i', which is zero in x_i, and in the additional one of x_i”. Hence, by Lemma <ref> and by the union bound over the valid bit strings, the probability to replace x_i is at most
1/en(1 + O(1/n) ) + n - i - 1/n^3(1 + O(1/n) ) = 1/en(1 + O(1/n) ).
Finally, if we choose x_i as a parent, then the distance from it to any valid bit string is 2, thus by Lemma <ref> the probability to generate any of them is at most
n - i/en^2(1 + O(1/n) )
and this probability is also at least
n - i/n^2(1 - 1/n)^n - 2≥n - i/en^2.
We now compute the probability to replace x_i with a valid bit string via the law of total probability. Let j be the index of the individual which we choose as a parent. Recall that we choose j u.a.r. from [0..n]. Uniting the considered cases we conclude that if x_i is different form x_i - 1 only in the hot position, then the probability that we replace x_i is at most
[|j - i| ≥ 3] ·n - i/en^3(1 + O(1/n) )
+ [|j - i| = 2] ·n - i/en^2(1 + O(1/n) )
+ [j = i - 1] ·n - i/en(1 + O(1/n) )
+ [j = i] ·n - i/en^2(1 + O(1/n) )
+ [j = i + 1] ·2/en(1 + O(1/n) )
≤(n - i/en^3 + 2(n - i)/en^3 + n - i/en^2 + n - i/en^3 + 2/en^2) (1 + O(1/n) )
= (n - i)(1 + 4/n) + 2/en^2(1 + O(1/n) ) = n - i + 2/en^2(1 + O(1/n) )
and at least
[j = i - 1] ·n - i/en = n - i/en(n + 1) = n - i/en^2(1 - 1/n + 1)
= n - i/en^2(1 - O(1/n)).
Otherwise this probability is at most
[|j - i| ≥ 3] ·n - i/en^3(1 + O(1/n) )
+ [|j - i| = 2] ·n - i/en^2(1 + O(1/n) )
+ [j = i - 1] ·1/en(1 + O(1/n) )
+ [j = i] ·n - i/en^2(1 + O(1/n) )
+ [j = i + 1] ·2/en(1 + O(1/n) )
≤(n - i/en^3 + 2(n - i)/en^3 + 1/en^2 + n - i/en^3 + 2/en^2) (1 + O(1/n) )
≤7/en^2(1 + O(1/n) ).
and at least
[j = i] ·n - i/en^2 = n - i/en^2(n + 1) = n - i/en^3(1 - O(1/n)).
We also show the similar lemma for the individuals with zero-bits in both cold and hot positions.
Consider individual x_i with zero-bits in both the hot and the cold positions (thus, i ≤ n - 2). If i = 0, then we cannot replace x_i with a different bit string. If i ≥ 1, then by replacing x_i with a different bit string we can only move the cold position to another place, but we can neither move the hot position nor find the optimal population.
If x_i + 1 is different from x_i only in the cold position (and has a one-bit in it), then the probability that we create an individual which can replace x_i without reducing the diversity is in
[i/en^2(1 - O(1/n)), i + 2/en^2(1 + O(1/n))].
Otherwise this probability is in
[i/en^3(1 - O(1/n)), 7/en^2(1 + O(1/n))].
We omit the proof, since it can be obtained from the proof of Lemma <ref> by swapping the zero-bits with one-bits.
For the last case, when there is a one-bit in the cold position and a zero-bit in the cold position, we split the analysis into two lemmas. The first one shows, what are the bit strings which can replace x_i and the second lemma estimates the probability to generate one of these strings.
Consider an individual x_i such that it has a zero-bit in the cold position and a one-bit in the hot position. Let x̃_i be the bit string which is different from x_i only in the hot and in the cold positions. Let S_0 and S_1 be the sets of positions of zero-bits and one-bits in x̃_i correspondingly.
Then if x_i is replaced by any string which has exactly one zero-bit in S_1 and exactly one one-bit in S_0 then the diversity stays the same. If it is replaced by x̃_i, then the diversity is improved (and thus becomes optimal). If it is replaced by any other bit string with i one-bits, the diversity is reduced.
If we replace x_i with x̃_i, then we reduce the number of ones in the hot position and increase the number of ones in the cold position and we do not change it in any other position. Hence, it gets us to a population with all positions balanced, thus it has the optimal diversity.
If we do not generate x̃_i, then in the valid bit string x_i' we either keep the hot position at the same place, or we move it to any position with a zero-bit in x_i, except for the cold position. Therefore, after we replace x_i with a valid bit string, the new hot position will be in S_0, and we will have a one-bit in this position. Similarly, the new cold position will be in S_1, and we will have a zero-bit in this position. Hence, any bit string with exactly one one-bit in S_0 and exactly one zero-bit in S_1 is valid.
If a bit string with i one-bits has at least two one-bits in S_0 (and therefore, it has at least two zero-bits in S_1), then after replacing x_i with x_i' we have at least four almost balanced positions, which reduces the diversity.
Let x_i be an individual with a zero-bit in the cold position and a one-bit in the hot position. Let x̃_i be a bit string, which is different from x_i only in the cold and in the hot positions. If either x_i - 1 or x_i + 1 is different from x̃_i in only one position, then the probability that we replace x_i in one iteration with a different bit string is in
[min{n - i, i}+ 1/en^2(1 - O(1/n)), 2/n(1 + O(1/n))]
and the probability that we reach the optimal diversity (by generating x̃_i) is at least 1/en^2(1 - O(1/n)). Otherwise, the probability that we replace x_i is
[1/2en^2(1 - O(1/n)), 13/n^2]
and the probability to find the optimal population is at least 1/en^3(1 - O(1/n)).
We use the same notation as in Lemma <ref> and call the bit strings which can replace x_i without reducing the diversity valid.
By Lemma <ref>, the set of valid bit strings is the set of bit strings with at most one one-bit in S_0 and at most one zero-bit in S_1. If we choose as a parent an individual which has k one-bits in S_0, then we need to flip at least k - 1 of these one-bits to create a bit string which could replace x_i. The probability of this is at most
k/n^k - 1(1-1/n) + 1/n^k≤k/n^k - 1,
which is monotonically decreasing in k for all k ∈ [1..n - 1].
Similarly, if we choose an individual with k zero-bits in S_1 as a parent, this probability is also at most k/n^k - 1.
All individuals x_j with j ≤ i - 3 have at least 3 zero-bits in S_1, as well as all individuals x_j with j ≥ i + 3 have at least 3 one-bits in S_0. Hence, if we choose x_j with |j - i| ≥ 3 as a parent, then the probability to generate a valid bit string is at most 3/n^2.
If we choose x_i ± 2 as a parent, then it either has at least two one-bits in S_0 or two zero-bits in S_1, hence the probability to create a valid bit string is at most 2/n.
If we choose x_i - 1 as a parent, then we have two options. First, if H(x_i - 1, x̃_i) = 1, then x_i - 1 has exactly one zero-bit in S_1 and no one-bits in S_0. Consequently, to create a valid bit string we must flip one of n - i + 1 zero-bits. By the union bound over all zero-bits, the probability of this event is at most n - i + 1/n. Also, to create a valid bit string it is enough to flip one of n - i + 1 zero bits and not to flip any other bits, the probability of which is
n - i + 1/n(1 - 1/n)^n - 1≥n - i + 1/en.
At the same time to generate x̃_i, which would give us the optimal diversity, we can flip the only zero-bit in S_1 and do not flip any other bit, the probability of which is
1/n(1 - 1/n)^n - 1≥1/en.
Second, if H(x_i - 1, x̃_i) > 1, then x_i - 1 has at least two zero-bits in S_1. Hence, the probability to create a valid bit string is at most 2/n.
If we choose x_i + 1 as a parent, then we also have two options. First, if H(x_i + 1, x̃_i) = 1, then x_i + 1 has exactly one one-bit in S_0 and no zero-bits in S_1. Consequently, to create a valid bit string we must flip one of i + 1 one-bits. By the union bound over all one-bits, the probability of this event is at most i + 1/n. Also, to create a valid bit string it is enough to flip one of i + 1 one-bits and not to flip any other bits, the probability of which is
i + 1/n(1 - 1/n)^n - 1≥i + 1/en.
At the same time to generate x̃_i, which would give us the optimal diversity, we can flip the only one-bit in S_0 and do not flip any other bit, the probability of which is
1/n(1 - 1/n)^n - 1≥1/en.
Second, if H(x_i + 1, x̃_i) > 1, then x_i + 1 has at least two one-bits in S_0. Hence, the probability to create a valid bit string is at most 2/n.
Finally, if we choose x_i as a parent then we create x̃_i with probability
1/n^2(1 - 1/n)^n - 2≥1/en^2,
since for this we can flip the two bits in the hot and in the cold positions and do not touch any other bits. To create a valid bit string different from x_i, we must either flip the bit in the cold position or flip the bit in the hot position. The probability of this is at most 2/n. To create a valid bit string it is also sufficient to either flip the bit in the hot position and any of n - i zero-bit and do not flip any other bits. We also can flip the bit in the cold position, one of i one-bits and do not flip any other bit to generate a valid bit string. The probability that at least one of these two events occur is at least
max{n - i/n^2(1 - 1/n)^n - 2, i/n^2(1 - 1/n)^n - 2}≥max{n - i, i}/en^2.
Now we compute the lower and upper bounds on the probability to replace x_i with a valid bit string. Let j be the index of an individual which we choose as the parent. We distinguish two cases.
Case 1: either x_i - 1 or x_i + 1 is in distance 1 from x̃_i. Then the probability that we replace x_i is at most
[|j - i| ≥ 3] ·3/n^2 + [j ∈{i - 2, i, i + 2}] ·2/n + [j = i ± 1] · 1
≤n - 4/n + 1·3/n^2 + 3/n + 1·2/n + 2/n + 1 = 2/n(1 + O(1/n)).
To compute the lower bound we define i' = i - 1, if x_i - 1 is in distance one from x̃_i and i' = i + 1 otherwise, hence we have H(x_i', x̃_i) = 1. Therefore, the probability to generate a valid bit string is at least
[j = i'] ·min{n - i + 1, i + 1}/en = min{n - i, i} + 1/en(n + 1)
= min{n - i, i} + 1/en^2(1 - O(1/n)).
The probability to generate x̃_i is at least
[j = i'] ·1/en≥1/en^2(1 - O(1/n)).
Case 2: when both x_i - 1 and x_i + 1 are in distance more than one from x̃_i, then the probability to create a valid bit string is at most
[|j - i| ≥ 3] ·3/n^2 + [|j - i| ≤ 2] ·2/n
= n - 4/n + 1·3/n^2 + 5/n + 1·2/n≤13/n^2,
and is at least
[j = i] ·1/2en≥1/2en^2(1 - O(1/n)).
The probability to create x̃_i in this case is at least the probability to create it from x_i, that is,
[j = i] ·1/en^2 = 1/en^3(1 - O(1/n)).
§ RUNTIME ANALYSIS
The main result of this section is the following theorem.
The expected runtime until the finds a population with an optimal diversity on starting from an almost balanced population is O(n^2) iterations.
From Section <ref> we see that the only way to find a population with optimal diversity is to replace individual x_i which has a zero-bit in the cold position and a one-bit in the hot position with x̃_i, which is different from x_i only in the two almost balanced positions (as defined in Lemma <ref>). To show the O(n^2) bound on the runtime of the last optimization step, we aim at proving that we often get to the state of the algorithm when there is a linear number of such individuals which allow us to find the optimal diversity and that we spend enough time in this state before leaving it. The main difficulty in this proof is that we might have individuals which can be replaced with a high probability, which does not allow us to stay in this state for long enough.
To track the population dynamics in the last optimization stage for an almost balanced population P we introduce the following notation.
* We denote by I_01(P) the set of indices i of individuals in P such that x_i has a zero-bit in the hot position and a one-bit in the cold position. By Lemma <ref>, such individuals cannot be replaced.
* We denote by I_00(P) the set of indices i of individuals in P such that x_i has zero-bits in both the cold and the hot positions. By Lemma <ref> replacing these individuals can only move the cold position to another place, but cannot improve the diversity. We also denote by J_00(P) the subset of I_00(P) of indices i such that x_i is different from x_i + 1 only in the cold position. By Lemma <ref> the probability to replace such x_i is Θ(i/n^2), which is asymptotically larger than the O(1/n^2) probability for the individuals in I_00(P) ∖ J_00(P) for any i = ω(1). Informally, this subset should be seen as a subset of indices of individuals which are too easy to replace.
* We denote by I_11(P) the set of indices i of individuals in P such that x_i has one-bits in both the cold and the hot positions. By Lemma <ref> replacing these individuals can only move the hot position to another place, but cannot improve the diversity. We also denote by J_11(P) the subset of I_11 of indices i such that x_i is different from x_i - 1 only in the hot position. By Lemma <ref> the probability to replace such x_i is Θ(n - i/n^2), which is asymptotically larger than the O(1/n^2) probability for the individuals in I_11(P) ∖ J_11(P) for any (n - i) = ω(1). Informally, this subset should be seen as a subset of indices of individuals which are too easy to replace.
* We denote by I_10(P) the set of indices i of individuals in P such that x_i has a one-bit in the hot position and a zero-bit in the cold position. By J_10(P) we denote the subset of I_10(P) of indices i such that with S_0 and S_1 defined as in Lemma <ref> either x_i - 1 has exactly one zero-bit in S_1 or x_i + 1 has exactly one one-bit in S_0. By Lemma <ref> the probability to replace such x_i is O(1/n) and the probability to find the optimal population by replacing it is Ω(1/n^2). Similar to J_00(P) and J_11(P), this subset should be seen as a subset of indices of individuals which are easy to replace, but they give us a good chance of finding the optimal population.
With the introduced notation we summarize the results of Lemmas <ref>-<ref> in Table <ref>. We also show the relation between the set sizes in the following lemma.
For any almost balanced population P we have |I_00(P)| = |I_11(P)| and |I_10(P)| = |I_01(P)| + 2. We also have |I_00(P)| ≤n - 1/2 and |I_11(P)| ≤n - 1/2, and 2 ≤ I_10(P) ≤n + 3/2.
The number of individuals with a one-bit in the hot position is
|I_10(P)| + |I_11(P)| = n + 3/2
and the number of individuals with a zero bit in the cold position is
|I_10(P)| + |I_00(P)| = n + 3/2.
Hence, |I_00(P)| = |I_11(P)|. The number of individuals with a zero-bit in the hot position is
|I_01(P)| + |I_00(P)| = n - 1/2.
Subtracting this equation from (<ref>) we obtain
|I_10(P)| - |I_01(P)| = 2.
This implies |I_10(P)| ≥ 2. We also have
|I_00(P)| = |I_11(P)| = |I_00(P)| + |I_11(P)|/2≤|P| - |I_10(P)|/2
≤n + 1 - 2/2 = n - 1/2.
Finally, we have I_10(P) ≤n + 3/2, since there are at most n + 3/2 individuals with a one-bit in the hot position.
For any almost balanced population P we additionally define (P) as the set of indices i such that x_i is different from x_i - 1 only in the hot position. The following lemma describes the relation between (P) and the previously introduced notation.
Let P be an almost balanced population. Then (P) is a superset of J_11(P) and it is a subset of J_11(P) ∪ J_10(P).
By the definition of J_11(P) for all i ∈ J_11(P) we have that x_i is different from x_i - 1 only in the hot position, thus i ∈(P). Hence, (P) ⊃ J_11(P).
Consider i ∈(P). There are two possible values of the bit in the cold position of x_i. If this bit is a one-bit, then x_i has one-bits in both almost balanced positions and therefore i belongs to J_11(P). Otherwise, if we have a zero-bit in the cold position, then i is in I_10(P) and x̃_i (as it was defined in Lemma <ref>) is different from x_i - 1 only in the cold position. Hence, i ∈ J_10(P).
We aim at showing that the algorithm is likely to obtain Ω(n) individuals with index in I_10(P) and it stays in this state for long enough to have a good probability to generate an individual which yields the optimal diversity. The individuals with index in J_00(P), J_11(P) and J_10(P), however, increase the probability to change population P, which reduces the time which we are expected to spend with |I_10(P)| = Ω(n). The individuals with index in J_10(P) also play an opposite (positive) role, since they increase the probability to improve the diversity. Hence the good state of the algorithm, from which we have a good probability to find the optimal diversity and in which we have a not too large probability to leave this state, is when we have small J_00(P) and small J_11(P) and we have large I_10(P). These observations lead us to distinguishing the following states of the algorithm with an almost balanced population P.
State 3: we have |I_10(P)| ≥n/32, |(P)| ≤ 19 and |J_00(P)| ≤ 9.
State 2: we are not in State 3 and have |(P)| ≤ 17.
State 1: all other possible situations.
For i = 1, 2, 3 we say that “the algorithm is in State i at iteration t” or just “population P_t is in State i” for brevity, if P_t satisfies the conditions of State i.
We also split the algorithm run into phases. We later show that each such phase ends in O(n) iterations in expectation and with probability at least Ω(1/n) it ends in the population with optimal diversity. The first phase starts at the first iteration. The current phase ends in the end of iteration t and the a new phase starts at the beginning of iteration t + 1 if populations P_t and P_t + 1 in these iterations are such one of the follwoing conditions is satisfied.
* P_t is in State 1 and P_t + 1 has an optimal diversity.
* P_t is in State 2 and P_t + 1 is not in State 3 and we either have |(P_t + 1)| > |(P_t)| or in iteration t we replace an individual with index I_10(P_t) with a different bit string (and probably, P_t + 1 has an optimal diversity)
* P_t is in State 3 and P_t + 1 is different from P_t (that is, any change in State 3, including finding the optimal diversity, ends the current phase).
From this definition of phases it follows that in the frames of one phase we cannot go from State 2 to State 1 (since increasing |(P_t)| ends the phase or moves us to State 3) and we also cannot go from State 3 to any other state (since any change in the population ends the phase). We illustrate the possible transitions during one phase in Figure <ref>. Note that although this illustration resembles a Markov chain, it is not one, since the transition probabilities can vary in time.
§.§ Analysis of State 3
In this section we estimate the time we spend in State 3, show the possible outcomes and their probabilities. By the definition of State 3, any change in the population results in the end of the phase, hence once we are in State 3, we cannot go to States 1 and 2 in the same phase, but we can finish the current phase in a population with optimal diversity. The following lemma estimates the expected time until a change in the population happens and the probability that at that change we find the optimal population.
Let the population P_τ of at some iteration τ be in State 3. Then the expected runtime until the end of the phase is at most 64en + O(1) and the probability that we find the optimal population in the end of the phase is at least 1 - O(1/n)/1156n = Ω(1/n).
The proof is based on Lemma <ref>. W.l.o.g. we assume that we start with iteration τ = 1. Then the iterations starting from τ can be considered as a sequence of trials {X_t}_t ∈. For each t the outcomes of X_t are ω_1, when P_t + 1 has an optimal diversity, ω_2, when P_t + 1 P_t, but its diversity is not optimal, and ω_3, when P_t + 1 = P_t. The phase ends at iteration T, which is the first t for which we have X_T ω_3 and the probability that P_T + 1 has an optimal diversity is the probability that X_T = ω_1.
For all t ∈ we define events A_t, B_t and C_t similar to Lemma <ref>, that is, A_t is when X_t = ω_1, B_t is when X_t = ω_2 and C_t is when all X_τ with τ < t are ω_3 (note that C_t is undefined for t = 1).
Events A_t and B_t depend only on the population P_t in the start of iteration t. Event C_t implies that in iteration t ≥ 2 we have the same population P_t as P_1, that is, P_t is in State 3.
Therefore, the probabilities p_t and q_t of events A_t and B_t conditional on C_t (or unconditional for t = 1) are the probabilities of these events conditional on P_t being in State 3.
We now estimate p_t + q_t and q_t/p_t. By Lemma <ref> the probability that we change an individual with index in I_10(P_t) is at least 1/2en^2(1 - O(1/n)). Conditional on C_t, we have |I_10(P_t)| ≥n/32 (since P_t belongs to State 3). Hence, the probability of A_t ∪ B_t in this case is at least the probability that we change one of these individuals, that is,
p_t + q_t ≥n/32·1/2en^2(1 - O(1/n)) = 1/64en(1 - O(1/n)) p.
To show an upper bound on q_t/p_t we use
q_t/p_t≤q_t + p_t/p_t = [A_t ∪ B_t | C_t]/[A_t | C_t].
These probabilities depend only on the population P_t, in which by condition C_t we have |J_00(P_t)| ≤ 9 and |J_11(P_t)| ≤ |(P_t)| ≤ 19 and |I_10(P_t)| ≥n/32. We denote m |I_10(P_t)| and m' |J_10(P_t)|. By Lemma <ref>, for each individual with index in J_10(P_t) the probability to replace it and get the optimal diversity is at least 1/en^2(1 - O(1/n)) and for each individual in I_10(P_t) ∖ J_10(P_t) this probability is at least 1/en^3(1 - O(1/n)). Hence,
p_t = [A_t | C_t] ≥m - m'/en^3(1 - O(1/n)) + m'/en^2(1 - O(1/n))
= (m - m'/en^3 + m'/en^2)(1 - O(1/n))
= (m/en^3 + m'/en^2)(1 - O(1/n)) = 1 - O(1/n)/en^3· (nm' + m),
where we hid the (-m'/en^3) term inside the O(1/n) term.
Event A_t ∪ B_t occurs when (and only when) we replace an individual with a different one. Let D_t^i be the event that we replace individual i. Since these events are disjoint, we have
[A_t ∪ B_t | C_t] = ∑_i = 0^n [D_t^i | C_t].
By using our estimates to replace an individual based on its index i obtained in Section <ref> and summarized in Table <ref>, and also the conditions on P_t implied by State 3, we have
[A_t ∪ B_t | C_t] ≤∑_i ∈ J_10(P_t)2/n(1 + O(1/n)) + ∑_i ∈ I_10(P_t) ∖ J_1013/n^2
+ ∑_i ∈ J_00(P_t)i + 2/en^2(1 + O(1/n)) + ∑_i ∈ I_00(P_t) ∖ J_007/en^2(1 + O(1/n))
+ ∑_i ∈ J_11(P_t)n - i + 2/en^2(1 + O(1/n)) + ∑_i ∈ I_11(P_t) ∖ J_117/en^2(1 + O(1/n))
≤ |J_10(P_t)| ·2/n(1 + O(1/n)) + |I_10(P_t)| ·13/n^2
+ |J_00(P_t) ∪ J_11(P_t)| ·1/en(1 + O(1/n))
+ |I_00(P_t) ∪ I_11(P_t)| ·7/en^2(1 + O(1/n))
≤(2m'/n + 13m/n^2 + 19 + 9/en + 7n/en^2)(1 + O(1/n))
≤1 + O(1/n)/en^2·(2em'n + 13em + 35n).
By (<ref>), we obtain
q_t/p_t ≤1 + O(1/n)/en^2·(2em'n + 13em + 35n)/1 - O(1/n)/en^3· (nm' + m)
= (n + O(1)) ·2em'n + 13em + 35n/nm' + m≤ (n + O(1)) (13e + 35n/m)
≤ (n + O(1)) (13e + 35 · 32) ≤ 1156n +O(1).
Therefore, by Lemma <ref> we have E[T] ≤1/p≤ 64en + O(1) and the probability that the phase ends in the optimal population, that is, X_T = ω_1, is at least 1/1 + 1156n + O(1) = 1 - O(1/n)/1156n = Ω(1/n).
§.§ Analysis of State 2
We proceed with considering the possible scenarios when we are in State 2. To ease the reading we introduce the following notation. For an almost balanced population P let I_H1(P) be the set of indices of individuals in P with a one-bit in the hot position and I_H0(P) be a set of indices of individuals with a zero-bit in the hot position. Let also I_C1(P) and I_C0(P) be the sets of indices of individuals with a one-bit and a zero-bit in the cold position correspondingly.
For an almost balanced population P we call a balanced position i a cold-candidate position, if at least n/16 individuals in P with index in I_C0(P) have a one-bit in position i. The following lemma shows that when |I_10(P)| ≤n/32, then moving the cold position to one of such positions would bring us to State 3 and it is also relatively easy to do.
Consider iteration t and assume that P_t is in State 2 and that |I_10(P_t)| ≤n/32. For all i ∈ [1..n] if position i is a cold-candidate, then the probability to move the cold position to i without moving the hot position in one iteration is at least 1/16en^2 and after this move we have |I_10(P_t + 1)| ≥n/32.
To move the cold position to i, we can choose one of the n/16 individuals which have a one in position i and zero in the cold position and flip these two bits (and only them) in it. The new individual has the same number of one-bits as its parent (and thus, it is on the same fitness level) and it also makes the cold position balanced and adds an extra zero-bit to position i if it replaces its parent in the population. Thus this replacement does not reduce the diversity and will be accepted by the . The hot position stays at the same place in this case.
The probability to choose one of these individuals is n/16(n + 1) and the probability to flip two particular bits and do not flip the others is 1/n^2(1 - 1/n)^n - 2. Therefore, the probability that we move the cold position to i and do not move the hot position is at least
n/16(n + 1)·1/n^2(1 - 1/n)^n - 2 = 1/16(n^2 - 1)(1 - 1/n)^n - 1≥1/16en^2.
Since position i is balanced and there are at least n/16 individuals with index in I_C0(P_t) which have a one-bit in position i, then at most n + 1/2 - n/16 individuals with index in I_C1(P_t) have a one-bit in this position. Since there are n - 1/2 one-bits in the cold position, we have |I_C1(P_t)| = n - 1/2. Therefore, the number of individuals with index in I_C1(P_t) and a zero-bit in position i is at least n - 1/2 - (n + 1/2 - n/16) = n/16 - 1. Some of these individuals might have a zero-bit in the hot position, and then their index is in I_01(P_t). By the lemma conditions and by Lemma <ref> we have |I_01(P_t)| = |I_10(P_t)| - 2 ≤n/32 - 2. Hence at least
(n/126 - 1) - (n/32 - 2) ≥n/32
individuals have a one-bit in the hot position and a zero-bit in position i. Since we cannot move the cold position to position i by replacing any of these individuals (we need to reduce the number of zero-bits in position i, thus the replaced individual must have a one-bit in position i), they all are included into I_10(P_t + 1) after moving the cold position. Therefore, this move gives us a population P_t + 1 with |I_10(P_t + 1)| ≥n/32.
The following lemma shows that there is always a linear number of cold-candidates in any almost balanced population, independently of the current state.
If n ≥ 5, then in any almost balanced population there are at least n/8 cold-candidate positions.
Consider an arbitrary almost balanced population P. There are n + 3/2 individuals with index in I_C0(P), and since they all have different fitness and therefore have a different number of one-bits in them, the total number of one-bits in these individuals is at least
∑_i = 0^n + 1/2 i = (n + 1)(n + 3)/8.
At most n + 3/2 of these ones are in the hot position and none of them are in the cold position. Hence, these individuals have at least
(n + 1)(n + 3)/8 - n + 3/2 = (n + 3)(n + 1 - 4)/8 = n^2 - 9/8
one-bits in balanced positions. Assume that there are s < n/8 cold-candidate positions. Since these positions are balanced, for each of these positions there are at most n + 1/2 individuals with index in I_C0(P) and a one-bit in this position. For the other (n - 2 - s) positions there are at most at most n/16 - 1 individuals with index in I_C0(P) and a one-bit in it (otherwise they would be cold-candidates). Therefore, by our assumption on s, the number of one-bits in individuals with index in I_C0(P) is at most
s·n + 1/2 + (n - 2 - s) ·n - 16/16 = s ·7n + 24/16 + (n - 2)(n - 16)/16
≤(n/8 - 1)(7n + 24) + (n - 2)(n - 16)/16
= 15/16n^2 - 3n + 4/8 = n^2 - 9/8 - n^2/16 + 3n - 13/8.
The last term is strictly positive, when n ≥ 5, hence this is strictly less than n^2 - 9/8, which is the lower bound on the number of zero-bits in these individuals. Hence, our assumption is wrong, and there are at least n/8 cold-candidates.
With these two lemmas we are in position to show the lower bound on the probability that P_t + 1 is in State 3 when we have P_t in State 2 and |I_10(P_t)| ≤n/32. Later we will show that with larger |I_10(P_t)| we can rely on finding the optimal population without going to State 3.
For any iteration t with P_t in State 2, and |I_10(P_t)| ≤n/32 the probability that P_t + 1 is in State 3 is at least 1/256en.
By Lemma <ref> we have at least n/8 cold-candidates positions, and since by the lemma conditions we have |I_10(P_t)| ≤n/32, then by Lemma <ref> the probability to move the cold positions to each of these positions is at least 1/16en^2 and it yields |I_10(P_t + 1)| ≥n/32.
We now show that the move of the cold position (without moving the hot position) cannot make |(P_t + 1)| > |(P_t)| + 2. Assume that we have moved the cold position by replacing individual x_i. Then for all j ∉{i, i + 1} we have j ∈(P_t + 1) if and only if j ∈(P_t), since the hot position is at the same place in P_t and P_t + 1 and both x_j and x_j - 1 have not changed in iteration t.
Therefore, the move of the cold position can only add two indices to (P_t + 1) (compared to (P_t)), that are, i and i + 1. Since in State 2 we have |(P_t)| ≤ 17, after the move of the cold position to any of the cold-candidate positions, we will have |(P_t + 1)| ≤ 19.
Hence, moving the cold position to any of the cold-candidates satisfies |I_10(P_t + 1)| ≥n/32 and |(P_t + 1)| ≤ 19. To get P_t + 1 in State 3 we also need to make |J_00(P_t + 1)| ≤ 9, and now we show that at least n/16 cold-candidates allow us to do that.
Consider an event when we move the cold position to position j (which is not necessarily a cold-candidate) by replacing individual x_i with x_i'. Consider also some index k ∈ I_H0(P_t) ⊃ J_00(P_t). Note that I_H0(P_t + 1) is equal to I_H0(P_t), since we do not move the hot position and we do not change the bit value in it in any individual, including x_i. Hence, J_00(P_t + 1) ⊂ I_H0(P_t). There are two cases, in which k belongs to J_00(P_t + 1). The first case is if k = i, then x_i' can be different from x_i + 1 only in position j, then it will be included into J_00(P_t + 1) by the definition of this set. In the second case, if k i and k i - 1, then k ∈ J_00(P_t + 1) only if x_k was different from x_k - 1 only in position j before the iteration. Note that k = i - 1 can never be added to J_00(P_t + 1), since x_i' must have a zero-bit in position j (moving the cold position to j implies that we increase the number of zero-bits in it), and thus x_i' cannot be different from x_i - 1 only in the new cold position.
With this observation, for each position j ∈ [1..n], except for the hot and the cold positions, we denote by J_j the set of indices k ∈ I_H0(P_t) such that in P_t individual x_k is different from x_k + 1 only in position j. Note that these are disjoint sets for different positions j and they are independent of the index i of the individual we replace. Then for all j ∈ [1..n] (except the two almost balanced positions) moving the hot position to j will give us |J_00(P_t + 1)| ≤ |J_j| + 1, where we add one to take into account the case when k = i, that is, when the index of the changed individual is added to J_00(P_t + 1).
Since J_j are disjoint subsets of I_H0(P_t), then we have ∑_j |J_j| ≤n - 1/2. If we assume that there are at least n/16 positions j that have |J_j| > 8, then we have ∑_j |J_j| > n/16· 8 = n/2 > n - 1/2, hence we must have less than n/16 such positions. Consequently, there are at least n/8 - n/16 = n/16 cold-candidates j with |J_j| ≤ 8, moving the cold position to which yields |J_00(P_t + 1)| ≤ 8 + 1 = 9. The probability to move the cold position to any of these n/16 cold-candidates is at least
n/16·1/16en^2 = 1/256en.
To bound the probability of finding the optimal population and the probability of ending the phase when we are in State 2 we also need the following auxiliary result.
For any iteration t with P_t in State 2, the probability that either we have |(P_t + 1)| > |(P_t)| or in iteration t we change an individual with index in I_10(P_t) is at most
15 + 2|J_10(P)|/en(1 + O(1/n)).
The probability that we find the optimal population in one iteration is at least
(1/en^2|J_10(P_t)| + 1/en^3|I_10(P_t) ∖ J_10(P_t)|) (1 - O(1/n)).
The lower bound on the probability to find the optimal population follows from Lemma <ref>. Since for any individual with index in J_10(P_t) the probability to replace it and obtain the optimal diversity is at least 1 - O(1/n)/en^2 and for any individual with index in I_10(P) ∖ J_10(P_t) this probability is at least 1 - O(1/n)/en^3 and since these are disjoint events for different individuals, we have that this probability is at least
(1/en^2|J_10(P_t)| + 1/en^3|I_10(P_t) ∖ J_10(P_t)|) (1 - O(1/n)).
Let A be the event when either we replace an individual with index in |I_10(P_t)| or we get |(P_t + 1)| > |(P_t)|. Event A it can be represented as a union of three disjoint events A = A_00∪ A_11∪ A_10, where A_10 is the event when we replace an individual with index in I_10(P_t) and A_00, A_11 are the events when we replace an individual with index in I_00(P_t) or I_11(P_t) correspondingly and get |(P_t + 1)| > |(P_t)|. Note that by Lemma <ref> we cannot replace an individual with index in I_01(P_t), hence we do not consider this as a part of event A. Therefore, we have [A] = [A_00] + [A_11] + [A_10]. We estimate each of the three probabilities separately.
Replacing an individual with index in I_11(P_t) is a super-event of A_11, hence its probability is not smaller than [A_11].
By Lemma <ref>, the probability to replace an individual with index in I_11(P_t) is at most 1/en(1 + O(1/n)), if its index is in J_11(P_t) and it is at most 7/en^2(1 + O(1/n)) otherwise. Since in State 2 we have |J_11(P_t)| ≤ |(P_t)| ≤ 17 and since by Lemma <ref> we always have |I_11(P_t)| ≤n - 1/2, then we have
[A_11] ≤(n - 1)/2·7/en^2(1 + O(1/n)) + 17 ·1/en(1 + O(1/n))
≤41/2en(1 + O(1/n)).
Similarly, by Lemma <ref>, the probability to replace an individual with index in I_10(P_t) is at most 13/n^2, if its index is not in J_10(P_t) and it is at most 2/n otherwise. Since |I_10(P_t)| ≤n + 3/2, the probability of A_10 is at most
[A_10] ≤ |I_10(P_t) ∖ J_10(P_t)| ·13/n^2 + |J_10(P_t)| ·2/n(1 + O(1/n))
≤13(n + 3)/2n^2 + |J_10(P_t)| ·2/n(1 + O(1/n))
= (13/2n + |J_10(P_t)| ·2/n)(1 + O(1/n)).
To estimate the probability of A_00, we use a more rigorous approach. First we note, that by replacing an individual with index in I_00(P_t) we do not move the hot position by Lemma <ref>. Hence, similar to our argument in Lemma <ref>, the only two indices which can be added to (P_t + 1) which are not in (P_t) are the index of the changed individual i or the index of its neighbour i + 1. However, there is a guarantee that i in this case cannot be included into (P_t + 1), since it is in I_00(P_t) and even after replacement new x_i has a zero-bit in the hot position. Hence, it will be different from x_i - 1 not only in the hot position and therefore, it is not in (P_t + 1).
To replace x_i with a bit string which is different from x_i + 1 only in the hot position we must generate a particular bit string x_i' in level i. For all j i the Hamming distance from this bit string to x_j is at least |i - j| and the distance to x_i is at least 2. Hence, by Lemma <ref>, the total probability over all parents we can choose (each with probability 1/n + 1) that we generate x_i' is at most
∑_j i1/n + 1 ·1/e(n - 1)^|j - i| + 1/n + 1·1/e(n - 1)^2
≤2/e(n + 1)(1/(n - 1)^2 + ∑_d = 1^n 1/(n - 1)^d)
≤2/e(n + 1)(1/(n - 1)^2 + 1/n - 1·1/1 - 1/n - 1)
= 2/en^2(1 + O(1/n)).
By the union bound over all i ∈ I_00(P_t), the probability that we replace any individual x_i with a bit string different from x_i + 1 only in the hot position is at most
|I_00(P)_t| 2/en^2(1 + O(1/n)) ≤n - 1/2·2/en^2(1 + O(1/n))
≤1/en(1 + O(1/n)),
which is an upper bound on [A_00].
Summing up the probabilities of A_00, A_11 and A_10, we have that the probability of A is at most
[A] ≤41/2en(1 + O(1/n)) + (13/2n + |J_10(P_t)| ·2/n)(1 + O(1/n))
+ 1/en(1 + O(1/n))
= (43 + 13 e/2en + |J_10(P)| ·2/n)(1 + O(1/n))
≤15 + 2|J_10(P)|/en(1 + O(1/n)),
since 43 + 13e/2e≈ 14.409.
We are now in position to prove the main result of this subsection.
For any iteration t with P_t in State 2 the expected time until the end of the current phase is at most 320en + O(1) iterations. The probability that at the end of the phase we have a population with optimal diversity at least 1/3551232en + O(1) = Ω(1/n).
Similar to the proof of Lemma <ref>, we aim at applying Lemma <ref>. W.l.o.g. we assume that we start at iteration t = 1 with P_1 in State 2. We consider the sequence of algorithm's iterations as a sequence {X_t}_t ∈ of trials. Each trial X_t has three possible outcomes. The first outcome ω_1 occurs either when we get P_t + 1 with optimal diversity or when we get P_t + 1 in State 3 and the phase ens in a population with optimal diversity. In the latter case we consider all iterations spent in State 3 as an auxiliary trial not belonging to the sequence {X_t}_t ∈. The second outcome ω_2 occurs when we either end the phase without getting P_t+ 1 with optimal diversity or when we get P_t + 1 in State 3 and then end a phase without finding the optimal population (in this case we also consider all iterations spent in State 3 as an auxiliary trial). The last outcome ω_3 occurs when neither the phase ends, nor we get P_t + 1 in State 3.
For all t ∈ we define events A_t and B_t as X_t = ω_1 and X_t = ω_2 correspondingly and we define C_t as an event when for all τ∈ [1..t - 1] we have X_t ∈τ (this is undefined for t = 1). Then if P_1 is in State 2 and for some t ≥ 2 event C_t occurs, then it means that we have P_t in State 2 as well, since we do not leave State 2 and we also do not end the phase in any iteration τ∈ [1..t - 1].
We define p_t = [A_t | C_t] and q_t = [B_t | C_t]. To apply Lemma <ref>, we need to estimate p_t + q_t and q_t/p_t.
We first estimate p_t + q_t. Since events A_t and B_t are disjoint, this is [A_t ∪ B_t | C_t]. If we have |I_10(P_t)| ≤n/32, then by Lemma <ref> the probability to go to State 3, which is a sub-event of A_t ∪ B_t, is at least 1/256en. Thus, we have [A_t ∪ B_t | C_t] ≥1/256en. Otherwise, if we have |I_10(P_t)| > n/32, then by Lemma <ref> the probability to replace an individual with index in |I_10(P_t)| is at least 1/2en^2(1 - O(1/n)). Replacing one of
at least n/32 such individuals is a sub-event of A_t ∪ B_t, and its probability is at least
n/32·1/2en^2(1 - O(1/n)) = 1/64en(1 - O(1/n)) > 1/256en,
if n is large enough. Hence, independently on |I_10(P_t)| we always have [A_t ∪ B_t | C_t] ≥1/256en p.
To find an upper bound on q_t/p_t we fix an arbitrary t and split the event A_t ∪ B_t into the following three disjoint sub-events (we avoid t in the notation, since t is fixed).
Let be the event when P_t + 1 has the optimal diversity, be the event when P_t + 1 is in State 3 and let be the event that we end the phase, but P_t + 1 is not in State 3 and it is not optimally diverse.
We denote the probabilities of these events , and respectively. If occurs, then iteration t results in X_t = ω_1. If occurs, then X_t = ω_2. If occurs, then by Lemma <ref> with probability at least 1 - O(1/n)/1156n we have X_t = ω_1 and otherwise we have X_t = ω_2.
We denote by q 1 - O(1/n)/1156n the lower bound on the probability of ω_1 conditional on . Then we have
q_t/p_t = q_t + p_t/p_t - 1 ≤ + + / + q - 1 = + (1 - q)/ + q.
We consider two cases depending on population P_t.
Case 1: we have either |I_10(P_t)| ≥n/32 or |J_10(P_t)| ≥ 1. If occurs, then we either have |(P_t + 1)| > |(P_t)| or we have changed an individual with index in |I_10(P_t)|. Similar to the notation used in Lemma <ref>, we denote m |I_10(P_t)| and m' |J_10(P_t)|. Then by Lemma <ref> we have
/ ≤12/n + 2/nm'/(m - m'/en^3 + m'/en^2)(1 - O(1/n))
= en^2/n·(12 + 2m')/(m/n + m')(1 + O(1/n))
= en ·(12 + 2m')/(m/n + m')(1 + O(1/n)),
where we hid the -m'/en^3 term in the denominator in the first line into the O(1/n) term. If we have m ≥n/32, then this is at most
en ·12 + 2m'/1/32 + m'(1 + O(1/n)) = 2en(1 + 6 - 1/32/1/32 + m')(1 + O(1/n))
≤ 386en + O(1).
If we have m' ≥ 1, then this is at most
en ·(12/m' + 2) (1 + O(1/n)) ≤ 14en + O(1).
In both cases this is at most 386en + O(1) ≈ 1049n, which is smaller than 1 - q/q = 1156n(1 - Θ(1/n))/1 - O(1/n) = 1156n ± O(1), when n is large enough. Hence, we have ≤1 - q/q, and by (<ref>) we obtain
q_t/p_t≤·1 - q/q + (1 - q)/ + q = 1 - q/q≤ 1156n + O(1).
Case 2: |I_10(P)| < n/32 and |J_10(P)| = 0. By Lemma <ref> we have ≥1/256en. By Lemma <ref> we have ≤12/n≤ 12 · 256e. Hence, by (<ref>) we have
q_t/p_t ≤3072e + (1 - q)/q ≤3072e + 1/q
= (3072e + 1)(1156n +O(1)) = 3551232en + O(1),
which is a larger bound than in case 1. Hence, in both cases we have q_t/p_t≤ 3551232en + O(1) α.
Consequently, by Lemma <ref> the probability that the current phase ends in the optimal population is at least 1/1 + α = 1/3551232en + O(1) and that the expected time until event A_t ∪ B_t occurs is at most 1/p≤ 256en. When A_t ∪ B_t occurs, then we either end the phase immediately or we go to State 3, where by Lemma <ref> we need in expectation another 64en + O(1) iterations to end the phase. Therefore, the expected time until the phase ends is at most 320en + O(1).
§.§ Analysis of State 1
In this section we show, that we spend in State 1 expected number of O(n) iteration before we leave it to get either to another state or to the optimal population. For this it is enough to show the Ω(1/n) probability to leave State 1, which we do in the following lemma.
Let n ≥ 5. If at iteration t we have P_t in State 1, then the probability that |(P_t + 1)| ≤ 17 is at least 1/256en.
This proof is very similar to the proof of Lemma <ref>, but it is easier, since we do not need to have |I_10(P_t + 1)| ≥n/32. We call a balanced position in P_t a hot-candidate (analogue to the cold-candidates positions in Subsection <ref>), if there are at least n/16 individuals with index in I_H1(P) which have a zero-bit in that position.
To move the hot position to a hot-candidate position i, we can choose one of the n/16 individuals that have a zero-bit in position i and a one bit in the hot position and flip these two bits without flipping any other bit. It will give us an individual with the same number of one-bits as in its parent and if it replaces the parent, then we balance the hot position and add an extra one-bit to position i, which makes it hot. Thus, this does not reduce the diversity and is accepted by the . The probability that we do that is at least
n/16·1/(n + 1)·1/n^2(1 - 1/n)^n - 2≥1/16en^2.
We now show that there are many (at least n/8) hot-candidates positions. Consider all the individuals with a one-bit in the hot position, that are, the individuals with index in I_H1(P_t). There are n + 3/2 such individuals, and since they all have a different fitness, they have at least
∑_i = 0^n + 1/2 i = (n + 1)(n + 3)/8
zero-bits in them. Since at most n + 3/2 of these bits are in the cold position, at least
(n + 1)(n + 3)/8 - n + 3/2 = n^2 - 9/8
zero-bits are in the balanced positions. Assume that there are s < n/8 hot-candidate positions. Since each of these positions is balanced, none of them can have more than n + 1/2 zero-bits in it. The other positions, since they are not hot-candidates, have at most n/16 - 1 zero-bits in them. Hence by the same arguments as in the proof of Lemma <ref>, the total number of zero-bits in the individuals with a one-bit in the hot position is
s ·n + 1/2 + (n - 2 - s) ·(n/16 - 1) < n^2 - 9/8,
when n ≥ 5. Hence, we must have s ≥n/8.
We now show that at least n/16 of these hot-candidates are such that when we move the hot position to them we get |(P_t + 1)| ≤ 17. Consider an event when we move the hot position to position j (which is not necessarily a hot-candidate) by replacing individual x_i with x_i'. Consider also some index k ∈ [0..n]. There are two cases, in which k belongs to (P_t + 1), namely, (i) if k = i and x_i' is different from x_i - 1 only in position j or (ii) if k i and k i + 1, and x_k was different from x_k - 1 only in position j in P_t. Note that k = i + 1 can never be in (P_t + 1), since x_i' must have a one-bit in position j (moving the hot position to j implies that we increase the number of one-bits in it), and thus it cannot be different from x_i + 1 only in the new hot position j.
With this observation, for each position j ∈ [1..n], except for the hot and the cold positions, we denote by J_j the set of indices k such that x_k is different from x_k - 1 only in position j in P_t. Note that these are disjoint sets for different positions j and they are independent from the index of the individual we replace. Then for all j ∈ [1..n] (except the two almost balanced positions) moving the hot position to j will give us |(P_t + 1)| ≤ |J_j| + 1, where we add one to take into account case (i), when the index of the changed individual is added to (P_t + 1).
Since J_j are disjoint subsets of [1..n] (note that k = 0 cannot be in J_j by the definition of J_j), then we have ∑_j |J_j| ≤ n. If we assume that there are at least n/16 positions j that have |J_j| > 16, then we have ∑_j |J_j| > n/16· 16 = n, hence we must have less than n/16 such positions. Consequently, there are at least n/8 - n/16 hot-candidates j with |J_j| ≤ 16, moving the hot position to which yields |(P_t + 1)| ≤ 16 + 1 = 17. The probability to move the hot position to any of these n/16 hot-candidates is at least
n/16·1/16en^2 = 1/256en.
We show the main result of this subsection in the following corollary.
Consider some iteration τ with P_τ in State 1. Then the expected time until the end of phase is at most 576en + O(1) iterations and the probability that the phase ends in an optimal population is at least 1/3551232en(1 - O(1/n)).
W.l.o.g. assume that τ = 1.
For all t ∈ let A_t be the event that either P_t has an optimal diversity or it is in State 2 or it is in State 3. Let T be the minimum t when A_t occurs. For all t < T we have P_t in State 1 and the phase does not end earlier than in the end of iteration T.
For all t event A_t occurs when we get |(P_t + 1)| ≤ 17. By Lemma <ref> the probability of this is at least 1/256en, therefore, T is dominated by the geometric distribution (1/256en), and thus E[T] ≤ 256en.
At iteration T there are three options. First, we might have P_T + 1 with the optimal diversity and therefore, the phase ends immediately and the probability that it ends in an optimal population is 1. Second, we can have P_T + 1 in State 2, then by Lemma <ref> we need another 320en +O(1) iterations in expectation to end the phase and the phase ends by finding a population with optimal diversity with probability at least 1/3551232en + O(1). The last option is that we have P_T+ 1 in State 3 and then by Lemma <ref> we need another 64en + O(1) iterations until we end the phase and in the end of the phase we find an optimal population with probability 1 - O(1/n)/1156n. In all three cases the expected number of iterations until we end the phase is at most 320en +O(1), thus, when we are in State 1, the expected number of iterations until the end of the phase is at most E[T] + 320en +O(1) ≤ 576en + O(1). The probability that the phase ends in an optimal population is at least 1/3551232en + O(1).
§.§ The Total Runtime
In this subsection we prove our main result, that is, Theorem <ref>.
Let T_i be the time of the i-th phase of the algorithm and let N be the number of the first successful phase which ends in an optimal population. Then the total runtime T of the algorithm is T = ∑_i = 1^N T_i.
We aim at showing that T is integrable and to bound its expectation from above. Since T is a sum of non-negative random variables, then it is also non-negative, hence we have E[|T|] = E[T], and it is enough to only give an upper bound on E[T], since it will also imply that T is integrable. By the law of total expectation we have
E[T] = ∑_k = 1^∞[N = k] E[∑_i = 1^N T_i | N = k]
= ∑_k = 1^∞[N = k] E[∑_i = 1^k T_i | N = k]
By Lemmas <ref> and <ref> and by Corollary <ref>, we have that conditional on N = k for all i ≤ k we have E[X_i | N = k] ≤ 576en + O(1) Δ. Hence, for every k ∈ we have
E[∑_i = 1^k T_i | N = k] = ∑_i = 1^k E[T_i | N = k] ≤ kΔ
Therefore, we have
E[T] ≤∑_k = 1^∞[N = k] kΔ = Δ E[N].
By Lemmas <ref> and <ref> and by Corollary <ref> we also have that N is dominated by the geometric distribution (1/3551232en + O(1)), thus E[N] ≤ 3551232en + O(1). Consequently, we have
E[T] ≤δ E[N] ≤ (576en + O(1))(3551232en + O(1)) = O(n^2).
This is a finite upper bound, which implies the correctness of Eqs. (<ref>), (<ref>) and then (<ref>) (in that order).
§ CONCLUSION
In this paper we have analysed the last stage of the optimization of the total Hamming distance on with . We have shown that the population of performs a random walk, and the rigorous study of this random walk reveals that a significant part of it is spent in a "good" region of the populations' space, where we have a lot of opportunities to make progress. This is in a big contrast with the previous study <cit.>, which pessimistically assumes that we always have the minimal number of such opportunities. We show that the pessimism is too critical in the last stage and increases the upper bound by a factor of Ω(n). This also indicates that the pessimism of <cit.> is too strong in the earlier stages, that is, during those stages the population is also likely to perform a random walk, which often visits good regions of the populations' space. Our study suggests the properties of the population which should be considered, shows rigorously how the population can change during the random walk and suggests some suitable (but not novel) methods for the analysis of this random walk, such as dividing it into phases. We are optimistic that these observations can be also helpful outside of the EDO context, e.g., in the analysis of population-based algorithms.
Intuition suggests that having more opportunities to improve the diversity in the earlier stages of the optimization should result into a coupon collector effect and give the expected runtime of O(n^2log(n)) iterations to optimize the diversity form scratch.
Taking into account that the finds a population covering the whole Pareto front of starting from a random point in O(n^2log(n)) expected time (which was shown in <cit.>, Theorem 3), this would suggest that optimizing the diversity is not asymptotically harder than covering the Pareto front (however, this statement also requires a lower bound on the time to cover the front).
In general we are optimistic that the analysis of random walks of complicated sets of solutions provided in this paper might be fruitful when studying the EDO on different problems. We also note that the leading constants in our results are unnaturally large due to the pessimistic assumptions we made in our proofs. However, our preliminary empirical study suggests that during a typical run we do not have many individuals in distance one from the individuals in the neighboring fitness levels, and therefore, in our analysis we could disregard the presence of individuals in J_10, J_00 and J_11. This would allow us to simplify our analysis and would significantly reduce the leading constants. For this reason a more rigorous empirical study of this problem is one of the most interesting future directions of this topic.
§ ACKNOWLEDGEMENTS
This work was supported by the Australian Research Council through grants DP190103894 and FT200100536.
10
DBLP:conf/alife/LehmanS08
Joel Lehman and Kenneth O. Stanley.
Exploiting open-endedness to solve problems through the search for
novelty.
In International Conference on the Synthesis and Simulation of
Living Systems, ALIFE 2008, pages 329–336. MIT Press, 2008.
DBLP:journals/corr/MouretC15
Jean-Baptiste Mouret and Jeff Clune.
Illuminating search spaces by mapping elites.
CoRR, abs/1504.04909, 2015.
DBLP:journals/ras/KimCD21
Seungsu Kim, Alexandre Coninx, and Stéphane Doncieux.
From exploration to control: Learning object manipulation skills
through novelty search and local adaptation.
Robotics Autonomous Systems, 136:103710, 2021.
DBLP:conf/gecco/RakicevicCK21
Nemanja Rakicevic, Antoine Cully, and Petar Kormushev.
Policy manifold search: exploring the manifold hypothesis for
diversity-based neuroevolution.
In Genetic and Evolutionary Computation Conference, GECCO
2021, pages 901–909. ACM, 2021.
DBLP:conf/ppsn/HaggAB18
Alexander Hagg, Alexander Asteroth, and Thomas Bäck.
Prototype discovery using quality-diversity.
In Parallel Problem Solving from Nature, PPSN 2018, Part I,
pages 500–511. Springer, 2018.
DBLP:journals/tciaig/AlvarezDFT22
Alberto Alvarez, Steve Dahlskog, José M. Font, and Julian Togelius.
Interactive constrained map-elites: Analysis and evaluation of the
expressiveness of the feature dimensions.
IEEE Transactions on Games, 14(2):202–211, 2022.
DBLP:conf/aaai/FontaineLKMTHN21
Matthew C. Fontaine, Ruilin Liu, Ahmed Khalifa, Jignesh Modi, Julian Togelius,
Amy K. Hoover, and Stefanos Nikolaidis.
Illuminating mario scenes in the latent space of a generative
adversarial network.
In AAAI Conference on Artificial Intelligence, AAAI 2021,
pages 5922–5930. AAAI Press, 2021.
DBLP:journals/ec/GaoNN21
Wanru Gao, Samadhi Nallaperuma, and Frank Neumann.
Feature-based diversity optimization for problem instance
classification.
Evolutionary Computation, 29(1):107–128, 2021.
DBLP:conf/gecco/DoBN020
Anh Viet Do, Jakob Bossek, Aneta Neumann, and Frank Neumann.
Evolving diverse sets of tours for the travelling salesperson
problem.
In Genetic and Evolutionary Computation Conference, GECCO
2020, pages 681–689. ACM, 2020.
DBLP:conf/foga/NikfarjamB0N21
Adel Nikfarjam, Jakob Bossek, Aneta Neumann, and Frank Neumann.
Computing diverse sets of high quality TSP tours by eax-based
evolutionary diversity optimisation.
In Foundations of Genetic Algorithms, FOGA 2021, pages
9:1–9:11. ACM, 2021.
DBLP:conf/gecco/NikfarjamBN021
Adel Nikfarjam, Jakob Bossek, Aneta Neumann, and Frank Neumann.
Entropy-based evolutionary diversity optimisation for the traveling
salesperson problem.
In Genetic and Evolutionary Computation Conference, GECCO
2021, pages 600–608. ACM, 2021.
DBLP:conf/gecco/BossekN021
Jakob Bossek, Aneta Neumann, and Frank Neumann.
Breeding diverse packings for the knapsack problem by means of
diversity-tailored evolutionary algorithms.
In Genetic and Evolutionary Computation Conference, GECCO
2021, pages 556–564. ACM, 2021.
DBLP:conf/gecco/Bossek021
Jakob Bossek and Frank Neumann.
Evolutionary diversity optimization and the minimum spanning tree
problem.
In Genetic and Evolutionary Computation Conference, GECCO
2021, pages 198–206. ACM, 2021.
DBLP:conf/gecco/NeumannB021
Aneta Neumann, Jakob Bossek, and Frank Neumann.
Diversifying greedy sampling and evolutionary diversity optimisation
for constrained monotone submodular functions.
In Genetic and Evolutionary Computation Conference, GECCO
2021, pages 261–269. ACM, 2021.
DBLP:conf/gecco/NeumannA022
Aneta Neumann, Denis Antipov, and Frank Neumann.
Coevolutionary pareto diversity optimization.
In Genetic and Evolutionary Computation Conference, GECCO
2022, pages 832–839. ACM, 2022.
DBLP:conf/gecco/Neumann23
Aneta Neumann, Sharoltte Goulder, Xiankun Yan, Gregory Sherman, Benjamin
Campbell, Mingyu Guo, and Frank Neumann.
Evolutionary diversity optimization for the detection and concealment
of spatially defined communication networks.
In Genetic and Evolutionary Computation Conference, GECCO
2023, pages 1436–1444. ACM, 2023.
DBLP:conf/gecco/UlrichT11
Tamara Ulrich and Lothar Thiele.
Maximizing population diversity in single-objective optimization.
In Genetic and Evolutionary Computation Conference, GECCO
2011, pages 641–648. ACM, 2011.
DBLP:conf/gecco/NeumannGDN018
Aneta Neumann, Wanru Gao, Carola Doerr, Frank Neumann, and Markus Wagner.
Discrepancy-based evolutionary diversity optimization.
In Genetic and Evolutionary Computation Conference, GECCO
2018, pages 991–998. ACM, 2018.
DBLP:conf/gecco/NeumannG0019
Aneta Neumann, Wanru Gao, Markus Wagner, and Frank Neumann.
Evolutionary diversity optimization using multi-objective indicators.
In Genetic and Evolutionary Computation Conference, GECCO
2019, pages 837–845. ACM, 2019.
DBLP:journals/telo/DoGNN22
Anh Viet Do, Mingyu Guo, Aneta Neumann, and Frank Neumann.
Analysis of evolutionary diversity optimization for permutation
problems.
ACM Transactions on Evolutionary Learning and Optimisation,
2(3):11:1–11:27, 2022.
DBLP:conf/ppsn/NikfarjamDN22
Adel Nikfarjam, Anh Viet Do, and Frank Neumann.
Analysis of quality diversity algorithms for the knapsack problem.
In Parallel Problem Solving from Nature, PPSN 2022, Part II,
pages 413–427. Springer, 2022.
DBLP:conf/gecco/DoerrGN16
Benjamin Doerr, Wanru Gao, and Frank Neumann.
Runtime analysis of evolutionary diversity maximization for
oneminmax.
In Genetic and Evolutionary Computation Conference, GECCO
2016, pages 557–564. ACM, 2016.
DangFKKLOSS16
Duc-Cuong Dang, Tobias Friedrich, Timo Kötzing, Martin S. Krejca,
Per Kristian Lehre, Pietro S. Oliveto, Dirk Sudholt, and Andrew M. Sutton.
Escaping local optima with diversity mechanisms and crossover.
In Genetic and Evolutionary Computation Conference, GECCO
2016, pages 645–652. ACM, 2016.
DangFKKLOSS18
Duc-Cuong Dang, Tobias Friedrich, Timo Kötzing, Martin S. Krejca,
Per Kristian Lehre, Pietro S. Oliveto, Dirk Sudholt, and Andrew M. Sutton.
Escaping local optima using crossover with emergent diversity.
IEEE Transactions on Evolutionary Computing, 22(3):484–497,
2018.
GielL10
Oliver Giel and Per Kristian Lehre.
On the effect of populations in evolutionary multi-objective
optimisation.
Evolutionary Computation, 18(3):335–356, 2010.
|
http://arxiv.org/abs/2307.04081v1 | 20230709014122 | Score-based Conditional Generation with Fewer Labeled Data by Self-calibrating Classifier Guidance | [
"Paul Kuo-Ming Huang",
"Si-An Chen",
"Hsuan-Tien Lin"
] | cs.CV | [
"cs.CV",
"cs.LG"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
====================================================================================
Score-based Generative Models (SGMs) are a popular family of deep generative models that achieves leading image generation quality. Earlier studies have extended SGMs to tackle class-conditional generation by coupling an unconditional SGM with the guidance of a trained classifier. Nevertheless, such classifier-guided SGMs do not always achieve accurate conditional generation, especially when trained with fewer labeled data. We argue that the issue is rooted in unreliable gradients of the classifier and the inability to fully utilize unlabeled data during training. We then propose to improve classifier-guided SGMs by letting the classifier calibrate itself. Our key idea is to use principles from energy-based models to convert the classifier as another view of the unconditional SGM. Then, existing loss for the unconditional SGM can be adopted to calibrate the classifier using both labeled and unlabeled data. Empirical results validate that the proposed approach significantly improves the conditional generation quality across different percentages of labeled data. The improved performance makes the proposed approach consistently superior to other conditional SGMs when using fewer labeled data. The results confirm the potential of the proposed approach for generative modeling with limited labeled data.
§ INTRODUCTION
Score-based Generative Models (SGMs) capture the underlying data distribution by learning the gradient function of the log-likelihood on data, also known as the score function. SGMs, when coupled with a diffusion process that gradually converts noise to data, can often synthesize higher-quality images than other popular alternatives, such as generative adversarial networks <cit.>. SGMs attracted research attention and demonstrated promising performance not only in image generation <cit.> but also in audio synthesis <cit.>, natural language generation <cit.>, and various other fields.
Many successful SGMs above focus on unconditional generation, which models the data distribution without considering other variables <cit.>. When aiming to generate data with some control, it is necessary to model the conditional distribution concerning another variable, such as the class label for generating images from a particular class.
Such conditional SGMs will be the focus of this paper. They have achieved cutting-edge performance for class-conditional generation <cit.>, image inpainting <cit.>, and audio upsampling <cit.>.
There are two major families of conditional SGMs. The family of Classifier-Free SGMs designs specific conditional network architectures with their losses derived from the conditional score functions <cit.>. Such SGMs are known to generate high-fidelity images in fully-supervised settings where all data are labeled. Nevertheless, they are often criticized for generating data with less diversity, favoring some easier classes while being inaccurate for some harder classes. Furthermore, their performance drops significantly as the proportion of labeled data decreases, making them less preferable in semi-supervised settings.
Classifier-Guided SGMs (CGSGMs) form another family of conditional SGMs that address the aforementioned issues by decomposing the conditional score function into a mixture of the unconditional score function and the gradient of an auxiliary classifier <cit.>.
For instance, the vanilla CGSVM <cit.> trains the unconditional SBM with the popular Denoising Score Matching (DSM) <cit.> technique that learns the score function from noise-perturbed data, and the classifier with the usual cross-entropy loss from labeled data. The additional classifier improves the accuracy of conditional generation and allows better control of the trade-off between generation diversity and fidelity <cit.>. Furthermore, because the unconditional SBM can be trained with either labeled or unlabeled data in principle, CGSGMs potentially fit the semi-supervised setting better by requiring fewer labeled data.
The quality of the auxiliary classifier is critical for CGSGMs. If the classifier is overly confident in its predictions, as often happens with cross-entropy loss <cit.>, the resulting conditional scores may be unreliable. This, in turn, leads to low generation accuracy, even if the unconditional scores are reliable enough to ensure decent generation fidelity.
Robust CGSGM <cit.> trains an adversarial robust classifier instead of a usual one to improve the quality of the auxiliary classifier. Somehow there is no theoretical guarantee that adversarial robustness is related to reliable conditional scores. Denoising Likelihood Score Matching <cit.> proposes to calibrate the classifier on the labeled data externally, leveraging the help of the unconditional SGM. Then, the training of the classifier is dependent on having a trained unconditional SGM first.
Our proposed approach is aligned with both techniques above to design a better loss to train the classifier. Still, it significantly differs from them by letting the classifier self-calibrate. Unlike the robust CGSGM, the self-calibration technique carries a sound theoretical guarantee by converting the classifier to another view of the unconditional SGM when reinterpreting the classifier through the angle of energy-based models. The novel view allows reusing DSM seamlessly to design a Self-Calibration (SC) loss (as illustrated with ℒ_SC in Fig. <ref>) that can be used on the classifier without dependence to the unconditional SGM. Furthermore, the SC loss can be effortlessly applied to both labeled and unlabeled data, resulting in immediate advantages in the semi-supervised setting.
We demonstrate the effect of self-calibration by visualizations on a toy data set. The results justify that our proposed CGSGM with the SC loss (CGSGM-SC) approach results in more accurate classifier gradients, thus enhancing the estimation of the conditional scores.
We further conduct thorough experiments on CIFAR-10 and CIFAR-100 datasets to validate the advantages of the proposed approach. The results confirm that CGSGM-SC is superior to the vanilla CGSGM and state-of-the-art techniques in the CGSGM family. Furthermore, in an extreme setting of having only 5% of the data being labeled, CGSGM-SC, which can use unlabeled data to self-calibrate the classifier, is significantly better than both classifier-guided and classifier-free SGMs, which cannot easily take the unlabeled data into account. The results confirm the potential of CGSGM-SC in scenarios where labeled data are costly to obtain.
§ BACKGROUND
Consider a data distribution p(x) where x∈ℝ^d. SGMs aim to generate samples from p(x) via the information contained in the score function ∇_xlog p(x), which is learned from data. We first introduce how the score function can be efficiently learned from data in Section <ref>, which is related to the derivation of our proposed loss. Then, we discuss how a diffusion process can be combined with learning a score function to effectively sample from p(x) in Section <ref>. Finally, we review studies that extend SGMs to conditional SGMs in Section <ref>.
§.§ Learning the score function
Learning the score function aims to choose the best function from a family of functions {s_θ(x)}_θ, such as deep learning models parameterized by θ, to approximate the score function ∇_x log p(x) of interest. The learning is based on some data {x_n}_n=1^N that are assumed to be sampled from p(x). It has been shown that the aim can be achieved by optimizing the in-sample version of the following score-matching loss over θ:
ℒ_SM=𝔼_p(x)[tr(∇_x s_θ(x))+1/2‖ s_θ(x)‖^2_2],
where tr(·) denotes the trace of a matrix and ∇_x s_θ(x)=∇^2_x log p(x) is the Hessian matrix of log-likelihood p(x). Somehow calculating the score-matching loss requires O(d) passes of computation for x ∈ℝ^d, which makes the optimization process computationally prohibitive on high-dimensional data.
Several previous studies <cit.> attempted to resolve the computational issue by approximating or transforming score matching into equivalent objectives. One standard approach nowadays is called Denoise Score Matching (DSM) <cit.>, which learns the score function of a noise-perturbed data distribution q(x̃) instead. DSM typically assumes that q(x̃) comes from the original distribution p(x)
injected with a pre-specified noise q(x̃|x). Then, it has been proved <cit.> that the score function can be learned by minimizing the in-sample version of
𝔼_q(x̃|x)p(x)[1/2‖ s_θ(x̃) - ∇_x̃log q(x̃|x)‖_2^2],
where ∇_x̃log q(x̃|x) is the score function of the noise distribution centered at x. DSM is generally more efficient than the original score matching and is scalable to high-dimensional data as it replaces the heavy computation on the Hessian matrix with simple perturbations that can be efficiently computed from data.
§.§ Generating from the score function by diffusion
Assume that we hope to sample from some unknown target distribution p(x) = p_0(x), and the distribution can be transited to a known prior distribution p_T(x) through a Markov chain that is described with
some stochastic differential equation (SDE) <cit.>:
dx(t)=f(x(t),t)dt+g(t)dw,
where the Markov chain is computed for 0 ≤ t < T using the drift function f(x(t),t) that describes the overall movement and the dispersion function g(t) that describes how the noise w from a standard Wiener process enters the system.
To sample from p(x) = p_0(x), the VE-SDE framework <cit.> proposes
to reverse the SDE from p_T(x) to p_0(x), which turns out to operate with another SDE <cit.>:
dx=[f(x(t),t)-g(t)^2 s(x(t), t)]dt+g(t)dw̅
where w̅ is a standard Wiener process when time-step flows from T back to 0 and s(x(t), t) ≡∇_xlog p_t(x(t))=s(x(t), t) denotes a time-dependent score function. If we can learn the score function s(x(t), t), the diffusion process in (<ref>) can then be used to take any instance sampled from the known p_T(x) to a sample from the unknown p(x) = p_0(x).
Learning the time-dependent score function s(x(t), t)) can be done by minimizing an time-generalized (in-sample) version of the DSM loss because the diffusion process can be viewed as one particular way of injecting noise. The extended DSM loss is defined as
ℒ_DSM(θ)=𝔼_t[λ(t)𝔼_x^(t),x^(0)[1/2‖ s_θ(x(t),t) - s_t(x^(t)|x^(0))‖_2^2]],
where t is selected uniformly between 1 and T, x^(0)∼ p_t(x), x^(0)∼ p_0(x), and s_t(x |x^(0)) denotes the score function of p_t(x | x^(0)), and λ(t) is a weighting function that balances the loss of different time steps.
In this paper, we take the same drift, dispersion, and weighting functions f(x,t), g(t), and λ(t) as the original VE-SDE framework <cit.>.
§.§ Related studies of conditional score-based generative models
In conditional SGMs, we are given some labeled data {(x_m, y_m)}_m=1^M in addition to the unlabeled data {x_n}_n=M+1^M+N, where y ∈{1, 2, …, K} denotes the class label. The case of N = 0 is called the fully-supervised setting, while we focus on the more challenging semi-supervised setting with N > 0 (and possibly N ≫ M) in this paper.
Conditional score-based generative models aim to learn the conditional score function ∇_x log p(x | y) from the data and then generate samples from p(x | y). Previous studies <cit.> showed how to decompose the conditional score function using Bayes' theorem:
∇_x log p(x|y) =∇_x[log p(x) + log p(y|x)- log p(y)]= ∇_xlog p(x) + ∇_xlog p(y|x).
The term log p(y) can be dropped because it is not a function of x and is thus of gradient 0. The decomposition shows that conditional generation can be achieved by an unconditional SGM that learns the score function ∇_x log p(x) plus an extra conditional gradient term ∇_xlog p(y|x).
The vanilla form of Classifier Guidance (CG) for SGM estimates ∇_xlog p(y|x) with an auxiliary classifier trained from the cross-entropy loss on the labeled data and learns the unconditional score function by the DSM loss ℒ_DSM that can in principle be applied on both the labeled and unlabeled data.
Nevertheless, the classifier within the vanilla CG approach is known to be potentially over-confident <cit.> on its predictions, which in term results in inaccurate gradients.
The issue can mislead the conditional generation process and decrease class-conditional generation quality.
<cit.> propose to address the issue by tuning the term ∇_xlog p(y|x) with a scaling parameter λ_CG≠ 1.
∇_x log p(x|y) = ∇_xlog p(x) + λ_CG∇_xlog p_ϕ(y|x),
where p_ϕ(y|x) is the posterior probability distribution outputted by a classifier parameterized by ϕ. Increasing λ_CG sharpens the distribution p_ϕ(y | x), guiding the generation process to produce less diverse but higher fidelity samples. While the tuning heuristic is effective in improving the vanilla CG approach, it is not backed by sound theoretical explanations.
<cit.> propose to resolve the issue differently by enhancing the adversarial robustness of the classifier. It is empirically observed that adversarially robust classifiers produce more interpretable and perceptually more aligned <cit.> gradients. Somehow it remains theoretically unclear whether robust classifiers are truly more accurate for capturing the true data distribution.
<cit.> propose the Denoising Likelihood Score Matching (CG-DLSM) approach that calibrates the classifiers to resolve the issues. The calibration is done by designing a loss computed from the outputs of a trained unconditional SGM to regularize the classifier during training. CG-DLSM achieves state-of-the-art performance within the CGSGM family in the fully-supervised setting. Somehow because of the design, the learning of unconditional SGM and the classifier needs to be done in sequential steps, losing the computational advantage of the original vanilla CGSGM of being able to train the two components in parallel. Furthermore, it is not clear whether the unlabeled data in the semi-supervised setting could be helpful in improving the classifier under the design.
The approaches above are all CGSGMs. Another popular approach for conditional SGM is Classifier-Free Guidance (CFG) <cit.>. The approach parameterizes its deep learning model with more sophisticated architectures such that the class labels y can be included as inputs to calculate the score. A null token y_nil is used to indicate unconditional score calculation, which is linearly combined with conditional score calculation for some specific y to form the final estimate of s(x | y). CFG is a state-of-the-art conditional SGM in the fully-supervised setting. Nevertheless, as we shall show in our experiments, its performance drops significantly in the semi-supervised setting, as the conditional parts of the architecture may not get enough labeled data during training. The disadvantages of CFG and other CGSGMs in the semi-supervised setting motivate us to design another CGSGM that (1) comes with theoretical justifications; (2) includes a classifier that can be trained in parallel to the unconditional SGM; (3) can leverage both the unlabeled and labeled data to achieve better performance in the semi-supervised setting.
§ SELF-CALIBRATION FOR CLASSIFIER GUIDANCE
§.§ Motivation
As mentioned in Section <ref>, inaccurate gradients of classifiers could potentially misguide the conditional generation process.
Therefore, we need an efficient way to calibrate the classifiers.
Motivated by JEM <cit.> where the classifiers are calibrated by being reinterpreted as an energy-based model (EBM), we propose to connect the EBM and SGM and calibrate the classifiers by interpreting them as EBMs in a similar approach.
To be more specific, we formulate a self-calibration loss that utilizes denoising score matching to calibrate the score function estimated by the classifier.
§.§ Formulation of self-calibration loss
In this work, we adopted the framework of score-based generative modeling using stochastic differential equations (SDEs) <cit.>. Given a target distribution p_0(x) and a known prior distribution p_T(x) (typically a Gaussian distribution) where the transition between them is a diffusion process with timestep 0≤ t< T, we can describe the diffusion process and its reverse process using SDEs. To incorporate the results of Section <ref> into this framework, we introduce the time-dependent version of ∇_x log p(x) and ∇_x log p(y|x). That is ∇_x log p_t(x(t)) and ∇_x log p_t(y|x(t)), respectively, where x(t)∼ p_t. Denoising score matching (DSM) <cit.> is often utilized to train the score-based model under this framework due to its close relationship with diffusion process modeling. A time-generalized cross-entropy loss is adopted o train the classifier.
Inspired by JEM <cit.>, we propose to improve CGSGM through self-calibration during the training stage. We reinterpret the classifier as a time-dependent EBM and obtain the score function by calculating the gradient. Since both energy function -log p(x) and score function ∇_x log p(x) are calculated from the log-likelihood function, we hypothesize that integrating EBM-related objectives into classifier training can be beneficial to CGSGM. To incorporate the energy function into our framework, we used a time-dependent version of the transformation described in JEM <cit.>:
E_ϕ,t(x) = -log∑_yexp(f_ϕ,t(x)[y])= -LogSumExp_y(f_ϕ,t(x)[y])
where f_ϕ,t(x)[y] is the output logits of the classifier. The score function can then be computed like the following:
s_ϕ(x,t) =∇_x LogSumExp_y(f_ϕ,t(x)[y])
To calibrate this score estimated by the classifier, we adopt DSM to calculate the Self-calibration Loss (SC loss):
ℒ_SC(ϕ)=𝔼_t[λ(t)𝔼_x_t,x_0[1/2‖ s_ϕ(x_t,t)-s_t(x_t|x_0)‖_2^2]]
where x_t∼ p_t, x_0∼ p_0, and s_t(x_t|x_0) denotes the score function of the noise centered at x_0. Fig. <ref> summarizes the calculation of the proposed SC loss. After the self-calibration loss is obtained, it is summed with the cross-entropy loss to train the classifier. The total loss can be written as:
ℒ_CLS(ϕ)=ℒ_CE(ϕ)+λ_SCℒ_SC(ϕ)
where ℒ_CE is the cross-entropy loss and λ_SC is a hyperparameter. By applying self-calibration, the classifier should be able to more accurately estimate the score function of the underlying data distribution, which implies the underlying data distribution itself is also more accurately estimated. As a result, the gradients of the classifiers should be more aligned with the ground truth as it is calculated from the estimated distribution.
After self-calibration, the classifier then can be used just like the original classifier to guide an unconditional SGM to achieve conditional generation. Note that since our method calibrates the classifier in training time and scaling classifier gradient is done in sampling time, we can easily combine the two methods to achieve better performance.
§.§ 2D toy dataset
We use a 2D toy dataset containing two classes to demonstrate the effects of the self-calibration loss. The data distribution is shown in Fig. <ref>, where the two classes are shown in two different colors. After training the classifiers on the toy dataset with (1) only cross-entropy loss and (2) both cross-entropy loss and self-calibration loss, we plot the gradients ∇_x log p(y|x) estimated by the classifiers and compare them with the ground truth. Also, we added the ground truth unconditional score to the estimated gradients just like CGSGM and compared the results with the real conditional score. Additional quantitative measurements of the toy dataset is included in Appendix <ref>.
Fig. <ref> shows the ground truth classifier gradient (Fig. <ref>) and the gradients estimated by classifiers trained on the toy dataset (1) without self-calibration (Fig. <ref>) and (2) with self-calibration (Fig. <ref>).
Uncalibrated classifiers produce gradients that contain rapid changes in magnitude across the 2D space, with frequent fluctuations and mismatches with the ground truth.
Such fluctuations can impede the convergence of the reverse diffusion process to a stable data point, leading SGMs to generate noisier samples.
Moreover, the divergence from the ground truth gradient can misguide the SGM, leading to generation of samples from incorrect classes.
Uncalibrated classifiers also tend to generate large gradients near the distribution borders and tiny gradients elsewhere.
This implies that when the sampling process is heading toward the incorrect class, such classifiers are not able to “guide" the sampling process back toward the desired class.
In contrast, the introduction of self-calibration results in estimated gradients that are more stable, continuous across the 2D space, and better aligned with the ground truth.
This stability results in a smoother generation process and contributes to the production of higher-quality samples.
§.§ Using self-calibration loss on semi-supervised learning
In this work, we also explore the benefit of self-calibration loss in semi-supervised setting where only a small proportion of data are labeled.
In original classifier guidance, the classifiers are solely trained from labeled data.
The lack of label in semi-supervised setting leads to more challenges to learn an unbiased classifier.
With self calibration, we are able to better utilize the large amount of unlabeled data by calculating the self calibration loss with all data.
To incorporate the loss and utilize the unlabeled samples during training time, we changed the way of calculating ℒ_CLS from Eq. <ref>. As illustrated in Fig. <ref>, the entire batch of data is used to calculate ℒ_SC, but only the labeled data is used to calculate ℒ_CE. During training, we observed that when the majority is unlabeled data, the cross-entropy loss does not converge to a low-and-steady stage if the algorithm randomly samples from all training data. We suspect this is due to the low percentage of labeled data in each batch. Therefore, we changed the way of sampling batches. We always ensure that half of the data is labeled while the other half is not. Appendix <ref> summarizes the semi-supervised training process of the classifier.
Note that even though the classifier is learning a time-generalized classification task, we can still make it perform as an ordinary classifier that classifies the unperturbed data by setting the input timestep t=0. Therefore, we can easily incorporate many other common semi-supervised classification methods like pseudo-labeling <cit.>, self-training, and noisy student <cit.>.
§ EXPERIMENTS
We have tested our method on a toy dataset (Section <ref>) to provide a high-level view of how self-calibration can improve classifiers in terms of producing accurate gradients. In this section, we present the experimental results on the CIFAR-10 and CIFAR-100 datasets to demonstrate the improvement of CGSGM after incorporating our method on different percentage of labeled data (Section <ref>). Randomly selected images of CGSGM before and after self-calibration on the dataset CIFAR-10 are shown in Appendix <ref>. For conditional metrics, we report the average scores across all classes. Results of individual classes on the CIFAR-10 dataset are included in Appendix <ref>.
§.§ Experimental setup
In the following sections, we tested our methods on the CIFAR-10 and CIFAR-100 datasets for image generation. We demonstrate that our methods are able to improve generation quality both conditionally and unconditionally with different percentage of labeled data.
Implementation details
We follow NCSN++ <cit.> to implement the unconditional score estimation model. We also adapted the encoder part of NCSN++ as the classifier used in CGSGM <cit.>.
Sampling method: We used Predictor-Corrector (PC) samplers <cit.> with 1000 sampling steps.
Evaluation metrics: Besides commonly used metrics Frechet Inception Distance (FID) <cit.> and Inception Score (IS) <cit.>, we also evaluated class-conditional performance of our methods using several different methods. This includes intra-FID, which measures the average FID for each class, and generation accuracy (on the CIFAR-10 dataset), which uses a pre-trained ViT <cit.> classifier to check whether the samples are generated in the correct class. The test accuracy of the pre-trained ViT is 98.52% on the CIFAR-10 dataset.
Baseline methods:
The baseline methods used in our work include:
* Cond: Adopts conditional SGMs by conditional normalization techniques <cit.> rather than classifier guidance.
* CFG-labeled: Classifier-free guidance<cit.> using only labeled data is applied.
* CFG-all: Classifier-free guidance<cit.> using only labeled data to train the conditional part of the model and all data to train the unconditional part of the model.
* CG: Vanilla classifier guidance.
* CG-DLSM: Classifier guidance with DLSM loss <cit.> applied.
§.§ Experiment Result
Table <ref> and Fig. <ref> present the performance of all methods when applied to varying percentages of labeled data.
Notice that it includes the fully-supervised setting when 100% of data are labeled.
CG-SC-labeled implies self-calibration is only applied on labeled data while CG-SC-all implies self-calibration is applied on all data.
Conditional SGMs vs Unconditional SGMs.
The first observation from our results is that conditional SGMs, including Cond, CFG-labeled, and CFG-all, consistently excel in generation accuracy.
However, when the quantity of labeled data decreases below 40%, a significant performance drop is witnessed in these models.
These conditional SGMs, while generating high-quality images, tend to lose diversity when working with fewer labeled data.
This occurs mainly because of the lack of labeled data in training phase, leading them to generate samples closely mirroring the distribution of the labeled data instead of all data.
In contrast, unconditional SGMs, such as CG, demonstrate superior performance when the majority of the data is unlabeled, as they are capable of leveraging both labeled and unlabeled data during training.
Classifier-Guided SGMs (CGSGMs) vs Conditional SGMs
Our experimental results align with our expectations that CGSGMs produce improved performance compared to conditional SGMs.
The CG method exhibits a consistent performance in terms of FID and inception scores across varying percentages of labeled data when evaluated using unconditional metrics.
Notably, when unlabeled data is in the majority, we observe a 16% drop in generation accuracy on the CIFAR-10 dataset.
Despite this, the intra-FID of CG significantly outperforms that of conditional SGMs on both datasets.
As for the proposed method, incorporating self-calibration with labeled data does not majorly affect unconditional metrics but substantially improves conditional metrics.
This process reduces intra-FID by 8.25 and 17.86 on the CIFAR-10 and CIFAR-100 dataset respectively and increases generation accuracy on CIFAR-10 by up to 23%.
The results demonstrate that with self-calibration, the classifier can better represent the class-conditional distribution even when labeled data is limited.
Leverage unlabeled data for semi-supervised conditional generation
Intuitively, incorporating unlabeled data into the computation of self-calibration loss would enhance the quality of conditional generation, because the classifier can exploit additional information from unlabeled data during the training phase.
As the proportion of labeled data decreases, this benefit of leveraging unlabel data should become more significant.
As our experimental results show, conditional metrics do not differ greatly when the proportion of labeled data ranges between 40% and 100%.
However, when the percentage of labeled data falls below 40%, the use of unlabeled data significantly improves intra-FID and generation accuracy.
Specifically, with just 5% labeled data, intra-FID improves by 12.22, and generation accuracy increases by 22.8% compared to the original CG.
These results affirm our expectation that as the quantity of labeled data decreases, the beneficial impact of utilizing unlabeled data increases.
§ CONCLUSION
In this work, we verify that the existing CGSGM approach results in a high generation fidelity but low accuracy. We hypothesize that the root cause lies in the unreliable scores produced by the classifiers and design a Self-Calibration Loss to enhance the classifier directly towards better scores without resorting to an external SGM. The Self-Calibration Loss is derived from rigorous principles when viewing the classifier as an energy-based model. We demonstrate three immediate benefits of the proposed Self-Calibrating CGSGM approach. Using the toy dataset, we show that the scores computed from the approach are indeed closer to the ground-truth scores. Secondly, across all percentages of labeled data, our proposed approach outperforms the existing CGSGM in the semi-supervised setting. Lastly, our empirical study justifies that our proposed approach can consistently reach the best intra-FID by seamlessly leveraging the power of unlabeled data, when compared to other conditional SGMs. The benefits establish the rich potential of the proposed approach.
§ LIMITATIONS
The major limitation of our work lies in the selection of datasets. We can only afford to conduct experiments on smaller and lower-resolution datasets (CIFAR-10 and CIFAR-100) because of limited computational resources. In particular, even with those smaller data, training, sampling, and testing a single approach on a single setting once requires up to 210 hours (more than a week) with 4 NVIDIA Tesla V100 GPUs. We understand that conducting more experiments on larger and higher-resolution datasets can further strengthen our claims, but those experiments are not affordable to us. While we tested on only two datasets, the observed results are consistent—our proposed approach achieves the best class-conditional performance in the semi-supervised setting with much fewer labeled data.
plainnat
§ DETAILED CLASS-CONDITIONAL GENERATION MEASUREMENTS OF CIFAR-10
Section <ref> contains the class-conditional measurements averaged among all classes of CIFAR-10. This section includes a more detailed result that contains the measurement of each class.
§ TRAINING ALGORITHM FOR SEMI-SUPERVISED SELF-CALIBRATING CLASSIFIER
[h]
Semi-supervised classifier training with self-calibration loss
§ QUANTITATIVE MEASUREMENTS OF TOY DATASET
Table <ref> shows the quantitative measurements of the methods on the toy dataset. First, we compared the gradients ∇_x log p(y|x) estimated by the classifiers with the ground truth by calculating the mean squared error (first column) and cosine similarity (second column). We observed that after self-calibration, the mean squared error of estimated gradients can be lowered by 18%, and tuning the scaling factor can further improve it to 36%. This improvement after scaling implies that the direction of gradients is more aligned with the ground truth, and scaling can further reduce the mismatch between the magnitude of the classifier and the ground truth. In terms of cosine similarity, self-calibration grants the classifiers an improvement of 42%. The numerical results agree with our previous observation that after self-calibration, classifiers align better with the ground truth in terms of both direction and magnitude.
Then, we add the unconditional score of the training data distribution to the classifier gradients to calculate the conditional scores and compare the results with the ground truth. As we can see, the classifiers are able to estimate conditional scores with a cosine similarity of 0.9175 even without self-calibration. The result shows that with a well-trained unconditional SGM, in which we use the ground truth unconditional score in this case, CGSGM is able to produce conditional scores pointing in the correct directions in most cases. This explains why the original CGSGM is able to generate samples with decent quality. After applying the self-calibration loss and scaling method, we can further improve the cosine similarity to 0.9689, which we believe can enhance the quality of class-conditional generation.
§ TUNING THE SCALING FACTOR FOR CLASSIFIER GUIDANCE
This section includes the experimental results of tuning the scaling factor λ_CG for classifier guidance with and without self-calibration under fully-supervised setting.
Fig. <ref> shows the result of tuning the scaling factor λ_CG for classifier guidance. While tuning λ_CG with and without self-calibration, we can see that self-calibration does not affect unconditional performance by much. However, when evaluated with conditional metrics, the improvement after incorporating self-calibration becomes more significant. The improvement in intra-FID is up to 7.9 while the generation accuracy can improve up to 13%.
§ IMAGES GENERATED BY CLASSIFIER GUIDANCE WITH AND WITHOUT SELF-CALIBRATION
This section includes images generated by classifier guidance with (first 6 images) and without (last 6 images) self-calibration after training on different percentage of labeled data. Each row corresponds to a class in the CIFAR-10 dataset.
|
http://arxiv.org/abs/2307.04461v1 | 20230710101657 | Multi-modal Graph Learning over UMLS Knowledge Graphs | [
"Manuel Burger",
"Gunnar Rätsch",
"Rita Kuznetsova"
] | cs.LG | [
"cs.LG"
] |
Deformations at Earth's dayside magnetopause during quasi-radial IMF conditions: Global kinetic simulations and soft X-ray imaging
Chi Wang
August 12, 2023
==================================================================================================================================
Clinicians are increasingly looking towards machine learning to gain insights about patient evolutions. We propose a novel approach named Multi-Modal UMLS Graph Learning (MMUGL) for learning meaningful representations of medical concepts using graph neural networks over knowledge graphs based on the unified medical language system. These representations are aggregated to represent entire patient visits and then fed into a sequence model to perform predictions at the granularity of multiple hospital visits of a patient. We improve performance by incorporating prior medical knowledge and considering multiple modalities. We compare our method to existing architectures proposed to learn representations at different granularities on the MIMIC-III dataset and show that our approach outperforms these methods. The results demonstrate the significance of multi-modal medical concept representations based on prior medical knowledge.
We provide our code here[<https://anonymous.4open.science/r/mmugl/>] and showcase some of our
results with an online demo available under this link[<https://mmugl.dnsalias.org>].
§ INTRODUCTION
Modern healthcare facilities record patient information as Electronic Health Records (EHR). EHR Datasets such as MIMIC-III <cit.>, HiRID <cit.>, and eICU <cit.> enable modeling of disease progressions within a single hospital visit, for example in Intensive Care Units (ICU) <cit.>, or progressions across multiple patient visits <cit.>. These progressions can be meaningfully encoded into patient representations using deep learning as shown by numerous prior works <cit.>. This large body of work highlights the value of strong patient representations which aggregate information across entire patient histories from multiple hospital stays, enabling clinicians to model potential risks in various predictive tasks regarding patients' evolution.
Further, we see advantages of recent multi-modal approaches in the ICU setting <cit.> and visit sequence modeling <cit.>. In multi-modal EHR representation learning <cit.>, we benefit from two modalities: structured EHR data (e.g., billing codes) and unstructured text information stored in rich clinical reports. Other modalities of medical data exist outside of in-hospital datasets, where a vast amount of prior medical knowledge is stored in static form in databases such as the Unified Medical Language System (UMLS <cit.>).
We identify two drawbacks of current UMLS based approaches <cit.>. First, the approaches do not consider a complete set of relational information stored in UMLS (considering multiple vocabularies) and solely use UMLS as a unified concept space. Second, prior solutions <cit.> specify the usage of hierarchical relations, which implies the use of an underlying graph in the form of a tree (single vocabulary). More complex graph structures inside and across vocabularies are thus omitted.
We introduce Multi-Modal UMLS Graph Learning (MMUGL) to overcome the previously stated limitations. MMUGL is a novel approach for learning representations over medical concepts extracted from the UMLS Metathesaurus in the form of a complex knowledge graph and relations; extracted using a simple and ambitious procedure considering a considerable set of vocabularies and all the relations across and within them.
We apply auto-encoder pretraining techniques (e.g., <cit.>). By training a shared latent space <cit.>, we bridge the modality gap between structured EHR codes and unstructured text. The approach includes rich prior knowledge important in the medical domain, deals with sample scarcity by relying on prior knowledge structure and pretraining techniques, and leverages multiple modalities as inputs.
§.§ Generalizable Insights about Machine Learning in the Context of Healthcare
The contributions of our work are threefold:
* In Section <ref>, we introduce a novel medical knowledge representation learning approach with graph neural networks (GNN) over knowledge graphs based on the UMLS Metathesaurus of previously unseen complexity. While modern machine learning techniques are unlocking amazing advancements in health-care, improved precision, early detection, personalized treatments, and democratized access, all by learning from large amounts of data, most of the accumulated medical knowledge often remains untouched by our algorithms. Prior work has considered to tap into this knowledge, but we go one step further and show, that we can extract large and complex knowledge graphs by considering a considerable amount of the entire UMLS Metathesaurus and build a strong structural prior into our machine learning model and gain performance in the process.
* We introduce a shared latent Concept Embedding (Sec. <ref>) space and a shared Visit Encoder (Sec. <ref>) to optimize the single latent space from any modality jointly in a parameter efficient manner. Prior work has established the importance of leveraging EHR records in their entirety, thus incorporating all the available modalities. In our work we show the benefits of grounding all modalities by the same prior knowledge and training a single latent space in end-to-end fashion for all input modalities (structured and unstructured EHR).
* In Section <ref> we demonstrate, that we strongly outperform prior graph-based works in pretraining and downstream tasks and can perform competitively with prior work trained at a much larger scale of data. We show the benefits of our large-scale knowledge graph, shared latent space from multiple modalities and tailored (pre-)training procedure.
§ RELATED WORK
In the following, we introduce related work in EHR modeling, knowledge graph learning, and graph learning in the context of EHRs.
EHR Various types of deep learning architectures have been proposed to learn representations at different granularities (patients, visits, histories, etc.) in EHR datasets. <cit.> propose EHR-specific visit sequence models. <cit.> propose to focus on the inherent structure of EHRs w.r.t. treatments, diagnosis, visits, and patients. <cit.> adapt the masked language modeling approach to learn medical concept embeddings.
Multi-Modality
Prior work has considered learning representations from either structured components of EHR data <cit.> or from unstructured clinical text reports <cit.>. <cit.> have proposed multi-modal architectures and <cit.> go a step further and introduce even stronger structural priors, while considering the two modalities of structured EHR data, as well as unstructured clinical reports.
Knowledge Graphs and GNNs
A vast amount of static prior medical knowledge often remains untouched in current modeling approaches. This prior knowledge can be extracted and transformed into knowledge graphs <cit.>. Existing work in natural language processing has established the benefits of knowledge graph representations to various downstream applications <cit.>; where the most recent approaches include GNNs <cit.>. We aim to leverage the recent success of GNNs, which apply graph convolutions over arbitrary graph structures to learn node (and edge) representations <cit.>.
Graph Learning in EHR
GRAM <cit.> proposed to include prior knowledge from medical ontologies such as the International Classification of Diseases (ICD).
To model structural and relational data explicitly, approaches have started to use GNNs. <cit.> proposed to use the Graph Attention <cit.> operator together with an architecture to pretrain embeddings over two ontologies.
Other works learn over heterogeneous graphs with different types of nodes <cit.>.
<cit.> construct a global graph of diseases, as well as dynamic local (within a single visit) subgraphs.
<cit.> focus on the EHR structure within a single visit.
Finally, <cit.> consider hyperbolic embeddings for medical ontologies. The learned embeddings can then be incorporated into task-specific architectures <cit.> to improve outcome predictions in different healthcare settings.
Previous approaches do consider dataset-specific structures such as the hierarchical organization of EHRs (patients, visits, etc.) and co-occurrence information or structure coming from ontologies. However, the explored set of ontologies is usually kept small and most of them are tree-like structures. To the best of our knowledge, no prior work has considered using a GNN directly on top of a complex large-scale ontology such as the UMLS Metathesaurus and the complete set of unstructured relational information within it.
Further, while previous work considered multiple modalities, they use fusion approaches to join modalities, which can require larger amounts of data to train effectively. Our work proposes to use the learned knowledge representations over the UMLS Metathesaurus as a single shared latent space for information coming from both the structured (billing codes) and unstructured modalities (clinical reports).
§ GLOSSARY
We consider an EHR dataset of multiple patients and present the following terminology:
* Patient: p_i indexed by i
* Visit: a patient p_i has one or multiple visits v_i,t indexed by t. A visit contains a set of medical concepts c ∈𝒞_i, t, the total set of medical concepts over the dataset is then 𝒞 = ∪_∀ i, t𝒞_i,t. A medical concept can be of different types and we distinguish them by index 𝒞(*):
* Disease: indexed by d s.t. 𝒞_i, t(d) and 𝒞(d) = ∪_∀ i, t𝒞_i,t(d) the total set of disease concepts
* Medication: (or prescriptions) with type m, similar to diseases we introduce 𝒞(m) and 𝒞_i,t(m)
* Concept from clinical reports: a set of medical concepts extracted from text data (clinical reports, Sec. <ref>). The total set of considered medical concepts from text 𝒞(n) = ∪_∀ i, t𝒞_i,t(n) where the set 𝒞_i,t(n) is collected from all reports at a specific visit t of patient i. The type is n for text note.
The vector representation of a visit considering data of a specific type * is 𝐯_𝐢,𝐭(*) ∈ℝ^k.
* Ontology: each of them has a vocabulary 𝒱_Ont and defines some relation amongst the members of the vocabulary using an edge set ℰ_Ont, which defines the ontology graph 𝒢_Ont = (𝒱_Ont, ℰ_Ont). We consider the following ontologies/databases:
* 𝒢_ICD (International Classification of Diseases) where 𝒞(d) ⊆𝒱_ICD
* 𝒢_ATC (Anatomical Therapeutic Chemical) where 𝒞(m) ⊆𝒱_ATC
* 𝒢_UMLS (Unified Medical Language System) where {𝒞(d) ∪𝒞(m) ∪𝒞(n)} = 𝒞⊆𝒱_UMLS
§ METHOD
The architecture consists of three main components and is derived from the work done by <cit.>; fig:architecture provides an overview.
* Concept embedding module f_θ(c): 𝒞 ↦ ℝ^k (parametrized by θ), which computes a representation for any given medical concept c.
* Visit encoding module Assume q = |𝒞_i, t| and r ∈{2, 3} the number of concept types considered (diseases and medications with or without concepts from text) then g_ψ(v): ℝ^q × k↦ℝ^r × k (parametrized by ψ), which, given all concept token representations of a single visit v computes single representations for each different type of tokens thereof.
* Predictor module which performs either a pretraining task on a single visit or a downstream fine-tuning task across a sequence of visits. In either case, this module receives representations for each visit of a patient from the previous visit encoding module.
In the following subsections, we introduce the Concept Embedding module (Sec. <ref>), present how we extract richer concepts from clinical reports (Sec. <ref>), encode the information (Sec. <ref>), and perform predictions (Sec. <ref>).
§.§ Concept Embeddings
We consider the following implementations of f_θ(c):
ICD/ATC Hierarchies
Based on the work done by <cit.>, we consider the two tree hierarchies ICD[We consider the 9th revision, as of working on MIMIC-III] for diseases and ATC for medications. In this case, we consider c ∈{𝒞(d) ∪𝒞(m)}. We compute the node embeddings 𝐍_* (⊕ for concatenation):
𝐍_𝒞(d) = GNN_θ_1(𝒢_ICD),
𝐍_𝒞(m) = GNN_θ_2(𝒢_ATC),
f_θ(c) = Lookup(𝐍_𝒞(d)⊕𝐍_𝒞(m))(c)
where we use a distinct (parametrized by θ_1 and θ_2) multi-layer GNN for each of the two hierarchies (Eqns. <ref>, <ref>) and then perform a lookup (retrieve nodes by index)
against the resulting node embeddings (Eqn. <ref>). In this case, we initialize all of the nodes with randomly initialized trainable embeddings. We refer to this approach to learn concept embeddings with ICD/ATC.
We can additionally consider co-occurrence information (e.g., <cit.>) to connect the two hierarchies. We refer to this approach with ICD/ATC-CO (details in Appendix <ref>).
MMUGL
We present our novel approach to rely on the UMLS Metathesaurus as a unified concept space to learn representations
for any general medical concept present in the database based on multiple modalities. Given that, we refer to our approach as Multi-Modal UMLS Graph Learning (MMUGL).
To constrain the number of concepts we consider from the database we use the set of clinical reports present in EHR datasets such as MIMIC-III <cit.>. Using an extraction pipeline (Sec. <ref>) we collect the set of medical concepts 𝒞(n); additionally, we ensure all of the concepts in the ICD and ATC hierarchies are present as well in our final vocabulary.
The final vocabulary {𝒞(d) ∪𝒞(m) ∪𝒞(n)} = 𝒞 = 𝒱_UMLS is used to construct 𝒢_UMLS by extracting all the edges in UMLS fully contained within the vocabulary. To simplify we consider all edges to be undirected.
In UMLS many concepts are annotated with a short natural language description. We use SapBERT <cit.>[<https://github.com/cambridgeltl/sapbert>], a pretrained language model fine-tuned to discriminate amongst UMLS concepts, to initialize the node embeddings from these descriptions. This contributes in two ways: (i) by not using trainable embeddings, we reduce the otherwise huge amount of free parameters given the large vocabulary 𝒱_UMLS (ii) we incorporate prior medical knowledge by considering the concept descriptions. We then train a multi-layer GNN on top of the extracted graph:
f_θ(c) = GNN_θ(𝒢_UMLS)(c)
To retrieve a concept, we return its computed node embedding. We additionally found it to be beneficial for performance to consider two distinct stacks of GNN layers over the same graph and perform a Max-Pooling operation after the final layer across the two stacks. This falls in line with using two distinct
GNNs in the simple ICD/ATC Hierarchy case presented in Eqn. <ref>.
§.§ Concept Extraction
The goal of our approach is to include data from additional modalities such as the clinical reports found in EHR datasets. MMUGL learns modality agnostic representations of medical concepts based on UMLS knowledge. It fuses discrete code information (e.g., ICD codes) with medical concepts extracted from text. The extraction with QuickUMLS <cit.> yields a set of medical concepts 𝒞_i,t(n) based on the collection of clinical reports of that particular visit. Further, we perform a rule-based negation extraction using NegEx <cit.>; for each concept, we extract a binary feature, whether it is negated or not, and concatenate it with its learned concept embedding (Eqn. <ref>). This is a crucial piece of information as clinical reports can both mention the existence or the absence of a certain condition.
§.§ Visit Encoder
We present implementations of the function g_ψ(v). In line with the work by <cit.>, we consider a multi-layer transformer without positional encodings. To aggregate a set of concepts into a single representation we use a learned token representation at the transformer output as the aggregate.
For each concept type in a given visit, we encode a separate representation using the same (weight-sharing) Transformer_ψ with parameters ψ where * ∈{d, m, n}. The aggregated representations g_ψ(v) for each modality are considered as the output of this module in MMUGL:
𝐯_𝐢,𝐭(*) = Transformer_ψ( f_θ( 𝒞(*)_i,t) ) [],
g_ψ(v) = ( 𝐯_𝐢,𝐭(d), 𝐯_𝐢,𝐭(m), 𝐯_𝐢,𝐭(n) )
we can also consider a case without the information from clinical reports (Eqn. <ref>), e.g., in cases where we
use a simpler graph such as ICD/ATC (Eqn. <ref>) or in MMUGL without 𝒞(n).
g_ψ(v) = ( 𝐯_𝐢,𝐭(d), 𝐯_𝐢,𝐭(m) )
§.§ Predictors and Training
In the following, we introduce the pretraining module and downstream fine-tuning modules.
§.§.§ Pretraining Module
We replicate the auto-encoding pretraining approach developed by <cit.> with the reconstruction loss ℒ_recon and perform four different predictions (from each of the two modalities, disease and prescription, as a source to either as the label) using distinct Multi-layer Perceptrons (MLP_∙→ * predicting type * from representations of type ∙) and attach a binary cross-entropy loss ℒ_BCE to model multi-label classification.
ℒ_recon = ∑_∙, * ∈{d, m}ℒ'(∙, *),
ℒ'(∙, *) = ℒ_BCE( MLP_∙→ *( 𝐯(∙) ) , 𝒞(*))
During pretraining, we additionally randomly mask and replace certain tokens at the input in Eqn. <ref> (same as <cit.>, inspired by masked language modeling <cit.>).
Weighted reconstruction pretraining
We consider a weighted version of Eqn. <ref>:
ℒ_recon = ∑_∙, * ∈{d, m} w_∙, * ℒ'(∙, *)
As some of the considered downstream tasks focus on disease diagnosis we consider a tailored disease-focused pretraining approach. In this setting, we omit the predictions (and loss signal) to medications and only predict diseases from either the visits aggregated disease or medication representation.
Meaning we set w_∙, d = 1 ∧ w_∙, m = 0.
The contributions to the performance of this adaption are presented in Section <ref> and Appendix <ref>.
Sum Aggregation Loss Due to the strong imbalance in the distribution of diseases and medications, we explore additional loss components to prevent the attention mechanism from overfitting to the most common tokens. Instead of taking the token representation we take the sum over all tokens excluding and again decode this unbiased aggregate using an MLP to predict the set of diseases or prescriptions (∖ for set difference):
ℒ_sum = ∑_* ∈{d, m}ℒ'(t),
ℒ'(*) = ℒ_BCE(MLP^ sum_* → *( 𝐯^sum(*) ) , 𝒞(*) ),
𝐯^sum(*) = ∑( Transf._ψ(f_θ( 𝒞(*) ) ) ∖{})
the idea is to ensure a more unbiased aggregation while still allowing the tokens to interact and impute masked or missing information. With this approach, we can induce a more dispersed distribution in the attention mechanism (Sec. <ref>).
Concepts from clinical reports
In our approach MMUGL we consider additional medical concepts extracted from text (clinical reports) and we concatenate the aggregated representation of these concepts for the respective visit 𝐯(n) to each of the two modalities at the input to the predictor MLP. For example in the case of ℒ_recon:
ℒ_recon = ∑_∙, * ∈{d, m}ℒ'(∙, *),
ℒ'(∙, *) = ℒ_BCE( MLP_∙→ *( 𝐯(∙) ⊕𝐯(n) ) , 𝒞(*) )
The final loss for pretraining ℒ_pre is a combination of ℒ_recon (Eqn. <ref>, <ref>, <ref>) and ℒ_sum (Eqn. <ref>):
ℒ_pre = ℒ_recon + λℒ_sum,
where ℒ_sum is configured as a regularizer with hyperparameter λ (for which we provide an ablation in Sec. <ref>).
§.§.§ Downstream Modules
In this work, we focus our contribution on learning concept representations over a knowledge graph from multiple modalities. We thus consider two prior architectures to perform time-series modeling and leave them mostly unchanged. It is intentional, that we do not propose a novel downstream architecture, but aim to show performance improvements alone through learning more robust and meaningful medical knowledge graph representations and aggregations thereof.
Average Pooling To compare to work by <cit.> in medication recommendation, we consider their downstream architecture. Given a patient history of visits (of which we get the representations using modules from Sec. <ref> and <ref>), we perform the same pooling scheme over the past and current visit to get a final representation which is used as input to an MLP to perform a predictive task.
RNN Based on the architecture by <cit.> given a patient and sequence of past visits (obtained by encoding in Sec. <ref>), we feed them through a GRU <cit.>. The hidden states at the output of the GRU are aggregated using a temporal attention mechanism where the query is a trainable embedding. We perform a minor modification here w.r.t. to the architecture by <cit.> and introduce a hyperparameter n_q, which refers to the number of trainable queries. If more than one query is used, we aggregate the different temporal aggregations of each query to get a single representation of the entire past of the patient. This representation is used to perform a prediction into the future using a MLP.
§ EXPERIMENTS
We perform our experiments on the MIMIC-III <cit.> dataset, using the , , tables.
Medications are mapped to the ATC hierarchy using the approach shared by <cit.>. For any of the approaches and baselines during pretraining we consider the training split of the respective baseline as well as any other patient in the dataset not present in the test or validation splits; this concerns especially patients with only a single visit (which are not usable for fine-tuning sequence tasks) in the dataset. We consider three different downstream tasks all trained using binary cross-entropy (binary/multi-label). Appendix <ref> shows data statistics for each of them. The result tables show standard deviations over three seeded training runs and we highlight the best results in bold font. In Appendix <ref>, <ref>, and <ref> we share training, architecture, and task details.
Medication Recommendation
To compare to the work by <cit.> (who have shown improvements over any previously published results on this task) we benchmark the medication recommendation task. We use their provided preprocessed patient data derived from MIMIC-III.
The multi-label prediction task was evaluated on a sample-averaged Area under the precision-recall curve AuPRC, as well as sample-averaged macro F1 score.
Heart Failure
This task has been benchmarked in CGL <cit.> (Collaborative Graph Learning), Chet <cit.> (Context-aware Health Event Prediction via Transition Functions), and Sherbet <cit.> (Self-Supervised Graph Learning With Hyperbolic Embedding for Temporal Health Event Prediction); who have performed extensive benchmarking against prior work.
We run their provided preprocessing and extract the used target code sets, as well as the computed patient splits. The binary classification is evaluated using F1 score and area under the receiver-operator curve AuROC.
Diagnosis
Similar to the previous heart failure task we compare to the results of CGL <cit.>, Chet <cit.>, and Sherbet <cit.>. We extract the target code sets and patient splits by running the provided preprocessing in each of the repositories to ensure comparability.
We consider thresholded weighted F1 (w-F1) score, and to be comparable to <cit.> we consider their adapted computation of F1. The variant is slightly inflated by considering the number of ground truth positive labels for each sample[<https://github.com/LuChang-CS/CGL/blob/main/metrics.py>]. This avoids the need to set a threshold, but leaks the number of ground-truth positives to the evaluation; we refer to it as w-F1 (infl.). We also report recall at top k predictions (according to model confidence); referred to as R@k (e.g. R@20).
§ RESULTS AND DISCUSSION
§.§ Pretraining: Sum Loss
In fig:pretraining-sum-loss-entropy we perform an ablation w.r.t. to the hyperparameter λ controlling the contribution of ℒ_sum (Eqn. <ref>) to the total pretraining loss.
<cit.> have computed the entropy of the distribution induced by the attention mechanism to analyze Transformer behavior. Similarly, we show the average (test set) entropy of the distribution induced by attention from the token to all the other tokens. For larger λ the entropy increases, hence the distribution is more dispersed, and we can see an improvement in pretraining performance (shown by the test set reconstruction loss ℒ_recon corresponding to improved test log-likelihood of our model). The idea is, that a more dispersed distribution is a better aggregator and generalizes better to rare diseases, which might otherwise be overlooked by a pointy (overfitted) attention distribution.
In Appendix <ref> and <ref> we provide further experimental results ablating pretraining and the different loss terms.
§.§ Medication Recommendation
We report our performance on the medication recommendation task (Sec. <ref>) using the average pooling architecture (Sec. <ref>) in tab:baseline-comparison-med. Our method and training approach can outperform the previously published state-of-the-art results by <cit.>, however, we note that the multi-modal approach with medical concepts from clinical reports cannot provide improvements on this task and data split (patients have high variation w.r.t. the richness of available clinical reports); also see Appendix <ref>.
§.§ Disease Tasks
In tab:baseline-comparison-diag we present benchmarking results on two disease-related tasks (Sec. <ref>) using the RNN architecture (Sec. <ref>). We train and evaluate our models on the patient splits and code sets extracted by considering three different prior work implementations, which have performed extensive benchmarking on previous state-of-the-art methods.
For Heart Failure we see that our approach can outperform any previous state-of-the-art published methods. Considering the diagnosis task our method outperforms CGL <cit.> (which considers unstructured text data), as well as Chet <cit.>; to be fair, neither considers a pretraining scheme. We also considerably outperform MedPath <cit.>, which considers personalized graphs to enhance the predictive performance of backbone time-series architectures for EHR. Our general method, considering pretraining tailored to encode both diseases and medications, performs on par with the hyperbolic approach Sherbet <cit.>, which performs pretraining too. However, if we tune our visit representations towards encoding disease-specific information (see Eqn. <ref> with w_∙, d = 1 ∧ w_∙, m = 0) we can also outperform this prior method.
§.§ Concept Embedding Ablation
In tab:concept-embedding-ablation we show ablations over different types of concept embeddings (Sec. <ref>) on a diagnosis task (Sec. <ref>). Our approach strongly benefits from richer multi-modal information coming from clinical reports and thus outperforms prior work (the multi-modal approach can also increase robustness w.r.t. missing and erroneous information, Appendix <ref>). We can see further improvements by tailoring our pretraining towards the downstream task by using disease-focused pretraining (Eqn. <ref> with w_∙, m = 0).
Note that in some cases (e.g. Heart Failure) using MMUGL w/o 𝒞(n) (i.e. w/o clinical reports) can slightly harm performance and be outperformed by more dataset-specific approaches such as using co-occurrence information and relying on simpler ontology structures with trainable embeddings, thus being able to adapt better to the dataset than the language model initialized MMUGL embeddings.
However, our approach enables the use of richer information coming from clinical reports and a larger concept vocabulary without introducing new parameters.
Further, our approach is more general, grounded by prior knowledge, and can hopefully be used to push transfer learning performance in the future. This is crucial in the medical data setting, where publicly available training data is scarce and sharing among institutions difficult to protect the privacy of individual patients.
We compare to two alternative approaches for learning concept embeddings by replacing the Concept Embedding <ref> module and performing the same proposed training procedure. As presented by <cit.> we additionally pretrain our knowledge graph concept embeddings using Node2Vec <cit.>. Secondly we compare to Cui2Vec <cit.>. Cui2Vec consists of medical concept embeddings pretrained on a large-scale corpus using a Word2Vec <cit.> style objective function. We show, that using our graph on the scale of 100'000 and around 30'000 patients for pretraining, we can compete with an approach that used training data on the order of 60 million patients, 20 million clinical notes, and 1.7 million biomedical journal articles.
§.§ Interpretability Analysis on Clinical Reports
We can use the attention mechanism to interpret the results on a patient level to rank diagnosis and medications, as well as general medical concepts from clinical reports w.r.t. their importance for the prediction using their respective attention score. See Figure <ref> where we show aggregated attention values for disease and prescription categories (Fig. <ref>), as well as the highest ranked concepts inside the highest ranked clinical reports (Fig. <ref>). We can also perform various dataset global analyses. We analyze the distribution of medical concepts extracted from clinical reports w.r.t. MIMIC-III report type and present the results in fig:text-attention-category-distribution; please mind the logarithmic y-scale (Appendix <ref> also shows a linear scale). After pretraining, we can see a very strong shift from the dataset's type distribution toward discharge summaries. This is sensible given the pretraining task is an auto-encoder, essentially training for summarizing the visit. By fine-tuning for specific tasks we can see slight shifts towards more specific report types, which can help provide more detailed insights for a given task; note for example how the focus in the respiratory category increases as we fine-tune for a general diagnosis, but decreases below the pretraining level for a heart failure prediction.
§.§ Limitations
Based on the previously shown results we can see the strong benefits of incorporating larger scale prior knowledge. We conclude the feasibility of extracting a complex graph from the large UMLS database using a fairly simple extraction pipeline and effectively learn strong medical knowledge representations over it.
We have proposed a simple extraction pipeline, where we extract an undirected graph from UMLS and ignore potential edge information. A more sophisticated extraction paired with an appropriate GNN should be able to handle the increased heterogeneity of different nodes, edges, and their respective features. However, this will come at a computational cost. One will have to navigate the complexities associated with the various node and edge types within the heterogenous set of subvocabularies present inside the UMLS Metathesaurus.
By creating a single shared latent space (our knowledge graph) for multiple modalities, we can achieve improved performance using much less data than prior art or outperform work using the same amounts of data. However, by reducing a clinical report to a set of medical concepts, which we can map onto our graph space, we neglect the natural language context and ordering. As we are already using a Transformer architecture inside our Visit Encoder (Sec. <ref>), we could include the remaining text (without concept matches) to provide the language context to obtain even finer grained final representations of patients and their visits.
§ CONCLUSION
We have introduced a novel way to train a unified latent space for general medical knowledge from multiple modalities. By grounding our representations with prior knowledge from the UMLS Metathesaurus, we have demonstrated improved performance on downstream tasks.
Our extended pretraining approach and the corresponding results emphasize its importance to tackle the supervised label scarcity in the medical domain. The more generalized approach to medical concept representations can aid in future designs and explorations of knowledge embedding transferability. Knowledge transfer is an important factor in the medical setting where publicly available training samples are scarce due to necessary regulations to protecting patient privacy.
Our results pave the way for future research to bridge the gap between within-visit modeling (e.g.,
ICU time-series models <cit.>) and across-visit modeling, such as we benchmarked against in this work. Whereas disease and medication codes are usually assigned post-visit (for billing or archival purposes), many clinical reports are generated during the patient stays. To provide richer context information, future within-visit models might include patient histories and the knowledge captured in our global concept representations.
This project was supported by grant #2022-278 of the Strategic Focus Area “Personalized Health and Related Technologies (PHRT)” of the ETH Domain (Swiss Federal Institutes of Technology).
Further, we would like to thank Hugo Yèche for his feedback during the revision process. Thanks go to Jonas Bokstaller and Severin Husmann whose theses have provided relevant insights.
§ EXPERIMENTAL DETAILS
§.§ Dataset and Split details
A small overview of data and task statistics are provided in tab:apd-data-statistics-disease. Splits and target code sets have been extracted from the respective repositories[<https://github.com/jshang123/G-Bert>,
<https://github.com/LuChang-CS/CGL>,
<https://github.com/LuChang-CS/Chet>,
<https://github.com/LuChang-CS/sherbet>]
§.§ Knowledge Graph Statistics
The extracted knowledge graph contains 87'445 nodes, 261'212 edges with node degrees of 5.97±20.91. The total vocabulary of all considered medical concepts is a subset of 21 UMLS Metathesaurus Vocabularies (percentages in brackets, some concepts belong to multiple): SNOMEDCT_US (46.75%), ICD9_CM (10.44%), CCPSS (7.71%), CSP (6.75%), FMA (6.15%), RXNORM (5.25%), DXP (4.21%), NCI_CDISC (4.13%), WHO (2.40%), ATC (2.12%), DRUGBANK (1.80%), CPT, NOC, BI, CCS, ICNP, NIC, ICF, CCC, PCDS, RAM.
Given a patient split we compute the coverage over our vocabulary during pretraining and downstream training. Inclusion criterias causing differences between the two are availability of medication (req. for pretraining) and multiple visits (req. for downstream training).
* All splits: 91.59% (pre), 71.24% (down)
* Train: 90.30% (pre), 68.21% (down)
* Validation: 16.45% (pre), 16.97% (down)
* Test: 37.68% (pre), 38.47% (down)
A percentage of concepts in the validation and test splits are unseen during training. Because of the graph structure, we can still learn meaningful representations for them:
* Validation: 0.78% (pre), 1.93% (down)
* Test: 3.11% (pre), 7.07% (down)
§.§ Architecture and Training
We perform early stopping based on the validation set loss both during pretraining and fine-tuning. The network is first fully pretrained until early stopped, the concept embedding (Sec. <ref>) backend is then frozen, the visit encoder (Sec. <ref>) is left trainable together with the downstream architecture to allow the attention mechanism to be fine-tuned to perform task-specific aggregations. We find a larger batch size (e.g. 32 or more) to be beneficial for better training stability.
Appendix <ref> shows an overview of the hyperparameters, which have been tuned w.r.t. validation set performance.
§.§.§ GNN Architecture
In Section <ref> we use a parametrized GNN in Eqns. <ref>, <ref>, and <ref>. We use Pytorch Geometric <cit.> to implement these networks and based on our hyperparameter searches in Appendix <ref> we settled on using the graph convolution operator GraphSAGE as introduced by <cit.>.
The ICD and ATC hierarchical ontologies or our complex UMLS based knowledge graph are passed to the GNN considering all edges as undirected. In the case of multiple GNN layers we use a non-linear ReLU activation after all but the last layer. The representations for each medical concept of an ontology or the knowledge graph at the GNN output are cached and used to retrieve concept embeddings for further processing by the Visit Encoder (Sec. <ref>) module.
§.§.§ GNN with Co-Occurrence
Similar to work done by <cit.> or <cit.> we can
additionally consider co-occurrence information present in our dataset. We construct a new graph 𝒢_ICD/ATC-CO which contains multiple sets of nodes and edges. The node sets are the ICD and ATC tree hierarchy nodes, while the edge sets consist of
the two ontologies and four co-occurrence edge sets; one for co-occurrence within each of the two ontologies and one (directed) from
each of the two to the other. We then compute a heterogeneous (nodes of different types) multi-layer GNN (see <cit.>) over these node and edge sets, where each edge set is associated with its own parametrized graph convolution operator. As a result, we compute multiple different embeddings for a given node in each layer, which are summed. Co-Occurrence edges can additionally be weighted by computing a count over the dataset (training split) and normalizing s.t. incoming edges sum to one. Such weights can be considered by the GNN by multiplying messages from neighboring nodes with the corresponding weight. Again we have c ∈{𝒞(d) ∪𝒞(m)}:
f_θ(c) = GNN_hetero(𝒢_ICD/ATC-CO)(c)
§.§ Hyperparameters
In tab:apd-hp-pretraining,tab:apd-hp-medication-recommend,tab:apd-hp-heart-failure,tab:apd-hp-diagnosis we present an overview of the model hyperparameters. Final choices based on validation set performances have been marked in bold font.
Hardware A typical training is finished in under a day. Depending on the task and set of considered input modalities it can be much faster. We trained our models using mostly GPUs with 11GB of dedicated GPU memory; some larger models, which included medical concepts extracted from text have been trained on GPUs with 24GB of dedicated GPU memory. We use 2-6 worker processes and around 32-64GB of main memory.
§.§ Tasks and Evaluation
In the following, we provide a more detailed overview of the benchmarked downstream tasks (Sec. <ref>) and the evaluation thereof.
§.§.§ Medication Recommendation
We benchmark the medication recommendation task based on preprocessed data by <cit.>. The task is to predict a set of medications (ATC level 4 codes) given a patient's history and the current diagnosis (assigned ICD codes). Given a patient i and a trained predictor ĥ we can formalize as follows:
𝒞̂_i, t(m) = ĥ( 𝒞_i, 0… t-1(*), 𝒞_i, t(d) )
where * ∈{d, m, n}
Given that this is a multi-label prediction we consider sample-averaged scores. Due to a significant imbalance in the distribution of medication codes, we use the F1 score for thresholded hard predictions and the area under the precision-recall curve (AuPRC) for unthresholded confidence scores. This is in line with the evaluation by <cit.>.
§.§.§ Heart Failure
This is a binary prediction task as already benchmarked by many prior works on the MIMIC-III <cit.> dataset. The task is to predict the risk of heart failure for a patient in a future visit given the patient's history. The label is extracted from the set of assigned ICD codes by matching with the prefix 428 after stripping the codes of any special characters. Let y_i, t be the target label and it is 1 if there exists a code c ∈𝒞_i, t(d) which has the prefix 428. For a patient i and trained predictor ĥ we can formalize as follows:
ŷ_i, t = ĥ( 𝒞_i, 0… t-1(*) )
where * ∈{d, m, n}
The task with mild label imbalance is evaluated using F1 score and area under the receiver-operator curve (AuROC) for untresholded performance evaluation; this is in line with work by <cit.> and others.
§.§.§ Diagnosis
This is a multi-label prediction over a set of diseases. Given a patient's history we predict the set of potential diseases for an upcoming visit . For a patient i and trained predictor ĥ we can formalize as follows:
𝒞̂_i, t(d) = ĥ( 𝒞_i, 0… t-1(*) )
where * ∈{d, m, n}
This task might not seem very sensible at first as we cannot expect to reliably predict accidents that cause a hospital visit based on past EHR records. However, this is useful to catch chronic diseases and re-occurring patient patterns. Such a model's predictions could serve as a high-level aggregation of all EHR records for a specific patient. A doctor can get a very quick assessment of the potential risks for a patient upon admission and can tailor further investigations to this.
Due to the extreme imbalance over the very large set of potential labels we use weighted-F1 score. To assess the unthresholded model confidence scores we use a popular metric from information retrieval. Recall at top k predictions (ranked by model confidence scores) can give an intuitive indication if the model can retrieve the desired ground truth diseases. The evaluation is in line with prior work e.g. by <cit.>.
§.§ Baselines
In this section, we provide a summary overview of the presented baselines and the key points of their architectures.
§.§.§ CGL: Collaborative Graph Learning
In this work, <cit.> propose a collaborative graph learning approach. They consider two graphs, one where patients and diseases are connected based on co-occurrence and one where only diseases are connected amongst each other based on the ICD ontology. GNN layers over the two edge sets and the shared set of nodes are run in an interleaved fashion (collaboratively). The computed embeddings for a certain disease are aggregated to represent patient visits and a sequence model performs task predictions.
§.§.§ Chet: Context aware Health Event Prediction via Transition Functions
The core contribution of this work by <cit.> is to consider a global disease graph, which connects diseases by co-occurrence and ontology relations, as well as a local graph (for each visit), which models the interactions of assigned disease codes within this specific visit. The architecture includes aggregation functions and sequence modeling to perform task-specific predictions.
§.§.§ Sherbet: Self- Supervised Graph Learning With Hyperbolic Embedding for Temporal Health Event Prediction
With Sherbet <cit.> propose to encode the structure of a disease ontology in hyperbolic space. The hyperbolic embeddings for the respective diseases are used to pretrain (using a patient history reconstruction task) and fine-tune a sequence model architecture to perform task-specific predictions.
§.§.§ MedPath: Augmenting Health Risk Prediction via Medical Knowledge Paths
With MedPath <cit.> propose to enhance the performance of existing EHR representation learning architectures by incorporating a personalized graph extracted using knowledge from Semantic MEDLINE <cit.>. The extracted graph is dataset and task-specific and can improve the performance of the backbone architecture. We transformed our data to adapt to their published pipeline and performed the heart failure prediction task using their implementations. We use HiTANet <cit.> as the backbone architecture, because it performed the best on the validation set in our hyperparameter search.
§.§.§ G-BERT: Pre-training of Graph Augmented Transformers for Medication Recommendation
<cit.> show performance improvements on a medication recommendation task by pretraining disease and medication code embeddings using GNNs over two ontologies. The pretraining objective is a reconstruction task of observed codes during a patient visit and borrows ideas from masked language modeling. The pretrained architecture includes a Transformer-based encoder, which outputs a encoding for each patient visit. The proposed downstream architecture performs a pooling scheme over patient histories and recommends medications for a current patient visit given the patient's history and the current diagnosis of diseases.
§.§.§ Embedding Matrix
In tab:concept-embedding-ablation we show a concept embedding ablation using an Embedding Matrix. This refers to a matrix of trainable parameters 𝐄∈ℝ^|𝒞| × k where |C| is the total number of considered medical concepts and k the embedding dimension. The embedding matrix replaces the Concept Embedding (Sec. <ref>) module and is pretrained and fine-tuned using the same procedure.
§.§.§ Concept Embeddings using Node2Vec
In tab:concept-embedding-ablation we show a concept embedding ablation using Node2Vec <cit.>. We consider our extracted complex UMLS-based knowledge graph and perform Node2Vec-style pretraining to obtain embeddings for each concept in our knowledge graph.
We then initialize an embedding matrix (which is used to retrieve concept embeddings by index lookup) and use it to replace our proposed GNN-based concept embeddings. To ensure fair comparison we then perform the same reconstruction pretraining as our proposed approach MMUGL to ensure the parameters of the Visit Encoder (Sec. <ref>) module are well pretrained too. Similarly, we apply the same pipeline as for our approach during fine-tuning for downstream tasks.
§.§.§ Concept Embeddings using Cui2Vec
Cui2Vec as introduced by <cit.> is a collection of pretrained medical concept embeddings mapped to the space of UMLS. Their training optimizes a Word2Vec <cit.> style objective over a large-scale corpus (60 million patient records, 20 million clinical notes, and 1.7 million full-text biomedical journal articles). We use the Cui2Vec embeddings to initialize a lookup matrix from which concept embeddings are retrieved by index and replace our GNN-based concept embeddings. To ensure fair comparison we apply the same pretraining (reconstruction) and fine-tuning procedure to obtain downstream task performance results.
§ TRAINING AND ARCHITECTURE ABLATIONS
§.§ Clinical Reports Performance Contribution
In this section, we would like to clarify our findings about why the additional modality of extracted concepts from unstructured text (i.e. clinical reports) cannot yield a performance improvement in all cases.
Overall, the billing codes represent an aggregate of information for an entire patient's visit to the hospital and the labels are defined based on them. Thus, the billing codes (ICD, ATC codes) are the strongest signal for our predictions. The additional medical concepts from clinical reports can help in two ways. First, they can help to deal with missing or noisy information from the billing codes (see also Appendix <ref>). Second, they can help the model to make more fine-grained predictions due to the higher level of detail.
Heart Failure Here the additional concepts from clinical reports do seem to help, but in most cases only marginally. We hypothesize this is due to the fact, that we are only performing a binary prediction and the finer details of the clinical reports cannot yield enough additional information in most cases to significantly improve our predictions.
Diagnosis This is a very complex classification task and here we see the strongest improvement after adding the concepts extracted from clinical reports. For this task, we can benefit from the higher level of detail present in the clinical reports compared to the billing codes.
Medication Recommendation On this task the strongest signal comes from the current set of diseases. The additional concepts from clinical reports are only present in the representation of the patient's history, where we do not seem to benefit from the more detailed content of the clinical reports. To avoid information leakage we cannot directly use all concepts from all reports of the current visit when performing the medication recommendation. To accommodate for this we would have to adapt the task to a within-visit online medication recommendation; predicting medication based on the patient's global (past hospital visits) as well as local (past time within current visit) history. This would enable the inclusion of already accumulated clinical reports in the local (current visit) context.
§.§ Ablation: SapBERT
We ablate the use of SapBERT <cit.> compared to training randomly initialized node embeddings. SapBERT performs better in pretraining (the selection criteria), where we see an increase from 49.38±0.49 to 61.77±0.44 AuPRC. The improvement carries over to the downstream performance, where for the diagnosis prediction we see an improvement of 25.46±0.50 to 26.19±0.30 in the F1 (inflated) score.
§.§ Pretraining Ablation
In tab:baseline-comparison-diag we can see, that prior work including pretraining schemes performs much stronger than the ones that don't.
In tab:pretraining-ablation we perform an ablation w.r.t. pretraining different concept embeddings and report performance on the Diagnosis task (Sec. <ref>) on pretrained and on randomly initialized networks. We note, the more structure bias we provide, the better the performance without pretraining.
In Appendix <ref> and <ref> we present further results on exploring modifications to the pretraining loss function.
§.§ Sum Aggregation Loss
We provide further empirical evidence for the contribution of the additional loss term introduced in Eqn. <ref> in tab:sum-loss-diagnosis.
tab:sum-loss-diagnosis shows results on the Diagnosis downstream task across different Concept Embedding implementations
and with different pretraining regimes. We show results without pretraining, pretraining on only the default reconstruction loss ℒ_recon (Eqn. <ref>) and including the additional introduced loss term ℒ_sum (Eqn. <ref>).
We can see that the additional loss component ℒ_sum during pretraining contributes to better pretrained representations as across different
downstream models we can see either at least the same performance or increased performance. This difference is especially notable and important for the
best-performing model implementation MMUGL, where w_∙, m = 0 (Eqn. <ref>, pretraining focused on recovering diseases only).
We hypothesize, that without the additional loss regularization, we experience stronger overfitting to the training distribution during pretraining, as we have more data
available (given that MMUGL includes additional rich information coming from medical concepts in clinical reports) and we have reduced the task complexity (as we set w_∙, m = 0 in the pretraining loss, Eqn. <ref>). We also observe a tendency to more consistent results under pretraining
including the ℒ_sum loss component as standard deviations tend to be lower.
This stays consistent also on a further task e.g. Heart Failure. For MMUGL with w_∙, m = 0 including ℒ_sum in pretraining
we observe a downstream heart failure prediction performance (on the CGL <cit.> patient split) of 87.60±0.40 where this drops to 86.93±0.13
if we pretrain without ℒ_sum.
§.§ Reconstruction Loss
In tab:recon-loss-ablation,tab:recon-loss-ablation-med we perform an ablation with respect to the different weights
in the weighted version of the pretraining reconstruction loss ℒ_recon (Eqn. <ref>).
The base version as introduced by <cit.> considers all weights w_∙, * = 1. This is flexible
in the sense that it does not enforce a bias towards encoding information relevant for disease or medication predictions.
However, by weighting (or fully disabling) the different terms, we can tailor our pretraining to different downstream scenarios.
Please also note, that the following experiments have been performed without the additional loss component ℒ_sum (Eqn. <ref>) to focus purely on the effects within
the reconstruction loss term ℒ_recon (Eqn. <ref>, <ref>).
Downstream Diagnosis tab:recon-loss-ablation shows this effect on the downstream Diagnosis task. We can see that while
having all loss terms active yields strong performance, in the case of a diagnosis prediction it is beneficial to
only pretrain on loss terms that are predictive for diseases i.e. w_∙, d = 1 ∧ w_∙, m = 0.
This is further supported by results shown in tab:baseline-comparison-diag and tab:concept-embedding-ablation,
where results on the full MMUGL model (including medical concepts from clinical reports) improve by pretraining with w_∙, m = 0.
Downstream Medication Recommendation tab:recon-loss-ablation-med shows the exact same behaviour when performing downstream medication recommendation. The best performance is achieved by only considering loss terms towards predicting the modality relevant for the downstream prediction task. We can conclude, that cross-modality pretraining is beneficial to learn embeddings that can be useful for a yet unspecified downstream application. However, if the nature of
the target modality of the downstream task is known and the cost of pretraining affordable, we can achieve
better performance by adapting the pretraining to the downstream scenario.
§ CLINICAL REPORT CONCEPT CATEGORY DISTRIBUTION
In fig:apd-text-attention-lin-log-comparison we show the plot discussed in Section <ref> with both logarithmic and linear scale. The plot with logarithmic scale in Figure <ref> is better suited to highlight the fine details and changes in categories such as Respiratory or Radiology. The linear scale in fig:apd-text-attention-lin shows the strong changes caused by the pretraining (compared to the actual token distribution per category) in e.g. the discharge summary type of reports.
One might notice a particularly large drop in tokens from reports of the respiratory type. First we would like to highlight that fig:apd-text-attention-log uses a logarithmic y-axis and thus the absolute number of tokens found in the respective report type is comparatively low. Still, we can observe a change over one order of magnitude. This can be explained by looking more in-depth at the reports of this specific type.
In MIMIC-III the clinical reports of type Respiratory are mostly highly structured status reports assessing a patient's state w.r.t. the respiratory system. Being a structured report, there is a large set medical concepts matched, which correspond to the field names of the structured report to be filled with patient information and further most of the provided assessments in the form do not vary much across patients. As such, many of the extracted medical concepts from these reports are not discriminative across patients and thus we observe a drop in attention to the tokens extracted from these reports after training the model.
§ HEART FAILURE PERFORMANCE DISENTANGLEMENT
Due to the chronic nature of heart failure, we disentangle the performance on the test set with a fixed model for patients with and without reported histories of heart failure (the target codes have appeared in the patient history). The results are shown in Table <ref>.
The model is naturally performing much better on the subset of patients with a reported history of heart failure and can exploit the chronic nature of the disease. However, we note that with our proposed multi-modal approach we see a notable performance improvement on the hard cases of patients without a reported history of heart failure. We conclude, using clinical report concepts backed by a knowledge graph, not just billing codes, aids in understanding disease progressions.
§ SINGLE PATIENT INTERPRETABILITY
In fig:apd-single-patient we present various ways how attention scores of our visit encoder (Sec. <ref>) can be used to provide interpretability of our predictions. We provide an example score analysis of visit 121518 by patient 1784 in the MIMIC-III <cit.> dataset. The patient was assigned the following set of codes:
* ICD: 519.1, 496.0, 414.01, 401.9, 443.9, V45.82
* ATC4: N05CD, A02BC, B01AB, A06AD, C07AB, B05CX, G04CA, A07EA
The scores can be used to highlight the most relevant diseases and medications (fig:apd-patient-overview). By grouping scores of individual codes and computing an aggregate for each group (e.g. 90th-percentile of scores) we can highlight the most relevant disease and medication categories for this patient at the given visit.
We can further extract which of the reports collected during the entire visit contain the most predictive identifiers by computing an aggregated score over the scores of all the matched concepts within each report (fig:apd-patient-overview).
In (fig:apd-patient-top-reports) we then highlight the concepts within the two highest-ranked reports with the largest attention scores.
We can see that the scores are consistent across different modalities, considering for example the high score given to the Respiratory category for the disease (ICD) codes (fig:apd-patient-overview), as well as high scores for concepts found in clinical reports (e.g. (Tracheomalacia) in or (Carinal reconstruction) in ; fig:apd-patient-top-reports) related to respiratory conditions. We can conclude that for this sample the unified concept latent space promotes consistency across modalities and can improve interpretability.
§ ROBUSTNESS W.R.T. MISSING INFORMATION
In fig:masking-progression we show the results of an experiment, where we progressively mask a larger percentage of input tokens of different modalities. This is done by replacing the respective token identifier with the token used during masked language modeling style pretraining <cit.>.
Tokens can either be masked randomly or we sort them with respect to the attention score assigned to them in the visit encoder. The y-axis shows the pretraining performance w.r.t to Eqn. <ref>; decoding to any of the two modalities (diseases, medications) from the visit representation of either.
The results show, that although the auto-encoding objective is only formulated w.r.t. the disease and medications tokens, the additional text information can successfully prevent stronger decay in performance and help impute the missing or incorrect information.
We can further see that masking tokens according to their attention scores results in a faster overall decrease in performance, highlighting the benefits of using an attention-based encoder, that can focus on relevant medical concepts when encoding a patient's current state.
|
http://arxiv.org/abs/2307.05149v1 | 20230711101122 | Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation | [
"Nadhir Ben Rached",
"Abdul-Lateef Haji-Ali",
"Shyam Mohan Subbiah Pillai",
"Raúl Tempone"
] | math.NA | [
"math.NA",
"cs.NA",
"stat.CO",
"60H35, 65C30, 65C05, 65C35"
] |
Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation
Nadhir Ben Rached Department of Statistics, School of Mathematics, University of Leeds ([email protected])., Abdul-Lateef Haji-Ali Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK ([email protected])., Shyam Mohan Subbiah Pillai Corresponding author; Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, RWTH Aachen University, Aachen, Germany([email protected]).
and Raúl Tempone Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia ([email protected]). Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Aachen, Germany ([email protected])..
This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. URF/1/2584-01-01 and the Alexander von Humboldt Foundation. This work was also partially performed as part of the Helmholtz School for Data Science in Life, Earth and Energy (HDS-LEE) and received funding from the Helmholtz Association of German Research Centres. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
empty
This work introduces a novel approach that combines the multi-index Monte Carlo (MC) method with importance sampling (IS) to estimate rare event quantities expressed as an expectation of a smooth observable of solutions to a broad class of McKean-Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator, previously introduced in our works <cit.>, to the multi-index setting. We formulate a new multi-index DLMC estimator and conduct a comprehensive cost-error analysis, leading to improved complexity results. To address rare events, an importance sampling scheme is applied using stochastic optimal control of the single level DLMC estimator. This combination of IS and multi-index DLMC not only reduces computational complexity by two orders but also significantly decreases the associated constant compared to vanilla MC. The effectiveness of the proposed multi-index DLMC estimator is demonstrated using the Kuramoto model from statistical physics. The results confirm a reduced complexity from _r^-4 for the single level DLMC estimator <cit.> to _r^-2 (log_r^-1)^2 for the considered example, while ensuring accurate estimation of rare event quantities within the prescribed relative error tolerance _r.
Keywords: McKean-Vlasov stochastic differential equation, importance sampling, multi-index Monte Carlo, decoupling approach, double loop Monte Carlo.
2010 Mathematics Subject Classification 60H35. 65C30. 65C05. 65C35.
§ INTRODUCTION
McKean-Vlasov stochastic differential equations (MV-SDEs) are a special class of SDEs whose drift and diffusion coefficients depend on the law of the solution itself <cit.>. These equations arise from the mean-field behaviour of stochastic interacting particle systems, which find applications in various fields such as pedestrian modeling <cit.>, animal behavior studies <cit.>, biology <cit.>, finance <cit.>, and chemistry <cit.>. Detailed analysis <cit.> and numerical treatment <cit.> of MV-SDEs can be found in the referenced works. In this work, our focus lies in estimating rare event quantities, which are expressed as expectations of observables associated with the solution to a broad class of MV-SDEs at a fixed terminal time T. Specifically, we develop a computationally efficient Monte Carlo (MC) estimator for G(X(T)), where G:ℝ^d →ℝ represents a sufficiently smooth rare event observable, and X:[0,T] Ω→ℝ^d denotes the stochastic McKean-Vlasov process.
Stochastic P-particle systems are commonly employed to approximate MV-SDEs. These systems consist of P coupled d-dimensional Itô SDEs, and they exhibit a mean-field limit as the number of particles approaches infinity <cit.>. Estimating expectations associated with particle systems by numerically solving the (P × d)-dimensional Kolmogorov backward equation quickly becomes computationally intractable. To address this challenge, Monte Carlo (MC) methods are used to obtain computationally feasible estimates. Previous works <cit.> have studied MC methods that utilize Euler-Maruyama time-discretized particle systems, assuming bounded, Lipschitz drift and diffusion coefficients, for smooth, non-rare observables. These methods achieve a complexity of ^-4 for a prescribed error tolerance . However, when it comes to rare events, MC methods become prohibitively expensive. This is due to the significant increase in the relative estimator complexity constant as the event becomes rarer <cit.>. To tackle this issue, importance sampling (IS) is employed as a variance reduction technique to mitigate the limitations of standard MC methods in the rare event regime <cit.>.
The initial study of importance sampling (IS) for rare events in the context of MV-SDEs was conducted in <cit.>. In <cit.>, the authors propose a decoupling approach that involves defining a decoupled MV-SDE. Here, they replace the McKean-Vlasov law in the drift/diffusion coefficients of the MV-SDE with an empirical realization of the law computed beforehand by running a particle system. Measure change for IS is performed on this decoupled process ensuring that IS is separated from the estimation of the McKean-Vlasov law. Building upon this decoupling approach, our previous work <cit.> introduces a double loop Monte Carlo (DLMC) estimator. Stochastic optimal control theory is then utilized to derive a zero-variance time- and pathwise-dependent IS control. Subsequently, we develop an adaptive DLMC algorithm with a complexity of ^-4, matching that of the MC estimator for non-rare observables in <cit.>. In the subsequent work <cit.>, we combine multilevel MC methods with the aforementioned IS scheme to reduce the computational complexity of the estimator compared to the single-level estimator introduced in <cit.>.
The multi-index Monte Carlo (MIMC) method was introduced in <cit.> to tackle problems involving multiple (two or more) discretization parameters. This approach builds upon the efficiency of multilevel MC but requires mixed regularity with respect to the discretization parameters. While the multi-index method has been applied to smooth, non-rare observables associated with particle systems in <cit.>, our goal in this study is to combine the multi-index technique with the IS scheme introduced in <cit.>. The objective is to develop an estimator for rare event expectations associated with MV-SDEs that is not only efficient but also surpasses the performance of the multilevel DLMC estimator <cit.>. Although the combination of multilevel MC with importance sampling has been studied in various contexts <cit.> before, this work represents, to the best of our knowledge, the first attempt to integrate IS with multi-index MC. We summarize the main contributions of this paper as follows:
* This paper extends the DLMC estimator introduced in <cit.> to the multi-index setting and proposes a novel multi-index DLMC estimator for MV-SDEs. We include detailed discussion on the proposed estimator's bias and variance, and devise a complexity theorem, showing improved complexity compared with multilevel DLMC. We also formulate a robust adaptive multi-index DLMC algorithm that iteratively determines optimal parameters.
* This paper proposes a novel combination of IS scheme with the multi-index DLMC estimator to handle rare event quantities. We apply the IS control developed for the DLMC estimator in <cit.> to all multi-indices in the multi-index DLMC estimator.
* This paper investigates numerically the effect of the IS scheme on the variance of this novel multi-index DLMC estimator. Numerical simulations confirm significant variance reduction due to this IS scheme, improving _r^-4 complexity, obtained in <cit.> for the considered example; to _r^-2 (log^-1)^2 in the multi-index setting, while allowing feasible rare event quantity estimation up to the prescribed relative error tolerance _r.
The paper is structured as follows. In Section <ref>, we provide an introduction to the MV-SDE, along with the associated notation and problem statement. Section <ref> presents the decoupling approach for MV-SDEs <cit.> and formulates a DLMC estimator based on this approach. Moving on to Section <ref>, we present the zero-variance IS control for the decoupled MV-SDE, derived using stochastic optimal control techniques. Additionally, we introduce the DLMC estimator with IS from our previous work <cit.>. In Section <ref>, we introduce the novel multi-index DLMC estimator and provide new complexity results for this estimator. We then propose an IS scheme for the multi-index DLMC estimator. Furthermore, we develop an adaptive multi-index DLMC algorithm, which enables the feasible estimation of rare event quantities associated with MV-SDEs. To validate our approach and verify the assumptions made in this work, Section <ref> applies the proposed methods to the Kuramoto model from statistical physics. We also provide numerical verification of the derived complexity rates for the multi-index DLMC estimator with respect to an appropriate rare event observable.
§ MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATION
We consider a broad class of McKean-Vlasov equations that arise from the mean-field limit of stochastic interacting particle systems with pairwise interaction kernels <cit.>. Consider the probability space {Ω,ℱ,{ℱ_t}_t ≥ 0,P}, where ℱ_t is the filtration of a standard Wiener process {W(t):t ∈ [0,T]}. For functions b:ℝ^d ℝ⟶ℝ^d, σ:ℝ^d ℝ⟶ℝ^d d, κ_1: ℝ^d ℝ^d ⟶ℝ and κ_2: ℝ^d ℝ^d ⟶ℝ, consider the following Itô SDE for the McKean-Vlasov stochastic process X: [0,T] ×Ω→ℝ^d
[left=, right = ,]equation
2
X(t) = b(X(t),∫_ℝ^d κ_1 (X(t),x) μ_t(x)) t
+ σ(X(t),∫_ℝ^d κ_2 (X(t),x) μ_t(x)) W(t), t>0
X(0) = x_0 ∼μ_0 ∈𝒫(ℝ^d) ,
where W:[0,T] Ω⟶ℝ^d is a d-dimensional Wiener process with mutually independent components; μ_t ∈𝒫(ℝ^d) is the mean-field law of X(t), where 𝒫(ℝ^d) is the space of all probability measures in ℝ^d; and x_0 ∈ℝ^d is a random initial state with distribution μ_0 ∈𝒫(ℝ^d). Functions b(·) and σ(·) are called drift and diffusion functions/coefficients, respectively. Under certain smoothness and boundedness conditions on b,σ,κ_1,κ_2, one can prove existence and uniqueness of solutions to (<ref>) <cit.>. The distribution of the McKean-Vlasov process X satisfies the following Fokker-Planck partial differential equation (PDE),
[left=, right = ,]alignat=2
-∂μ(s,x;t,y)/∂s - ∑_i=1^d ∂/∂x_i (b_i(x,∫_ℝ^d κ_1(x,z) μ(s,z;t,y) z ) μ(s,x;t,y))
+ ∑_i=1^d ∑_j=1^d 1/2∂^2/∂x_i ∂x_j ( . ∑_k=1^d σ_ik σ_jk (x,∫_ℝ^d κ_2(x,z) μ(s,z;t,y) z )
μ(s,x;t,y) . ) = 0, (s,x) ∈(t,∞) ℝ^d
μ(t,x;t,y) = δ_y(x) ,
where μ(s,x;t,y) denotes the conditional distribution of X(s) given that X(t) = y; and δ_y(·) denotes the Dirac measure at y. (<ref>) is a non-linear integral differential PDE with non-local terms. The cost of numerical approximation of (<ref>) up to relative error tolerances increases exponentially with the number of dimensions d, motivating study of strong approximations to (<ref>).
Consider following system of P exchangeable Itô SDEs, also known as a stochastic interacting particle system, with pairwise interaction kernels comprising P particles <cit.>. For p=1, …, P, we have the following SDE for the process X^P_p:[0,T]Ω→ℝ^d,
[left=, right = ,]alignat=2
X^P_p(t) = b(X^P_p(t), 1/P ∑_j=1^P κ_1(X^P_p(t),X^P_j(t)) ) t
+ σ(X^P_p(t), 1/P ∑_j=1^P κ_2(X^P_p(t),X^P_j(t)) ) W_p(t), t>0
X^P_p(0) = (x_0)_p ∼μ_0 ∈𝒫(ℝ^d) ,
where {(x_0)_p}_p=1^P are independent and identically distributed (iid) random variables sampled from the initial distribution μ_0; {W_p}_p=1^P are mutually independent d-dimensional Wiener processes, also independent of {(x_0)_p}_p=1^P. Equation (<ref>) approximates the mean-field law μ_t from (<ref>) by an empirical law based on particles {X^P_p(t)}_p=1^P
μ_t( x) ≈μ_t^P( x) = 1/P∑_j=1^P δ_X^P_j(t) ( x) ·
Strong convergence of particle systems has been proven for a broad class of b(·),σ(·) <cit.>. The high dimensionality of the corresponding Fokker-Planck equation, satisfied by the particle system's joint probability density, motivates the use of MC methods, which do not suffer from the curse of dimensionality.
§.§ Running Example: Kuramoto oscillator model
The methodology introduced in this work is tested on the simple, one-dimensional Kuramoto model <cit.>, which is used to describe synchronization in statistical physics to help model behavior of large sets of coupled oscillators. It is a system of P fully connected, synchronized oscillators. The state of each oscillator is represented by process X_p^P:[0,T]Ω→ℝ that has the following Itô SDE dynamics
[left=, right = ,]alignat=2
X^P_p(t) = (ξ_p + 1/P ∑_q=1^P sin(X^P_p(t) - X^P_q(t))) t + σW_p (t) , t>0
X^P_p(0) = (x_0)_p ∼μ_0 ∈𝒫(ℝ) ,
where {ξ_p}_p=1^P are iid random variables sampled from a prescribed distribution; diffusion σ∈ℝ is constant; {(x_0)_p}_p=1^P are iid random variables sampled from μ_0 ∈𝒫(ℝ); {W_p}_p=1^P are mutually independent one-dimensional Wiener processes; and {ξ_p}_p=1^P, {(x_0)_p}_p=1^P, {W_p}_p=1^P are mutually independent. The oscillator system (<ref>) reaches mean-field limit as the number of oscillators tends to infinity, where each particle now behaves according to the following MV-SDE,
[left=, right = ,]equation
2
X(t) = (ξ+ ∫_ℝ sin(X(t)-x) μ_t (x) ) t + σW(t), t>0
X(0) = x_0 ∼μ_0 ∈𝒫(ℝ) ,
where X denotes the mean-field process; ξ is a random variable sampled from some prescribed distribution; and μ_t is the mean-field law of X at time t.
§.§ Problem Setting
Consider the McKean-Vlasov process X:[0,T] Ω→ℝ^d as defined in (<ref>), where T>0 is some finite terminal time. Let G: ℝ^d ⟶ℝ be a given scalar observable function. Our objective is to build a computationally efficient estimator 𝒜 of G(X(T)) that satisfies relative error tolerance _r > 0 in the following sense
𝒜-G(X(T))/G(X(T))≥_r≤ν,
for a given confidence level determined by 0 < ν≪ 1.
The high-dimensionality of the KBE corresponding to the stochastic particle system (<ref>) makes it challenging to numerically approximate G(X(T)) using standard PDE solvers for a given relative tolerance _r. To overcome this issue, it becomes necessary to employ Monte Carlo (MC) methods, which mitigate the curse of dimensionality. Additionally, the combination of MC with importance sampling (IS), is essential for obtaining feasible estimates of rare events. In Section <ref>, we introduce the decoupling approach for MV-SDEs, present the associated notation, and introduce the DLMC estimator <cit.>.
§ DOUBLE LOOP MONTE CARLO ESTIMATOR
The decoupling approach, as presented in <cit.>, involves a two-step process. Firstly, an empirical approximation of the McKean-Vlasov law (<ref>) is used to define the decoupled MV-SDE. Secondly, a measure change is applied to this decoupled SDE. This approach separates the estimation of the mean-field law from the necessary change in probability measure for importance sampling (IS). By decoupling these two aspects, we can treat the decoupled MV-SDE as a standard SDE with random coefficients, allowing for a well-formulated change of measure <cit.>. In this section, we introduce the general scheme of the decoupling approach, which consists of the following steps, as outlined in <cit.>.
* The mean field law {μ_t:t ∈ [0,T]} is approximated using an empirical measure {μ_t^P: t ∈ [0,T]}, derived using (<ref>) with one realization of the P-particle system (<ref>) with particles {X^P_p}_p=1^P.
* Given {μ_t^P: t ∈ [0,T]}, we define the decoupled MV-SDE. Consider the decoupled McKean-Vlasov process X^P:[0,T] ×Ω→ℝ^d with the following dynamics
[left=, right = ,]alignat=2
X^P(t) = b(X^P(t), 1/P ∑_j=1^P κ_1(X^P(t),X^P_j(t)) ) t
+ σ(X^P(t), 1/P ∑_j=1^P κ_2(X^P(t),X^P_j(t)) ) W(t), t ∈[0,T]
X^P(0) = x_0 ∼μ_0, x_0 ∈ℝ^d ,
where superscript P indicates that the process is conditioned on an empirical law computed with a P-particle system; W is a standard d-dimensional Wiener process that is independent of Wiener processes {W_p}_p=1^P driving the corresponding particle system (<ref>); and x_0 ∈ℝ^d is a random initial state sampled from μ_0 and independent from {(x_0)_p}_p=1^P used in (<ref>). Thus, (<ref>) is a standard SDE for a given realization of the empirical law {μ_t^P: t ∈ [0,T]}.
* Suppose the particle system (<ref>) is defined on the probability space (Ω,ℱ,ℙ). Consider a copy space (Ω,ℱ,ℙ) <cit.>; and we define (<ref>) on the product space (Ω,ℱ,ℙ) (Ω,ℱ,ℙ). One can look at ℙ as a probability measure induced by random variables driving the particle system (<ref>) and ℙ as the measure induced by random variables in (<ref>) conditioned on empirical law {μ_t^P:t∈ [0,T]}.
* Re-express and approximate the quantity of interest as
G(X(T)) ≈𝔼_ℙ⊗ℙ[G(X^P(T))]
= 𝔼_ℙ[𝔼_ℙ[ G(X^P(T)) |{μ_t^P: t ∈ [0,T]}]] ·
Henceforth, we denote G(X^P(T))≡𝔼_ℙ⊗ℙ[G(X^P(T))] for convenience.
By utilizing the decoupling approach, we represent our quantity of interest as the nested expectation in (<ref>). However, directly approximating the inner expectation 𝔼_ℙ̅[G(X^P(T))|{μ_t^P: t ∈ [0,T]}] by numerically solving the Kolmogorov backward equation for each realization of the empirical law for a relative error tolerance is computationally infeasible. To address this challenge, we employ a nested Monte Carlo average with importance sampling (IS) to estimate the nested expectation (<ref>) in the rare events regime <cit.>. Algorithm <ref> outlines the procedure for this estimation. In the algorithm, ω_1:P^(i) represents the i^th realization of the P sets of random variables (Wiener increments, initial conditions etc.) that drive the dynamics of the interacting P-particle system in (<ref>). Thus, {μ^P_t:t ∈ [0,T]}(ω_1:P^(i)) denotes i^th realization of the empirical law. Furthermore, let ω̃^(i) denote the i^th realization of the random variables that drive the dynamics of the decoupled MV-SDE (<ref>), conditioned on an empirical law realization.
§ IMPORTANCE SAMPLING FOR MV-SDES
Within this section, we present a stochastic optimal control-based importance sampling (IS) measure change for the decoupled MV-SDE (<ref>). Subsequently, we integrate this IS approach into the DLMC Algorithm <ref>.
§.§ Importance sampling using stochastic optimal control for decoupled MV-SDE
In <cit.>, we applied an optimal measure change to the decoupled MV-SDE that provides a zero-variance DLMC estimator via stochastic optimal control theory. First, we formulate the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) that yields the optimal control for the decoupled MV-SDE.
Let decoupled McKean-Vlasov process X^P satisfy (<ref>). Consider following Itô SDE for the controlled process X^P_ζ: [0,T] Ω→ℝ^d with control ζ(t,x):[t,T] ℝ^d →ℝ^d,
[left=, right = ·]alignat=2
X^P_ζ(t) = ( . b(X^P_ζ(t), 1/P ∑_j=1^P κ_1(X^P_ζ(t),X^P_j(t)) )
+ σ(X^P_ζ(t), 1/P ∑_j=1^P κ_2(X^P_ζ(t),X^P_j(t)) ) ζ(t,X^P_ζ(t)) . ) t
+ σ(X^P_ζ(t), 1/P ∑_j=1^P κ_2(X^P_ζ(t),X^P_j(t)) ) W(t), 0<t<T
X^P_ζ(0) = X^P(0) = x_0 ∼μ_0 .
Here, the corresponding realization of the empirical law {μ^P_t:t ∈ [0,T]} is computed beforehand from the particle system (<ref>). The value function u(t,x) that minimizes the second moment (see <cit.> for derivation) of the MC estimator of G(X̅^P(T)) |{μ^P_t:t ∈ [0,T]} with IS is written as
u(t,x) = min_ζ∈𝒵𝔼[.G^2(X_ζ^P(T)) exp-∫_t^T ζ(s,X_ζ^P(s))^2 - 2 ∫_t^T ⟨ζ(s,X_ζ^P(s)), W(s) ⟩
|X_ζ^P(t) = x, {μ_t^P: t ∈ [0,T]}.] ·
Assume u(t,x) is bounded, smooth, and non-zero throughout its domain. Define γ(t,x) such that u(t,x) = exp-2 γ(t,x). Then γ(t,x) satisfies the non-linear HJB equation
[left=, right = ,]alignat=2
∂γ/∂t + ⟨b(x, 1/P ∑_j=1^P κ_1(x,X^P_j(t)) ), ∇γ⟩+ 1/2 ∇^2 γ: (σσ^T) (x, 1/P ∑_j=1^P κ_2(x,X^P_j(t)) )
- 1/4 σ^T ∇γ(x, 1/P ∑_j=1^P κ_2(x,X^P_j(t)) )^2 = 0, (t,x) ∈[0,T) ℝ^d
γ(T,x) = - logG(x), x ∈ℝ^d ,
with optimal control
ζ^*(t,x) = - σ^T (x, 1/P∑_j=1^P κ_2(x,X^P_j(t)) ) ∇γ(t,x ),
which minimizes the second moment of the IS estimator conditioned on {μ_t^P: t ∈ [0,T]}.
See Appendix B in <cit.>.
In proposition <ref>, ⟨·,·⟩ is the Euclidean dot product between two ℝ^d vectors; ∇· is the gradient vector of a scalar function; ∇^2 · is the Hessian matrix of a scalar function; · : · is the Frobenius inner product between two matrix-valued functions; and · is the Euclidean norm of an ℝ^d vector. Previous works <cit.> have shown that (<ref>) leads to a zero variance estimator of the inner expectation G(X^P(T)) |{μ^P_t:t ∈ [0,T]}, provided G(·) does not change sign. Using the transformation u(t,x) = v^2(t,x), one can thus recover the linear KBE corresponding to (<ref>),
[left=, right = ,]alignat=2
∂v/∂t + ⟨b(x, 1/P ∑_j=1^P κ_1(x,X^P_j(t)) ), ∇v ⟩
+ 1/2 ∇^2 v : (σσ^T) (x, 1/P ∑_j=1^P κ_2(x,X^P_j(t)) ) = 0 , (t,x) ∈[0,T) ℝ^d
v(T,x) = G(x), x ∈ℝ^d ,
with optimal control
ζ^*(t,x) = σ^T (x, 1/P∑_j=1^P κ_2(x,X^P_j(t)) ) ∇log v (t,x) ·
To calculate the control using (<ref>) and (<ref>), a realization of the empirical law is required. However, in order to avoid the computational burden of computing the optimal control for each realization of {μ_t^P: t ∈ [0,T]} within the DLMC algorithm, we independently obtain an accurate realization of the empirical law offline, by using a sufficiently large number of particles and time steps, as described in Algorithm 2 of our previous work <cit.>. This approach is motivated by the fact that it is sufficient to solve (<ref>) roughly but cheaply to get efficient variance reduction in the IS estimator. We numerically demonstrate this in Section <ref>.
§.§ DLMC estimator with importance sampling
We briefly outline the DLMC estimator for a given IS control ζ:[0,T] ℝ^d →ℝ^d.
* Consider uniform discretization 0=t_0<t_1<t_2<…<t_N = T of time domain [0,T] with N equal time steps of the particle system (<ref>), i.e., t_n = n Δ t, n=0,1,…,N and Δ t = T/N. Let us denote by X_p^P|N the discretized version of particle X_p^P corresponding to (<ref>) with P particles.
* Define the discrete law obtained from the time-discretized particle system by μ^P|N as
μ^P|N(t_n) = 1/P∑_j=1^P δ_X_j^P|N(t_n), ∀ n=0,…,N ·
* Consider the same time discretization as the particle system (<ref>) for the controlled decoupled MV-SDE (<ref>) with N equal time steps. Let us denote by {X^P|N_ζ(t_n)}_n=1^N the Euler-Maruyama time-discretized version of the controlled, decoupled McKean-Vlasov process X^P|N_ζ, conditioned on empirical law μ^P|N (<ref>)
* Thus, we can approximate our quantity of interest with IS as
G(X(T))≈G(X^P|N(T)) = G(X_ζ^P|N(T))𝕃^P|N,
where the likelihood factor 𝕃^P|N (see <cit.> for derivation) is
𝕃^P|N = ∏_n=0^N-1exp-1/2Δ t ζ(t_n,X_ζ^P|N(t_n))^2 - ⟨Δ W(t_n), ζ(t_n,X^P|N_ζ(t_n)) ⟩,
and {Δ W(t_n)}_n=0^N-1∼𝒩(0,√(Δ t)𝕀_d) are the Wiener increments used in the time-discretized decoupled MV-SDE (<ref>).
* Let M_1 be the number of realizations of μ^P|N in the DLMC estimator. Let ω_1:P^(i) denote the i^th realization of ω_1:P. For each realization of μ^P|N, let M_2 be the number of sample paths for the decoupled MV-SDE for each μ^P|N. Let ω̃^(j) denote the j^th realization of ω̃. Then, the DLMC estimator 𝒜_MC is defined as
𝒜_MC = 1/M_1∑_i=1^M_11/M_2∑_j=1^M_2 G (X_ζ^P|N(T) ) 𝕃^P|N( ω_1:P^(i), ω̃^(j)) ·
In <cit.>, we demonstrated that the DLMC estimator with IS in (<ref>), achieves an optimal complexity of 𝒪(_r^-4) for prescribed relative error tolerance _r. Moreover, the use of IS ensures that the constant associated with the complexity (<ref>) decreases drastically compared to naive MC, making the computation of rare event probabilities feasible. Furthermore, in our subsequent work <cit.>, we extended the aforementioned DLMC estimator to the multilevel setting. This extension allows for the estimation of expectations of rare event Lipschitz observables up to a relative tolerance of _r, while achieving a reduced complexity of 𝒪(_r^-3) for the considered Kuramoto example (<ref>).
The main contribution of this work is represented in Section <ref>, where we extend this estimator to the multi-index setting to obtain even better complexity.
§ MULTI-INDEX DOUBLE LOOP MONTE CARLO
Following <cit.>, we introduce the multi-index DLMC discretization. As seen in Section <ref>, two discretization parameters (P,N) are used to generate sample paths of the decoupled MV-SDE. We introduce multi-index α = ( α_1, α_2 ) ∈ℕ^2 and τ = ( τ_1, τ_2 ) ∈ℕ^2 that define the discretization parameters,
P_α_1 = P_0 τ_1^α_1,
N_α_2 = N_0τ_2^α_2·
P_0 and N_0 are the minimum number of particles and time steps used to generate approximate sample paths of the decoupled MV-SDE. Henceforth, we use τ_1 = τ_2 = τ for easier presentation. Let G = G(X(T)) and its corresponding discretization G_α = G(X̅^P_α_1|N_α_2(T)). Following <cit.>, we define the first order mixed difference for this setting
Δ G_α = ( G_(α_1,α_2) - G_(α_1-1,α_2)) - ( G_(α_1,α_2-1) - G_(α_1-1,α_2-1)) ·
The multi-index MC method is based on the following telescoping sum
G = ∑_α∈ℕ^2Δ G_α,
with G_(-1,0) = 0, G_(0,-1) = 0 and G_(-1,-1) = 0. Let Δ𝒢_α be a random variable such that Δ𝒢_α=Δ G_α. In the trivial case, Δ𝒢_α = Δ G_α. One can also choose Δ𝒢_α cleverly such that Δ𝒢_α≪Δ G_α. Each of the expectations in (<ref>) is approximated using a DLMC estimator, giving rise to the multi-index DLMC estimator,
G≈𝒜_MIMC(ℐ) = ∑_α∈ℐ1/M_1,α∑_m_1=1^M_1,α1/M_2,α∑_m_2=1^M_2,αΔ𝒢_α( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)) ,
where ℐ∈ℕ^2 is an appropriately chosen index-set and {M_1,α,M_2,α} are integer number of samples in the inner and outer loops of the DLMC estimator for each α∈ℐ. Following <cit.>, we construct an antithetic estimator Δ𝒢_α defined as
Δ𝒢_α( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)) = ( . ( G_(α_1,α_2) - 𝒢_(α_1-1,α_2))
- ( G_(α_1,α_2-1) - 𝒢_(α_1-1,α_2-1)) . ) ( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)),
where 𝒢_(α_1-1,α_2) is highly correlated to G_(α_1,α_2) and is defined as
𝒢_(α_1-1,α_2)( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)) = 1/τ∑_a=1^τ G_(α_1-1,α_2)( ω_(a-1)P_α_1-1+1:aP_α_1-1^(α,m_1), ω̃^(α,m_2)) ·
In (<ref>), we split the P_α_1 sets of random variables into τ iid groups of size P_α_1-1 each and then use each group to independently generate a realization of the empirical law. For each of the τ realizations of the empirical law we generate approximate sample paths of X̅^P_α_1-1| N_α_2 using the same ω̃ as for G_(α_1,α_2), before averaging the quantity of interest over the τ groups. From <cit.>, we know that the computational cost of the DLMC estimator with P number of particles, N number of time steps, M_1 number of outer loop samples and M_2 number of inner loop samples in Algorithm <ref> is M_1 N^γ_2 P^1+γ_1 + M_1 M_2 N^γ_2 P^γ_1. Here γ_1 > 0 is the computational complexity rate of estimating the empirical measure in the drift and diffusion coefficients and γ_2 > 0 is the computational complexity rate of the time discretization scheme. With this, we can express the total computational cost of the multi-index DLMC estimator,
𝒲[𝒜_MIMC(ℐ)] ≲∑_α∈ℐ M_1,α N_α_2^γ_2 P_α_1^1+γ_1 + M_1,α M_2,α N_α_2^γ_2 P_α_1^γ_1·
Notation a ≲ b means that there exists a constant c independent of b such that a < cb. Note that Δ G_α requires 4 evaluations of G at different discretization parameters. However, the largest work among them corresponds to the index α = (α_1,α_2). Due to independence of DLMC estimators for each α in (<ref>), the multi-index DLMC estimator variance can be written as
𝒜_MIMC(ℐ) = ∑_α∈ℐ1/M_1,α1/M_2,α∑_m_2=1^M_2,αΔ𝒢_α( ω_1:P_α_1^(α,·), ω̃^(α,m_2))·
Using the law of total variance,
𝒜_MIMC(ℐ) = ∑_α∈ℐ1/M_1,α( Δ𝒢_α|ω_1:P_α_1^(α,·)_=V_1,α + 1/M_2,αΔ𝒢_α|ω_1:P_α_1^(α,·)_=V_2,α)
= ∑_α∈ℐ( V_1,α/M_1,α + V_2,α/M_1,α M_2,α) ·
Here V_1,α and V_2,α are essentially conditioned on the discretized empirical law μ^P_α_1| N_α_2 which are described by the set of random variables ω_1:P_α_1^(α,·). Our aim is to build an efficient multi-index DLMC estimator that satisfies (<ref>). We bound the relative error of 𝒜_MIMC as
G - 𝒜_MIMC/G≤G - 𝒜_MIMC/G_=ϵ_b, Relative bias + 𝒜_MIMC - 𝒜_MIMC/G_=ϵ_s, Relative statistical error·
We split the accuracy between relative bias and statistical errors using parameter θ∈ (0,1) and impose following stricter constraints
ϵ_b ≤ (1-θ) _r,
ϵ_s ≥θ_r≤ν·
Throughout this work, θ is assumed to be given and fixed. Detailed analysis of the role of θ can be found in <cit.>. Using the asymptotic normality of the multi-index estimator <cit.>, the statistical error constraint (<ref>) can be approximated by the following constraint on the variance of the estimator
𝒜_MIMC(ℐ) = ∑_α∈ℐ( V_1,α/M_1,α + V_2,α/M_1,α M_2,α) ≤( θ_rG/C_ν)^2,
where C_ν is the (1 - ν/2)- quantile of the standard normal distribution. For a given index set ℐ, we optimize total computational cost (<ref>) with respect to M_1,α∈ℝ_+ and M_2,α∈ℝ_+ for all α∈ℐ subject to the statistical error constraint (<ref>).
[left=, right = ·]equation
2
min_{M_1,α,M_2,α}_α∈ℐ ∑_α∈ℐ M_1,α N_α_2^γ_2 P_α_1^1+γ_1 + M_1,α M_2,α N_α_2^γ_2 P_α_1^γ_1
s.t. (∑_α∈ℐ V_1,α/M_1,α + V_2,α/M_1,α M_2,α ) ≈( θ_r G/C_ν )^2
Solution to (<ref>) yields
ℳ_1,α = ( C_ν/θ_rG)^2 √(V_1,α/N_α_2^γ_2 P_α_1^1+γ_1)∑_β∈ℐ( √(V_1,β N_β_2^γ_2 P_β_1^1+γ_1) + √(V_2,β N_β_2^γ_2 P_β_1^γ_1)),
ℳ̃_α = ℳ_1,αℳ_2,α
= ( C_ν/θ_rG)^2 √(V_2,α/N_α_2^γ_2 P_α_1^γ_1)∑_β∈ℐ( √(V_1,β N_β_2^γ_2 P_β_1^1+γ_1) + √(V_2,β N_β_2^γ_2 P_β_1^γ_1)) ·
In practice, we use natural numbers for {ℳ_1,α,ℳ_2,α}_α∈ℐ. For this reason, and to guarantee at least one ℳ_1,α and ℳ_2,α for each α, we use following number of samples
M_1,α = ⌈ℳ_1,α⌉, M_2,α = ⌈ℳ̃_α/⌈ℳ_1,α⌉⌉·
Using (<ref>) we bound the estimator cost as
𝒲[𝒜_MIMC(ℐ)] ≲∑_α∈ℐ( (ℳ_1,α+1) N_α_2^γ_2 P_α_1^1+γ_1 + (ℳ_1,α+1) ( ℳ̃_α/⌈ℳ_1,α⌉+1 ) N_α_2^γ_2 P_α_1^γ_1)
≤∑_α∈ℐ( ℳ_1,α N_α_2^γ_2 P_α_1^1+γ_1 + ℳ̃_α N_α_2^γ_2 P_α_1^γ_1)_=W_1(ℐ) + ∑_α∈ℐ( P_α_1^1+γ_1 N_α_2^γ_2 + P_α_1^γ_1 N_α_2^γ_2) _=W_2(ℐ), cost of one sample per multi-index
+ ∑_α∈ℐℳ_1,α N_α_2^γ_2 P_α_1^γ_1_=W_3(ℐ) + ∑_α∈ℐℳ̃_α/⌈ℳ_1,α⌉ N_α_2^γ_2 P_α_1^γ_1_=W_4(ℐ)·
Using P_α_1 > 0 and γ_1 > 0, it is easy to see that,
W_3(ℐ) = ∑_α∈ℐℳ_1,α N_α_2^γ_2 P_α_1^γ_1≤∑_α∈ℐℳ_1,α N_α_2^γ_2 P_α_1^1+γ_1≤ W_1(ℐ),
W_4(ℐ) = ∑_α∈ℐℳ̃_α/⌈ℳ_1,α⌉ N_α_2^γ_2 P_α_1^γ_1≤∑_α∈ℐℳ̃_α/max(1,ℳ_1,α) N_α_2^γ_2 P_α_1^γ_1
≤∑_α∈ℐℳ̃_α N_α_2^γ_2 P_α_1^γ_1≤ W_1(ℐ) ·
Hence, we can rewrite (<ref>) as follows,
𝒲[𝒜_MIMC(ℐ)] ≲ W_1(ℐ) + W_2(ℐ) ·
Substituting (<ref>) in (<ref>), we get
𝒲[𝒜_MIMC(ℐ)] ≲( C_ν/θ_rG)^2 ( ∑_α∈ℐ√(V_1,α N_α_2^γ_2 P_α_1^1+γ_1) + √(V_2,α N_α_2^γ_2 P_α_1^γ_1))^2
+ ∑_α∈ℐ( P_α_1^1+γ_1 N_α_2^γ_2 + P_α_1^γ_1 N_α_2^γ_2) ·
We make the following assumptions in this work.
For all α∈ℐ, the absolute value of the expected value of Δ𝒢_α satisfies
Δ𝒢_α≤ Q_B τ^-α_1 b_1 - α_2 b_2,
for constants Q_B > 0 and b_1,b_2 > 0.
For all α∈ℐ, the variance terms in (<ref>) satisfy
V_1,α = Δ𝒢_α|ω_1:P_α_1^(α,·) ≲τ^-α_1 w_1 -α_2 w_2,
V_2,α = Δ𝒢_α|ω_1:P_α_1^(α,·) ≲τ^-α_1 s_1 -α_2 s_2 ,
for constants w_i > 0 and s_i >0 for i=1,2.
(On Assumptions <ref> and <ref>)
Assumption <ref> is derived from the bias assumptions made in our previous work <cit.>, which are based on the well-known propagation of chaos in McKean-Vlasov SDEs (where the empirical law converges to the mean-field law) and the standard weak convergence of the Euler-Maruyama time discretization scheme for SDEs. Assumption <ref> is motivated by the convergence of variance in the multilevel DLMC estimator introduced in our previous work <cit.>. Currently, there are no formal proofs of these assumptions for particle systems, but we validate them numerically for the Kuramoto model (<ref>) in Section <ref>.
Under Assumptions <ref>, <ref>, total computational cost (<ref>) can be estimated as
𝒲[𝒜_MIMC(ℐ)] ≲( C_ν/θ_rG)^2 ( ∑_α∈ℐ( exp𝐠̅·α + exp𝐠̅̅̅·α)_=W̃_1(ℐ))^2 + ∑_α∈ℐexpλ·α_=W̃_2(ℐ) ,
where the vectors 𝐠̅,𝐠̅̅̅,λ∈ℝ^2 are defined as
𝐠̅ = log(τ) [1+γ_1-w_1/2,γ_2-w_2/2],
𝐠̅̅̅ = log(τ) [γ_1-s_1/2, γ_2-s_2/2],
λ = log(τ) [1+γ_1, γ_2 ] ·
It should be noted that W̃_2(ℐ) represents the total computational cost required to generate one sample for each multi-index α∈ℐ. This is the minimum cost for a DLMC estimator, and it is essential to ensure that it does not dominate the first term in the bound (<ref>). For now, we assume that the first term indeed dominates over the second term, allowing us to focus solely on W̃_1(ℐ). In the subsequent theorem, we explicitly outline the conditions that guarantee the dominance of the first term.
§.§ Optimal index set
One of the main objectives of this work is to motivate a choice for the set of multi-indices ℐ = ℐ(_r) that minimizes 𝒲[𝒜_MIMC(ℐ)], subject to the following bias constraint
ϵ_b (ℐ) = 1/G∑_α∉ℐΔ𝒢_α≤1/G∑_α∉ℐΔ𝒢_α≤ (1-θ) _r·
Using (<ref>), we rewrite (<ref>) as
B̃(ℐ) = ∑_α∉ℐexp-ρ·α≤(1-θ) _rG/Q_B,
where we define the vector ρ = log(τ) [b_1, b_2 ] ∈ℝ^2. B̃(ℐ) can be seen as an upper bound on the global bias of 𝒜_MIMC. To find optimal ℐ, we solve the following optimization problem
min_ℐ∈ℕ^2𝒲[𝒜_MIMC(ℐ)] such that ϵ_b(ℐ) ≤ (1-θ) _r·
To simplify the problem we work with the upper bound on the bias (<ref>) and the work (<ref>). We assume that the first term in (<ref>) dominates the second. Subsequently, we solve the following simplified problem to get quasi-optimal set ℐ
min_ℐ∈ℕ^2W̃_1(ℐ) such that B̃(ℐ) ≤(1-θ) _rG/Q_B·
Note that W̃_1(ℐ) is defined in (<ref>) and B̃(ℐ) is defined in (<ref>). For convenience, we refer to the objective W̃_1 as the "work" and the constraint function B̃ as the "error" in the rest of this section. To solve the optimization problem (<ref>), we closely follow the procedure in <cit.>. First, we write the total error associated with index set ℐ from (<ref>) as follows
B̃(ℐ) = ∑_α∉ℐexp-ρ·α = ∑_α∉ℐℰ_α·
Here ℰ_α = exp-ρ·α denotes the "error" contribution of multi-index α. Next, we write the total "work" using (<ref>) as follows
W̃_1(ℐ) = ∑_α∈ℐ( exp𝐠̅·α + exp𝐠̅̅̅·α) = ∑_α∈ℐϖ_α·
Here ϖ_α = exp𝐠̅·α + exp𝐠̅̅̅·α denotes the "work" contribution of multi-index α. Next, we assign a "profit" indicator to each multi-index and only the most profitable multi-indices are added to the index set ℐ. Let us define the profit of a multi-index α as follows
𝒫_α = ℰ_α/ϖ_α·
Using (<ref>),(<ref>), we can approximate the profits by
𝒫_α = exp-ρ·α/exp𝐠̅·α + exp𝐠̅̅̅·α·
Following <cit.>, we define optimal index set for given level v ∈ℝ_+ as the following level set
ℐ(v) = {α∈ℕ^2: 𝒫_α≥ v}·
Inserting (<ref>) into (<ref>), we get the optimal index set for some L ∈ℝ_+ for the multi-index DLMC estimator
ℐ(L) = {α∈ℕ^2: exp(𝐠̅ + ρ)_=δ̅·α + exp(𝐠̅̅̅+ρ)_=δ̅̅̅·α≤ L }·
Here δ̅ = [δ̅_1, δ̅_2],δ̅̅̅ = [δ̅̅̅_1, δ̅̅̅_2] ∈ℝ_+^2 are normalised weighting vectors written explicitly as follows
δ̅ = [ ( 1 + γ_1 - w_1/2 + b_1 )/C_δ̅ , ( γ_2 - w_2/2 + b_2 )/C_δ̅], C_δ̅ = ( 1 + γ_1 - w_1/2 + b_1 ) + ( γ_2 - w_2/2 + b_2 ),
δ̅̅̅ = [ ( γ_1 - s_1/2 + b_1 )/C_δ̅̅̅ , ( γ_2 - s_2/2 + b_2 )/C_δ̅̅̅], C_δ̅̅̅ = ( γ_1 - s_1/2 + b_1 ) + ( γ_2 - s_2/2 + b_2 ) ·
To ensure admissible index sets according to the classical multi-index method <cit.>, we need that δ̅_1, δ̅_2,δ̅̅̅_1, δ̅̅̅_2 > 0. This leads to the following condition on the convergence rates
2b_1 ≥max (w_1-1,s_1)-γ_1, 2b_2 ≥max(w_2,s_2) - γ_2 ·
For the following theorem, let us introduce the following notation.
χ_11 = C_δ̅log (τ) max( 1 + γ_1 - w_1/1 + γ_1 - w_1 + 2 b_1, γ_2 - w_2/γ_2 - w_2 + 2 b_2),
χ_12 = C_δ̅log (τ) max( γ_1 - s_1/1 + γ_1 - w_1 + 2 b_1, γ_2 - s_2/γ_2 - w_2 + 2 b_2),
χ_21 = C_δ̅̅̅ (log) τmax( 1 + γ_1 - w_1/γ_1 - s_1 + 2 b_1, γ_2 - w_2/γ_2 - s_2 + 2 b_2),
χ_22 = C_δ̅̅̅log (τ) max( γ_1 - s_1/γ_1 - s_1 + 2 b_1, γ_2 - s_2/γ_2 - s_2 + 2 b_2),
η_1 = C_δ̅log (τ) min( 2 b_1/1 + γ_1 - w_1 + 2 b_1, 2 b_2/γ_2 - w_2 + 2 b_2),
η_2 = C_δ̅̅̅log (τ) min( 2 b_1/γ_1 - s_1 + 2 b_1, 2 b_2/γ_2 - s_2 + 2 b_2),
e_1 =
2, 1+γ_1-w_1/2b_1 = γ_2-w_2/2b_2
1, otherwise,
e_2 =
2, γ_1-s_1/2b_1 = γ_2-s_2/2b_2
1, otherwise,
ℵ_1 =
2, γ_1 - s_1/1 + γ_1 - w_1 + 2b_1 = γ_2-s_2/γ_2 - w_2 + 2b_2
1, otherwise, ℵ_2 =
2, 1 + γ_1 - w_1/γ_1 - s_1 + 2b_1 = γ_2-w_2/γ_2 - s_2 + 2b_2
1, otherwise,
d_1 =
2, w_1 = 1+γ_1 and w_2 = γ_2
1, w_1 = 1+γ_1 or w_2 = γ_2
0, otherwise,
d_2 =
2, s_1 = γ_1 and s_2 = γ_2
1, s_1 = γ_1 or s_2 = γ_2
0, otherwise,
ς = min( max(0, χ_11/η_1, χ_12/η_1), max( 0, χ_21/η_2, χ_22/η_2) ),
ϱ = max( d_1, d_2 ), ς = 0
(e_1 - 1) ( 1 + χ_11/η_1) , ς = χ_11/η_1
(ℵ_1 - 1) + (e_1-1) χ_12/η_1, ς = χ_12/η_1
(ℵ_2 - 1) + (e_2-1) χ_21/η_2, ς = χ_21/η_2
(e_2 - 1) ( 1 + χ_22/η_2), ς = χ_22/η_2,
Ψ = min( 1 + γ_1/1+γ_1 - w_1 + 2 b_1, γ_2/γ_2 - w_2 + 2 b_2, 1 + γ_1/γ_1 - s_1 + 2 b_1, γ_2/γ_2 - s_2 + 2 b_2) ·
(Optimal multi-index DLMC complexity)
Consider the optimal multi-index set given by (<ref>) with optimal vectors δ̅ and δ̅̅̅ as defined in (<ref>) and satisfying the condition (<ref>). Under Assumptions <ref> and <ref>, let the optimal L̅∈ℝ_+ satisfy the bias constraint (<ref>) in the following sense
lim__r↓ 0B̃(ℐ(L̅))/(1-θ) _rG/Q_B≤ 1 ·
Then, the total computational cost of the multi-index DLMC estimator (<ref>) 𝒲[𝒜_MIMC(ℐ(L̅))], subject to constraint (<ref>) satisfies the following
lim sup__r↓ 0𝒲[𝒜_MIMC(ℐ(L̅))]/_r^-2-2ς( log_r^-1)^2ϱ≤ C_work < ∞,
given that the following condition holds
Ψ≤ 1 + ς·
See Appendix <ref>.
(On isotropic directions)
To show the superiority of the multi-index DLMC estimator over the multilevel DLMC estimator from <cit.>, we look at the case when γ_1=γ_2=γ > 0, w_1=w_2=s_1=s_2=s>0 and b_1=b_2=b>0. That is, the complexity, bias and variances converge at the same rates in both P and N directions. One such case is the Kuramoto setting (<ref>) with the smooth, non-rare observable G(x) = cos(x). From Theorem <ref>, we have asymptotically as _r→ 0
𝒲[𝒜_MIMC] = _r^-2, s > 1+γ
_r^-2( log_r^-1)^2, s = 1 + γ
_r^-2-1+γ-s/b, s < 1 + γ·
On the other hand, the multilevel DLMC estimator <cit.> shows the following complexity rates
𝒲[𝒜_MLMC] = _r^-2, s > 1+2γ
_r^-2( log_r^-1)^2, s = 1 + 2γ
_r^-2-1+2γ-s/b, s < 1 + 2γ·
Comparing (<ref>) and (<ref>), one notices that multi-index DLMC has better complexity rates than multilevel DLMC in all regimes of variance convergence rate s with respect to the complexity rate γ. Also notice that the condition (<ref>) for the multi-index DLMC estimator simplifies in this setting to 2b ≥ s. For the multilevel case, we have the more restrictive condition 2b ≥min(s,1+2γ) <cit.>. One also deduces that the complexity of the multi-index DLMC estimator is the same as that of a multilevel DLMC estimator with only P as the discretization parameter (as if we generate exact sample paths of the particle system and decoupled MV-SDE). However, the multi-index DLMC requires mixed regularity (in sense of Assumptions <ref>,<ref>) with respect to P and N while multilevel DLMC only requires ordinary regularity.
(Kuramoto Model)
We know that γ_1=1 for a naive estimation method of the empirical mean in the drift/diffusion coefficients of the Kuramoto model (<ref>). We also know that γ_2=1 for the Euler-Maruyama scheme with uniform time grid for the type of drift/diffusion coefficients in (<ref>). In this setting with the mollified indicator observable G(x) = 1/2( 1 + tanh (3(x-K)) ), we show numerically in Section <ref> that b_1=b_2=1, w_1=w_2=s_1=2 and s_2=1.5. Using Theorem <ref> with these values, one can see that the complexity rate of the multilevel DLMC estimator would be _r^-2( log_r^-1)^2 compared to _r^-3 complexity of the multilevel DLMC estimator <cit.>.
§.§ IS scheme for multi-index DLMC estimator
One of the main objectives of this work is to incorporate IS into a multi-index estimator. In this study, we achieve this by computing control ζ once, through the solution of the control PDE (<ref>), as derived in Section <ref>. This computation is performed using a single realization of the particle system (<ref>) with a large number of particles P̅ and time steps N̅. Once this control is obtained, it is applied uniformly across all multi-indices α∈ℐ in the multi-index estimator (<ref>). Using IS, we rewrite the quantity of interest as follows
G≈∑_α∈ℐΔ G_α = ∑_α∈ℐΔ G^IS_α,
where the new IS sample of the mixed difference is defined as,
Δ G^IS_α = ( G^ζ_(α_1,α_2)𝕃_(α_1,α_2) - G^ζ_(α_1-1,α_2)𝕃_(α_1-1,α_2))
- ( G^ζ_(α_1,α_2-1)𝕃_(α_1,α_2-1) - G^ζ_(α_1-1,α_2-1)𝕃_(α_1-1,α_2-1)) ·
Here, G^ζ_(α_1,α_2) = G ( X̅_ζ^P_α_1| N_α_2 (T) ) and the discretized likelihood factor at multi-index α is given by
𝕃_α = ∏_n=0^N_α_2-1exp{. -1/2T/N_α_2ζ(t_n,α,X^P_α_1|N_α_2_ζ(t_n,α))^2 - ⟨Δ W(t_n,α), ζ(t_n,α,X^P_α_1|N_α_2_ζ(t_n,α)) ⟩. }·
X^P_α_1|N_α_2_ζ is the discretized, controlled, decoupled MV-SDE process (<ref>), {Δ W(t_n,α)}_n=1^N_α_2∼𝒩( 0,√(T/N_α_2)𝕀_d ) are the Wiener increments driving the process X^P_α_1|N_α_2_ζ and { t_n,α}_n=1^N_α_2 are the corresponding time discretization grid points. The proposed multi-index DLMC estimator with IS is written as
G≈𝒜^IS_MIMC(ℐ) = ∑_α∈ℐ1/M_1,α∑_i=1^M_1,α1/M_2,α∑_j=1^M_2,αΔ𝒢_α^IS (ω_1:P_α_1^(α,i),ω̃^(α,j)) ·
Here, the antithetic IS sampler is defined as
Δ𝒢^IS_α( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)) = ( . ( G^IS_(α_1,α_2) - 𝒢^IS_(α_1-1,α_2))
- ( G^IS_(α_1,α_2-1) - 𝒢^IS_(α_1-1,α_2-1)) . ) ( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)),
where G^IS_(α_1,α_2) = G^ζ_(α_1,α_2)𝕃_(α_1,α_2) and
𝒢^IS_(α_1-1,α_2)( ω_1:P_α_1^(α,m_1), ω̃^(α,m_2)) = 1/τ∑_a=1^τ G^IS_(α_1-1,α_2)( ω_(a-1)P_α_1-1+1:aP_α_1-1^(α,m_1), ω̃^(α,m_2)) ·
In this work, the computation of the IS control ζ is performed offline and independent of the multi-index α. In our previous study <cit.>, we demonstrate that ζ leads to a zero-variance estimator for G(X̅^P(T)) |{μ^P_t: t ∈ [0,T]}. However, it should be emphasized that this does not imply optimal minimization of Δ G_α. The task of obtaining an α-dependent control using stochastic optimal control, which minimizes Δ G_α, is left for future research.
Algorithm <ref> outlines the implementation of above IS scheme to estimate Δ G_α for each α∈ℐ, that is required for the multi-index DLMC estimator (<ref>).
§.§ Adaptive multi-index DLMC algorithm with importance sampling
The purpose of this section is to develop an adaptive multi-index DLMC algorithm that sequentially selects the index set ℐ and determines the optimal number of samples {M_1,α,M_2,α}_α∈ℐ to satisfy the error constraints specified in (<ref>) and (<ref>). To accomplish this, the adaptive algorithm relies on cost-effective yet reliable estimations of the relative bias and variances {V_1,α,V_2,α}_α∈ℐ. This algorithm is a modified version of the MIMC algorithm presented in <cit.>.
§.§.§ Estimating bias of index set ℐ
The optimal index set (<ref>) is defined using the parameter or threshold, L. Given that ℐ(L) ⊂ℐ(L+1), we use the following heuristic estimate <cit.> for the absolute bias corresponding to index-set ℐ(L)
G - 𝒜_MIMC^IS(ℐ(L))≈∑_α∈∂ℐ(L)Δ G_α^IS,
where ∂ℐ(L) is the outer boundary of the index set ℐ(L) and is defined as follows
∂ℐ(L) = {α∈ℐ(L): α + (1,0) ∉ℐ(L) 𝐨𝐫 α + (0,1) ∉ℐ(L)}·
Since ∂ℐ(L) ⊂ℐ(L), we use already computed nested averages using optimal {M_1,α,M_2,α}_α∈∂ℐ(L) in (<ref>) to estimate each of the expectations in (<ref>).
§.§.§ Estimating {V_1,α,V_2,α}
One needs cheap and robust empirical estimates of {V_1,α,V_2,α} for all α∈ℐ(L) to compute the optimal number of samples required to satisfy the variance constraint (<ref>) of the estimator using (<ref>). For this, Algorithm <ref> with appropriately chosen M̃_1,M̃_2 is used.
Estimating {V_1,α,V_2,α} for all α∈ℐ(L) using Algorithm <ref> can become computationally burdensome when ℐ(L) is even moderately large. To alleviate this computational overload, we leverage Assumption <ref> and employ an extrapolation approach to estimate {V_1,α,V_2,α} for deeper multi-indices. Specifically, we utilize Algorithm <ref> solely to estimate {V_1,α,V_2,α} for the small full tensor index set {0,1,2}{0,1,2}. Then, we employ the extrapolation Algorithm <ref>, making use of Assumption <ref>, to estimate {V_1,α,V_2,α} for the remaining multi-indices. To further alleviate computational burden, we only estimate {V_1,α,V_2,α} for the newly added multi-indices in each iteration, i.e. for α∈ℐ(L+1) ∖ℐ(L). As ℐ(L) ⊂ℐ(L+1) by construction, we can reuse the estimates of {V_1,α,V_2,α} for multi-indices from previous iterations.
§.§.§ Relative error control
To meet the relative error constraints (<ref>) and (<ref>) for a given relative error tolerance _r, the adaptive algorithm requires a heuristic estimate of the quantity of interest G itself. In our algorithm, we update this estimate at each iteration L. For L=0, we utilize Algorithm <ref> with appropriately selected M̅_1 and M̅_2 to obtain an initial estimate G̅ for G_(0,0). In subsequent iterations, we employ the multi-index estimator (<ref>) with optimal values of {M_1,α,M_2,α}_α∈ℐ(L) to update G̅ as well as the absolute error tolerances. Combining all the aforementioned components, we present the adaptive multi-index DLMC algorithm <ref> for evaluating rare event observables associated with MV-SDEs. The IS control ζ in Algorithm <ref> is computed offline by generating one realization of the empirical law μ^P̅|N̅ with some large P̅,N̅ and then numerically solving control PDE (<ref>) given μ^P̅|N̅.
In Algorithm <ref>, it is evident that in order to determine the optimal index set ℐ(L) using equation (<ref>), it is necessary to estimate the rates {b_1,b_2}, {w_1,w_2}, {s_1,s_2}, {γ_1,γ_2} associated with Assumptions <ref> and <ref>. To obtain these estimates, reliable pilot runs are required. It is important to note that such pilot runs are not necessary for the adaptive DLMC algorithm <cit.> or the multi-level DLMC algorithm <cit.> described in our previous works.
§ NUMERICAL RESULTS
This section provides numerical evidence for the assumptions and complexity rates derived in Section <ref>. We also numerically validate the efficiency of the IS scheme introduced in Section <ref> for the multi-index DLMC estimator. The results outlined below focus on the Kuramoto model (see Section <ref>) with the following parameter settings: σ = 0.4, T = 1, x_0 ∼𝒩(0,0.2), and ξ∼𝒰(-0.2,0.2). We fix parameters τ=2, θ=0.5, and ν=0.05. We set
P_α_1 = 5 × 2^α_1, N_α_2 = 4 × 2^α_2 .
We implement the proposed multi-index DLMC method for the following rare-event observable, also known as the "mollified" indicator function
G(x) = 1/2( 1 + tanh( 3 (x - K) ) ),
where K ∈ℝ is a threshold parameter that determines how rare the event is. In the following results, we use K = 3.5, which corresponds to G≈ 2.04 10^-5. We test two methods: multilevel DLMC as outlined in <cit.> and multi-index DLMC (<ref>) with the index set defined in (<ref>). For both methods, we use an IS control ζ obtained from (<ref>) described in Section <ref> by simulating a particle system (<ref>) offline with P̅ = 1000 particles and N̅ = 100 time steps. The solution to the offline problem (<ref>) was obtained using finite differences with linear interpolation over the entire domain. Our initial analysis focused on verifying the reduction in variance for the DLMC estimators of the mixed difference Δ G_α. Figure <ref> illustrates the squared coefficient of variation for the estimator of Δ G_α with and without importance sampling, plotted as a function of M_1 with a fixed M_2. The results in Figure <ref> demonstrate a significantly reduced squared coefficient of variation (around one order of magnitude) for the fixed α = (2,2). It is worth noting that both variances decrease with M_1^-1, and IS decreases the associated constant. Subsequently, we examine the impact of our IS scheme on all multi-indices α∈ℕ^2. To evaluate this, we analyze the following ratio:
ℛ(𝒜) = 𝒜_IS/𝒜_MC·
Here, 𝒜_IS and 𝒜_MC denote the DLMC estimators of some quantity of interest (G_α or Δ G_α) with and without IS respectively. On the left (Figure <ref>), we present a contour plot illustrating the values of the ratio ℛ for the DLMC estimator of G_α across different α∈ℕ^2, representing the quantity of interest approximated with discretization parameters defined by α. One sees in Figure <ref> that there is significant and uniform variance reduction for the DLMC estimator of G_α for all α. This is expected since our IS control ζ has been constructed to minimize the variance of the corresponding estimator <cit.>. On the right (Figure <ref>), we depict the same plot for the DLMC estimator of Δ G_α across different α∈ℕ^2. Even though there is still good variance reduction achieved by the same IS control ζ, we observe that this effect diminishes as we delve into deeper multi-indices for the mixed differences estimator.
Figures <ref> and <ref> provide confirmation of Assumptions <ref> and <ref> while also numerically determining the convergence rates {b_1,b_2},{w_1,w_2},{s_1,s_2} for the above-considered Kuramoto example. In Figure <ref>, the convergence of Δ G_α is plotted to obtain the rates b_1 = 1 and b_2 = 1. Additionally, Figures <ref> and <ref> showcase the convergence of V_1, α and V_2, α for the mixed differences, respectively, yielding the rates w_1 = 2, w_2 = 2, s_1 = 2, and s_2 = 1.5. These convergence rates are compared to those of the level differences, Δ G_ℓ = G_ℓ - G_ℓ-1, utilized in the multilevel DLMC estimator with IS <cit.>. Notably, the mixed differences of the multi-index DLMC estimator exhibit higher convergence rates compared to the level differences of the multi-level estimator. For computational cost analysis, we know that γ_1 = 1 since the cost of naively estimating the empirical mean in the drift and diffusion coefficients of the particle system (<ref>) is P with respect to the number of particles. Additionally, we know γ_2 = 1 as the computational complexity of the uniform Euler-Maruyama time discretization scheme is N.
Figure <ref> additionally shows that Assumptions <ref> and <ref> are indeed satisfied for sufficiently fine discretizations. Using the rates obtained in Figures <ref> and <ref> we can explicitly write down the optimal index set for the considered Kuramoto example
ℐ(L) = {α∈ℕ^2 : exp( 2/3α_1 + 1/3α_2 ) + exp( 2/5α_1 + 3/5α_2 ) ≤ L }·
We plug the above rates in Theorem <ref> and we expect the work complexity of multi-index DLMC with the index set (<ref>) to be _r^-2( log_r^-1)^2. Also, the condition (<ref>) for multi-index DLMC is satisfied. For the corresponding multilevel convergence rates obtained in Figure <ref>, we expect work complexity of _r^-3 for the multilevel DLMC estimator <cit.>.
We plot the index-set (<ref>) for different values of L in Figure <ref>. We compare this to the index-set obtained with actual numerical profit computed as follows
ℐ(v) = {α∈ℕ^2: 𝒫_α = ℰ_α/ϖ_α≈Δ G_α/√(V_1,α P_α_1^2 N_α_2) + √(V_2,α P_α_1 N_α_2)≥ v}·
We use DLMC estimates of Δ G_α, V_1,α, V_2,α computed in Figure <ref> to approximate the numerical profits in (<ref>) to produce Figure <ref>. We see that (<ref>) is a reasonable approximation to (<ref>), for sufficiently fine discretizations.
Figures <ref>-<ref> depict the results of running the adaptive multi-index DLMC Algorithm <ref> with IS and comparing them with the results obtained from the adaptive multilevel algorithm in <cit.>. We employ the parameter values {M̅_1, M̅_2} = {10^3, 10^2} and {M̃_1, M̃_2} = {25, 100}. Algorithm <ref> and the corresponding multilevel Algorithm, are independently executed five times each, and the combined results are plotted. Figure <ref> displays the exact computational relative error achieved by our proposed multi-index DLMC estimator with IS for various prescribed relative tolerances _r, as estimated using a reference multi-index DLMC approximation with _r = 1%. Each marker represents a separate run of the corresponding adaptive algorithm. Figure <ref> confirms that both the multi-index and multilevel estimators satisfy the prescribed _r, in the sense of (<ref>). Figure <ref> illustrates the maximum level of discretization required for both the number of particles and time steps in order for both the multilevel and multi-index estimators to satisfy the relative bias constraint (<ref>). In the above Kuramoto model setting, we observe that the maximum number of particles is approximately the same for both the multilevel and multi-index estimators.
In Figure <ref>, we present a plot of the various error estimates computed during the execution of Algorithm <ref>. Figure <ref> displays the estimated relative bias at the final iteration of Algorithm <ref>, which is calculated using (<ref>). Furthermore, Figure <ref> showcases the estimated relative statistical error of our estimator 𝒜_MIMC(ℐ). As per the algorithm's design, both relative errors should remain below _r/2 for Algorithm <ref> to terminate. Moreover, Figure <ref> verifies that the number of samples chosen in (<ref>), for our estimator 𝒜_MIMC(ℐ), is indeed optimal.
In Figure <ref>, we present the average computational runtime of Algorithm <ref> for various relative error tolerances. Additionally, Figure <ref> displays the average computational cost of both estimators for different values of _r, computed using (<ref>). That is,
Computational Cost[𝒜_MIMC(ℐ)] ≈∑_α∈ℐ(L)( M_1,α N_α_2 P_α_1^2 + M_1,α M_2,α N_α_2 P_α_1) ·
Figure <ref> also highlights that the adaptive multilevel algorithm failed to produce estimates of the quantity of interest up to _r = 1%, while the multi-index algorithm achieved it within a fixed computational budget. Furthermore, Figures <ref> and <ref> provide further evidence supporting the superiority of the proposed multi-index DLMC estimator over the multilevel DLMC estimator. The multi-index estimator demonstrates a complexity reduction of approximately one order (with logarithmic terms) compared to the multilevel estimator in the Kuramoto setting considered. Moreover, Figure <ref> also indicates that the complexity rates derived in this work are reasonably accurate.
§ CONCLUSION
This paper provides theoretical and numerical evidence, under certain verifiable assumptions, of the superiority of our multi-index DLMC estimator over the multilevel DLMC estimator <cit.> when approximating rare event quantities of interest in the MV-SDE context. These quantities are expressed as expectations of sufficiently smooth observables of solutions to stochastic particle systems in the mean field limit. We utilize the IS scheme introduced in our previous work <cit.> uniformly for all mixed difference estimators within the proposed multi-index DLMC estimator. Numerical results confirm significant variance reduction and computational gains. For the example considered, the proposed multi-index DLMC estimator achieves a complexity of _r^-2 (log^-1)^2, which is one order less than that of the multilevel DLMC estimator, where _r is the prescribed relative error tolerance. Integrating the IS scheme into the multilevel MC estimator significantly reduces the constant associated with its complexity compared to the multi-index MC estimator for smooth, non-rare observables introduced in <cit.>.
Future research directions involve extending the IS scheme to higher-dimensional problems by employing model reduction techniques or utilizing stochastic gradient-based learning methods to solve the associated higher-dimensional stochastic optimal control problem. Further optimization of the multi-index DLMC algorithm could involve finding optimal parameters τ and θ <cit.> or integrating a continuation-type multi-index algorithm <cit.>. Furthermore, extending the present analysis to numerically handle non-smooth rare event observables, such as indicator functions for computing rare event probabilities, is an area for exploration.
§ PROOF OF THEOREM <REF>
We begin by writing the optimal index set (<ref>) explicitly for given level v ∈ℝ_+
ℐ(v) = {α∈ℕ^2: exp-ρ·α/exp𝐠̅·α + exp𝐠̅̅̅·α≥ v }·
We make use of the following optimality result to prove Theorem <ref>.
(Optimal Index Sets)
The set ℐ(v) = {α∈ℕ^2: exp-ρ·α/exp𝐠̅·α + exp𝐠̅̅̅·α≥ v } is optimal in the sense that any other set ℐ̅ such that B̃(ℐ(v)) = B̃(ℐ̅) will be a set with larger work, i.e. W̃_1(ℐ(v)) ≤W̃_1(ℐ̅).
We have that for any α∈ℐ(v) and α̂∉ℐ(v)
exp-ρ·α/exp𝐠̅·α + exp𝐠̅̅̅·α≥ v,
exp-ρ·α̂/exp𝐠̅·α̂ + exp𝐠̅̅̅·α̂ < v ·
Let us divide ℕ^2 into the following disjoint sets:
𝒥_1 = ℐ(v) ∩ℐ̅^C, 𝒥_2 = ℐ(v) ∩ℐ̅,
𝒥_3 = ℐ(v)^C ∩ℐ̅, 𝒥_4 = ℐ(v)^C ∩ℐ̅^C,
where ℐ(v)^C is the complement of the set ℐ(v). By construction of ℐ̅, we have that
B̃(ℐ(v)) = B̃(ℐ̅)
∑_α∈𝒥_3 ∪𝒥_4exp-ρ·α = ∑_α∈𝒥_1 ∪𝒥_4exp-ρ·α
∑_α∈𝒥_3exp-ρ·α = ∑_α∈𝒥_1exp-ρ·α·
Note that 𝒥_1 and 𝒥_3 cannot be empty, for sets which satisfy B̃(ℐ(v)) = B̃(ℐ̅), unless ℐ(v) = ℐ̅. Consider the difference between the work of the two sets,
W̃_1(ℐ(v)) - W̃_1(ℐ̅) = ∑_α∈𝒥_1 ∪𝒥_2exp𝐠̅·α + exp𝐠̅̅̅·α - ∑_α∈𝒥_2 ∪𝒥_3exp𝐠̅·α + exp𝐠̅̅̅·α
= ∑_α∈𝒥_1exp𝐠̅·α + exp𝐠̅̅̅·α - ∑_α∈𝒥_3exp𝐠̅·α + exp𝐠̅̅̅·α·
We know from (<ref>) and (<ref>) that
∀α∈𝒥_1, exp𝐠̅·α + exp𝐠̅̅̅·α≤1/vexp-ρ·α and,
∀α∈𝒥_3, - exp𝐠̅·α - exp𝐠̅̅̅·α < - 1/vexp-ρ·α·
Inserting (<ref>),(<ref>) into (<ref>), we get
W̃_1(ℐ(v)) - W̃_1(ℐ̅) ≤1/v( ∑_α∈𝒥_1exp-ρ·α - ∑_α∈𝒥_3exp-ρ·α) _=0 from (<ref>)
⇒W̃_1(ℐ(v)) ≤W̃_1(ℐ̅) ·
Since the index set ℐ(v) is optimal in the sense of Lemma <ref>, we bound W̃_1(ℐ(v)) by the optimal work done by an index set that can be analysed following the classical multi-index setting <cit.>. From the definition of the optimal index set (<ref>), there are two obvious candidates for such index sets.
ℐ_δ̅(L̅) = {α∈ℕ^2: δ̅·α≤L̅},
ℐ_δ̅̅̅(L̅) = {α∈ℕ^2: δ̅̅̅·α≤L̅}·
We now analyse the optimal work of the two index sets separately.
Set 1: ℐ_δ̅
Following <cit.>, we know there exists an L̅_1 ∈ℝ_+ such that the following bias constraint is satisfied.
lim__r↓ 0B̃(ℐ_δ̅(L̅_1))/(1-θ) _rG/Q_B≤ 1 ·
For L̅ that satisfies (<ref>), let us estimate the work of the corresponding index set.
W̃_1(ℐ_δ̅(L̅_1)) = ∑_{α∈ℕ^2: δ̅·α≤L̅_1 }( exp𝐠̅·α + exp𝐠̅̅̅·α)
= ∑_{α∈ℕ^2: δ̅·α≤L̅_1 }exp𝐠̅·α_= W̃_11 + ∑_{α∈ℕ^2: δ̅·α≤L̅_1 }exp𝐠̅̅̅·α_= W̃_12·
From <cit.>, we know that asymptotically as _r→ 0,
W̃_11≲( log_r^-1)^d_1 , χ_11≤ 0
_r^-χ_11/η_1( log_r^-1)^j_11, χ_11 > 0
,
and
W̃_12≲( log_r^-1)^d_2 , χ_12≤ 0
_r^-χ_12/η_1( log_r^-1)^j_12, χ_12 > 0
,
where we have the following additional notation.
j_11 = (e_1 - 1) ( 1 + χ_11/η_1),
j_12 = (ℵ_1 - 1) + (e_1-1) χ_11/η_1·
Set 2: ℐ_δ̅̅̅
Following <cit.>, we know there exists an L̅_2 ∈ℝ_+ such that the bias constraint (<ref>) is satisfied. The work of the corresponding index set is written as
W̃_1(ℐ_δ̅(L̅_2)) = ∑_{α∈ℕ^2: δ̅̅̅·α≤L̅_2 }( exp𝐠̅·α + exp𝐠̅̅̅·α)
= ∑_{α∈ℕ^2: δ̅̅̅·α≤L̅_2 }exp𝐠̅·α_= W̃_21 + ∑_{α∈ℕ^2: δ̅̅̅·α≤L̅_2 }exp𝐠̅̅̅·α_= W̃_22·
From <cit.>, we know that asymptotically as _r→ 0,
W̃_21≲( log_r^-1)^d_1 , χ_21≤ 0
_r^-χ_21/η_2( log_r^-1)^j_21, χ_21 > 0
,
and
W̃_22≲( log_r^-1)^d_2 , χ_22≤ 0
_r^-χ_22/η_2( log_r^-1)^j_22, χ_22 > 0
,
where we have the following additional constants.
j_21 = (ℵ_2 - 1) + (e_2-1) χ_21/η_2,
j_22 = (e_2 - 1) ( 1 + χ_22/η_2) ·
From Lemma <ref>, we can bound the optimal work of the index set (<ref>) as follows
W̃_1(ℐ(v_1)) ≤min( W̃_1(ℐ_δ̅(L̅_1)), W̃_1(ℐ_δ̅̅̅(L̅_2)) ) ·
Inserting (<ref>), (<ref>) into (<ref>) and (<ref>), (<ref>) into (<ref>) and combining all into (<ref>), we obtain
W̃_1(ℐ(v_1)) ≲_r^-ς( log_r^-1)^ϱ·
Let us also analyse W̃_2(ℐ(v_1)) since we need to ensure that this work does not dominate the first term in (<ref>). Following the same analysis of the work as for W̃_1, we have
W̃_2(ℐ(v_1)) = ∑_{α∈ℕ^2: expδ̅·α + expδ̅̅̅·α≤ L }expλ·α
≤min( ∑_{α∈ℕ^2: δ̅·α≤L̅}expλ·α , ∑_{α∈ℕ^2: δ̅̅̅·α≤L̅}expλ·α) ·
From <cit.> we know that
∑_{α∈ℕ^2: δ̅·α≤L̅}expλ·α≲_r^-Γ_1/η_1( log_r^-1)^m_1,
∑_{α∈ℕ^2: δ̅̅̅·α≤L̅}expλ·α≲_r^-Γ_2/η_2( log_r^-1)^m_2·
where we define the following constants,
Γ_1 = 2 C_δ̅logτmin( 1 + γ_1/1+γ_1 - w_1 + 2 b_1, γ_2/γ_2 - w_2 + 2 b_2),
Γ_2 = 2 C_δ̅̅̅logτmin( 1 + γ_1/γ_1 - s_1 + 2 b_1, γ_2/γ_2 - s_2 + 2 b_2),
g_1 =
2, 1+γ_1/1+γ_1-w_1+2b_1 = γ_2/γ_2-w_2+2b_2
1, otherwise,
g_2 =
2, 1+γ_1/γ_1-s_1+2b_1 = γ_2/γ_2-s_2+2b_2
1, otherwise,
m_1 = (g_1 - 1) + (e_1-1) Γ_1/η_1 ,
m_2 = (g_2 - 1) + + (e_2-1) Γ_2/η_2·
Inserting (<ref>),(<ref>) in (<ref>), we get
W̃_2(ℐ(v_1)) ≲_r^-2 Ψ( log_r^-1)^ϰ,
where
ϰ =
(g_1 - 1) + (e_1-1) Γ_1/η_1, Ψ = 1 + γ_1/1+γ_1 - w_1 + 2 b_1 or γ_2/γ_2 - w_2 + 2 b_2
(g_2 - 1) + + (e_2-1) Γ_2/η_2 , Ψ = 1 + γ_1/γ_1 - s_1 + 2 b_1 or γ_2/γ_2 - s_2 + 2 b_2·
Inserting (<ref>) and (<ref>) into (<ref>), we get
𝒲[𝒜_MIMC(ℐ(v_1))] ≲_r^-2-2ς( log_r^-1)^2ϱ + _r^-2 Ψ( log_r^-1)^ϰ·
To ensure that the first term in (<ref>) dominates the second term, we need to satisfy the condition (<ref>). This concludes the proof.
plainnat
authoryear,open=[,close=]
|
http://arxiv.org/abs/2307.04081v1 | 20230709014122 | Score-based Conditional Generation with Fewer Labeled Data by Self-calibrating Classifier Guidance | [
"Paul Kuo-Ming Huang",
"Si-An Chen",
"Hsuan-Tien Lin"
] | cs.CV | [
"cs.CV",
"cs.LG"
] |
Age of FGK Dwarfs Observed with LAMOST and GALAH: Considering the Oxygen Enhancement
Jinghua Zhang
Received August 12, 2023; accepted August 12, 2023
====================================================================================
Score-based Generative Models (SGMs) are a popular family of deep generative models that achieves leading image generation quality. Earlier studies have extended SGMs to tackle class-conditional generation by coupling an unconditional SGM with the guidance of a trained classifier. Nevertheless, such classifier-guided SGMs do not always achieve accurate conditional generation, especially when trained with fewer labeled data. We argue that the issue is rooted in unreliable gradients of the classifier and the inability to fully utilize unlabeled data during training. We then propose to improve classifier-guided SGMs by letting the classifier calibrate itself. Our key idea is to use principles from energy-based models to convert the classifier as another view of the unconditional SGM. Then, existing loss for the unconditional SGM can be adopted to calibrate the classifier using both labeled and unlabeled data. Empirical results validate that the proposed approach significantly improves the conditional generation quality across different percentages of labeled data. The improved performance makes the proposed approach consistently superior to other conditional SGMs when using fewer labeled data. The results confirm the potential of the proposed approach for generative modeling with limited labeled data.
§ INTRODUCTION
Score-based Generative Models (SGMs) capture the underlying data distribution by learning the gradient function of the log-likelihood on data, also known as the score function. SGMs, when coupled with a diffusion process that gradually converts noise to data, can often synthesize higher-quality images than other popular alternatives, such as generative adversarial networks <cit.>. SGMs attracted research attention and demonstrated promising performance not only in image generation <cit.> but also in audio synthesis <cit.>, natural language generation <cit.>, and various other fields.
Many successful SGMs above focus on unconditional generation, which models the data distribution without considering other variables <cit.>. When aiming to generate data with some control, it is necessary to model the conditional distribution concerning another variable, such as the class label for generating images from a particular class.
Such conditional SGMs will be the focus of this paper. They have achieved cutting-edge performance for class-conditional generation <cit.>, image inpainting <cit.>, and audio upsampling <cit.>.
There are two major families of conditional SGMs. The family of Classifier-Free SGMs designs specific conditional network architectures with their losses derived from the conditional score functions <cit.>. Such SGMs are known to generate high-fidelity images in fully-supervised settings where all data are labeled. Nevertheless, they are often criticized for generating data with less diversity, favoring some easier classes while being inaccurate for some harder classes. Furthermore, their performance drops significantly as the proportion of labeled data decreases, making them less preferable in semi-supervised settings.
Classifier-Guided SGMs (CGSGMs) form another family of conditional SGMs that address the aforementioned issues by decomposing the conditional score function into a mixture of the unconditional score function and the gradient of an auxiliary classifier <cit.>.
For instance, the vanilla CGSVM <cit.> trains the unconditional SBM with the popular Denoising Score Matching (DSM) <cit.> technique that learns the score function from noise-perturbed data, and the classifier with the usual cross-entropy loss from labeled data. The additional classifier improves the accuracy of conditional generation and allows better control of the trade-off between generation diversity and fidelity <cit.>. Furthermore, because the unconditional SBM can be trained with either labeled or unlabeled data in principle, CGSGMs potentially fit the semi-supervised setting better by requiring fewer labeled data.
The quality of the auxiliary classifier is critical for CGSGMs. If the classifier is overly confident in its predictions, as often happens with cross-entropy loss <cit.>, the resulting conditional scores may be unreliable. This, in turn, leads to low generation accuracy, even if the unconditional scores are reliable enough to ensure decent generation fidelity.
Robust CGSGM <cit.> trains an adversarial robust classifier instead of a usual one to improve the quality of the auxiliary classifier. Somehow there is no theoretical guarantee that adversarial robustness is related to reliable conditional scores. Denoising Likelihood Score Matching <cit.> proposes to calibrate the classifier on the labeled data externally, leveraging the help of the unconditional SGM. Then, the training of the classifier is dependent on having a trained unconditional SGM first.
Our proposed approach is aligned with both techniques above to design a better loss to train the classifier. Still, it significantly differs from them by letting the classifier self-calibrate. Unlike the robust CGSGM, the self-calibration technique carries a sound theoretical guarantee by converting the classifier to another view of the unconditional SGM when reinterpreting the classifier through the angle of energy-based models. The novel view allows reusing DSM seamlessly to design a Self-Calibration (SC) loss (as illustrated with ℒ_SC in Fig. <ref>) that can be used on the classifier without dependence to the unconditional SGM. Furthermore, the SC loss can be effortlessly applied to both labeled and unlabeled data, resulting in immediate advantages in the semi-supervised setting.
We demonstrate the effect of self-calibration by visualizations on a toy data set. The results justify that our proposed CGSGM with the SC loss (CGSGM-SC) approach results in more accurate classifier gradients, thus enhancing the estimation of the conditional scores.
We further conduct thorough experiments on CIFAR-10 and CIFAR-100 datasets to validate the advantages of the proposed approach. The results confirm that CGSGM-SC is superior to the vanilla CGSGM and state-of-the-art techniques in the CGSGM family. Furthermore, in an extreme setting of having only 5% of the data being labeled, CGSGM-SC, which can use unlabeled data to self-calibrate the classifier, is significantly better than both classifier-guided and classifier-free SGMs, which cannot easily take the unlabeled data into account. The results confirm the potential of CGSGM-SC in scenarios where labeled data are costly to obtain.
§ BACKGROUND
Consider a data distribution p(x) where x∈ℝ^d. SGMs aim to generate samples from p(x) via the information contained in the score function ∇_xlog p(x), which is learned from data. We first introduce how the score function can be efficiently learned from data in Section <ref>, which is related to the derivation of our proposed loss. Then, we discuss how a diffusion process can be combined with learning a score function to effectively sample from p(x) in Section <ref>. Finally, we review studies that extend SGMs to conditional SGMs in Section <ref>.
§.§ Learning the score function
Learning the score function aims to choose the best function from a family of functions {s_θ(x)}_θ, such as deep learning models parameterized by θ, to approximate the score function ∇_x log p(x) of interest. The learning is based on some data {x_n}_n=1^N that are assumed to be sampled from p(x). It has been shown that the aim can be achieved by optimizing the in-sample version of the following score-matching loss over θ:
ℒ_SM=𝔼_p(x)[tr(∇_x s_θ(x))+1/2‖ s_θ(x)‖^2_2],
where tr(·) denotes the trace of a matrix and ∇_x s_θ(x)=∇^2_x log p(x) is the Hessian matrix of log-likelihood p(x). Somehow calculating the score-matching loss requires O(d) passes of computation for x ∈ℝ^d, which makes the optimization process computationally prohibitive on high-dimensional data.
Several previous studies <cit.> attempted to resolve the computational issue by approximating or transforming score matching into equivalent objectives. One standard approach nowadays is called Denoise Score Matching (DSM) <cit.>, which learns the score function of a noise-perturbed data distribution q(x̃) instead. DSM typically assumes that q(x̃) comes from the original distribution p(x)
injected with a pre-specified noise q(x̃|x). Then, it has been proved <cit.> that the score function can be learned by minimizing the in-sample version of
𝔼_q(x̃|x)p(x)[1/2‖ s_θ(x̃) - ∇_x̃log q(x̃|x)‖_2^2],
where ∇_x̃log q(x̃|x) is the score function of the noise distribution centered at x. DSM is generally more efficient than the original score matching and is scalable to high-dimensional data as it replaces the heavy computation on the Hessian matrix with simple perturbations that can be efficiently computed from data.
§.§ Generating from the score function by diffusion
Assume that we hope to sample from some unknown target distribution p(x) = p_0(x), and the distribution can be transited to a known prior distribution p_T(x) through a Markov chain that is described with
some stochastic differential equation (SDE) <cit.>:
dx(t)=f(x(t),t)dt+g(t)dw,
where the Markov chain is computed for 0 ≤ t < T using the drift function f(x(t),t) that describes the overall movement and the dispersion function g(t) that describes how the noise w from a standard Wiener process enters the system.
To sample from p(x) = p_0(x), the VE-SDE framework <cit.> proposes
to reverse the SDE from p_T(x) to p_0(x), which turns out to operate with another SDE <cit.>:
dx=[f(x(t),t)-g(t)^2 s(x(t), t)]dt+g(t)dw̅
where w̅ is a standard Wiener process when time-step flows from T back to 0 and s(x(t), t) ≡∇_xlog p_t(x(t))=s(x(t), t) denotes a time-dependent score function. If we can learn the score function s(x(t), t), the diffusion process in (<ref>) can then be used to take any instance sampled from the known p_T(x) to a sample from the unknown p(x) = p_0(x).
Learning the time-dependent score function s(x(t), t)) can be done by minimizing an time-generalized (in-sample) version of the DSM loss because the diffusion process can be viewed as one particular way of injecting noise. The extended DSM loss is defined as
ℒ_DSM(θ)=𝔼_t[λ(t)𝔼_x^(t),x^(0)[1/2‖ s_θ(x(t),t) - s_t(x^(t)|x^(0))‖_2^2]],
where t is selected uniformly between 1 and T, x^(0)∼ p_t(x), x^(0)∼ p_0(x), and s_t(x |x^(0)) denotes the score function of p_t(x | x^(0)), and λ(t) is a weighting function that balances the loss of different time steps.
In this paper, we take the same drift, dispersion, and weighting functions f(x,t), g(t), and λ(t) as the original VE-SDE framework <cit.>.
§.§ Related studies of conditional score-based generative models
In conditional SGMs, we are given some labeled data {(x_m, y_m)}_m=1^M in addition to the unlabeled data {x_n}_n=M+1^M+N, where y ∈{1, 2, …, K} denotes the class label. The case of N = 0 is called the fully-supervised setting, while we focus on the more challenging semi-supervised setting with N > 0 (and possibly N ≫ M) in this paper.
Conditional score-based generative models aim to learn the conditional score function ∇_x log p(x | y) from the data and then generate samples from p(x | y). Previous studies <cit.> showed how to decompose the conditional score function using Bayes' theorem:
∇_x log p(x|y) =∇_x[log p(x) + log p(y|x)- log p(y)]= ∇_xlog p(x) + ∇_xlog p(y|x).
The term log p(y) can be dropped because it is not a function of x and is thus of gradient 0. The decomposition shows that conditional generation can be achieved by an unconditional SGM that learns the score function ∇_x log p(x) plus an extra conditional gradient term ∇_xlog p(y|x).
The vanilla form of Classifier Guidance (CG) for SGM estimates ∇_xlog p(y|x) with an auxiliary classifier trained from the cross-entropy loss on the labeled data and learns the unconditional score function by the DSM loss ℒ_DSM that can in principle be applied on both the labeled and unlabeled data.
Nevertheless, the classifier within the vanilla CG approach is known to be potentially over-confident <cit.> on its predictions, which in term results in inaccurate gradients.
The issue can mislead the conditional generation process and decrease class-conditional generation quality.
<cit.> propose to address the issue by tuning the term ∇_xlog p(y|x) with a scaling parameter λ_CG≠ 1.
∇_x log p(x|y) = ∇_xlog p(x) + λ_CG∇_xlog p_ϕ(y|x),
where p_ϕ(y|x) is the posterior probability distribution outputted by a classifier parameterized by ϕ. Increasing λ_CG sharpens the distribution p_ϕ(y | x), guiding the generation process to produce less diverse but higher fidelity samples. While the tuning heuristic is effective in improving the vanilla CG approach, it is not backed by sound theoretical explanations.
<cit.> propose to resolve the issue differently by enhancing the adversarial robustness of the classifier. It is empirically observed that adversarially robust classifiers produce more interpretable and perceptually more aligned <cit.> gradients. Somehow it remains theoretically unclear whether robust classifiers are truly more accurate for capturing the true data distribution.
<cit.> propose the Denoising Likelihood Score Matching (CG-DLSM) approach that calibrates the classifiers to resolve the issues. The calibration is done by designing a loss computed from the outputs of a trained unconditional SGM to regularize the classifier during training. CG-DLSM achieves state-of-the-art performance within the CGSGM family in the fully-supervised setting. Somehow because of the design, the learning of unconditional SGM and the classifier needs to be done in sequential steps, losing the computational advantage of the original vanilla CGSGM of being able to train the two components in parallel. Furthermore, it is not clear whether the unlabeled data in the semi-supervised setting could be helpful in improving the classifier under the design.
The approaches above are all CGSGMs. Another popular approach for conditional SGM is Classifier-Free Guidance (CFG) <cit.>. The approach parameterizes its deep learning model with more sophisticated architectures such that the class labels y can be included as inputs to calculate the score. A null token y_nil is used to indicate unconditional score calculation, which is linearly combined with conditional score calculation for some specific y to form the final estimate of s(x | y). CFG is a state-of-the-art conditional SGM in the fully-supervised setting. Nevertheless, as we shall show in our experiments, its performance drops significantly in the semi-supervised setting, as the conditional parts of the architecture may not get enough labeled data during training. The disadvantages of CFG and other CGSGMs in the semi-supervised setting motivate us to design another CGSGM that (1) comes with theoretical justifications; (2) includes a classifier that can be trained in parallel to the unconditional SGM; (3) can leverage both the unlabeled and labeled data to achieve better performance in the semi-supervised setting.
§ SELF-CALIBRATION FOR CLASSIFIER GUIDANCE
§.§ Motivation
As mentioned in Section <ref>, inaccurate gradients of classifiers could potentially misguide the conditional generation process.
Therefore, we need an efficient way to calibrate the classifiers.
Motivated by JEM <cit.> where the classifiers are calibrated by being reinterpreted as an energy-based model (EBM), we propose to connect the EBM and SGM and calibrate the classifiers by interpreting them as EBMs in a similar approach.
To be more specific, we formulate a self-calibration loss that utilizes denoising score matching to calibrate the score function estimated by the classifier.
§.§ Formulation of self-calibration loss
In this work, we adopted the framework of score-based generative modeling using stochastic differential equations (SDEs) <cit.>. Given a target distribution p_0(x) and a known prior distribution p_T(x) (typically a Gaussian distribution) where the transition between them is a diffusion process with timestep 0≤ t< T, we can describe the diffusion process and its reverse process using SDEs. To incorporate the results of Section <ref> into this framework, we introduce the time-dependent version of ∇_x log p(x) and ∇_x log p(y|x). That is ∇_x log p_t(x(t)) and ∇_x log p_t(y|x(t)), respectively, where x(t)∼ p_t. Denoising score matching (DSM) <cit.> is often utilized to train the score-based model under this framework due to its close relationship with diffusion process modeling. A time-generalized cross-entropy loss is adopted o train the classifier.
Inspired by JEM <cit.>, we propose to improve CGSGM through self-calibration during the training stage. We reinterpret the classifier as a time-dependent EBM and obtain the score function by calculating the gradient. Since both energy function -log p(x) and score function ∇_x log p(x) are calculated from the log-likelihood function, we hypothesize that integrating EBM-related objectives into classifier training can be beneficial to CGSGM. To incorporate the energy function into our framework, we used a time-dependent version of the transformation described in JEM <cit.>:
E_ϕ,t(x) = -log∑_yexp(f_ϕ,t(x)[y])= -LogSumExp_y(f_ϕ,t(x)[y])
where f_ϕ,t(x)[y] is the output logits of the classifier. The score function can then be computed like the following:
s_ϕ(x,t) =∇_x LogSumExp_y(f_ϕ,t(x)[y])
To calibrate this score estimated by the classifier, we adopt DSM to calculate the Self-calibration Loss (SC loss):
ℒ_SC(ϕ)=𝔼_t[λ(t)𝔼_x_t,x_0[1/2‖ s_ϕ(x_t,t)-s_t(x_t|x_0)‖_2^2]]
where x_t∼ p_t, x_0∼ p_0, and s_t(x_t|x_0) denotes the score function of the noise centered at x_0. Fig. <ref> summarizes the calculation of the proposed SC loss. After the self-calibration loss is obtained, it is summed with the cross-entropy loss to train the classifier. The total loss can be written as:
ℒ_CLS(ϕ)=ℒ_CE(ϕ)+λ_SCℒ_SC(ϕ)
where ℒ_CE is the cross-entropy loss and λ_SC is a hyperparameter. By applying self-calibration, the classifier should be able to more accurately estimate the score function of the underlying data distribution, which implies the underlying data distribution itself is also more accurately estimated. As a result, the gradients of the classifiers should be more aligned with the ground truth as it is calculated from the estimated distribution.
After self-calibration, the classifier then can be used just like the original classifier to guide an unconditional SGM to achieve conditional generation. Note that since our method calibrates the classifier in training time and scaling classifier gradient is done in sampling time, we can easily combine the two methods to achieve better performance.
§.§ 2D toy dataset
We use a 2D toy dataset containing two classes to demonstrate the effects of the self-calibration loss. The data distribution is shown in Fig. <ref>, where the two classes are shown in two different colors. After training the classifiers on the toy dataset with (1) only cross-entropy loss and (2) both cross-entropy loss and self-calibration loss, we plot the gradients ∇_x log p(y|x) estimated by the classifiers and compare them with the ground truth. Also, we added the ground truth unconditional score to the estimated gradients just like CGSGM and compared the results with the real conditional score. Additional quantitative measurements of the toy dataset is included in Appendix <ref>.
Fig. <ref> shows the ground truth classifier gradient (Fig. <ref>) and the gradients estimated by classifiers trained on the toy dataset (1) without self-calibration (Fig. <ref>) and (2) with self-calibration (Fig. <ref>).
Uncalibrated classifiers produce gradients that contain rapid changes in magnitude across the 2D space, with frequent fluctuations and mismatches with the ground truth.
Such fluctuations can impede the convergence of the reverse diffusion process to a stable data point, leading SGMs to generate noisier samples.
Moreover, the divergence from the ground truth gradient can misguide the SGM, leading to generation of samples from incorrect classes.
Uncalibrated classifiers also tend to generate large gradients near the distribution borders and tiny gradients elsewhere.
This implies that when the sampling process is heading toward the incorrect class, such classifiers are not able to “guide" the sampling process back toward the desired class.
In contrast, the introduction of self-calibration results in estimated gradients that are more stable, continuous across the 2D space, and better aligned with the ground truth.
This stability results in a smoother generation process and contributes to the production of higher-quality samples.
§.§ Using self-calibration loss on semi-supervised learning
In this work, we also explore the benefit of self-calibration loss in semi-supervised setting where only a small proportion of data are labeled.
In original classifier guidance, the classifiers are solely trained from labeled data.
The lack of label in semi-supervised setting leads to more challenges to learn an unbiased classifier.
With self calibration, we are able to better utilize the large amount of unlabeled data by calculating the self calibration loss with all data.
To incorporate the loss and utilize the unlabeled samples during training time, we changed the way of calculating ℒ_CLS from Eq. <ref>. As illustrated in Fig. <ref>, the entire batch of data is used to calculate ℒ_SC, but only the labeled data is used to calculate ℒ_CE. During training, we observed that when the majority is unlabeled data, the cross-entropy loss does not converge to a low-and-steady stage if the algorithm randomly samples from all training data. We suspect this is due to the low percentage of labeled data in each batch. Therefore, we changed the way of sampling batches. We always ensure that half of the data is labeled while the other half is not. Appendix <ref> summarizes the semi-supervised training process of the classifier.
Note that even though the classifier is learning a time-generalized classification task, we can still make it perform as an ordinary classifier that classifies the unperturbed data by setting the input timestep t=0. Therefore, we can easily incorporate many other common semi-supervised classification methods like pseudo-labeling <cit.>, self-training, and noisy student <cit.>.
§ EXPERIMENTS
We have tested our method on a toy dataset (Section <ref>) to provide a high-level view of how self-calibration can improve classifiers in terms of producing accurate gradients. In this section, we present the experimental results on the CIFAR-10 and CIFAR-100 datasets to demonstrate the improvement of CGSGM after incorporating our method on different percentage of labeled data (Section <ref>). Randomly selected images of CGSGM before and after self-calibration on the dataset CIFAR-10 are shown in Appendix <ref>. For conditional metrics, we report the average scores across all classes. Results of individual classes on the CIFAR-10 dataset are included in Appendix <ref>.
§.§ Experimental setup
In the following sections, we tested our methods on the CIFAR-10 and CIFAR-100 datasets for image generation. We demonstrate that our methods are able to improve generation quality both conditionally and unconditionally with different percentage of labeled data.
Implementation details
We follow NCSN++ <cit.> to implement the unconditional score estimation model. We also adapted the encoder part of NCSN++ as the classifier used in CGSGM <cit.>.
Sampling method: We used Predictor-Corrector (PC) samplers <cit.> with 1000 sampling steps.
Evaluation metrics: Besides commonly used metrics Frechet Inception Distance (FID) <cit.> and Inception Score (IS) <cit.>, we also evaluated class-conditional performance of our methods using several different methods. This includes intra-FID, which measures the average FID for each class, and generation accuracy (on the CIFAR-10 dataset), which uses a pre-trained ViT <cit.> classifier to check whether the samples are generated in the correct class. The test accuracy of the pre-trained ViT is 98.52% on the CIFAR-10 dataset.
Baseline methods:
The baseline methods used in our work include:
* Cond: Adopts conditional SGMs by conditional normalization techniques <cit.> rather than classifier guidance.
* CFG-labeled: Classifier-free guidance<cit.> using only labeled data is applied.
* CFG-all: Classifier-free guidance<cit.> using only labeled data to train the conditional part of the model and all data to train the unconditional part of the model.
* CG: Vanilla classifier guidance.
* CG-DLSM: Classifier guidance with DLSM loss <cit.> applied.
§.§ Experiment Result
Table <ref> and Fig. <ref> present the performance of all methods when applied to varying percentages of labeled data.
Notice that it includes the fully-supervised setting when 100% of data are labeled.
CG-SC-labeled implies self-calibration is only applied on labeled data while CG-SC-all implies self-calibration is applied on all data.
Conditional SGMs vs Unconditional SGMs.
The first observation from our results is that conditional SGMs, including Cond, CFG-labeled, and CFG-all, consistently excel in generation accuracy.
However, when the quantity of labeled data decreases below 40%, a significant performance drop is witnessed in these models.
These conditional SGMs, while generating high-quality images, tend to lose diversity when working with fewer labeled data.
This occurs mainly because of the lack of labeled data in training phase, leading them to generate samples closely mirroring the distribution of the labeled data instead of all data.
In contrast, unconditional SGMs, such as CG, demonstrate superior performance when the majority of the data is unlabeled, as they are capable of leveraging both labeled and unlabeled data during training.
Classifier-Guided SGMs (CGSGMs) vs Conditional SGMs
Our experimental results align with our expectations that CGSGMs produce improved performance compared to conditional SGMs.
The CG method exhibits a consistent performance in terms of FID and inception scores across varying percentages of labeled data when evaluated using unconditional metrics.
Notably, when unlabeled data is in the majority, we observe a 16% drop in generation accuracy on the CIFAR-10 dataset.
Despite this, the intra-FID of CG significantly outperforms that of conditional SGMs on both datasets.
As for the proposed method, incorporating self-calibration with labeled data does not majorly affect unconditional metrics but substantially improves conditional metrics.
This process reduces intra-FID by 8.25 and 17.86 on the CIFAR-10 and CIFAR-100 dataset respectively and increases generation accuracy on CIFAR-10 by up to 23%.
The results demonstrate that with self-calibration, the classifier can better represent the class-conditional distribution even when labeled data is limited.
Leverage unlabeled data for semi-supervised conditional generation
Intuitively, incorporating unlabeled data into the computation of self-calibration loss would enhance the quality of conditional generation, because the classifier can exploit additional information from unlabeled data during the training phase.
As the proportion of labeled data decreases, this benefit of leveraging unlabel data should become more significant.
As our experimental results show, conditional metrics do not differ greatly when the proportion of labeled data ranges between 40% and 100%.
However, when the percentage of labeled data falls below 40%, the use of unlabeled data significantly improves intra-FID and generation accuracy.
Specifically, with just 5% labeled data, intra-FID improves by 12.22, and generation accuracy increases by 22.8% compared to the original CG.
These results affirm our expectation that as the quantity of labeled data decreases, the beneficial impact of utilizing unlabeled data increases.
§ CONCLUSION
In this work, we verify that the existing CGSGM approach results in a high generation fidelity but low accuracy. We hypothesize that the root cause lies in the unreliable scores produced by the classifiers and design a Self-Calibration Loss to enhance the classifier directly towards better scores without resorting to an external SGM. The Self-Calibration Loss is derived from rigorous principles when viewing the classifier as an energy-based model. We demonstrate three immediate benefits of the proposed Self-Calibrating CGSGM approach. Using the toy dataset, we show that the scores computed from the approach are indeed closer to the ground-truth scores. Secondly, across all percentages of labeled data, our proposed approach outperforms the existing CGSGM in the semi-supervised setting. Lastly, our empirical study justifies that our proposed approach can consistently reach the best intra-FID by seamlessly leveraging the power of unlabeled data, when compared to other conditional SGMs. The benefits establish the rich potential of the proposed approach.
§ LIMITATIONS
The major limitation of our work lies in the selection of datasets. We can only afford to conduct experiments on smaller and lower-resolution datasets (CIFAR-10 and CIFAR-100) because of limited computational resources. In particular, even with those smaller data, training, sampling, and testing a single approach on a single setting once requires up to 210 hours (more than a week) with 4 NVIDIA Tesla V100 GPUs. We understand that conducting more experiments on larger and higher-resolution datasets can further strengthen our claims, but those experiments are not affordable to us. While we tested on only two datasets, the observed results are consistent—our proposed approach achieves the best class-conditional performance in the semi-supervised setting with much fewer labeled data.
plainnat
§ DETAILED CLASS-CONDITIONAL GENERATION MEASUREMENTS OF CIFAR-10
Section <ref> contains the class-conditional measurements averaged among all classes of CIFAR-10. This section includes a more detailed result that contains the measurement of each class.
§ TRAINING ALGORITHM FOR SEMI-SUPERVISED SELF-CALIBRATING CLASSIFIER
[h]
Semi-supervised classifier training with self-calibration loss
§ QUANTITATIVE MEASUREMENTS OF TOY DATASET
Table <ref> shows the quantitative measurements of the methods on the toy dataset. First, we compared the gradients ∇_x log p(y|x) estimated by the classifiers with the ground truth by calculating the mean squared error (first column) and cosine similarity (second column). We observed that after self-calibration, the mean squared error of estimated gradients can be lowered by 18%, and tuning the scaling factor can further improve it to 36%. This improvement after scaling implies that the direction of gradients is more aligned with the ground truth, and scaling can further reduce the mismatch between the magnitude of the classifier and the ground truth. In terms of cosine similarity, self-calibration grants the classifiers an improvement of 42%. The numerical results agree with our previous observation that after self-calibration, classifiers align better with the ground truth in terms of both direction and magnitude.
Then, we add the unconditional score of the training data distribution to the classifier gradients to calculate the conditional scores and compare the results with the ground truth. As we can see, the classifiers are able to estimate conditional scores with a cosine similarity of 0.9175 even without self-calibration. The result shows that with a well-trained unconditional SGM, in which we use the ground truth unconditional score in this case, CGSGM is able to produce conditional scores pointing in the correct directions in most cases. This explains why the original CGSGM is able to generate samples with decent quality. After applying the self-calibration loss and scaling method, we can further improve the cosine similarity to 0.9689, which we believe can enhance the quality of class-conditional generation.
§ TUNING THE SCALING FACTOR FOR CLASSIFIER GUIDANCE
This section includes the experimental results of tuning the scaling factor λ_CG for classifier guidance with and without self-calibration under fully-supervised setting.
Fig. <ref> shows the result of tuning the scaling factor λ_CG for classifier guidance. While tuning λ_CG with and without self-calibration, we can see that self-calibration does not affect unconditional performance by much. However, when evaluated with conditional metrics, the improvement after incorporating self-calibration becomes more significant. The improvement in intra-FID is up to 7.9 while the generation accuracy can improve up to 13%.
§ IMAGES GENERATED BY CLASSIFIER GUIDANCE WITH AND WITHOUT SELF-CALIBRATION
This section includes images generated by classifier guidance with (first 6 images) and without (last 6 images) self-calibration after training on different percentage of labeled data. Each row corresponds to a class in the CIFAR-10 dataset.
|
http://arxiv.org/abs/2307.04692v1 | 20230710164454 | Spoofing-Resilient LiDAR-GPS Factor Graph Localization with Chimera Authentication | [
"Adam Dai",
"Tara Minda",
"Ashwin Kanhere",
"Grace Gao"
] | eess.SP | [
"eess.SP",
"cs.RO",
"cs.SY",
"eess.SY"
] |
Spoofing-Resilient LiDAR-GPS Factor Graph Localization with Chimera Authentication
The views expressed are those of the authors and do not reflect the official guidance or position of the United States Government, the Department of
Defense or of the United States Air Force. Statement from DoD: The appearance of external hyperlinks does not constitute endorsement by the United States
Department of Defense (DoD) of the linked websites, or the information, products, or services contained therein. The DoD does not exercise any editorial,
security, or other control over the information you may find at these locations.
Adam Dai
Electrical Engineering
Stanford University
Stanford, USA
[email protected]
Tara Mina
Electrical Engineering
Stanford University
Stanford, USA
[email protected]
Ashwin Kanhere
Aeronautics and Astronautics
Stanford University
Stanford, USA
[email protected]
Grace Gao
Aeronautics and Astronautics
Stanford University
Stanford, USA
[email protected]
August 12, 2023
===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
fancy
Many vehicle platforms typically use sensors such as LiDAR or camera for locally-referenced navigation with GPS for globally-referenced navigation.
However, due to the unencrypted nature of GPS signals, all civilian users are vulnerable to spoofing attacks, where a malicious spoofer broadcasts fabricated signals and causes the user to track a false position fix.
To protect against such GPS spoofing attacks, Chips-Message Robust Authentication (Chimera) has been developed and will be tested on the Navigation Technology Satellite 3 (NTS-3) satellite being launched later this year.
However, Chimera authentication is not continuously available
and may not provide sufficient protection for vehicles which rely on more frequent GPS measurements.
In this paper, we propose a factor graph-based state estimation framework which integrates LiDAR and GPS while simultaneously detecting and mitigating spoofing attacks experienced between consecutive Chimera authentications.
Our proposed framework combines GPS pseudorange measurements with LiDAR odometry to provide a robust navigation solution.
A chi-squared detector, based on pseudorange residuals, is used to detect and mitigate any potential GPS spoofing attacks.
We evaluate our method using real-world LiDAR data from the KITTI dataset and simulated GPS measurements, both nominal and with spoofing.
Across multiple trajectories and Monte Carlo runs, our method consistently achieves position errors under 5 m during nominal conditions, and successfully bounds positioning error to within odometry drift levels during spoofed conditions.
GPS, spoofing, LiDAR, sensor fusion, Chimera, factor graphs
§ INTRODUCTION
Localization is a fundamental task for vehicle-related applications, such as autonomous driving or precision farming. Currently, state-of-the-art vehicle localization relies on sensor fusion, as various sensors possess different tradeoffs and advantages. Centimeter-level localization of self-driving cars has been demonstrated with fusion of LiDAR (Light Detection and Ranging), vision, and GPS, with evaluation on a real-world fleet of cars <cit.>. In particular, LiDAR and GPS have complementary advantages. LiDAR localization and odometry works well in structured environments, but struggles in empty areas lacking spatial features. Conversely, GPS struggles in structured environments due to signal blockage and multipath, but excels in open-sky conditions.
However, GPS is vulnerable to spoofing, in which an attacker transmits fabricated GPS signals at higher power than the real signals, causing the victim to lock on to the fake signals. The attacker can then induce arbitrary errors to the victim’s GPS position estimate. For a vehicle running sensor fusion with GPS, these errors will propagate to the localization solution, compromising the safety of humans onboard or near the vehicle. Indeed, this vulnerability has been demonstrated in recent work, in which a well-designed GPS spoofing attack is able to cause an autonomous vehicle to crash with 97% success rate <cit.>.
As a countermeasure to GPS spoofing, the Air Force Research Lab (AFRL) has proposed the Chips-Message Robust Authentication (Chimera) signal enhancement for the GPS L1C signal <cit.>. The Chimera signal enhancement punctures the L1C spreading code in the pilot channel with encrypted markers, which cannot be predicted beforehand, but can be verified via a digital signature provided to the user with a short latency. For standalone receivers, authentication is available every 3 minutes through the slow channel, while users with access to secure Internet connection or an augmentation system can receive authentication every 1.5 or 6 seconds through the fast channel. The time duration between consecutive Chimera authentications is referred to as the Chimera epoch.
Nevertheless, for either the slow or fast channel, the Chimera authentication service is not continuously available. For applications such as self-driving, 1.5 seconds can easily make the difference between staying safe and crashing. Furthermore, for users relying on the slow channel, an attacker would have a large window of time to introduce spoofing errors. Prior works have addressed this issue through spoofing mitigation between Chimera authentications <cit.>. These works use IMUs (inertial measurement units) and wheel encoders as trusted (i.e. unaffected by spoofing) sensors for fusion with GPS. However, the problem of spoofing detection and mitigation has yet to be explored for LiDAR-GPS sensor fusion.
§.§ Contributions
In this work, we develop a novel spoofing detection and mitigation framework for LiDAR-GPS sensor fusion. This problem has received little attention in prior literature, and to the best of our knowledge, our solution is the first to examine the problem in the context of Chimera signal enhancement.
When combined with Chimera authentications, our approach mitigates the localization error induced by a spoofer over the Chimera epoch, which we experimentally validate using real LiDAR and simulated GPS measurements.
The key contributions of this work are:
* We perform tightly-coupled factor graph optimization with LiDAR odometry and GPS pseudoranges for accurate vehicle localization within the Chimera epoch.
* During the Chimera epoch, we use a chi-squared detector to determine the authenticity of GPS measurements. When measurements are deemed unauthentic, we mitigate the effects of the spoofing attack by relying on LiDAR odometry and excluding GPS.
* We validate our approach experimentally for the 3-minute Chimera slow channel, using real-world LiDAR measurements from the KITTI dataset and simulated GPS measurements. During nominal conditions, our approach maintains accuracy comparable to baseline methods. During spoofed conditions, our approach demonstrates consistent detection and mitigation of the attack across various trajectories and spoofing attacks.
To the best of our knowledge, we believe this is the first spoofing detection and mitigation approach for tightly-coupled GPS factor graph optimization.
§.§ Paper Organization
The remainder of this paper is organized as follows. Section <ref> surveys relevant literature to this work. Section <ref> introduces the problem statement and notation, and provides background on pose representation and factor graph optimization. Section <ref> details our factor graph optimization and spoofing detection and mitigation framework. Section <ref> describes the setup and parameters for experimental validation, Section <ref> presents the experimental results, and Section <ref> concludes this paper.
§ RELATED WORK
Our work bridges the areas of LiDAR-GPS sensor fusion and spoofing detection and mitigation in the context of Chimera GPS. In this section, we discuss existing approaches for LiDAR-GPS sensor fusion, followed by prior works addressing spoofing detection and resilient estimation.
§.§ LiDAR-GPS Sensor Fusion
LiDAR-GPS sensor fusion approaches can be broadly separated in two main categories: filtering-based and graph-based.
Filtering-based approaches rely on recursive Bayesian estimation as the underlying state estimation and fusion framework.
The most notable examples of Bayesian filters are the Kalman filter (KF), Extended Kalman Filter (EKF), and the Particle Filter (PF).
Several current state-of-the art LiDAR-GPS sensor fusion approaches rely on filtering <cit.>.
Graph-based approaches encode vehicle states and sensor observations into a graph data structure, and employ graph optimization to solve for the optimal trajectory.
Over the past decade, there has been continually growing interest in factor graph optimization (FGO) for sensor fusion localization.
Recently, Wen et al. <cit.> compared EKF and FGO localization approaches in a GPS challenged environment, and found that the FGO outperformed the EKF for the tightly-coupled case, in which GPS pseudoranges are incorporated directly into the graph.
The authors also showed that tightly-coupled FGO outperforms the loosely-coupled alternative, in which GPS position measurements are used as factors rather than pseudoranges.
Successful graph-based integrations of LiDAR and GPS have also been explored.
Chen et al. <cit.> present a Bayesian graph for fusion of LiDAR, GPS, and 3D building maps in order to localize a UAV in an urban environment.
The authors demonstrate significant improvement over a GPS-only Kalman filter approach, but the method relies on map matching with existing 3D building models to achieve accurate localization.
He et al. <cit.> also leverage graph optimization to fuse LiDAR, IMU and GPS.
The authors evaluate their method on the KITTI dataset <cit.>, outperforming state-of-the-art LiDAR odometry approaches with meter-level accuracy, while also demonstrating their algorithm can be run in real-time at low latency.
§.§ Spoofing Detection and Resilient State Estimation
However, none of the above LiDAR-GPS sensor fusion works address the vulnerability of GPS to spoofing attacks.
In 2020, Shen et al. <cit.> demonstrated a spoofing method which is able to exploit the sensor fusion algorithm of <cit.>, and induce large lateral deviations to the vehicle's state estimate, and consequently to the actual trajectory, during periods of low confidence.
With just 2 minutes of attack time, the spoofing algorithm is able to induce dangerous vehicle behavior with a 97% success rate.
Outside of sensor fusion, GPS spoofing attack methods and detection strategies have received much attention <cit.>.
However, many detection strategies make assumptions about receiver capabilities or require additional functionality, such as multiple antennas.
Chimera is the first proposed authentication service for GPS signals and is set to be tested onboard the NTS-3 (Navigation Technology Satellite 3) platform scheduled for launch in 2023 <cit.>.
Very recently, some works have begun to address the problem of spoofing-resilient GPS sensor fusion with Chimera.
Mina et al. <cit.> present a spoofing-resilient filter for continuous state estimation between Chimera authentications, which leverages IMU and wheel encoders as self-contained sensors to determine the trustworthiness of received GPS signals.
Kanhere et al. <cit.> use FGO to combine GPS, IMU and wheel odometry to perform spoofing mitigation with Chimera authentication.
The authors model the authentication state as switchable constraints <cit.> in the graph, and test their method on simulated trajectories using the fast channel authentication period of 6 seconds.
Our approach extends upon these prior works, integrating elements of the chi-squared detection scheme from prior works <cit.> into a graph formulation.
Furthermore, while the factor graph approach of <cit.> only performs implicit mitigation, and is only evaluated for fast-channel application on a short (12 s) trajectory, our approach performs explicit detection and mitigation and is evaluated on trajectories spanning the slow-channel Chimera epoch of 3 minutes.
Additionally, our approach uses a tightly-coupled factor graph with GPS pseudorange factors, which has been found to outperform the loosely-coupled version in localization accuracy in prior work <cit.>.
Finally, we incorporate LiDAR as a new sensor in the realm of sensor fusion under Chimera.
§ PRELIMINARIES
In this section, we present our problem statement and objective, discuss notation and models used in the paper, and cover relevant background on Lie groups and factor graph optimization.
§.§ Problem Statement
We consider a vehicle equipped with a GPS receiver and LiDAR moving through an environment with continuous GPS availability.
During operation, the vehicle may be subject to GPS spoofing attacks which induce arbitrary bias error to the GPS measurements.
However, the GPS receiver has access to slow channel Chimera authentication every 3 minutes.
Within the 3-minute Chimera epoch, our objective is to perform spoofing-resilient localization.
In particular, we wish to determine when to leverage the available, but not-yet-authenticated GPS measurements and when to fall back on the LiDAR measurements only.
In this way, we seek to improve localization performance when GPS is likely authentic, while remaining resilient to an experienced spoofing attack.
§.§ Notation
We model time as discrete, with Δ t denoting the discretization interval in seconds. The variable k is used to denote the current time index, while the variable i is used to denote an arbitrary time index.
At time k, if a LiDAR measurement is available, we obtain a point cloud P_k ∈^N_points× 3 where N_points is the number of points in the point cloud.
Likewise, if GPS is available at time k, we obtain a set of pseudorange measurements ρ_k = (ρ_k^(1), …, ρ_k^(m)) ∈^m, where ρ_k^(j) is the measured pseudorange to satellite j, and m is the number of visible satellites.
Recall that the time duration between successive authentications is referred to as the Chimera epoch.
The length of the Chimera epoch for the slow channel, in discretized timesteps, is then N_epoch 180/Δ t.
refers to the identity matrix, and to the matrix of zeros.
Scalars are denoted with lowercase italics, vectors with lowercase boldface, and matrices with uppercase boldface.
§.§ GPS Pseudorange Error Model
We model the distribution of authentic, i.e. unspoofed, GPS pseudorange error as a zero-mean Gaussian (0, σ), where σ is the standard deviation of the pseudorange error.
Thus we can write
ρ_k = ρ̅_k + ϵ_k, ϵ_k ∼(0, σ),
where ρ_k is the measured range and ρ̅_k is the true range.
We assume that clock bias effects have been removed from the measurements.
§.§ (3) and (3) Lie groups
We present a brief background on the rotation and rigid body transformation Lie groups (3) and (3), as we use the corresponding 3D representations for the vehicle's 3D pose.
More extensive coverage of these topics can be found in <cit.>.
The vehicle's 3D pose at time k is denoted as _k ∈(3), where (3) is the Special Euclidean group of dimension 3.
A pose in (3) consists of a rotation R∈(3), where (3) is the Special Orthogonal group of dimension 3, and a translation t∈^3.
(3) is defined as
(3) = {R∈^3× 3 | RR = , R = 1},
i.e. the set of all rotation matrices (orthogonal matrices with determinant 1).
(3) can then be represented as the Cartesian product of (3) with ^3, i.e., (3) ∼(3) ×^3.
We can represent _k with a transformation matrix
T_k = [ R_k t_k; 1 ]∈^4× 4,
where R_k ∈^3× 3 is a rotation matrix representing orientation in the global frame, and t_k ∈^3 is a translation vector representing position in the global frame.
The Lie groups (3) and (3) have associated Lie algebras denoted (3) and (3), with dimensionality 3 and 6 respectively.
The Lie algebra can be thought of as the tangent space (linearization) of the manifold at the identity element, and are linear spaces upon which optimization may be done conveniently.
More precisely, there is an isomorphism from (3) to ^3, and from (3) to ^6, and any R∈(3) can be represented with a vector ω∈^3, and similarly any T∈(3) can be represented with a vector ν∈^6.
The exponential map exp: (3) ↦(3) maps from the tangent space (3) to (3) (from ν to T), while the logarithmic map log: (3) ↦(3) maps from (3) to its tangent space (3) (from T to ν).
Details on how the exponential and logarithmic map are computed can be found in <cit.>.
For x, y∈(3), the “ominus" operator is defined as y⊖x = log(x^-1y) ∈(3) <cit.>.
This operator allows us to compute the “difference" of poses in (3) in linearized tangent space coordinates, and will be used later in defining the LiDAR odometry residual for our factor graph.
§.§ Factor Graph Optimization
Our state estimation framework relies on factor graph optimization (FGO), in which a graph encoding vehicle poses and sensor measurements is optimized to determine the estimated trajectory.
In this section, we provide a brief background on general factor graph formulation.
More details can be found in <cit.> and <cit.>.
A factor graph consists of a set of nodes = {_1,…,_N} which represent poses or states, and a set of edges or factors which represent sensor measurements which constrain the graph.
A sensor observation linking nodes _i and _j is denoted _i,j with associated information matrix Ω_i,j, which is defined as the squared inverse of the measurement covariance matrix: Ω_i,jΣ_i,j^-1.
Each sensor has an associated measurement model _i,j(_i, _j), which is used to define a residual _i,j(_i, _j) = _i,j - _i,j(_i, _j) for each factor.
Optimizing the factor graph consists of minimizing the following objective
F() = ∑_(i,j)∈_ijΩ_ij_ij
which is the sum of information-normalized squared error of the residuals.
This objective represents the negative log-likelihood of the vehicle poses given the sensor measurements.
Thus, solving the optimization problem
^* = _F().
yields the optimal set of poses ^* given our measurements.
The optimization is done by linearizing 𝐅 and iteratively solving for updates to the state .
For each edge (i,j)∈, the gradient b_ij and Hessian H_ij are computed as
b_ij = _ijΩ_ijJ_ij,
H_ij = J_ijΩ_ijJ_ij,
where J_ij is the Jacobian of _ij().
The individual gradients and Hessians are then accumulated to form the gradient and Hessian for the entire graph, b = ∑b_ij and H = ∑H_ij, and the linear system
HΔ^* = -b
is solved with sparse Cholesky factorization to find the optimal update Δ^*, which is applied to the state + Δ^*.
§ APPROACH
We now describe the details of our spoofing-resilient LiDAR-GPS factor graph approach.
Fig. <ref> shows a high-level block diagram of the framework.
§.§ LiDAR-GPS Factor Graph
Our approach revolves around maintaining a tightly-coupled factor graph which integrates LiDAR and GPS for both localization and spoofing detection and mitigation, the structure of which is shown in Fig. <ref>.
The nodes of our factor graph are vehicle poses _i ∈(3) as described in Section <ref>.
The measurement models (Equations <ref> and <ref>) and residuals (Equations <ref> and <ref>) for GPS pseudoranges and LiDAR odometry factors are detailed in the following subsections.
§.§.§ GPS Pseudorange Factors
Given pose _i with position component t_i and satellite j at position _i^(j) at time i, the expected GPS pseudorange measurement from satellite j to node i is
ρ̂_i^(j)(_i) = t_i - _i^(j)_2.
Now, given received pseudorange measurement ρ_i^(j), the GPS residual function can be defined as
_i,j(_i) = ρ_i^(j) - ρ̂_i^(j)(_i).
As the residual is a scalar in this case, the information matrix is also a scalar, (σ)^-2, where σ is the standard deviation of authentic GPS pseudoranges from Section <ref>.
In this work, we do not consider the effect of clock bias states in our pseudorange factors and simulated pseudorange measurements, but we will incorporate this into our measurement model in our future works.
§.§.§ LiDAR Odometry Factors
For LiDAR, we first use point-to-plane ICP (Iterative Closest Point) <cit.> to register successive point clouds and produce an odometry measurement.
Given point clouds P_i and P_i+1 at times i and i+1, ICP produces an odometry measurement T_i^i+1:
T_i^i+1icp(P_i, P_i+1).
T_i^i+1 is a rigid body transformation in (3), and can be written as
T_i^i+1 = [ R_i^i+1 t_i^i+1; 1 ]∈^4× 4,
where R_i^i+1∈^3× 3 is the relative rotation between poses at times i and i+1 and t_i^i+1∈^3 is the relative translation between poses at times i and i+1.
The LiDAR odometry measurement model is the expected transformation between poses i and i+1:
T̂_i,i+1(_i, _i+1) = _i+1 (_i)^-1.
The LiDAR odometry residual function is then defined as
_i,i+1(_i,_i+1) = T_i^i+1⊖T̂_i,i+1,
where ⊖ is the “ominus" operator defined in Section <ref>.
Intuitively, this residual is the difference between the expected and measured odometry transformation in (3) tangent space coordinates.
Thus, _i,i+1(_i,_i+1) ∈^6, and the information matrix for LiDAR odometry measurements is denoted by Ω∈^6× 6.
§.§.§ Optimization
We optimize the graph in a sliding window fashion.
The window size is denoted by N, and as new nodes and measurements are added to the graph, previous nodes and edges are removed in order to maintain the maximum number of nodes in the graph as N.
The objective for our factor graph over a single window can thus be written as
F() =
∑_i=1^N∑_j=1^m (_i,j) (σ)^-2_i,j
+ ∑_i=1^N-1 (_i,i+1)Ω_i,i+1
The optimization carried out as described in Section <ref>.
§.§ Spoofing Detection and Mitigation
To perform detection between Chimera authentication times, we design a chi-squared spoofing detector within our factor graph framework.
Our detector computes a test statistic q_k at time k based on the information-normalized residuals of the GPS factors over the current window:
q_k = ∑_i=i^N ∑_j=1^m (_i,j) (σ)^-2_i,j
Note that we do not include a normalization term based on the state estimate uncertainty (as typically done in the chi-squared detector with Kalman filter) as the factor graph does not maintain an estimate of state uncertainty.
We then compare q_k with a threshold τ, which is pre-computed based on user-specified false alarm requirements.
If q_k > τ, then it is determined that spoofing is present in the measurements, otherwise the measurements are deemed authentic.
We now derive the computation of the threshold τ.
When the received GPS measurements are authentic and follow the nominal distribution with zero-mean Gaussian noise as shown in Equation (<ref>), the GPS residuals _i,j are distributed according to (0, σ) (assuming the estimated positions from FGO are close to ground-truth).
Then, since the test statistic q_k is computed from squaring the residuals (of which there are Nm) and normalizing by (σ)^-2, q_k follows a central chi-squared distribution with n = Nm degrees of freedom.
Thus, given a desired false alarm rate α to remain under, i.e., (detection | not spoofed) ≤α, we desire (q_k ≤τ) = 1 - α for nominal conditions.
Therefore, we compute τ as
τ = Φ^-1(1-α; n=Nm)
where Φ^-1 is the inverse cumulative distribution function (CDF) of the chi-squared distribution with n=Nm degrees of freedom.
If spoofing is detected at time k, i.e., q_k > τ, then any future GPS measurements are deemed unauthentic and the FGO henceforth proceeds with LiDAR only.
Additionally, the current window is also re-processed with LiDAR measurements only, and GPS measurements discarded.
§.§ Chimera Authentication
After N_epoch timesteps have passed, we receive Chimera authentication, which indicates whether the GPS measurements in the past Chimera epoch are authentic or unauthentic.
If the Chimera authentication determines the GPS measurements to be authentic, we leverage this information and rely on the received measurements within our factor graph for N measurements, which corresponds to the window size of our factor graph.
However, if authentication fails, then we perform the same mitigation steps outlined in Section <ref>, where we discard GPS measurements and rely only on the LiDAR sensor.
At this point, the spoofed victim could discontinue nominal operations and proceed according a fail-safe protocol, such as safely pulling over to the side of road, the specifics of which are outside the scope of this work.
§ EXPERIMENTS
We now describe details of our experimental validation, including the dataset used, spoofing attacks considered, baselines which we compare to, and parameters choices for our implementation.
§.§ KITTI Dataset
We evaluate our approach using LiDAR data from the KITTI dataset <cit.> and simulated GPS pseudorange measurements based on the ground-truth positions and satellite ephemeris.
In order to test our algorithm's ability to detect and mitigate spoofing attacks over a 3-minute slow-channel Chimera epoch, we select all sequences from the raw data recordings of duration 3 minutes or longer.
Table <ref> lists the four sequences used in our experiments, their total duration in seconds, and the abbreviations used to refer to each one throughout the remainder of the paper.
For all sequences, we consider a 200 second segment of the trajectory, which contains a full 180 second Chimera slow channel epoch.
We simulate the Chimera authentication as occurring successfully at the first time step of the trajectory.
As a result, the second Chimera authentication time occurs 180 seconds into the trajectory, and we simulate this as a failed authentication for each of the simulated GPS spoofing test scenarios described in Section <ref>.
For all tested trajectories, we use the synced and rectified data in order to handle LiDAR motion distortion effects.
To simulate GPS pseudorange measurements, we compute GPS satellite positions over time _i^(j) using ephemeris data pulled from the location and timestamps of each sequence and follow the measurement model outlined in Section <ref>.
For ground-truth reference trajectories, we use the OXTS ground-truth positions and orientations provided by KITTI.
§.§ Spoofing Attacks
We simulate spoofing attacks on the vehicle by generating a spoofed reference trajectory with added biases, then computing spoofed GPS pseudoranges based on the spoofed reference trajectory.
Specifically, we consider a ramping attack that begins between the Chimera authentications, in which the spoofer introduces a bias which starts small and steadily ramps up to a large error.
This type of attack is typically the most difficult to detect, as the spoofer can gradually induce error without any sudden jumps to alert a standard RAIM solution.
And although the bias may start small, a spoofing victim under this attack can still experience significant positioning error over a sufficient time window, such as the 3 minute slow channel Chimera epoch.
For our experiments, we use a ramping bias which starts at 0 m, and begins linearly increasing at rate r m/s from time T onward.
We add the spoofing bias to the ENU positive x (East) direction, and choose T = 100 s for a total spoofing duration of 100 seconds.
We run experiments for ramp rates of r = 0.5 m/s, 1.0 m/s, and 2.0 m/s, for maximum bias of 50 m, 100 m, and 200 m respectively.
Fig. <ref> shows the reference trajectories for each sequence, and spoofed trajectories for the chosen ramp rates.
§.§ Metrics and Baselines for Comparison
In our experiments we compare our approach with two baseline approaches.
The first baseline is “Odometry only," in which only LiDAR odometry is used to localize the vehicle between Chimera authentications, and GPS measurements are only used at the slow channel 3 minute interval.
The second baseline is “Naive FGO," in which LiDAR-GPS factor graph optimization produces a fused state estimate but no spoofing detection or mitigation is employed.
Finally, “SR FGO" refers to our spoofing-resilient LiDAR-GPS factor graph optimization approach presented in Section <ref>.
For characterizing performance, we consider L^2 norm position error, which is calculated as
e_k = t_k - t_k_2 for each time index k of the trajectory, where t_k is the reference trajectory position at time k and t_k is the estimated trajectory position at time k.
In addition, we consider two metrics: mean L^2 norm position error and maximum L^2 norm position error, which are simply computed as
e = mean_k(e_k)
and
e_max = max_k(e_k)
respectively, and hereafter referred to more concisely as mean error and max error.
§.§ Parameters
We discretize time with Δ t = 0.1 s, and use LiDAR point clouds from the KITTI sequences taken at 10 Hz.
We use Σ = diag(0.01, 0.01, 0.01, 0.05, 0.05, 0.05) as the standard deviation of LiDAR ICP odometry measurements, where the 0.01 values correspond to the rotational components of (3) (equivalent to 0.01 rad std.) and the 0.05 values correspond to the translational components (0.05 m std.).
We simulate GPS measurements at 1 Hz and take σ = 7.0 m according to the typical User-equivalent Range Error (UERE) for a single-frequency receiver <cit.>.
We choose a window size of N = 100, and shift the window by 10 steps per iteration.
α=0.001 is chosen as the false alarm rate for our detector.
For point-to-plane ICP LiDAR registration, we use the Open3D <cit.> function with parameter class and threshold value of 1.0.
We use SymForce <cit.>, a recently developed state-of-the-art symbolic computation library for robotics applications, as the factor graph optimization backend for our method.
Our code is available online at our GitHub repository[<https://github.com/Stanford-NavLab/chimera_fgo>].
§ RESULTS
Now we present the experimental validation results of our spoofing-resilient factor graph algorithm.
We run our algorithm along with the two baselines (Section <ref>) on four KITTI sequences (Table <ref>), for both nominal GPS measurements and various ramping GPS spoofing attacks (shown in Fig. <ref>).
We compare performance in terms of L^2 norm error over time, as well as mean and max L^2 norm error, and also include case studies on window size variation and detection statistics.
§.§ Comprehensive Comparison
We begin by presenting a comprehensive comparison of our approach against the two baselines considered, across the different KITTI sequences and multiple spoofing ramp rates.
These results are illustrated in Fig. <ref>.
We see that for all sequences, the mean and max L^2 norm error of our SR FGO approach remains under that of LiDAR odometry only.
In particular, as the spoofing attack rate increases, the Naive FGO mean and max errors increase, and in some cases eventually exceed the levels of odometry drift, whereas the SR FGO errors are successfully mitigated in all cases and remain bounded under odometry.
§.§ Comparison of Errors over Time under Spoofed Conditions
Next, we focus on the spoofed case of r = 2.0 m/s, and compare the L^2 position error over time of our SR FGO approach against the two baselines Odometry only and Naive FGO.
Fig. <ref> shows a comparison plot for each KITTI trajectory.
In each plot, 20 Monte Carlo runs of Naive and SR FGO are shown, and the start of the spoofing attack at 100 seconds is indicated by the vertical red dashed line.
For each trajectory, we see that LiDAR odometry suffers from significant drift over time, on the order of 100s to 200s of meters of final L^2 norm position error after 200 seconds.
For both Naive FGO and SR FGO, L^2 position errors remain under 5.0 m for the first 100 seconds during authentic conditions.
However, after the start of the attack at 100 seconds, the Naive FGO approach is heavily influenced by the spoofing attack, and its L^2 norm position error diverges, with final error exceeding that of LiDAR odometry in sequences 0018 and 0028.
On the other hand, our SR FGO approach is able to consistently detect and mitigate the spoofing attack, and keep position errors bounded to under odometry drift levels.
§.§ Window Size Comparison
Now, we perform a case study to analyze to effect of varying window size on the performance of our algorithm.
Table <ref> shows a comparison of mean and max L^2 norm position error as well as average iteration time across a range of increasing window sizes for each sequence.
As expected, for all sequences, the average iteration time increases with window size, as the factor graph optimization must be done over a larger window with more measurements.
We also observe high rate of false detection for the smallest window size of 20.
This is to be expected, as the test statistic will be more sensitive to measurement noise and small errors in the trajectory estimates for a smaller window, and thus this window size behaves similarly to LiDAR odometry only in performance.
For all sequences, we also notice that improvement in mean and max error saturates as we increase the window size, occurring at N = 50 for sequence 0018, N = 100 for sequence 0027, N = 200 for sequence 0028, and N = 100 for sequence 0034.
This is most likely due to that fact that, as window size increases, a larger window of the trajectory is re-processed when spoofing is detected.
If spoofing is detected for a window with majority authentic measurements but some spoofed measurements towards the end, then we may discard more authentic measurements which may adversely affect the overall positioning performance.
The results of this case study validate our general choice of window size N = 100 for our experiments.
§.§ Detection Statistics
Finally, we examine the detection statistics for our algorithm, first running 100 Monte Carlo simulations for the nominal, unspoofed case to test the false alarm rate of our detector. These runs are plotted along with the corresponding detection threshold in Fig. <ref>.
The parameter α = 0.001 corresponds to the false alarm probability of a single trial, so across a 180 second long Chimera epoch with 180 trials (one for every GPS measurement at 1 Hz), the probability of a false alarm occurring during the Chimera epoch is 1 - (1 - α)^180 = 0.165.
Across the 100 Monte Carlo runs, there were a total of 9 false alarms, for an empirical per run false alarm rate of 0.09, and 15 total individual trial false alarms out of 18000 individual trials for an empirical per trial false alarm rate of 0.000833.
Thus, we see that our detector satisfies the desired false alarm rate requirements set by the user.
We also perform Monte Carlo simulation for the spoofed case for an attack with r = 1.0 m/s, shown in Fig. <ref>.
In each of the 10 runs, the test statistic crosses the threshold shortly after the start of the attack, successfully detecting it, with a average time to detect of 11.2 seconds.
§ CONCLUSION
In this work, we present a new framework for spoofing-resilient LiDAR-GPS factor graph fusion for Chimera GPS, which provides continuous and secure state estimation between Chimera authentication times.
Our approach fuses LiDAR and GPS measurements with factor graph optimization, and computes a test statistic for spoofing detection based on the GPS factor residuals.
From this test statistic, our approach determines when to leverage the unauthenticated GPS measurements during the Chimera epoch, in order to improve localization performance when GPS is likely authentic.
We evaluate our approach with real-world data from the KITTI self-driving dataset, using sequences which span the Chimera slow channel 3-minute epoch.
Our results demonstrate rapid detection and effective mitigation of spoofing attacks during vulnerable periods between authentications.
This work contributes towards the problem of designing LiDAR-GPS factor graph localization that is robust to GPS spoofing attacks. Our approach is designed around the Chimera signal enhancement, which will be a critical utility to authenticating GPS measurements against spoofing. Between Chimera authentications, we utilize the LiDAR sensor measurements to validate and strategically leverage GPS measurements to improve localization performance during authentic conditions, while maintaining resilience against experienced attacks during spoofing. Our work addresses the research gap for LiDAR-GPS fusion platforms, and takes an important step towards ensuring continuous navigation security for users of the future Chimera-enhanced GPS.
§ ACKNOWLEDGMENT
This material is based upon work supported by the Air Force Research Lab (AFRL) under grant number FA9453-20-1-0002.
We would like to thank the AFRL for their support of this research.
We would also like to thank Shubh Gupta for reviewing this paper.
IEEEtran
|
http://arxiv.org/abs/2307.03888v1 | 20230708034255 | Spectral radius, fractional $[a,b]$-factor and ID-factor-critical graphs | [
"Ao Fan",
"Ruifang Liu",
"Guoyan Ao"
] | math.CO | [
"math.CO",
"05C50, 05C35"
] |
Spectral radius, fractional [a,b]-factor and ID-factor-critical graphs[Supported by National Natural Science Foundation of China
(Nos. 11971445 and 12171440),
Henan Natural Science Foundation (No. 202300410377) and
Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. NJZY22280).]
Ao Fan^a, Ruifang Liu^aCorresponding author.
E-mail addresses: [email protected], [email protected], [email protected]., Guoyan Ao^a, b
^a School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China
^b School of Mathematics and Physics, Hulunbuir University, Hailar, Inner Mongolia 021008, China
==========================================================================================================================================================================================================================================================================================================================================
Abstract
Let G be a graph and h: E(G)→ [0,1] be a function.
For any two positive integers a and b with a≤ b, a fractional [a,b]-factor of G with the indicator function h is
a spanning subgraph with vertex set V(G) and edge set E_h such that a≤∑_e∈ E_G(v)h(e)≤ b for any vertex v∈ V(G),
where E_h = {e∈ E(G)|h(e)>0} and E_G(v)={e∈ E(G)| e v G}.
A graph G is ID-factor-critical if for every independent set I of G whose size has the same parity as |V(G)|, G-I has a perfect matching.
In this paper, we present a tight sufficient condition based on the spectral radius for a graph to contain a fractional [a,b]-factor,
which extends the result of Wei and Zhang [Discrete Math. 346 (2023) 113269].
Furthermore, we also prove a tight sufficient condition in terms of the spectral radius for a graph with minimum degree δ to be ID-factor-critical.
Keywords: Spectral radius, Fractional [a,b]-factor, ID-factor-critical, Minimum degree
AMS Classification: 05C50; 05C35
§ INTRODUCTION
Let G be a finite, undirected and simple graph with vertex set V(G) and edge set E(G).
The order and size of G are denoted by |V(G)|=n and |E(G)|=e(G), respectively.
We denote by δ(G), i(G) and o(G) the minimum degree,
the number of isolated vertices and the number of odd components of G, respectively.
We use K_n and I_n to denote the complete graph of order n and the complement of K_n.
For a vertex subset S of G, let G[S] be the subgraph of G induced by S.
Let G_1 and G_2 be two vertex-disjoint graphs.
We denote by G_1+G_2 the disjoint union of G_1 and G_2.
The join G_1∨ G_2 is the graph obtained from G_1+G_2 by adding all possible edges between V(G_1) and V(G_2).
For undefined terms and notions, one can refer to <cit.>.
Given a graph G of order n, the adjacency matrix of G is the 0-1 matrix A(G)=(a_ij)_n× n indexed
by the vertex set V(G) of G, where a_ij=1 when v_i and v_j are adjacent and a_ij=0 otherwise.
The eigenvalues of A(G) are also called the eigenvalues of G.
Note that A(G) is a real nonnegative symmetric matrix. Hence its eigenvalues are real,
which can be arranged in non-increasing order as λ_1(G)≥λ_2(G) ≥⋯≥λ_n(G).
The largest eigenvalue of A(G), denoted by ρ(G), is called the spectral radius of G.
Let g and f be two integer-valued functions defined on V(G) such that 0≤ g(v)≤ f(v) for each vertex v in V(G).
A (g,f)-factor of G is a spanning subgraph F of G satisfying g(v)≤ d_F(v)≤ f(v) for any vertex v in V(G).
Let a and b be two positive integers with a≤ b.
A (g,f)-factor is called an [a,b]-factor if g(v)≡ a and f(v)≡ b for any v∈ V(G).
An [a,b]-factor is called a 1-factor (also called a perfect matching) if a=b=1.
Let h : E(G)→ [0,1] be a function and E_G(v)={e∈ E(G)| e v G}.
If g(v)≤∑_e∈ E_G(v)h(e)≤ f(v) holds for any vertex v∈ V(G),
then we call a subgraph F with vertex set V(G) and edge set E_h a fractional (g,f)-factor of G with indicator function h,
where E_h = {e∈ E(G)|h(e)>0}.
A fractional (g,f)-factor is called a fractional [a,b]-factor if g(v)≡ a and f(v)≡ b.
In particular, for a positive integer k, a fractional [k, k]-factor of a graph G is called a fractional k-factor of G.
A fractional 1-factor is also called a fractional perfect matching.
Note that if G contains a (g,f)-factor, then it also contains a fractional (g,f)-factor.
However, if G has a fractional (g,f)-factor, G may not have a (g,f)-factor.
We start with the following well-known fractional (g,f)-factor theorem.
Let G be a graph and g,f: V(G)→ Z^+ be two integer functions such that g(v)≤ f(v) for all v∈ V(G).
Then G has a fractional (g,f)-factor if and only if for any subset S⊆ V(G), we have
f(S)-g(T)+∑_v∈ Td_G-S(v)≥0,
where T={v|v∈ V(G)-S d_G-S(v)<g(v)}.
If g(v)≡ a and f(v)≡ b, then by Theorem <ref>, we obtain the following result.
Let G be a graph and let a, b be two positive integers with a≤ b.
Then G has a fractional [a,b]-factor if and only if for any subset S⊆ V(G), we have
b|S|-a|T|+∑_v∈ Td_G-S(v)≥0,
where T={v|v∈ V(G)-S d_G-S(v)<a}.
There are many sufficient conditions which can assure a graph to have a fractional [a,b]-factors
(see for example, <cit.>).
Cho, Hyun, O and Park <cit.> posed the spectral version conjecture for the existence of [a,b]-factors in graphs.
Fan, Lin and Lu <cit.> proved that the conjecture holds for the case n≥ 3a+b-1.
Very recently, Wei and Zhang <cit.> confirmed the full conjecture.
Let a, b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1.
If ρ(G)>ρ(K_a-1∨(K_n-a+K_1)) and na≡ 0 (mod 2) when a=b, then G has an [a,b]-factor.
It is well known that if G contains an [a,b]-factor, then it contains a fractional [a,b]-factor.
Inspired by the work of Wei and Zhang <cit.>,
we obtain a tight sufficient condition in terms of the spectral radius for a graph to contain a fractional [a,b]-factor.
Let a, b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1.
If
ρ(G)≥ρ(K_a-1∨(K_n-a+K_1))
and na≡0 (mod 2) when a=b, then G has a fractional [a,b]-factor
unless G≅ K_a-1∨(K_n-a+K_1).
Note that 4+√(32a^2+24a+5)> a+1. Our Theorem <ref> improves the following result.
Let b≥ a≥ 1 be two integers, and let G be a graph of order n≥ 4+√(32a^2+24a+5).
If ρ(G)≥ρ(K_a-1∨(K_n-a+K_1)), then G has a fractional [a,b]-factor unless G≅ K_a-1∨(K_n-a+K_1).
A graph G is independent-set-deletable factor-critical, shortly ID-factor-critical, if for every
independent set I of G whose size has the same parity as |V(G)|, G-I has a perfect matching.
Let S_n, k be the join of a clique on k vertices with an independent set of n-k vertices for n>k. That is to say, S_n, k=K_k∨ I_n-k.
A graph G has a perfect matching if and only if o(G-S)≤|S| for every S⊆ V(G).
The following theorem is a direct consequence of Tutte's Theorem.
A graph G is ID-factor-critical if and only if o(G-I-S)≤|S| for every independent set I such that |I| has the same parity as |V(G)| and every subset S⊆ V(G)-I.
Using Theorem <ref>, we prove a tight spectral condition for a graph with minimum degree δ to be ID-factor-critical.
Let G be a graph of order n with minimum degree δ≥3r+1, where r≥1 is an integer.
If n≥ max{20δ+r+8, δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2} and
ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
then G is ID-factor-critical unless G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
§ PROOF OF THEOREM <REF>
Before presenting our proof, we introduce some necessary lemmas.
Let G be a graph of order n≥3. If e(G)≥n-12+1, then G has a Hamilton path.
Although the following Lemma <ref> can be obtained directly from Theorem 2 in <cit.>,
here we can present a much simpler proof of Lemma <ref> for a fractional [a,b]-factor.
Let a and b be two positive integers with a≤ b, and let G be a graph of order n≥ a+1 and minimum degree δ≥ a. If
e(G)≥n-12+a+1/2
and na≡0 (mod 2) when a=b, then G has a fractional [a,b]-factor.
For any two disjoint vertex subsets S and T in G, let
φ(S,T)=b|S|-a|T|+∑_v∈ Td_G-S(v).
Suppose to the contrary that G has no fractional [a,b]-factor. By Corollary <ref>,
there exist two disjoint subsets S and T of V(G) such that
φ(S,T)≤-1,
where T={v|v∈ V(G)-S d_G-S(v)<a}.
n≥ a+2 and b≥2.
Note that δ≥ a. If n=a+1, then G is a complete graph. It is well known that the complete graph contains an [a,b]-factor,
and hence G contains a fractional [a,b]-factor, a contradiction. So we have n≥ a+2.
If b=1, then a=b=1, and thus e(G)≥n-12+1.
By Lemma <ref>, G has a Hamilton path.
Note that n is even. Then G contains a 1-factor, and hence G contains a fractional 1-factor, a contradiction.
Hence b≥2.
S≠∅
Assume that S=∅.
Note that G-S=G and δ(G)≥ a. Then δ(G-S)≥ a. Recall that T={v|v∈ V(G)-S d_G-S(v)<a}.
Then T=∅, and thus
φ(∅,∅)=0,
which is contrary to (<ref>).
Next we will evaluate the value of |T|.
Case 1. 0≤|T|≤ b.
Note that δ≥ a. Then
φ(S,T) = b|S|-a|T|+∑_v∈ Td_G-S(v)
= b|S|-a|T|+∑_v∈ Td_G(v)-e_G(S,T)
≥ b|S|-a|T|+a|T|-|T||S|
= (b-|T|)|S|
≥ 0,
which contradicts (<ref>).
Case 2. |T|≥ b+1.
Since S and T are two disjoint subsets of V(G), n≥|S|+|T|≥|S|+b+1.
By the assumption e(G)≥n-12+a+1/2, there exist at most n-1-a+1/2 edges which are not in E[V(G-T-S),T]∪ E(G[T]).
Hence
∑_v∈ Td_G-S(v)≥(n-1-|S|)|T|-2(n-1-a+1/2).
Subcase 2.1. a<b.
Combining Claim <ref>, we have
φ(S,T)
= b|S|-a|T|+∑_v∈ Td_G-S(v)
≥ b|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+1/2)
= (n-1-|S|-a)|T|+b|S|-2n+a+3
≥ (n-1-|S|-a)(b+1)+b|S|-2n+a+3
= (b-2)n+n-|S|-ab-b+2
≥ (b-2)n+(|S|+b+1)-|S|-ab-b+2
= (b-2)n-ab+3
≥ (b-2)(a+2)-ab+3
= 2b-2a-1
≥ 1,
which is contrary to (<ref>).
Subcase 2.2. a=b.
Recall that n≥|S|+b+1=|S|+a+1 and na ≡0 (mod 2). If a is odd, then n is even.
By Claim <ref>, we have n≥ a+3 and a≥3.
Then
φ(S,T) = a|S|-a|T|+∑_v∈ Td_G-S(v)
≥ a|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+1/2)
= (n-1-|S|-a)|T|+a|S|-2n+a+3
≥ (n-1-|S|-a)(a+1)+a|S|-2n+a+3
= (a-2)n+n-|S|-a^2-a+2
≥ (a-2)n+(|S|+a+1)-|S|-a^2-a+2
= (a-2)n-a^2+3
≥ (a-2)(a+3)-a^2+3
= a-3.
≥ 0,
a contradiction.
Next we consider that a is even. Since e(G)≥n-12+a+1/2, we obtain that e(G)≥n-12+a+2/2,
and hence ∑_v∈ Td_G-S(v)≥(n-1-|S|)|T|-2(n-1-a+2/2).
By Claim <ref>, we have n≥ a+2. Then
φ(S,T) = a|S|-a|T|+∑_v∈ Td_G-S(v)
≥ a|S|-a|T|+(n-1-|S|)|T|-2(n-1-a+2/2)
= (n-1-|S|-a)|T|+a|S|-2n+a+4
≥ (n-1-|S|-a)(a+1)+a|S|-2n+a+4
= (a-2)n+n-|S|-a^2-a+3
≥ (a-2)n+(|S|+a+1)-|S|-a^2-a+3
= (a-2)n-a^2+4
≥ (a-2)(a+2)-a^2+4
= 0,
which contradicts (<ref>).
Let A=(a_ij) and B=(b_ij) be two n× n matrices.
Define A≤ B if a_ij≤ b_ij for all i and j, and define A< B if A≤ B and A≠ B.
Let A=(a_ij) and B=(b_ij) be two n× n matrices with the spectral radii λ(A) and λ(B).
If 0≤ A≤ B, then λ(A)≤λ(B).
Furthermore, if B is irreducible and 0≤ A < B, then λ(A)<λ(B).
We will use the following lemma in the proof of Theorem <ref>.
Let G be a graph with minimum degree δ. Then
ρ(G)≤δ-1/2+√(2e(G)-δ n+(δ+1)^2/4).
[Hong, Shu and Fang <cit.>, Nikiforov <cit.>]
For graph G with 2e(G)≤ n(n-1), the function
f(x)=x-1/2+√(2e(G)-nx+(x+1)^2/4)
is decreasing with respect to x for 0≤ x≤ n-1.
Proof of Theorem <ref>. Let G be a graph of order n≥ a+1.
Note that the minimum degree of K_a-1∨ (K_n-a+K_1) is a-1.
Let h: E(G)→ [0,1] be a function. Then for v∈ V(K_1), we have ∑_e∈ E_G(v)h(e)≤ a-1.
By the definition of a fractional [a, b]-factor, then K_a-1∨ (K_n-a+K_1) has no fractional [a, b]-factor.
Assume that G K_a-1∨ (K_n-a+K_1) (see Fig. <ref>). It suffices to prove that G contains a fractional [a, b]-factor.
First we prove the following claim.
δ≥ a.
If δ≤ a-1, then there exists a vertex v∈ V(G) such that d(v)≤ a-1.
This means that G is a subgraph of K_a-1∨ (K_n-a+K_1).
By Lemma <ref>, we have
ρ(G)≤ρ(K_a-1∨ (K_n-a+K_1)).
By the assumption ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1)), we have G≅ K_a-1∨ (K_n-a+K_1), a contradiction.
Hence δ≥ a.
We distinguish the proof into the following two cases.
Case 1. a=1.
By the assumption, we have ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1))= ρ(K_n-1+K_1)=n-2.
By Claim <ref>, Lemma <ref> and Proposition <ref>, we obtain that
n-2≤ρ(G)≤√(2e(G)-n+1).
It follows that e(G)≥n-12+1/2, and hence e(G)≥n-12+1.
By Lemma <ref>, then G contains a fractional [a,b]-factor.
Case 2. a≥ 2.
Note that K_n-1 is a proper subgraph of K_a-1∨ (K_n-a+K_1).
By the assumption and Lemma <ref>, we have ρ(G)≥ρ(K_a-1∨ (K_n-a+K_1))>ρ(K_n-1)=n-2.
By Claim <ref>, Lemma <ref> and Proposition <ref>, we have
n-2<ρ(G)≤a-1/2+√(2e(G)-an+(a+1)^2/4).
It follows that e(G)>n-12+a/2. That is to say, e(G)≥n-12+a+1/2.
By Lemma <ref>, then G contains a fractional [a,b]-factor.
§ PROOF OF THEOREM <REF>
By the Perron-Frobenius Theorem, ρ(G) is always a positive number (unless G is an empty graph),
and there exists an unique positive unit eigenvector corresponding to ρ(G), which is called the Perron vector of G.
Let n=∑_i=1^tn_i+s. If n_1≥ n_2≥⋯≥ n_t≥ p and n_1<n-s-p(t-1), then
ρ(K_s∨(K_n_1+ K_n_2 + ⋯ + K_n_t))<ρ(K_s∨(K_n-s-p(t-1)+ (t-1)K_p)).
Graph S_δ+r,δ∨(K_n-2δ-r-1+I_δ+1) is not ID-factor-critical.
Let G= S_δ+r,δ∨(K_n-2δ-r-1+I_δ+1) (see Fig. <ref>).
Suppose to the contrary that G is ID-factor-critical.
By the definition of an ID-factor-critical graph,
we have G-I has a perfect matching for any independent set I of G whose size has the same parity as |V(G)|.
Note that S_δ+r, δ=K_δ∨ I_r. However, if we take I=I_r and let H=G-I.
Then
H≅ K_δ∨(K_n-2δ-r-1+I_δ+1).
Note that the vertices of I_δ+1 are only adjacent to the vertices of K_δ. Hence H has no perfect matching, a contradiction.
Now, we are in a position to present the proof of Theorem <ref>.
Proof of Theorem <ref>.
Assume that G is not ID-factor-critical. According to Theorem <ref>, there exists some independent set I such that |I| has the same parity as
|V(G)|=n , we have o(G-I-S)≥ |S|+1 for some subset S⊆ V(G)-I. Let |I|=r and |S|=s. Then o(G-I-S)≥ s+1.
Since n-r is even, o(G-I-S) and s have the same parity. Hence we have o(G-I-S)≥ s+2.
It is clear that G is a spanning subgraph of G'=I_r∨ (K_s∨(K_n_1+ K_n_2+⋯+K_n_s+2))
for some odd integers n_1≥ n_2≥⋯≥ n_s+2>0 with ∑_i=1^s+2n_i=n-r-s. Then we have
ρ(G)≤ρ(G'),
where equality holds if and only if G≅ G'.
Let G”=S_s+r, s∨(K_n-2s-r-1+I_s+1). By Lemma <ref>, we obtain that
ρ(G')≤ρ(G”),
where equality holds if and only if (n_1, n_2, … ,n_s+2)=(n-2s-r-1,1,… ,1).
Case 1. s=δ.
Combining (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”)=ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
By the assumption ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
we have G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
By Lemma <ref>, S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1) is not ID-factor-critical.
Hence G≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Case 2. s≥δ+1.
Recall that G”=S_s+r, s∨(K_n-2s-r-1+I_s+1).
The vertex set of G” can be divided into V(G”)=V(K_s)∪ V(I_s+1)∪ V(I_r)∪ V(K_n-2s-r-1),
where V(K_s)={u_1, u_2, … ,u_s},
V(I_s+1)={v_1, v_2, … ,v_s+1}, V(I_r)={w_1, w_2, … ,w_r}
and V(K_n-2s-r-1)={z_1, z_2, … ,z_n-2s-r-1}.
Let E_1={v_iz_j|δ+2≤ i≤ s+1, 1≤ j≤ n-2s-r-1}
∪{v_iv_j|δ+2≤ i≤ s,i+1≤ j≤ s+1}
and E_2={u_iv_j|δ+1≤ i≤ s,1≤ j≤δ+1}.
Let G^*=G”+E_1-E_2. Obviously, G^*≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Let x be the perron vector of A(G”), and let ρ” =ρ(G”).
By symmetry, x takes the same value on the vertices of V(K_s), V(I_s+1), V(I_r) and V(K_n-2s-r-1), respectively.
It is easy to see that
A(G”)=[
[ (J-I)_s× s J_s× (s+1) J_s× r J_s× (n-2s-r-1); J_(s+1)× s O_(s+1)× (s+1) J_(s+1)× r O_(s+1)× (n-2s-r-1); J_r× s J_r× (s+1) O_r× r J_r×(n-2s-r-1); J_(n-2s-r-1)× s O_(n-2s-r-1)× (s+1) J_(n-2s-r-1)× r (J-I)_(n-2s-r-1)×(n-2s-r-1) ]].
We denote the entry of x by x_1, x_2, x_3 and x_4 corresponding to the vertices in the above four vertex sets, respectively.
By A(G”)x=ρ” x, we have
ρ” x_2=sx_1+rx_3,
ρ” x_3=sx_1+(s+1)x_2+(n-2s-r-1)x_4,
ρ” x_4=sx_1+rx_3+(n-2s-r-2)x_4.
Observe that n≥2s+r+2. According to (<ref>) and (<ref>), we obtain that x_4≥ x_2.
By (<ref>) and (<ref>), we have ρ” x_3-ρ” x_4=(s+1)x_2-rx_3+x_4.
It follows that x_4=(ρ”+r)x_3-(s+1)x_2/ρ”+1≥ x_2.
Then we have x_3≥ρ”+s+2/ρ”+rx_2.
Note that s≥δ+1 and δ≥3r+1. Then ρ”+s+2≥ρ”+δ+3>ρ”+r, and hence x_3>x_2.
Combining (<ref>), we have
x_2>sx_1/ρ”-r.
Recall that G^*≅ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1).
Note that G^* contains K_n-2δ-r-1 as a proper subgraph. Then ρ^*>n-2δ-r-2.
Similarly, let y be the perron vector of A(G^*), and let ρ^* =ρ(G^*).
By symmetry, y takes the same value (say y_1, y_2, y_3 and y_4) on the vertices of
V(K_δ), V(I_δ+1), V(I_r) and V(K_n-2δ-r-1).
By A(G^*)y=ρ^*y, we have
ρ^*y_2=δ y_1+ry_3,
ρ^*y_4=δ y_1+ry_3+(n-2δ-r-2)y_4.
Combining (<ref>) and (<ref>), we have
y_4=ρ^*y_2/ρ^*-(n-2δ-r-2).
Note that n≥2s+r+2. Then δ+1≤ s≤n-r-2/2. Since G” is not a complete graph, ρ”< n-1.
ρ”<ρ^*.
Suppose that ρ”≥ρ^*. By x_4≥ x_2, (<ref>) and (<ref>), we have
y^T(ρ^*-ρ”)x
= y^T(A(G^*)-A(G”))x
= ∑_i=δ+2^s+1∑_j=1^n-2s-r-1(x_v_iy_z_j+x_z_jy_v_i)+∑_i=δ+2^s∑_j=i+1^s+1(x_v_iy_v_j+x_v_jy_v_i)-∑_i=δ+1^s∑_j=1^δ+1(x_u_iy_v_j+x_v_jy_u_i)
= (n-2s-r-1)(s-δ)(x_2y_4+x_4y_4)+(s-δ-1)(s-δ)x_2y_4-(s-δ)(δ+1)(x_1y_2
+x_2y_4)
≥ (s-δ)[2(n-2s-r-1)x_2y_4+(s-δ-1)x_2y_4-(δ+1)x_2y_4-(δ+1)x_1y_2]
= (s-δ)[(2n-3s-2δ-2r-4)x_2y_4-(δ+1)x_1y_2]
> (s-δ)[(2n-3s-2δ-2r-4)·sx_1/ρ”-r·ρ^*y_2/ρ^*-(n-2δ-r-2)-(δ+1)x_1y_2]
= (s-δ)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-3s-2δ-2r-4)sρ^*-(δ+1)(ρ”-r)(ρ^* .
.-(n-2δ-r-2))]
= (s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[ρ^*(2n-2δ-3s-2r-4)·s/δ+1-(ρ”-r)(ρ^*.
.-(n-2δ-r-2))].
Note that s≥δ+1, ρ”≥ρ^* and ρ^*>δ-1≥3r. Then
y^T(ρ^*-ρ”)x
> (s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[ρ^*(2n-2δ-3s-2r-4)-ρ”ρ^*+ρ”(n-2δ-r-2).
.+rρ^*-r(n-2δ-r-2)]
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-2δ-3s-2r-4)-ρ”+ρ”/ρ^*·(n-2δ-r-2).
.+r-r/ρ^*·(n-2δ-r-2)]
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))[(2n-2δ-3s-2r-4)-ρ”+(n-2δ-r-2).
.+r-1/3·(n-2δ-r-2)]
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(8/3n-10/3δ-3s-5/3r-16/3-ρ”).
Since K_s⊂ G” and δ≥3r+1, ρ”> ρ(K_s)=s-1≥δ >r.
Note that s≤n-r-2/2, ρ”< n-1, ρ^*>n-2δ-r-2 and n≥ 20δ+r+8. Then
y^T(ρ^*-ρ”)x
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(8/3n-10/3δ-3·n-r-2/2-5/3r-16/3-ρ”)
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(7/6n-10/3δ-1/6r-7/3-ρ”)
= ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))(1/6n-10/3δ-1/6r-4/3+(n-1)-ρ”)
> ρ^*(s-δ)(δ+1)x_1y_2/(ρ”-r)(ρ^*-(n-2δ-r-2))·n-20δ-r-8/6
≥ 0.
This implies that ρ^*>ρ”, which contradicts the assumption ρ”≥ρ^*.
By Claim <ref>, (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”)<ρ(G^*)=ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
which contradicts ρ(G)≥ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
Case 3. s<δ.
Recall that G'=I_r∨ (K_s∨(K_n_1+ K_n_2+⋯+K_n_s+2)).
Then d_G'(v)=n_s+2+s+r-1≥δ for v ∈ V(K_n_s+2), and hence n_s+2≥δ-s-r+1.
Let G”'=I_r∨(K_s∨(K_n-s-r-(s+1)(δ-s-r+1)+(s+1)K_δ-s-r+1)).
By Lemma <ref>, we have
ρ(G')≤ρ(G”'),
where equality holds if and only if (n_1,n_2,…,n_s+2)=(n-s-r-(s+1)(δ-s-r+1),δ-s-r+1,…,δ-s-r+1).
Let ρ”'=ρ(G”').
ρ”'< n-r-1-(s+1)(δ-s+1).
Suppose to the contrary that ρ”'≥ n-r-1-(s+1)(δ-s+1).
Let x be the perron vector of A(G”'). By symmetry, x takes the same values x_1, x_2, x_3 and x_4
on the vertices of K_s, (s+1)K_δ-s-r+1, I_r and K_n-s-r-(s+1)(δ-s-r+1), respectively.
According to A(G”')x=ρ”' x, we obtain that
ρ”'x_1=(s-1)x_1+(s+1)(δ-s-r+1)x_2+rx_3+(n-s-r-(s+1)(δ-s
-r+1))x_4,
ρ”'x_2=sx_1+(δ-s-r)x_2+rx_3,
ρ”'x_3=sx_1+(s+1)(δ-s-r+1)x_2+(n-s-r-(s+1)(δ-s-r+1))x_4,
ρ”'x_4=sx_1+rx_3+(n-s-r-1-(s+1)(δ-s-r+1))x_4.
By (<ref>) and (<ref>), we have
x_3=(ρ”'+1)x_1/ρ”'+r.
Substituting (<ref>) into (<ref>) and (<ref>), we have
x_2=sx_1+r(ρ”'+1)/ρ”'+rx_1/ρ”'-δ+s+r,
x_4=sx_1+r(ρ”'+1)/ρ”'+rx_1/ρ”'-[n-s-r-1-(s+1)(δ-s-r+1)].
Since n≥δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2, we have
ρ”'≥ n-r-1-(s+1)(δ-s+1)>δ-r+1.
Substituting (<ref>), (<ref>) and (<ref>) into (<ref>), we have
ρ”'+1 = s+(s+1)(δ-s-r+1)(s+r(ρ”'+1)/ρ”'+r)/ρ”'-δ+s+r+r(ρ”'+1)/ρ”'+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r(ρ”'+1)/ρ”'+r)/ρ”'-(n-s-r-1-(s+1)(δ-s-r+1))
≤ s+(s+1)(δ-s-r+1)(s+r)/ρ”'-δ+s+r+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r)/ρ”'-(n-s-r-1-(s+1)(δ-s-r+1))
< s+(s+1)(δ-s-r+1)(s+r)/(δ-r+1)-δ+s+r+r
+[n-s-r-(s+1)(δ-s-r+1)](s+r)/[n-r-1-(s+1)(δ-s+1)]-(n-s-r-1-(s+1)(δ-s-r+1))
= s+(s+r)(δ-s-r+1)+r+[n-s-r-(s+1)(δ-s-r+1)](s+r)/s-sr-r
= n-r-1-(s+1)(δ-s+1)-1/sr-s+r[(sr+2r)n+(2r-1)s^3+(2r^2-2δ r
+δ+1)s^2+(r^3-δ r^2-r^2-3δ r-2r+1)s+r^3-δ r^2-3r^2-2δ r-3r].
Let f(n)=(sr+2r)n+(2r-1)s^3+(2r^2-2δ r+δ+1)s^2+(r^3-δ r^2-r^2-3δ r-2r+1)s+r^3-δ r^2-3r^2-2δ r-3r.
We assert that f(n)≥ 0. Suppose that f(n)<0. Then
n<1/sr+2r[(-2r+1)s^3+(-2r^2+2δ r-δ-1)s^2+(-r^3+δ r^2+r^2+3δ r+2r-1)s-r^3+δ r^2+3r^2+2δ r+3r].
Note that 0≤ s<δ, -2r+1<0, -2r^2+2δ r-δ-1>0 and -r^3+δ r^2+r^2+3δ r+2r-1>0. Then
n < 1/sr+2r[(-2r+1)s^3+(-2r^2+2δ r-δ-1)s^2+(-r^3+δ r^2+r^2+3δ r+2r-1)s
-r^3+δ r^2+3r^2+2δ r+3r]
< 1/2r[(-2r^2+2δ r-δ-1)δ^2+(-r^3+δ r^2+r^2+3δ r+2r-1)δ-r^3+δ r^2+3r^2
+2δ r+3r]
= 1/2r[(2r-1)δ^3+(-r^2+3r-1)δ^2+(-r^3+2r^2+4r-1)δ-r^3+3r^2+3r]
< 1/2r[2rδ^3+(-r^2+3r)δ^2+(-r^3+2r^2+4r)δ-r^3+3r^2+3r]
= δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2,
which contradicts n≥δ^3-r-3/2δ^2-r^2-2r-4/2δ-r^2-3r-3/2. Hence f(n)≥ 0.
Then
ρ”'+1 < n-r-1-(s+1)(δ-s+1)-1/sr-s+rf(n)
< n-r-1-(s+1)(δ-s+1)
≤ ρ”',
a contradiction. Therefore, we have ρ”'<n-r-1-(s+1)(δ-s+1).
By Claim <ref> and s< δ, we obtain that
ρ”' < n-r-1-(s+1)(δ-s+1)
= n-δ-r-1-[(δ-s)s+1]
< n-δ-r-1.
Note that K_n-δ-r⊂ S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1). Then
n-δ-r-1=ρ(K_n-δ-r)<ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)).
Combining (<ref>) and (<ref>), we have
ρ(G)≤ρ(G')≤ρ(G”')<n-δ-r-1<ρ(S_δ+r, δ∨(K_n-2δ-r-1+I_δ+1)),
which contradicts the assumption, as desired.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
99
Anstee1990 R.P. Anstee, Simplified existence theorems for (g,f)-factor, Discrete Appl. Math. 27 (1990) 29–38.
Berman1979 A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.
Bondy2008 J.A. Bondy, U.S.R. Murty, Graph Theory, Grad. Texts in Math. vol. 244, Springer, New York, 2008.
Cho2021 E. Cho, J. Hyun, S. O, J. Park, Sharp conditions for the existence of an even [a,b]-factor in a graph,
Bull. Korean Math. Soc. 58 (2021) 31–46.
Fan D.D. Fan, H.Q. Lin, Spectral conditions for k-extendability and k-factors of bipartite graphs, arXiv:2211.09304.
Fan2021 D.D. Fan, S. Goryainov, X.Y. Huang, H.Q. Lin, The spanning k-tree, perfect matchings and spectral radius of graphs,
Linear Multilinear Algebra 70 (2022) 7264–7275.
Fan2022 D.D. Fan, H.Q. Lin, H.L. Lu, Spectral radius and [a,b]-factors in graphs, Discrete Math. 345 (2022) 112892.
Hong2001 Y. Hong, J.L. Shu, K.F. Fang, A sharp upper bound of the spectral radius of graph,
J. Combin. Theory, Ser. B 81 (2001) 177–183.
Horn1986 R.A. Horn, C.R. Johnson, Matrix analysis, Cambridge University Press, New York, 1986.
Liu2001 G.Z. Liu, L.J. Zhang, Fractional (g,f)-factors of graphs, Acta Math. Sci. 21 (2001) 541–545.
Liu2008 G.Z. Liu, L.J. Zhang, Toughness and the existence of fractional k-factors of graphs, Discrete Math. 308 (2008) 1741–1748.
Lu2013 H.L. Lu, Simplified existence theorems on all fractional [a,b]-factors,
Discrete Appl. Math. 161 (2013) 2075–2078.
Nikiforov2002 V. Nikiforov, Some inequalities for the largest eigenvalue of a graph,
Combin. Probab. Comput. 11 (2002) 179–189.
Tutte1947 W.T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107–111.
Wang2023 J.J. Wang, J.X. Zheng, Y.L. Chen, Spectral radius conditions for fractional [a, b]-covered graphs,
Linear Algebra Appl. 666 (2023) 1–10.
Wei2023 J. Wei, S.G. Zhang, Proof of a conjecture on the spectral radius condition for [a, b]-factors,
Discrete Math. 346 (2023) 113269.
|
http://arxiv.org/abs/2307.04254v1 | 20230709193700 | Relativistic time dilation as a quantum mechanism | [
"Esteban Martínez-Vargas"
] | quant-ph | [
"quant-ph"
] |
[email protected]
Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona) Spain
We propose a mechanism for time dilation using quantum systems. We introduce
a family of operators that are sensitive to the changes of quantum states from different
frames of reference. The change between reference frames is done via a Galilean
transformation, therefore, the source of the dillation in our case comes
from the observable. These observables grow linearly in time and depending
on the reference frame of the state the linear growth changes its slope,
therefore it takes longer to grow to the same point.
Such mechanism implies a different view from the usual understanding
of spacetime.
Relativistic time dilation as a quantum mechanism
Esteban Martínez Vargas
August 12, 2023
=================================================
One of the most obvious questions in contemporary physics is that of quantum gravity.
A universal phenomenon such as gravitation should have a description in terms of the most
fundamental theory at hand: quantum theory <cit.>.
Despite big efforts, it has remained a mystery how to build a successful quantum
theory of gravitation <cit.>.
As it is universally known, the most accurate description of gravity is given by
Einstein's General theory of relativity, of which special Relativity is a particular case <cit.>.
Although there have been successful theories that have merged the phenomenology of special
relativity with quantum mechanics as Quantum Field Theory and Dirac's equation <cit.>, it has remained
elusive how to do the proper in the general theory. Much of it has to do with the fact
of how such theories treat the concept of spacetime, which is central to general relativity.
For example, one of the challenges resides in the fact that quantum phenomena are taken with an absolute
causal structure whereas General Relativity does not have one <cit.>.
This could point toward the abandonment of the use of absolute time frames
to describe phenomena in quantum systems <cit.>.
If absolute time is not necessary it is however a powerful concept to describe quantum
phenomena. In any case, this asks for a build-up of a theory of spacetime for quantum systems.
The usual ontology of spacetime is some kind of field that permeates
the universe which interacts with matter slowing down time and expanding space
accordingly. This is the viewpoint of approaches
that stem from particle physics like Loop Quantum gravity <cit.> and
the AdS/CFT correspondence <cit.>.
Recently, there has been an interest in using the tools from
low energy quantum theory and quantum information theory
to study relativistic effects <cit.>.
However, these low-energy studies are inside the
same ontological point of view. In all these approaches Relativity theory has a superior
status in some sense: it has some symmetry properties that quantum systems
are constrained to fulfill.
Here, we propose a different point of view, space
and time can be correlated in the interactions between quantum systems in different frames of
reference. In other words, from the perspective of special relativity: time dilation can be explained as
a quantum phenomenon. What we show here is that flat spacetime can be described
using quantum theory.
Explicitly, we take a quantum system in a frame of reference and another in another reference frame. We can
apply a Galilean transformation to see how the second quantum system would be observed in the
first reference frame.
The Galilean transformation for quantum states was previously introduced in <cit.>
and extended in <cit.>.
We show that there exist observables that grow linearly in time which, in
the other reference frame transform in a way that makes time dilate. Specifically, the
expectation value would grow linearly but with a flatter slope that accounts for Lorentz
time dilation.
Quantum devices have been
used to create such regular motion that produces ticks as we have seen in atomic clocks <cit.>
and also in time crystals <cit.>.
However, the second part of the clock, the record is as important.
The problem is that, if no record is taken, then what means regularity in
the first place? We have to remember that a regular motion has been previously
in such a place.
This is a crucial point here, as we will not describe a quantum clock but a quantum time register.
This means the mechanism that indicates time passage in accumulation, similar to a sand
clock. The concept of time register will be formalized in the next section, it is based
on the theory of Sequential Analysis <cit.>. Afterward, we extend this
concept to Quantum Mechanics. Using a Galilean transform we observe that the
ground state of a harmonic oscillator is viewed as a coherent state for a suitable
α in another reference frame. For observables that grow linearly in time we observe that the slope is
given by the expectation value of some observable Ŝ. The main theorem is then
stated: there exist observables Ŝ that account for the Lorentz time dilation.
The Sequential Probability Ratio Test (SPRT) is the optimal protocol in terms of the resources needed for
the acquisition of information to decide between two hypotheses and is the central protocol
studied by Wald <cit.>. Specifically, given an error threshold, it is the protocol
that minimizes the needed resources to make a decision. Surprisingly, the central object
in this protocol is not deterministic at all: it is a random walk <cit.>.
In the important Gaussian case, it is a Wiener process <cit.> that
will stop until a threshold is reached. However, this process happens in time, which
means, it is correlated with the passage of time.
Therefore, we can correlate Sequential Analysis
not as a clock but as a time register. This means, if we assume that the samples
detected are exactly correlated with the ticks of a perfect clock, we can use the random walk
as evidence of the passage of time.
In Fig. (<ref>) we have 1000 iterations of a Wiener process with N=100 that
is biased. The information about one hypothesis or another comes from this bias. The
average of the random walkers is a line with a specific slope that depends on the
sampling distribution.
We can write the evolution of Z as a Wiener process so it
obeys a dynamical equation,
Z(t+dt)-Z(t)=√(δ^2dt)N_t^t+dt(θ,1/2).
Where N_t^t+dt(θ,1/2) represents a Gaussian with mean θ and
variance 1/2.
Observe that this is an instance of a stochastic differential equation that obeys
different rules from usual calculus <cit.>.
When the Martingale crosses a threshold we can call it a tick
that will be registered.
The previous paragraph suggests that for the acquisition of information one should use a
random walker. However, it should be a biased one, so that the general trend evidences
the acquisition of information. Moreover, from Fig. (<ref>) we see that the
trend is that the mean value of the Martingale grows linearly in time, ϵ t for
some constant ϵ. Can we extend the previous concept of time registers for
quantum systems? The quantum analog of Z would be an observable that evolves with n.
In contrast with time crystals the value of the observable should always
grow. Suppose that we have a clock that is perfectly regular and that at each tick it
yields a quantum state |ψ_0⟩. This state travels and a measurement is performed
on the observable Ẑ(t) as depicted in Fig. (<ref>). As quantum measuring is a stochastic process, we would
have something similar to Fig. (<ref>). The bias should be such that it
has to be proportional to the passage of time. Following the analogy with the martingales of Fig. (<ref>)
we define a Quantum Time Register (QTR) as any observable Ẑ(t) such that the expectation
value with respect to |ψ_0⟩ is proportional to time, considering the difference with
an initial t_0 i.e.
⟨ψ_0|(Ẑ(t)-Ẑ(t_0))|ψ_0⟩=ϵΔ t.
Suppose we have a QTR, i.e. an observable that grows linearly in time
and we measure states coming from a source in a different reference frame, as depicted
in Fig. (<ref>).
A quantum state depends on the reference
frame used to describe it <cit.>.
We will consider a specific type of QTRs for a family of states. Suppose we measure
in the reference frame 𝒪 and the source of the state is in the reference
frame 𝒪^' that moves at constant velocity v respect to 𝒪
as in Fig. (<ref>).
We know from <cit.>
that to change a quantum particle description to a different reference at constant
velocity we do the transformation
|ψ^'⟩=e^i/ħvĜ|ψ_0⟩,
where v is the velocity between reference frames and Ĝ:=p̂t-mx̂ with m
the mass of the particle |ψ_0⟩.
Is there a QTR that is consistent with relativistic time dilation? This means,
one for which time dilates according to the Lorentz transformations.
Such observable Ẑ would have the expectation value
⟨ψ^'|(Ẑ(t)-Ẑ(t_0))|ψ^'⟩=ϵ/γΔ t,
where
γ := 1/√(1-v^2/c^2)
is the Lorentz factor. In what follows we will take units where c=1.
We are searching for a mechanism in the interaction that gives rise to time dilation
as illustrated in Fig. (<ref>). An operator with expectation value
that grows
the proper time frame would behave as in the red line until it reaches the value A.
In a different reference frame, the growth would take longer, as with the blue and green lines. If the speed is
too high the expectation value would never reach the value A.
The most general description of the observable
Ẑ according to Quantum Mechanics is given by an open system dynamics with the Quantum
Langevin equations <cit.>
⟨dẐ/dt⟩ = ⟨ĉ^†Ẑĉ-1/2ĉ^†ĉẐ-1/2Ẑĉ^†ĉ+i[Ĥ,Ẑ]⟩,
for some Hamiltonian Ĥ and anhilation operator ĉ.
Let us define
Ŝ≡dẐ/dt.
Let us assume dŜ/dt=0, then our problem at hand can be reduced to finding
Ŝ such that
⟨ψ_0|Ŝ|ψ_0⟩ =ϵ
⟨ψ^'|Ŝ|ψ^'⟩ =ϵ/γ.
We express |ψ_0⟩ in terms of the eigenstates of the harmonic oscillator.
This is convenient
as we can write the operator e^i/ħvĜ as a displacement
one. Observe that
e^i/ħv(tp̂-mx̂) = e^αâ^†-α^*â,
introducing the frequency ω,
and defining
α≡-v√(m/2ħ)(t√(ω)+i).
For |ψ_0⟩=|0⟩ we have that |ψ^'⟩=e^αâ^†-α^*â|0⟩=|α⟩.
This will be, of course, a harmonic oscillator with mass.
Observe that the condition (<ref>) implies that for
all v<1 there
is a constant operator Ŝ such that ⟨Ŝ⟩=ϵ/γ(v)
with the states |ψ^'(v)⟩.
There exists always such operator Ŝ as proven in the following
theorem
There exist a constant operator Ŝ such that
⟨α|Ŝ|α⟩ = ϵ/γ,
for all v<1, where γ is the Lorentz factor and α is given by
equation (<ref>).
Let Ŝ be written in the Harmonic oscillator basis as
Ŝ=∑_n,mS_n,m|n⟩⟨m|.
Then we would have
⟨α|Ŝ|α⟩ = e^-|α|^2∑_n,mS_n,mα^*nα^m/√(n!m!).
In order to fulfill Eq. (<ref>) the condition
ϵ√(1-v^2)e^|α|^2 = ∑_n,mS_n,mα^*nα^m/√(n!m!),
should be fulfilled for some tensor S_n,m.
Observe that ϵ√(1-v^2)e^|α|^2=ϵ√(1-v^2)e^v^2β for
a constant β. This is a holomorphic function, except at
v=1 because it is not differentiable. Therefore, there exists a Maclaurin series for v<1.
ϵ√(1-v^2)e^v^2β= ∑_n=0^∞a_nv^n.
Therefore, we can always find
an operator Ŝ which is diagonal in the Harmonic oscillator basis such that
∑_n=0^∞ S_n,n|α|^2n/n!=ϵ√(1-v^2)e^v^2β.
However, there can be more combinations as α∝ v, so we can have
a more general hermitian operator Ŝ that fulfills Eq. (<ref>).
We have thus shown the existence of QTRs where time dilates according to
Lorentz transformations. We have done it for the special case of the Harmonic Oscillator.
The ground state of a Harmonic Oscillator looks like a nontrivial coherent
state in a different reference frame that moves at constant speed v with respect to
the one where it is measured. Time dilation would thus be a phenomenon caused by the
distortion of states in different reference frames.
Can this be elevated to a fundamental mechanism?
Observe that time dilation is something usually associated with spacetime as a
field, outside matter itself. In this approach, any quantum observable that depends on time
would have to behave in such a way that its time coordinate is dilated.
In order to elevate this to a fundamental mechanism we would need to create a theory
where all observables that depend on time are sensitive to the distortion that
comes from changing from reference frame. Moreover, the expectation values would have
to change according to Lorentz transformations, i.e. f(t)→ f(t/γ).
This all stems from a fundamental change of perspective: we consider the acquisition
of information as the source of time: change produces time.
In other approaches, time is an intrinsic property
of the universe that would cause the change: exactly the opposite.
The latter is the perspective of Pauli or
the relational physics approach <cit.>.
We are proposing here a deviation from the usual spacetime picture.
We propose that the mixture happens in the interaction:
time measured from a source that comes from another frame would become distorted.
This has an advantage in the sense that it provides a clear explanation of spacetime
from the perspective of Quantum Mechanics. However, the long-range aim of these inquiries is
to provide a coherent theory of Quantum Gravitation. In the known theory of General
Relativity the concept of spacetime as a field is fundamental. The main question
that stems from this research is how to build a consistent theory of Gravitation
without using spacetime as a field. Can General Relativity be rebuilt from Quantum systems
without the spacetime field? There is reason to believe it can be done, as the
equivalence principle is independent of spacetime. In other words,
the statement that the laws of physics in a coordinate system in
uniform acceleration and a stationary coordinate system in a homogeneous
gravitational field are equivalent does not imply that physics happens
in 4-dimensional spacetime.
|
http://arxiv.org/abs/2307.06195v1 | 20230712143440 | Relativistic second-order viscous hydrodynamics from kinetic theory with extended relaxation-time approximation | [
"Dipika Dash",
"Sunil Jaiswal",
"Samapan Bhadury",
"Amaresh Jaiswal"
] | nucl-th | [
"nucl-th",
"hep-ph",
"hep-th"
] |
[email protected]
School of Physical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Jatni 752050, India
[email protected]
Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400005, India
Department of Physics, The Ohio State University, Columbus, Ohio 43210-1117, USA
[email protected]
Institute of Theoretical Physics, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Krakow, Poland
[email protected]
School of Physical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Jatni 752050, India
We use the extended relaxation time approximation (ERTA) for the collision kernel, which incorporates a particle-energy dependent relaxation time, to derive second-order viscous hydrodynamics from the Boltzmann equation for a system of massless particles. The resulting transport coefficients are found to be sensitive to the energy dependence of the relaxation time and have significant influence on the fluid's evolution. Using the derived hydrodynamic equations, we study the evolution of a fluid undergoing (0+1)-dimensional expansion with Bjorken symmetry and investigate the fixed point structure inherent in the equations. Further, by employing a power law parametrization to describe the energy dependence of the relaxation time, we successfully reproduce the stable free-streaming fixed point for a specific power of the energy dependence. The impact of the energy-dependent relaxation time on the processes of isotropization and thermalization of an expanding plasma is discussed.
Relativistic second-order viscous hydrodynamics from kinetic theory with extended relaxation-time approximation
Amaresh Jaiswal
August 12, 2023
===============================================================================================================
§ INTRODUCTION
The relativistic Boltzmann equation is a transport equation that governs the space-time evolution of the single-particle phase-space distribution function. It is capable of accurately describing the collective dynamics of the system in the limit of small mean free path and therefore has been employed extensively to formulate the theory of relativistic hydrodynamics <cit.>. However, solving the Boltzmann equation directly is challenging due to the complicated integro-differential nature of the collision term, which involves the integral of the product of distribution functions. Over several decades, various approximations have been proposed to simplify the collision term in the linearized regime. In 1969, following earlier works by Bhatnagar-Gross-Krook <cit.> and Welander <cit.>, Marle introduced a relaxation time approximation for non-relativistic systems <cit.>. However, Marle's version was not applicable to massless particles and was ill-defined in the relativistic limit. Anderson and Witting resolved these issues by generalizing Marle's model to the relativistic regime, qualitatively recovering the results obtained using Grad's method of moments in the relativistic limit <cit.>. These models, introduced by Marle and Anderson-Witting, incorporate a collision time scale known as the relaxation time. The Anderson-Witting model requires the relaxation time to be independent of particle momenta, making it straightforward to apply in the formulation of relativistic dissipative hydrodynamics.
The Anderson-Witting model achieves enormous simplification by approximating that collisions drive the system towards local equilibrium exponentially without explicitly describing the interaction mechanism of the microscopic constituents. This approximation provides a highly accurate description of the collective dynamics for systems close to equilibrium. In the following, we will refer to the Anderson-Witting model as the relaxation-time approximation (RTA). Despite its simplistic nature, RTA and its variations have proven to be immensely useful and have been extensively employed in formulating relativistic dissipative hydrodynamics as well as in deriving transport coefficients <cit.>. Recently, it has also been applied to study the domain of applicability of hydrodynamics <cit.>. This simple model appears to capture effective microscopic interactions across a wide range of theories.
When deriving dissipative hydrodynamic equations from kinetic theory using the RTA approximation, it is typically assumed that the relaxation time is independent of particle energy (or momentum). Additionally, one is constrained to work in the Landau frame to ensure the preservation of macroscopic conservation laws. However, in realistic systems, the collision time scale generally depends on the microscopic interactions <cit.>. Introducing an energy-dependent relaxation time leads to a violation of microscopic conservation laws in the Landau frame. As a result, there has been considerable interest in developing a consistent formulation of relativistic dissipative hydrodynamics with an energy-dependent relaxation-time approximation for the Boltzmann equation that satisfies both microscopic and macroscopic conservation laws <cit.>.
Relativistic viscous hydrodynamics-based multistage dynamical models have demonstrated success in accurately describing a broad spectrum of soft hadronic observables in heavy-ion collisions <cit.>. The hydrodynamics stage of the evolution encompasses the deconfined quarks and gluons regime at high temperatures, the phase transition, and the hadron gas phase <cit.>. The dynamical properties of the evolving non-equilibrium nuclear matter are governed by a set of transport coefficients, such as the shear and bulk viscosities <cit.>. These transport coefficients play a crucial role in explaining the hadronic observables in heavy-ion collisions. Thus, a major goal of heavy-ion phenomenology is to extract the temperature dependence of these transport coefficients for the evolving nuclear matter, and considerable efforts have been made to determine these coefficients from various aspects. Most phenomenological studies adopt parameterized forms for the shear and bulk viscosities <cit.>. Recent studies have employed Bayesian methods to obtain these parameters and have provided bounds on the transport coefficients <cit.>. However, since these parameterizations do not stem from microscopic considerations, the predictability of such models is limited. Additionally, the second-order transport coefficients utilized in these hydrodynamic models are obtained for specific interactions, and, as a result, they may fail to accurately capture the system's behavior during its evolution.
In our recent work <cit.>, we presented a framework for the consistent derivation of relativistic dissipative hydrodynamics from the Boltzmann equation, incorporating a particle energy-dependent relaxation time by extending the Anderson-Witting relaxation-time approximation[
In a recent work <cit.>, the transseries structure of ERTA was explored.].
Within this extended RTA (ERTA) framework, we derived the first-order hydrodynamic equations and demonstrated that the hydrodynamic transport coefficients can exhibit significant variations with the energy dependence of the relaxation time. Notably, the ERTA framework allows for the adjustment of interaction characteristics by tuning the energy dependence of the relaxation time, enabling a partial description of the transition from deconfined quark-gluon plasma at high temperatures to a weakly interacting gas of hadrons at lower temperatures. While the formulation presented in Ref. <cit.> successfully incorporates an energy-dependent relaxation time into the RTA, it still suffers from the well-known issue of acausality in first-order relativistic hydrodynamics within the Landau frame <cit.>. Consequently, there is a need for a second-order theory that addresses this issue <cit.>, allowing for its application in heavy-ion collision simulations.
In the present study, we employ the ERTA framework to derive second-order hydrodynamic equations in the Landau frame for a conformal system without conserved charges, incorporating an energy-dependent relaxation time. The second-order transport coefficients are found to be sensitive to the energy dependence of the relaxation time. We focus on a boost-invariant flow in (0+1) dimensions and investigate the fixed point structure of the hydrodynamic equations. Our analysis reveals that the location of the free-streaming fixed points is influenced by the energy dependence of the relaxation time. By employing a power law parametrization to describe this energy dependence, we successfully reproduce the stable free-streaming fixed point for a specific power of the energy dependence. Furthermore, we explore the impact of the energy-dependent relaxation time on the processes of isotropization and thermalization of a boost invariant expanding plasma.
This paper is organized as follows: In Sec. <ref> we review the basic hydrodynamic equations for a conformal, chargeless fluid. Sec. <ref> briefly summarizes the results of Ref. <cit.> and outlines the steps necessary to derive second-order hydrodynamic equations, which we present in Sec. <ref>. Appendix <ref> contains the derivation of the results stated in Sec. <ref>. In Sec. <ref>, we consider Bjorken flow and study the effect of the energy dependence of the relaxation time on systems' thermalization. We summarize our results in Sec. <ref>.
§ OVERVIEW
The energy-momentum tensor for a system of massless particles with no net conserved charge can be expressed in terms of the single-particle phase–space distribution function, f(x,p), as
T^μν = ∫dP p^μ p^ν f = u^μ u^ν -Δ^μν + π^μν ,
where dP = d^3 p⃗/[(2π)^3 E_p] is the invariant momentum-space integration measure with E_p representing the particle energy which is equal to the magnitude of the particle three-momenta for massless particles, E_p=|p|. The projection operator Δ^μν=g^μν-u^μ u^ν is orthogonal to the hydrodynamic four-velocity u^μ defined in the Landau frame: u_ν T^μν= u^μ, where is the energy density. In the above equation, 𝒫 is the thermodynamic pressure and π^μν is the shear viscous stress. We work with the Minkowskian metric tensor g^μν≡diag(+,-,-,-).
The energy-momentum conservation ∂_μ T^μν =0 yields the fundamental evolution equations for and u^μ as,
+ (+)θ - π^μνσ_μν = 0,
(+)u̇^α - ∇^α + Δ^α_ν∂_μπ^μν = 0,
Here we use the standard notation Ȧ=u^μ∂_μ A for co-moving derivatives, θ≡∂_μ u^μ for the expansion scalar, σ^μν≡1/2(∇^μ
u^ν+∇^ν u^μ)-1/3θΔ^μν for the velocity stress tensor, and ∇^α=Δ^μα∂_μ for space-like derivatives.
We consider the equilibrium momentum distribution function to have the Maxwell-Boltzmann distribution in the local rest frame of the fluid, f_ eq=exp[-(u· p)/T ]. The equilibrium energy density then takes the form,
_0 = u_μ u_ν∫dP p^μ p^ν f_ eq = 3T^4/π^2.
For an out-of-equilibrium system, the temperature T is an auxiliary quantity which we define using the matching condition ≡_0.
Also, the thermodynamic pressure and entropy density are given by,
= -1/3Δ_μν∫dP p^μ p^ν f_ eq = T^4/π^2 ,
S = +/T =4T^3/π^2 .
The evolution of temperature is obtained from the hydrodynamic equations of motion (<ref>) and (<ref>),
β̇ =βθ/3-β/3(+)π^μνσ_μν ,
∇^μβ = -βu̇^μ-β/+Δ^μ_α∂_νπ^αν ,
where β=1/T.
The non-equilibrium phase-space distribution function can be written as f=f_ eq+Δ f, where Δ f represents the out-of-equilibrium correction to the distribution function. Using Eq. (<ref>) the shear stress tensor π^μν can be expressed in terms of Δ f as
π^μν = Δ^μν_αβ∫dP p^α p^β Δ f ,
where Δ^μν_αβ≡1/2(Δ^μ_αΔ^ν_β + Δ^μ_βΔ^ν_α) - 1/3Δ^μνΔ_αβ is a doubly symmetric and traceless projection operator orthogonal to u^μ as well as Δ^μν. The evolution of the shear stress tensor depends on the evolution of the distribution function. In this work, we consider the evolution of the distribution function to be governed by the Boltzmann equation with the collision term, C[f], in the Extended Relaxation Time Approximation (ERTA) <cit.>,
p^μ∂_μ f = C[f] = - (u· p)/(x,p)(f - f_ eq^*) ,
where the relaxation time, (x,p), may depend on the particle momenta. The equilibrium distribution function is considered to be of the Maxwell-Boltzmann form in the `thermodynamic frame', f_ eq^* = exp[-(u^* · p)/T^*]. Here the thermodynamic frame is defined to be the local rest frame of a time-like four vector u_μ^* which need not necessarily correspond to the hydrodynamic four-velocity u_μ, and T^* is the temperature in the local rest frame of u_μ^* (see Ref. <cit.> for a detailed discussion).
We briefly review the derivation of first-order shear stress from the above kinetic equation in the next section.
§ FIRST-ORDER HYDRODYNAMICS
We employ Chapman-Enskog-like expansion about hydrodynamic equilibrium
[We shall refer to f_ eq = exp[-(u· p)/T ] as the hydrodynamic equilibrium distribution function with u^μ being the fluid four-velocity and T the local fluid temperature in the local rest frame of u^μ.]
to iteratively solve the ERTA Boltzmann equation (<ref>),
f= f_ eq+ δ f_(1)+δ f_(2)+δ f_(3)+ ⋯
Here δ f_(i) represents the ith order gradient correction to the hydrodynamic equilibrium distribution function. The correction to the distribution function to the first order is
δ f_(1) = δ f^*_(1) +/Tp^μ p^ν/u· pσ_μν f_ eq ,
where we have replaced the derivatives of temperature with the derivatives of fluid velocity using Eqs. (<ref>,<ref>) consistently keeping terms till first order in gradients, and have defined δ f^* ≡ f_ eq^* - f_ eq. Defining T^* ≡ T+δ T and u^μ_* ≡ u^μ + δ u^μ, we obtain the first-order correction δ f_(1)^* by Taylor expanding f_ eq^* about u^μ and T,
δ f^*_(1) = (- (δ u · p)/T + (u· p) δ T/T^2)f_ eq .
Using equations (<ref>) and (<ref>), the quantities δ u^μ and δ T are obtained by imposing the Landau frame conditions, u_ν T^μν = u^μ, and the matching condition, =_0. We find that these quantities vanish for a system of massless and chargeless particles at first-order in gradients, and the resulting first-order correction is given by
δ f_(1) = τ_ R/Tp^μ p^ν/u· pσ_μν f_ eq .
It can be easily checked that the microscopic conservation of energy-momentum at first order holds by taking the first momentum-moment of the Boltzmann equation (<ref>) with f↦ f_(1)=f_ eq + δ f_(1),
∂_μ∫dP p^μ p^ν f_(1) = - ∫dP u· p/τ_ R p^ν(f_(1) - f_ eq^* )
∂_μ T^μν_(1) = - σ_αβ/T∫dP p^ν p^α p^β f_ eq =0 .
Using δ f_(1) obtained in Eq. (<ref>), the expression of shear stress tensor from the definition (<ref>) is obtained to be <cit.>,
π^μν = 2ησ^μν,
where η = K_3,2/T is the coefficient of shear viscosity. We have defined the integrals
K_n,q≡1/(2q+1)!!∫dP τ_ R(x,p) (u · p)^n-2q (Δ_αβ p^α p^β)^q f_ eq .
We will now derive the second-order constitutive relation (and evolution equation) for the shear stress tensor in the next section.
§ SECOND-ORDER HYDRODYNAMICS
The non-equilibrium correction to the distribution function till second order can be written as
f= f_ eq + δ f_(1) + δ f_(2) + 𝒪(∂^3) = f_ eq + Δ f_(2) + 𝒪(∂^3),
where we define Δ f_(2)≡δ f_(1) + δ f_(2) representing the non-equilibrium correction till second order. Using the kinetic equation (<ref>), and employing the Chapman-Enskog expansion, we obtain Δ f_(2) as,
Δ f_(2) = Δ f^*_(2) - /u· p p^μ∂_μδ f^*_(1) - /u· p p^μ∂_μ f_ eq
+ /u· p p^μ p^ν∂_μ( /u· p∂_ν f_ eq) .
Here δ f^*_(1) is out-of-equilibrium correction at first order.
As discussed in the previous section, the first-order contribution of δ u^μ and δ T vanishes, and therefore they have contributions starting from second-order. Keeping terms till second-order in gradients, the first term on the right-hand side (r.h.s.) of the above equation is,
Δ f^*_(2) = (-(δ u · p)/T + (u· p)δ T/T^2) f_ eq.
The second term on r.h.s. of Eq. (<ref>),
-/u· p p^μ∂_μδ f^*_(1) = 𝒪(∂^3),
has correction starting from third-order in gradients because it involves derivatives of δ u^μ and δ T which are at least second-order. The third term on the r.h.s. simplifies to,
-/u· p p^μ∂_μ f_ eq = /T[p^μ p^ν/u· pσ_μν -
4/3u· p/+π^μνσ_μν.
.- 1/+( p^μ∇_νπ^ν_μ
- p^μπ^ν_μu̇_ν) ] f_ eq .
In deriving, we have kept all terms till second order when replacing derivatives of temperature with derivatives of fluid velocity using Eqs. (<ref>) and (<ref>). The last term on the r.h.s. of Eq. (<ref>) is given by,
/u· p p^μ p^ν∂_μ(/u· p∂_ν f_ eq) =
- /T[ τ̇_ Rp^μ p^ν/u· pσ_μν
+ (∇_α) p^α p^μ p^ν/(u· p)^2σ_μν] f_ eq
- ^2/T[ 2θ/3p^μ p^ν/u· pσ_μν
+p^μ p^ν/u· pσ̇_μν.
. + p^α p^μ p^ν/(u· p)^2 (∇_ασ_μν)
-2 p^α p^μ p^ν/(u· p)^2σ_μνu̇_α
- (1/T+1/u· p)(p^μ p^νσ_μν)^2/(u· p)^2] f_ eq .
Therefore, the complete non-equilibrium correction till second order from Eqs. (<ref>)-(<ref>) is given by,
Δ f_(2)= [(u· p)δ T/T^2-(δ u · p)/T-/T{(τ̇_ R-1) p^μ p^ν/u· pσ_μν+(∇_α)p^μ p^ν p^α/(u· p)^2σ_μν+1/+(4/3 (u· p) π^μνσ_μν+ p^μ∇_νπ^ν_μ- p^μπ^ν_μu̇_ν) }.
- ^2/T{2 p^μ p^ν/3(u· p)θσ_μν
+p^μ p^ν/u· pσ̇_μν
+p^α p^μ p^ν/(u· p)^2 (∇_ασ_μν) .
- . . 2 p^α p^μ p^ν/(u· p)^2σ_μνu̇_α-(1/T+1/u· p)(p^μ p^νσ_μν)^2/(u· p)^2}] f_ eq .
§.§ Imposing Landau frame conditions
We note that the second-order correction to the equilibrium distribution function given by Eq. (<ref>) has the undetermined quantities δ u^μ and δ T. We determine these by imposing the Landau frame condition (u_μ T^μν = u^ν) and matching condition ( = _0) with f↦ f_(2)= f_ eq + Δ f_(2) (see Appendix <ref> for derivation),
δ u^μ = 5 K_3,2/T(+)^2(π^μνu̇_ν - ∇_νπ^μν -π^αβσ_αβ u^μ)
+ 2L_3,2/T(+)(2σ^μνu̇_ν + ∇_νσ^μν + σ^αβσ_αβ u^μ) ,
δ T = 5/3K_3,2/(+)^2π^μνσ_μν
+ 1/+( L_3,2 -L_4,2/3 T)σ^μνσ_μν .
The L_n,q integrals appearing in the above expressions are defined as,
L_n,q≡1/(2q+1)!!∫dP τ_ R^2(x,p) (u · p)^n-2q (Δ_αβ p^α p^β)^q f_ eq .
In the derivation, we have used the relation between the integrals,
X_n,q= -(1/2q+1) X_n,q-1 ,
which holds for all integrals defined in this article. We note that when the relaxation time does not depend on particle energy, the ERTA approximation of the collision term reduces to the Anderson-Witting RTA approximation, and consequently δ u^μ and δ T vanishes (see Appendix <ref>).
§.§ Verification of microscopic conservation
To verify microscopic energy-momentum conservation up to the second order, we show that the first momentum-moment of the collision kernel is at least third-order in gradients. To this end, we consider the first moment of the collision kernel in the Boltzmann equation (<ref>) and substitute f↦ f_(2) = f_ eq + Δ f_(2), where Δ f_(2) is given by Eq. (<ref>),
∫dP p^νC[f] = -∫dP (u · p)/τ_ R p^ν(Δ f_(2)-Δ f^*_(2)) .
Using the expression of Δ f_(2)-Δ f^*_(2) from Eq. (<ref>),
∫dP p^νC[f] =∫dP p^μ p^ν∂_μ f_ eq-∫dP p^ν p^α p^β∂_α(/u· p∂_β f_ eq).
The first term in the right-hand-side of the above equations is simplified as
∫dP p^μ p^ν∂_μ f_ eq = u̇_μπ^μν -∇_μπ^μν .
Similarly, the second term is simplified as
∫dP p^ν p^α p^β∂_α(/u· p∂_β f_ eq) = ∂_α∫dP/u· p p^ν p^α p^β∂_β f_ eq
= ∂_μ( 2 K_3,2/Tσ^μν) +𝒪(δ^3)
= u̇_μπ^μν-∇_μπ^μν+ 𝒪(δ^3).
In the last step, we have used the first-order constitutive relation (<ref>). Using Eqs. (<ref>) and (<ref>) in Eq. (<ref>), we obtain
∫dP p^νC[f] = 𝒪(δ^3).
This demonstrates the preservation of microscopic energy-momentum conservation up to second order. It is noteworthy that δ u^μ and δ T did not appear in the equations during the verification of microscopic conservation. This outcome is specific to the case of massless and chargeless particles and does not happen in general. The contribution from these quantities becomes essential to ensure the conservation of energy-momentum and net current in systems involving massive or charged particles.
§.§ Shear stress till second-order
The expression for shear stress tensor till second order in terms of the hydrodynamic fields is obtained by integrating Δ f_(2) in definition (<ref>),
π^μν = 2 ησ^μν - 2 ητ_π( σ̇^⟨μν⟩ + 1/3σ^μνθ) - 4/7L_4,2/T^2σ_γ^⟨μσ^ν⟩γ
+ 4 L_3,2/Tσ_γ^⟨μω^ν⟩γ ,
where η = K_3,2/T is the first order transport coefficient and we have defined τ_π≡ L_3,2/K_3,2. The equation presented above is consistent with the one derived in Ref. <cit.> under the assumption of conformal symmetry. It is worth noting that the above equation retains its conformal invariance regardless of the specific functional dependence of the relaxation time on the particle energy.
One can rewrite Eq. (<ref>) as a relaxation-type equation for the evolution of shear stress tensor by replacing σ^μν→π^μν/(2 K_3,2/T)[
In deriving, we used the relation
σ̇^⟨μν⟩ = π̇^⟨μν⟩/2 η -(T K_3,2+ Q_3,2- K_4,2/3 η T^2) σ^μνθ ,
where the Q_n,q integral is defined as
Q_n,q≡1/(2q+1)!!∫dP∂τ_ R/∂β(u · p)^n-2q (Δ_αβ p^α p^β)^q f_ eq .
Further, we used the relation, Q_n,q = K_n+1,q -(n+1) T K_n,q , and expressed Q_n,q in terms of K_n,q integral.
],
π̇^⟨μν⟩ +π^μν/τ_π = 2 β_πσ^μν-4/3π^μνθ + 2 π_γ^⟨μω^ν⟩γ- 𝒞π_γ^⟨μσ^ν⟩γ ,
where β_π≡η/τ_π = (K_3,2)^2/T L_3,2 and 𝒞≡2/7L_4,2/T L_3,2. It is straightforward to verify that when the relaxation time is independent of particle energies, τ_π→, β_π→ (+)/5, and 𝒞→ 10/7, which agrees with the previous results <cit.>. It is interesting to note that there is one new integral K_3,2 (corresponding to η) in first-order, and two new integrals, L_3,2 and L_4,2 (corresponding to τ_π and 𝒞, respectively), in the second order.
As an illustration, we shall consider the following parametrization of the relaxation time <cit.>,
(x,p) = τ_ eq(x) (u· p/T)^ℓ,
where τ_ eq(x) represents the particle energy-independent part of relaxation time and scales as 1/T for conformal systems. We consider τ_ eq(x) = κ/T, where κ is a dimensionless constant. Note that the exponents ℓ may depend on the space-time coordinates.
With this parametrization, the coefficient of shear viscosity is obtained to be <cit.>,
η = K_3,2/T = 4κ T^3/5π^2[Γ(5+ℓ)/24] ,
iff ℓ >-5.
Also, the coefficients τ_π, β_π, and 𝒞 appearing in Eq. (<ref>) can be determined analytically to have the form,
β_π ≡(K_3,2)^2/T L_3,2 = 4 T^4/5π^2[Γ (ℓ+5)^2/24 Γ (2 ℓ+5)] ,
τ_π ≡L_3,2/K_3,2 = κ/T[ Γ (2 ℓ+5)/Γ (ℓ+5)] ,
𝒞 ≡2/7L_4,2/T L_3,2 = 10+4ℓ/7,
with the condition ℓ >-5/2. The above results will be employed in the next section to study the evolution of a plasma undergoing boost-invariant expansion.
§ BJORKEN FLOW
We shall now study the hydrodynamic equation obtained for a fluid undergoing Bjorken expansion <cit.>. Bjorken symmetries enforce translational and rotational symmetry in the transverse (x,y) plane, boost invariance along the z (longitudinal) direction, and reflection symmetry z → -z. The symmetries are manifest in Milne coordinate system (τ, x, y, η_s), where τ= √(t^2 - z^2) is the proper time and η_s=tanh^-1(z/t) the space-time rapidity. In these coordinates the fluid appears to be static, u^μ = (1,0,0,0), irrotational (ω^μν = 0) and unaccelerated (u̇^μ = 0), but has a non-zero local expansion rate, θ=1/τ. Symmetries further constrain the shear tensor to be diagonal and space-like in Milne coordinates, leaving only one independent component which we take to be the η_sη_s component: π^xx=π^yy= -τ^2 π^η_sη_s/2 ≡π/2.
The hydrodynamic equations for evolution of energy density (<ref>) and the shear tensor (<ref>) in Milne coordinates takes the form,
d/dτ =-1/τ(+-π) ,
dπ/dτ =-π/τ_π + 4/3β_π/τ - (4+𝒞/3) π/τ .
The above equations can be transformed into an equation for the quantity <cit.>,
g ≡τ/∂/∂τ = π/ - 4/3 .
When the energy density exhibits power law behavior, g corresponds to the exponent of that specific power law (i.e., if ∼τ^a, then g=a). Equations (<ref>) and (<ref>) can be written as a non-linear, first-order, differential equation in g as,
-ℬ(g) = g^2 + (8+𝒞/3) g
+4/3(4/3+𝒞/3-β_π/)
+ τ/τ_π( g +4/3),
where we have defined ℬ(g) ≡τ (d g/dτ). Note that 𝒞 and β_π/ are dimensionless.
The hydrodynamic regime is reached when the scattering rate exceeds the expansion rate, i.e., τ_π≪τ. The last term in the above equation is dominant in this regime, and the hydrodynamic fixed point g_* is given by:
g_* = - 4/3 .
In the collisionless regime, the expansion rate far exceeds the scattering rate (τ_π≫τ), and the function ℬ(g) in Eq. (<ref>) is dominated by the terms that do not depend on τ/τ_π. The zeros of this function correspond to the free-streaming fixed points,
g_ fp = -(8+𝒞/6) ±√(4β_π/3 + 𝒞^2/36) ,
with the positive root corresponding to the free-streaming stable fixed point. For a plasma undergoing Bjorken expansion, it has been shown that the stable free-streaming fixed point of the exact kinetic solution corresponds to vanishing longitudinal pressure, or g=-1 <cit.>[
Although this was shown for the RTA Boltzmann equation, it holds true even for the ERTA case since the collision term vanishes in free-streaming.].
Using the parametrization (<ref>) for the relaxation time and the corresponding values of the transport coefficients given in Eqs. (<ref>) and (<ref>), we observe that the value of the stable fixed point in the exact kinetic equation (g=-1) can be recovered from Eq. (<ref>) for ℓ≈ 0.763. Extending the domain of Israel-Stewart-type hydrodynamic theories requires the hydrodynamic equations to accurately capture the location of the stable free-streaming fixed point, as emphasized in Ref. <cit.>. Therefore, the evolution Eq. (<ref>), or analogously Eqs. (<ref>) and (<ref>), is expected to provide a good description of the underlying weakly coupled microscopic theory with ℓ≈ 0.763, even in far-off-equilibrium regimes. It is worth mentioning that this value of ℓ is not arbitrary; many microscopic theories lie in the range ℓ =[0,1] <cit.>.
To illustrate the impact of the ERTA framework, we show the comparison of the second-order hydrodynamic equations obtained from the RTA approximation (ℓ =0) with those derived from the ERTA approximation setting ℓ =0.5 and compare them with BAMPS results <cit.> in Fig <ref>. The initial temperature is set to be 500MeV at an initial time of 0.4fm/c with a vanishing initial shear stress. Further, we fix the values of κ appearing in Eqs. (<ref>) and (<ref>) such that η/s is set to different values as mentioned in the figure[
For RTA approximation (ℓ=0), κ = {1/4, 1, 2, 15} for η/s = {0.05, 0.2, 0.4, 3.0}, respectively. Similarly for ERTA approximation with ℓ=0.5, κ = {0.11, 0.46, 0.92, 6.9} for η/s = {0.05, 0.2, 0.4, 3.0}, respectively.].
As can be seen from the figure, the solid curves representing the ERTA approximation with ℓ =0.5 are in an overall better agreement with the BAMPS solution than the dashed curves <cit.>[
We note that the second-order hydrodynamic equations obtained from the RTA approximation (dashed curve) perform better than the one obtained from ERTA approximation (solid curve) for η/s=0.2.
].
In Figure <ref>, we present the evolution of three quantities: g, the pressure anisotropy P_L/P_T = (P-π)/(P+π/2), and the Knudsen number τ_π/τ. The initial temperature is set to be 500MeV at an initial time of 0.4fm/c. Additionally, we consider the parameter κ appearing in Eqs. (<ref>) and (<ref>) to have the value 5/(4π)[
The value κ= 5/(4π) implies η/s=1/(4π) when the relaxation time is independent of the particle energies (ℓ=0), i.e. when ERTA reduces to Anderson-Witting RTA.].
In all three panels, the solid curves represent cases where P_L is initialized at 0, corresponding to π=/3, while the dashed curves are initialized with a vanishing initial shear stress, π =0. The blue, orange, green, and red curves correspond to different values of ℓ = -1, 0, 1/2, 1, respectively.
The left panel displays the evolution of the quantity g, with the gray dashed line representing the hydrodynamic fixed point g_*. It can be observed from the systematic trend of blue, orange, green, and red curves that the system remains out of equilibrium for a longer duration as the values of ℓ are increased. This feature is also visible in the middle panel where the evolution of pressure anisotropy, P_L/P_T is shown – approach to P_L/P_T =1 is delayed for the orange, green, and red curves compared to the blue curve, indicating a slower isotropization. Also, in the left and middle panels we observe that the solid and dashed curves, representing different initial shear stress, overlap earlier for smaller values of ℓ. Interestingly, the evolution of Knudsen number shown in the right panel is not strongly dependent on the initial values of shear stress but has a strong dependence on the strength of the momentum-dependence of the relaxation time i.e. on ℓ; the solid and dashed curves largely overlap during the entire evolution. It is worth noting that increasing the value of ℓ enhances the initial gradient strength (as τ_π increases), and smaller values of ℓ drive the system towards thermalization at a faster rate, which is evident from the middle panel.
In Figure <ref>, we show the evolution of the temperature normalized with the ideal temperature evolution, T_ id = T_ in (τ_ in/τ)^1/3. It is observed that at a given time, the fluid maintains a higher temperature when the initial shear stress has a large positive value (solid curves), in contrast to when the initial shear stress is vanishing (dashed curves). An interesting observation is that increasing values of ℓ also lead to higher temperatures of the medium, as indicated by the trend of the differently colored curves. This may be understood from the right panel of Fig. <ref>, where we observe that an increase in the values of ℓ results in a larger Knudsen number. Consequently, this leads to increased dissipation, resulting in a slower fall of temperature compared to ideal evolution. Moreover, the interplay between the initial conditions for shear stress and the various medium interactions (characterized by different values of ℓ) is intriguing, and can provide insights towards constraining the initial conditions for hydrodynamic simulation of heavy-ion collisions. Further, the various curves in Fig. <ref> crossing the temperature surface of 155MeV (represented by the black dotted curve) at different proper times suggests that a constant temperature particlization surface can be reached at different times with varying anisotropies. This can be seen more clearly in Fig. <ref>, where the evolution of the pressure anisotropy P_L/P_T with τ/τ_π is shown. The evolution of the curves is stopped when the temperature of the plasma reaches 155MeV during the expansion (at times when the different curves cross the black dotted curve in Fig. <ref>). In Fig. <ref>, we see that the pressure anisotropy across the different colored curves differs significantly. We also note that the evolution of P_L/P_T for different curves in the near-equilibrium regime (τ≳ 5τ_π) is nearly the same, but differs substantially in the far-off-equilibrium regime.
§ SUMMARY AND OUTLOOK
To summarize, we have derived relativistic second-order hydrodynamics from the Boltzmann equation using the extended relaxation time approximation for the collision kernel, incorporating an energy-dependent relaxation time. The transport coefficients are shown to explicitly depend on the microscopic relaxation rate. We investigated the fixed point structure of the hydrodynamic equations for a plasma undergoing Bjorken flow and showed that the location of the free-streaming fixed points depends on the energy dependence of the relaxation time. Additionally, we employed a power law parametrization to describe the energy dependence of the relaxation time and examined its impact on the thermalization process of the expanding plasma. We demonstrated that the plasma's approach to equilibrium is affected by the relaxation time's dependence on different powers of energy; the plasma remains in the out-off-equilibrium regime and at a higher temperature for longer duration as larger positive values of ℓ are considered.
While the derivation in the present article is done for a conformal system without conserved charges, it can be extended for non-conformal systems with conserved charges and quantum statistics by following the steps outlined in the article. It is also desired to have typical relaxation rates for the energy dependence of the relaxation time across different stages of the evolution of the nuclear matter formed in heavy-ion collisions. Such parameterization of the relaxation time can have parameters which may depend, for example, on the temperature of the medium[
It remains to be explored if some of the essential features of a strongly coupled fluid can be captured in this framework by parameterizing the relaxation time.
]. Incorporating these rates will make the full hydrodynamic equations with the associated transport coefficients more suitable for a (3+1)-dimensional hydrodynamic simulation. It should be noted that the functional form of the first-order transport coefficients, such as η, is determined within the framework. Furthermore, such an analysis may also provide insights into the form of distribution function at particlization. These aspects will be investigated in future studies.
S.J. thanks Richard J. Furnstahl and Ulrich Heinz for insightful comments and suggestions on the manuscript. He acknowledges the kind hospitality of NISER, Bhubaneswar where part of this work was done. A.J. is supported in part by the DST-INSPIRE faculty award under Grant No. DST/INSPIRE/04/2017/000038. S.J. is supported by the NSF CSSI program under award no. OAC-2004601 (BAND Collaboration).
§ DERIVATION OF
In this Appendix, we obtain δ u^μ and δ T by imposing Landau frame and matching conditions. The Landau Frame condition, u_μ T^μν = u^ν, with the matching = _0, for a non-equilibrium distribution f=f_ eq + δ f can be written as
u_μ∫dP p^μ p^ν (f_ eq +δ f) = u^ν∫dP (u· p)^2 f_ eq
u_μ∫dP p^μ p^νδ f = u^ν u_α u_β I^αβ - u_μ I^μν = 0.
where I^αβ = ∫dP p^α p^β f_ eq. Replacing δ f ↦δ f_(2) obtained in Eq. (<ref>), and performing the integrals in the local rest frame of u^μ, the above equation reduces to
I_3,1δ u^μ - I_3,0δ T/T u^μ + 5 K_3,2/(+)(π^μνu̇_ν - ∇_νπ^μν)
+ (6 L_3,2 + 2 M_4,2 - 2 N_3,2/T) σ^μνu̇_ν + 2L_3,2∇_νσ^μν
+ ( 10 L_3,2 - 2L_4,2/T + 2M_4,2) σ^αβσ_αβ u^μ = 0 .
Note that the term δ u · u = (δ u·δ u)/2 ∼𝒪( δ^4), since δ u^μ is at least second order (see discussion in Section <ref>), and has been ignored in the derivation. Further, we have defined the thermodynamic integrals,
I_n,q ≡1/(2q+1)!!∫dP (u · p)^n-2q (Δ_αβ p^α p^β)^q f_ eq ,
M_n,q ≡1/(2q+1)!!∫dPτ_ R∂τ_ R/∂ (u· p) (u · p)^n-2q (Δ_αβ p^α p^β)^q f_ eq ,
N_n,q ≡1/(2q+1)!!∫dP τ_ R∂τ_ R/∂β (u· p)^n-2q (Δ_αβ p^α p^β)^q f_ eq .
We note that M_n,q and N_n,q integrals can be expressed in terms of the L_n,q integrals through the relations,
M_n,q = 1/2T L_n,q - n+1/2 L_n-1,q , iff n > -1 ,
2 N_n,q = L_n+1,q - n T L_n,q .
Using these relations, Eq. (<ref>) simplifies to,
I_3,1δ u^μ- I_3,0δ T/T u^μ+ 5 K_3,2/(+)( π^μνu̇_ν- ∇_νπ^μν) + 2 L_3,2
×( 2 σ^μνu̇_ν+∇_νσ^μν) + ( 5 L_3,2-L_4,2/T)σ^αβσ_αβ u^μ = 0 .
Similarly, using the matching condition:
u_μu_ν T^μν = _0
u_μ u_ν∫dP p^μ p^νδ f = 0 ,
and replacing δ f ↦δ f_(2) obtained in Eq. (<ref>), we obtain
- I_3,0δ T/T + 5 K_3,2/+π^μνσ_μν + ( 3 L_3,2 -L_4,2/T)σ^μνσ_μν = 0 .
Noting that I_3,0 = 3T (+) and solving for δ T, we obtain
δ T= 5/3K_3,2/(+)^2π^μνσ_μν
+ 1/+( L_3,2 -L_4,2/3 T)σ^μνσ_μν .
The expression for δ u^ν is obtained by using Eq. (<ref>) in Eq. (<ref>),
δ u^μ = 5 K_3,2/T(+)^2(π^μνu̇_ν - ∇_νπ^μν -π^αβσ_αβ u^μ)
+ 2L_3,2/T(+)(2σ^μνu̇_ν + ∇_νσ^μν + σ^αβσ_αβ u^μ) ,
where we have used the relation I_3,1 = -T (+).
In the case when the relaxation time is independent of the particle energies, the integrals K_n,q→ I_n,q and L_n,q→ I_n,q = K_n,q. Using these, and noting that = 5 η/(+), Eq. (<ref>) simplifies to
δ u^μ = 5 K_3,2/T(+)^2(π^μνu̇_ν - ∇_νπ^μν -π^αβσ_αβ u^μ)
+ 5 K_3,2/T(+)^2 2η(2σ^μνu̇_ν + ∇_νσ^μν + σ^αβσ_αβ u^μ)
= 5 K_3,2/T(+)^2(3 π^μνu̇_ν - ∇_νπ^μν + 2η∇_νσ^μν)
=0 .
In transitioning to the second equality, we employed the first-order relation π^μν = 2 ησ^μν. Furthermore, we used the relation ∇_νη = -3Tη∇_νβ = 3 ηu̇_ν in the last equality.
Similarly, Eq. (<ref>) reduces to
δ T = 1/+[5/3K_3,2/(+)π^μνσ_μν+ ^2( I_3,2-5/3 I_3,2)σ^μνσ_μν]
= 1/+[5/3K_3,2/(+)π^μνσ_μν - 5η/(+)2/3 K_3,2σ^μνσ_μν]
=0 .
The fact that δ u^μ and δ T vanish when the relaxation time is independent of particle energy is anticipated since ERTA reduces to the Anderson-Witting RTA, thus providing a consistency validation.
|
http://arxiv.org/abs/2307.03967v1 | 20230708124657 | End-to-End Supervised Multilabel Contrastive Learning | [
"Ahmad Sajedi",
"Samir Khaki",
"Konstantinos N. Plataniotis",
"Mahdi S. Hosseini"
] | cs.CV | [
"cs.CV"
] |
Impact of noise on inverse design: The case of NMR spectra matching
O. Anatole von Lilienfeld
August 12, 2023
===================================================================
Multilabel representation learning is recognized as a challenging problem that can be associated with either label dependencies between object categories or data-related issues such as the inherent imbalance of positive/negative samples. Recent advances address these challenges from model- and data-centric viewpoints. In model-centric, the label correlation is obtained by an external model designs (e.g., graph CNN) to incorporate an inductive bias for training. However, they fail to design an end-to-end training framework, leading to high computational complexity. On the contrary, in data-centric, the realistic nature of the dataset is considered for improving the classification while ignoring the label dependencies. In this paper, we propose a new end-to-end training framework–dubbed KMCL (Kernel-based Mutlilabel Contrastive Learning)–to address the shortcomings of both model- and data-centric designs. The KMCL first transforms the embedded features into a mixture of exponential kernels in Gaussian RKHS. It is then followed by encoding an objective loss that is comprised of (a) reconstruction loss to reconstruct kernel representation, (b) asymmetric classification loss to address the inherent imbalance problem, and (c) contrastive loss to capture label correlation. The KMCL models the uncertainty of the feature encoder while maintaining a low computational footprint. Extensive experiments are conducted on image classification tasks to showcase the consistent improvements of KMCL over the SOTA methods. PyTorch implementation is provided in <https://github.com/mahdihosseini/KMCL>.
§ INTRODUCTION
Learning from multilabel representation is a common practice that is considered in both computer vision <cit.> and medical image <cit.> application domains. Images usually contain more than one object for classification, where they can be semantically related to each other. The idea is to create an embedded feature space that can capture label dependencies to improve the classification task <cit.>. However, effectively learning such embedded space is known to be a challenging problem and various methods have been proposed over the past few years, including sequence-to-sequence modeling <cit.>, graph approaches <cit.>, and new loss-function designs <cit.>. Generally, there are two main approaches to addressing the multilabel representation learning problem: the data-centric approach and the model-centric approach. The data-centric approach focuses on addressing data-related issues like inherent imbalance <cit.>, impartial label training <cit.>, and hierarchical relationships <cit.> while ignoring label dependencies. On the contrary, the model-centric approach aims to capture label interactions for semantic embedding such as graph convolutional networks <cit.>, attention mechanisms <cit.>, and transformer-based learning <cit.>. Despite the benefits, they fail to design an end-to-end learning framework due to their high computational costs or the laborious task of capturing heuristic label dependencies like using correlation matrices. These limitations make them challenging to implement, optimize, and interpret.
In this paper, we aim to combine the benefits of both data-centric and model-centric approaches while addressing their potential drawbacks. The solution lays on the foundation of asymmetric loss <cit.> which tackles the imbalance between positive and negative samples in multilabel classification. Our design augments this loss function by capturing the semantic relationships between labels using a kernel-based contrastive loss. This is achieved through two steps: (a) leveraging a Kernel Mixture Module (KMM) to explore the epistemic uncertainty of the feature encoder (see Figs. <ref> and <ref>). This is done by converting the embedded features of multilabel images into a Gaussian Reproducing Kernel Hilbert Space (RKHS) ℋ, and (b) employing a contrastive learning framework on the Gaussian RKHS to capture label dependencies through a weighted loss-function design (see Fig. <ref>). The resulting loss is trainable from end-to-end, providing high numerical stability during training. The following summarizes the contribution of the paper:
[C1]: We propose a novel end-to-end framework –dubbed KMCL– to strike a balance between model-centric and data-centric approaches using a new contrastive loss augmented on asymmetric classification loss from <cit.>. KMCL is capable of capturing both the epistemic uncertainty of the model and label dependencies between classes simultaneously.
[C2]: We introduce a KMM block design within the KMCL framework to generate a mixture of exponential kernels in Gaussian RKHS to model the uncertainty of the feature encoder and improve the robustness of the classification task. To reconstruct the mixture kernels from data, we propose a loss function ℒ_REC (in Eq. <ref>) as an alternative to the negative log-likelihood loss that addresses the numerical instabilities mentioned in <cit.>.
[C3]: We construct the ℒ_KMCL (in Eq. <ref>) as a complementary loss to ℒ_ASL <cit.> to capture label dependencies and enhance classification performance. We utilize the Bhattacharyya coefficient (ρ) as a similarity metric between two kernel representations to pull together similar classes (positive) from a pair of multilabel images while contrasting dissimilar ones (negative) in Gaussian RKHS.
[C4]: We consistently improve classification performance on both computer vision and medical imaging tasks with low computational footprints. Our loss design yields robust behavior toward a range of hyperparameters that are fixed across all experiments.
§.§ Related Work
Multilabel Image Representation. Multilabel image representation problems have been extensively studied, focusing on exploiting label dependencies within semantically aware regions. Previous approaches include RNN-CNN models for sequence-to-sequence modeling <cit.>, transforming the problem into a multi-instance problem <cit.>, and using recurrent attention reinforcement learning <cit.>. Later, efforts were made to incorporate linguistic embedding of training labels into graph neural network designs <cit.>. However, graph-based approaches assume the presence of coexisting label dependencies, which may not hold true when labels co-occur infrequently. Attention mechanisms have been introduced in dynamic graph modeling networks to address this issue <cit.>. Despite their effectiveness, these approaches often result in complex models with heavy computational requirements and limited generalization in different domains. A residual attention mechanism was introduced <cit.> to reduce such complexities by augmenting independent class feature scores using a class-agnostic average pooling method for aggregation scoring. Recent developments in this field emphasize the realistic nature of multilabel data representation. For example, the design proposed in <cit.> introduces an asymmetric loss function to balance the frequency of positive and negative classes. Other approaches include class-aware loss design for impartial label training <cit.> and exploring hierarchical relationships of multilabel data in a contrastive learning framework <cit.>. In this paper, we leverage both data- and model-centric approaches to reduce the above-mentioned complexities.
Contrastive Learning. Self-supervised learning methods primarily focus on contrastive learning, which involves capturing inter-relational object information in image representation. This is achieved through the use of contrastive loss functions, either in unsupervised contrastive learning where labels are absent <cit.>, or in supervised contrastive learning where labels are available <cit.>. The framework has been extended to multilabel representation learning <cit.> by considering shared label images as positive and unshared label images as negative. The existing multilabel contrastive loss designs rely on hard-coded features and lack flexibility in representing semantically aware objects and their label dependencies. However, we propose transforming embedded features into a mixture of exponential kernels in Gaussian RKHS to account for the potential uncertainty of model parameters and accordingly relax the embeddings.
§ BACKGROUND ON BHATTACHARYYA COEFFICIENT BETWEEN EXPONENTIAL KERNELS
The Bhattacharyya coefficient is a widely used metric to measure the similarity between probability distributions in various fields, including computer vision, pattern recognition, and statistical analysis <cit.>. Normal distributions are commonly evaluated using this metric to determine class separability in transfer learning <cit.>, perform point cloud instance segmentation <cit.>, and employ pseudo-labels for semi-supervised classification <cit.>. However, the Gaussian probability may not always be the best option for estimating the target variable due to normality assumptions which leads to numerical instabilities such as singularity <cit.>. A mixture of exponential kernels can be used as a reliable alternative to estimate the relative likelihood of the target variable, especially when the distribution is unknown or multimodal. In such cases, the Bhattacharyya coefficient ρ between the normalized versions of the kernel components can assess the geometric similarity and degree of overlap. Compared to Kullback-Leibler divergence <cit.> or L_p norms, ρ takes values in the range of [0, 1], which makes it a practical choice for comparing two statistical samples. In the following remark, we will elaborate on the closed-form expression of ρ between two exponential kernels.
Let p(𝐱):= K_Σ_p(𝐱, μ_p) = exp(-1/2𝐱 - μ_p^2_Σ_p^-1) and q(𝐱) := K_Σ_q(𝐱, μ_q) = exp(-1/2𝐱 - μ_q^2_Σ_q^-1) be anisotropic multivariate squared exponential kernels that define a Gaussian RKHS ℋ <cit.>. Then, the Bhattacharyya coefficient between the normalized p(𝐱) and q(𝐱)
is:
ρ(p(𝐱), q(𝐱) ) = ∫(p(𝐱)∫p(𝐱) d𝐱)^1/2(q(𝐱)∫q(𝐱) d𝐱)^1/2d𝐱 = |Σ_p|^1/4|Σ_q|^1/4/|Σ|^1/2exp(-1/8μ_p-μ_q^2_Σ^-1),
where, μ_p-μ_q^2_Σ^-1 = (μ_p-μ_q)^TΣ^-1(μ_p-μ_q) and Σ = Σ_p+Σ_q/2. The μ_p, μ_q∈ℝ^M and Σ_p, Σ_q∈𝕊_++^M are the mean vectors and the covariance matrices, respectively, and the operation |·| represents the determinant of a matrix. The proof of Remark <ref> is provided in Supplementary material.
The Bhattacharyya coefficient, also known as the Hellinger affinity <cit.>, measures the normalized correlation between the square roots of kernels over the entire space. This similarity metric compares p(𝐱) and q(𝐱) by projecting their square roots onto a unit hypersphere and measuring the cosine of the angle between them in the complete inner product space ℋ.
A careful examination of Equation <ref> reveals that the Bhattacharyya coefficient between normalized p(𝐱) and q(𝐱) consists of two terms: a scale factor and an exponential component. The scale factor measures overlap by comparing the generalized variances of the kernels, which are determined by the determinant of their covariance matrices. The scale factor converges to one when the covariance matrices of the two kernels are similar, indicating an overlap between them. The generalized variance of a kernel is related to its entropy and power entropy <cit.>, which measure uncertainty and spread. This allows the scale factor to consider differences in information content and orientation, resulting in separability due to covariance dissimilarity. On the other hand, the second term measures the similarity between the means μ_p and μ_q weighted by the precision matrix Σ^-1, providing separability based on positional differences. This exponential component represents the Mahalanobis kernel similarity <cit.> between μ_p and μ_q with respect to Σ^-1. The following corollary will further elucidate the connection of the Bhattacharyya coefficient with the Mahalanobis and Gaussian similarities.
Let p(𝐱) := K_Σ_p(𝐱, μ_p) and q(𝐱) := K_Σ_q(𝐱, μ_q) be multivariate kernels defined in Remark <ref>. The Bhattacharyya coefficient between normalized p(𝐱) and q(𝐱) can be reduced to either the Mahalanobis or the RBF kernel similarity, depending on the covariance matrices:
(i) The Mahalanobis kernel similarity, Sim_M(p(𝐱), q(𝐱)), is obtained when the covariance matrices are homoscedastic, i.e., Σ_p = Σ_q = Σ. It has the following closed-form expression:
Sim_M(p(𝐱), q(𝐱)) = ρ(K_Σ(𝐱, μ_p), K_Σ(𝐱, μ_q)) = exp(-12 (2)^2μ_p-μ_q^2_Σ^-1).
The described Mahalanobis metric evaluates the similarity between p(𝐱) and q(𝐱) based on their mean difference and relative positions (see Fig. <ref>d).
(ii) The Gaussian kernel similarity, Sim_G(p(𝐱), q(𝐱)), is obtained when the covariance matrices are equal and isotropic, meaning Σ_p = Σ_q = σ^2I. The closed-form expression will be:
Sim_G(p(𝐱), q(𝐱)) = ρ(K_Σ(𝐱, μ_p), K_Σ(𝐱, μ_q)) = exp(-μ_p-μ_q^28σ^2).
In cases where two kernels have similar means but different covariance matrices, the Mahalanobis and Gaussian kernel similarities often exhibit a perfect correlation that may not precisely reflect true similarities (Figs. <ref>a and c). Instead, the Bhattacharyya coefficient evaluates the generalized variances of the kernels and identifies similarities in their orientation, shape, and means (Figs. <ref>a and c). Therefore, it is often a superior metric to the Mahalanobis and the Gaussian kernel similarities.
The process of computing the final value of the closed-form expression between high-dimensional kernels can be time-consuming and resource-intensive. This problem can be alleviated by imposing constraints on the mean vectors and/or the covariance matrices. Following <cit.>, we will cover how specific constraints can be applied to improve computational efficiency in a subsequent corollary.
Let p(𝐱) := K_Σ_p(𝐱, μ_p) and q(𝐱) := K_Σ_q(𝐱, μ_q) be two multivariate kernels as defined in Remark <ref>. The following statements hold:
(i) If the covariance matrices are diagonal, meaning that
Σ_p = diag(σ_p,1^2, ⋯, σ_p,M^2) and Σ_q = diag(σ_q,1^2, ⋯, σ_q,M^2), the Bhattacharyya coefficient between normalized p(𝐱) and q(𝐱) will be
ρ(p(𝐱), q(𝐱)) = (∏_i=1^M(σ_p,i^2+σ_q,i^2/2σ_p,iσ_q,i)^-1/2)exp(-14∑_i=1^M(μ_p,i -μ_q,i)^2/σ_p,i^2+σ_q,i^2). (Anisotropic)
(ii) If the mean vectors have identical values across all dimensions (μ_p = μ_p1, μ_q = μ_q1, where 1 = [1, ⋯, 1]^T∈ℝ^M is the one vector), and the covariance matrices are diagonal with homogeneous variances (Σ_p = σ_p^2I, Σ_q = σ_q^2I, where I∈𝕊^M_++ is the identity matrix), then the Bhattacharyya coefficient between two normalized isotropic kernels p(𝐱) and q(𝐱) can be calculated as
ρ(p(𝐱), q(𝐱)) = (σ_p^2+σ_q^2/2σ_pσ_q)^-M/2exp(-M4(μ_p -μ_q)^2/σ_p^2+σ_q^2). (Isotropic)
§ PROPOSED METHOD
[13]R0.68
< g r a p h i c s >
Overview of KMCL framework. The training pipeline comprises a feature encoder that feeds into the KMM, which outputs the parameters of a mixture model in the Gaussian RKHS ℋ. These parameters then define the objective function that captures label correlation to aid in training the model for the multi-label classification.
The multi-label classification task involves assigning multiple labels to an image 𝐱^n from sample space 𝐗. These labels are typically correlated with each other and represented by a multi-hot binary vector 𝐲^n∈{0,1}^K, where K denotes the number of labels. In this section, we propose an end-to-end multi-label learning framework–dubbed Kernel-based multi-label Contrastive Learning (KMCL), that captures label correlations to improve recognition performance. Given an input batch of data, we first propagate it through the encoder network to obtain the feature embedding. The embedding is then inputted into a novel fully connected layer called the Kernel Mixture Module (KMM), which produces a Gaussian Reproducing Kernel Hilbert Space ℋ. The Gaussian RKHS embedding can handle higher-order statistics of the features and has a complete inner product that enables linear geometry, making it richer than the deterministic feature embedding. Finally, we compute the loss function using the KMM outputs on space ℋ to capture label correlation and train the model for multi-label classification. Figure <ref> provides a visual explanation.
§.§ KMCL Framework
The main components of the KMCL framework are:
[2]R0.23
< g r a p h i c s >
Internal architecture of KMM.
Feature Encoder.The encoder network takes two samples from the input batch separately and generates corresponding feature representation vectors 𝐟∈ℝ^M. The dimension of the feature vector depends on the encoder type.
KMM.
Most feature encoders produce deterministic results that do not quantify or control uncertainty, leading to low confidence in robust multi-label classification tasks and errors in interpreting the output predictions. Uncertainty in deep learning arises from two sources: epistemic uncertainty (model uncertainty), resulting from uncertainty in model parameters, and aleatoric uncertainty (data uncertainty), which stems from the inherent noise in data and label ambiguity. In this study, we propose the Kernel Mixture Module (KMM) to estimate epistemic uncertainty in predictions. The KMM takes the feature vector 𝐟 from the encoder network and generates a mixture of exponential kernels within the Hilbert space, each corresponding to a specific class in an image. Specifically, the fully connected layer in the KMM utilizes learnable weights and biases to produce three outputs for each unimodal exponential kernel component: the mixture coefficient π_k, mean vector μ_k, and covariance matrix Σ_k (Fig. <ref>). The parameters π_k, μ_k, and Σ_k quantify the existence, relative spatial positioning, and relative statistical complexities (measures of spread and uncertainty) of the kth class membership. These parameters are then used to model the label representation of a given sample 𝐱^n associated with a class vector 𝐲^n using the following expression:
𝒢_𝒮(𝐟^n) := ∑_k ∈𝒮π_k^n g_k(𝐟^n) =
∑_k ∈𝒮π_k^nexp(-‖𝐟^n - μ_k^n1‖^2/2(σ_k^n)^2),
where, 𝒮 = {k: y_k^n = 1} and 𝐟^n is the extracted feature vector of the input sample. The component g_k(𝐟^n) := K_Σ_k^n(𝐟^n, μ_k^n) is an isotropic exponential kernel where μ_k^n = μ_k^n1, Σ_k^n = (σ^n_k)^2I, and π_k^n∈ [0, 1]. These adaptive parameters i.e., θ_k^n = [μ_k^n, (σ^n_k)^2, π_k^n] are calculated through forward propagation, using suitable activation functions to ensure that the parameters adhere to their constraints. The sigmoid activation function is used to normalize the mixture coefficient for efficient multi-label classification, accurately predicting the likelihood of multiple labels. The modified version of the exponential linear unit (ELU) <cit.> is also used as an activation function for variances, ensuring their semi-positivity. The detailed architecture of KMM can be found in Fig. <ref> and Supplementary material.
§.§ Multi-label Learning with KMCL
Building upon the KMCL framework, we aim to provide insights into the learning process of multi-label tasks. To achieve this, we introduce the details of our objective function, which comprises three components: reconstruction loss, classification loss, and contrastive loss. Throughout this paper, we use N and K to denote the mini-batch size and the total number of classes, respectively.
Reconstruction Loss.
[12]R0.3
< g r a p h i c s >
Relative frequency histograms of class distributions in four datasets show that most images have 2, 2, 4, and 1 labels in the Pascal-VOC, MS-COCO, ADP, and ChestX-ray14, respectively.
It is straightforward to compute the mixture model defined in Equation <ref> using the KMM output parameters, which provide 3K values for each input sample. Following this calculation, the model can be used to learn label-level representations in Hilbert space ℋ by minimizing its negative log-likelihood. Therefore, we introduce to optimize the following reconstruction loss over the data batch to train the mixture model
ℒ_REC = 1/N∑_n=1^N-log𝒢_𝒮(𝐟^n)/𝒢_𝒴(𝐟^n),
where, 𝒢_𝒴(𝐟^n) := ∑_k∈𝒴={1, ⋯, K}π_kg_k and 𝒢_𝒮(𝐟^n) denotes the kernel mixture associated with image 𝐱^n defined in Equation <ref>. The log-ratio term in Equation <ref> is always negative i.e. 𝒢_𝒮(𝐟^n)≤𝒢_𝒴(𝐟^n), where the loss is led by the supervised labels for reconstruction. We propose this as an alternative choice for reconstruction loss, which is commonly used in the literature <cit.>. Our new loss function ℒ_REC exhibits robust behavior without relying on numerical tricks for stabilization.
Classification Loss.
The analysis in Figure <ref> reveals that despite varying statistical and conceptual properties across datasets, most images have only a fraction of labels, causing a significant imbalance between positive and negative samples. This imbalance can lead to poor training accuracy as gradients from positive labels may be underemphasized. To mitigate this issue, we use ASL <cit.> as a classification loss function that adjusts the contributions of positive and negative samples by down-weighting easy negative samples and focusing on the hard ones. Therefore, given the predictive mixture of coefficients π^n from KMM and the ground-truth multi-hot label vector 𝐲^n, the classification loss for a batch is obtained as
ℒ_ASL = 1/N∑_n=1^N∑_k=1^K -y_k^n(L_k^n)_+-(1-y_k^n)(L_k^n)_-,
where, (L_k^n)_+ = (1-π_k^n)^γ_+log (π_k^n), and (L_k^n)_- = (max(π_k^n-m, 0))^γ_-log (1-max(π_k^n-m, 0)) represent the positive and negative loss parts, respectively, such that γ_+, γ_-, and m are the hyper-parameters used to balance the loss. For additional information on ℒ_ASL, please refer to <cit.>.
Kernel-based Contrastive Loss.
The ASL loss function classifies labels independently, making it difficult to capture correlations between co-occurring semantic labels. Moreover, it fails to account for uncertainty in predictions, which can undermine decision-making confidence. To address these limitations, we propose a new loss function, ℒ_KMCL, which incorporates label correlation and epistemic uncertainty into supervised contrastive learning to improve representation.
The objective of kernel-based multi-label contrastive loss ℒ_KMCL is to pull together the kernel representations of positive images that have shared classes with the anchor image 𝐱^n in the embedding space ℋ, while pushing apart negative samples that do not share any classes. This approach differs from deterministic supervised contrastive losses <cit.> as ℒ_KMCL constructs the positive and negative pairs using similarity measures that consider the uncertainty of kernel representations. The similarity is measured by a Bhattacharyya coefficient discussed in Corollary <ref> (isotropic), which determines the overlap between these exponential kernels and their confidence in proximity. Essentially, the kernel-based contrastive loss optimizes the similarity of frequently co-occurring labels and captures their statistical dependencies, making it a valuable complement to ASL. The contrastive loss is defined for the entire minibatch as follows:
ℒ_KMCL = 1/N∑_n=1^N -1/|𝒜(n)| ∑_m∈𝒜(n)J(n, m) (∑_k∈𝒦(n, m) logexp(ρ_k,k^n,m/τ)/∑_i∈{N\n}exp(ρ_k,k^n,i/τ)),
where, ρ_k,l^n,m:=ρ(g_k(𝐟^n), g_l(𝐟^m)) indicates the Bhattacharyya coefficient between the normalized exponential kernels g_k(𝐟^n) and g_l(𝐟^m) (see Corollary <ref>) and τ is the temperature parameter. The positive set 𝒜(n) = {m ∈{N \ n}: 𝐲^n·𝐲^m≠ 0, where · is a dot product.} includes samples that share at least one label with the anchor image 𝐱^n, while 𝒦(n,m)= {k∈𝒴: y_m^k = y_n^k = 1} represents the indices of shared labels between 𝐱^n and 𝐱^m. The Jaccard index J(n, m)=𝐲^n·𝐲^m/𝐲^n^2+𝐲^m^2-𝐲^n·𝐲^m serves as a weighting factor for positive samples based on the number of shared labels with the anchor. It measures the intersection over union (IOU) of the label vectors between the anchor and positive image, taking into account object co-occurrences. In this way, ℒ_KMCL prioritizes positive samples with a high Jaccard index for a given anchor while downplaying samples with few shared labels.
[14]R0.44
< g r a p h i c s >
(a) Training loss over different epoch training. Plots show the normalized total loss ℒ as well as different normalized sub-losses, and (b) Training accuracy of KMCL pipeline over different epoch training
Objective Function. The overall training loss of the KMCL is the augmented Lagrangian of the three aforementioned losses, which can be expressed as:
ℒ = ℒ_REC + λ_1 ℒ_ASL + λ_2 ℒ_KMCL,
where λ_1 and λ_2 are the Lagrangian multipliers used to balance the gradients of ℒ_ASL and ℒ_KMCL, respectively. We use an end-to-end pipeline to incorporate contrastive learning into supervised classification, which simultaneously trains the feature encoder and classification parts. This approach is different from previous methods that use contrastive losses <cit.>. In those methods, the encoder is trained with a contrastive loss and then frozen before being transferred to the classifier for tuning. Instead, the KMCL framework combines these training regimes into one formulation, enabling us to learn multi-label classification and label correlations with data-driven techniques.
§.§ KMCL Algorithm
[15]R0.57
The pseudo-code of the proposed KMCL framework is outlined in Algorithm <ref>, which takes a set of batches and a specified number of epochs as inputs. The pair of anchor images and their positive set are fed through the network depicted in Figure <ref> to obtain the feature vectors and parameters of the corresponding kernel mixtures (lines <ref>-<ref>). The overall loss is then computed as an augmented Lagrangian of the ℒ_REC, ℒ_ASL, and ℒ_KMCL using the KMM parameters (lines <ref>-<ref>). Finally, the objective function is back-propagated through the KMM and the feature encoder for each iteration to update the weights based on the gradients associated with the subsequent forward pass (line <ref>). This iterative process continues until convergence is reached.
Figures <ref> (a) and (b) demonstrate the results of implementing the KMCL framework with TResNet-L <cit.> as the encoder network on the Pascal-VOC dataset <cit.>. Fig. <ref> (b) displays the objective loss behavior along with the evolution of the three loss terms for the training and test sets; whereas The mean average precision (mAP) accuracy is presented in <ref> (a). The losses decrease with different multiplicative factors due to the tuned Lagrangian multipliers. The convergence speed of the method on multi-label tasks is impressive, reaching 96.2% mAP accuracy in fewer than 30 epochs.
§ EXPERIMENTS
In this section, we present the experimental setup and demonstrate the superior performance of KMCL in both general computer vision and medical imaging domains. To ensure robust feature extraction, we utilized TResNet-M and TResNet-L <cit.>, state-of-the-art architectures designed for different image resolutions (224 and 448, respectively). The features are then passed through the KMM to obtain the mixture parameters π, μ, and Σ. Additional information regarding the encoders, KMM, datasets, evaluation metrics, and training details can be found in Supplementary material.
Datasets. We evaluate the KMCL's performance on popular computer vision datasets, PASCAL-VOC <cit.> and MS-COCO <cit.>, as well as on medical datasets, ADP <cit.> and ChestX-ray14 <cit.>.
Evaluation Metrics.
Following SOTA <cit.>, we report the standard metrics of mean average precision (mAP), average overall precision (OP), recall (OR), and F1 score (OF1) in addition to per-class precision (CP), recall (CR), and F1 score (CF1). We considered the number of parameters (M) and GMAC as measures of computational costs. Finally, for the ChestX-ray14 dataset <cit.>, we reported per-class AUC scores to assess model discriminability for specific classes.
Training Details.
We implemented the KMCL framework using PyTorch, following Alg. <ref>. The backbone feature encoders were initialized with pre-trained architectures, while the mixture parameters were initialized by applying a uniform distribution to π and μ and setting Σ to a constant value of 1. In all experiments, we assign fixed values of 0.1 and 0.3 to λ_1 and λ_2 respectively, as specified in Eq. <ref>. The Adam optimizer <cit.> was used with an initial learning rate of 2e-4, and the OneCycleLR scheduler <cit.> for 40 epochs. Standard augmentations from RandAugment policy <cit.> were applied to the training data. Experiments were conducted on four NVIDIA GeForce RTX 2080Ti GPUs.
How does KMCL compare to SOTA methods on computer vision datasets? We evaluate KMCL with SOTA methods on computer vision datasets in Table <ref> and Fig. <ref>. KMCL outperforms the best competitors on PascalVOC and MS-COCO, achieving superior performance with a margin of 0.4% and 0.2% in mAP score, respectively. In particular, KMCL excels in challenging classes on PascalVOC, such as the sofa and bus classes, with an improvement of over 3.0%. On MS-COCO, KMCL demonstrates significant improvements across multiple metrics, including mAP, OF1, and CF1. Using the TResNet-M encoder at resolution 224, we achieve state-of-the-art results with a 5.0% increase in mAP compared to the best method. Similarly, with TResNet-L at a resolution of 448, KMCL surpasses other methods in overall and per-class metrics. These achievements are attained by integrating the proposed contrastive learning with ASL classification loss, to capture label correlation and enhance prediction accuracy. This is illustrated through the Top3-metrics on MS-COCO, where our 3 classes are better selected by considering label correlation when ranking the predictions.
How well KMCL generalizes to medical imaging datasets?
[8]r7cm
Comparisons with state-of-the-art methods on the ADP dataset.
!
2pt1pt1pt
22emMethod 7c|Performance 2cComplexity
2-10
mAP OP OR OF1 CP CR CF1 Parameters (MM) GMAC
2pt1pt1pt
ML-GCN (Binary) <cit.> 94.9 92.0 86.9 89.7 91.8 87.0 89.3
44.90 31.39
ASL (TResNet-L) <cit.> 96.1 92.1 90.7 91.4 92.5 89.2 90.8
44.14 35.28
TDRG <cit.> 95.5 94.3 86.2 90.5 94.6 84.8 89.4
75.20 64.40
CSRA <cit.> 96.1 93.0 89.7 91.7 93.1 88.6 90.8
42.52 31.39
KMCL (TResNet-M) 95.1 94.2 91.0 90.4 94.7 88.9 89.8
29.41 5.74
KMCL (TResNet-L) 96.5 92.7 92.9 92.8 92.6 92.0 92.3
44.20 35.28
2pt1pt1pt
We evaluate KMCL against SOTA methods on medical imaging datasets presented in Tables <ref> and<ref>. The recall is a crucial factor in these datasets, as it reflects the likelihood of missing a medical diagnosis. The proposed method achieves a superior tradeoff between precision and recall by significantly improving recall metrics while maintaining competitive precision scores, including SOTA mAP. On the ADP dataset, KMCL outperforms the surveyed SOTA with margins of 0.4%, 2.2%, and 2.8% for mAP, OR, and CR, respectively. Similarly, on the ChestX-ray14 dataset, both TResNet-M and TResNet-L models exhibit significant improvements, with our best model surpassing SOTA results by 5.2%, 7.0%, and 11.6% in mAP, OR, and CR, respectively. In comparison, competing methods such as ML-GCN <cit.> use label correlation but suffer from increased computational complexity and a multi-stage approach, as shown in Table <ref>. However, our method surpasses the SOTA while maintaining a small model size and low GMAC scores. These findings highlight the advantage of KMCL in computationally constrained environments.
How KMCL's performance varies with different similarity measurements? In this ablation study, we examine the impact of changing the Battacharya coefficient to either Mahalanobis kernel similarity or Gaussian kernel similarity in the KMCL framework (Corrolary <ref> (i) and (ii)). Under the Mahalanobis kernel similarity, the performance decreases across the PascalVOC and ADP, as indicated in Table <ref>. This is likely due to the constraint that the variance must be identical across all classes, leading to an inability to capture entropy and uncertainty as reported in Section <ref>.
[7]r7cm
Ablative comparison for similarity measures and kernel representation cases.
!
2pt1pt1pt
2c||Configuration 7c|ADP PascalVOC 2cComplexity
Similarity Metric Case mAP OP OR OF1 CP CR CF1 mAP Params(MM) GMAC
2pt1pt1pt
Bhattacharyya Anisotropic 95.4 94.0 92.7 90.6 94.8 90.7 90.5 95.4
104.91 5.81
Bhattacharyya Isotropic 95.1 94.2 91.0 90.4 94.7 88.9 89.8 95.2
29.41 5.74
Mahalanobis - 94.7 92.0 92.4 90.9 92.6 90.5 90.4 95.1
71.34 5.78
Gaussian Kernel - 94.5 91.5 89.7 90.6 92.3 86.5 89.3 95.0
29.40 5.74
2pt1pt1pt
Similarly, when utilizing Gaussian kernel similarity, the performance further deteriorates because the model is constrained to learn a single variance value that applies to both the label classes and feature dimensions. Therefore, it is more meaningful to use the Bhattacharyya coefficient since it evaluates the generalized variances of the kernels and identifies similarities in their orientation, shape, and means (Eq. <ref>). We further investigate the assumptions from both isotropic and anisotropic cases of the exponential kernel representations in KMCL framework as discussed in Corrolary <ref>. While the anisotropic case leads to an improved performance as shown in Table <ref>, but results in an increase in learnable parameters at the cost of higher computational complexity. By incorporating variances over the feature dimension, we better capture epistemic uncertainty and achieve enhanced overall results. Thus, if computational resources are available, one could best leverage our framework in the anisotropic case to achieve sota results.
[11]r0.7
< g r a p h i c s >
Reduced t-SNEs for ASL (left) and KMCL(Center) on PascalVOC color-coded by user-defined super-classes in the legend; (Right) ground truth correlation matrix for PascalVOC.
Intuitive Visualizations.
KMCL presents an end-to-end framework for contrastive learning that has achieved quantitatively significant results compared to existing methods. In this section, we visualize how the learned feature representation incorporates label correlation and epistemic uncertainty. Figure <ref> shows a reduced t-SNE <cit.> visualization of the feature representation for ASL and KMCL on the Pascal VOC dataset. Both methods accurately discriminate between different classes, as seen from the plotted centroids of each cluster. Notably, both methods exhibit a clustering pattern based on user-defined super-classes (e.g., car and bus are both forms of Transportation). Upon analyzing the ground truth correlation matrix, it becomes apparent that KMCL captures label correlation more effectively. Specifically, the sofa class exhibits the highest correlation with the chair class, resulting in their closer proximity in the t-SNE visualization for KMCL compared to ASL.
[8]r0.65
< g r a p h i c s >
GradCam visualization of KMCL and competing SOTA method. Bolded class labels indicate instances where KMCL outperforms SOTA by a large margin.
Figure <ref> showcases the GradCam visualization for KMCL and a competing SOTA method. KMCL effectively distinguishes the sofa and chair classes, consistent with the t-SNE visualization results. Moreover, by capturing epistemic uncertainty from the kernel representation, our method accurately identifies the correct classes in the ADP sample with minimal extraneous activations. For more visualizations, please refer to the Supplementary material.
§ BROADER IMPACT
KMCL provides an end-to-end supervised contrastive learning framework for multilabel datasets. It requires fewer resources for the design and implementation of downstream tasks such as classification. Contrastive learning methods like <cit.> typically involve two stages of encoder training and fine-tuning for the task, which can take several hundred epochs. In contrast, KMCL only requires one stage of training with significantly fewer epochs. This translates into a much smaller carbon emission footprint, as highlighted in <cit.> for using more compact models for training. Although KMCL has been successfully applied in computer vision and medical imaging domains, its effectiveness has not yet been tested for segmentation/detection tasks or in other modalities like natural language processing. In future work, we will consider broadening our experiments for further validation. Additionally, we believe that society can benefit from the theoretical analysis of the similarity metrics presented in this paper, which can be adapted to different application domains.
§ ACKNOWLEDGMENT
Authors would like to thank Rahavi Selvarajan, Xiao Hu and Jiarui Zhang for their assistant and helpful discussion.
ieee_fullname
§ APPENDIX
§.§ Proof of Remark 1.
The Bhattacharyya coefficient between the normalized p(𝐱):= K_Σ_p(𝐱, μ_p) = exp(-1/2𝐱 - μ_p^2_Σ_p^-1) and q(𝐱):= K_Σ_q(𝐱, μ_q) = exp(-1/2𝐱 - μ_q^2_Σ_q^-1) is defined as
ρ(p(𝐱), q(𝐱) ) = ∫_𝒳(p(𝐱)∫_𝒳 p(𝐱) d𝐱)^1/2(q(𝐱)∫_𝒳 q(𝐱) d𝐱)^1/2d𝐱 = ∫_𝒳p(𝐱)^1/2q(𝐱)^1/2d𝐱/√(∫_𝒳p(𝐱) d𝐱)√(∫_𝒳q(𝐱) d𝐱).
To begin, we expand the integrand part of the enumerator, i.e., √(p(𝐱)q(𝐱)) as follows:
exp(-14𝐱^T(Σ_p^-1+Σ_q^-1)𝐱+12(Σ_p^-1μ_p+Σ_q^-1μ_q)^T𝐱 -14(μ_p^TΣ_p^-1μ_p + μ_q^TΣ_q^-1μ_q )).
In order to overcome the challenge of integrating the derived integrand in Equation <ref>, we will introduce a new approach. We will represent √(p(𝐱)q(𝐱)) as the product of a constant value, denoted as h(μ_p, μ_q, Σ_p, Σ_q), and a newly defined anisotropic multivariate squared exponential kernels, denoted as r(𝐱):= K_Σ_r(𝐱, μ_r). This formal representation can be expressed as follows:
√(p(𝐱)q(𝐱)) = h(μ_p, μ_q, Σ_p, Σ_q)r(𝐱).
We defined the new exponential kernel of Equation <ref> as
r(𝐱) := K_Σ_r(𝐱, μ_r) = exp(-1/2𝐱 - μ_r^2_Σ_r^-1) = exp(-12(𝐱-μ_r)^TΣ_r^-1(𝐱-μ_r)),
where Σ_r≜(12Σ_p^-1+12Σ_q^-1)^-1 and μ_r≜Σ_p(12Σ_p^-1μ_p+12Σ_q^-1μ_q). Once the values of Σ_r and μ_r are replaced in Equation <ref>, the kernel r(𝐱) will be
r(𝐱) = exp(-14𝐱^T(Σ_p^-1+Σ_q^-1)𝐱 + 12(Σ_p^-1μ_p+Σ_q^-1μ_p)^T𝐱 -14(Σ_p^-1μ_p+Σ_q^-1μ_p)^T
+(Σ_p^-1+Σ_q^-1)^-1(Σ_p^-1μ_p+Σ_q^-1μ_p)).
By substituting Equations <ref> and <ref> into Equation <ref>, we obtain the closed-form expression of h(μ_p, μ_q, Σ_p, Σ_q) as presented below.
exp(-1/4(
μ_p^T(Σ_p^-1-Σ_p^-1(Σ_p^-1+Σ_q^-1)^-1Σ_p^-1)μ_p+
μ_q^T(Σ_q^-1-Σ_q^-1(Σ_p^-1+Σ_q^-1)^-1Σ_q^-1)μ_q
-μ_p^T(Σ_p^-1(Σ_p^-1+Σ_q^-1)^-1Σ_q^-1)μ_q
-μ_q^T(Σ_q^-1(Σ_p^-1+Σ_q^-1)^-1Σ_p^-1)μ_p
))
Given the fact that Σ_p^-1-Σ_p^-1(Σ_p^-1+Σ_q^-1)^-1Σ_p^-1 = Σ_q^-1-Σ_q^-1(Σ_p^-1+Σ_q^-1)^-1Σ_q^-1 = Σ_p^-1(Σ_p^-1+Σ_q^-1)^-1Σ_q^-1 = Σ_q^-1(Σ_p^-1+Σ_q^-1)^-1Σ_p^-1 = (Σ_p+Σ_q)^-1 <cit.>, we can simplify Equation <ref> and derive
exp(-1/4μ_p^T(Σ_p+Σ_q)^-1μ_p+μ_q^T(Σ_p+Σ_q)^-1μ_q-μ_p^T(Σ_p+Σ_q)^-1μ_q-μ_q^T(Σ_p+Σ_q)^-1μ_p),
where can be further simplified to yield the following expression:
h(μ_p, μ_q, Σ_p, Σ_q) = exp(-18(μ_p-μ_q)^TΣ^-1(μ_p-μ_q)),
where Σ
= Σ_p+Σ_q/2. Ultimately, by utilizing the definition of the Bhattacharyya coefficient, Equation <ref>, and Equation <ref>, we can deduce the following conclusion:
ρ(p(𝐱), q(𝐱)) = ∫_ℝ^Mp(𝐱)^1/2q(𝐱)^1/2d𝐱/√(∫_ℝ^Mp(𝐱) d𝐱)√(∫_ℝ^Mq(𝐱) d𝐱) =∫_ℝ^Mh(μ_p, μ_q, Σ_p, Σ_q)r(𝐱)d𝐱/√(∫_ℝ^Mp(𝐱) d𝐱)√(∫_ℝ^Mq(𝐱) d𝐱)
= h(μ_p, μ_q, Σ_p, Σ_q) ∫_ℝ^M|2πΣ_r|^1/2𝒩(𝐱;μ_r, Σ_r)d𝐱/√(∫_ℝ^M|2πΣ_p|^1/2𝒩(𝐱;μ_p, Σ_p) d𝐱)√(∫_ℝ^M|2πΣ_q|^1/2𝒩(𝐱;μ_q, Σ_q) d𝐱)
= |Σ_r|^1/2/|Σ_p|^1/4|Σ_q|^1/4h(μ_p, μ_q, Σ_p, Σ_q) = |2Σ_p(Σ_p+Σ_q)^-1Σ_q|^1/2/|Σ_p|^1/4|Σ_q|^1/4h(μ_p, μ_q, Σ_p, Σ_q)
(a)= |Σ_p|^1/2|Σ_q|^1/2|Σ|^1/2exp(-18(μ_p-μ_q)^TΣ^-1(μ_p-μ_q)),
where, Σ
= Σ_p+Σ_q/2 and (a) is followed by the probability property that the total area underneath a probability density function is 1. The notation 𝒩(𝐱;μ, Σ) represents a multivariate Gaussian probability distribution in M dimensions, characterized by a mean vector μ and a covariance matrix Σ. This completes the proof of Remark 1.
§.§ Forward Propagation in KMM.
The KMM (Kernel Mixture Module) takes the feature vector 𝐟^n∈ℝ^M as input from the encoder network and produces the parameters for each exponential kernel component in the kernel mixture model. This transformation converts the feature vector into 3K values, where each K represents the parameters for the kth kernel component (existing class), such as μ_k^n∈ℝ, σ_k^n∈ℝ^+, π_k^n∈ [0, 1]. The adaptive parameters are computed through forward propagation, employing suitable activation functions to ensure that the parameters satisfy their respective constraints. The activations corresponding to the parameters of the kth component for the KMM ((a_k^μ)^n,(a_k^σ^2)^n, (a_k^π)^n) are used to accomplish this, and they are calculated through the forward propagation of a fully connected layer by
(a_k^μ)^n = 𝐰_k^μ𝐟^n + b^μ_k, (a_k^σ^2)^n = 𝐰_k^σ^2𝐟^n + b^σ^2_k, (a_k^π)^n = 𝐰_k^π𝐟^n + b^π_k,
where, {𝐰_k^μ, 𝐰_k^σ^2, 𝐰_k^π}∈ℝ^M are the weights, and {b^μ_k, b^σ^2_k, b^π_k}∈ℝ represent the biases associated with {(a_k^μ)^n, (a_k^σ^2)^n, (a_k^π)^n}, respectively. We make a minor revision to the idea of using nonlinear activation from <cit.> by replacing softmax with sigmoid to normalize the mixture of coefficients and address multilabel issues. In the following, we define the nonlinear and linear transformations applied to (ak^μ)^n, (ak^σ^2)^n, (a_k^π)^n using
π_k^n = 11+exp(-(a_k^π)^n),
μ_k^n = (a_k^μ)^n, (σ_k^n)^2 = ELU((a_k^σ^2)^n)+2+ϵ,
where ELU(·) and ϵ are the exponential linear unit function <cit.> and the hyperparameter used to ensure training stability, respectively. We use a modified ELU function rather than the exponential function as the activation on (a_k^σ^2)^n in order to ensure that variances remain non-negative ((σ_k^n)^2≥ 0). This modification is necessary because the vanilla exponential function exhibits rapid growth for larger values, which can lead to training instability, particularly when dealing with high-variance datasets. It is important to note that there is no constraint on the mean μ_k^n, as it is obtained directly from the activation (a_k^μ)^n.
§ DATASETS
PASCAL-VOC
The PASCAL Visual Object Classes Challenge (2007) <cit.> is a common computer vision dataset used in multi-label classification. It contains a total of 9963 images over 20 classes, including 'cat', 'bottle', and 'person'. Being consistent with the state of the art, we trained our architecture on the trainval set and evaluated it on the test set with a total of 5011 and 4952 images in each set, respectively. Referencing the relative frequency in the main paper, we can see that the number of classes per image to the total number of classes is heavily unbalanced, with the majority of images having only 2-4 classes.
MS-COCO
The Microsoft COCO dataset <cit.> is another common computer vision dataset used in multi-label classification. This dataset includes 82,081 training and 40,504 validation images across 80 different classes including 'person', 'bicycle', and 'elephant'. Following the state of the art, we test our method on the validation dataset making it comparable with competitive approaches.
ADP
The Atlas of Digital Pathology for Hisotological Tissue Type Classification <cit.> is composed of digital histology images taken from several organ tissues, including the colon, brain, stomach, etc. These images were generated via a Whole Slide Image (WSI) scanner. This database includes 17,668 image patches that are multilabel in nature. The training, validation, and test sets contain 14,134, 1767, and 1767 images respectively. This labeling scheme follows a three-tier hierarchy: L1 (9 labels), L2 (11 labels), and L3 (22 labels). As we progress down the levels, the features being annotated gradually progress from coarse to fine detail. The highest level (L1) contains classes that amalgamate several lower-level classes.
For example, Dense Regular Connective (C.D.R) is an L3 precise label that falls under the more coarse L1 category of Connective (C). For the purpose of our work, we have selected L1 as it seems to be the most statistically significant selection with a better balance of per-class distribution.
ChestXray-14
The ChestX-Ray 14 dataset contains hospital-scale frontal-view chest X-ray images from 30,805 unique patients. Each image either contains multiple common thoracic illnesses including ‘cardiomegaly’ or ‘pneumonia’ or is designated ‘normal’ indicating no illness. The released version of the dataset catalogs 14 common illnesses to date, as opposed to the original 8 that was released at the time of publication.
§.§ Hyperparameters & Tuning
In this section, we list all the necessary parameters for the reproducibility of our method. We have categorized our hyperparameters depending on which part of the pipeline they relate to (i.e., Training Optimization refers to any parameters used in setting up the training phase). A special note is made for the Loss Development λ values. In order to best tune our method, we sampled a 15-point log-random search in a subset of the provided range to best adapt our model to the given datasets. See Table <ref>.
§.§ Additional Information on Metrics
Being consistent with state-of-the-art methods, we calculate the average overall precision (OP), recall (OR), and F1 score (OF1), in addition to the average per-class precision (CP), recall (CR), and F1 score (CF1), as metrics for evaluating the different methods on the datasets <cit.>. Overall these metrics challenge the model’s ability to accurately discriminate the class of interest in terms of measuring false positives and false negatives. Superior OF1 and CF1 indicate that the model is well-tuned for class discrimination as this metric encompasses both recall and precision in the calculation. For some experiments, we include the following computational complexity measures: Parameters (MM) to indicate model size, and GMAC to indicate the forward computational resource required. The motivation behind these metrics is to illustrate that performance is not only measured through how well the method discriminates classes but also through the complexity of deploying said method in the real world. Finally, due to the increased difficulty of the ChestX-ray14 dataset, we additionally report per class AUC scores to identify model discriminability for the class of interest, this has been a common trend in papers that have cited results on this dataset <cit.>.
§.§ Additional Visualizations
To further augment the main paper visualizations, we attach supplemental visualizations on the two additional datasets: MS-COCO and ChestXray-14. As can be seen, by the visualizations, our model is more precise at localizing the correct features. Due to capturing the epistemic uncertainty from the kernel representation, our method is able to focus the activation on the correct class, limiting extraneous false positive results. See Figure <ref>.
|
http://arxiv.org/abs/2307.05073v1 | 20230711071528 | Knowledge-wh and False Belief Sensitivity: A Logical Study (An Extended Abstract) | [
"Yuanzhe Yang"
] | cs.LO | [
"cs.LO"
] |
Retrieval-augmented GPT-3.5-based Text-to-SQL Framework with Sample-aware Prompting and Dynamic Revision Chain
Chunxi Guo, Zhiliang Tian (), Jintao Tang, Shasha Li, Zhihua Wen,
Kaixuan Wang and Ting Wang ()
August 12, 2023
==============================================================================================================
In epistemic logic, a way to deal with knowledge-wh is to interpret them as a kind of mention-some knowledge (MS-knowledge).
But philosophers and linguists have challenged both the sufficiency and necessity of such an account:
some argue that knowledge-wh has, in addition to MS-knowledge, also a sensitivity to false belief (FS);
others argue that knowledge-wh might only imply mention-some true belief (MS-true belief).
In this paper, we offer a logical study for all these different accounts.
We apply the technique of bundled operators, and introduce four different bundled operators -
^x ϕ := ∃ x (ϕ∧ϕ),
^x ϕ := ∃ x (ϕ∧ϕ) ∧∀ x (ϕ→ϕ),
^x ϕ := ∃ x ϕ and
^x ϕ := ∃ x ϕ∧∀ x (ϕ→ϕ) -,
which characterize the notions of MS-true belief, MS-true belief with FS, MS-knowledge and MS-knowledge with FS respectively.
We axiomatize the four logics which take the above operators (as well as ) as primitive modalities on the class of S4.2-constant-domain models,
and compare the patterns of reasoning in the obtained logics, in order to show how the four accounts of knowledge-wh differ from each other, as well as what they have in common.
§ INTRODUCTION
In standard epistemic logic, for the most time, we deal with propositional knowledge (or knowledge-that):
that is, an agent knows that ϕ, where ϕ is a certain proposition.
However, this clearly does not exhaust our daily use of the notion of “knowledge”.
Besides knowledge-that, we also frequently talk about various kinds of knowledge-wh:
for example, I know how to ride a bike, I know who proved the incompleteness theorems,
I know when a certain meeting is held, I know where to buy a certain book, I know what is the password of my computer, I know why a certain event happens, etc.
Thus, besides standard propositional knowledge, knowledge-wh also seems to be an interesting subject for epistemic logic to study.
There are already a number of logical studies of various kinds of knowledge-wh (e.g. know whether in <cit.>, know why in <cit.>, know how in <cit.>,<cit.>, <cit.>, <cit.> and <cit.>, just to name a few),
and a more general framework for logics of knowledge-wh is also proposed in <cit.>.
In this paper, following <cit.>, we will also focus mainly on the general logical structures shared by various kinds of knowledge-wh.
As suggested in <cit.> (following the philosophical stance of the so-called “intellectualism” initiated in <cit.>), in many cases, knowledge-wh can be interpreted as a kind of mention-some knowledge (MS-knowledge for short):
for example, I know how to prove a theorem, iff there exists some proof such that I know that this proof is a proof for the theorem;
I know where to buy newspapers, iff there exists some place where I know I can buy newspapers, etc.
Then, in such cases, it seems that the logical structure of knowledge-wh can be formally captured by the first-order modal formula ∃ x ϕ(x).[
However, as it is also noted in <cit.>, in some other situations, it seems more natural to interpret knowledge-wh in terms of mention-all, rather than mention-some, knowledge.
For example, when I say “I know who came to the meeting yesterday”, it may mean that I know all the people who came to that meeting, which should probably be formalized as, for example, ∀ x (ϕ(x) →ϕ(x)) or ∀ x (ϕ(x) ∨ϕ(x)).
We will not deal with the mention-all reading of knowledge-wh in this paper, since the behavior of mention-all knowledge is rather different from mention-some knowledge, and it thus seems better to study it independently elsewhere.
In fact, axiomatization of mention-all knowledge in terms of ∀ x (ϕ(x) ∨ϕ(x)) has been studied in <cit.>, an unpublished undergraduate thesis.
]
However, while it is quite clear that in many situations, knowledge-wh does involve some kind of mention-some structure, it is not as clear whether MS-knowledge really is the right account for knowledge-wh in these situations.
In fact, both the sufficiency and necessity of such an account are challenged.
For example,
as it is argued in <cit.>, <cit.>, <cit.> and <cit.>,
knowledge-wh may not only involve mention-some knowledge, but also involve false belief sensitivity (FS for short).
Let's consider the following scenario, adapted from one offered in <cit.>, to illustrate this point.
There are two stores, Newstopia and Paperworld.
Newstopia sells newspapers, while Paperworld sells only stationery.
Now, Alice knows that Newstopia sells newspapers,
but also believes erroneously that Paperworld sells newspapers.
In such a scenario, it is natural to judge that that Alice does not know where to buy newspapers (psychological experiments conducted in <cit.> also show that such an intuition is shared by many people):
even though she has a MS-knowledge concerning where to buy newspapers,
it seems that her false belief that Paperworld sells newspapers would corrupt her knowledge-where.
Hence, maybe knowledge-wh should be sensitive to false belief:
that is, even under an MS-reading, maybe MS-knowledge should not be characterized by ∃ x ϕ(x) alone,
but should rather be characterized by ∃ x ϕ(x) ∧∀ x (ϕ(x) →ϕ(x)).
On the other hand, the necessity of the MS-knowledge account is also doubted.
For example, as it is argued in <cit.>, it seems that knowledge-wh is subject to a kind of epistemic luck which is not consistent with propositional knowledge.
Let's consider the following scenario, adapted from one offered in <cit.>, to illustrate this point.
Suppose that Bob believes that w is a way to change light bulbs, and w is indeed a reliable way to do so.
His belief is obtained by reading an instruction in a book.
However, unknown to him, all other contents in the book are erroneous, and it is merely due to a very rare print error that the instruction he read is correct.
In this case, Bob's true belief that w is a way to change light bulbs is too lucky to be counted as his knowledge;
but nevertheless, it still seems natural to judge that Bob knows how to change light bulbs.
Then, it seems that sometimes a mention-some true belief (MS-true belief for short), i.e. ∃ x (ϕ(x) ∧ϕ(x)), is enough for knowledge-wh.
(In philosophical discussions, such a stance is sometimes called “revisionary intellectualism”,
which is first proposed in <cit.>, in contrast to intellectualism.)
Of course, none of the arguments presented above is decisive.
But they do reveal an enormous complexity in the question concerning the nature of knowledge-wh.
Hitherto, no consensus concerning this question is reached in philosophical discussions,
and nor will we offer a determinate answer here.
On the contrary, in this paper, we will study all the accounts mentioned above in a formal way.
In order to do so, we apply the technique of “bundled operators”[
For a detailed introduction of such an idea, see <cit.> and <cit.>.
].
The general idea is that we pack a complex first-order modal formula (e.g. ∃ x ϕ(x) ∧∀ x (ϕ(x) →ϕ(x))) into the semantics of a single operator,
and study the logic which takes such an “bundled operator” as primitive modality.
By working in such languages with limited expressivity, we can focus on the behavior of the epistemic notion in which we are really interested,
without being distracted by irrelevant notions which can also be expressed in a stronger language.
Moreover, with the help of bundled operators, we can study the complex notions in a compact manner.
In this paper, then, we will study the following four different bundled operators:[
The bundled operator ^x is first introduced in <cit.> (the notation used there is ^x, though);
later, further study concerning its decidability and complexity is presented in <cit.>, and axiomatization in <cit.>.
The result presented in this paper concerning this operator (namely, the axiomatization on S4.2), however, is new.
On the other hand, , and are all novel bundled operators that have not yet been studied in literature.
]
^x ϕ(x) := ∃ x (ϕ(x) ∧ϕ(x))
^x ϕ(x) := ∃ x (ϕ(x) ∧ϕ(x)) ∧∀ x (ϕ(x) →ϕ(x))
^x ϕ(x) := ∃ x ϕ(x)
^x ϕ(x) := ∃ x ϕ(x) ∧∀ x (ϕ(x) →ϕ(x)).
We will axiomatize the logics which take these operators plus an operator for propositional knowledge as primitive modalities on the class of S4.2-models, a class of models which characterizes knowledge, belief and their interactions in a reasonable way.
Completeness results will also be presented.
Moreover, we will compare the obtained logics,
in order to show the differences and commonalities in the ways we reason about knowledge-wh, propositional knowledge and belief, which are logically implied by the different accounts of knowledge-wh.
§ FIRST-ORDER S4.2-MODELS
First, we introduce the models we use to characterize knowledge and belief on the semantic level.
Since first-order quantifiers are involved in the notions of MS-knowledge, MS-true belief and FS,
we will use first-order Kripke models as the semantic basis.
We fix a set of predicates 𝒫.
A first-order Kripke model, then, is defined as follow:[
In this paper, we will not introduce function symbols and constants to our language.
Hence, we will also not consider functions and constants in the following definition.
]
A first-order Kripke model is a 4-tuple = (W, D, R, ρ), where
* W ≠∅ is the set of epistemically possible worlds of the model;
* D ≠∅ is the domain of the model;
* R ⊆ W^2 is the accessibility relation among the possible worlds, which characterizes epistemic indistinguishability;
* ρ: 𝒫× W →℘ (D^<ω) assigns each n-ary predicate an n-ary relation on each possible world.
(We may abbreviate the term “first-order Kripke model” simply as “model” in the following discussions.)
Note that such a model can be interpreted rather freely on the conceptual level, so that it can characterize various kinds of knowledge-wh.
For example, if we want to characterize the knowledge-how of an agent, then the elements in D can be interpreted as different methods or devices available for the agent in question,
and a predicate P ∈𝒫 can be interpreted as a certain goal, while a ∈ρ(P, w) reads “at the epistemically possible world w, a is a way to achieve P”.
Similarly, if we want to characterize knowledge-where, then the elements in D can be interpreted as different locations accessible for the agent,
while predicates in 𝒫 are interpreted as properties of these locations.
Of course, in a similar fashion, different models can also be used to characterize knowledge-who, knowledge-when or knowledge-what.
Also note that we only consider constant-domain models here:
all possible worlds in a model share the same domain.
This is mainly in order to avoid technical and conceptual complexities,
and we believe this is indeed a reasonable (though inevitably idealized) assumption.
Of course, since we use first-order Kripke models to characterize the epistemic states of an agent, the Kripkean part of such models should also possess certain frame properties.
It is a popular choice to use S5-models to characterize an agent's knowledge, but we will not use such models in this paper.
This is mainly because we need to deal with both knowledge and belief, as well as the interactions between them (moreover, in our discussion, the notion of belief should be interpreted in a rather strong sense, so we would prefer interaction principles like ϕ→ϕ to hold),
and we must also allow the possibility for false belief, in order for the notion of FS to make any sense at all.
This, however, seems to be a difficult task when knowledge is characterized by S5-models.
Hence, we will use S4.2-models instead -
that is, models which are reflexive, transitive and strongly convergent.[
Here, we use the notion of strong convergence to define S4.2-models;
but elsewhere, when defining S4.2-models,
the notion of weak convergence might be used instead (A frame (W, R) is weakly convergent, iff for all w, v, v' ∈ W s.t. wRv and wRv', there is some u ∈ W s.t. vRu and v'Ru).
Standard modal logic cannot distinguish these two kinds of models (as noted in <cit.>), but some of the languages studied in this paper are strong enough to distinguish them.
We choose the stronger notion of convergence here, because it seems more favorable both technically and conceptually.
This is also in accordance with Stalnaker's note in <cit.>.
]
The formal definition is as follow:
A frame (W, R) is strongly convergent, iff for all w ∈ W, there is some u ∈ W s.t. for all v ∈ W, if wRv, then vRu.
A model based on a reflexive, transitive and strongly convergent frame is called an S4.2-model.
We find such models attractive, because the class of S4.2-models validates the logic of knowledge 𝐒4.2, in which belief can be reasonably defined in terms of knowledge by the definition ϕ := ϕ (as explained in <cit.>, the underlying idea is that, if one knows that she does not know something, then she would not believe it; and if she does not believe something, then she would know by introspection that she does not know it).
Moreover, the logic for the belief defined in this way is 𝐊𝐃45, and we also have many intuitive interaction principles between knowledge and belief (e.g. ϕ→ϕ, ϕ→ϕ, ϕ→ϕ, ϕ→ϕ).
(It is Lenzen who first proposed 𝐒4.2 as a logic for knowledge in <cit.> and <cit.>, from a syntatic perspective.
Later, Stalnaker also studied 𝐒4.2 from a more semantic perspective in <cit.>.)
Moreover, as it is noted by Stalnaker in <cit.>, in an S4.2-frame (W, R), we can define the following relation , which corresponds to the notion of belief defined in terms of knowledge:
Given a frame (W, R),
⊆ W^2 is the relation which satisfies that for all w, u ∈ W, w u iff for all v ∈ W s.t. wRv, vRu.
It is not hard to check that if (W, R) is an S4.2-frame, then (W, ) is KD45.
Moreover, after we formally introduce the languages and their semantics,
it will be easy to check that corresponds to
in exactly the way R corresponds to .
§ LANGUAGES AND SEMANTICS
Now, we introduce the languages for the bundled operators, as well as their exact semantics.
We first fix a set of variables 𝐗.
Then, for any ∈{, , , },
the corresponding language () (and also _≈()) is defined as follow:
()-formulas are defined recursively as follow:
ϕ ::= P(y_1, ..., y_n) |ϕ|ϕ∧ϕ|ϕ|^x ϕ
where P ∈𝒫, n ≥ 0 and x, y_1, ..., y_n ∈𝐗.
ϕ is denoted as ϕ;
ϕ is an abbreviation for ϕ.
∨, → and ↔ are defined in the usual way.
Moreover, let _≈() be the language obtained by further adding an identity relation ≈ to ().[
In the following discussions, we will be working in the language () when we do not specifically mention the language in which we are working.
We will make it clear whenever we switch our working language to _≈().
]
Corresponding to our definition of the bundled operators,
we define the semantics for the above languages recursively as follow:
Given a model = (W, D, R, ρ), a w ∈ W and an assignment σ from 𝐗 to D:
,w,σ P(x_1, ..., x_n) (σ(x_1), ..., σ(x_n)) ∈ρ(P,w)
,w,σ x ≈ y σ(x) = σ(y)
,w,σϕ ,w,σϕ
,w,σϕ∧ψ ,w,σϕ and ,w,σϕ
,w,σϕ For all v ∈ W, if wRv, then ,v,σϕ
,w,σ^x ϕ There is some a ∈ D, s.t. ,w,σ[x ↦ a] ϕ∧ϕ
2*,w,σ^x ϕ 2* (i) There is some a ∈ D, s.t. ,w,σ[x ↦ a] ϕ∧ϕ
(ii) For all b ∈ D, ,w,σ[x ↦ b] ϕ→ϕ
,w,σ^x ϕ There is some a ∈ D, s.t. ,w,σ[x ↦ a] ϕ
2*,w,σ^x ϕ 2* (i) There is some a ∈ D, s.t. ,w,σ[x ↦ a] ϕ
(ii) For all b ∈ D, ,w,σ[x ↦ b] ϕ→ϕ
where σ[x ↦ a] is the assignment which maps x to a, and agrees with σ on any other point.
Note that we need not introduce an independent operator for belief, since ϕ is already defined by ϕ in the languages given above.
It is also not hard to check that on S4.2-models, the semantics for ϕ defined this way is indeed the following one:
,w,σϕ For all v ∈ W, if w v, then ,v,σϕ
§ THE LOGICS
Then, we introduce the four logics, corresponding to our four accounts of knowledge-wh respectively.
Their axiomatizations are all obtained in a similar fashion:
generally speaking, we start from a standard 𝐒4.2 system for ,
and then add axioms and rules to describe the behaviors of the bundled operators.
Below is a list of schemas of axioms and rules that will be used to offer the axiomatizations (in which the operator should be substituted by , , or in the corresponding logics):[
Note that when we use the notation ϕ[y/x] to denote the formula obtained by replacing every free occurrences of x in ϕ with y, we also implicitly assume that y is admissible for x in ϕ:
that is, y does not appear in the scope of any operator of the form ^y in ϕ.
]
4cAxioms
𝚃𝙱𝚝𝚘𝙺_𝚠𝚑 (ϕ∧ϕ)[y/x] →^x ϕ
𝙺𝚝𝚘𝙺_𝚠𝚑 ϕ[y/x] →^x ϕ
𝙺_𝚠𝚑𝚝𝚘𝙵𝚂 ^x ϕ→ (ϕ→ϕ) [y / x]
𝙱𝚝𝚘𝙱𝙺_𝚠𝚑 ϕ[y/x] →^x ϕ
4cRules
2*𝙺_𝚠𝚑𝚝𝚘𝚃𝙱 3l2* ⊢ψ_0 →(ψ_1 →⋯ (ψ_n → (ϕ∧ϕ)) ⋯ )/⊢ψ_0 →(ψ_1 →⋯ (ψ_n →^x ϕ) ⋯ )
2*𝙺_𝚠𝚑𝚝𝚘𝙺 3l2* ⊢ψ_0 →(ψ_1 →⋯ (ψ_n →ϕ) ⋯ )/⊢ψ_0 →(ψ_1 →⋯ (ψ_n →^x ϕ) ⋯ )
2*𝙵𝚂&𝙱𝚝𝚘𝙺_𝚠𝚑 3l2* ⊢ψ_0 → (ψ_1 →⋯ (ψ_n → (ϕ→ϕ)) ⋯ )/⊢ψ_0 → (ψ_1 →⋯ (ψ_n → (ϕ [y/x] →^x ϕ)) ⋯ )
2*𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑 3l2* ⊢ψ_0 → (ψ_1 →⋯ (ψ_n → (ϕ→ϕ)) ⋯ )/⊢ψ_0 → (ψ_1 →⋯ (ψ_n → (ϕ [y/x] →^x ϕ)) ⋯ )
4c(In all the rules above, n can be any natural number, and we require that x ∉⋃_i ≤ n FV(ψ_i))
By using rules like 𝙺_𝚠𝚑𝚝𝚘𝚃𝙱 or 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑, we have sacrificed some intuitiveness for technical reasons, but the underlying idea is straightforward:
for example, 𝙺_𝚠𝚑𝚝𝚘𝚃𝙱 essentially says ^x ϕ→∃ x (ϕ∧ϕ), and 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑 says ∀ x (ϕ→ϕ) ∧ϕ[y/x] →^x ϕ, in languages where the existential and universal quantifiers are not available.
With the help of the above axioms and rules, then, we can give the following four logics:
𝐒4.2^ 𝐒4.2^⊕{𝚃𝙱𝚝𝚘𝙺_𝚠𝚑, 𝙺_𝚠𝚑𝚝𝚘𝚃𝙱}
𝐒4.2^ 𝐒4.2^⊕{𝙺_𝚠𝚑𝚝𝚘𝙵𝚂, 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑, 𝙺_𝚠𝚑𝚝𝚘𝚃𝙱, 𝙵𝚂&𝙱𝚝𝚘𝙺_𝚠𝚑}
𝐒4.2^ 𝐒4.2^⊕{𝙺𝚝𝚘𝙺_𝚠𝚑, 𝙺_𝚠𝚑𝚝𝚘𝙺}
𝐒4.2^ 𝐒4.2^⊕{𝙺_𝚠𝚑𝚝𝚘𝙵𝚂, 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑, 𝙺_𝚠𝚑𝚝𝚘𝙺, 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑}
Moreover, for any ∈{, , , }, when we work in the language _≈(), let 𝐒4.2_≈^ be the logic defined as follows:
𝐒4.2_≈^ 𝐒4.2^⊕{x ≈ x, x ≈ y → (ϕ[x/z] →ϕ[y/z]), x ≉y → (x ≉y)}
Note that all the logics given here are non-normal, since they are all non-aggregative:
that is, ^x ϕ∧^x ψ→^x(ϕ∧ψ) is not an inner theorem of 𝐒4.2^ (or 𝐒4.2_≈^) for any ∈{, ,
, }
(in fact, in all these logics, ^x Px ∧^x Px is consistent).
Moreover, some of the logics are even non-monotone, as we will see below.
Then, we show the completeness theorem for these logics.
Since we are now dealing with bundled operators with more complex structures, the strategy to prove completeness theorems for the case of in <cit.> and <cit.> cannot be directly applied here (moreover, axiomatization of the logic of on S4.2 has also not yet been studied).
Hence, we will develop a new strategy to prove completeness theorems for all the above logics in a uniform way.
𝐒4.2^ (as well as 𝐒4.2_≈^) is sound and strongly complete w.r.t. the class of S4.2-constant-domain models,
where ∈{, , , }.
We only sketch the general idea of the proof here.
A detailed proof for the case of 𝐒4.2^ can be found in the appendix.
Generally, we use maximal consistent sets (MCS) of formulas which also contain certain witness formulas to construct the canonical model.
The main difficulty is to ensure at the same time that (i) every MCS in the model contains all the witness formulas we need, (ii) every formula of the form ϕ in an MCS has some accessible MCS containing ϕ as its witness, and
(iii) the canonical model is an S4.2-constant-domain model.
In order to construct such a model, we use a step-by-step method.
We start from a consistent set Γ_0,
and extend consistent sets to MCS, add new formula sets as witnesses for formulas of the form ϕ,
and add witness formulas to formula sets during the same process.
The key is to ensure, at each step of the construction, that every formula set except Γ_0 is finite,
and all the information contained in a set is recorded in its predecessor with a formula of the form ϕ.
This ensures that we can always add witness formulas to formula sets using rules like 𝙺_𝚠𝚑𝚝𝚘𝚃𝙱 and 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑.
Then, after countably many steps, we obtain a model which satisfies both (i) and (ii),
and is also an S4-constant-domain model.
Finally, we add another set of MCSs to the model to make it strongly convergent, so that we can obtain an S4.2-model.
The above logics also have some interesting technical aspects.
For example, it is shown in <cit.> that the language () cannot distinguish constant-domain and increasing-domain models in general.
However, when we confine the models to S4.2-ones, () can distinguish constant-domain and increasing-domain models,
and consequently, 𝐒4.2^ is not sound w.r.t. the class of S4.2-increasing-domain models (e.g. ^x ϕ→^x ϕ is an inner theorem of 𝐒4.2^, but is not valid on S4.2-increasing-domain models).
In fact, for all ∈{, , ,
}, 𝐒4.2^ is not sound w.r.t. S4.2-increasing-domain models.
Another interesting fact is that 𝐒4.2^ and 𝐒4.2^ are able to distinguish S4.2-models (defined in terms of strong convergence) and models which are reflexive, transitive but only weakly convergent.
The axiom 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑: ϕ[y/x] →^x ϕ does the trick.
When = or , on the other hand, we also have ϕ[y/x] →^x ϕ as an inner theorem of 𝐒4.2^,
but in this case, the formula does not have the power to distinguish strong and weak convergence,
and consequently, 𝐒4.2^ and 𝐒4.2^ are also sound w.r.t. the class of reflexive, transitive and weakly convergent models.
§ COMPARISONS
Now, we have the formal ground to compare the different accounts of knowledge-wh.
§.§ Differences
An interesting difference among the different accounts of knowledge-wh concerns the ways these accounts interact with propositional knowledge.
For example, consider positive introspection.
Since we take 𝐒4.2 to be the underlying logic for propositional knowledge, which is stronger than 𝐒4, it is clear that propositional knowledge satisfies positive introspection:
ϕ→ϕ is an inner theorem of 𝐒4.2^ for any ∈{, , , }.
However, does knowledge-wh also have positive introspection?
To put it more formally, is ^x ϕ→^x ϕ an inner theorem of 𝐒4.2^?
The answer is as follow:
𝐒4.2^⊢^x ϕ→^x ϕ,
but 𝐒4.2^⊬^x ϕ→^x ϕ when ∈{, , }.
The underlying reason for the failure of positive introspection in 𝐒4.2^, 𝐒4.2^ and 𝐒4.2^ is similar.
Essentially, this is because these accounts may involve true beliefs (the MS-true belief in ^x ϕ or ^x ϕ, or a true belief required by the FS condition in ^x ϕ or ^x ϕ), but positive introspection requires knowledge rather than mere true belief, while the latter in general does not imply the former in an 𝐒4.2 system.
The following proposition helps us make this point clear on the formal level.
Note that in the formulation of (a part of) the following proposition, we will also need the identity relation ≈ and the logic 𝐒4.2_≈^ which involves the axioms for ≈.
We have the following identities between logics:
𝐒4.2^⊕^x ϕ→^x ϕ = 𝐒4.2^⊕ϕ∧ϕ→ϕ
𝐒4.2^⊕^x ϕ→^x ϕ = 𝐒4.2^⊕ϕ∧ϕ→ϕ
𝐒4.2_≈^⊕^x ϕ→^x ϕ = 𝐒4.2_≈^⊕ x ≉y → (ϕ∧ϕ→ϕ)
In other words, under our 𝐒4.2 setting for propositional knowledge, requiring ^x ϕ and ^x ϕ to satisfy positive introspection is in effect the same as requiring true belief to imply knowledge.
The case for ^x ϕ, on the other hand, is a bit more complex:
when ^x ϕ satisfies positive introspection, either true belief implies knowledge, or there is at most one element in the domain (in which case the notion of FS is clearly trivialized).
A similar phenomenon also appears in the case of the formula ^x ϕ→^x ϕ.
Intuitively, the formula says that knowledge-wh offers the agent a way to obtain propositional knowledge:
for example, if we interpret in terms of knowledge-how, then the formula says that if an agent knows how to achieve ϕ, then she also knows how to make herself know that ϕ.
In fact, Proposition <ref> and <ref> still hold after we substitute every occurrences of ^x ϕ in these propositions with ^x ϕ,
since
^x ϕ↔^x ϕ is in fact an inner theorem in 𝐒4.2^ for all ∈{, , , }.
A more interesting difference among the different accounts of knowledge-wh concerns the monotonicity of knowledge-wh.
We say our notion of knowledge-wh is monotone if the following rule is admissible in the corresponding logic:
𝙼𝙾𝙽𝙾 ⊢ϕ→ψ/⊢^x ϕ→^x ψ
The rule says that if ψ follows logically from ϕ,
then if an agent has knowledge-wh of ϕ, then she automatically also has knowledge-wh of ψ.
Note that in order for this to hold, we need to assume that the agent we consider is logically omniscient;
and we have indeed assumed so in our underlying logic for propositional logic, 𝐒4.2^.
However, even such a logically omniscient agent still may not have a monotone notion of knowledge-wh, when FS is involved in our account of knowledge-wh.
The propositions below show how FS influences the monotonicity of knowledge-wh.
(Note that we need the identity relation ≈ to formulate Proposition <ref>.)
𝙼𝙾𝙽𝙾 is admissible in 𝐒4.2^ and 𝐒4.2^,
but inadmissible in 𝐒4.2^ and 𝐒4.2^.
The following equivalences holds:
𝐒4.2_≈^⊕𝙼𝙾𝙽𝙾 = 𝐒4.2_≈^⊕ x ≉y → (ϕ→ϕ)
𝐒4.2_≈^⊕𝙼𝙾𝙽𝙾 = 𝐒4.2_≈^⊕ x ≉y → (ϕ→ϕ)
As we can see, FS corrupts the monotonicity of knowledge-wh.
In fact, as it is shown in Proposition <ref>, if we force ^x ϕ and ^x ϕ to be monotone, then either the agent can have no false belief at all, or there is only one element in the domain of the model which characterizes her knowledge and belief -
in both cases, the notion of FS is completely trivialized.
In this sense, we may say that FS is incompatible with the monotonicity of knowledge-wh in quite an essential way:
in order to retain the monotonicity of knowledge-wh, we have to give up FS completely.
§.§ Commonalities
As we have seen, different accounts of knowledge-wh behave rather differently when interacting with propositional knowledge.
However, when interacting with belief,
their behaviors are much more similar.
For example, the following proposition shows some inner theorems shared by all the logics presented above:[
Note that (iii) in the proposition below is in fact an axiom in 𝐒4.2^ and 𝐒4.2^.
]
For all ∈{, , , },
the following are 𝐒4.2^-theorems:
(i) ϕ[y/x] →^x ϕ (ii) ϕ[y/x] →^x ϕ
(iii) ϕ[y/x] →^x ϕ
(iv) ^x ϕ∨^x ϕ
If we interpret ^x ϕ in terms of knowledge-how, then (i) and (ii) say that if an agent believes / does not believe that some certain y is a way to achieve ϕ, then she knows how to make herself believe / not believe that ϕ;
(iii) says that if the agent believes that some y is a way to achieve ϕ, then she also believes that she knows how to achieve ϕ;
and (iv) says that an agent is “confident” concerning her own epistemic state:
for any ϕ, she either believes that she knows how to ϕ, or believes that she does not knows how to ϕ. Note that in 𝐒4.2^, we also have the interaction principles ϕ→ϕ, ϕ→ϕ and ϕ→ϕ and ϕ∨ϕ between propositional knowledge and belief;
hence, we may say that when interacting with propositional belief (rather than knowledge),
our accounts of knowledge-wh show more aspects that resemble propositional knowledge.
Also note that from (i) and (iii),
we can deduce the following two formulas, respectively:
(v) ^x ϕ→^x ϕ (vi) ^x ϕ→^x ϕ
As we can see, though ^x ϕ→^x ϕ and ^x ϕ→^x ϕ cannot be deduced in 𝐒4.2^ when ∈{, , }, when the operator is relaxed to , we obtain (v) and (vi), which are inner theorems of
𝐒4.2^ for all ∈{, , , }.
Another interesting commonality shared by all our logics (which also has to do with the interaction between knowledge-wh and belief) concerns what logic of knowledge-wh our agent believes.
In section <ref>,
we have already shown some complexities in the reasoning about knowledge-wh:
for example, concerning positive introspection and monotonicity, different accounts yield different behaviors of knowledge-wh.
These complexities, however, only appear when we reason about the knowledge-wh of an agent from an external perspective;
when the agent herself reasons about her own knowledge-wh from within, all such complexities evaporate.
To put this point more rigidly, we introduce the following notion:
For any logic 𝐋, let 𝐋_𝖡 = {ϕ|ϕ∈𝐋}.
Intuitively, for a logic 𝐋, 𝐋_𝖡 collects all the formulas which 𝐋 says that an agent believes.
In this sense, if 𝐋 characterizes the epistemic states of an agent,
then 𝐋_𝖡 characterizes the epistemic logic believed by this agent.
Then, with the help of this new notation,
we can formulate the following theorem:
For all ∈{, , , },
𝐒4.2^_𝖡 can be axiomatized by the following system:
𝐒5^ All axioms and rules of an 𝐒5 system for
𝙺𝚝𝚘𝙺_𝚠𝚑 ϕ[y/x] →^x ϕ
2*𝙺_𝚠𝚑𝚝𝚘𝙺^0 2*⊢ϕ→ψ/⊢^x ϕ→ψ (where x ∉ FV(ψ))
It is also not hard to check that this system is equivalent to the system 𝖲𝖬𝖫𝖬𝖲𝖪 presented in <cit.>, a system in the language () which is sound and strongly complete w.r.t. the class of S5-models.
Hence, conceptually, the above theorem says that no matter which account of knowledge-wh we choose, it makes no difference for our agent:
the agent always believes that her knowledge-wh behaves in exactly the same way as MS-knowledge, and the logic for the underlying propositional knowledge is as strong as 𝐒5.
In such a logic, of course knowledge-wh is monotone and satisfies positive introspection;
moreover, it even satisfies negative introspection: ^x ϕ→^x ϕ is in 𝐒4.2^_𝖡 for all ∈{, , , }.
On the other hand, all the subtle differences among the different accounts of knowledge-wh, generated from the gap between mere true belief and knowledge, as well as the peculiar behavior of the FS condition, are all invisible for the agent in question.
§ CONCLUSION
In this paper, we studied four bundled operators: , , and , which correspond to the four different accounts of knowledge-wh.
We axiomatized the logics which take them (as well as ) as primitive modalities on the class of S4.2-constant-domain models, and compared the ways we reason about knowledge-wh in different logics.
There many potential future works that can be done based on our work.
For example, we can further study the four bundled operators introduced in this paper.
We have only studied their behavior on S4.2-models, which characterize knowledge and belief in a highly idealized way;
our study of the obtained logics is also far from complete.
Hence, it seems interesting to study the logics obtained in this paper in greater detail,
or to study the behavior of the bundled operators on other reasonable models for knowledge and belief (of course, we need not confine ourselves to Kripke models).
This may offer us a deeper understanding of the different accounts of knowledge-wh,
and may eventually help us decide which account is indeed the right one.
Moreover, the kind of step-by-step proof method applied in this paper can be generalized to study other complex epistemic notion.
For example, there are cases where it is better to understand knowledge-wh in terms of mention-all knowledge,
and there are also various competing accounts of these kinds of knowledge-wh, e.g.the weakly exhaustive reading (first proposed in <cit.>), the strongly exhaustive reading (first proposed in <cit.>), and the intermediately exhaustive reading (first raised, but soon rejected, in <cit.>, and later proposed again in <cit.>), which can be formalized as ∀ x (ϕ(x) →ϕ(x)), ∀ x (ϕ(x) ∨ϕ(x)) and ∀ x (ϕ(x) →ϕ(x)) ∧∀ x (ϕ(x) →ϕ(x)), respectively.
Using the technique developed in this paper, we can easily pack these complex notions into bundled operators, and study their behavior.
Speaking on a more general level, the step-by-step method used in this paper can at least be generalized to any logic equipped with a set of ordinary modal operators {_a}_a ∈τ plus a set of bundled operators of the form ▪^x ϕ := ∃ x α [ϕ / p] ∧∀ x β [ϕ/p],
where α and β are propositional modal formulas containing only one propositional symbol p, boolean connectives and operators in {_a}_a ∈τ.
Our trick works no matter how complicated the structures of α and β are, so a great deal of complex first-order modal notions can be handled in this way.
eptcs
§ ACKNOWLEDGEMENT
I owe the very idea of studying knowledge-wh with FS using bundled operators to Yanjing Wang, and
I would also like to thank him for his insightful advice on this paper.
I would also like to thank Yimei Xiang for introducing works concerning MS-knowledge and FS in linguistics to me,
and Xun Wang for discussing matters concerning the ∃-bundled fragment with me.
I should also thank three anonymous reviewers from TARK 2023 for their helpful advice on the writing of this paper,
and for identifying the typos and grammar mistakes in the paper.
Finally, I would like to thank the support of NSSF grant 19BZX135.
§ APPENDIX
In the appendix, we show how to prove theorem <ref>, Proposition <ref> and Proposition <ref>.
First, we consider theorem <ref>.
We only prove the case for 𝐒4.2^, since the other cases can be proved in a similar way.
Moreover, for most of the time, we will be working in the language (), since our proof can easily be generalized to the case of _≈() with the help of some slight modifications.
We will demonstrate how to do so along the proof.
First, we check that the soundness result holds.
𝐒4.2^ is sound w.r.t. the class of S4.2-constant-domain models.
We only prove that 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑 is valid on the class of S4.2-constant-domain models,
and 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑 preserves validity on such models.
For 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑:
Let = (W, R, D, ρ) be a S4.2-model, let w ∈ W be arbitrary, and let σ be an arbitrary assignment.
Assume that ,w,σϕ[y/x].
Then, for all v ∈ W s.t. w v, ,v,σϕ[y/x].
Then, let v ∈ W be arbitrary, and assume that w v.
We first show that ,v,σϕ[y/x].
This is clear, since for all u ∈ W s.t. vRu, it is easy to check that w u,
and thus ,u,σϕ[y/x].
Then, we show that for all a ∈ D, ,v,σ[x ↦ a] ϕ→ϕ.
This is also clear: since (W, ) is KD45, v is -reflexive.
Hence, it is easy to see that ,v,σ^x ϕ,
and thus ,w,σ^x ϕ.
For 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑:
Let = (W, D, R, ρ) be an arbitrary S4.2-model, and assume that
ψ_0 →(ψ_1 →⋯(ψ_n → (ϕ→ϕ)) ⋯ ) is valid on ,w, where n is an arbitrary natural number;
and let x be an variable s.t. x ∉⋃_i ≤ n FV(ψ_i).
Then, let σ be an arbitrary assignment, and suppose (towards a contradiction) that ,w,σψ_0 →(ψ_1 →⋯(ψ_n → (ϕ[y/x] →^x ϕ)) ⋯ ).
Then, there is some w_0, w_1, ..., w_n ∈ W,
s.t. w = w_0 R w_1 R ⋯ R w_n,
,w_i,σψ_i for all i ≤ n,
and ,w_n,σϕ[y/x] →^x ϕ.
By the validity of ψ_0 →(ψ_1 →⋯(ψ_n → (ϕ→ϕ)) ⋯ ),
and since x ∉⋃_i ≤ n FV(ψ_i),
for all a ∈ D,
,w_n,σ[x ↦ a] ϕ→ϕ.
But then, since ,w_n, σϕ[y/x],
,w_n, σ[x ↦σ(y)] ϕ, and thus it should follow that ,w_n,σ^x ϕ,
causing a contradiction.
It is also not hard to check that 𝐒4.2^ has the following inner theorems, which will be used in our completeness proof.
𝙽𝙱𝙺_𝚠𝚑𝚝𝚘𝙱𝙽𝙺_𝚠𝚑 ^x ϕ→^x ϕ
𝙱𝙺_𝚠𝚑𝚝𝚘𝙺_𝚠𝚑𝙱 ^x ϕ→^x ϕ
𝚁^ ^x ϕ↔^y ϕ[y/x] (where y does not appear in ϕ)
Now, we are ready to prove the completeness theorem.
As preparation, we first define the language ^+(), which is obtained by adding countably many new variables to ().
We use 𝐗^+ to denote the set of variables of ^+().
Then, we use a step-by-step method to prove the completeness theorem.
We first define the notion of a network.
Note that when constructing such networks, the states will all be taken from a set of states {w_i | i ∈ω},
which we fix in advance.
A network is a triple = (W, R, ν), where
* {w_0}⊆ W ⊆{w_i | i ∈ω};
* R ⊆ W^2, and (W, R) forms a tree where w_0 is the root;
* ν assigns each element in W a set of ^+()-formulas.
We also define the following two properties for the formula sets in a network:
(𝙼𝚂-property)
An ^+()-formula set Δ has 𝙼𝚂-property, iff for all ϕ∈^+() and x ∈𝐗^+, if ^x ϕ∈Δ, then there is some y ∈𝐗^+ s.t. ϕ [y/x] ∈Δ.
(𝙵𝚂-property)
An ^+()-formula set Δ has 𝙵𝚂-property, iff for all ϕ∈^+() and x, y ∈𝐗^+, if ^x ϕ, ϕ[y/x] ∈Δ, then there is some z ∈𝐗^+ s.t. (ϕ∧ϕ)[z/x] ∈Δ.
Then, we define the notion of coherence and saturation for networks:
A network = (W, R, ν) is coherent, iff the following conditions are satisfied:
(i) W is finite;
(ii) For all w ∈ W, ν(w) is 𝐒4.2^-consistent; and for all w ∈ W ∖{w_0}, ν(w) is finite;
(iii) For all w, v ∈ W s.t. wRv, there is some ψ s.t. ⊢ψ↔⋀ν(v) and ψ∈ν(w);
(iv) There are countably many variables in 𝐗^+ which do not appear in ν(w) for any w ∈ W.
A network = (W, R, ν) is saturated, iff for all w ∈ W and ϕ∈^+(), the following holds:
(i) ν(w) is a MCS of ^+()-formulas;
(ii) If ϕ∈ν(w), then for all v ∈ W s.t. wRv, ϕ∈ν(v);
(iii) If ϕ∈ν(w), then there is some v ∈ W s.t. wRv and ϕ∈ν(v);
(iv) ν(w) has the 𝙼𝚂-property;
(v) ν(w) has the 𝙵𝚂-property.
Then, corresponding to the requirements of saturation, we also introduce the following notion of defects:
The possible kinds of defects we may find on a state on a w ∈ W in a network = (W, R, ν) are as follow:
(d1) ϕ∉ν(w) and ϕ∉ν(w)
(d2) ϕ∈ν(w), but there is some v ∈ W s.t. wRv and ϕ∉ν(v)
(d3) ϕ∈ν(w), but there is no v ∈ W s.t. wRv and ϕ∈ν(v)
(d4) ^x ϕ∈ν(w), but there is no y ∈𝐗^+ s.t. ϕ[y/x] ∈ν(w)
(d5) ^x ϕ, ϕ[y/x] ∈ν(w), but there is no z ∈𝐗^+ s.t. (ϕ∧ϕ)[z/x] ∈ν(w)
where w ∈{w_i | i ∈ω}, ϕ∈^+() and x ∈𝐗^+.
Then, we prove the repair lemma,
which shows how to repair defects in a coherent network, while maintaining its coherence.
(Repair lemma)
For any coherent network = (W, R, ν) and any defect (d) of ,
then there is a coherent network ' = (W', R', ν') s.t. W ⊆ W', R ⊆ R',
ν(w) ⊆ν'(w) for all w ∈ W, and ' does not has (d).
Let = (W, R, ν) be a coherent network, and assume that has a defect (d) for some w_m ∈ W and ϕ∈^+().
Since (W, R) forms a tree where w_0 is the root, there is a unique path w_0 = v_0 R v_1 R ⋯ R v_n = w_m in for some n ∈ω.
Then, since is coherent, for all 1 ≤ i ≤ n, let ψ_i stand for the formula s.t. ⊢ψ_i ↔⋀ν(v_i) and ψ_i ∈ν(v_i-1).
Then, it is easy to see that
ν(v_0) ⊢(ψ_1 ∧ (ψ_2 ∧⋯(ψ_n-1∧ψ_n) ⋯))
We then consider five cases.
Case 1: (d) is of the kind (d1).
That is, ϕ∉ν(v_n) and ϕ∉ν(v_n).
Then, it is easy to check that
ν(v_0) ⊢(ψ_1 ∧ (ψ_2 ∧⋯(ψ_n ∧ϕ) ⋯ )) ∨(ψ_1 ∧ (ψ_2 ∧⋯(ψ_n ∧ϕ) ⋯ ))
Then, at least one of the disjuncts is consistent with ν(v_0) = ν(w_0).
We only consider the case where the former is consistent with ν(w_0), since the other case is similar.
In this case, let
ν' = {(w, ν(w)) | w ≠ v_i for all i ≤ n}
∪ {(v_n, ν(v_n) ∪{ϕ})}
∪ {(v_i, ν(v_i) ∪{(ψ_i+1∧⋯(ψ_n ∧ϕ) ⋯ )}) | i < n}
and let ' = W, R, ν'.
It is then easy to check that ' is coherent, and does not have the defect (d).
Case 2: (d) is of the kind (d2).
That is, ϕ∈ν(v_n), but there is some u ∈ W s.t. v_nRu and ϕ∉ν(u).
Since is coherent, there is some ψ_u s.t. ⊢ψ_u ↔⋀ν(u) and ψ_u ∈ν(v_n).
Hence, it is easy to check that
ν(v_0) ⊢(ψ_1 ∧(ψ_2 ∧⋯(ψ_n ∧(ψ_u ∧ϕ)) ⋯ ))
Then, let
ν' = {(w, ν(w)) | w ≠ v_i for all i ≤ n}
∪ {(u, ν(u) ∪{ϕ})}
∪ {(v_i, ν(v_i) ∪{(ψ_i+1∧⋯(ψ_n ∧ϕ) ⋯ )}) | i ≤ n}
It is easy to check that ' = (W, R, ν') is still coherent, and does not have the defect (d).
Case 3: (d) is of the kind (d3).
That is, ϕ∈ν(v_n), but there is no u ∈ W s.t. v_nRu and ϕ∈ν(u).
Since W is finite, there is some {w_i | i ∈ω}∖ W ≠∅.
Then, let w_k be the element in {w_i | i ∈ω}∖ W with the least index number,
and let W' = W ∪{w_k}, R' = R ∪{(v_n, w_k)}, and ν' = ν∪{(w_k, {ϕ})}.
It is easy to check that = (W', R', ν') is coherent, but does not have (d).
Case 4: (d) is of the kind (d4).
That is, ^x ϕ∈ν(v_n), but there is no y ∈𝐗^+ s.t. ϕ[y/x] ∈ν(v_n).
Then, let y ∈𝐗^+ be a variable that does not appear in ν(w) for any w ∈ W,
and suppose (towards a contradiction) that
ν(v_0) ⊢(ψ_1 →(ψ_2 →⋯(ψ_n →ϕ[y/x]) ⋯ ))
Then, by 𝙺_𝚠𝚑𝚝𝚘𝙺 (and 𝚁^), we have
ν(v_0) ⊢(ψ_1 →(ψ_2 →⋯(ψ_n →^x ϕ) ⋯ ))
which contradicts the fact that ν(v_0) = ν(w_0) is consistent.
Hence, (ψ_1 ∧(ψ_2 ∧⋯(ψ_n ∧ϕ[y/x]) ⋯ )) is consistent with ν(v_0) = ν(w_0).
Hence, let
ν' = {(w, ν(w)) | w ≠ v_i for all i ≤ n}
∪ {(v_n, ν(v_n) ∪{ϕ[y/x]})}
∪ {(v_i, ν(v_i) ∪{(ψ_i+1∧⋯(ψ_n ∧ϕ[y/x]) ⋯ )}) | i < n}
It is easy to check that ' = (W, R, ν) is still coherent, and does not have the defect (d).
Case 5: (d) is of the kind (d5).
That is, ^x ϕ∈ν(v_n) and ϕ[y/x] ∈ν(v_n), but there is no z ∈𝐗^+ s.t. (ϕ∧ϕ)[z/x] ∈ν(v_n).
Then, let z ∈𝐗^+ be a variable that does not appear in ν(w) for any w ∈ W,
and suppose (towards a contradiction) that
ν(v_0) ⊢(ψ_1 →(ψ_2 →⋯(ψ_n → (ϕ[z/x] →ϕ[z/x])) ⋯ ))
Then, by 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑 (and 𝚁^), we have
ν(v_0) ⊢(ψ_1 →(ψ_2 →⋯(ψ_n → (ϕ[y/x] →^x ϕ)) ⋯ ))
which contradicts the fact that ν(v_0) = ν(w_0) is consistent.
Hence, (ψ_1 ∧(ψ_2 ∧⋯(ψ_n ∧ (ϕ∧ϕ)[z/x]) ⋯ )) is consistent with ν(v_0) = ν(w_0).
Hence, let
ν' = {(w, ν(w)) | w ≠ v_i for all i ≤ n}
∪ {(v_n, ν(v_n) ∪{(ϕ∧ϕ)[z/x]})}
∪ {(v_i, ν(v_i) ∪{(ψ_i+1∧⋯(ψ_n ∧ (ϕ∧ϕ)[z/x]) ⋯ )}) | i < n}
It is easy to check that ' = (W, R, ν) is still coherent, and does not have the defect (d).
Then, we can easily show that every coherent network can be extended into a saturated network.
For any coherent network = (W, R, ν),
there exists a saturated network ' = (W', R', ν') s.t. W ⊆ W', R ⊆ R' and ν(w) ⊆ν'(w) for all w ∈ W.
Let = (W, R, ν) be a coherent network.
It is not hard to see that there are only countably many possible defects.
Hence, we can enumerate them as (d)_1, (d)_2, (d)_3, …
Then, we define a countable sequence of networks _i = (W_i, R_i, ν_i) (i ∈ω) recursively as follow:
* _0 =;
* Given a coherent network _k, let (d)_m be the defect of _k with the least index number (note that according to our definition of coherence, _k necessarily has defects), and let _k+1 = (W_k+1, R_k+1,
ν_k+1) be a coherent network which does not has (d)_m, and also satisfies that W_k ⊆ W_k+1, R_k ⊆ R_k+1,
ν_k(w) ⊆ν_k+1(w) for all w ∈ W_k.
The existence of such a network is guaranteed by lemma <ref>.
Then, let ' = (W', R', ν'), where
* W' = ⋃_i ∈ω W_i;
* R' = ⋃_i ∈ω R_i;
* For all w ∈ W, ν'(w) = ⋃_i ≥ kν_i(w), where k is the least number s.t. w ∈ W_k.
It is not hard to see that ' is a saturated network s.t. W_0 ⊆ W, R_0 ⊆ R and ν_0(w) ⊆ν(w) for all w ∈ W_0.
Then, we show how to induce a canonical model from a saturated network.
Given a saturated network = (W, R, ν), ^c_ = (W_^c, R_^c, D_^c, ρ_^c) is the model induced from , where
* W_^c = {ν(w) | w ∈ W}∪ FC,
where FC = {Θ|Θ is a MCS in ^+(), {ϕ|ϕ∈ν(w_0)}⊆Θ};[
FC stands for Final Cluster.
In fact, we can show that for all Δ∈ W^c and Θ∈ FC, Δ R^c Θ, which justifies our naming.
]
* D_^c = 𝐗^+;
* R_^c satisfies that for all Δ, Θ∈ W^c, Δ R^c Θ iff for all ϕ∈^+(), if ϕ∈Δ, then ϕ∈Θ;
* ρ_^c satisfies that for all Δ∈ W_^c, x̅∈ (D^c_)^<ω and P ∈𝒫, x̅∈ρ^c(P, Δ) iff Px̅∈Δ.
We may drop the subscript when the context is clear.
If we are working in the language _≈(), then we let D^c_ = {[x] | x ∈𝐗^+}, where [x] = {y ∈𝐗^+| x ≈ y ∈ν(w_0)}.
We then show that a model induced from a saturated network is indeed S4.2,
and also has all the properties we need.
For any saturated network , ^c_ satisfies the following:
(i) ^c_ is an S4.2-model;
(ii) For all Δ∈ W^c and ϕ∈Δ, there is some Δ' ∈ W^c s.t. Δ R^c Δ' and ϕ∈Δ';
(iii) For all Δ∈ W^c, Δ has the 𝙼𝚂-property and the 𝙵𝚂-property.
Let ^c_ be an arbitrary model induced from a saturated network = (W, R, ν).
For item (i):
By the definition of R^c and the canonicity of 𝚃^ and 4^, it is easy to see that ^c_ is reflexive and transitive.
We then show that ^c_ is strongly convergent.
Clearly FC ≠∅,
since ⊤∈ν(w_0).
Then, we show that for all Δ∈ W^c and Θ∈ FC, Δ R^c Θ.
Let Δ∈ W^c and Θ∈ FC be arbitrary.
We consider two cases:
Case 1: there is some w ∈ W s.t. Δ = ν(w).
Let ϕ∈ν(w) be arbitrary.
It is easy to see that ν(w_0) R^c ν(w);
hence, ϕ∈ν(w_0), i.e. ϕ∈ν(w_0).
Hence, by definition, ϕ∈Θ.
Thus, ν(w) R^c Θ.
Case 2: Δ∈ FC.
Let ϕ∈Δ be arbitrary.
Then, since Δ∈ FC, ϕ∈ν(w_0), i.e. ϕ∈ν(w_0).
Then, by 𝚃^, ϕ∈ν(w_0), i.e. ϕ∈ν(w_0).
Hence, ϕ∈Θ and
thus, Δ R^c Θ.
Therefore, _^c is strongly convergent.
For item (ii):
Since is saturated, we only need prove that for all Θ∈ FC and ϕ∈Θ, there is some Θ' ∈ W^c s.t. Θ R^c Θ' and ϕ∈Θ'.
Let Θ∈ FC, ϕ∈Θ be arbitrary.
Then, since Θ∈ FC, ϕ∈ν(w_0), i.e. ϕ∈ν(w_0).
Hence, by 4^, ϕ∈ν(w_0), i.e. ϕ∈ν(w_0),
and thus, there is some Θ' ∈ FC s.t. ϕ∈Θ'.
Then, as we have already proved, Θ R^c Θ'.
For item (iii):
Again, since is saturated, we only need to prove that every Θ∈ FC has the 𝙼𝚂-property and the 𝙵𝚂-property.
Let Θ∈ FC and ϕ∈^+() be arbitrary.
First, assume that ^x ϕ∈Θ.
Then, ^x ϕ∈ν(w_0),
and thus, by 𝙽𝙱𝙺_𝚠𝚑𝚝𝚘𝙱𝙽𝙺_𝚠𝚑, ^x ϕ∈ν(w_0).
Then, by 𝙱𝙺_𝚠𝚑𝚝𝚘𝙺_𝚠𝚑𝙱, ^x ϕ∈ν(w_0).
Then, since is saturated, ν(w_0) has the 𝙼𝚂-property, and thus there is some y ∈𝐗^+ s.t. ϕ[y/x] ∈ν(w_0).
Hence, ϕ[y/x] ∈ν(w_0), and thus ϕ[y/x] ∈ν(w_0).
Hence, ϕ[y/x] ∈Θ.
Next, assume that ^x ϕ∈Θ.
Then, ^x ϕ∈ν(w_0),
and thus for all y ∈𝐗^+,
ϕ[y/x] ∈ν(w_0) by 𝙱𝚝𝚘𝙱𝙺_𝚠𝚑.
Hence, for all y ∈𝐗^+, ϕ[y/x] ∈ν(w_0),
and thus ϕ[y/x] ∉Θ.
Then, it is routine to prove the truth lemma:
For all ^c_ induced from a saturated network ,
for all Δ∈ W^c and ϕ∈^+(), _^c,Δ,σ^c ϕϕ∈Δ,
where σ^c is the assignment s.t. σ^c(x) = x for all x ∈𝐗^+.
If we are working in the language _≈(), then in the formulation of the above lemma, we let σ^c be the assignment s.t. σ^c(x) = [x] for all x ∈𝐗^+.
Finally, notice that for any 𝐒4.2^-consistent set Γ of ()-formulas Γ_0,
({w_0}, ∅, {(w_0, Γ_0)}) is a coherent network.
Hence, it can be extended into a saturated network ', from which we can induce a canonical model ^c_', such that ^c_',ν'(w_0),σ^c Γ_0.
Hence, we have the following completeness theorem:
𝐒4.2^ (as well as 𝐒4.2_≈^) is sound and strongly complete w.r.t. the class of S4.2-constant-domain models.
Then, we consider Proposition <ref>.
The cases for ^x ϕ and ^x ϕ are relatively easy, since it is easy to see that x ∉ FV(ϕ), ^x ϕ and ^x ϕ are equivalent to ϕ∧ϕ.
Hence, we only prove the following proposition here:
The following equivalence holds:
𝐒4.2_≈^⊕^x ϕ→^x ϕ = 𝐒4.2_≈^⊕ x ≉y → (ϕ∧ϕ→ϕ)
We first show that x ≉y → (ϕ∧ϕ→ϕ) can be deduced in 𝐒4.2_≈^⊕^x ϕ→^x ϕ.
It is easy to check that ⊢ϕ∧ x ≉y →^z (x ≈ z →ϕ[z/x]), where z is a fresh variable.
Then, by positive introspection, ⊢ϕ∧ x ≉y →^z (x ≈ z →ϕ[z/x]),
and by 𝙺_𝚠𝚑𝚝𝚘𝙵𝚂,
⊢ϕ∧ x ≉y → ( (x ≈ x →ϕ) → (x ≈ x →ϕ)).
Hence, ⊢ϕ∧ x ≉y → (ϕ→ϕ),
and thus ⊢ϕ∧ x ≉y → (ϕ→ϕ).
Hence, ⊢ x ≉y → (ϕ∧ϕ→ϕ).
Then, we show that ^x ϕ→^x ϕ can be deduced in 𝐒4.2^_≈⊕ x ≉y → (ϕ∧ϕ→ϕ).
Equivalently, we show that ^x ϕ∧^x ϕ can be deduced.
Since ⊢ x ≉y → (ϕ∧ϕ→ϕ) for some fresh y,
by 𝙺_𝚠𝚑𝚝𝚘𝙺,
we have ⊢^y(x ≉y) → (ϕ∧ϕ→ϕ).
Then, we first show that ⊢^x ϕ→ (ϕ→ϕ).
On the one hand, it is easy to check that we have ⊢^y (x ≉y) → z ≈ x (where z is a fresh variable), and thus
⊢ϕ[z/x] ∧^y (x ≉y) → (ϕ→ϕ).
Hence, ⊢^x ϕ∧^y (x ≉y) → (ϕ→ϕ) by 𝙺_𝚠𝚑𝚝𝚘𝙺 (and 𝚁^).
On the other hand, ⊢^x ϕ∧^y (x ≉y) → (ϕ∧ϕ→ϕ),
and thus ⊢^x ϕ∧^y (x ≉y) → (ϕ→ϕ) by 𝙺_𝚠𝚑𝚝𝚘𝙵𝚂.
Hence, ⊢^x ϕ→ (ϕ→ϕ).
Then, by 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑 and 4^,
⊢^x ϕ→ (ϕ→^x ϕ),
and thus ⊢^x ϕ→^x ϕ by 𝙺_𝚠𝚑𝚝𝚘𝙺.
Finally, for Proposition <ref>,
we only prove the case for , since the case for is similar.
That is, we prove the following proposition:
The following equivalence holds:
𝐒4.2_≈^⊕𝙼𝙾𝙽𝙾 = 𝐒4.2_≈^⊕ x ≉y → (ϕ→ϕ)
We first show that x ≉y → (ϕ→ϕ) can be deduced in 𝐒4.2_≈^⊕𝙼𝙾𝙽𝙾.
Clearly ⊢ (x ≈ y) ∧ (x ≉y) →ϕ, i.e. ⊢ x ≈ y → (x ≉y →ϕ).
Then, by 𝙼𝙾𝙽𝙾, ⊢^y(x ≈ y) →^y (x ≉y →ϕ).
It is also easy to check that ⊢^y(x ≈ y).
Hence, ⊢^y (x ≉y →ϕ).
Then, by 𝙺_𝚠𝚑𝚝𝚘𝙵𝚂, ⊢ (x ≈ y ∨ϕ) → (x ≉y →ϕ).
Hence, clearly ⊢ϕ→ (x ≉y →ϕ), i.e. ⊢ x ≉y → (ϕ→ϕ).
Then, we show that 𝙼𝙾𝙽𝙾 is admissible in 𝐒4.2_≈^⊕ x ≉y → (ϕ→ϕ).
Since we have x ≉y → (ϕ→ϕ) for some fresh y,
by 𝙺_𝚠𝚑𝚝𝚘𝙺, ^y (x ≉y) → (ϕ→ϕ).
Assume that ⊢ϕ→ψ.
We first prove that ⊢^x ϕ→ (ψ→ψ).
On the one hand, ⊢^y (x ≉y) → z ≈ x (where z is a fresh variable),
and thus ⊢ϕ[z/x] ∧^y (x ≉y) →ϕ.
Then, since ⊢ϕ→ψ,
⊢ϕ[z/x] ∧^y (x ≉y) →ψ,
and thus ⊢ϕ[z/x] ∧^y (x ≉y) → (ψ→ψ).
Then, by 𝙺_𝚠𝚑𝚝𝚘𝙺 and 𝚁^
⊢^x ϕ∧^y (x ≉y) → (ψ→ψ).
On the other hand, clearly ⊢^x ϕ∧^y (x ≉y) → (ψ→ψ).
Hence, ⊢^x ϕ→ (ψ→ψ).
And since ⊢ϕ→ψ,
we also have ⊢ϕ→ψ.
Hence, by 𝙵𝚂&𝙺𝚝𝚘𝙺_𝚠𝚑,
⊢^x ϕ∧ϕ→^x ψ,
and thus ⊢^x ϕ→^x ψ by 𝙺_𝚠𝚑𝚝𝚘𝙺.
|
http://arxiv.org/abs/2307.10220v1 | 20230714213401 | Pandemic, Hybrid Teaching & Stress: Examining Indian Teachers' Sociotechnical Support Practices in Low-income Schools | [
"Akanksha Y. Gavade",
"Annie Sidotam",
"Rama Adithya Varanasi"
] | cs.HC | [
"cs.HC",
"J.4; K.4"
] |
Examining Indian Teachers' Sociotechnical Support Practices in Low-income Schools]Pandemic, Hybrid Teaching & Stress: Examining Indian Teachers' Sociotechnical Support Practices in Low-income Schools
A.Gavade]Akanksha Y. Gavade
Computer Science
Lehigh University
Bethlehem
P.A.
U.S.A.
A.Sidotam]Annie Sidotam
Education, Practice, and Society
University College London
London
U.K.
R.Varanasi]Rama Adithya Varanasi
Information Science
Cornell University
New York
N.Y.
U.S.A.
Support plays a vital role in the teaching profession. A good support system can empower teachers to regulate their emotions and effectively manage stress while working in isolation. The COVID-19 pandemic has ushered in a hybrid form of education, necessitating the acquisition of new skills by teachers and compelling them to adapt to remote teaching. This new development further amplifies the sense of isolation prevalent amongst the teaching community. Against this backdrop, our study investigates the availability of sociotechnical support infrastructures for teachers in low-income schools while also looking into the support practices embraced by this class of teachers following the pandemic. Through 28 qualitative interviews involving teachers, management and personnel from support organizations, we demonstrate how teachers have largely taken the initiative to establish their own informal support networks in the absence of formal support infrastructures. Smartphones have significantly augmented these support practices, serving as both a valuable source of support as well as a medium for facilitating support practices. However, in comparison to other forms of support received from these sources, the availability of emotion-focused support for teachers have proven to be inadequate, creating imbalances in their support seeking practices. Our paper provides different contextual ways to reduce these imbalances and improve the occupational well-being of teachers.
<ccs2012>
<concept>
<concept_id>10003120.10003121.10011748</concept_id>
<concept_desc>Human-centered computing Empirical studies in HCI</concept_desc>
<concept_significance>500</concept_significance>
</concept>
</ccs2012>
[500]Human-centered computing Empirical studies in HCI
[
[
August 12, 2023
===================
§ INTRODUCTION
“COVID-19 came and it forced us to use technology for teaching. No one asked us if we wanted to teach using technology. No one asked us if we could teach comfortably using technology …Now COVID-19 is gone but the technology still remains. I am yet to see someone care about us, reach out to us, and ask us how we are doing and not just ask us to implement another technology for teaching.”
– P09, Government Teacher.
The COVID-19 pandemic had a significant impact on essential workers worldwide, including individuals working in the teaching profession <cit.>. In a remarkably short period, teachers' work practices were disrupted and transformed. A key transformation was the extensive integration of technologies (e.g., smartphones, laptops) and the inculcation of remote-based learning by schools <cit.>. This transition has presented significant challenges for teachers, especially in low-income communities within the Global South, where technology-enabled work practices are just beginning to gain traction <cit.>. Previously considered optional, these technologies have now become critical infrastructure for curriculum preparation, delivery, and other administrative tasks. The top-down integration of technology during the pandemic has further exacerbated the already demanding nature of the teaching profession, contributing to various forms of stress <cit.>. Understanding these stressors and mitigating them, especially for teachers in low-income communities within the context of hybrid technology-based teaching, is crucial for ensuring teachers' well-being and productivity in the classroom <cit.>.
Among other strategies, seeking support has proven to be effective for mitigating stress, thereby highlighting the the need for understanding how teachers in low-income communities engage in support-seeking practices <cit.>. By gaining insights into these practices we can identify potential solutions to alleviate stress and promote more efficient and sustainable teaching approaches in resource-constraint settings. However, most of the prior research on teacher-support has focused on educators in western contexts. There is extremely limited understanding around the sociotechnical support-seeking infrastructures available for the teachers in low-resource schools in the Global South. In order to fill this gap, we studied the support seeking practices of teachers within low-income schools within India.
In particular, we answer the following research questions: RQ-1 What kind of support do teachers seek in a post-pandemic educational setting? RQ-2 What is the the role of technology in facilitating such support? To answer these questions we conducted a qualitative interview study with 28 participants from low-income private and public schools in India, comprising teachers, higher management, and personnel from support organizations assisting the schools (education-focused non-profits and companies). To examine the prevalent support practices manifesting in everyday work, we used a widely accepted framework within support literature that recognizes two types of support seeking practices <cit.>. First, problem-focused support practices which aim to regulate or alter a problematic situation that is contributing to the stressful situation <cit.>. Second, emotion-focused support practices that are directed towards regulating emotional response to an often intangible issue <cit.>.
Our findings show that teachers lacked access to significant form of official support infrastructures to regulate the stress stemming from the transformed teaching practices. Instead, teachers relied on support infrastructures that they developed on their own to deal with new stressors. Smartphone technology played two key roles in this effort. Firstly, the smartphone acted as a source of support for some teachers who were hesitant to reach out to their networks. Secondly, several teachers used smartphones as a medium to receive and provide social support from colleagues at work. We also found asymmetries in the support practices of teachers which indicated that they were skewing heavily towards problem-focused methods. At the same time, teachers also faced challenges in receiving substantial emotion-focused support that could help them regulate their emotions in times of adversity. Our resultant discussion provides three critical ways to reduce the asymmetries in teachers' support seeking practices, namely focusing on the lived experiences of the teachers, reducing power differentials, and improving the sustainability of support structures. Overall, our study contributes to the research of support in HCI context in the following manner:
* We explore two major ways in which smartphones are becoming integral for bottom-up support seeking infrastructures set up by teachers.
* We present asymmetries in these support infrastructures, contributing to skewed support practices.
* We propose three ways to reduce the asymmetries and improve occupational well-being of teachers.
§ RELATED WORK
§.§ Support Practices: Coping Mechanism against Stress
When individuals experience distress (negative stress, hereafter referred to as stress), they attempt to mitigate the effects through cognitive and affective actions <cit.>. Coping theory put forward by <cit.> recognizes such actions as coping or support mechanisms that are employed by individuals in response to the challenges in their well-being. Support mechanisms can manifest in different ways in our daily lives. They can take the form of individualistic responses such as disconnecting from the source of stress and voluntarily regulating one's emotions. They can also manifest socially when individuals rely on other people, groups, or organizations to deal with their stress. <cit.> , in their seminal work, define this as social support theory, a process of “exchanging resources” between individuals where either the provider or the receiver has an explicit intention of improving the well-being of the receiver.
Three key factors play a significant role in the success of support mechanisms <cit.>. Firstly, the structural dimension of support <cit.>. It is determined by the individual's preference for seeking support through artifacts (e.g., smartphone, book), through their relationships with individuals (i.e., peer coping) or through organizations and institutions (i.e., institutional coping).
Secondly, the function of support in an individual's life <cit.>. Support can be problem-focused, directed at “managing or altering the problem causing the stress” <cit.>. It can be informational, sharing different types of inputs to help with stress-management. It can also be instrumental or tangible, providing individuals with explicit assistance to reduce stress. Support can also be emotion-focused, directed at “regulating emotional response to the problem” <cit.>. One such example is emotional support that can provide an individual with emotional warmth and care to reduce their stress <cit.>. Other examples include network support, creating a sense of belonging among people with similar interests <cit.> and esteem support, referring to expressions of “regard for one's skills and abilities” <cit.>. Third factor is perceptual which takes into consideration how an individual perceives and evaluates their social support <cit.>. Prior studies have established a strong positive relationship between social support and overall well-being <cit.>. Multiple studies show that perceived social support by individuals helps in reducing stress and improving their well-being in personal <cit.> and professional contexts <cit.>.
Role of Technology in Social Support
Increasing development of internet and everyday technology has created new avenues for support through online-mediated communications in the form of emails <cit.>, forums <cit.>, (micro-) blogs <cit.> and social networking sites <cit.>. Early research on social support in online communication in the early internet-age has shown contradictory results, skepticism on one side <cit.>, about whether increased internet use corresponds to reduced online connections and therefore reduced support, and other side demonstrating promising outcomes for social support <cit.>. Recent research on ubiquitous devices like smartphones indicate them to be a crucial buffer for people. These technologies have the potential to increase an individual's social presence in online spheres thereby increasing their opportunities to avail support <cit.>. However, there have been several inconsistencies in research about which kind of social support is really effective on online media; informational support <cit.>, emotional support <cit.>, or network support <cit.>.
These inconsistencies have encouraged a narrower focus on individual elements in online spaces to understand their impact on social media <cit.>. A subset of research has focused exclusively on specific ecosystems within the broader internet medium. For instance, researchers have shown how certain applications (e.g., Snapchat) were associated with lower levels of social support than others <cit.>. Another set of studies has focused on more specific features and cues of online communities (e.g., likes, reactions, and emojis), known as paralinguistic digital affordances <cit.>, that provide social connections and support for individuals <cit.>. Research has also focused on diverse actions associated with design features of technology and their affordances to understand how users feel supported. One such instance would be to look at how people perceived minor user actions <cit.>, like receiving a 'like' on their social media post, to be supportive <cit.>. Taken together, the overall research in this area indicates a need for further studies that establish a clear understanding of ubiquitous technology's role in seeking support.
§.§ Support in Work
Though we lack a clear understanding of ubiquitous technologies in the context of support, it is a fact that they have actively proliferated into individuals' work settings. Resultant forms of sociotechnical interactions through these technologies in work is contributing to stress in individuals, leading to new strands of HCI research to study and design technology-supported infrastructures to combat work stress. Some of these early HCI studies focused on applying established theoretical frameworks to measure the amount of support that employees perceived to be receiving in their work environments. Three such popular frameworks are the job demands-resource model that explores work resources of employees in response to job demands <cit.>, conservation of resources theory that explores how workers use social support as a job resource <cit.>, and social exchange theory that explores how individuals reciprocate social support practices <cit.>. More recently, scholars in HCI have focused on examining support practices in the context of specific elements of work life like employee upskilling <cit.>, performance of specific work functions <cit.> and finding new jobs <cit.>. For instance, <cit.> in organizational work contexts found that weaker ties provide better support by bringing informational and instrumental support when individuals are coping with job loss.
Insights from these studies has motivated HCI researchers to study problems inherent in support infrastructures within under-resourced, emotional-labor-intensive job roles such as hospitality industry workers <cit.>, health care workers <cit.>, social workers <cit.>, gig workers <cit.> and teachers <cit.>. For instance, <cit.> developed a computer-mediated intervention focused on peer-support for home care workers called “sharing circles”. The intervention allowed workers to leverage storytelling practice to collectively reflect on their home care experiences and support each other. For the remaining of the section, we will focus on the teaching community around whom this study is situated.
§.§ Support in Teaching Contexts
Teaching is an emotionally demanding and labour-intensive job that requires teachers to spend a significant amount of time in silos while carrying out teaching responsibilities in limited resource environments <cit.>. As a result, teachers experience stress and burnout, contributing to higher attrition rates <cit.>. Technology-induced stress (or technostress) has become an additional source contributing to burnout due to the rapid integration of technology in teaching <cit.>. Post COVID-19, the shift to hybrid form of teaching has made the teaching experience more isolated and stress inducing. For instance, <cit.> show how female teachers who did not have prior experience with technology experienced the most stress. Having an adequate support system in these contexts can be a crucial resource for teachers to improve not only their occupational well-being but also their professional development <cit.>. Most early research in this area has focused on source, type, issues, and impacts of different types of social support. <cit.>, in their work, pose this as a question of “who gives what to whom regarding which issues”. For instance, one strand of research that studied peer interaction between teachers has argued that such practices, both offline and online, facilitated informational, instrumental, and emotional support, thereby reducing the overall stress levels of teachers. <cit.>. Another strand of research studied how traditional support practices (e.g., emotional, instrumental) in offline teaching translated to technology-enabled online peer interactions in the online forums <cit.>. However, these comprehensive support seeking practices have been mostly shaped based on practices broadly resembling that of the Global North.
With proliferation of smartphones and cheap internet in Global South, teaching communities in low-resource settings have also started adopting technology in their profession <cit.>. However, their technology-mediated teaching practices vary significantly when compared to the teachers in the Global North. Firstly, teachers in the Global South mainly rely on smartphones for teaching purposes, as compared to laptops and desktops in the Global North, due to their cheap cost and ease of adoption in resource-constrained teaching environments <cit.>. Secondly, the smartphones used by teachers are often personal devices employed in work contexts, which add complexities that are not applicable to teachers teaching primarily through school-based technological infrastructure (e.g., smartboards, laptops) <cit.>. In addition, the pandemic pushed teachers, who had barely taught any classes remotely, to embrace hybrid teaching through smartphones <cit.>. Early studies indicate how this push has contributed to reduced confidence, burnout, and attrition <cit.>. In this backdrop of changed landscape of teaching, it is essential to understand (1) the kind of sociotechnical support infrastructures that are prevalent and (2) the support practices that such infrastructures encourage teachers to take up in response to their stressors. Our work contributes to this research by exploring the support practices of teachers in low-income schools in post-pandemic settings. In particular, our work sheds light on the role of smartphones in facilitating the support activities.
§ METHODS
To answer our research questions, we conducted a qualitative interview study spanning five months with diverse stakeholders in low-income schools. We describe the IRB-approved research in detail.
§.§ Recruitment
We recruited a total of 28 participants, including fourteen teachers, five higher management personnel from private and government schools that catered to low-income communities, and nine individuals from different education companies and non-profit organizations, which assist low-income schools in various capacities (hereafter collectively referred to as support organizations, see Table-<ref>). Following the self-selection sampling technique <cit.>, we advertised our study on WhatsApp groups, social media groups (e.g., Facebook), and offline social networks that were used by teachers and higher management from low-income schools. We also arranged informal information sessions with individuals who reached out with potential interest. We recruited final-study participants from among the individuals who signed up for our study. To be eligible for the interview, both teachers and higher management had to have been actively working in the last three months. To provide additional detail and context to the perspectives of teachers and higher management, we also recruited support organization personnel who worked in the participants' schools. To recruit personnel, we leveraged our long-standing networks and reached out to individuals who were either working or had recently worked in the participants' schools. Shortlisted organization personnel had worked for a minimum of three years (avg.=6.1 years, max.=10 years, S.D.=2.6 years) with the school faculty.
Procedure Interviews were conducted in-person or remotely over phone, WhatsApp call, and Zoom, based on the interviewee's preference. A brief pre-interview call was set up to inquire about the participants' health, well-being, availability for a conversation and their preferred mode of communication. Participants were always provided with the option to reschedule or cancel the interview. The interview protocol was divided into three parts: we began the interview with high-level questions that acted as ice-breaking questions, encouraging the participants to share their current standing in their work, including their experiences and challenges in the last two months (e.g., “How has your work experience been over the last two months?”). The next set of questions were intended to capture the different mechanisms that teachers employed daily to cope with the stress that they experienced at work, as well as to understand the immediate challenges they encountered around such practices (e.g., “Can you describe a recent example when you asked help from someone through the smartphone?”). We also asked follow-up questions to better understand the support mechanisms employed by teachers (e.g., “Can you explain an instance where you had sought help from the management?”). The final set of questions dealt with the different factors that influenced participants' support practices (e.g., “What issues did you experience from your peers on [WhatsApp] while asking for help?”). For the interview sessions with higher management and support organization personnel, we extended this inquiry with a set of questions to triangulate the teachers' practices (e.g., “What kind of medium did teachers prefer to receive support from you? ”) . In addition, we also inquired about their motivations and hesitations regarding the coping mechanisms employed by the teachers. All the questions were open-ended, free of technical jargon, and designed to be neutral rather than leading. Interviews were conducted in Hindi, Telugu, or Marathi by the authors.
Data Collection & Analysis
Our data consisted of 32 hours of audio-recorded interviews and 45 pages of field notes. All the interviews were translated into English, transcribed and analyzed using thematic analysis in MAXQDA. The process of analysis involved all authors taking several passes on the transcribed interviews to familiarize themselves with the participants' experience and internalize their narratives. All authors conducted open-coding on the transcription without any preconceived theoretical assumptions. The authors established credibility by engaging with the data over several weeks. Significant disagreements between authors during the process were resolved through multiple rounds of peer-debriefing <cit.>. The authors chose the final codes carefully by merging the overlapping codes. The final codebook consisted of 54 codes. Example codes included “informal agreements”, “trust issue”, and “support temporality”. The codes were further mapped and categorized into appropriate themes. Example themes included “support apprehensions”, “support layers”, and “dual nature of support”.
Ethical Considerations
Our overall interpretations in the paper are shaped by our education and research experience. While all authors have received education in western contexts, they also have significant experience teaching in India, with two authors spending over four years teaching in these settings. All the authors are of Indian origin. All but one are women. The interviews were conducted by male and female authors to balance individual biases and capture diverse perspectives.
§ FINDINGS
Our overall findings reveal significant inconsistencies across different types of support available to teachers. The majority of the participants (n=22) reported a lack of any formal support, whether remote or in-person, both before and after the pandemic. Instead, teachers were expected to develop and rely on their own informal support structures. This outlook stemmed from the managements' expectation that teachers should be resilient and self-reliant. Several principals and school board members used words like “selfless” and “community-focused” to characterize the teaching profession, while also placing the responsibility on teachers to reach out and seek the needed support. As P12, a government school principal pointed out, “Teachers know when they need help. We are always available. If they have any issues, they can always reach out to us”. Due to the lack of explicit support infrastructures, we observed a great variability in support activities and the associated role of smartphones in them. In the following sections, we present two major ways in which teachers utilized smartphones to develop their own bottom-up support practices.
§.§ Seeking Support in Smartphone Apps: A Sign of Self-reliance & Strength
A fraction of teachers nurtured their own support infrastructures by relying on smartphones as a source of support. For a subset of teachers (n=3), this motivation stemmed from their hesitation to reach out for social support from their professional network. They perceived reaching out for support as a sign of imperfection or weakness in an otherwise idealistic job – a job that is expected to provide selfless emotional labor. These teachers conformed to this ideal image by solely providing assistance instead of asking for it from their colleagues, management, or family members. A few teachers also feared that seeking support could be perceived by the management as a sign of incompetency. For example, P25, a Math teacher at a low-income private school with a teaching experience of over twelve years, spoke of the challenge of adapting to the changing curriculum post-pandemic. Despite facing difficulties with the new curriculum, he hesitated to seek support as he believed it might undermine his expertise gained over the years. Instead, he focused on finding solutions on his own by watching YouTube videos of others teaching the same curriculum. Another teacher P18, teaching English in a government school, shared how she used smartphone as a source of informational support and a mode of distraction:
“I can’t take support from others as many people depend on me for all kinds of support …they think I am strong because I give good advice and help them. So, I don’t feel like showing them my vulnerable side. Most of the times, I try to handle my problems myself …In school or at home, I sit in a corner and watch video on how to teach difficult poems to kids. Sometimes, if I have a lot of stress, I have an app to write daily diary, but it's not possible every time. Then I just listen to music or read Shayari (Urdu poetry) on phone. Expressing myself by writing status on WhatsApp gives me great relief too.”
Similarly, interviewees shared how the onset of COVID-19 pushed teachers to work in isolation while trying to understand and incorporate new pedagogical norms. They were also taking on the additional workload of teachers who were leaving their jobs. The resultant pressure left them too exhausted to make the emotional and physical effort to seek out social support. Moreover, teachers shared how they realized that their peers were in a similar situation, fearing rejection if they reached out for any form of support. Post COVID-19, teachers experienced similar challenges when schools reopened. They had to attend to critical issues like helping students recover from longitudinal learning gaps that emerged during the pandemic, in addition to embracing new forms of hybrid education. It meant that teachers had to forfeit their time for socialization, like lunch- and tea-breaks, to take up full-day teaching schedules, leaving fewer opportunities for social support.
§.§.§ Problem-focused Support through Smartphones
The reasons outlined above made it easier for a subset of teachers to seek support directly from smartphones instead of reaching out to their social networks. In particular, several teachers relied on their smartphones for problem-focused support. To cope with the pressure of unanticipated pedagogical changes, such as introduction of remote teaching, several teachers relied on Google Answers and YouTube videos to find solutions to their problems with technology-mediated teaching practices. For instance, P28, a Science teacher in a low-income private school used YouTube videos in her local language to receive informational and instrumental support and to find solutions for technological issues that she was facing while teaching an online class:
“I don't ask for help…I don't think the other teachers can help me. They are struggling themselves…During COVID-19, I had issues playing video in Zoom, there was no audio. Google and YouTube were the only place for solutions when I was breaking my head over the problem. YouTube is effective because it is like someone is telling me where to click without me having to be burden to someone else…The downside is that it is either a hit or it is a miss which consumes a lot of time”
Despite the significant time it took to find solutions from the videos, the accessibility of smartphone-based interactions made teachers like P28 feel autonomous and self-sufficient. The assistance was similar to being helped by another peer, without having to feel that they were being a "burden" on colleagues and family. Such context-based online resources also allowed teachers to not feel privy to the judgment and potential gossip between their peers, that would create further stress.
§.§.§ Emotion-focused Support through Smartphones
Several teachers also utilized smartphones for emotion-focused support. One commonly used strategy by teachers for emotion-focused support was technological distraction. When teachers found an issue overwhelming and stressful, they distracted themselves from their negative stressors for brief periods of time by engaging in digital activities on their smartphones. These distractions included switching to personal tasks such as browsing Instagram, watching YouTube videos, or listening to songs.
Another emotion-focused strategy was the usage of smartphones to distance themselves from the person or the event contributing to their stress. A common strategy included changing indicators that made them invisible online while working from home on apps like Zoom, WhatsApp, Google Meet, and Microsoft Teams. It helped them avoid stressful engagements with peers and management by distancing themselves in the online space while continuing their own work. In extreme cases, strategies were also aimed at completely disconnecting themselves from work through custom apps designed to silence WhatsApp, text messages, and call notifications from the management. P14, the principal of a government school, shared how teachers in her school used such mechanisms frequently to disconnect from troublesome parents who negatively impacted their work lives and made them feel “helpless”.
§.§ Seeking Support From Social Connections: Smartphone as a Medium
While a few teachers used smartphones as a medium for self-reliance, most teachers (n=11) relied upon smartphones to be a medium through which they could build their informal support infrastructures and seek social support from their professional (peers, management and support organizations) and personal (family members and friends) connections. While teachers already had the option of seeking support from their networks through smartphones prior to COVID-19, it was only after the onset of the pandemic that they embraced the option completely, given the challenge of remote learning. For these informal infrastructures to be successful, the teachers who were seeking support and the individuals who were providing support had to rely on common cues that both the former and the latter understood. The first category of cues were subtle changes in work-related attitudes or behavior of an individual, including lack of attention, increased temperament, careless mistakes and/or nonchalant attitude towards their students. For example, when a peer who is otherwise punctual and meticulous, began sharing irregular attendance sheets, made silly calculation errors, and seemed distracted on calls during lunch breaks, P19, a Hindi language teacher in a government school, saw these actions as cues that prompted her to reach out to her peer to provide assistance.
The second category of cues included smartphone-mediated actions that indicated signs of stress. These cues included unusual WhatsApp and Facebook status changes, abnormal profile pictures updates, melancholic forwards, repeated deactivation and reactivation of profiles on social media, leaving work WhatsApp groups, and being offline for lengthy periods of times. P09, a veteran government Science teacher shared how she became concerned about her peer's well-being and reached out to offer support when her peer suddenly removed her husband's photo from her WhatsApp profile picture and regularly went offline for long periods of time.
§.§.§ Problem-focused Support from Peers through Smartphones
Teachers frequently sought problem-focused support from their networks, utilizing smartphones as the primary medium. These behaviors arose as a means to address the overall work stress stemming from various issues, particularly the shift to online teaching during the onset of the COVID-19 pandemic. Within this support category, we observed a few distinct patterns of support-seeking behavior among teachers.
Information Support
One of the biggest challenges during the COVID-19 pandemic was the information asymmetry experienced by teachers working in low-income private schools in comparison to government and high-income schools. Critical information, required for teachers to fulfill their responsibilities, come from the management headquarters that oversee the functioning of the schools and their curriculum. These include school schedules (e.g., “Is the government going to open the schools or not?”), examination schedules, guidelines, assessment instructions, attendance weightage, extracurricular requirements, and professional development opportunities. This information flowed through many top-down channels before reaching the low-income schools. While teachers in government schools had strong information channels (e.g., state-based WhatsApp groups <cit.>) that helped them get guidance during and after the COVID-19 pandemic, most teachers in low-income private schools had to rely on information support from different sources. One way to fill their information gap was to establish connections and seek support from teachers in governments schools who received the information faster. P16, an English teacher in a low-income school shared:
“We have [name omitted] Ma'am, who has worked in different schools before. She has excellent network with the teachers from other schools. When we were all struggling in COVID-19, she got the syllabus and exams schedule information very fast from the government school teachers on WhatsApp. Problem is that we are a low-income school …By the time our principal gets information and shares with us, we have very less time to do anything and that puts pressure on us …She also later introduced me to one of the teachers in the government school …now I have my own network. I ask that new teacher.”
”
Information support extended beyond managing information asymmetries to cope up with managements' increased demands and overall loss of teacher workforce. The management introduced several disparate tools that were supposed to act as a `unified learning management system'. For example, one combination that a few schools tried was using Zoom for teaching, Google Forms for capturing attendance and conducting exams, and WhatsApp for coordinating corrections and classroom management. Teachers found it extremely stressful to adjust with several new tools and online pedagogical practices at the same time. As a result, they relied on personnel from support organizations who were in touch with them during the COVID-19 pandemic. Even though the support personnel were not present in-person, they provided much required information support to fill the gaps in teachers' mental models on how to use the tools.
To manage the unrealistic work expectations from managements, teachers also requested their peers to take up the role of information intermediaries to share their own online work load. P11, a Social Science teacher in a government school, shared how he sought information support from his peer to attend administrative meetings with the management during an extremely busy work day, while he combined his class with his peer's class and taught everyone together. The teachers sought and acted on the meeting information that they received from their peers later. The unpredictability of the COVID-19 pandemic, beyond work life, also extended teachers' information support seeking practices with their peers and support organizational personnel to their personal lives. One of the most common modes of information support sought by teachers working in affordable private schools was career advice as these schools were drastically reducing teacher workforce or even closing down altogether due to lack of fee payment by the parents. Several teachers reached out to support personnel for information, seeking leads on teaching opportunities, strategies to upskill themselves, and feedback on their application material.
Instrumental Support
The post-COVID-19 hybrid teaching practices also put significant pressure on the teachers' work lives. Consequently, several teachers relied on instrumental support in the preparation phase of their work to learn and troubleshoot the new technology-based work practices, including figuring out how to conduct classroom management activities such as disciplining elementary school students through screens, coordinating structured responses, encouraging participation, and seeking support of parents in the process. Teaching through technology required them to design learning materials for remote teaching and make formative assessments. For example, P15, a new Mathematics teacher who joined a low-income private school during the pandemic, constantly struggled with taking online attendance and managing a classroom of thirty children on Zoom. P15 shared how she took the support of their peers, who hopped on to WhatsApp or a Zoom call to help her take attendance while she managed the classroom and focused on teaching. The teachers used this information to follow-up with the absent students. P27, who helped teachers similar to P15 shared:
“I also support her [a teacher] through virtual mode. She does not know much about computers and technology …but she reaches out to me and messages me about making a PowerPoint presentation and Google Forms. I made it and sent it to her WhatsApp and she used it in her classroom. If we have to enter the marks, admin staff from school guides us. We just follow the same process. If we are not in school, then we take the help of a peer or our brother or sister and do it.”
Unlike informational support, we found that teachers sought instrumental support from individuals in both professional and personal networks. For example, teachers took help of their friends and family members for support around technological features and issues of smartphones while they sought support from their colleagues on how to apply the technology to pedagogical practices. Common examples in this category included, “how to present materials to the classroom”, “how to manage young students”, “how to plan and execute online in-class activities” remotely. Balancing instrumental support between professional and personal connections helped teachers deal with the increased technostress during the height of the pandemic. In extreme situations, COVID-19 pushed several teachers' families into health and economic volatility compelling teachers to seek financial assistance from colleagues working in governments schools who had relatively stable jobs.
While teachers successfully reached out for information support both before and during the COVID-19 pandemic, teachers faced substantial setback in receiving instrumental support during the pandemic. This was because instrumental support often had to be sought remotely through smartphones and often the support was for learning how to use smartphone for teaching, creating an additional layer of complexity. For example, teacher P18
described what the instrumental support for teachers before the COVID-19 pandemic entailed:
“I used to stay with the new teacher during their initial classes, introduced the new teacher to students and sat there along with the teacher throughout the first class to build the trust and bond between them. …Post class, I used to give them feedback about their teaching style and their learning materials and provide assistance when they were teaching by managing the classroom, like motivating children to answer the questions. These small but important practices have become extremely difficult to carry out after COVID-19. ”
In-person instrumental support was more effective as the focus of the support was entirely on providing assistance around teaching and classroom management. The same teacher explained how her instrumental support practices shifted to technology troubleshooting and low-level coordination activities that made providing support less rewarding,
“The help has shifted to telling the teacher that the child has muted the mic, you have to call the child's name, ask them to unmute. If a student who is not part of the subject joins your class by mistake, then how to remove the child. I have to show them two or three times before they can start doing themselves …help is more about about non-teaching stuff.”
To overcome this challenge, many teachers began providing a hybrid form of instrumental support for technology-enabled teaching. Teachers initiated this support by sending informational resources about the topic in question through WhatsApp or email. Once the teacher reviewed the information resource, they shaped their understanding by providing contextual information over a call or in-person meeting. For example, when a support organization personnel, P01, saw that an English teacher was struggling to teach the concept of simile and was stressed because of the pressure from the principal, she suggested teaching resources that could help her learn new techniques. She then called her and then walked her through example classroom scenarios and ways implement those resources.
§.§.§ Emotion-focused Support from Peers through Smartphones
Teachers' informal support infrastructures were skewed towards receiving and providing problem-focused support. Compared to these practices, we found limited development of emotion-focused support practices in both offline and online mediums. A key hesitation for the teachers in seeking emotion-focused support was the lack of confidentiality and nurturing environment that the workplace provided. In the beginning of the pandemic, top-down hierarchies promoted teacher productivity to improve student outcomes, often at the cost of their own personal space. Teachers were encouraged to compete with their peers to demonstrate improved outcomes in their subjects. Surveillance structures, such as “teacher monitors”, were established to ensure that teachers were performing up to their potential without any distractions. Teachers and support organization personnel shared how these practices created a culture of gossip and rumors that brought a sense of apprehension and curtailed teachers from sharing problems that were emotion-focused with their peers. <cit.>, in their recent study, showed similar top-down structures that didn't have any provision of safe spaces, contributing to emotional stress amongst teachers and pushing them to engage in emotion-focused support practices. New hybrid education mediums moved these competitive structures and surveillance practices online during and after the COVID-19 pandemic, increasing apprehensions to seek support.
Support organizations, that reached out to different schools to assist teachers in transitioning to hybrid teaching during the pandemic, had to put in a lot of effort to help teachers overcome their fear of sharing their emotional experiences in anticipation of retaliation from management and peers. Support person P06, working in a start-up focused on overall school capacity development, shared the required effort in her own words:
“I really had to convince teachers that I was on their team and I wasn't a spy for school management. They are definitely wary of the school management because the management also tends to treat these teachers like kids. They scold the teachers and say, `this is not done right' or `sit straight in a zoom call' …COVID-19 has brought out a weird way of disciplining teachers in the school …But when the teachers realized that the conversations that we were having were indeed confidential that I haven't gone and complained, which meant no repercussions for them. So being patient really helped me build that trust.”
Unlike problem-focused support, emotion-focused support practices required teachers to establish profound trust and foster strong bonds before seeking assistance. Even after cultivating the necessary trust, teachers adhered to distinct practices that set emotion-focused support practices apart from other forms of support. For instance, some teachers favored seeking support offline or via voice calls instead of text messages, apprehensive that written communication could be used as evidence and potentially lead to repercussions from management.
Network & Esteem Support
Within the limited instances of emotion-focused support practices that the teachers shared, network support was quite prominent (n=4). Network support is a type of emotion-focused support that focuses on providing oneself with a sense of belonging with individuals with similar interests or commonalities <cit.>. Teachers who were uncomfortable or new to technology-mediated teaching received assistance from support organization personnel in forming offline groups. They achieved this by calling teachers in their neighborhood districts to find other teachers who were struggling to share their challenges. While these groups consisted of teachers who didn't know each other previously, the groups provided a sense of solidarity and encouragement to come together and talk about the challenges that they were encountering in their preparation and teaching process. P21, who taught Telugu language in a low-income private school, shared how she, along with seven more teachers in other schools across the district, curated and maintained a contact list of senior teachers who joined their calls and provided the required emotional support to new teachers joining the network. Teachers who were struggling with the administrative aspects of their jobs, such as not receiving regular salaries in the early periods of the COVID-19 lockdown, also formed common WhatsApp groups to find a sense of togetherness and reduce anxiety around the lack of information and clarity in their professional lives.
In addition, support organizations personnel also provided the much needed esteem support for teachers, validating the teachers' skills and bolstering their intrinsic sense of purpose <cit.>. Examples of this mainly included supporting teachers in instances of failure or when they made mistakes. Support personnel shared how school managements rarely promoted learning-by-failure <cit.>, a key concept in education where teachers are provided a safe space to learn through failing and to grow from their mistakes. Support organizations gave the appropriate space for teachers to reach out in their moments of failures and receive validation of their teaching skills. Common situations included when teachers struggled to teach a post-pandemic classroom with large variability in learning skills or lost confidence in their own pedagogical and content knowledge. P28 shared how a personnel from a support organization used to “encourage” her remote teaching ideas and skills even though she was struggling with it. Another place where teachers used to receive their esteem support from was over impromptu online meetings with teachers from other schools whom they had not met in a long time. Teachers shared how such unexpected encounters initiated formal conversations that were focused on solidarity, where teachers came together to bolster their own confidence in their teaching skills and to encourage each other to keep on working.
§ DISCUSSION
The lack of formal support infrastructures for low-income teachers in HCI4D settings is a cause for concern, as indicated by our research findings. Prior studies have highlighted the positive impact of a proper support system on teachers' overall stress management capability, particularly in relation to technology-induced stress <cit.>. The absence of formal support further hinders teachers' ability to effectively alleviate stress. This becomes particularly challenging in a post-pandemic setting within India, where teachers are grappling with the demands of hybrid teaching. In this challenging context, teachers' own grassroot level informal support networks helped regulate some of these stressors. However, these networks have their limitations and cannot fully address the issue. For instance, teachers tend to be hesitant in seeking emotion-focused support, resulting in imbalanced support structures. In this section, we present our initial recommendations to bridge this gap and improve teachers’ occupational well-being. To achieve this, we seek inspiration from <cit.>’s framework to root our recommendations in the local contexts of teachers. <cit.> define their framework for localized technology-mediated forms of care where sociotechnical practices are grounded in the lived experiences of individuals who need care. It is focused on reducing power differentials and promoting structures that are sustainable by being considerate of community contexts and constraints.
§.§ Tapping Lived Experiences: Exposing Teachers' Collective Narratives
A key reason for the absence of official support structures from the management is the lack of a clear visibility of teachers' issues from their perspective. With hybrid work, activities are often hidden from the direct purview of the employer <cit.>. This also means that the stressors arising from work are also invisible, leading to an inaccurate impression that the employees do not need support. To rectify this situation and provide better support, it is essential to increase the visibility of the lived experiences <cit.>.
A common solution to this issue in the western-work contexts has been the implementation of monitoring-based solutions to increase worker visibility. However, such solutions are often misused by the management to enforce status quo, giving rise to the risk of surveillance practices that can in turn create new sources of stress <cit.>. An alternative approach involves bringing to light the collective narratives of teachers’ lived experiences, allowing for a more nuanced understanding of their challenges while avoiding the pitfalls of surveillance. For instance, collecting and disseminating oral histories at school or district level can expose issues that were not previously visible to the managements’ attention. Such initiatives can also challenge the structural assumptions that teaching roles are inherently noble and selfless which push teachers to disregard any kind of hardships that arise in their work. This shift in perspective is essential to foster a supportive and empowering environment for teachers.
§.§ Reducing Power-differentials: Support Organizations as Mediators of Trust
To develop collective narratives, it is essential for teachers to trust their peers, higher management, and the overall work ecosystem. Our findings indicate a sense of eroding trust on the side of teachers. In particular, teachers found their peers to be less reliable and the management to lack openness to seek different forms of emotion-focused support. Cultivating these qualities is crucial for creating a high-trust environment in schools <cit.>. But in reality, we saw how informal but critical support structures that required high levels of trust, were not utilized as much. To fill this gap, personnel from support organizations can serve as invaluable resources. By operating outside the traditional hierarchical structures of the school ecosystem <cit.>, they occupy a neutral position in schools, enabling them to provide support in unconventional ways. For example, support personnel can have more flexibility than the management to engage in meaningful conversations with the teachers outside of regular school hours, where teachers may have more time and feel more comfortable seeking emotional support. Furthermore, support personnel can act as crucial intermediaries between teachers and management, as well as peers, by offering anonymous and collective feedback on the challenges teachers face. This process can reduce burden on the management, allowing them to provide support for certain issues (e.g., problem-focused support), complementing the support that organization personnel can provide (e.g., emotion-focused support)
§.§ Sustainable Sociotechnical Support Structures: Hybrid Solutions for Hybrid Future
In this final section, we present recommendations for researchers and designers who can play an active role in developing sociotechnical interventions and can better assist teachers’ emotion-focused support requirements. Popular technology-based interventions (e.g., chatbot assistance <cit.>) that work well to provide support in western contexts do not work in resource-constrained settings. Moreover, these interventions can add to the cognitive stress experienced by teachers who are already grappling with the challenges of technology-mediated hybrid teaching. To address these concerns, previous researches have emphasized the importance of carefully considering technologies in such contexts to mitigate technostress and burnout among teachers <cit.>. Building on this line of research, we propose the notion of collective hybrid support interventions that balance subject matter expertise and the localized contexts to deliver effective support. This approach involves multiple actors working in synergy to deliver support. For instance, in the case of emotion-focused support, collective hybrid support interventions can bring together teacher experts who have extensive experience in handling emotion-focused issues but lack local contexts and support personnel who posses teachers’ trust and are aware of the local contexts and norms (e.g., cultural taboos associated with mental health) but lack the expertise in mental health to augment each other and provide holistic mental health support for teachers.
§ LIMITATIONS AND CONCLUSION
Overall, our study provides exploratory insights into how teachers in low-resource settings establish and nurture their sociotechnical support infrastructures in post-pandemic work settings. In the absence of top-down support infrastructures, teachers often took it upon themselves to establish their own practices. Smartphones played a key role in these practices by acting both as a source of support as well as a medium to connect teachers with their networks to seek social support. Out of all types of support, teachers struggled the most to find adequate emotion-focused support as compared to problem-focused support, presenting an excruciating need for sociotechnical interventions that can integrate support organizations to provide contextual emotion-focused support. We end our paper by providing key recommendations to fulfill this need. Our work has usual limitations posed by qualitative research such as the sample size being small. Also, the primary focus of our study being the experiences of teachers in India, limits its generalizability beyond the Global South.
§ ACKNOWLEDGMENTS
We are grateful to our participants who provided valuable time and experiences for this study. We thank Nitya Agarwala who contributed to this research in its early stages. We also appreciate the guidance and feedback provided by Dr. Nicola Dell to make this study a success. Finally, we also thank the external reviewers for value feedback on the paper.
ACM-Reference-Format
|
http://arxiv.org/abs/2307.04788v1 | 20230710180002 | Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations | [
"Linhao Li",
"Masaki Oshikawa",
"Yunqin Zheng"
] | cond-mat.str-el | [
"cond-mat.str-el",
"hep-th"
] |
equationsection
claim [linecolor=black!0,backgroundcolor=black!10]
==
figure[1][]
[#1]
[linecolor=black!0,backgroundcolor=black!1]
ℝ
ℤ
𝒞
U
SU
#1#1
Im
𝕀id
triv
#1#2#2_[#1]
#1#1
#1⟨#1⟩
#1#1
𝒜
ℱ
ℒ
𝒜
ℬ
𝒞
𝒟
ℰ
ℱ
𝒢
ℋ
ℐ
𝒥
𝒦
ℒ
ℳ
𝒩
𝒪
𝒫
𝒬
ℛ
𝒮
𝒰
𝒱
𝒲
𝒳
𝒴
𝒵
shapes.multipart
decorations.pathmorphing
snake it/.style=decorate, decoration=snake
tikzmark
#1#1
#1#1
#1#1
#1#1
#1#1
#1#1
3cm
Intrinsically/Purely Gapless-SPT
from Non-Invertible Duality Transformations
1cm
Linhao Li^1, Masaki Oshikawa^1,2, and Yunqin Zheng^1,2
1cm
^1 Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan
^2 Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
1cm
The Kennedy-Tasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work <cit.> on a ring by sacrificing the unitarity, and should be understood as a non-invertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations.
§ INTRODUCTION AND SUMMARY
Overview of gapless SPT:
Gapless symmetry protected topological (SPT) systems have received intensive attention recently <cit.>, which are a family of gapless systems exhibiting topological features analogous to the gapped SPT phases.
So far, there are a number of constructions of gapless SPT systems, which we briefly review here. The authors of <cit.>, for the first time, constructed the gapless SPT (gSPT) using the decorated defect construction<cit.>, for example, with G× H global symmetry. The construction applies to any dimensions. The idea is to decorate the G defect of the G symmetric gapless system/CFT (with one vacuum) with a H gapped SPT. The decorated defect construction introduces a gapped sector, under which both H and G act. However, H does not act on the low energy gapless sector. A common feature of the gapless SPT of this kind is that the same topological features (including non-trivial ground state charge under twisted boundary conditions) can also be realized in a gapped G× H SPT, hence is not “intrinsic" to gapless SPT <cit.>. It has been shown that this type of gSPT states can exist at the critical points/regime between spontaneously symmetry breaking (SSB) phases and gapped SPT phases. Recent research proposed that gSPT states at criticality between the Haldane phase and various SSB phases can have a possible experimental realization in the lattice systems with bosons <cit.> and fermions <cit.>.
In <cit.>, the authors proposed a systematic construction beyond the previous one, where the non-trivial topological features do not have a counter part in the gapped SPT, hence is termed the intrinsically gapless SPT (igSPT). The schematic idea is as follows. The total symmetry is Γ, fitting into the extension 1→ H→Γ→ G→ 1. One starts with a G symmetric gapless system/CFT (with a unique vacuum) with a G self anomaly ω_G. One further stacks an H SPT on top of G defects. However, due to the non-trivial group extension, the induced gapped sector has an opposite G anomaly -ω_G. This cancels the anomaly in the gapless sector, and the combined system is Γ anomaly free. The combined system is termed igSPT. By construction, the igSPT also contains a gapped sector coming from the decorated defect construction. Moreover, the topological features can not be realized in a Γ gapped SPT, which justifies the name “intrinsic". In <cit.>, we also analyzed concrete analytically tractable spin models of the gSPT and the igSPT in detail, with _2×_2 symmetry and _4 symmetry respectively in (1+1)d. These two models will be crucial for the discussions in the main text below. Recently, the authors of <cit.> discussed a more systematic construction. Besides, the authors of <cit.> constructed a spin-1 model which hosts both gSPT and igSPT simultaneously. A 2+1 dimensional igSPT example at the deconfined transition between a quantum spin Hall insulator and a superconductor is discussed in more detail in <cit.>.
Both the gSPT and the igSPT contain a gapped sector by construction, which results in the exponential decaying energy splitting of edge modes. A natural question is whether there are gapless SPTs without gapped sectors, hence is purely gapless SPT? In <cit.>, the authors studied one interesting model with time reversal symmetry, and demonstrated that this model does not have a gapped sector by noting that the energy splitting under OBC is polynomial. Moreover, the ground state also exhibits non-trivial topological features that admit a gapped counterpart. Following <cit.> we name it as purely gapless SPT (pgSPT). Moreover, in <cit.>, the authors also put forward an open question about the existence of an intrinsic gapless SPT without gapped sector, i.e. intrinsically purely gapless SPT. In the present work, we answer this question in the affirmative. Finally, one may naturally wonder if there is a more systematic construction of a family of pgSPT and ipgSPT analogue to the decorated defect construction of the gSPT and the igSPT. We confirm this by constructing all the above gapless SPTs using the Kennedy-Tasaki transformation, for simple symmetries. A more systematic exploration with general symmetries will be left to the future.
We summarize the current understanding of gapless SPT in Table <ref>.
Kennedy Tasaki transformation:
The Kennedy-Tasaki (KT) transformation was originally found to map the Haldane's spin-1 chain in the _2×_2 symmetry spontaneously broken (SSB) phase to a “hidden _2×_2 symmetry breaking phase" <cit.>. Such a transformation was first found to have a simple compact form
U_KT=∏_i>jexp(iπ S^z_i S^x_j)
by one of the authors of the present work <cit.>. It is a unitary and highly non-local operator. The “hidden _2×_2 symmetry breaking phase" is now well-known as the Haldane phase or _2×_2 gapped SPT phase <cit.>. It is one of the earliest realizations of symmetry protected topological phenomena in condensed matter physics.
The KT transformation (<ref>) discussed in <cit.> was for spin-1 systems with open boundary condition, and how to define it on a ring remains open until recently.
In our recent work <cit.>, we proposed a way to realize the KT transformation on a ring, and discussed both spin-1 and spin-1/2 chains, which were proven equivalent.
We note that,
the KT transformation generalized to a spin-1/2 in two dimensions with a subsystem symmetry was discussed in <cit.>,
apparently while being unaware of the original KT transformation found in 1990s.
Interestingly, the KT transformation on a ring is implemented by the non-unitary operators and obeys the non-invertible fusion rule, while it becomes a unitary transformation under a suitable boundary condition on an interval. We thus name the KT transformation as a non-invertible duality transformation. The effect of KT transformation is to gauge the _2×_2 with certain topological twists which we specify in the main text. [The topological operator implementing the twisted gauging has been discussed extensively in terms of duality defects. See the recent reviews, e.g.<cit.>, and references therein for more details. We emphasize that the KT transform in the present work maps one theory to another, and is not a symmetry of a single theory. Relatedly, the Kramers-Wannier (KW) duality transformation between two theories and the KW duality operator within one theory have been discussed in <cit.> and <cit.> respectively. ]
For convenience, we will focus on spin-1/2 systems throughout.
Constructing (gapped and gapless) SPTs from Kennedy Tasaki transformation:
In this work, we propose to use the KT transformation to construct known examples of gSPT with global symmetry _2×_2 and igSPT with global symmetry _4, and show that our construction automatically gives rise to decorated defect construction in <cit.> when gapped sectors exist. We also apply the KT transformation to construct possibly the first examples of pgSPT and ipgSPT with only the on-site global symmetries (and not time reversal symmetry). The similar role of the original KT transformation in the gSPT of the spin-1 system is also discussed in the work <cit.>.
We summarize the main results as follows.
_2^σ SSB + _2^τ SSB ⟺ ^σ_2×^τ_2 gapped SPT
_2^σ SSB + _2^τ trivial ⟺ _2^σ SSB + _2^τ trivial
_2^σ trivial + _2^τ trivial ⟺ _2^σtrivial + _2^τtrivial
_2^σ Ising CFT + _2^τ SSB ⟺ ^σ_2×^τ_2 gapless SPT
_2^σ Ising CFT + _2^τ Ising CFT ⟺ SPT-trivial critical point
_2^σ SSB + _4^τ free boson CFT ⟺ ^Γ_4 intrinsically gapless SPT
_2^σ free boson CFT + _2^τ free boson CFT ⟺ ^σ_2×^τ_2 purely gapless SPT
_2^σ free boson CFT + _4^τ free boson CFT ⟺ ^Γ_4 intrinsically purely gapless SPT
Eqs. (<ref>), (<ref>), and (<ref>) covering gapped phases of _2 ×_2 symmetric systems
are already discussed in Ref. <cit.>, but included here for the sake of completeness and illustration.
In particular, the _2^σ×_2^τ gapped SPT can be obtained by starting with decoupled _2^σ SSB phase and _2^τ SSB phase and
perform the KT transformation.
These mappings between gapped phases were essentially identical to those in the earlier works <cit.>
on the KT transformation, although the present formulation <cit.> on S=1/2 chain is more convenient for construction of gapless phases.
First, replacing the ^σ_2 SSB phase in the left-hand side of (<ref>) with the _2^σ Ising CFT,
we obtain the _2^σ×_2^τ gSPT after the mapping (<ref>).
If we further replace the other ^τ_2 SSB phase with the _2^σ Ising CFT (<ref>), by the KT transformation
we obtain the gapless theory corresponding to the critical point between the _2^σ×_2^τ gapped SPT and trivial phases,
as can be seen from (<ref>) and (<ref>).
While this is also gapless, it has an emergent symmetry which has a mixed anomaly with the original symmetry protecting the gapped SPT <cit.> and does not belong to
gapless SPT phases we focus on in this paper.
Furthermore, replacing the _2^τ SSB in (<ref>) by _4^τ
symmetric free boson CFT (realized by XX chain on the lattice), we obtain the _4^Γ intrinsically gapless SPT
where _4^Γ is generated by the product of generators of _2^σ and _4^τ.
Replacing both _2^σ and _2^τ SSB by _2^σ and _2^τ free boson CFTs respectively, we obtain the _2^σ×_2^τ pgSPT.
Finally replacing _2^σ SSB in (<ref>) by _2^σ free boson CFT, and _2^τ SSB
in (<ref>) by _4^τ free boson CFT, we obtain _4^Γ ipgSPT.
Advantages of the present construction of gapless SPT phases using the KT transformation:
The common feature of the above constructions is that we start with a decoupled system, which can be either gapped or gapless, and KT transformation will map it to a coupled system with interesting topological features. This construction enables us to construct not only the known models of gSPT and igSPT, but also the new models of pgSPT and ipgSPT.
Furthermore, it allows us to study the stability of various gapless SPTs from (<ref>) to (<ref>) under certain symmetric perturbations.
In particular, if the perturbation of the gapless SPT is such that by undoing KT transformation the theory is still decoupled, we can analytically investigate the topological features of the decoupled theory on both a ring and an interval, and then use the KT transformation to trace these topological features back to gapless SPTs of interest.
Indeed, this leads to an analytical understanding of the phase diagram of a nontrivial model,
which we were only able to investigate numerically in <cit.>.
Gapless SPT phases are often characterized by edge states, which appear as low-energy states in the energy spectrum of an open chain
and can be distinguished from low-energy gapless excitations in the bulk.
Although such distinction between the gapless excitations and the edge states are possible, it is more subtle compared to identification
of edge states in gapped SPT phases.
By the KT transformation, we can often relate the low-energy states due to the edge states to the quasi-degeneracy of finite-size
ground states due to spontaneous symmetry breaking.
This clarifies the identification of the edge states in gapless SPTs and their stability, which also underscores the analysis of the phase
diagram as discussed above.
Finally, we also remark that, as the theories on the left hand side of (<ref>) to (<ref>) admit field theory descriptions,
we are able to derive the field theory description of the gapless SPTs of interest, using the KT transformation.
Outlook and the structure of the present paper:
Although we will only study the gapless SPTs with simple global symmetries like _2×_2 or _4, the KT transformation and the construction of these interesting gapless topological systems can be straightforwardly generalized to more general symmetry groups, as well as to higher dimensions <cit.>. It is also interesting to explore what condition we should impose on the two decoupled theories so that under (suitably generalized) KT transformation they give rise to gapless SPTs. We will leave these questions to future studies.
The plan of this paper is as follows. In Section <ref>, we review the basic properties of KT transformation from <cit.>. In Section <ref>, we revisit how to use the KT transformation to construct _2^σ×_2^τ gapped SPT starting from two decoupled _2 SSB systems. In Section <ref>, Section <ref> and Section <ref>, we construct the gapless SPT, intrinsically gapless SPT and purely gapless SPT as well as intrinsically purely gapless SPT respectively. We discuss how to use KT transformation to construct these models, how to probe the topological features, and how to analytically study the phase diagrams under certain symmetric perturbations.
§ REVIEW OF KENNEDY-TASAKI TRANSFORMATION
In this section, we review the Kennedy-Tasaki (KT) transformation defined in <cit.>, which is well-defined under both closed and open boundary conditions. By definition, this new KT transformation is defined by implementing STS on a _2×_2 symmetric system, where both _2's are anomaly free. Here S is gauging of both _2's, and T is stacking the system with a _2×_2 bosonic gapped SPT phase.
§.§ Definition of the KT transformation for spin-1/2 chains
Let us consider a spin chain with L sites and L links. Each site supports one spin-1/2, spanning a two dimensional local Hilbert space |s^σ_i⟩, where s^σ_i=0,1 and i=1, ..., L. Moreover, each link also supports one spin-1/2 spanning a two dimensional local Hilbert space |s^τ_i-1/2⟩, where s^τ_i-1/2=0,1 for i=1, ..., L. Hence each unit cell contains two spin-1/2's. The local states can be acted upon by Pauli operators,
σ_i^z|s^σ_i⟩= (-1)^s^σ_i|_i⟩, σ^x_i|_i⟩= |1-_i⟩
τ^z_i-1/2|_i-1/2⟩ = (-1)^_i-1/2|_i-1/2⟩, τ^x_i-1/2|_i-1/2⟩= |1-_i-1/2⟩.
The _2×_2 symmetry is generated by U_σ and U_τ respectively, where
U_σ= ∏_i=1^L σ^x_i, U_τ= ∏_i=1^L τ^x_i-1/2.
The symmetry and twist sectors are labeled by (u_σ, u_τ, t_σ, t_τ). Here (-1)^u_σ, (-1)^u_τ are the eigenvalues of U_σ, U_τ respectively, and t_σ, t_τ label the boundary conditions _i+L= _i+t_σ, _i-1/2+L= _i-1/2+t_τ. The KT transformation is then defined by the following action on the Hilbert space basis state <cit.>
_KT|{_i, _i-1/2}⟩ = 1/2^L-1∑_{_i, _i-1/2}(-1)^∑_j=1^L (_j + _j)(_j-1/2 + _j+1/2 + _j-1/2+ _j+1/2) + (_1/2+ _1/2)(t_σ + t'_σ)|{_i, _i-1/2}⟩
= 1/2^L-1∑_{_i, _i-1/2}(-1)^∑_j=1^L (_j-1/2+ _j-1/2)(_j-1 + _j + _j-1+ _j) + (_L+ _L)(t_τ + t'_τ)|{_i, _i-1/2}⟩
where we have presented two equivalent expressions, which will be convenient for the applications later.
It is useful to emphasize that the original KT transformation is defined for spin-1 systems, while this KT transformation is defined for a spin chain with two spin-1/2 per unit cell. Although in <cit.> we have shown that they are equivalent, it turns out to be more convenient to use the latter set up for the entire discussions, which we will assume throughout this work.
§.§ Properties of KT trnasformation
In <cit.>, various properties of (<ref>) are examined, including the mapping between symmetry and twist sectors, the fusion rule of the non-invertible defects, the definition on open boundary conditions and the relation to the original KT transformations in the spin-1 models <cit.>. We briefly review the results and refer interested readers to <cit.> for details.
Mapping between symmetry-twist sectors: Suppose a state is within the symmetry-twist sector labeled by [(u_σ, t_σ),(u_τ, t_τ)], then under the KT transformation, the resulting state is within the symmetry-twist sector labeled by
[(u_σ', t_σ'),(u_τ', t_τ')] = [(u_σ, t_σ+u_τ), (u_τ, t_τ+u_σ)].
In the sections below, we will frequently use the following result. Suppose the Hamiltonian H' is obtained from H by a KT transformation, i.e. H' = H. If |ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ is an eigenstate of H in the symmetry-twist sector [(u_σ,t_σ),(u_τ, t_τ)] with energy E^H_[(u_σ,t_σ),(u_τ, t_τ)], then |ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ is an eigenstate of H' in the symmetry-twist sector |ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ with the same energy E^H_[(u_σ,t_σ),(u_τ, t_τ)]:
H'|ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ = H |ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ = E^H_[(u_σ,t_σ),(u_τ, t_τ)]|ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩.
Note that |ψ_[(u_σ,t_σ),(u_τ, t_τ)]⟩ sits in the symmetry-twist sector [(u'_σ,t'_σ),(u'_τ, t'_τ)], hence
E_[(u'_σ,t'_σ),(u'_τ, t'_τ)]^H' = E^H_[(u_σ,t_σ),(u_τ, t_τ)] = E^H_[(u'_σ, t'_σ+u'_τ),(u'_τ, t'_τ+u'_σ)].
We will use (<ref>) repeatedly in the subsequent sections.
Fusion rules:
The fusion rules involving the operator implementing the KT transformation, , and the _2×_2 symmetry operators U_σ, U_τ are
× U_σ = (-1)^t_τ+ t'_τ,
× U_τ = (-1)^t_σ+ t'_σ,
× = 4(1+ (-1)^t_σ+t'_σ U_τ) (1+ (-1)^t_τ+t'_τ U_σ).
In particular, the last fusion rule shows that _KT is non-invertible, and the transformation is non-unitary.
All the above discussions are on a ring.
We finally note that on an open interval, the KT transformation is a unitary transformation, hence preserves the energy eigenvalues of the Hamiltonian <cit.>.
§ GAPPED SPT FROM KT TRANSFORMATION
The KT transformation was designed to map a _2×_2 symmetry spontaneously broken (SSB) phase to a _2×_2 symmetry protected topological (SPT) phase. It is straightforward to check at the level of partition function that STS transformation relates the two. We will review how the SPT phase can be generated from the KT transformation, and this will be the first example of using the KT transformation to generate exotic models with interesting topological features.
Gapped SPT from KT transformation:
The Hamiltonian for the _2×_2 SSB phase is
H_SSB= -∑_i=1^L (σ^z_i-1σ^z_i+ τ^z_i-1/2τ^z_i+1/2)
where the degrees of freedom charged under two _2's are decoupled. Under the KT transformation, the operators are mapped as follows
σ^z_j-1σ^z_j = σ^z_j-1τ^x_j-1/2σ^z_j, τ^z_i-1/2τ^z_i+1/2 = τ^z_j-1/2σ^x_jτ^z_j+1/2
for j=1, ..., L. Note that the boundary condition is encoded in the states/operators already. For instance, the boundary condition s^σ_L=s^σ_0+t_σ induces σ_0^z= (-1)^t_σσ_L^z. The resulting Hamiltonian is precisely the cluster model describing the _2×_2 gapped SPT <cit.>
H_SPT= -∑_j=1^L ( σ^z_j-1τ^x_j-1/2σ^z_j + τ^z_j-1/2σ^x_jτ^z_j+1/2).
Let's also comment on the field theory of the gapped SPT. We start with the _2^σ×_2^τ SSB phase, whose partition function is Z_SSB[A_σ, A_τ]:= δ(A_σ)δ(A_τ). We define the topological manipulations S and T on a generic quantum field theory with _2^σ×_2^τ symmetry as
S: Z_S[A_σ, A_τ] = ∑_a_σ, a_τ Z_[a_σ, a_τ]e^iπ∫_X_2 a_σ A_τ- a_τ A_σ,
T: Z_T[A_σ, A_τ] = Z_[A_σ, A_τ] e^iπ∫_X_2 A_σ A_τ.
In <cit.>, it was found that the KT transformation is STS, under which the SSB partition function Z_SSB[A_σ, A_τ] is mapped to
Z_SPT[A_σ, A_τ] = ∑_a_σ, a_τδ(a_σ)δ(a_τ) e^i π∫_X_2 a_σã_τ + a_τã_σ + ã_σã_τ + ã_σ A_τ + ã_τ A_σ= e^iπ∫_X_2 A_σ A_τ
which is merely an invertible phase in terms of the background fields. This is commonly known as the field theory description of the gapped SPT <cit.>. [The construction of gapped SPT from STS is somewhat round about, since T itself is stacking a gapped SPT (or equivalently domain wall decoration). However, this construction will be more useful when constructing gapless SPT in later sections. ]
Ground state charge under TBC:
A key feature of the SPT is that the ground state carries a non-trivial charge under twisted boundary conditions. To see this, we note that every two terms in (<ref>) commute, hence the ground state should be a common eigenstate of each local operator in (<ref>). Under PBC for both _2's, the ground state satisfies
σ_i-1^z τ^x_i-1/2σ^z_i |ψ⟩_PBC= |ψ⟩_PBC, τ^z_i-1/2σ^x_iτ^z_i+1/2|ψ⟩_PBC= |ψ⟩_PBC
for i=1, ..., L, where τ^z_L+1/2=τ^z_1/2, and σ^z_0= σ^z_L. Hence
U_σ|ψ⟩_PBC = ∏_i=1^L σ^x_i |ψ⟩_PBC = ∏_i=1^L τ^z_i-1/2τ^z_i+1/2|ψ⟩_PBC =|ψ⟩_PBC,
U_τ|ψ⟩_PBC =∏_i=1^L τ^x_i-1/2|ψ⟩_PBC = ∏_i=1^L σ^z_i-1σ^z_i|ψ⟩_PBC =|ψ⟩_PBC
where we have used σ^z_0=σ^z_L and τ^z_1/2= τ^z_L+1/2 for PBC. Hence the ground state under PBC is neutral under _2^σ×_2^τ.
Under TBC of _2^σ, if writing the Hamiltonian in terms of the Pauli operators supported within i=1, ..., L, the sign of the term σ_0^z τ^x_1/2σ^z_1 = -σ_L^z τ^x_1/2σ^z_1 changes sign, and the ground state in the TBC of _2^σ satisfies
σ_i-1^z τ^x_i-1/2σ^z_i |ψ⟩_TBC_σ= |ψ⟩_TBC_σ, i=2, ..., L, σ_L^z τ^x_1/2σ^z_1 |ψ⟩_TBC_σ=- |ψ⟩_TBC_σ,
τ^z_i-1/2σ^x_iτ^z_i+1/2|ψ⟩_TBC_σ= |ψ⟩_TBC_σ, i=1, ..., L-1, τ^z_L-1/2σ_L^x τ^z_1/2|ψ⟩_TBC_σ= |ψ⟩_TBC_σ.
Hence
U_σ|ψ⟩_TBC_τ = ∏_i=1^L σ^x_i |ψ⟩_TBC_τ = ∏_i=1^L τ^z_i-1/2τ^z_i+1/2|ψ⟩_TBC_τ =|ψ⟩_TBC_τ,
U_τ|ψ⟩_TBC_σ =∏_i=1^L τ^x_i-1/2|ψ⟩_TBC_σ = ∏_i=1^L σ^z_i-1σ^z_i|ψ⟩_TBC_σ =-|ψ⟩_TBC_σ
where we have used σ^z_0=-σ^z_L, and τ^z_1/2= τ^z_L+1/2 for TBC of _2^σ. Hence the ground state under TBC of _2^σ is _2^σ even and _2^τ odd.
This is a key feature of the gapped SPT. Similarly, one can also show that the ground state under the TBC of _2^τ is _2^σ odd and _2^τ even.
It is useful to see how the topological features of (<ref>) discussed above can be uncovered from the KT transformation without solving (<ref>). We begin by analyzing the ground states of the SSB phase (<ref>) under various boundary conditions. Because (<ref>) is a classical model, its energy spectrum is straightforward to find. Concretely, we have
E_(u, t)^σ= E_(u, t)^τ = -L+2 [t]_2=
-L, (u,t)=(0,0),(1,0),
-L+2, (u,t)=(0,1),(1,1)
where E^σ_(u,t) is the ground state energy of the sigma spin in the symmetry-twist sector (u,t), and [t]_2 is the mod 2 value of t. Similar for E^τ_(u,t). Then the ground state energy of the SPT Hamiltonian (<ref>) in the symmetry-twist sector [(u_σ,t_σ),(u_τ,t_τ)] is
E^SPT_[(u_σ,t_σ),(u_τ,t_τ)]= E^σ_(u_σ, t_σ+u_τ) + E^τ_(u_τ, t_τ+u_σ)= -2L+ 2 [t_σ+u_τ]_2 +2 [t_τ+u_σ]_2
where we have used (<ref>) in the first equality, and (<ref>) in the second equality. The energy (<ref>) is minimized if
t_σ=u_τ, t_τ=u_σ.
This means that the ground state
in the _2^σ twisted sector carries non-trivial _2^τ charge, and the ground state
in the _2^τ twisted sector carries non-trivial _2^σ charge. This reproduces the key topological features of the gapped SPT phase reviewed above.
We would like to emphasize the power of the latter method. Typically, the symmetry properties of the system before KT transformation are much easier to analyze, and by (<ref>) we automatically know the symmetry properties of the system after KT transformation. Below, we will encounter systems which are difficult to analyze after KT transformation, hence the latter method becomes much more powerful.
String order parameter:
The string order parameters follow from the local order parameters under KT transformation. In the SSB phase, the long range order is given by the conventional correlation functions
⟨σ^z_i σ^z_j|,⟩⟨τ^z_i-1/2τ^z_j-1/2|,⟩ i<j,
both of which are of order 1 in the vacuum. Under KT transformation, σ^z_i σ^z_i+1 is mapped to σ^z_iτ^x_i+1/2σ^z_i+1, and τ^z_i-1/2τ^z_j-1/2 is mapped to τ^z_i-1/2σ^x_iτ^z_i+1/2, thus the above two conventional correlation functions become string order parameters
⟨σ^z_i (∏_k=i^j-1τ^x_k+1/2 )σ^z_j|,⟩⟨τ^z_i-1/2
(∏_k=i^j-1σ^x_k)τ^z_j-1/2|.⟩
Both the string order parameters develop a non-trivial order 1 vacuum expectation value (VEV) in the ground state of SPT.
§ GAPLESS SPT FROM KT TRANSFORMATION
In Section <ref>, we constructed the gapped SPT phase from two decoupled copies of _2 symmetry breaking phases. Recent years have witnessed extensive studies of gapless SPT states <cit.>. It is then natural to consider whether such states admit constructions from the KT transformation. In this section, we will confirm this possibility and show that the gapless SPT state first found in <cit.> (see also <cit.>) can be constructed in this way.
§.§ Constructing the gapless SPT
Instead of starting with two decoupled _2 SSB models, we start with a _2 gapless model (i.e. the transverse field Ising model) and a decoupled _2 SSB model. The Hamiltonian is
H_Ising+SSB = -∑_i=1^L (τ^z_i-1/2τ^z_i+1/2 + σ_i-1^z σ^z_i + σ^x_i).
The _2^τ is SSB, and the degrees of freedom charged under _2^σ are gapless. As usual, we encode the boundary condition in the Hilbert space, and the Hamiltonian applies to arbitrary boundary conditions. To apply the KT transformation, we still use the operator maps (<ref>) and also the map
σ^x_i= σ^x_i
for i=1,...,L. The resulting Hamiltonian is
H_gSPT=-∑_j=1^L (τ^z_i-1/2σ^x_i τ^z_i+1/2 + σ_i-1^z τ^x_i-1/2σ^z_i + σ^x_i).
Note that the first term commutes with the last two terms, hence the ground state |ψ⟩ should satisfy τ^z_i-1/2σ^x_i τ^z_i+1/2|ψ⟩= |ψ⟩. This condition is actually also satisfied by the low excited states as well, because violating it would cost energy of order 1, while the excitation gap is only of order 1/L. See <cit.> for more detailed discussions on this point. Hence within the low energy sector, σ^x_i can be safely replaced by τ^z_i-1/2τ^z_i+1/2, and the Hamiltonian (<ref>) is equivalent to
H_gSPT≃ -∑_j=1^L (τ^z_i-1/2σ^x_i τ^z_i+1/2 + σ_i-1^z τ^x_i-1/2σ^z_i + τ^z_i-1/2τ^z_i+1/2).
This is exactly the Hamiltonian for the gapless SPT originally constructed in <cit.> and later revisited in <cit.>. [In <cit.> and <cit.>, the role of σ and τ are exchanged. ] (<ref>) and (<ref>) are also related by Kramers-Wannier (KW) transformation for both _2^σ×_2^τ.
§.§ Field theory of gapless SPT
The KT transformation also allows us to write down the field theory for the gapless SPT. We start with the partition function for the Ising CFT + SSB phase,
Z_Ising [A_σ] Z_SSB[A_τ]
where the partition function of the Ising CFT can be conveniently written as a Wilson-Fisher fixed point,
Z_Ising [A_σ] := ∫ϕexp(i ∫_X_2 (D_A_σϕ)^2 + ϕ^4), D_A_σϕ = dϕ - i π A_σϕ
and the partition function of the _2^τ SSB phase is simply a delta function restricting its background field to zero,
Z_SSB[A_τ] = δ(A_τ).
We then perform KT transformation, i.e. a STS transformation, changing the partition function to
Z_gSPT[A_σ, A_τ] = ∑_a_σ, a_τ, ã_σ, ã_τ Z_Ising [a_σ] Z_SSB[a_τ] e^i π∫_X_2 a_σã_τ + a_τã_σ + ã_σã_τ + ã_σ A_τ + ã_τ A_σ
= ∑_a_σ, a_τ Z_Ising [a_σ] δ(a_τ) e^i π∫_X_2 (a_σ + A_σ)(a_τ+A_τ)= ∑_a_σ Z_Ising [a_σ] e^i π∫_X_2 a_σ A_τ + A_σ A_τ
⟷ Z_Ising[A_σ] e^iπ∫_X_2 A_σ A_τ.
In the last line, we used the Kramers-Wannier duality which identifies the gauged Ising CFT with the Ising CFT itself. Comparing the head and tail of
(<ref>) shows that the _2^σ×_2^τ gapless SPT is simply an Ising CFT stacked with a _2^σ×_2^τ gapped SPT, which matches the construction in <cit.>.
§.§ Topological features of gapless SPT
We proceed to study the topological features of the gapless SPT directly from the Hamiltonian (<ref>). We will focus on the symmetry charges of the ground state under the TBCs, as well as the degeneracy under the open boundary condition.
The discussion here follows <cit.>.
§.§.§ Symmetry charge of ground state under TBC
For definiteness, we will consider the Hamiltonian (<ref>), although (<ref>) is equivalent.
The discussion is similar to that for the gapped SPT in Section <ref>. Under PBC, since τ^z_i-1/2σ^x_i τ^z_i+1/2 commutes with the remaining terms, the ground state |ψ⟩_PBC must be an eigenstate of τ^z_i-1/2σ^x_i τ^z_i+1/2 for all i,
τ^z_i-1/2σ^x_i τ^z_i+1/2|ψ⟩_PBC = |ψ⟩_PBC, i=1, ..., L-1, τ^z_L-1/2σ^x_L τ^z_1/2|ψ⟩_PBC = |ψ⟩_PBC.
In particular, this means that |ψ⟩_PBC is neutral under _2^σ,
U_σ|ψ⟩_PBC = ∏_i=1^L σ^x_i |ψ⟩_PBC = (∏_i=1^L-1τ^z_i-1/2τ^z_i+1/2) τ^z_L-1/2τ^z_1/2|ψ⟩_PBC = |ψ⟩_PBC.
However, the above method does not fix the _2^τ charge of the ground state. By exact diagonalization, we confirmed that the ground state is _2^τ even. Furthermore, exact diagonalization also shows that there is only one ground state under PBC, which is the desired property of gapless SPT <cit.>.
We proceed to the TBC of _2^τ. The ground state |ψ⟩_TBC_τ satisfies
τ^z_i-1/2σ^x_i τ^z_i+1/2|ψ⟩_TBC_τ = |ψ⟩_TBC_τ, i=1, ..., L-1, τ^z_L-1/2σ^x_L τ^z_1/2|ψ⟩_TBC_τ =- |ψ⟩_TBC_τ.
This means that |ψ⟩_TBC_τ is _2^σ odd,
U_σ|ψ⟩_TBC_τ = ∏_i=1^L σ^x_i |ψ⟩_TBC_τ =- (∏_i=1^L-1τ^z_i-1/2τ^z_i+1/2) τ^z_L-1/2τ^z_1/2|ψ⟩_TBC_τ =- |ψ⟩_TBC_τ.
One can again numerically check that |ψ⟩_TBC_τ is even under _2^τ.
We finally consider the TBC of _2^σ. The two Hamiltonians under PBC and TBC of _2^σ are related by conjugating by τ^z_1/2,
H_gSPT^TBC_σ=τ^z_1/2 H_gSPT^PBCτ^z_1/2.
Hence their ground states are also related,
|ψ⟩_TBC_σ=τ^z_1/2|ψ⟩_PBC.
As a consequence, the _2^σ charge of |ψ⟩_TBC_σ and |ψ⟩_PBC are the same, while their _2^τ charge are the opposite.
As we see from the above, the discussion depends heavily on the form of the Hamiltonian. In particular, we repeatedly used the fact that the first term in the Hamiltonian (<ref>) commutes with the rest of the terms. This will be no longer true if one adds a generic symmetric perturbation, for example
- h ∑_i=1^Lτ^x_i-1/2.
In this situation, the analysis in the current subsection does not work, and one has to apply numerical computation to find the ground state charge. However, in Section <ref>, we will re-derive the above results using the KT transformation, and the result holds under perturbation as well hence is more powerful.
§.§.§ Degeneracy under open boundary condition
We proceed to discuss the topological features under OBC. There are many different open boundary conditions, depending on how one truncates the lattice, and what types of local interactions are added to the boundary. For simplicity, we focus on one particular boundary condition, where only the sites i and i-1/2 for i=1, ..., L belong to the lattice. The Hamiltonian is chosen such that only the terms fully supported on the lattice are preserved. The Hamiltonian is
H_gSPT^OBC=-∑_i=1^L-1τ^z_i-1/2σ^x_i τ^z_i+1/2 -∑_i=2^Lσ_i-1^z τ^x_i-1/2σ^z_i -∑_i=1^L σ^x_i.
Then it is easy to check that the following terms commute with the Hamiltonian
τ^z_1/2, τ^z_L-1/2σ^x_L, U_σ, U_τ.
The first two terms are localized on the boundaries, and the last two terms are symmetry operators. Because {τ^z_1/2, U_τ}= {τ^z_L-1/2σ^x_L, U_τ}=0, the irreducible representation of the algebra is two dimensional. Hence there are two degenerate ground states.
Under the bulk perturbation (<ref>), the boundary terms τ^z_1/2, τ^z_L-1/2σ^x_L no longer commute with the Hamiltonian, and the degeneracy from the above are lifted. However, by perturbation theory analysis, the gap between the two lowest states decays exponentially with respect to the system size (See Section 2.4.1 in <cit.> for further details). This exponential edge degeneracy of gSPTs is also discussed by the decorated domain wall argument in references <cit.>.
§.§ Topological features from KT transformation
In this subsection, we reproduce the results in Section <ref> using the KT transformation. We first analyze the topological features of the decoupled system (<ref>) as well as its perturbations. Since the decoupled system is relatively simple, we know the symmetry properties even under perturbation. We then use the KT transformation to relate the symmetry properties of the decoupled system (<ref>) to the gapless SPT (<ref>). This will enable us to determine the symmetry properties of the ground states even after perturbation.
We first study the symmetry properties of the ground states before the KT transformation, i.e. (<ref>), with a symmetric perturbation -h ∑_i=1^L τ^x_i-1/2. We will assume h≪ 1 in this subsection. Hence the Hamiltonian is simply a decoupled critical Ising Hamiltonian plus a transverse field Ising model with a small transverse field (hence in deep SSB phase),
H_Ising+SSB+pert = H_Ising + H_SSB+pert
where
H_Ising= -∑_i=1^L ( σ_i-1^z σ^z_i + σ^x_i ), H_SSB+pert= -∑_i=1^L (τ^z_i-1/2τ^z_i+1/2 + h τ^x_i-1/2).
The symmetry properties of the ground states of the above two models are well-known. Let us denote the ground state energy of the critical Ising model H_Ising as E_(u_σ, t_σ)^σ. It is well-known that they satisfy the following relations
E_(0,0)^σ1/L< E_(1,0)^σ = E_(0,1)^σ1/L< E_(1,1)^σ.
The symbol E_1 1/L< E_2 means that the difference between the energies on its two sides E_2-E_1 is of order 1/L. The equality E_(1,0)^σ = E_(0,1)^σ is ensured by the Kramer-Wannier self-duality of the Ising model where the KW exchanges (u_σ, t_σ) ↔ (t_σ, u_σ).
The low energy spectrum of the SSB phase is also well-known. When h=0, there are two exactly degenerate ground states under PBC, |u_τ=0,1⟩. When h>0, the degeneracy is lifted, and where the gap between 1/√(2)(|u_τ=0⟩+ |u_τ=1⟩) and 1/√(2)(|u_τ=0⟩- |u_τ=1⟩) decays exponentially with respect to the system size. Moreover, the ground state energy in the twisted sector is roughly the energy of the domain wall excitation, which is of order 1. Denote the ground state energy of the model H_SSB+pert as E_(u_τ, t_τ)^τ, then they satisfy the relation
E_(0,0)^τe^-L< E_(1,0)^τ1< E_(0,1)^τ1/L^2< E_(1,1)^τ.
See Appendix <ref> for a concrete derivation using Jordan Wigner transformation.
We proceed to perform the KT transformation on (<ref>). Since τ^x_i-1/2 is mapped to itself, i.e. τ^x_i-1/2= τ^x_i-1/2, the perturbation in -h∑_i=1^L τ^x_i-1/2 is preserved under KT transformation. Hence we get the gapless SPT with perturbation (<ref>),
H_gSPT+pert=-∑_j=1^L (τ^z_i-1/2σ^x_i τ^z_i+1/2 + σ_i-1^z τ^x_i-1/2σ^z_i + σ^x_i + h τ^x_i-1/2).
Denote the ground state energy of the Hamiltonian (<ref>) in the symmetry-twist sector as E^gSPT_[(u_σ, t_σ), (u_τ, t_τ)]. By (<ref>) and using E^σ_(u_σ,t_σ) and E^τ_(u_τ, t_τ), we obtain
E^gSPT_[(u_σ, t_σ), (u_τ, t_τ)]= E^σ_(u_σ,t_σ+u_τ)+ E^τ_(u_τ, t_τ+u_σ)
from which we are able to determine the charge of the ground state in each twist sector. Let us discuss them case by case.
* t_σ=0, t_τ=0: Both sigma and tau spins obey PBC. The energy (<ref>) reduces to
E^gSPT_[(u_σ, 0), (u_τ, 0)] = E^σ_(u_σ,u_τ)+ E^τ_(u_τ, u_σ)
= E^σ_(0,0)+ E^τ_(0,0)+
0, (u_σ, u_τ)=(0,0)
1/L+ 1, (u_σ, u_τ)=(1,0)
1/L+ e^-L, (u_σ, u_τ)=(0,1)
1/L+ 1/L+ 1 + 1/L^2. (u_σ, u_τ)=(1,1)
The minimal energy is achieved in the symmetry sector (u_σ, u_τ)=(0,0).
* t_σ=1, t_τ=0: The sigma spins obey TBC, and tau spins obey PBC. The energy (<ref>) reduces to
E^gSPT_[(u_σ, 1), (u_τ, 0)] = E^σ_(u_σ,1+u_τ)+ E^τ_(u_τ, u_σ)
= E^σ_(0,0)+ E^τ_(0,0)+
1/L, (u_σ, u_τ)=(0,0)
1/L+ 1/L+1, (u_σ, u_τ)=(1,0)
e^-L, (u_σ, u_τ)=(0,1)
1/L+ 1 + 1/L^2. (u_σ, u_τ)=(1,1)
The minimal energy is achieved in the symmetry sector (u_σ, u_τ)=(0,1).
* t_σ=0, t_τ=1: The sigma spins obey PBC, and tau spins obey TBC. The energy (<ref>) reduces to
E^gSPT_[(u_σ, 0), (u_τ, 1)] = E^σ_(u_σ,u_τ)+ E^τ_(u_τ, 1+u_σ)
= E^σ_(0,0)+ E^τ_(0,0)+
1, (u_σ, u_τ)=(0,0)
1/L, (u_σ, u_τ)=(1,0)
1/L+1+1/L^2, (u_σ, u_τ)=(0,1)
1/L+ 1/L+ e^-L. (u_σ, u_τ)=(1,1)
The minimal energy is achieved in the symmetry sector (u_σ, u_τ)=(1,0).
* t_σ=1, t_τ=1: The sigma spins obey TBC, and tau spins obey TBC. The energy (<ref>) reduces to
E^gSPT_[(u_σ, 1), (u_τ, 1)] = E^σ_(u_σ,1+u_τ)+ E^τ_(u_τ, 1+u_σ)
= E^σ_(0,0)+ E^τ_(0,0)+
1/L+1, (u_σ, u_τ)=(0,0)
1/L+ 1/L, (u_σ, u_τ)=(1,0)
1+1/L^2, (u_σ, u_τ)=(0,1)
1/L+ e^-L. (u_σ, u_τ)=(1,1)
The minimal energy is achieved in the symmetry sector (u_σ, u_τ)=(1,1).
In the above, we only write down the schematic scaling behavior of energy with respect to the system size L. In summary, under the _2^σ or _2^τ twisted boundary condition, the ground state carries nontrivial _2^τ or _2^σ charge respectively. These results are not only consistent with, but also significantly generalize the discussions in Section <ref> since we also allow a perturbation -h∑_i τ^x_i-1/2 here and the method in Section <ref> no longer applies.
Furthermore, let us focus on the OBC where the KT transformation is unitary. When h≪ 1, the τ spins that remain in the _2 SSB phase exhibit a two-fold (exponential) degeneracy of the ground states. After the KT transformation, this implies that the gSPT has two-fold exponential “edge” degeneracy, which still survives under the perturbation.
Let us make some comments.
* The key property that we are able to determine the symmetry properties after the perturbation is that before the KT transformation the system (after perturbations) is decoupled and we know its structure well. A generic perturbation typically mixes the σ and τ degrees of freedom after undoing KT transformation, and we will need numerics.
* Since the KT transformation implements a twisted gauging, i.e. STS, the qualitative feature such as the location of the phase transition in terms of the perturbation h can not change. Before the KT transformation, turning on a small perturbation h does not trigger a phase transition, hence the system after KT transformation is also stable under turning on a small h. This means that the gapless SPT (<ref>) is stable under the small perturbation (<ref>). [We would like to emphasize that in <cit.>, we used the Hamiltonian (<ref>) as the gapless SPT. Although (<ref>) and (<ref>) share the same low energy spectrum before turning on (<ref>), adding such a perturbation would make the low energy spectrum different since the first term in (<ref>) does not commute with the perturbation. Hence the discussion for the stability of gapless SPT under a small perturbation no longer applies to the discussion in Section 2.4 in <cit.>. Indeed, as one of the referees of <cit.> pointed out, the perturbation there would gap out the gapless SPT.
]
We will study the phase diagram in the following subsection.
* Discrete gauging also does not change the existence of gapped sectors. Since the system before the KT transformation contains a gapped sector, i.e. the SSB associated with the τ spins, after the KT transformation, there is still a gapped sector, even after a small perturbation. The existence of the gapped sector in the gapless SPT has been emphasized in <cit.>, which comes from the gapped degrees of freedom decorating the domain wall, and they are crucial in protecting the non-trivial topological properties of the gapless SPT.
* In general, the gapped sector can be seen from the exponential decaying energy splitting of edge modes under OBC. However, it is very difficult to prove the stability of such exponential decaying behavior under symmetric perturbations. With the help of KT transformation, we provide an analytical proof of this stability for gSPTs (<ref>) and igSPTs (<ref>) in appendix <ref>.[Although such degeneracy under OBC is stable under symmetric perturbation, the system can be driven to the SSB phase or gapped SPT phase when the bulk perturbation can open a gap for the gapless systems before KT transformation.]
§.§ Phase diagram
In Section <ref>, we discussed the topological features of the gapless SPT using the KT transformation, and also showed that the gapless SPT is stable under a small perturbation h. In this subsection, we would like to understand the structure of the phases when one increases h, and determine the phase diagram. We will end up commenting on the string order parameter.
Phase diagram:
Since the qualitative structure of the phase diagram is not affected by (twisted) gauging of finite groups, the phase diagram and the location of phase transitions can be inferred from the system before the KT transformation. Before the KT transformation, the system is simply a critical Ising model for σ spin and a transverse field Ising model for τ spin, it is clear that there is only one phase transition at h=1 and the total system is in the Ashkin-Teller (AT) university class <cit.>. Hence the gapless SPT is stable as long as the perturbation h is smaller than 1.
Let us determine the symmetry properties of the ground state within each phase and at the phase transition. When h<1, the symmetry properties of the ground state in different sectors have been discussed in Section <ref>. When h>1, by using the same method as in Section <ref>, we find that after the KT transformation, the ground states under four boundary conditions are all _2^σ
even and _2^τ even. Indeed, in the Hamiltonian (<ref>) in the large h limit, the last term dominates, and the ground state satisfies τ^x_i-1/2≃ 1. Then the Hamiltonian simplifies to an Ising paramagnetic (trivially gapped phase) for the τ spin and a critical Ising model for the σ spin. Indeed, in such a model, the ground state under any boundary condition is even for both _2^σ and _2^τ. We plot the phase diagram as in Figure <ref>. The system in the entire phase diagram is gapless. The transition is when the gapped sector becomes gapless in the gapless system.
The transition h=1 is described by free boson CFT in low energy with central charge c=1, while the central charge away from h=1 is c=1/2. This phase transition is the KT dual theory of two decoupled lsing criticality. However, it is also a phase transition between the SPT and trivial phases, which is an anomalous theory<cit.> and is distinct from gSPT models constructed using decoupled free boson CFTs in Eq.(<ref>) and Eq.(<ref>) [Since the gapless SPT for h<1 can be understood as stacking an Ising criticality with a gapped SPT phase, the transition at h=1 can also be intuitively understood as an Ising criticality stacked with phase transition between the SPT and trivial phases. As lsing criticality is anomaly-free, the anomaly of the transition at h=1 only comes from phase transition between the SPT and trivial phases].
String order parameter:
We finally comment on the string order parameter in the gapless SPT. We start with the local order parameter for the decoupled Hamiltonian (<ref>) before the KT transformation,
⟨σ^z_i σ^z_j|∼⟩1/|i-j|^2Δ
, ⟨τ^z_i-1/2τ^z_j-1/2|∼⟩(1), h<1
e^-ξ L, h>1
where Δ is the scaling dimension of σ^z.
After the KT transformation, the local order parameters become string order parameters and obey the same scaling behavior,
⟨σ^z_i (∏_k=i^j-1τ^x_k+1/2 )σ^z_j|∼⟩1/|i-j|^2Δ, ⟨τ^z_i-1/2
(∏_k=i^j-1σ^x_k)τ^z_j-1/2|∼⟩(1), h<1
e^-ξ L. h>1
§ INTRINSICALLY GAPLESS SPT FROM KT TRANSFORMATION
In Section <ref>, we constructed the gapless SPT by applying KT transformation on a decoupled critical Ising model stacked with a _2 SSB phase. In this section, we will apply the KT transformation to construct the intrinsically gapless SPT (igSPT) which was first found in <cit.>, and later revisited in <cit.>.
The intrinsically gapless SPT states is a class of gapless systems which exhibit an emergent anomaly of the low energy symmetries. Although the entire global symmetry G is anomaly free, G does not act faithfully on the low energy gapless degrees of freedom. There is a normal subgroup H of G which only acts on the gapped sector, hence the quotient G/H acts faithfully on the low energy sector. Because G is a nontrivial extension of G/H by H, G/H then has a nontrivial 't Hooft anomaly <cit.>, named the emergent anomaly in <cit.>. In <cit.>, building upon <cit.>, the gapped sector was rephrased in terms of the anomalous SPT in the modified decorated domain wall construction. The emergent anomaly of G/H is canceled by the anomalous SPT with H symmetry, hence the total symmetry G is anomaly free. The gapped sector turns out to be crucial to protect the nontrivial topological properties of the intrinsically gapless SPT.
In this section, we will construct an example of intrinsically gapless SPT with G=_4 and H=_2, which was originally discussed in <cit.>. We will find that this can be achieved by starting with a decoupled stacking of XX chain with a _2 SSB phase, and then performing the KT transformation. The KT transformation also allows us to analytically determine the phase transition under the perturbation of igSPT, which we were unable to determine by small-scale numerical calculation in <cit.>.
§.§ Constructing the intrinsically gapless SPT
Let us start with an Ising SSB Hamiltonian stacked with an XX chain. The Hamiltonian is
H_SSB+XX =- ∑_i=1^L(τ^z_i-1/2τ^z_i+1/2 + τ^y_i-1/2τ^y_i+1/2 + σ^z_i-1σ^z_i).
This Hamiltonian actually has a larger U(1)^τ×_2^σ global symmetry, where _2^σ is generated by U_σ defined in (<ref>), and U(1)^τ is generated by
∏_i=1^L e^i α/2 (1-τ^x_i-1/2), α≃α+2π.
The _2^τ normal subgroup of U(1)^τ is generated by U_τ in (<ref>). We will instead consider the _4^τ subgroup of U(1)^τ, where the generator of _4^τ is
V_τ= ∏_i=1^L e^i π/4 (1-τ^x_i-1/2)
satisfying V_τ^2= U_τ. We will be interested in the _4^τ×_2^σ symmetry, generated by V_τ and U_σ.
Let us apply the KT transformation, by using the _2^τ×_2^σ symmetry generated by U_τ and U_σ. Under KT transformation, we have
τ^z_i-1/2τ^z_i+1/2 = τ^z_i-1/2σ^x_i τ^z_i+1/2,
τ^y_i-1/2τ^y_i+1/2 = τ^y_i-1/2σ^x_i τ^y_i+1/2,
σ^z_i-1σ^z_i = σ^z_i-1τ^x_i-1/2σ^z_i.
Hence the Hamiltonian after KT transformation is precisely the intrinsically gapless SPT (or strong SPTC) found in <cit.>
H_igSPT = - ∑_i=1^L(τ^z_i-1/2σ^x_iτ^z_i+1/2 + τ^y_i-1/2σ^x_iτ^y_i+1/2 + σ^z_i-1τ^x_i-1/2σ^z_i).
Since the KT transformation commutes with both σ^x_i and τ^x_i-1/2, the resulting Hamiltonian (<ref>) also has _4^τ×_2^σ symmetry, generated by V_τ and U_σ respectively.
In <cit.>, the Hamiltonian (<ref>) was claimed to be the intrinsically gapless SPT protected by _4^Γ. This _4^Γ is generated by the product U_σ V_τ. Here, we would like to emphasize that not only the combination U_σ V_τ commutes with the Hamiltonian, but both U_σ and V_τ separately commutes with (<ref>) as well. This accidental symmetry allows us to construct it using the KT transformation.
§.§ Mapping between symmetry-twist sectors
Since the intrinsic gapless SPT has an accidental anomaly-free symmetry _4^τ×_2^σ, we first consider the symmetry and twist sectors. The symmetry sectors are labeled by the eigenvalues of the operator V_τ and U_σ, which are e^i π/2 v_τ for v_τ=0, 1, 2,3 and (-1)^u_σ for u_σ=0,1 respectively. The twist sector (i.e. the boundary condition) is defined by
|s_i+L^σ⟩ = ∑_s_i^σ=0,1[(σ^x_i)^t_σ]_s_i+L^σ, s_i^σ|s_i^σ⟩ = |s_i^σ+ t_σ⟩
|s_i-1/2+L^τ⟩=∑_s_i-1/2^τ=0,1[e^iπ r_τ/4 (1-τ^x_i-1/2)]_s_i-1/2+L^τ, s_i-1/2^τ|s^τ_i-1/2⟩ = 1/2∑_s_i-1/2^τ=0,1(1+ e^iπ/2r_τ + i π (s_i-1/2^τ+ s_i-1/2+L^τ)) |s_i-1/2^τ⟩
where t_σ≃ t_σ+2, and r_τ≃ r_τ+4.
In particular, when r_τ=2 t_τ, the second equality in the above simplifies to |s_i-1/2+L^τ⟩= |s_i-1/2^τ+ t_τ⟩. Hence the symmetry and twist sectors are labeled by
[(u_σ, t_σ), (v_τ, r_τ)], u_σ, t_σ∈_2, v_τ, r_τ∈_4.
To show how the _4^τ×_2^σ symmetry and twist sectors are mapped under the KT transformation, we need to duplicate the same discussion as in Section 6.2 of <cit.>. However, as we know that the KT transformation implements the twisted gauging STS, it turns out that it is much easier to obtain the map from the partition function directly. Relegating the derivation to the Appendix <ref>, we find the mapping between symmetry and twist sectors as
[(u'_σ, t'_σ),(v'_τ, r'_τ)] = [(u_σ, t_σ+v_τ),(v_τ, r_τ+2u_σ)].
Indeed, the symmetry sectors u_σ and v_τ are unchanged under the KT transformation, which directly follows from the observation below (<ref>) that the symmetry operators U_σ and V_τ are unchanged under the KT transformation.
Note that the intrinsic gapless SPT is protected by _4^Γ, rather than _2^σ×_4^τ. Since the gapless SPTs appear after the KT transformation, we denote the symmetry-twist sectors of _4^Γ by (u'_Γ, t'_Γ) where the prime stands for the sectors after the KT transformation according to (<ref>). Note that U_Γ= U_σ V_τ, their eigenvalues are thus related by e^iπ/2 u'_Γ= (-1)^u'_σ e^iπ/2 v'_τ. Hence
u'_Γ= 2 u'_σ+ v'_τ 4.
To see how the twist sectors are related, we see that _4^Γ twisted boundary condition is determined by
|σ^σ_i+L, s^τ_i-1/2+L⟩ = ∑_s^σ_i, s^τ_i-1/2=0,1[(σ^x_i)^t'_Γ]_s_i+L^σ, s_i^σ[e^iπ t'_Γ/4 (1-τ^x_i-1/2)]_s_i-1/2+L^τ, s_i-1/2^τ|σ^σ_i, s^τ_i-1/2⟩.
Comparing with (<ref>), we find
t'_σ= t'_Γ 2, r'_τ = t'_Γ 4.
Indeed, the _2^σ×_4^τ charge completely determines the _4^Γ charge. However, not every _2^σ×_4^τ twist sector gives rise to a consistent _4^Γ twist sector. (<ref>) implies that a consistent _4^Γ twist sector exists only when t'_σ=r'_τ 2.
How are the _4^Γ symmetry-twist sectors determined in terms of the sectors before the KT transformation? From (<ref>) and (<ref>), we find that the _2^σ×_4^τ symmetry and twist sectors before KT the transformation are related to the _4^Γ symmetry and twist sectors after the KT transformation as
(u'_Γ, t'_Γ) = (2u'_σ+v'_τ, r'_τ)= (2u_σ+v_τ, r_τ+2u_σ).
From the previous paragraph, the first equality in (<ref>) holds only when t'_σ=r'_τ 2, or equivalently t_σ= v_τ+ r_τ 2. Note that the twist parameter r_τ+2u_σ on the right hand side of (<ref>) can not be consistently written in terms of a _4^Γ twist. This means that in order to obtain consistent _4^Γ untwisted and twisted sectors after KT transformation, we should consider the entire _2^σ×_4^Γ untwisted and twisted sectors with the _2^σ twist parameter fixed by t_σ= v_τ+ r_τ 2 before the KT transformation, rather than the _4^Γ untwisted and twisted sectors before the KT transformation. In short, the _4^Γ (un)twisted sectors are not invariant under KT transformation.
§.§ Field theory of the intrinsically gapless SPT
The KT transformation also allows us to write down the field theory for the intrinsically gapless SPT. We start with the partition function for the free boson CFT + SSB phase as the low energy theory of Eq. (<ref>) <cit.>
Z_free boson [A_τ] Z_SSB[A_σ],
where the partition function of the free boson CFT is given by
Z_free boson [A_τ] := ∫θexp(i ∫_X_21/2π(D_A_τθ)^2), D_A_τθ = dθ - i π/2 A_τθ.
Here A_τ is a _4 1-cocycle.
And the partition function of the _2^τ SSB phase is also simply a delta function,
Z_SSB[A_σ] = δ(A_σ).
We then perform a STS transformation, changing the partition function to
Z_igSPT[A_σ, A_τ] =∑_a_σ=0,1,
a_τ=A_τ 2 Z_SSB[a_σ] Z_free boson [a_τ]e^i π/2∫_X_2 (A_τ - a_τ)(A_σ + a_σ)
=∑_a_τ=A_τ 2 Z_free boson [a_τ]e^i π/2∫_X_2 (A_τ A_σ - a_τ A_σ)
where the detail of derivation is shown in appendix <ref>. Finally, the igSPT in <cit.> was defined with respect to the symmetry _4^Γ. Denoting the _4^Γ background field as A_Γ, from (<ref>), we identify A_τ=A_Γ 4, A_σ=A_Γ 2. The partition function is then
Z_igSPT[A_Γ]= ∑_a=A_Γ 2 Z_free boson [a]e^i π/2∫_X_2 (A_Γ- a) A_Γ.
We proceed to see the relation between (<ref>) and the decorated domain wall construction in <cit.>. To see this, we decompose the _4^Γ background field A_Γ into A_Γ= 2B+A with the constraint δ B = A^2 2, and the right hand side of (<ref>) can be precisely factorized into the form of Eq.(6) in <cit.>,
Z_igSPT[A_Γ]= Z_low[A] Z_gapped[A,B]
where
Z_low[A]= ∑_b=0,1, δ b= A^2Z_free boson [2b+A]e^iπ∫_X_2 bA, Z_gapped[A,B]= e^iπ∫_X_2 AB.
Note that the low energy partition function Z_low[A] depends only on the quotient _2 background field, and the term e^iπ∫_X_2 BA plays the role of anomalous domain wall decoration. Note that both Z_low[A] and Z_gapped[A,B] are anomalous. The former is anomalous due to the constraint δ b=A^2 and the latter is due to the constraint δ B=A^2.
§.§ Topological features of intrinsically gapless SPT
We proceed to study the topological features of the intrinsically gapless SPT directly from the Hamiltonian (<ref>). The content of this subsection already appeared in Section 3 of <cit.>, which we briefly review here. We will focus on _4^Γ symmetry charge of the ground state under TBC of _2^τ and _4^Γ. Note that the _4^Γ symmetry charge operator is given by the product U_σ V_τ.
As we found in <cit.>, under PBC, the number of ground states depends on the number of sites. This is potentially due to the effective twisted boundary condition when the system size is of a certain type. We will focus on the sequence of system size where the ground state is unique under PBC. We also found that the _4^Γ charge of the ground state depends on the system size as well, even when the ground state is unique. However, the _2^τ normal subgroup of _4^Γ, generated by U_τ, is always trivial. This can be seen as follows. Note that the last term in (<ref>) commutes with the first two terms, the ground state |ψ⟩_PBC must be an eigenstate of each of them,
σ^z_i-1τ^x_i-1/2σ^z_i|ψ⟩_PBC = |ψ⟩_PBC, i=2, ..., L
, σ^z_Lτ^x_1/2σ^z_1|ψ⟩_PBC = |ψ⟩_PBC.
In particular, this means that |ψ⟩_PBC is neutral under _2^τ normal subgroup of _4^τ,
U_τ|ψ⟩_PBC = ∏_i=1^Lτ^x_i-1/2|ψ⟩_PBC =
( ∏_i=2^Lσ^z_i-1σ^z_i) σ^z_L σ_1^z |ψ⟩_PBC.
We proceed to the TBC of _2^τ. We first consider twisting by _2^τ normal subgroup. By restricting all the Pauli operators within the physical lattice, i.e. i=1, ..., L, we find
H_igSPT^_2^τ = -∑_i=1^L-1( τ^z_i-1/2σ^x_i τ^z_i+1/2 + τ^y_i-1/2σ^x_i τ^y_i+1/2 + σ^z_iτ^x_i+1/2σ^z_i+1) + τ^z_L-1/2σ^x_L τ^z_1/2 + τ^y_L-1/2σ^x_L τ^y_1/2 - σ^z_Lτ^x_1/2σ^z_1
= σ^z_L H_igSPTσ^z_L.
The above relation immediately implies that the relative _4^Γ charge between the ground state under the _2^τ TBC |ψ⟩__2^τTBC and the ground state under PBC |ψ⟩_PBC is 2 4. Namely, we have
__2^τTBC⟨ψ|U_Γ|ψ⟩__2^τTBC = - _PBC⟨ψ|U_Γ|ψ⟩_PBC.
We finally consider the TBC of _4^Γ, under which the Hamiltonian becomes
H_igSPT^_4^Γ = - ∑_i=1^L-1( τ^z_i-1/2σ^x_i τ^z_i+1/2 + τ^y_i-1/2σ^x_i τ^y_i+1/2 + σ^z_iτ^x_i+1/2σ^z_i+1) - τ^z_L-1/2σ^x_L τ^y_1/2 + τ^y_L-1/2σ^x_L τ^z_1/2 + σ^z_L τ^x_1/2σ^z_1.
There are actually more than one ground state, but
it is then straightforward to show that all the ground states |ψ⟩__4^ΓTBC carry _2^τ charge 1 2. By comparing the _2^τ charge of the PBC ground state in (<ref>), we again find that the relative charge between the _4^Γ TBC ground state and the PBC ground state is 1 2, namely we have
__4^ΓTBC⟨ψ|U_τ|ψ⟩__4^ΓTBC = - _PBC⟨ψ|U_τ|ψ⟩_PBC.
Similar to the discussion in Section <ref>, the above discussion heavily depends on the form of the Hamiltonian. In particular, we repeatedly used the fact that the last term in the Hamiltonian (<ref>) commutes with the remaining terms. Under a generic perturbation, for instance,
-h ∑_i=1^L (σ^x_i + τ^x_i-1/2)
would destroy this feature. In this situation, the analysis in the current subsection does not work. In Section 3.4 of <cit.>, we used small scale exact diagonalization to study the ground state charge under various boundary conditions as well as the energy gap as a function of perturbation (<ref>). We found that when the perturbation h is small, the relative symmetry charge discussed in the previous paragraphs are unchanged until h increases to some critical value h_c. As h further increases, the charges are then subjected to oscillations until around h≃ 2, after which the relative charge becomes trivial. The value of h_c changes with respect to the system size, and it was not clear the behavior of h_c in the thermodynamics limit L→∞. Below, we will use the KT transformation to analytically study the phase diagram under the perturbation (<ref>).
§.§ Topological features from the KT transformation
§.§.§ Symmetry properties before the KT transformation
In this subsection, we reproduce the results in Section <ref> using the KT transformation. We first analyze the topological features of the decoupled system (<ref>) as well as its perturbation (<ref>). In this subsection, we will assume h≪ 1, and will discuss the phase diagram for finite h in the next subsection. The Hamiltonian is
H_XX+ SSB+pert= H_XX+pert + H_SSB+pert
where
H_XX+pert= -∑_i=1^L( τ^z_i-1/2τ^z_i+1/2 + τ^y_i-1/2τ^y_i+1/2 + h τ^x_i-1/2),
H_SSB+pert= -∑_i=1^L(σ^z_i-1σ^z_i + hσ^x_i).
The symmetry properties of the ground states of the above two models are well-known. In Appendix <ref>, we discuss the ground state properties of H_XX+pert, and find that the ground state energy in each symmetry-twist sector is
E^τ_(v_τ,r_τ) = 2π√(1-h^2/4)/L[ 1/4(min([v_τ]_4,4-[v_τ]_4))^2 + 1/16(min([r_τ]_4,4-[r_τ]_4))^2 ].
Hence
E_(0,0)^τ1/L<E_(0,1)^τ
E_(0,3)^τ1/L<E_(1,0)^τ
E_(3,0)^τ
E_(0,2)^τ1/L<E_(1,1)^τ
E_(3,1)^τ
E_(1,3)^τ
E_(3,3)^τ1/L<E_(1,2)^τ
E_(3,2)^τ1/L<
E_(2,0)^τ1/L<E_(2,1)^τ
E_(2,3)^τ1/L<
E_(3,3)^τ
where the energies on each column are equal. The symmetry properties of H_SSB+pert has been already considered in (<ref>), which we reproduce here
E^σ_(0,0)e^-L< E_(1,0)^σ1< E_(0,1)^σ1/L^2< E_(1,1)^σ.
The ground state energy of the Hamiltonian H_XX+ SSB+pert in the symmetry-twist sector [(u_σ,t_σ),(v_τ,r_τ)] is given by
E_[(u_σ,t_σ),(v_τ,r_τ)] = E^τ_(v_τ, r_τ) + E^σ_(u_σ, t_σ).
§.§.§ Symmetry properties after the KT transformation
Under the KT transformation, the decoupled Hamiltonian (<ref>) becomes the perturbation of igSPT,
H_igSPT+pert = - ∑_i=1^L(τ^z_i-1/2σ^x_iτ^z_i+1/2 + τ^y_i-1/2σ^x_iτ^y_i+1/2 + σ^z_i-1τ^x_i-1/2σ^z_i + h σ^x_i + h τ^x_i-1/2).
Note that the perturbation is the same as in the decoupled system (<ref>) because σ^x and τ^x commute with KT transformation. By combining the ground state symmetry properties of the decoupled system before gauging in Section <ref> and how the symmetry-twist sectors are mapped under KT transformation in Section <ref>, we are able to determine the ground state properties of the perturbed igSPT Hamiltonian (<ref>) after the KT transformation. In particular, we would like to determine the _4^Γ symmetry charge of the ground state under each _4^Γ twisted boundary condition.
Concretely, by (<ref>), the energy in the symmetry-twist sector after the KT transformation E^Γ_(u_Γ, t_Γ) is equal to the energy before the KT transformation E_[(u_σ,t_σ),(v_τ,r_τ)] which is further equal to E^τ_(v_τ, r_τ) + E^σ_(u_σ, t_σ) by (<ref>). Here, (u_Γ, t_Γ) is determined by [(u_σ,t_σ),(v_τ,r_τ)] via the relation (u_Γ, t_Γ)= [(u_σ,t_σ),(v_τ,r_τ)]. In summary, we have
E^Γ_(u_Γ, t_Γ)(<ref>)= E_[(u_σ,t_σ),(v_τ,r_τ)](<ref>)=E^σ_(u_σ, t_σ) + E^τ_(v_τ, r_τ)(<ref>)= E^σ_(u_σ, u_Γ+t_Γ) + E^τ_(u_Γ+2u_σ, t_Γ+ 2 u_σ).
In the last equality, we used (<ref>) to solve v_τ=u_Γ+ 2u_σ 4 and r_τ= t_Γ + 2 u_σ 4, and also used the condition t_σ 2= v_τ+r_τ = u_Γ+t_Γ 2 which is imposed by (<ref>). For a fixed twisted boundary condition t_Γ, we search for the minimal energy over all possible u_σ and u_Γ by comparing the energy hierarchy (<ref>) and (<ref>). The selected u_Γ is the _4^Γ charge of the ground state in the _4^Γ twisted boundary condition specified by t_Γ. We will discuss the four twisted boundary conditions separately.
* t_Γ=0 4: By (<ref>), the energy of the ground state in the symmetry-twist sector is
E^Γ_(u_Γ, 0)=E^σ_(u_σ, u_Γ) + E^τ_(u_Γ+2u_σ, 2 u_σ).
We would like to minimize the right hand side over all possible u_σ and u_Γ. It is obvious that u_σ=0 and u_Γ=0 yield the lowest energy. We will only be interested in the relative symmetry charge between under TBC (t_Γ =1,2,3) and PBC (t_Γ=0).[The energy (<ref>) is obtained in the continuum, and does not capture everything on the lattice. In particular, the absolute charge of the ground state under PBC from the continuum is always trivial, while on the lattice the ground state charge depends on L 8. But we anticipate that the relative charge on the lattice and in the continuum match if we focus on L=0 8. ]
* t_Γ=1 4: By (<ref>), the energy of the ground state in the symmetry-twist sector is
E^Γ_(u_Γ, 1)=E^σ_(u_σ, u_Γ+1) + E^τ_(u_Γ+2u_σ, 1+2 u_σ).
To minimize the energy, we need u_Γ=1 2, because otherwise there is an energy cost of order 1 from the sigma spins. This implies that both the symmetry and twist parameters of the tau spin are odd. From (<ref>), all the energies of this kind degenerate. The minimal energy is thus achieved by choosing u_σ=0 2 and u_Γ=1, 3 4. In summary, the _4^Γ charge under t_Γ=1 TBC is u_Γ=1,3 4. This is consistent with the discussion in the unperturbed case h=0 in Section <ref>, where it has been shown that the ground state energy under _4^Γ twist has a non-trivial relative _2^τ⊂_4^Γ charge.
* t_Γ=2 4: By (<ref>), the energy of the ground state in the symmetry-twist sector is
E^Γ_(u_Γ, 2)=E^σ_(u_σ, u_Γ) + E^τ_(u_Γ+2u_σ, 2+2 u_σ).
By similar analysis, the minimal energy is achieved by choosing u_Γ=2 4 and u_σ=1 2. Hence the _4^Γ charge under t_Γ=2 TBC is u_Γ=2 4. This is again consistent with the discussion in the unperturbed case h=0 in Section <ref>, where it has been shown that the ground state energy under _2^τ twist has a relative _4^Γ charge 2 4.
* t_Γ=3 4: By (<ref>), the energy of the ground state in the symmetry-twist sector is
E^Γ_(u_Γ, 3)=E^σ_(u_σ, u_Γ+3) + E^τ_(u_Γ+2u_σ, 3+2 u_σ).
By similar analysis, the minimal energy is achieved by choosing u_Γ=1,3 4 and u_σ=0 2. Hence the _4^Γ charge under t_Γ=3 TBC is u_Γ=1,3 4. This is again consistent with the discussion in the unperturbed case h=0 in Section <ref>, where it has been shown that the ground state energy under odd _4^Γ twist has a relative _2^τ charge 1 2.
In summary, we have reproduced the _4^Γ ground state charge under various _4^Γ twisted boundary conditions as in Section <ref> which was also discussed in <cit.>. The novel feature here is that the method used here also works after turning on perturbation h, while the method in Section <ref> does not apply for non-trivial perturbation even when h is infinitesimally small. Since the non-trivial relative _4^Γ charge of the ground state under _4^Γ twisted boundary conditions is a key feature of the intrinsically gapless SPT, we take it as a strong support that the igSPT is robust under small perturbation of the kind (<ref>).
§.§ Phase diagram
In the previous subsection, we only studied a small perturbation and determined the ground state symmetry properties under various twisted boundary conditions. The results suggested that the igSPT at h=0 is robust for small h. In this subsection, we proceed to increase h and study the phase diagram. We finally comment on the string order parameter.
Phase diagram:
We first determine the critical h where the phase transitions of (<ref>) take place. Because the location of the phase transition is not changed under discrete gauging, the value of h_c can be inferred from the model (<ref>) before the KT transformation. Since before the KT transformation the system is decoupled, i.e. H_XX+ SSB+pert= H_XX+pert + H_SSB+pert, the phase transition can be determined separately for each part. We summarize the structure of two parts below.
* For the XX model with a transverse field, as discussed in Appendix <ref>, when 0≤ h <2, the model is a free massless boson (or equivalently a free massless Dirac fermion). The Fermi velocity decreases as h increases, and eventually goes to zero at h=2. When h>2, the transverse field takes place and the model is trivially gapped. The phase transition between a gapless free boson and a trivially gapped phase occurs at h=2. This is the Lifshitz transition <cit.>.
* For the _2^σ Ising model with a transverse field, when 0≤ h< 1, the model is in a gapped _2^σ SSB phase. When h>1, the model is in a trivially gapped phase. The transition occurs at h=1, i.e. the Ising phase transition.
Combining the above, we identified two phase transitions of the igSPT with perturbation (<ref>) at h=1 and h=2 respectively. The schematic phase diagram is shown in Figure <ref>.
To see the phases in each regime, we discuss the topological features in 0≤ h <1, 1<h<2 and h>2 respectively.
* 0≤ h <1: The topological features have been determined in Section <ref>, which represents the intrinsically gapped SPT phase.
* 1<h<2: We proceed to discuss the topological features in the regime 1<h<2, by repeating the same analysis in Section <ref>. Before the KT transformation, the energy hierarchy of the XX model with a transverse field is the same as (<ref>), while that of the transverse field Ising model is modified to
E^σ_(0,0)e^-L< E_(0,1)^σ1< E_(1,0)^σ1/L^2< E_(1,1)^σ.
This follows from the fact that the energy in 1<h<2 is related to 1/2<h<1 via a Kramers-Wannier transformation, which exchanges the symmetry-twist sectors by (u,t)↔ (t,u). Then we combine (<ref>), (<ref>) and (<ref>) to find the _4^Γ charge of the ground state for various _4^Γ twisted boundary conditions. From (<ref>), we find that u_σ=0 in order to avoid the order one energy cost (according to (<ref>)),
E^Γ_(u_Γ, t_Γ)=E^σ_(0,t_Γ+ u_Γ) + E^τ_(u_Γ, t_Γ).
The energy is then dominated by the τ spin.
From (<ref>), we further observe that for any t_Γ, the energy E^τ_(u_Γ, t_Γ) is minimized by u_Γ=0 4. This concludes that the relative _4^Γ charge is always trivial. This implies that the Hamiltonian (<ref>) with 1<h<2 is a trivial gapless SPT!
* h>2: Only the transverse field term -h ∑_i=1^L (σ^x_i + τ^x_i-1/2) dominates in this regime, and hence the theory is in the trivially gapped phase.
We summarize the above discussion in the phase diagram in Figure <ref>. We note that the above phase diagram is also supported by exact numerical diagonalization studied in <cit.>, for example, the transition at h=2 is clearly indicated in Figure 5 of <cit.>. The jumps of the charges for h<2 in the numerical results in <cit.> are essentially due to the subtleties of L≠ 0 8 and various finite size effects. The new KT transformation thus provides an analytic understanding of the stability of intrinsically gapless SPT under perturbation.
String order parameter:
We finally comment on the string order parameter in the igSPT. Before the KT transformation, the decoupled Hamiltonian has the conventional correlation function:
⟨σ^z_i σ^z_j|∼⟩(1), h<1
e^-ξ_σ L, h>1
, ⟨τ^z_i-1/2τ^z_j-1/2|∼⟩1/|i-j|^2Δ, h<2
e^-ξ_τ L, h>2
where Δ and ξ_τ are the scaling dimension and the correlation length of τ^z respectively. Similar for ξ_σ.
After the KT transformation, the conventional correlation function becomes string order parameters:
⟨σ^z_i (∏_k=i^j-1τ^x_k+1/2 )σ^z_j|∼⟩(1), h<1
e^-ξ_σ L, h>1
, ⟨τ^z_i-1/2
(∏_k=i^j-1σ^x_k)τ^z_j-1/2|∼⟩1/|i-j|^2Δ, h<2
e^-ξ_τ L. h>2
§ PURELY GAPLESS SPT AND INTRINSICALLY PURELY GAPLESS SPT FROM KT TRANSFORMATION
In Section <ref> and Section <ref>, we discussed the construction of gapless SPT and intrinsically gapless SPT from the KT transformation, starting from the decoupled _2^σ SSB phase and a gapless theory (Ising CFT and XX model for gSPT and igSPT respectively). Since both cases contain a gapped sector (_2^σ SSB phase) before the KT transformation, the resulting gSPT and igSPT always contain a gapped sector. This is confirmed, for example by computing the energy splitting of the edge modes on an open chain <cit.>. The gapped sectors can also be seen by the decorated domain wall construction explored in <cit.>. A natural question is can we construct a gapless SPT with no gapped sector?
In this section, we will construct both gapless SPT and intrinsically gapless SPT that do not contain gapped sectors, and follow <cit.> to denote them as purely gapless SPT (pgSPT) and intrinsically purely gapless SPT (ipgSPT) respectively.
In particular, we begin with two decoupled gapless XXZ chains, and perform a KT transformation.
§.§ Constructing pgSPT and ipgSPT
Let us start with two decoupled XXZ chains, The Hamiltonian is
H^h_XXZ+XXZ=- ∑_i=1^L( σ^z_iσ^z_i+1 + σ^y_iσ^y_i+1 + h σ^x_iσ^x_i+1 + τ^z_i-1/2τ^z_i+1/2 + τ^y_i-1/2τ^y_i+1/2 + h τ^x_i-1/2τ^x_i+1/2).
For pgSPT, we will focus on _2^σ×_2^τ, and for ipgSPT we will focus on the _2^σ×_4^τ symmetry. Here _2^σ and _2^τ are defined in (<ref>) and _4^τ is defined in (<ref>). Below, we will mainly focus on the ipgSPT and the symmetry _2^σ×_4^τ. The pgSPT is obtained by the same theory and only changes the symmetry to be the normal subgroup _2^σ×_2^τ.
Constructing ipgSPT with _4^Γ symmetry:
Applying the KT transformation (associated with _2^σ×_2^τ where the latter is the normal subgroup of _4^τ), the decoupled Hamiltonian becomes
H_ipgSPTpert^h= -∑_i=1^L( σ^z_iτ^x_i+1/2σ^z_i+1 + σ^y_iτ^x_i+1/2σ^y_i+1 + h σ^x_iσ^x_i+1 + τ^z_i-1/2σ^x_iτ^z_i+1/2 + τ^y_i-1/2σ^x_iτ^y_i+1/2+h τ^x_i-1/2τ^x_i+1/2).
Since the KT transformation commutes with both σ^x_i and τ^x_i-1/2, the resulting Hamiltonian (<ref>) also has _2^σ×_4^τ symmetry, generated by U_σ and V_τ. We will again focus on the _4^Γ subgroup, generated by U_σ V_τ.
The Hamiltonian (<ref>) depends on a parameter h, which can be taken to be either positive or negative. Below, we will study the _4^Γ symmetry properties of the ground state of (<ref>) under various _4^Γ twisted boundary conditions to determine the regime of h where the Hamiltonian is topologically non-trivial, hence is ipgSPT. It will become clear in the following subsections that the ipgSPT corresponds to 0<h<1.
Before determining the topological properties of the phases, it is useful to see where the phase transitions can take place, which can be inferred from the decoupled theory (<ref>) before the KT transformation.
From Appendix <ref>, we know that the phase transition occurs at h=-1 and h=1. Later we will also see that there is a more subtle phase transition at h=0.
pgSPT with symmetry _2^σ×_2^τ:
The Hamiltonian of pgSPT is given by the same Hamiltonian (<ref>). The only difference is the choice of symmetry, which is taken to be _2^σ×_2^τ. Here _2^τ is the normal subgroup of _4^τ. It turns out that the non-trivial pgSPT also corresponds to the parameter regime 0<h<1.
§.§ Field theory description
Field theory of ipgSPT:
Before studying the ground state property, we first comment on the field theory description of the lattice ipgSPT Hamiltonian (<ref>). As reviewed in Appendix <ref>, the field theory of the XXZ model is a free boson <cit.>. Hence we begin with the partition function
Z_boson^h[A_τ] Z_boson^h[A_σ]
where A_τ is a _4 cocycle, and A_σ is a _2 cocycle. More explicitly the two decoupled partition functions are
Z_boson^h[A_τ]:= ∫θ_τexp( i ∫_X_21/2π K_h (D_A_τθ_τ)^2), D_A_τθ_τ= dθ_τ - i π/2 A_τθ_τ
Z_boson^h[A_σ]:= ∫θ_σexp( i ∫_X_21/2π K_h (D_A_σθ_σ)^2), D_A_σθ_σ= dθ_σ - i π A_σθ_σ .
We then perform a STS transformation, changing the partition function to
Z_ipgSPTpert^h[A_σ, A_τ]= ∑_a_σ=0,1
a_τ= A_τ 2 Z_boson^h[a_σ] Z_boson^h[a_τ] e^iπ/2∫_X_2 (A_τ-a_τ)(A_σ+a_σ)
which is a gauged version of two decoupled free bosons. By further restricting the _2^σ×_4^τ symmetry to _4^Γ, whose background fields are identified as A_τ=A_Γ 4, and A_σ=A_Γ 2, we have
Z_ipgSPTpert^h[A_Γ]= ∑_a_σ=0,1
a_τ= A_Γ 2 Z_boson^h[a_σ] Z_boson^h[a_τ] e^iπ/2∫_X_2 (A_Γ-a_τ)(A_Γ+a_σ).
It is also useful to see why the construction in <cit.> does not apply here. Let us decompose A_Γ as A_Γ= 2B+A, with δ B=A^2 2, then the partition function simplifies to
Z_ipgSPTpert^h[A_Γ]= ∑_a,b=0,1
δ b=A^2 2 Z_boson^h[a] Z_boson^h[2b+A] e^iπ∫_X_2 ab + aB + bA+ AB.
Note that the partition function can not be factorized into Z_low[A] Z_gapped[A,B] where the low energy sector only depends on the quotient _2. Hence the entire _4 symmetry couples to the low energy degrees of freedom and there is no obvious gapped sector.
Field theory of pgSPT:
The field theory of pgSPT can be obtained by starting from Z_boson^h[A_τ] Z_boson^h[A_σ] where both A_τ and A_σ are _2 valued background fields, and perform a KT transformation. The resulting partition function is
Z_pgSPTpert^h[A_σ, A_τ]= ∑_a_σ, a_τ=0,1 Z_boson^h[a_σ] Z_boson^h[a_τ] e^iπ∫_X_2 (A_τ+a_τ)(A_σ+a_σ).
Again there is also no decoupled gapped sector because both A_σ and A_τ couple to the gapless sectors via magnetic coupling.
§.§ Topological features of pgSPT and ipgSPT from KT transformation and phase diagram
In this subsection, we proceed to study the topological features of (<ref>). Since there is no term in (<ref>) which commutes with the rest of the terms, the method from <cit.> (which were also reviewed in Section <ref> and Section <ref>) does not work. Hence we directly apply the KT transformation to study the topological features. We will be mainly discussing the topological features and the phase diagram of ipgSPT as a function of h. We will also briefly comment on the phase diagram of pgSPT.
We begin by studying the topological features of two copies of XXZ chain with anisotropy parameter h. Note that the global symmetry of the sigma spin is _2^σ, by repeating the discussion in Appendix <ref>, we find the ground state energy in the _2^σ symmetry-twist sector E_(u_σ, t_σ)^σ is
E_(u_σ, t_σ)^σ(h)= 2π/ L[ 1/4K_h [u_σ]_2 ^2 + K_h/4 [t_σ]_2^2].
Note that [u_σ]_2 is u_σ modulo 2, and similar for [t_σ]_2. For the τ spin, the global symmetry is _4^τ, and the ground state energy in the _4^τ symmetry-twist sector is given by (<ref>),
E_(v_τ, r_τ)^σ(h)= 2π/L[ 1/4K_h (min ([v_τ]_4, 4-[v_τ]_4))^2 + K_h/16 (min ([r_τ]_4, 4-[r_τ]_4))^2].
Here K_h is related to h as follows,
-1<h<0 ⇔1/2<K_h<1, 0<h<1 ⇔ K_h>1.
Note that in the regime h<-1 and h>1, the XXZ model is gapped and spontaneously breaks the ^y_2 symmetry which is generated by the ∏_i σ^y_i <cit.>. Hence the cosine terms in the Sine-Gordon model can not be ignored as they are relevant operators.
We proceed to the system (<ref>) after the KT transformation, and consider the ground state in each _4^Γ symmetry-twist sector E^Γ_(u_Γ, t_Γ)(h). The energy is the same as (<ref>). Substituting (<ref>) and (<ref>) into (<ref>), we obtain
E^Γ_(u_Γ, t_Γ)(h) = E^σ_(u_σ, u_Γ+t_Γ)(h) + E^τ_(u_Γ+2u_σ, t_Γ+2u_σ)(h)
= L/2π[ 1/4K_h [u_σ]_2 ^2 + K_h/4 [u_Γ+t_Γ]_2^2 + 1/4K_h (min ([u_Γ+2u_σ]_4, 4-[u_Γ+2u_σ]_4))^2
+ K_h/16 (min ([t_Γ+2u_σ]_4, 4-[t_Γ+2u_σ]_4))^2 ].
The structure of the minimal energy spectrum is as follows.
* When t_Γ=0, the lowest energy is achieved by u_Γ=0 and u_σ=0.
* When t_Γ=1, the lowest energy is achieved by (u_σ,u_Γ)=(0, 0) if K_h<1, and (u_σ,u_Γ)=(0, 1) or (0,3) if K_h>1.
* When t_Γ=2, the lowest energy is (u_σ,u_Γ)=(0,0) if K_h<1, and (u_σ,u_Γ)=(1,2) if K_h>1.
* When t_Γ=3, the lowest energy is achieved by (u_σ,u_Γ)=(0, 0) if K_h<1, and (u_σ,u_Γ)=(0, 1) or (0,3) if K_h>1.
Combining the correspondence between h and K_h, we find that when -1<h<0, the ground state of the Hamiltonian (<ref>) always has trivial _4^Γ charge u_Γ=0 under any twisted boundary condition, hence it is in the trivial gapless phase. When 0<h<1, the ground state of the Hamiltonian (<ref>) has non-trivial
relative _4^Γ charge under _4^Γ twisted boundary conditions, hence it is in the topologically non-trivial ipgSPT phase. We summarize the phase diagram in Figure <ref>.
We summarize the properties of the various phase transitions.
* h=-1: The phase transition at h=-1 can be described by the Sine-Gordon model at K_h=1/2. This phase transition is described by the SU(2)_1 WZW CFT, where the cosine term cos(2φ) becomes marginal, between the irrelevant h>-1 (K_h>1/2) to relevant h<-1 (K_h<1/2).
* h=1: The phase transition at h=1 corresponds to K_h→∞, and is not described within the free boson description. But from the lattice model, the transition can be understood intuitively as the XX term dominates over the other terms and triggers the system to a gapped phase where _2^y is SSB.
* h=0: Unlike the transitions at h=± 1, this transition is not directly implied from the two decoupled XXZ chains before the KT transformation because both K_h>1 and K_h<1 are described by the free bosons with trivial topological features. However, after the KT transformation, the topological feature becomes non-trivial for K_h>1 while still remains trivial for K_h<1. This implies that there is a topological phase transition at K_h=1, i.e. h=0.
String order parameter:
We comment on the string order parameter in the gapless region. Before the KT transformation, each decoupled Hamiltonian has two quasi long-range orders when -1<h<1. One is the correlation function of Pauli Z operators and the other is the disorder order parameter of Pauli X operators:
⟨τ^z_i-1/2τ^z_j-1/2|=⟩⟨σ^z_i σ^z_j|∼⟩1/|i-j|^1/2K_h
, ⟨∏^j_k=iτ^x_k-1/2|=⟩⟨∏^j_k=iσ^x_k |∼⟩1/|i-j|^K_h/2.
Thus, after the KT transformation, the first two quasi long-range orders become string order parameters
⟨τ^z_i-1/2(∏^j-1_k=iσ^x_k) τ^z_j-1/2|=⟩⟨σ^z_i (∏_k=i^j-1τ^x_k+1/2 ) σ^z_j|∼⟩1/|i-j|^1/2K_h
, ⟨∏^j_k=iτ^x_k-1/2|=⟩⟨∏^j_k=iσ^x_k |∼⟩1/|i-j|^K_h/2.
Note that string order parameters carry a nontrivial symmetry charge on the end while the disorder charge does not. When 1<K_h (0<h<1), the string order parameters decay lower than disorder parameters, while when 1/2<K_h<1 (-1<h<0), the string order parameters decay faster than disorder parameters. Here we remark that this result is consistent with ground state charge under twisted boundary conditions due to the state operator correspondence <cit.>.
We also comment on the boundary degeneracy under OBC. Since the pgSPT is obtained from two decoupled XXZ chains, the low-energy spectrum of an XXZ chain under OBC in the gapless region, similar to that under PBC, scales as (1/L).
Based on the mapping by the KT transformation, we expect that the spectrum of pgSPT constructed in this Section
also scales as (1/L), under either PBC or OBC.
This curious observation should be contrasted from the (1/L^14) finite size gap decaying behavior under OBC of the pgSPT with a different anti-unitary symmetry—_2×_2^T <cit.>, which clearly differs from (1/L) under PBC. Nevertheless, various other signatures including non-trivial symmetry charge under TBC as well as the symmetry charge of the string order parameter suggest non-trivial SPT features, hence we still consider our model as a different type of pgSPT.
We will leave a more elaborated discussion on pgSPT to the future.
Phase diagram of pgSPT:
We finally comment on the pgSPT. The discussions are mostly in parallel to the ipgSPT, and we will not give the details here. The topological features of the purely gapless SPTs are such that the _2^σ (_2^τ) symmetry charge of ground state under _2^τ (_2^σ) twisted boundary condition is non-trivial. It is also interesting to note that the decaying behavior of the string order parameter does not change qualitatively for the trivial gapless phase and the pgSPT since they both decay polynomially, hence the phase transition is subtle as we have seen above from a separate perspective. The final phase diagram is shown in Figure <ref>.
§ ACKNOWLEDGEMENT
We would like to thank Philip Boyle Smith, Weiguang Cao, Yoshiki Fukusumi and Hong Yang for useful discussions. L.L. is supported by Global Science Graduate Course (GSGC) program at the University of Tokyo. Y.Z. is partially supported by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. This work was supported in part by MEXT/JSPS KAKENHI Grants No. JP17H06462 and No. JP19H01808, and by JST CREST Grant No. JPMJCR19T2.
§ ENERGY SPECTRUM OF TRANSVERSE FIELD LSING MODEL UNDER _2 TBC
The Hamiltonian of transverse field lsing model is
H_lsing=-∑^L_i=1(σ^z_iσ^z_i+1+hσ^x_i)
with _2 twisted boundary condition: σ^z_i+L=-σ^z_i, σ^x_i+L=σ^x_i.
We apply the Jordan-Wigner (JW) transformation which maps the spin operator to the fermion operator
σ^x_i=(-1)^n_i=1-2f^†_i f_i, σ^z_i=∏_j=1^i-1(-1)^n_j(f^†_i+f_i)
where n_i:= f_i^† f_i is the fermion density operator. Note that when i=1, we simply have σ_1^z= f_1^† + f_1.
Applying the JW transformation to the lsing model, we can rewrite (<ref>) in terms of the fermions,
H_lsing=-hL-∑_i=1^L( -2hf^†_i f_i+(f^†_i-f_i)(f^†_i+1+f_i+1))
with boundary condition
f_i+L=(-1)^Ff_i, F=∑^L_j=1 n_j .
After Fourier transformation and Bogoliubov transformation, this Hamiltonian is diagonal
H_lsing= ∑_kω_k ( c^†_k c_k-1/2)
where ω_k=2√(1-2hcos k+h^2). Moreover, the fermion parity after transformation is given by
(-1)^∑_0<k <π c^†_k c_k+c^†_-k c_-k=(-1)^∑_0<k<πf^†_kf_k+f^†_-kf_-k ,
(-1)^ c^†_0 c_0=(-1)^f^†_0f_0sign(h-1), (-1)^ c^†_πc_π=(-1)^f^†_πf_π
When h=1, there is a zero mode with k=0 and whether it is realizable depends on the boundary condition. Moreover, the fermion parity of the k=0 modes after Bogoliubov transformation changes depends on h. We discuss them separately.
§.§ Energy spectrum of SSB phase (0≤ h<1)
If (-1)^F=1, the fermion chain has PBC. This means k=2π j/L where j=0,⋯,L-1. Therefore, we have k=0 mode when j=0. As the total fermion parity after Bogoliubov transformation changes, the ground states are: c^†_π|VAC⟩_PBC. The ground state energy is
E^PBC_GS=-1/2∑_k=2π j/Lω_k+2|1-h|.
If (-1)^F=-1, the fermion chain has anti-periodic boundary condition (ABC) where k=(2j+1)π/L and there is no k=0 mode. The ground states are c^†_-π/L|VAC⟩_ABC and c^†_π/L|VAC⟩_ABC with ground state energy:
E^ABC_GS=-1/2∑_k=(2j+1)π/Lω_k+2√(1+h^2-2hcosπ/L).
When L is large, the first term of E^ABC_GS and E^PBC_GS has a gap of order e^-L and the second term has a gap of order 1/L^2. Thus we have E^ABC_GS>E^PBC_GS and the gap is of order 1/L^2.
In summary, the ground state of the transverse field lsing model when h<1 is unique. Since ∏^L_j=1σ^x_j=(-1)^F, the ground state is always in the even sector while the first excited states are in the odd sector and the finite size gap is 1/L^2.
§.§ Energy spectrum of trivial phase (h>1)
If (-1)^F=1, the fermion chain has PBC, then k=2π j/L where j=0,⋯,L-1. When j=0, we have k=0 mode. Since the total fermion parity after Bogoliubov transformation is invariant, the ground state is |VAC⟩_PBC
with ground state energy:
E^PBC_GS=-1/2∑_k=2π j /Lω_k.
If (-1)^F=-1, the fermion chain has ABC where k=(2j+1)π/L. And the total fermion parity after the Bogoliubov transformation is also invariant. Since (-1)^F=-1, the ground states are c_-π/L|VAC⟩_ABC and c_π/L|VAC⟩_ABC with ground state energy:
E^ABC_GS=-1/2∑_k=(2j+1)π/Lω_k+2√(1+h^2-2hcosπ/L).
Therefore, we obtain that E^PBC_GS<E^ABC_GS and the gap is finite in thermodynamic limit.
Then the ground state of the transverse field lsing model when h>1 is unique and gapped. Moreover, the ground state is always in the even sector while the first excited states are in the odd sector.
§.§ Energy spectrum of critical point (h=1)
If (-1)^F=1, the fermion chain has PBC, then k=2π j/L where j=0,⋯,L-1. We also have k=0 mode when j=0. Since the total fermion parity after Bogoliubov transformation is invariant, the ground state is |VAC⟩_PBC
with ground state energy:
E^PBC_GS=-1/2∑_k=2π j /Lω_k=-2∑_k=2π j /L|cos(k/2)|=-2(π/2L).
If (-1)^F=-1, the fermion chain has ABC where k=(2j+1)π/L. And the total fermion parity after the Bogoliubov transformation is also invariant. Since (-1)^F=-1, the ground states are c_(2L-1)π/2L|VAC⟩_ABC and c_(2L+1)π/2L|VAC⟩_ABC with ground state energy:
E^ABC_GS=-1/2∑_k=(2j+1)π/Lω_k+4sinπ/2L=-2/sin(π/2L)+4sinπ/2L.
Therefore, we obtain that E^PBC_GS<E^ABC_GS and the finite size gap is of order 1/L. Then the ground state of the lsing model at the critical point under TBC is always in the _2 even sector while the first excited states are in the _2 odd sector and the finite size gap is of order 1/L.
§ STABILITY OF EDGE MODE OF GSPT AND IGSPT PHASE ON THE OBC
In this appendix, we will discuss the stability of the finite size gap of gapless SPT and intrinsically gapless SPT on the OBC under the symmetric boundary perturbation. We aim to prove that there are at least two nearly degenerate ground states with finite size gap of order e^-L and this (exponential) degeneracy is stable under any symmetric boundary perturbation.
Let's first consider the gapless SPT phase in section <ref>, and focus on the two decoupled systems before KT transformation. One is a _2 SSB model with τ spin and the other is the critical transverse field Ising model lsing model with σ spin. Thus the low energy states are:
|even_τ⟩⊗|ψ^σ_j⟩, |odd_τ⟩⊗|ψ^σ_j⟩
where |even_τ⟩ and |odd_τ⟩ are nearly degenerate ground states of τ spins with even and odd ^τ_2 charge and the finite size gap E^τ_odd-E^τ_evenis of order e^-L. ψ^σ_j is the energy eigenstate of the gapless σ spin model which is labeled by the positive integer j. In this basis, the low energy effective Hamiltonian is blocked diagonal:
H^low_SSB+gapless=([ E^τ_even+H^σ_gapless 0; 0 E^τ_odd+H^σ_gapless ])
.
Moreover, any other excited state of τ spin has a finite gap Δ_SSB in the thermodynamic limit. It is obvious that this system has at least two ground states with exponential energy splitting. Due to the unitarity of KT transformation on an open chain, this implies the exponential finite size gap between edge modes.
Now let us add a ^σ_2×^τ_2 symmetric perturbation hV in the gapless SPT Hamiltonian. Since symmetry operators are both invariant under KT transformation, the perturbation in the decoupled system before KT transformation is also ^σ_2×^τ_2 symmetric. As the KT transformation is unitary on the open chain, the energy spectrum before and after the KT transformation are the same and we will focus on the energy spectrum of the decoupled system with the perturbation above.
At first, we can decompose hV as follows:
hV=h∑^N_i=1V^i_σV^i_τ
where N can be polynomial in L. V^i_σ and V^i_τ only act on finite range of the σ and τ spin respectively. When h≪Δ_SSB, we consider the first perturbation theory, namely we only need to consider how this perturbation acts on the low energy states. Since hV^i_τ is symmetric, it is diagonal for |even_τ⟩ and |odd_τ⟩ :
hV^i_τ=([ ha^i_even 0; 0 ha^i_odd ])
,
where a^i_even=⟨even_τ|V^i_τ|even_τ⟩ and a^i_odd=⟨odd_τ|V^i_τ|odd_τ⟩ is finite independent of system size. Thus the low energy effective Hamiltonian with this perturbation is
H_SSB+gapless+hV=([ E^τ_even+H^σ_gapless+h∑^N_i=1a^i_evenV^i_σ 0; 0 E^τ_odd+H^σ_gapless+h∑^N_i=1a^i_oddV^i_σ ])
.
Moreover, as h≪Δ_SSB, the τ spin chain after adding hV^i_τ is still in the SSB phase and the finite size gap ha^i_even-ha^i_odd is of order e^-L.
In the next step, we denote the ground states of H^σ_gapless+h∑^N_i=1a^i_evenV^i_σ and H^σ_gapless+h∑^N_i=1a^i_oddV^i_σ as |GS_σ⟩ and |GS'_σ⟩ with energy E_1 and E'_1 respectively.
Without loss of generality, we can assume E_1≤ E'_1. Then we notice that
⟨GS_σ|H^σ_gapless+h∑^N_i=1a^i_oddV^i_σ-H^σ_gapless-h∑^N_i=1a^i_evenV^i_σ|GS_σ⟩∝⟨GS_σ|e^-L∑^N_i=1V^i_σ|GS_σ⟩∝ e^-L.
Note that the sum over i does not change exponential decaying behaviour of finite size gap.
On the other hand, we also have
⟨GS_σ|H^σ_gapless+h∑^N_i=1a^i_oddV^i_σ-H^σ_gapless-h∑^N_i=1a^i_evenV^i_σ|GS_σ⟩
= ⟨GS_σ|H^σ_gapless+h∑^N_i=1a^i_oddV^i_σ|GS_σ⟩-E_1
≥ (E'_1-E_1)≥0
Thus, we can obtain that E'_1-E_1 is of order e^-L. Since E^τ_odd-E^τ_even is also of order e^-L, the total finite size gap between |even_τ⟩⊗|GS_σ⟩ and |odd_τ⟩⊗|GS'_σ⟩ is of order e^-L, which finishes our proof.
For the intrinsically gapless SPT phase in section <ref>, if we add a ^Γ_4 symmetric perturbation, this perturbation may be mapped to a nonlocal perturbation under KT transformation. For example, σ^z_L(τ^z_L-1/2τ^z_L+1/2-τ^y_L-1/2τ^y_L+1/2)→ (∏_j<Lτ^x_j+1/2)σ^z_Lσ^x_L(τ^z_L-1/2τ^z_L+1/2-τ^y_L-1/2τ^y_L+1/2). Thus, we will consider ^σ_2×^τ_4 symmetric perturbations. which is still local and ^σ_2×^τ_4 symmetric under the KT transformation. Then the proof of the stability of edge modes is similar to that of the gapless SPT phase above and we don't repeat it here.
§ MAPPING _4^Τ×_2^Σ SYMMETRY-TWIST SECTORS UNDER THE KT TRANSFORMATION
In this appendix, we derive the mapping between _4^τ×_2^σ symmetry-twist sectors under KT transformation directly from the partition function.
We start with a theory with an anomaly-free _4^τ×_2^σ symmetry, and denote their background fields as A_τ and A_σ. The partition function is Z_[A_σ, A_τ]. The KT transformation is the twisted gauging of the _2^σ and _2^τ normal subgroup of _4^τ.
Since only the normal subgroup of _4^τ participates in gauging, it is useful to first decompose the background field into A_τ= 2 B_τ+C_τ, where both B_τ and C_τ are _2 valued 1-cochains, satisfying the condition
δ A_σ=0 2, δ C_τ= 0 2, δ B_τ= C_τ^2 2.
Under KT transformation, the partition function of the resulting theory is
∑_a_σ, b_τ, a_σ, b_τ Z_[a_σ, 2b_τ+C_τ] e^iπ∫_X_2 a_σb_τ+ b_τa_σ + b_τa_σ + a_σ B_τ + b_τ A_σ
= ∑_a_σ, b_τ Z_[a_σ, 2b_τ+C_τ] e^iπ∫_X_2 (b_τ+B_τ)(a_σ + A_σ).
In the second line, we summed over a_σ and b_τ.
What symmetry does the resulting theory have? To see this, we check whether the resulting partition function (<ref>) depends on the 3d bulk. The dynamical part Z_ is clearly independent of the 3d bulk since is anomaly free. By promoting the remaining part to the 3d integral using using derivative, and applying the bundle constraints δ a_σ=0 2, δ C_τ= 0 2, δ b_τ= C_τ^2 2 which follow from (<ref>), the 3d dependence is
e^iπ∫_X_3 (C_τ^2 + δ B_τ)(a_σ + A_σ) + (b_τ + B_τ)δ A_σ.
For the resulting theory to be an absolute theory, we need to demand that all terms involving dynamical fields to vanish. This in particular requires
δ B_τ = C_τ^2 2, δ A_σ=0 2.
Once these conditions are imposed, all the 3d dependence is trivialized. One can then introduce again a _4^τ connection, such that
This shows that after KT transformation, the theory still has a _2^σ×_4^τ symmetry, and they are also anomaly free.
Denoting the partition function after the KT transformation as Z_[A_σ, A_τ], it is related to the partition function of via
Z_[A_σ, A_τ]= ∑_a_σ=0,1,
a_τ=A_τ 2 Z_[a_σ, a_τ] e^i π/2∫_X_2 (A_τ - a_τ)(A_σ + a_σ).
In terms of holonomies around the time and space directions, the partition function can be rewritten as
Z_[(W_t^σ, W^σ_x), (W_t^τ, W_x^τ)]:= Z_[A_σ, A_τ]
where W_t^σ, W^σ_x∈_2 and W_t^τ, W_x^τ∈_4.
As discussed in Section <ref>, the partition function can also be labeled by [(u_σ, t_σ),(v_τ, r_τ)]. Hence we denote the partition function as Z_^((u_σ, t_σ),(v_τ, r_τ)). Its relation with Z_[(W_t^σ, W^σ_x), (W_t^τ, W_x^τ)] is
Z_^((u_σ, t_σ),(v_τ, r_τ))= 1/8∑_w_t^σ=0,1
w_t^τ=0,1,2,3Z_[(w_t^σ, t_σ), (w_t^τ, r_τ)] e^iπ w_t^σ u_σ + iπ/2 w_t^τ v_τ
and the converse relation is
Z_[(W_t^σ, W^σ_x), (W_t^τ, W_x^τ)] = ∑_u_σ=0,1
v_τ=0,1,2,3 Z_^((u_σ, w_x^σ),(v_τ, w_x^τ)) e^iπ w_t^σ u_σ - iπ/2 w_t^τ v_τ.
By combining (<ref>), (<ref>), (<ref>) and (<ref>), we find the relation between Z_^((u_σ, t_σ),(v_τ, r_τ)) and Z_^((u_σ, t_σ),(v_τ, r_τ)),
Z_^((u_σ, t_σ),(v_τ, r_τ)) = Z_^((u_σ, t_σ + v_τ),(v_τ, r_τ+ 2 u_σ))≡ Z_^((u_σ, t_σ),(v_τ, r_τ)).
Hence the symmetry and twist sectors are related as
[(u_σ, t_σ),(v_τ, r_τ)] = [(u_σ, t_σ-v_τ),(v_τ, r_τ+2u_σ)].
§ XX MODEL WITH A TRANSVERSE FIELD, XXZ MODEL, AND FREE BOSON
In this appendix, we discuss the XX model with two types of perturbations, and discuss the symmetry properties of their ground states under various twisted boundary conditions. These results will be useful in Section <ref> and <ref>.
§.§ XX model and free boson
We begin with the XX model, whose Hamiltonian is
H_XX= -∑_i=1^Lσ^z_iσ^z_i+1 + σ^y_iσ^y_i+1.
We introduce the ladder operators as σ_i^±= (σ_i^z± i σ_i^y)/2, the Hamiltonian can be rewritten as
H_XX= -∑_i=1^L 2 σ^+_iσ^-_i+1 + 2 σ^-_i σ^+_i+1 .
It has a U(1) global symmetry, but we will only focus on its _4 subgroup. The symmetry operator is
U= ∏_i=1^L e^iπ/4(1-σ^x_i)
whose eigenvalue is e^iπ/2u, where u=0,1,2,3.
The symmetry operator U acts on the ladder operators as U^†σ^±_i U= e^±iπ/2σ^±_i. Thus
the twisted boundary condition is specified by
σ^±_i+L = e^±iπ/2tσ^±_i, t=0,1,2,3.
We would like to consider the continuous limit of this theory. It is well-known that via Jordan-Wigner (JW) transformation, the XX model is equivalent to a free fermion model. To see this, we consider the JW transformation[Note that for convenience we changed the sign in the first relation compared to (<ref>).]
σ^x_i= 1-2 f_i^† f_i, σ^+_i= ∏_j=1^i-1 (-1)^f_j^† f_j f_j, σ^-_i= ∏_j=1^i-1 (-1)^f_j^† f_j f_j^†.
The Hamiltonian then becomes a free fermion
H_fer = -∑_i=1^L 2f^†_i f_i+1 + 2 f_i+1^† f_i.
After taking the Fourier transformation, the fermion Hamiltonian is diagonalized, which takes the form
H_fer = -∑_k 4cos(2π/L k) f_k^† f_k
where k≃ k+L, and the fractional value of k depends on the boundary conditions of the fermion, which will not be important for our purpose.
The energy spectrum is E_k= - 4cos(2π/L k), which intersects the Fermi surface at two
Fermi points k≃±L/4. The ground state is given by filling all electrons in the band within k∈ [-L/4, L/4], and the low energy excitations are all localized around the two Fermi points. After linearizing the energy spectrum around the Fermi points, we find a right moving Weyl fermion at k=L/4 and a left moving Weyl fermion at k=-L/4. The two Weyl fermions compose into a Dirac fermion. At the field theory level, the bosonization of a free Dirac fermion gives rise to a c=1 free boson. The Lagrangian of the free boson is
= 1/2π (∂_μθ)^2= 1/2π[(∂_tθ)^2 - (∂_x θ)^2].
Upon canonical quantization, we have P_θ= ∂_t θ/π, and [θ, P_θ]= i. It is customary to introduce the dual variable φ such that P_θ= ∂_x φ/2π, in terms of which the Hamiltonian becomes
H_boson=∫ dx (P_θθ̇ -)= 1/2π∫ dx [1/4(∂_x φ)^2 + (∂_x θ)^2 ].
In terms of the lattice variables, we have σ^±_i≃ e^± iθ, and the _4 symmetry operator is
U= e^-i/4∫ dx ∂_x φ.
Indeed, using the BCH formula, U^† e^iθ U = e^i θ+ iπ/2, which is consistent with the commutation relation on the lattice U^†σ_i^+ U = e^π i/2σ_i^+.
The fields φ(x,t) and θ(x,t) are subjected to the twisted boundary condition,
φ(x+L,t) = φ(x,t) + 2π m, θ(x+L,t)= θ(x,t) + 2π n.
After mode expansion, the fields are decomposed into zero modes and oscillator modes. Hence we have
φ(x,t) ≃ 2π m x/L + ⋯, θ(x,t) ≃ 2π n x/L + ⋯
where m,n are constrained by the twisted boundary conditions for φ and θ respectively, which will consequently be determined by the charge and symmetry twists on the lattice.
The ground state energy only receives a contribution from the zero modes, which gives 2π/L[1/4m^2 + n^2].
To see how m,n are constrained, we first notice that the eigenvalue of U= e^-i/4∫ dx ∂_x φ is e^i π/2 u. Substituting (<ref>) into U, we find
m= -u+ 4m'
where m' is an integer. On the other hand, the twisted boundary condition σ^+_i+L = e^π i/2tσ^+_i on the lattice implies the twisted boundary boundary condition of θ(x,t) in the continuum, hence
n= 1/4t + n'
where n' is an integer. This implies that the zero mode energy is
E^m',n'_(u,t) = 2π/L[ 1/4(4m'-u)^2 + (n'+t/4)^2 ].
Let us denote the ground state in the symmetry-twist sector as E_(u,t), obtained by minimizing (<ref>) overall m', n', we have
E_(u,t) = 2π/L[ 1/4(min([u]_4,4-[u]_4))^2 + 1/16(min([t]_4,4-[t]_4))^2 ].
where [u]_4 stands for the u mod 4, and similar for [t]_4. In particular, for any t, the minimal ground state always carries trivial _4 charge, i.e. u=0.
§.§ Adding a transverse field
We further perturb the XX model (<ref>) by a transverse field -h ∑_i=1^L σ^x_i, such that the total Hamiltonian is[Naively, one may attempt to simply add h ∂_x φ to the Lagrangian (<ref>), but this does not work. The reason is that turning on h changes the location of the Fermi point, while h σ^x_i ∼h/π∂_x φ holds only near the original Fermi point h∼L/4, hence is expanding around the wrong vacuum. ]
H_XXpert = -∑_i=1^L σ^z_iσ^z_i+1 + σ^y_iσ^y_i+1 + h σ^x_i
To see the property of the energy spectrum, it is again useful to perform a JW transformation, which shows that the energy is E_k = -4cos(2π/Lk) + 2h. The transverse field plays a role of shifting energy of the entire band by 2h. As long as h<2, the band still intersects E=0 with two Fermi points, hence the system is still gapless. The effective degrees of freedom are still two Weyl fermions at the two Fermi points, but the only difference is that they are at closer momenta, and the Fermi velocities are reduced (determined by dE_k/dk). We then re-bosonize at the field theory level, and obtain a free boson, whose Lagrangian is
= 1/2π (∂_μθ)^2= 1/2π[1/v_h(∂_tθ)^2 - v_h (∂_x θ)^2], v_h= √(1-h^2/4).
Indeed, when h=0, v_0=1, which is consistent with (<ref>). When h→ 2, the Fermi velocity reduces to zero, corresponding to the point where the band of the fermion is tangential to the E_k=0 axis, i.e. the two Fermi-points merge. When h>2, the Fermi velocity is imaginary showing that the description (<ref>) breaks down. Indeed, whe h>2, the term -hσ^x_i dominates, driving the system to a trivially gapped phase. Because the transition at h=2 is associated with a quadratic band, the transition has dynamical exponent z=2.
We proceed to discuss the ground state properties. In the free boson representation, the Hamiltonian is
H_boson= v_h/2π∫ dx [1/4(∂_x φ)^2 + (∂_x θ)^2 ].
The energy is exactly the same as (<ref>) except for an overall normalization by the Fermi velocity. Thus the symmetry charges of the ground state under various twisted boundary conditions are the same as the unperturbed case h=0, as long as h<2.
§.§ XXZ model and free boson
We can alternatively perturb the XX model (<ref>) by -h∑_i=1^L σ^x_iσ^x_i+1. The total Hamiltonian is
H_XXZ = -∑_i=1^L σ^z_iσ^z_i+1 + σ^y_iσ^y_i+1 + h σ^x_iσ^x_i+1.
When h=1 or -1, the Hamiltonian is the ferromagnetic/anti-ferromagnetic Heisenberg chain. When h=0, the Hamiltonian reduces to the XX model. For other values of h, the Hamiltonian is the XXZ model.
The continuum field theory for the XXZ model is well known. When -1<h<1, it is the gapless Luttinger liquid whose Hamiltonian is [More precisely, the low energy effective theory also contains a cos(2ϕ) term. Such a term is irrelevant for K_h>1/2 hence we ignore it in the low energy. However this term is relevant for K_h<1/2, which gaps out the Hamiltonian to obtain a SSB phase. ]
H_LL = 1/2π∫ dx [ 1/4K_h (∂_x φ)^2 + K_h (∂_x θ)^2 ].
Here the Luttinger parameter is K_h=π/2arccos h <cit.>. Therefore when h<0, the system is described by K_h<1 free boson, while when h>0 the system is described by K_h>1 free boson.
When h>1 or h<-1, the σ^x σ^x term in (<ref>) dominates <cit.>, and there are two nearly degenerate ground states with an exponentially decaying gap. Note that the two degenerate ground states spontaneously break the _2 generated by ∏_iσ^z_i or ∏_iσ^y_i.
The energy coming from the zero mode is almost the same as (<ref>), but with the Luttinger parameter adjusted,
E_(u,t) = 2π/L[ 1/4K_h(min([u]_4,4-[u]_4))^2 + K_h/16(min([t]_4,4-[t]_4))^2 ].
In particular, for any t, the minimal ground state always carries trivial _4 charge, i.e. u=0.
ytphys
=.95
|
Subsets and Splits