\section{Introduction and Motivation}
Research has shown that language-capable robots hold unique persuasive power over humans. Robots are not only capable of influencing humans to comply with their requests and commands~\cite{bartneck2010influence,cormier2013would,rea2017wizard}, but can also exert moral influence over human moral norm systems~\cite{jackson2019language,jackson2018robot,williams2018bayesian}.
It is thus important for robots to use their persuasive power in a way that is going to help regulate people's behavior and preserve a better moral ecosystem, by rejecting unethical commands~\cite{briggs2021and,jackson2019language,wen2022teacher} and calling out norm-violating behavior~\cite{winkle2022norm,winkle2021boosting,jung2015using}.
For example, imagine a scenario where a robot and a team of humans are working together on a collaborative task, and one of the human teammate verbally abuses another team member who accidentally made a mistake. To exert positive moral influence, the robot might be expected to call out the norm violation (i.e., the insult) in order to helpfully mediate the team's dynamics.

Moreover, research has shown that robots are more persuasive when they act in a socially interactive~\cite{fong2003survey}, sociable~\cite{breazeal2004designing} or socially agentic manner~\cite{jackson2021theory}, by leveraging human-like social cues~\cite{chidambaram2012designing,ghazali2019assessing} and strategies~\cite{srinivasan2016help}. This is critical when considering how robots should be leveraging their persuasive power in moral context. When rejecting immoral commands, for example, if robots only focus on explaining the morality of actions without considering the social strategies they could be using during that communication, their communication could be less effective, and they themselves could be viewed less favorably. For instance, Jackson et al.~\cite{jackson2019tact} found that if robots give miscalibrated norm violation responses (i.e., the harshness of the responses does not match the actual norm violation severity), they will be perceived less favorably.

To appropriately calibrate the harshness of their norm violation responses, robots can use the same types of strategies that humans use, including social norms like \textit{Politeness}. In Brown and Levinson's Politeness Theory~\cite{brown1987politeness}, humans regularly negotiate the level of threat to one another's \textit{Face}, which is the public image that a person wants to preserve and enhance in front of others~\cite{brown1987politeness}. Face includes Positive Face, which is a person's wish for a desirable self-image, and Negative Face, which is a person's wish to be free from imposition and to have freedom of action~\cite{brown1987politeness}.

From the Politeness Theory perspective, calling out a norm violation can be highly face-threatening, as it threatens both Positive Face (by making the violator looks bad in front of others) and Negative face (by appearing to intend to control the violator's behavior).
Thus, it is crucial for robots to address norm violations in a way that the moral value (i.e., how much positive moral influence their response could exert) is proportional to the social value (i.e., how much face threat might be imposed by their utterance), in order to exert positive moral influence and avoid exerting unintentional negative moral influence.

Indeed, proportionality is one of the fundamental and universal moral motives underlying human social-relational psychology~\cite{rai2011moral}, which is ``the motive for rewards and punishments to be proportionate to merit, benefits to be calibrated to contributions, and judgments to be based on a utilitarian calculus of costs and benefits''.
To the best of our knowledge, however, there has been no previous work on computational modeling of how humans reason about proportionality during communication -- or how robots could do the same.

In this paper, we begin to consider how such a computational model might be developed, and show the substantial complexity and nuance behind what might at first glance seem a simple problem. We present a simple (naive) mathematical model of proportionality which could explain how moral and social considerations should be balanced in multi-agent norm violation response generation. We then analyze the components of this model and procedurally explain how each is more complex than might immediately meet the eye. Through this process, we identify key research directions that must be explored in order to develop socially and morally competent language-capable robots.

\section{A Preliminary Model of Proportionality}
To begin, let us consider how we might model the utility of a candidate speech act that could be used to address a norm violation in a multi-agent context.
Given our desire for proportionality, we can start by assuming that overall utility should include components for both the moral benefits of the response, and the social benefits (that is, the negation of the social harms). This model can be represented as:

$$\mathcal{U_{A}} = \mathcal{U_{MA}} + \mathcal{U_{SA}}$$

where $\mathcal{U_{A}}$ denotes the utility of a speech act $\mathcal{A}$, $\mathcal{U_{MA}}$ denotes the \textit{moral} utility of a speech act $\mathcal{A}$, and $\mathcal{U_{SA}}$ denotes the \textit{social} utility of a speech act $\mathcal{A}$.

\textit{Moral utility} is positively related to moral benefits. In potentially multi-agent contexts in which robots need to address norm violations in front of zero or more human observers, the moral benefits of the robots' response come from accurately correcting misconceptions for each observer (including the violator).
Thus, the calculation for moral utility can be represented as:

$$\mathcal{U_{MA}} = \sum\limits_{i=1}^{\mid O\mid} (|S_a - S_{i}| - |S_a - S_c|)$$

Here, $\mid O \mid$ denotes the total number of observers (including the norm violator), $S_a$ denotes the actual norm violation severity.
This could be learned from human data~\cite{sarathy2017learning,wen2020dempster}, learned in one shot through language, or calculated through some type of utilitarian analysis.
$S_c$ denotes the norm violation severity that the speech act $\mathcal{A}$ is trying to convey. This could be estimated from empirical evidence of human perceptions or other computational models of politeness theory~\cite{danescu2013computational}).
Finally,  $S_i$ denotes the norm violation severity that is perceived by observer $O_i$ (the $i$-th observer).
Each observer's prior beliefs about a norm violation's severity could be estimated from prior behaviours from that specific observer.

We might also expand this to:

$$\mathcal{U_{MA}} = \sum\limits_{i=1}^{\mid O\mid} ((|S_a - S_{i}| - |S_a - S_c|) - \beta|S_a - S_c|).$$

Here, $\beta$ denotes the weight of the dishonesty penalty, which might be affected by the robot's role. For example, dishonesty might be more greatly penalized for a teacher robot than a tour guide robot.
To maximize moral utility, robots thus need to be honest with the norm violation severity: the norm violation severity they are trying to convey should be exactly (or very close to) the actual norm violation severity.

\textit{Social utility} is negatively related to the amount of face threat. The more people there are to observe a face threat (and the more the violator cares about their perception by each observer) the great the social penalty they will suffer from a harsh norm violation response.
Thus, the calculation for social utility can be represented as:
$$\mathcal{U_{SA}} = - \sum\limits_{i=1}^{\mid O \mid} (I (O_i,v) \times F(v,\mathcal{A})).$$

Here, $I (O_i,v)$ denotes the importance for a norm violator $v$ to maintain a positive social image in front of the observer $O_i$.
This could be estimated from the relational/organizational hierarchy or perceived power dynamic between the violator and the observer.
$F(v,\mathcal{A})$ denotes the amount of face threat the speech act $\mathcal{A}$ imposes to the violator $v$. This could be estimated from empirical evidence of human perceptions of language of different types, or through computational models of politeness theory~\cite{danescu2013computational}).

\section{Additional Sources of Complexity}
The simple model presented in the preceding section aims to balance moral benefits against the loss of social benefits.
While this model seems, on its face, to capture the basics of proportionality, it obscures a wealth of complex considerations that would need to be addressed in practice. In this section, we will discuss these hidden complexities, and speculate as to how these considerations might need to be captured in a more complex model.

\subsection{Different types of observers}
In the proposed model, we treat every observer equally, in terms of contributing to the total moral and social utilities. In real-life scenarios, there may be different types of observers, who may need to be treated differently from other observers. In addition to bystanders as naively considered in the preliminary model, we should also consider other types of special observers:
\begin{itemize}
    \item The norm violator;
    \item People who are (or will be) negatively impacted by the norm violation, (i.e., victims); and
     \item People who might not be aware of the norm and have facilitated or conducted the same norm violation.

\end{itemize}

By considering the different types of observers, we can understand the ways our model might need to be adjusted accordingly.
For example, it may be more important to focus on correcting the violator than to worry about correcting (likely unobserved) misconceptions that could be held by other observers. If this is the case, we might need to assign different weights on the benefits of correcting different observers.

We also may need to consider face threat to observers beyond the violator. For example, when a norm violation is called out, some observers may also feel face threaten or experience other kinds of social awkwardness if they are not aware of the norm. In this case, the total amount of face threat a speech act may cause should also include the potential face threat that the observers may receive.

Finally, victims are also important to consider when we calculate the utility of a norm violation response. For instance, calling out a norm violation, in general, may help the victims to reduce (or even avoid) the potential harm that the norm violation may cause to them. However, victims may not want somebody else to speak for them in certain cases, which might hurt both their positive and negative face by making them look incompetent and taking away their chances to defend themselves. Therefore, besides the correction benefits, the model should take all of these dimensions into account for calculating moral and social utilities.

\subsection{Scaling of face threat with the number of observers}
In the proposed model, the total amount of face threat is simply the summation of values over all observers. However, in reality, the amount of face threat does not tend to increase linearly with the number of observers.
For example, losing face in front of 102 people and 112 people might be different, but the size of this difference is likely not as large  as the difference between losing face in front of two people and losing face in front of twelve people.
Given the fact that face threat likely scales nonlinearly with the number of observers, models of proportionality may need to include discount factors into their utility models.

\subsection{Potential benefits of face threat}
When calculating the social utility of a speech act, we consider all face threats to the violator as negative impacts and try to minimize face threat to increase utility. While this may seem reasonable at first glance, it may not always be truly beneficial to minimize face threat to a violator. First, if the violator has already caused harm, some face threat to the violator might be beneficial to everybody emotionally, for seeing the violator being called out (especially in the presence of the victims). Also, from the Confucian Ethical perspective, receiving face-threatening responses (e.g., blame-laden moral rebukes) for norm violations could help violators to cultivate their ``heart of shame''~\cite{zhu2020blame}, which is one of the key components of Confucian moral self-cultivation. Moreover, from a pedagogical perspective, harsh responses might create a stronger impression and thus be more effective in helping people learn and grow.

While a certain amount of face threat to the norm violator could potentially be beneficial, adding it to the moral utility model is still challenging. It is unclear how much moral benefit this face threat provides; and indeed, over-weighting this benefit could have clear negative effects (e.g., robots that intentionally seek opportunities for public shaming). Models of proportionality may require careful calibration of ostensible benefits of violator-directed face threats, and how this calibration may depend on various relational and social contextual factors.

\section{Discussion}

So far we have discussed a list of possible source of complexity in the modeling of proportionality. In fact, those complexities are not only important for generating norm violation responses, but also for other type of moral communication and other aspects of moral competence, such as moral reasoning and decision making. Given the relevance to those broader topics, it is thus worth discussing the more general concerns and challenges of enabling moral competence in social robots.

In this study, we assumed that robots already had a substantial amount of prior knowledge about moral and social norms (i.e., some ground truth). However, where these norms come from and how robots should acquire this prior knowledge is an open question.
Because people who have different culture backgrounds often have adhere to different moral and social norms (or do so in different ways),
one potential solution is to use participatory design to collect information about which specific norms \textit{particular communities} would follow in different situations.
Such an approach would be compatible with recent calls to explore Design Justice~\cite{costanza2020design,ostrowski2022ethics} and Engineering Justice~\cite{leydens2017engineering,williams2021human} approaches to robot design.
Even people from the same culture may have different moral and social beliefs due to individual life experience. In some cases, these differences might cause misunderstandings and conflicts, while in other cases, they might not negatively affect interpersonal communication. Thus, it is a challenge that robots should not only understand these differences, but more importantly, know when to seek common ground and when to preserve differences.

Additionally, in this work, our intention is to consider how we might enable robot competence to exert positive moral influence in order to help preserve and cultivate more harmonious human-robot eco-systems. While having robots calling out norm violations could be an effective approach to achieve our goals, it also bring concerns about robots ``norm policing''. A long term goal of this research area must be to balance the real need to call out norm violation responses without further turning robots into surveillance machines or developing tools that could be misused by law enforcement and other forms of state oppression.

\section{Summary}
In this paper, we presented a simple (naive) mathematical model of proportionality which could explain how moral and social considerations should be balanced in multi-agent norm violation response generation. We then used this model as a starting point to consider the hidden complexity of modeling proportionality. These considerations discussed above represent key research directions that must be explored in order to develop socially and morally competent language-capable robots.

\section{Acknowledgments}
This work was funded in part by Young Investigator award FA9550-20-1-0089 from the United States Air Force Office of
Scientific Research.


\section{Introduction}
\vspace{-2mm}
\label{sec:intro}
In contrast to the traditional single-label learning and multi-label learning, label distribution learning (LDL)~\cite{geng2016label} aims to characterize instance polysemy with labels of different description degrees.
Specifically, LDL extends discrete labels into the form of a continuous probability distribution (description degree).
The descriptive degree is deemed as the relevance of the label to the sample, and the sum of the descriptive degree of all labels is 1, showing the relationship between the instance and the sample.
For example, a person's facial expression~\cite{zhao2021robust,DBLP:conf/icassp/SiWPX22,DBLP:journals/tii/ChenGXZYL22} may involve multiple emotions, including happiness, sadness, surprise, etc.
LDL can convey such sophisticated and ambiguous semantic information.

Since researchers built label distribution datasets by using manual annotation and label enhancement algorithms~\cite{zhaofusion,DBLP:journals/corr/abs-2004-03104,DBLP:conf/icml/0009SLG20,DBLP:journals/tkde/XuLG21}, the label space of the dataset is usually disturbed by the introduction of uncertainty and noise.
On the one hand, the manually annotated datasets have a high degree of subjectivity and ambiguity.
On the other hand, the inductive bias of the label enhancement algorithm cannot guarantee the accuracy of the label set.
The uncertainty of the label set further affects the modeling of label relations.
To clear this obstacle, we model the uncertainty of label relations to boost the performance of the LDL algorithm.

Inspired by the idea of modeling the implicit label distribution proposed by~\cite{zhengLABEL}, our proposed model also implicitly characterizes the label relationship's uncertainty.
So far, we develop a novel approach to learning label distributions via a label correlation grid (LCG), which can model the relationships between labels and learn the implied uncertainty between pairs of labels.
In particular, we first use an MLP (multi-layer perceptron) to extract the deep semantics of the input information in order to perform learning label distribution.
Then, we use the predicted label distribution to compute a covariance matrix between the labels, with the help of which we generate an LCG.
Each cell in the grid is generated by sampling the element values in the covariance matrix using a Gaussian distribution function.
Note that this grid is constrained by the grid generated from the label space of the samples with the help of the $L_{1}$ norm.

In addition, since deep networks are prone to overfitting when learning tabular data, we propose a label distribution projection algorithm as a regularization term that performs on datasets with numerous labels.
The label distribution projection algorithm aims to extract the depth semantics of the label space (in this paper, we use KPCA~\cite{scholkopf1998nonlinear}), similar to the perceptual loss~\cite{gatys2016image}, which improves the generalization ability of the whole model.
Compared to traditional LDL algorithms, our model can execute acceleration on GPU (millisecond level speed on one inference), which is unparalleled competitive in terms of efficiency and accuracy.
Extensive experiments (including the standard accuracy experiment and the noise immunity experiment) have confirmed the effectiveness of our approach.
\textbf{Our main contribution} includes three folds:
\textbf{i)}: We propose a new method to learn label distributions via a label correlation grid that models the uncertainty of the label relationship.
\textbf{ii)}: We design a robust depth model that combines efficiency (ms level) and accuracy on a tabular dataset.
\textbf{iii)}: To alleviate the overfitting of the model, we design a label distribution projection algorithm on the dataset with a large number of labels.
\begin{figure*}[htbp]
	\centerline{\includegraphics[width=0.7\textwidth]{model.eps}}
	\caption{\textbf{Our architecture}. Our algorithm aims to regress the label distribution of a sample using an MLP, where there are \textbf{three learning targets}, one is to learn the true label distribution, one is to learn the projection of the true label distribution, and one is to learn the LCG.}
	\label{fig}
	\vspace{-4mm}
\end{figure*}

\vspace{-4mm}
\section{related work}
\label{sec:related work}
\vspace{-2mm}
Geng et al.~\cite{geng2016label} first proposed a new machine learning paradigm: LDL, which focuses on the relative importance of each label to an instance.
More researchers have since devoted themselves to LDL \cite{DBLP:conf/emnlp/ZhouZZZG16,DBLP:conf/mm/ZhouXG15,DBLP:conf/ijcai/WangG19}.
In these researches, label correlation is usually applied to boost the generalization ability of the model.
LDL-LCLR~\cite{ren2019label} proposed to utilize both the global and local relevance among labels to train an LDL algorithm.
Wang et al.~\cite{wang2021label} tried the underlying manifold structure of label distribution to encode the correlations among labels.
Jia et al.~\cite{jia2021label} introduced a ranking loss in the LDL model to maintain the label ranking relationship.
Qian et al.~\cite{qian2022feature} proposed an algorithm to generate the label correlations that uses neighborhood granularity to explore feature similarity while generating the label correlation with the correlation coefficient.
In contrast to the explicit modeling learning algorithm described above, we focus on implicitly representing correlations between labels, while taking into account the uncertainty factor carried by the dataset itself.

\vspace{-2mm}
\section{PROPOSED METHOD}
\vspace{-2mm}
\label{sec:PROPOSED METHOD}
The architecture of our method is shown in Fig.~\ref{fig}.
MLP learns the label distribution based on the deep semantics of the input features.
Among them, we conduct three sub-learning targets, a key target is to learn \textit{the true label distribution} by using $L_{1}$ loss, followed by learning \textit{the LCG} and another one (label distribution projection loss) is to learn \textit{the deep semantics of the label distribution}.
Of these, the elements of the LCG come from the Gaussian distribution and the depth semantics of label distribution are from KPCA.

\noindent \textbf{Notations.}
Given a particular example, the goal of our method is to learn the degree to which each label describes that instance.
Input matrix $X\in R^{m \times n}$, where $m$ is the number of instances and $n$ is the dimension of features.
We define the \textit{i-th} instance in the dataset as $x_{i}$.
The label distribution is defined $Y\in R^{m \times t} $, then the \textit{j-th }label is defined $y_{j} $. For each instance $x_{i}$, we define its label distribution $ D_{i}=\left\{ {d_{x_{i}}^{y_{1}},d_{x_{i}}^{y_{2}},... ,d_{x_{i}}^{y_{j}} }\right\} $ where $ d_{x_{i}}^{y_{j}} $ is the degree of description of the label $y_{j} $ for the sample $x_{i} $.
The $ d_{x_{i}}^{y_{j}} $ is constrained by ${{d_{x_{i}}^{y_{j}}}}\in[0,1] $ and $\sum_{j=1}^{j}{d_{x_{i}}^{y_{j}}}=1$.

\noindent \textbf{Network Architecture.}
Early studies introduced neural networks into LDL, such as AA-BP~\cite{geng2016label}.
We refer to this network architecture to regress a label distribution.
Firstly, the input instance $x_{i}$ is fed into an MLP with two linear layers.
In the first hidden layer, we used 256 neurons to learn the latent features.
The number of linear layer neurons in the last layer is the number of labels.
Note that the number of neurons is set to the power of 2 in favor of acceleration on the GPU shader.
Two non-linear layers (sigmoid and softmax) are added after the hidden layer and the output layer, respectively.
The MLP has the ability to learn the latent semantic and global information of the input examples and generates accurate label distribution values.
The predicted label description degree $\tilde{ d_{x_{i}}^{y_{j}} }$ of each instance $x_{i}$ is
\begin{equation}
	\begin{aligned}
	\tilde{d_{x_{i}}^{y_{j}} }=\frac{\text{exp}(\text{MLP}(x_{i})) }{ \sum_{k=1}^{j}{\text{exp}(\text{MLP}(x_{k}))}}.
		\label{equ1}
	\end{aligned}
\end{equation}
We optimize the MLP by minimizing the $L_{1}$ distance between the predicted label distribution and the ground truth as follows:
\begin{equation}
	\begin{aligned}
L_{\text{1}}\left(d_{i}, \tilde{d_{i}} \right)=\sum_{i=1}^{T}        \left| \left| {  d_{x_{i}}^{y_{j}}-\tilde{ d_{x_{i}}^{y_{j}}    }   } \right| \right|	,
		\label{equ7}
	\end{aligned}
\end{equation}
where $\tilde{d_{i}}$ is the predicted label distribution and $d_{i}$ is the true label distribution, and \textit{T} denotes the number of labels.
The following describes some key regularization terms.

\noindent  \textbf{Label Correlation Grid.}
Implicit representation~\cite{zhengLABEL} has nice properties due to its powerful expressiveness and for this reason, we develop the scheme of implicitly representing label relations.
To robustly model label uncertainty, we build a tensor that can accommodate high-dimensional information.
Summing up the above thesis, we propose an LCG $y\_grid \in R^{t \times t \times t}$.
$y\_grid$ is initially composed of a covariance matrix computed from the label space, where each element is sampled by a Gaussian function to create a vector.
The Gaussian distribution has two hyperparameters and we use each element of the covariance matrix as the mean and the variance is set fixed at 0.5.
We also try other values \{0.1, 0.2, ... ,0.9\} as the variance, but the results are not good as seen from the extensive experimental results.
The LCG is shown as:
\begin{equation*}
	\begin{bNiceMatrix}[name=a]
		y_{1}^{1} & \cellcolor{blue!20} y_{1}^{2} &\cdots&y_{1}^{t} \\
		y_{2}^{1} & y_{2}^{2} &\cdots& y_{2}^{t} \\
		\vdots&\vdots& \ddots&\vdots \\
		y_{t}^{1} & y_{t}^{2}&\cdots & y_{t}^{t}
	\end{bNiceMatrix}\qquad
	\begin{bNiceMatrix}[name=b]
	q_{1}, q_{2}, ..., \tikzmarknode[anchor=south]{upper right}{\strut q_{l}}
	\end{bNiceMatrix}
	\tikz[remember picture, overlay]
	\draw[->, red] (a-1-2) to[in=180, out=10] (b-1-1);
\end{equation*}
in which $y_{i}^{j}=[q_{1},q_{2},...,q_{l}]$ indicates the information between the \textit{i-th} label and the \textit{j-th} label.
The final LCG is optimized by calculating the $L_{1}$ distance.
A large number of papers use Gaussian priors for uncertainty estimation of unknown information. So we use the Gaussian process regression~\cite{wilson2011gaussian} to generate the description information.

\noindent \textbf{Label Distribution Projection Algorithm.}
To boost the performance of the model, we propose a label distribution projection algorithm as a regularization term in the network training process.
We use a sub-MLP to aggregate feature information into a lower dimension in the network.
The sub-MLP consists of one linear layer with sigmoid and softmax as activation functions.
For example,  the 68-dimensional raw label distribution is mapped to a the 32-dimensional label distribution space by sub-MLP for the \textit{Human Gene} dataset.
The formulaic expression form is as follows:
\begin{equation}
	\begin{aligned}
 \tilde{ d_{x_{i}}^{y_{j}}} '= \text{softmax}(\text{sub-MLP}(d_{x_{i}}^{y_{j}})).
		\label{equ5}
	\end{aligned}
\end{equation}

For true low-dimensional label distributions, we use Kernel-based principal component analysis (KPCA) to project the raw true label distribution.
KPCA draws on a nonlinear approach to extract the abstract semantics of the label distribution, and in addition, has the effect of reducing noise.
The lower label distribution is calculated as ${d_{x_{i}}^{y_{j}}}'$.
We use $L_{1}$ to minimize the output of the network and the true low-dimensional label distribution.

\noindent \textbf{Loss Function.}
The network parameters are optimized by combining $L_{1}$, projection loss ($L_{ldp}$) and label correlation grid loss ($L_{lcg}$).
Combined with the content of the above, our target function is constructed as follows:
\begin{equation}
	\begin{aligned}
Loss=\lambda_{1} \times L_{1}+\lambda_{2} \times L_{ldp}+\lambda_{3} \times L_{lcg},
		\label{equ4}
	\end{aligned}
\end{equation}
where $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3} $ are parameters. In the four label distribution datasets, we select  $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3} $ to be 1, 0.1 and 0.05, respectively.
\begin{table}[htbp] \footnotesize
	\centering
	\caption{Statistics of four real-world datasets.}
    \setlength{\tabcolsep}{5mm}{
	\begin{tabular}{c|ccc}
		\toprule
		Dataset & Examples & Features & Labels \\
		\midrule
		Human Gene & 17892 & 36    & 68 \\
		Natural Scene & 2000  & 294   & 9 \\
		Yeast-alpha  & 2465   & 24   & 18 \\
		Movie & 7755  & 1869  & 5 \\
		\bottomrule
	\end{tabular}%
    }
	\label{tab:1}%
\end{table}%
\begin{table}[htbp] \footnotesize
	\centering
	\caption{Parameters on the four datasets.}
	\begin{tabular}{c|cccc}
		\toprule
		Dataset & Batch size & Learning rate  & Epoch & LDP loss \\
		\midrule
		Human Gene & 500   & 0.0005 & 400   & 32 \\
		Natural Scene & 500   & 0.0005 & 300   & \textbf{-} \\
		Yeast-alpha  & 500   & 0.0005 & 400   & 16 \\
		Movie & 2000  & 0.0005 & 100   & \textbf{-} \\
		\bottomrule
	\end{tabular}%
	\label{tab:2}%
\end{table}%
\vspace{-2mm}
\section{EXPERIMENTS}
\label{sec:EXPERIMENTS}
\textbf{Datasets.}
To validate the efficiency of our model, we conduct experiments on four LDL datasets.
Details of the datasets from Wang et al.~\cite{wang2021label} are presented in Table~\ref{tab:1}.
The datasets involve different types, including biological experiment datasets, natural scene datasets, movie category datasets, and large-scale biomedical research datasets.
The evaluation is based on six indicators proposed by~\cite{geng2016label}, including Chebyshev distance (Chebyshev), Clark distance (Clark), Canberra distance (Canberra), KL divergence (K-L), Cosine similarity (Cosine), and Intersection similarity (Intersection).

\noindent \textbf{Comparative Algorithms.}
We compared the results of our model with five LDL algorithms: DDH-LDL~\cite{DBLP:journals/spl/ZhangZLX22}, BFGSLLD \cite{geng2016label}, LDL-LRR~\cite{jia2021label}, LDLSF~\cite{DBLP:conf/ijcai/RenJLCL19} and LALOT~\cite{DBLP:conf/aaai/ZhaoZ18a}.
Except for DDH-LDL, the indicators for all datasets of the models are from ~\cite{jia2021label}. Our experiments in DDH-LDL are based on the tips in the code website corresponding to the article.\\
\begin{table*}[htbp]
	\centering
	\caption{Experimental results on four datasets and the best results are bolded.}
	\resizebox{\textwidth}{50mm}{
	\begin{tabular}{c|c|cccccc}
	\toprule
	\textbf{Dataset} & \textbf{Algorithm} & \multicolumn{1}{c|}{\textbf{Chebyshev $\downarrow$}} & \multicolumn{1}{c|}{\textbf{Clark $\downarrow$}} & \multicolumn{1}{c|}{\textbf{Canberra $\downarrow$}} & \multicolumn{1}{c|}{\textbf{K-L $\downarrow$}} & \multicolumn{1}{c|}{\textbf{Cosine $\uparrow$}} & \textbf{Intersection $\uparrow$} \\
	\midrule
    \multirow{6}[2]{*}{Movie} & Ours  & \textbf{0.1145$\pm$0.0073} & \textbf{0.5054$\pm$0.0027} & \textbf{0.9746$\pm$0.0055} & 0.1178$\pm$0.0029 & \textbf{0.9446$\pm$0.0092} & \textbf{0.8384$\pm$0.0090} \\
& DDH-LDL & 0.1543$\pm$0.0634 & 0.6427$\pm$0.2526 & 1.2094$\pm$0.0051 & 0.1451$\pm$0.0127 & 0.9026$\pm$0.0048 & 0.7915$\pm$0.0913 \\
& LDL-LRR & 0.1151$\pm$0.0016 & 0.5235$\pm$0.0065 & 0.9996$\pm$0.0093 & \textbf{0.0982$\pm$.0031} & 0.9353$\pm$0.0020 & 0.8355$\pm$0.0023 \\
& BFGS-LLD & 0.1397$\pm$0.0223 & 0.5856$\pm$0.0494 & 1.1309$\pm$0.1098 & 0.1378$\pm$0.0373 & 0.9108$\pm$.0227 & 0.8053$\pm$0.0263 \\
& LALOT & 0.2854$\pm$0.0903 & 1.431$\pm$0.9006 & 2.9023$\pm$0.8996 & 3.7216$\pm$0.9542 & 0.7021$\pm$0.1415 & 0.6015$\pm$0.2114 \\
& LDLSF & 0.1271$\pm$0.0027 & 0.6227$\pm$0.0112 & 1.0166$\pm$0.0158 & 0.1954$\pm$0.0257 & 0.9220$\pm$0.0028 & 0.8180$\pm$0.0034 \\
\midrule
\multirow{6}[2]{*}{Yeast-alpha} & Ours  & \textbf{0.0133$\pm$0.0025} & 0.2100$\pm$0.0083 & \textbf{0.6708$\pm$0.2624} & 0.0062$\pm$0.0194 & \textbf{0.9947$\pm$0.0099} & \textbf{0.9626$\pm$0.0056} \\
& DDH-LDL & 0.1627$\pm$0.0529 & 0.5228$\pm$0.1937 & 1.0236$\pm$0.3634 & 0.1177$\pm$0.0638 & 0.8794$\pm$0.0936 & 0.8128$\pm$0.9355 \\
& LDL-LRR & 0.0134$\pm$0.0002 & \textbf{0.2093$\pm$0.0031} & 0.6791$\pm$0.0100 & \textbf{0.0054$\pm$0.0001} & 0.9946$\pm$0.0001 & 0.9625$\pm$0.0005 \\
& BFGS-LLD & 0.0135$\pm$0.0001 & 0.2110$\pm$0.0030 & 0.6865$\pm$0.0106 & 0.0055$\pm$0.0001 & 0.9949$\pm$0.0001 & 0.9629$\pm$0.0006 \\
& LALOT & 0.0165$\pm$0.0003 & 0.2608$\pm$0.0043 & 0.8544$\pm$0.0150 & 0.0084$\pm$0.0003 & 0.9917$\pm$0.0003 & 0.9526$\pm$0.0008 \\
& LDLSF & 0.0139$\pm$.0002 & 0.2164$\pm$0.0021 & 0.6874$\pm$0.0111 & 0.0058$\pm$0.0001 & 0.9943$\pm$0.0001 & 0.9613$\pm$.00004 \\
\midrule
\multirow{6}[2]{*}{Natural Scene} & Ours  & 0.3298$\pm$0.0020 & \textbf{1.8424$\pm$0.0004} & \textbf{4.3583$\pm$0.0125} & 2.332$\pm$0.00178 & \textbf{0.7607$\pm$0.0075} & \textbf{0.5749$\pm$0.0078} \\
& DDH-LDL & 0.4051$\pm$0.0022 & 2.1792$\pm$0.0037 & 5.5492$\pm$1.3133 & 1.6338$\pm$0.4791 & 0.5603$\pm$0.0067 & 0.4228$\pm$0.0055 \\
& LDL-LRR & \textbf{0.2997$\pm$0.0070} & 2.4268$\pm$0.0106 & 5.7143$\pm$0.0126 & \textbf{0.7314$\pm$0.0070} & 0.7532$\pm$0.0044 & 0.5670$\pm$0.0022 \\
& BFGS-LLD & 0.3511$\pm$0.0158 & 2.4756$\pm$0.0217 & 5.8445$\pm$0.0932 & 0.9067$\pm$0.0556 & 0.6765$\pm$0.0211 & 0.4778$\pm$0.0196 \\
& LALOT & 0.3873$\pm$0.0033 & 2.4999$\pm$0.0030 & 5.9962$\pm$0.0251 & 1.2529$\pm$0.0145 & 0.5473$\pm$0.0046 & 0.3600$\pm$0.0029 \\
& LDLSF & 0.3196$\pm$0.0092 & 2.4622$\pm$0.0159 & 6.6739$\pm$0.0139 & 0.9981$\pm$0.0328 & 0.7297$\pm$0.0074 & 0.5397$\pm$0.0074 \\
\midrule
\multirow{6}[2]{*}{Human gene} & Ours  & \textbf{0.0528$\pm$0.0074} & \textbf{2.1000$\pm$1.245} & 14.3673$\pm$0.1252 & \textbf{0.2271$\pm$0.0347} & \textbf{0.8352$\pm$0.0082} & \textbf{0.7849$\pm$0.0085} \\
& DDH-LDL & 0.0597$\pm$0.0752 & 2.8830$\pm$0.8514 & 19.9391$\pm$0.9627 & 0.3815$\pm$0.3870 & 0.7313$\pm$0.8355 & 0.6878$\pm$0.8845 \\
& LDL-LRR & 0.0532$\pm$0.0011 & 2.1114$\pm$0.0122 & \textbf{13.5681$\pm$0.0025} & 0.2365$\pm$0.0049 & 0.8346$\pm$0.0020 & 0.7844$\pm$0.0014 \\
& BFGS-LLD & 0.0539$\pm$0.0009 & 2.1270$\pm$0.0141 & 14.5633$\pm$0.1107 & 0.2398$\pm$0.0038 & 0.8328$\pm$0.0018 & 0.7828$\pm$0.0014 \\
& LALOT & 0.0573$\pm$0.0012 & 4.0703$\pm$3.6917 & 17.8198$\pm$3.7167 & 0.2956$\pm$0.0059 & 0.8055$\pm$0.0024 & 0.7578$\pm$0.0016 \\
& LDLSF & 0.0533$\pm$0.0009 & 2.1295$\pm$0.0209 & 14.5681$\pm$0.0055 & 0.2395$\pm$0.0054 & 0.8332$\pm$0.0028 & 0.7828$\pm$0.0022 \\

		\bottomrule
	\end{tabular}%
}
	\label{tab:3}%
\end{table*}%
\noindent \textbf{Experimental Setting.}
For our algorithm, we put the customized selection of parameters for each dataset in Table~\ref{tab:2}.
The parameters that may be used include batch size, learning rate, and epoch.
Since the label space of the three datasets \textit{Natural Scene}, \textit{Movie} is small, there is not enforced LDP loss function in the training phase.
Here, 32 indicates that we squeeze the raw label dimension from 68 to 32 using KPCA, followed immediately by normalization using softmax.
Besides, we use the PyTorch framework and AdamW optimizer to train and test a deep network on the GPU shader.

\noindent \textbf{Results and Discussion.}
The experimental results for each dataset are summarized in Table~\ref{tab:3}, where we take the results of the 10 times 5-fold cross-validation.
The experimental results are reported in the form of ``mean$\pm$std''.
$\uparrow$ indicates that the higher the value of the metric, the better.
$\downarrow$ indicates that the smaller the value of the metric, the better.

Our algorithm performs optimally on six parameters across the four datasets, particularly on the metric \textit{Clark}.
Our approach also achieves competitive results in the remaining metrics.
This is mainly due to the ability of the algorithm to expand label relationships and mine feature semantic information.
Besides, we observe that algorithms that exploit label relations, like LDL-LRR and ours, generally outperform other algorithms, which demonstrates the importance of label relations in label distribution.
Our method has a better performance compared to LDL-LRR by taking into account the uncertainty of the label relationship.
Although our modeling has an expensive cost, a GPU acceleration can be enforced in the PyTorch to alleviate this problem.
We evaluate the inference speed of our model on CPU (AMD 8-core processor) and GPU (RTX3080) as \textbf{22ms} and \textbf{1ms}, respectively.

\noindent \textbf{Ablation Studies.}
We conduct ablation studies to demonstrate the effectiveness of LCG and label distribution projection algorithms on dataset \textit{Human gene}.
The results of simple ablation experiments are shown in Table~\ref{tab:4}.
w/o grid is an algorithm for removing the LCG and w/o $ L_{dpa}$ is defined when the loss term $ L_{dpa}$  is removed.
The variation of the data illustrates the effectiveness of the LCG and the label distribution projection algorithms.
We conduct 10 times 5-fold cross-validation on the dataset of the ablation studies.
The \textcolor{gray}{gray} markers represent optimal performance.

\noindent \textbf{Noise Interference.}
To verify the robustness of our algorithm against the noise and uncertainty in label space, we experimented on the dataset \textit{Movie} with Gaussian noise.
We synthesize a dataset on \textit{Movie} by Gaussian function to enforce different degrees of noise, and the labels with noise are finally normalized by softmax.
For label space on the dataset, the variance of the Gaussian distribution function is selected from (0.1, 0.2, ..., 1).

\begin{table}[htbp]
	\centering
	\caption{Results on Ablation studies.}
	\resizebox{1\columnwidth}{!}{
		\begin{tabular}{c|ccc}
			\toprule
			\textbf{Measures
			} & \multicolumn{1}{c|}{Ours} & \multicolumn{1}{c|}{w/o grid } & w/o  $ L_{dpa}$ \\
			\midrule
			Chebyshev $\downarrow$ & \cellcolor{lightgray}0.0528$\pm$0.0074 & 0.5346$\pm$0.0021 & 0.5349$\pm$0.0132 \\
			Clark $\downarrow$ & \cellcolor{lightgray}2.1000$\pm$1.2455 & 2.1011$\pm$1.1352 & 2.1073$\pm$0.0047 \\
			Canberra $\downarrow$ & \cellcolor{lightgray}14.3673$\pm$0.1252 & 14.3798$\pm$0.1368 & 14.3689$\pm$0.0054 \\
			K-L $\downarrow$ & \cellcolor{lightgray}0.2271$\pm$0.0347 & 0.2274$\pm$0.0052 & 0.2283$\pm$0.0032 \\
			Cosine $\uparrow$ & \cellcolor{lightgray}0.8352$\pm$0.0082 & 0.8350$\pm$0.0086 & 0.8313$\pm$0.0861 \\
			Intersection $\uparrow$ & \cellcolor{lightgray}0.7849$\pm$0.0085 & 0.7845$\pm$0.0074 & 0.7841$\pm$0.0015 \\
			\bottomrule
		\end{tabular}%
	}
\vspace{-5mm}
	\label{tab:4}%
\end{table}%
It is widely believed that the higher the variance, the more dispersed the data, and the more noise.
We conduct 10 times 5-fold cross-validation on the synthetic dataset.
Our algorithm yield 0.1257$\pm$0.0071 (Chebyshev), 0.5713$\pm$0.0027 (Clark), 1.0854$\pm$0.0055 (Canberra), 0.1452$\pm$0.0033 (K-L), 0.9250$\pm$0.0092 (Cosine), 0.8203$\pm$0.0037 (Intersection) respectively on the data set (the variance is 1) with Gaussian noise has a variance of 1 and mean of 0.
However, the accuracy of the label distribution predicted by other algorithms decreases by \textbf{50\% on average}, and our model is still very competitive.

\vspace{-2mm}
\section{CONCLUSION}
\vspace{-2mm}
Ambiguity and uncertainty of label space in LDL become major challenges, limiting the performance of LDL models.
In this paper, we propose a new algorithm for label distribution by using an LCG.
In standard evaluation experiments, our method achieves competitive results, especially on the metric of vector similarity rather than distance.
Furthermore, our method has a striking performance on the mini-test of noise resistance, thanks to the gains from uncertainty modeling.
Our model can execute GPU acceleration on processing LDL data, balancing accuracy and efficiency.

\vfill\pagebreak

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