LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Authors: Xiuyi Jia
Jinchi Li
Yunan Lu
Weiwei Li
Venue: ICML 2025 poster (2025)
Keywords: Label distribution learning
local interpretability
label dependency
Primary area: social_aspects->accountability_transparency_and_interpretability
TL;DR: LIME for LDL

Abstract: Label distribution learning (LDL) is a novel machine learning paradigm that can handle label ambiguity. This paper focuses on the interpretability issue of label distribution learning. Existing local interpretability models are mainly designed for single-label learning problems and are difficult to directly interpret label distribution learning models. In response to this situation, we propose an improved local interpretable model-agnostic explanations algorithm that can effectively interpret any black-box model in label distribution learning.
To address the label dependency problem, we introduce the feature attribution distribution matrix and derive the solution formula for explanations under the label distribution form. Meanwhile, to enhance the transparency and trustworthiness of the explanation algorithm, we provide an analytical solution and derive the boundary conditions for explanation convergence and stability. In addition, we design a feature selection scoring function and a fidelity metric for the explanation task of label distribution learning. A series of numerical experiments and human experiments were conducted to validate the performance of the proposed algorithm in practical applications. The experimental results demonstrate that the proposed algorithm achieves high fidelity, consistency, and trustworthiness in explaining LDL models.

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Xiuyi Jia Jinchi Li"
Yunan Lu" 12 Weiwei Li
Abstract
data mining in recent years. There are currently several paradigms for dealing with label ambiguity, such as multi- label learning (Tsoumakas & Katakis, 2007), multi-instance learning (Carbonneau et al., 2018), and label distribution learning (LDL) (Geng, 2016). Among these, LDL stands out an innovative learning paradigm that not only an- which labels relevant to the sample but also fur- ther characterizes the relative importance of labels to the sample. LDL handles multi-label samples where labels exhibit co-occurrence patterns (simultancous changes) and dependency propagation (one label's presence affects others' distributions). Given that LDL can provide richer semantic information, has been widely applied in practical applica- tion scenarios such as disease diagnosis (Wang et al., 2022), emotion recognition (Jia al., 2019), etc. Existing LDL research primarily focuses on the design and optimization of algorithms, but the interpretability of LDL algorithms have been neglected.
Label distribution learning (LDL) is novel ma- chine learning paradigm that can handle label am- biguity. This paper focuses on the interpretability issue of label distribution learning. Existing local interpretability models are mainly designed for single-label learning problems and are difficult to directly interpret label distribution learning mod- els. In response to this situation, we propose an improved local interpretable model-agnostic ex- planations algorithm that can effectively interpret any black-box model in label distribution learning. To address the label dependency problem, we in- troduce the feature attribution distribution matrix and derive the solution formula for explanations under the label distribution form. Meanwhile, to enhance the transparency and trustworthiness of the explanation algorithm, we provide an analyt- ical solution and derive the boundary conditions for explanation convergence and stability. In ad- dition, we design feature selection scoring func- tion and a fidelity metric for the explanation task of label distribution learning. A series of numer- ical experiments and human experiments were conducted to validate the performance of the pro- posed algorithm practical applications. The ex- perimental results demonstrate that the proposed algorithm achieves high fidelity, consistency, and trustworthiness in explaining LDL models.
Interpretability plays crucial role in understanding the Aecision-making logic of models on given data, helping re- searchers optimize models more effectively, and promoting the application of models in real-world scenarios. Within the existing field of interpretable machine learning, the LIME (local interpretable model-agnostic explanations) al- gorithm (Ribeiro et al., 2016) is widely used to enhance the interpretability of models. This is achieved by gener- ating simple models to approximate the local behavior of black-box models. However, the LIME algorithm is de- signed for single-label learning tasks and focuses the local region of single label, making it unsuitable for direct application to LDL tasks that involve multiple interdepen- dent real-valued labels. One intuitive approach is to apply the LIME algorithm in parallel to LDL tasks, but this ap- proach faces significant challenges: (1) In LDL tasks, there are complex dependencies and interactions among labels. In such cases, the single-label explanations provided by LIME can be misleading due the disregard for inter-label corre- lations. For example, in medical diagnostic tasks, "pneumo- nia" and "acute respiratory distress syndrome" often need to addressed simultaneously; a parallel explanation might overestimate the risk of pneumonia alone and underestimate the risk when both pncumonia and acute respiratory distress syndrome are present. (2) The approach encounters compu-
1. Introduction
Learning with ambiguity has emerged as an important research direction in the fields of machine learning and
'School of Computer Seience and Engineering, Nanjing Univer- sity of Science and Technology. China 'Deparmment of Computing, The Hong Kong Polytechnic University, China 'College of School of Computer Science and Technology. Nanjing University of Aero- nauties and Astronautics, China. Correspondence to: Weiwei Li <liweiwei@ quaa.edu.cn..
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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LIMEFLDI: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
tational complexity issues; when the dataset is very large, generating large number of perturbed samples for a single label and training local models becomes extremely time-
Our main contributions can be summarized as follows.
We propose the LIMEFLDL method, which is suit- able for interpretability tasks of LDL algorithms. This method takes into account the label distribution in the local region, generating local linear models to approx- imate the overall behavior of the black-box model's output label distribution. It can be widely applied to interpret complex models in various LDL. tasks.
consuming. (3) The local solution of parallelized LIME tends to violate the constraints of label distribution, which can lead to invalid label description degrees and unreliable explanations. For instance, when the true description degree of a label is very low, it can be predicted as a negative value, an invalid label description degree.
To address these challenges, we propose an improved LIME method for the local interpretability issues in LDL, named LIMEFLDL. Within this interpretive framework, (1) to ex- plain the dependencies among labels, we design a feature attribution matrix that forms a feature attribution distribu- tion for each feature. This distribution considers the impact of features on the entire label distribution and can be used to explain the inter-label dependencies, resulting in more consistent explanations. (2) To reduce computational com- plexity, we minimize the distance between the black-box model and the explanation model's output within the gen- erated local region in the form of a label distribution. This process requires only one local sampling and does not neces- sitate generating local region for each label, significantly reducing computational complexity. Additionally, we also provide a feature selection scoring function suitable for the label distribution format to accelerate the efficiency of expla- nation generation. (3) To avoid violating label distribution constraints, we incorporate linear constraints and penalty function constraints within the local model, ensuring that the local model's predictions meet the label distribution constraints while remaining clear and transparent.
We conduct extensive theoretical proofs and property analyses of the proposed LIMEFLDL method, ensuring that the interpretations provides in LDL tasks are reliable and effectively approximate the true decision- making process. We also validate the reliability of the proposed method through various experiments.
2. Related Works
Label distribution learning. LDL aims to learn a pre- dictive function that maps features to the sample's 's label distribution, where each value in the distribution represents the degree to which the corresponding label describes the sample. (Geng, 2016) first introduced maximum entropy model to represent the prediction function, laying the foun- dation for LDL. Subsequently, many studies have extended traditional learning algorithms to adapt to LDL tasks. For example, (Geng & Hou, 2015) proposed LDSVR by ex- tending the support vector machine to handle LDL problem. (Xing et al., 2016) applied the LogitBoost model to LDL tasks. (Shen et al., 2017) 7) developed LDLF by extending the drivable decision tree model for LDL. To further improve the generalization ability of label distribution learning mod- els, some research has focused on mining the correlations between labels, aiming to capture the interdependencies among labels (Jia et al., 2018; Zhao & Zhou, 2018; Zheng et al., 2018). In addition, there are also efforts that enhance the expressive power of models by mapping labels to a low- dimensional space through label embedding methods (Peng et al., 2018; Wang & Geng, 2018; Xu et al., 2019). Al- though these approaches have significantly advanced the performance of the LDL model, the integration of LDL with interpretability techniques remains largely unexplored area, presenting opportunities for future research.
In addition to designing the LIMEFLDL model specifically for the interpretability issues of LDI. algorithms, we have also conducted proof analysis and experimental validation of the related theories of this model. First, we have pro- vided analytical solution formulas for the improved inter- pretation algorithm under certain assumption conditions and proved the boundary conditions for the explanations to reach convergence and stability. Second, we have analyzed the properties of the interpretation algorithm, including lo- cal accuracy, dummy features, etc., and provided proofs, which enhances the theoretical completeness of the algo- rithm and also increases its usability and trustworthiness. Third, most traditional fidelity measures are only applica- ble to single-label cases. However, for label distribution scenarios, these measures cannot be directly applied. To address this, we have introduced metrics for LDL as fidelity measures. These measures can quantify the fidelity of the explanation model's output to the actual model's decisions. Finally, through numerical experiments and human exper- iments, we have verified that the proposed method offers high fidelity, consistency, and trustworthiness in the inter- pretation of LDL models.
Improvements to LIME. LIME is one of the most widely used techniques for interpreting black-box models, and many studies have also made improvements to it. For ex- ample, (Shi et al., 2020) proposed a modification to the perturbation sampling method to handle the correlations between features, thereby improving the accuracy of the ex- planations. Like other explanation methods, LIME relies on reference points (also known as baseline inputs) to generate samples. In (Kapishnikov et al., 2019), both black and white
2

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
references are used. In contrast, (Fong & Vedaldi, 2017) employed a combination of constant, noisy, and gaussian blur references. DLIME (Zafar & Khan, 2019), ALIME (Shankaranarayana & Rumje, 2019) and GLIME (Tan et al., 2023) applied auto-encoders, deterministic methods, and integration of the locality weighting function, respectively, into the sampling process methods to refine the weighting function, improving the accuracy and stability of the lo- cal surrogate model. It is worth mentioning that there are also some works that apply LIME to multi-label classifica- tion tasks (Kakogeorgiou & Karantzalos, 2021; Belal et al., 2023; Vinogradova& & Myers, 2023). These methods directly decompose the multi-label classification task into a series of binary classification tasks and then use LIME to seek explanations for each of them. After reviewing the existing research related to LIME, there has been no direct work on interpretability for LDL tasks. This paper will attempt to address this issue.
several individual boxes based on entropy algorithm (Gar- reau & von Luxburg, 2022). For the generated j-th local sampling instance, 2jg = 1 indicates that the q-th feature is in the same box as the value of the q-th feature of 24, and Zjg = O indicates that the q-th feature is in a different box as the value of the q-th feature of zy. In summary, for the to-be-explained instance zi, we generate the corresponding local sampling instance matrix Z based on the Bernoulli distribution.
Additionally, since each feature in LDL corresponds to mul- tiple labels, we generate an attribution vector a, for each fea- ture, the length of a; corresponds to the labeling sequence, with a, E R=*1, where each element dt denotes the at- tribution value of the i-th feature to the j-th label. These feature attribution values are then combined form the feature attribution matrix A. We uses uniform initialization as initialization. The local linear model for L.LL can thus be expressed as follows:
3. Proposed Interpretability Algorithm
G ( = ZA +b,
(I)
3.1. Problem formation
where Z E {0, 1)mx////////// is the distribution matrix sampled
from the instances to be interpreted, A E RTxx is the fea- ture attribution matrix, and b E R=* is the bias term. We can simplify the equation as follows:
Let X = (I1, ...,, Rn), where X' E Rax repre- sents the input space, zj denotes the i-th instance, is the number of instances, and is the feature dimension. D {DI, D2.... DnJ. where D E Ra represents the output space, and r is the number of labels. Each D: = d 2...... ..... d is the label distribution for Ii, where d represents the description degree of the label y/ in the label distribution of instance F4, satisfying d E [0. 1] and Σド= d =1 1. The goal of LDL is to learn a mapping function & from X' to D that can predict the label distribu- tions for unseen instances. Furthermore, in order to inter- pretate the LDL model, we aim to approximate the local behavior of the complex LDL mapping by simple linear model, i.e., min || ISsimplee - -|l where Ssimple is the simple linear model.
G = 1 A b = ATA
(2)
However, each locally sampled instance differs from the one to be interpreted, for each instance z; in Z, we assign a sampling weight π (2z). and the sampling weight matrix II is then defined for the entire locally sampled set Z.
((;) =exp (-o z:11)) II= diag( (3)
where is the kernel width parameter, and 1 denotes the all-one vector as the interpretable representation of the term to be interpreted. In this framework, we use the L2 norm by default. However, other metrics such as the L, norm, cosine, or euclidean distance can also be applied. Typically, cosine distance is used for image data, while the L1 norm is preferred for tabular data. For additional analysis, please refer to the Appendix A.3.
3.2. Main framework
The main goal of our work is to train a local linear model to approximate the overall behavior of a black-box model in LDL. For a given instance z; that needs to be explained, we generate m local sampling instances for it and trans- form them into interpretable feature representation ma- trix Z = (2jy) E {0,1) 1J where Zjq represents the q-th interpretable feature representation of the j-th local sam- pling instance. If the instance I; is an image, we can use a segmentation algorithms, such as Quickshift (Vedaldi & Soatto, 2008), to divide it into f blocks. For the generated j-th local sampling instance, Zig 1 indicates the q-th block using the original image block, and zjy 0 indicates the g-th block using the average of the channels. If x; is an instance from tabular data, each column is divided into
We restore the logical values in Z back to the correspond- ing original input h ZZ) through a process of inverse dis- cretization, and then input into the original black-box model F to obtain the label distribution on locally sampled in- stances, F(h(Z)) To ensure interpretability and fidelity, our objective is to minimize the distance between G (Z) and F F(h(Z) while still satisfying the LDL constraints. Since not all interpretations of A' are simple and smooth, we use A'I to constrain the complexity of the interpretation. We use the alternating direction method of multipliers (ADMM) (Boyd et al., 2011) to solve objective functions with lin- ear constraints. Ultimately, we express the LIMEFLDI
3

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
generation as an explanation.
first provide a simplified form. All proofs for theorems and properties presented in this section be found in Appendix B.
A
min 2m
I' F(h(Z)))
+ 2m
(4)
S.L. A'A x rx1 = -1 mxx: Z'A 2 0 0mx--
z' IIZ' = B, PZ Z =C
(FF(()(2)) = D,
zT Fuul L,
Equation (4) can be solved by the following alternative methods iteration t:
1x1 x 1ixr = E.
PZ 1mxilixe = O,
A arg min AÇT 2m
II I' AT
F()(Z))
B m x(x)((Z))) (zyR =
(9)
+ P
Iz' A TT
x 1xx1 - 1mx111
|AAT1
Set the gradient to zero to obtain the iterative form of A',
2m
2m
+( m
(t))
Z' A TT 1,x1 1mx1)
= (D- CA'
(10)
V
Σ max(- - -Z'A!" ,0),
(5)
where |B S (f+1))22 ||||| E [0, (m | r/m. From this we give an iterative form for A'.
2m
u((1) =0 P (Z' A x 1,x1 1mxi),
(6)
Theorem 3.2. If Assumption 3.1 holds and Z' follows the 1 distribution, ignoring the regularization con- straints, the condition for convergence is given by: p < 2- 2-1(J ping Principle (Banach fixed-point principle), there exists an iterative fixed point A' that minimizes the loss.
where represents the number of sampled instances, u E pmx1 is the Lagrange multiplier, and p and are the penalty factors. For the non-negative constraint, we introduce penalty tern and provide Assumption 3.1 to ensure that the non-negative constraint satisfied.
- By the Contraction Map-
Assumption 3.1. For a given >
0, EN > 0,
However, Theorem 3.2 only shows that the feature attribu- tion matrix has iterative fixed points, but it does not provide a specific form due to the iterative structure. We find that this caused by the L2 norm constraints on vectors with sum of 1. Building Theorem 3.2, derive an approxi- mate analytic solution, we apply linear constraints using the lagrange multiplier method and the KKT conditions. This allows us to obtain an analytic solution for A'.
" Σ max(- Z' AT, 0) < holds for any m m
1 > N, we
assume that the nonnegative constraint is satisfied.
Here, we use the limited-memory quasi-Newton method (L-BFGS) for optimization. The computation of L-BFGS is mainly related to the first-order gradient, which can be obtained by:
VA = Z TTI (Z'A - F(h(Z))) + ZT u1
A
(Z IIZ' (Z IIF()(Z)) Z s.t. ATA x 1,x1 = 1mx1-
uull ullxx))
+I" p (Z A, 1,x1 1mx1) 1,x1-
(7)
To ensure that the final interpretation model is concise and efficient, we must select the most important features. There- fore, we design new scoring function S, which accurately evaluates the contribution of each feature to the overall dis- tribution of the labels.
(11)
We divide the solution into two parts: U'' A. IIF (h(Z)) Z" ZTTll uulxr r, which allows us to determine the constraint ranges for both the numerical and desired solutions of A'.
KL (G (Zc). ((((()))) KL (Y mean. (((h (Z)))
S = 1
(8)
T m
A -
T - T +
m
where (() represents the output of the black-box model, and G (Ze) denotes the output of the local linear model based on the selected features. Ymean is the uniform distribution. KL divergence are used to measure the distance between these two outputs. Based on the value ranking of S, we select the most explanatory features.
-1
[mllF
(12)
where Am represents the computed value of A obtained
through the model, Á denotes the expected value. Similarly,
I I' corresponds to the computed value of T, I denotes the expected value. A = A-IT, m represents the computed value of ATT, À'T denotes the expected value.
3.3. Analytical solution analysis
To better understand the specific form of the feature attri- bution matrix A' when the constraints are minimized, we
From Appendix B.1 and the form of A, we can conclude each element of Am A falls within the range [-
4

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Using the matrix Hoeffding inequality (Tropp, 2015), for any > have,
Top-20 Jocsaar Tncles
Top-20 Jaccard Indle
1.50
0.50
mt²² A 21) S exp
2.45
(13)
8/2
2.0
0.0
Similarly, given (() S 1]mrr, <--) E [0, 1]. and
1005
3210
S m)xx the form of r ensures that each element of
avg. abs, diff. per element
Sumples (log scale)
(2 bb models)
m T is bounded by [-1, 1]. Thus, for any > 0, we have,
(b)
PP ([|m [mmm-T] F/r 21 25 exp
ml?2
Figure 1. Convergence and stability visualization. (a): Conver- gence visualization, where x-axis denotes the exponentially in- creasing number of samples (log scale), and y-axis denotes the Top Jaccard Index. It demonstrates convergence as Jaccard Index f and stabilized. (b): Stability visualization, where x-axis denotes the decreasing divergence between two black-box models, and y-axis denotes the Top 20 Jaccard Index. It shows Jaccard Index f when predictive distribution divergence between black-box models (x-axis).
(T882)
(14)
8f2
where IP represents the probability that the Frobenius norm of each matrix satisfies the given condition.
Theorem 3.3 (Convergence). Based on Theorem 3.2 Ve
M(max(2522f(F then have P(I m - A' < < 1-6, and A' lim Am where represents the lower bound on the number of in- stances required for this problem.
0, E (0,1), if
1) - c- In 1f 2 2
Algorithm LIMEFLDL
Input: instance be interpreted x. LDL model F, sampling sample size m., weight kernel x.
Initialization: Ao, (9) p.
We aim not only to ensure the solution converges, but also to ensure that minor changes to the black-bo box model do not affect the interpretation. That is the stability of the interpretation.
Output: A.
1: II, Z 0::
2: for i 21,2,..., m}
3: 21 f- sample around(x):
4: *(Z1) t #4i
Equation (3)
Theorem 3.4 (Stability). Based on Theorem 3.3, for
5: Z+ Z II T(2);
any two similar black-box models P and Q define (h and T (h(Z)) of
6: end for
T ( (Z)) = (h each element 1. (h()) e, there exists a constant E 0, 2- such that ||A A/ A
7: while the stopping criterion not satisfied do
8: Ze + S(2);
b. Equation (8)
9: update Ac+1;
Equation (5)
10: update u (++1);
b Equation (6)
11: t +;1;
The mathematic expectation of the explanation is calculated according to the analytical solution formula as the number of sampling instances increases exponentially. The expla- nations obtained from the model are then compared, and the Top-20 Jaccard index is computed for both. As shown in Figure (a)) as the number of sampling instances grows, the two explanations become more similar and eventually converge. This confirms the correctness of convergence. However, due to the inherent rounding errors in the numer- ical computation of the inverse matrix, the calculation of A-1 may instead result in (A + -A)-1, where is a small error matrix. This introduces inaccuracies, affecting the final index results. As shown in Figure 1(b), the difference gradually decreases, and the interpretations of the two black- box model outputs become increasingly consistent. This demonstrates the correctness of stability. Therefore, we con- clude that our explanation algorithm is stable in LDL task and can effectively approximate the real decision- maasinnnnmaingg process. The overall flow of our algorithm is outlined in Algorithm 1.
12: end while
13: return
3.4. Analysis of algorithm properties
To make the interpretation algorithm easy to understand, we highlight some important properties that aid in its use. First, we show that the explanatory model consistent with additive feature attribution (Lundberg Lee, 2017). We present Property 3.5 to illustrate this.
Property 3.5. Let be the instance to be interpreted, and & be its interpretable data representation. &, represents the i-th interpretable feature representation. Let F denote the black-box model and G represent the locally interpretable model. Then, we have
F(x) R G G = -000 + Clansaa ... .. Gair
(15)
where do (K(()) is the output corresponding to an
5

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LIMEFLDI: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
4. Experiments
Top-20 Jaceoard Incleo
In this section, we will evaluate the proposed method on eight real-world datasets, test the fidelity and robustness of the explanation algorithm, and conduct human experiments to assess whether meaningful explanations are provided to end-users.
Fidelity
12
20
20
2000
Number of Added Features
Test Set Index
(a)
(b)
4.1. Numerical data sets
We conduct experiments on eight label distribution datasets from different kinds of real-world tasks to eval- uate our approach. Yeast alpha (alpha) and Yeast cold (cold) (Geng, 2016) are collected from biological ex- periments. JAFFE (Lyons et al., 1998), SBU_3DFE (3DFE) (Yin et al., 2006), and Emotion6 (Emo) (Peng et al., 2015) are collected from sentiment mining tasks. Natu- ral Scene(Scene) (Geng, 2016) is generated from a natural scene recognition task. JAFFE and Emotionó have both image data and processed tabular data, For the tabular data in these six datasets, we employ table-based interpretation approaches. For the JAFFE ((JAFFE) and Emotion6 (iEmo) image datasets, we apply image interpretation methods di- rectly on visual data. The details of all eight data sets are summarized in Table 1.
Figure 2. Local accuracy and dummy features visualization. (a): Local accuracy in multiple forms of fidelity, where x-axis denotes the number of added features, and y-axis denotes the forms of fidelity. (b): Dummy features, where x-axis denotes the index of test set example, and y-axis denotes the top-20 Jaccard indices.
all-O vector. This represents the output when no features are selected, which is consistent with the architecture of additive feature attribution.
Property 3.6. For any interpretable representation &. let its k-th feature be S or Eg. where S # c If the interpre- tation model G(St, Entx) G(g Eyye), then the feature attribution of the k-th is O. indicating that the feature is dummy.
Table Statistics from eight real-world datasets.
ID
Dataset
# examples
# features
# labels
Yeast alpha(alpha)
2465
24
18
Property 3.7. For any interpretable representation É, let its i-th and j-th features be & and E,, where & * Ej- If these two feature values are swapped to form 5'. and the explanatory model satisfies G' (E) = G (E'), then the feature attribution for the i-th and j-th features remains the same.
Yeast colddco)l(
2465
24
4
3
JAFFE
213
243
6
4
SBU_3DFF (3DFF)
2500
243
6
5
Emotion6 (Emo)
1980
168
7
6
Natural Scene (Scene)
2000
294
9
Some features contribute positively or negatively the model's output, while others do not. To empirically val- idate these observations. Initially, we first apply a mask to the image based on the mean value of cach superpixel block. This mask has impact on the interpretation model's predictions. Next, the highest-ranker ranked features are gradually added. As shown in Figure 2(a), the number of features increases, the fidelity of the local model improves consis- tently, indicating that some features contribute positively or negatively to the model's output and the local accuracy is both reliable and precise. To demonstrate that certain features do not affect model outputs, for each test image, the lowest-ranked feature is marked with a black smudge and outlined in green. Clean and processed images are evaluated under the same black-box model. Top-20 Jaccard indices are computed, with a dashed blue line showing the mean and a shaded area for standard deviation. As shown in Fig- ure 2(b) explanations remain unchanged for some images, exhibit minor variations for others, and undergo significant changes in a small subset.
7 Image-JAFFE (iAFFFE
213
243
6
Image-Emotion6 (iEmo)
1980
168
7
4.2. Numerical experimental evaluations
Fidelity is one of the most widely used metrics in inter- pretable artificial intelligence to evaluate how well an inter- pretation model mimics black-box decisions. Depending on the type of interpretation, as analyzed in (Guidotti et al., 2019), 9), fidelity can be specialized various forms. In our setup, we evaluate the fidelity using six distinct metrics, grouped into two categories. The first group includes Can- berra distance, Clark distance, Kullback-Leible divergence, and Chebyshev distance metrics, which measure the dis- tance between two distributions. Lower values (4) of these metrics indicate better fidelity. The second group comprises intersection similarity and cosine similarity metrics, which assess the similarity between distributions. Higher values (t) of these metrics represent better performance. For the
6

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Table 2. The effectiveness of two interpretation algorithms is evaluated on different datasets, using fidelity and consistency metrics under black-box modeling with RBF-LDL-LRR Results are marked with e for better outcomes and for anomalous data.
Algurithms
Chebyshev 4
Clark +
Canberra
KL &
Cosine 1
Intersection
Jaccard
PULIME 0009 + .000
0137=.000
04522 =0002
0001 + .0000
.9999 - 00000
.975 -,000
6422
LIMEFLDL
.0006 .0000
0108 L.
.0358
.003
.0000
.0000
1.000 L.000
9980 1. 2.00
.5044 1. 1.342
2
PULIME
0035 +
0089 + 0001
.1553 = .0002
0
0001 + 0001
9999 + -,000
9961 -0000
.3608 22175
LIMEFLDI.
0033 E
0096 + .000
.0165 = 00001
0001
+ .0000
9999 -000
.960 20000
62000 2..64
PULIME
DM61 002a
.990 + .107
.3763 = o2010
-300
.0.00
9903 .001
.403 .0006
.5790 d0007
LIMEFLDL
0338 + 0018
1493 = 0074
2249 00127
0078
+ 0009
9935 1355..00 0005
9538 = .0200
., 7080
PULIME 0583 0008
.1166 .o00
.3601 1 .033
0013
.870 .0003
.9611 1: .0007
.4318 1.077
LIMEFLDL
0348 0005
.1328 + 0017
.2613
= .0044
0075
0002
-9002
.9599 = .0006
.4675 = .0540
PULIME 0001 .000
.001 10000
.0001
=0000
-0000
.0000
1.000 1.000000000
.9999 10000
.1556 1245
LIMEFLDL
0001 .0011
.0027
.0782 .0028
0000
.0000
1.000 1.000
.9999 = .0000
.3804 2000
PULIME
0010 +: .000
.051 + 0002
.0398 =00005
.01
+ 0001
.997 +
.9986 =
.2846 #.550
LIMEFLDE
0008 0000
.0154 + 0001
.0395 = 0006
0007
0001
.9999 +: 0000
.9989 =: .0001
.3487
2.99
PULIME
0040 +
0142 = -o002
.2889 =.000s
.000
= .001
9998 -0000
.9951 =.000
.7470 1:470=..0. $.220
LIMEFLDL
0003 1
.0166 1: .002
.223 d: 0005
00000 1
.9999 1: 0000
.9960 1: .0000
.7903 1004
PULIME
0000 + .000
.032 1022=.000 .0001
.0069
= .000
-0000
= .0000
.999 - -0000
.999 50000
.1591 1299
LIMEFLDI. DI.
0033 2: 0000
0116:_.0002
.0336
= .000s
0001 + .0000
.9999 10000
.9960 + _0001
. .7903 2064
instances to be explained, we calculate the distance between the label distributions predicted by the black-box model and those output by the local linear model. This serves as the fidelity metric for the interpretation model.
proved problematic. Apart from the target sample, all other local samples received weight of zero. To address this, we use cosine distance for local sampling in image datasets. This approach ensures that each local sample is assigned a weight proportional to its distance from the target sample. For tabular datasets, we retain the L22 norm as a distance metric. Detailed parameter configurations for LIMEFLDL and PULIME are documented in the Appendix A. These settings align with the implementations used in our numeri- cal experiments.
Consistency is fundamental requirement for any trustwor- thy method. Here, we compute the average top-K Jaccard index (JI) for interpretations generated using 10 different random seeds. Let wi...., Wio represent the inteppretations obtained from these seeds. For the first K highest-ranked features in wi, the set is denoted as R, We then calculate the average Jaceard index between R,KK and R, defined as: JI,R,,K, R) ,c A JI value closer to indicates greater stability interpretation consistency. The instances used in this analysis are randomly selected from the test set, ensuring diverse coverage across random seeds.
4.4. Human experiments
KUR,
In addition to the numerical experiments, we conduct human interpretability experiments to evaluate whether LIMEFLDL offers more meaningful interpretations for users. The exper- iment has two parts, with 10 participants in each. Details of the experimental process are provided below.
4.3. Experimental setup for numerical experiments
The proposed interpretation algorithm, in conjunction with the parallel use of the traditional LIME algorithm (PULIME), is applied to widely used LDL methods, includ- ing LDL-SCL (Zheng et al., 2018), LDL-LRR (Jia et al., 2023), AA-KNN and the Maximum Entropy Model (MEM). Since the feature extraction module in the LDL-LRRR model relatively simple, we incorporate a Gaussian kernel func- tion to enhance feature extraction during training (RBF- LDL-LRR). This improves the model's performance while also increasing its complexity. All codes are provided by the original authors and we adopt the recommended parameters reported in their respective studies. For image datasets, us- ing the Le norm as a distance metric during local sampling
Can LIMEFLDL improve the understanding model predic- tions? To evaluate this, we select images that the black-box model predicted correctly. The participants are shown both the original image and the generated interpretation. They are then asked to assess the degree of agreement between the interpretation and their intuitive understanding of each label. For each label, the participants first evaluate whether the provided reason is reasonable. If so, they then rate the degree of agreement on a scale of 1 to 5, where 1 indicates no match and 5 indicates perfect agreement.
Can LIMEFLDL help identify modeling errors? To exploit this, we select images where the -ox model showed a significant difference between the predicted and actual label
7

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Table The effectiveness of two interpretation algorithms is evaluated on different datasets, using fidelity and consistency metrics under black-bor box modeling with LDL-SCL. Results are marked with for better outcomes and for anomalous data.
Algurithms
Chebyshev 4
Clark +
Canberra
KL ل
Cosine
Intersection
Jaccard
PULIME
0008 + .000
0131 00000
0431 =0000
C -0001 + .0000
.9999 + .0000
9976 2
6446
LIMEFLDL
.009 .000
.1500 0.002
.0501 200000
.0000
.0000
.9999 1.0000
.972 2.000
.6434
1732
2
PULIME
0034 +
0087 + 0000
.1449 = 0.0022
0 -0000 +
9999 -0000
9962 -0000
.3924 2.006
LIMEFLDI.
0037 +
0097 + 0001
.0668 = 2.002
0001 + .000
9999 + #.000
.9588 2.000
7,558 i
PULIME
U523 0022
.2336 .0091 .091
.222
01911
0368
+. 0088
9885 - .000
.9332 .0001
.0331
.5687 2.822
LIMEFLDL
0337 2 0018
1468 + 0088
2903
- 00170
0077 = 0008
1936-.0000
.544 = .0025
6508 - 0560
PULIME
0326 0005
.1235 .0117
.2298 .0031
o00o
.950 .0001
.6133 1: 1.00
.581 1804
LIMEFLDL
0240 0005
0019 = .016
.1778
= .0332
0034
0001
9966 -0001
.700 = .0005
6-462 2. 1501
PULIME 1966 0023
.952 + .0022
1.997 0,030
-0000
0284
.9129 .0022
.7106 #.0881
4464 1: .0006
LIMEFLDL 1317 .022
= .00r9
1.800 000214
.7059
.0099
99132 .o02a
8058 = .0033
12281
PULIME 3584 +. 0000
1.995 + .025
5.330 =.010
-,512
+
.7999 +
.1487 .o009
.4877 #.040
LIMEFLDL
1692 0030
1.694 + .005
+4.139
.300
1.200
.590
8840 +: 0033
.7141 =: .0036
.5481 +: 0085
PULIME
0051 + 0001
.092 - 0000
.0385 -.000s
.0003 + .0022
9998 - -o000
* .993 -.000
8661
2.002
LIMEFLDL
.049 1 0001
.017 1: 0010
.0429 d .0200
00001 1:
9998 o000
.9931 1: .00a
8292 1,000
PULIME .003 0001
.001
.U753
= =0015
0003
= 0001
9997 0000
.9112 _0001 .000
2149:..1000
LIMEFIDI.
.081 +: 0001
0006 .000
.1033 - 20013
.0066 +: .0000
9995
.9881 =0001
18740
distribution. The participants are shown both the original image and the generated interpretations. They are then asked to assess the degree of deviation in the individual label interpretations and provide reasons for the model's incorrect predictions based on these interpretations, along with their own reasoning. The participants first evaluate whether the reasons reasonable. If so, they then rate the usefulness of the explanation on scale from 1 to 5, where I meant that the explanation offered no insight into the model's 's incorrect prediction and 5 meant the explanation is extremely helpful in understanding the incorrect prediction.
these datasets are derived from subjective labeling or rank- ing, making them more representative. The results from the two image datasets show better performance compared to the tabular datasets. This is due to the image dataset being characterized by pixel blocks, while the tabular dataset has feature columns with a partial range, leading higher error in the latter. (2) Our interpretation algorithm generally out- performs PULIME in terms of consistency metric (Jaccard index), which aligns with the stability shown in the theoreti- cal section of this paper. (3) Anomalous data are observed in the KL metric under the PULIME, with its KL divergence heing negative. This occurs because the PULIME fits the la- bel distribution individually. Although this approach better captures the description degree of each label, it can result in the sum of the description degree exceeding 1. This leads to the possibility of P Σ, ( which causes the final Σ3 (,(()
4.5. Results and discussion
Tables 2 and 3 present the detailed experimental results for the fidelity and consistency metrics of the LDL algorithm. These results are based on six tabular datasets and two image datasets. For each data set, we performed 10-fold cross- validations and recorded the mean and standard deviation for each evaluation metric.
KL divergence to be negative (see the Appendix B.6 for the overall proof), where ()) represents the descriptive mea- sure of each label in the fit. Experiments two additional black-box models are provided in the Appendix A.8.
Based on the results of the numerical experiments, the fol- lowing observations can be made: (1) alpha and cold dataset shows better results both on multiple fidelity metrics and with different black-box modeling algorithms. The rank- ings of the two interpretation algorithms have their own strengths because the label distributions of these datasets are approximately uniformly distributed. This makes it easy to approximate whether the parallel interpretation or the inter- pretation distribution. For data sets JAFFE, 3DFE, Emo and Scene, regardless of the LDL algorithm used, LIMEFLDL outperforms PULIME, In addition, the label distributions in
Figure 3 presents a qualitative evaluation method based on human visual perception to assess the consistency of the interpretation results with human understanding. In Figure 3, purple (a blend of blue and pink) denotes the cases where LIME LIMEFLDL:; blue denotes the cases where LIME > LIMEFLDL; pink denotes the cases where LIMEFLDI. > LIME. Participants are shown images with accurate model predictions along with interpretations of PULIME and LIMEFLDL. On average, PULIME receives a score of 2.1, while LIMEFLDL receives a score of 3.1. Thus, LIMEFLDI. outperforms PULIME by 1.0 points.
8

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LIMEFLDI: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Impact Statement
This paper makes a significant contribution to the field of interpretable machine learning, which is crucial for the de- velopment of LDL. We show that our methods are versatile for any LDL black models (covering at least two data types- tabular data and images), Our research provides the theo- retical foundation for further advancements in interpretable LDL. Facilitating the broader adoption of Al technologies in sensitive and impactful domains.
Genper
(a)
(b)
Figure 3. Human-interpretabiinn results. Purple (a blend of blue and pink) denotes the cases where LIME LIMEFLDI blue denotes the cases where LIME LIMEFLDL; pink denotes the cases where LIMEFLDL > LIME. (a): intereraittrr results for accurate predictions. (h): Human-interpretability results for incorrect predictions.
References
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Meanwhile, when participants review images with incorrect model predictions and corresponding interpretations from both methods, PULIME receives an average score of 2.3, while LIMEFLDL scores 3.1. Again, LIMEFLDL achieves higher score, with difference of 0.8 points. These re- sults strongly indicate that LIMEFLDL is more effective in explaining model behavior compared to PULIME.
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5. Conclusions
Garreau, D. and von Luxburg, U. Looking deeper into tabular LIME. arXiv, 2008.11092, 2022.
In this paper, we propose a local interpretable model- agnostic explanation approach for LDL, called LIMEFLDL. This algorithm adapts the original explanation approach to fulfill the constraints of the label distribution learning paradigm. We introduce a new constraint on the form of label distribution and the corresponding feature attribution matrix solution based on the alternating direction method of multipliers. Besides, we give new feature selection functions to improve the interpreting efficiency of model. We then provide theoretical analysis of the solutions, convergence behaviors, and stability cases of the model. Furthermore, we present a series of properties of our proposed explana- tion algorithm to facilitate the practical utilization. Finally, we demonstrate the effectiveness of LIMEFLDL by both theoretical analysis and experimental studies.
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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
A. More discussions
A.1. Details parameter settings
In the fidelity-metiii experiments, we set the number of sampled instances m to 100, and select the top 10 ranked features for evaluation on all test sets. For the consistency experiments, to ensure the explanatory model captures sufficient information, increase m 1000 and do not impose restrictions on feature rankings. Furthermore, in all experiments, the lagrange multiplier (0) is initialized an all-zero vector. The penalty coefficients X, and p are 0.01.
Parallel Utilization of LIME (PULIME) decomposes label distribution into series of single-label. For each label, the explanation task is performed independently. Specifically, the sampling size m and weighting parameters align with LIMELLDL. Other hyperparameters retain its default settings.
Feature transformation. For image data, the first step to crop each image to a size of (256, 256, 3). The image is then segmented into superpixels using the Quickshift method from tilite-mmae, with the following parameters: kernel-size=4, max-dist=200, ratio=0.2,, and random-seed=200 for the modified random seed. This ensures that the same images produce the same superpixels, reducing computational instability in the interpretation process. However, different images are still segmented in different ways. For tabular data, the parameters are set as follows: discretizer=" "entroppr""" "ntropy"." tree method used to adaptively bin feature columns. Fixed random seeds ensure consistency while guaranteeing that each feature column is involved in the binning process.
These settings align with the default configuration in LIME, except
random--eed=2024, and searetizeeconiinoouss
The decision
Computing Explanations. For image explanations, the masking operation follows the original LIME settings. The hide-color parameter is set to None, meaning that the average value of each superpixel serves its reference when that superpixel is removed. For distance weights, cosine is used for image datasets, while the L2 norm is applied for tabular datasets. To balance accuracy and efficiency, the maximum number of iterations for L-BFGS is set to 50. In addition, all random seed parameters are fixed to ensure reproducibility and consistency across runs.
A.2. Supplementary notes on human experimentation
We conducted the human evaluation with full adherence to ethical standards. Here is the required information: Our design mitigated bias via 10 postgraduate students participated in the study, with 7 specializing in machine learning and the remaining focusing on other areas within computer science. The evaluation was conducted through in in-person scoring sessions. Each participant independently evaluated 20 randomly selected cases from our test set. All participants signed informed consent forms outlining data usage. No personally identifiable information was collected. Raw responses were aggregated prevent individual identification. Random recruitment (ML/non-ML backgrounds), independent evaluation of 20 randomized interpretation results, and high inter-rater agreement despite participant diversity.
A.3. Sampling weight
For image data, sampled instance 24. differ significantly from the original image. For instance, if most of the elements of 2ig zero, 2; will be close the mean. When constructing a local model, each sampled instance differs slightly from the instance to be interpreted. Therefore, we treat the proximity weight T22).
(=lens 2i) 0²
V1 siism m,
T(Zi) = -exp
(16)
where a > is the kernel width parameter and does the cosine distance.
u 4
Yu, C Rd. dooe (2,, v) | ||
(17)
We can find that the weights T<z)) depend solely on the number of hyperpixel blocks, each taking the mean of its corresponding channel. For example, if z; has exactly elements equal to zero, we have 2, = m - 8, and
| = vm - s.
So |||| = /m, according to Equation (16), we deduce that (24) = p(_)),
((1 (11- vI 0)
Va E [0,1], p(a) = exp
(18)

---

LIMEFLDL: A Local Interpretable Modcl-Agnostic Explanations Approach for Label Distribution Learning
Algorithm 2 Get Summary Statistics: Getting summary statisties from training data
Input: Train set X' = (S....., GnJ, number of bins
1: for to do
2: for b to p do
3:
Ujh < Quantile S.ij://....sibppp)
4:
end for
5:
for = to p do
6:
- [Gi s.1. Gjb-1
< Gi S 9no)
7:
11g MSa
8:
J Var(S)
9:
end for
10:
end for
11:
return 9jb, Myb and 00 for 1 Sij and Sbsp
Tabular data sampling begins by dividing the input space of each column of features in the training set into bins. These bins used to create interpretable features. The method not only uses the ranges of the features, but also computes the mean and variance of the training set data once transformed into interpretable features. These values are then used to generate subsequent sampling instances. For detailed specifics, please refer to Algorithm 2.
Boundary extraction. The boundaries are determined by taking the quantiles at level for b E {0, ........... pp. other words, the ljb are such that
m 1 Ela-
1
(19)
m
where, for each < f. 930 = minigism Si.j. and dpp = maxisisi St.j On the other hand, the indices of the bins are used sample new samples 21, 2m For 1 S i m, each new sample z; is sampled dimension-wise independently: Zjg follows the distribution of truncated Gaussian random variable with parameters Gjb,5-1 9jb,, Pibi, and Jb,." So, we condition, when falling into the b-th bin, the probability density function of 2jg is given by the following expression.
((==)a)² exp
20
Pzb(t)
(20)
Jnb² 2t
+ (13b))
$ (l,b)
where we let ljs- random variable.
9.6 191 and r'zb
-:0 Pob and is the cumulative distribution function of standard Gaussian Ø₁
0yb
In the tabular data, the weights of the newly sampled instances are given by:
1
zz 02
T(Zi)= = exp
= exp
1 Aee
(21)
0
where ||-|| denotes the L2 norm, and a is the kernel width parameter. This weight calculation based on how many coordinates the bins the sampled instance differ from those the instance to be explained, followed by the application of exponential scaling. If all bins are identical, the weight 1. On the other hand, if the sampled instance is far from the instance to be explained, N (zz) can be very small. Here, far means that Z does not belong the same bins as the instance to be explained, and the bandwidth parameter is fixed at v0.75f.
Limitations. If two interpretations instances A and A₂ are very close, they may share the same bin indices and, consequently, the same A. On the other hand, if A₁ and A2 are close but do not share the same bin indices, their explanations may differ significantly. From practical perspective, if one believes that the exact values of A are important, increasing bins a solution. This results in thinner bins per dimension. In the limit, setting p = 0, no discretization is also an option.
12

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
A.4. Convergence of LIMEFLDI.
Figure 4 extends the experimental analysis in the main text by incorporating two additional common LDL black-box models: AA-KNN and MEM. In these experiments, all data sets from the main text used. The number of sampling instances increases sequentially from 100, 500, 1000, to 5000, with the top-20 average Jaccard index serving the evaluation metrie. Notably, the results in Figure 4 align closely with those in Figure (()). As the number of sampling instances increases, LIMEFLDL consistently converges, regardless of the dataset or the choice of black-box model.
A.5. Stability of LIMEFLDL
To introduce variability in the black-box model's predictions while avoiding overly biased distribution, we adjusted the number of iteration rounds. For RBF-LDL-LRR and LDL-LRR, the iteration rounds were varied from 10 to 500. For the AA-KNN model, the number of clusters was modified from 1 to 5. Additionally, we calculated the average difference in the descriptiveness of each label between each prediction and the best prediction. The top-20 average Jaccard index was used as the evaluation metric. As shown in Figure 5, the results align closely with those in Figure 1(b). As the average difference in label descriptiveness between each model's predictions and the best model's predictions decreases, the explanations generated by the models become increasingly consistent with those of the best model. These findings demonstrate that smaller perturbations in the black-box model lead to more consistent interpretations across different models.
A.6. Attribution of additive features in LIMEFLDL
To ensure the additive feature attribution holds, we add features one at time. For image data, we first mask each superpixel using the mean of the three color channels. Secondly, based on the interpretation results of the original image, we compute the feature attribution distribution for each superpixel block. We take the absolute values of these attributions and sum them. A larger value indicates a greater impact on the overall label distribution. Thirdly, we select the top 10 superpixel blocks and incrementally add the highest-rankes The masking operation flow for this process is shown in Figure 11. Fourthly, we observe the distance between the local prediction and the original image's 's local prediction, which is represented using fidelity. For the tabular dataset, we compute the top 10 features of the instances to be interpreted and add the features one time. For unselected features, we set the corresponding column to O, indicating it was not chosen for the bin. The results in Figure 7, show that, as features are gradually added, the individual fidelity metrics increase, meaning the predictions get closer to the original instance. This indicates that LIMEFLDL' additive feature attribution is correct.
block from the original image, starting with a mean mask for each block.
A.7. Dummy features of LIMEFLDL
Due to the harsh conditions of the virtual features, it is difficult to find feature attribution distribution that do not affect the whole label distribution in any way. However, it is possible to find the feature with the least impact based on the feature attribution ranking. Therefore, we choose to focus on the feature with the lowest ranks. For the image data, we first identify the lowest-ranked superpixel block. To demonstrate that the feature is almost false, we apply a complex process: we mask the superpixel, and add a green border. A specific example is shown in Figure 6. We then evaluate the fidelity and consistency of both the original and processed images. Consistency results in Figure 8 and Figure 9, and fidelity result is in Figure 10.
A.8. Comparison of fidelity and consistency metries in AA-KNN and MEM
Our goal is to enable explanation algorithm to interpret complex black-box box models. However, if the explanations for simpler models do not provide sufficient fidelity, it is challenging trust explanations for more complex black-box models. Therefore, in addition to comparing the fidelity and consistency rankings of our algorithm with PULIME on RBF-LDL-LRR and LDL-SCL black-box models, we also conducted experiments on simpler black-box models such as AA-KNN and Maximum Entropy. These models have relatively simpler processes. AA-KNN integrates a clustering algorithm within the model, while MEM employs KI, divergence as its loss function. Experimental results show that our algorithm performs better on simpler black-box box models than on more complex ones. Furthermore, in most datasets, our method outperforms PULIME in both fidelity and consistency metrics.
13

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LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
A.9. The ranking information in the comparison results between LIMEFLDL and PULIME
We give ranking statistics illustrate the extent to which LIMEFLDL ahead of PULIME in terms of evaluation metrics, the results are in Table 4, from the result, we see that LIMEFLDL leads the rankings for the vast majority of measures.
Table 4. The ranking information in the comparison results between LIMEFLDL. and PULIME.
Model
Algorilum
Chebyshev
Clark
Camberra
KL.
Cosine
Intersection
Jaccard
LIME
1.75
1.625
1.625
1.655
L875
RBF-LDL-LRR
1.25
1.375
1.375
1.125
LIME
1.625
1.5
1,875
1.5
1.5
1.625
LDL-SCL
LIMEFLDI
1.375
1.5
1.5
1,125
1.5
1.5
1.375
LIME
1.625
1.625
1,625
1.5
1.625
1.625
1.75
AA-KNN
LIMEFLDL
1.375
1.375
1.375
1.375
1,375
1.25
LIME
1.75
1.75
1.75
1,875
1.75
1.75
1.75
MEM
LIMEFLDL
1.25
1,25
1.125
1.25
A.10. The computational complexity of the LIMEFLDL algorithm compared to PULIME
We analyze the computational complexity LIMEFLDL and PULIME. For LIMEFLDL, sampling and forming weight matrix IT requires (m)) Matrix multiplication contributes (m,.r), black-box model inference accounts for (mrk). Element matrix subtraction and F-paradigm operations each add ((m)), with regularization terms contributing O(fr). The total complexity is Ommfr mrk + mrr + mr + fr + m). PULIME's per-label workflow involves sampling (0(m)), vector operations (m( for subtraction/squaring and (m for multiplication), black-box inference (O(mrk)), regularization (O())). With parallel computation, its total complexity O(mr mkr2 + +r++me m +r+r PULIME's ridge regression solver has O + m²) complexity, LIMEFLDL uses L-BFGS optimization with O(T(zr + mff) cost, where 2 R 5-20 (stored gradient pairs) and T denotes iterations.
A.11. Re-score aligned with LIME metric and expanded baselines
conducted supplementary experiments using R2-score aligned with LIME metric and expanded baselines to GLIME and DLIME, all evaluated under default parameters. The result Table can be seen that LIMEFLDI leads in the original fidelity metrics for most of the table and image datasets.
Table 5. R²-score aligned with LIME metric and expanded basclines.
Model
Index
LIME
DLIME
GLIME
LIMEFLDI.
0.15
2
3
0.48
0.35
0.34
0.58
4
0.39
0.33
0.34
0.40
LDI.
5
0.34
0.25
0.25
0.30
6
0.37
0.34
0.32
0.38
7
0.94
0.95
0.99
8
0.89
0.89
0.98
0.17
0.17
0.16
0.99
3 0.34
0.33
0.33
4
0.25
0.31
0.29
0.36
RBF-IDDL-L.R
5
0.26
0.47
0.36
1.00
6
0.52
0.49
0.47
0.99
7
0.93
0.96
0.99
8
0.67
0.94
0.99
14

---

LIMEFLDL: A Local Interpretable Model-Agostic Explanations Approach for Label Distribution Learning
0.44
----
Ema
--X-
Emo
Top-20 Jaccard Lndex
0.42
IJAFFE
Top- 20 Jaccard Index
0.40
cold
- alpha
3
iJAFFE
0.40
0.38
0.38
0.36
0.36
0.34
0.34
10a
10a
10
10a
Samples (log scale)
Samples (log scale)
(a) Top20-jaccard metrics for L.DI-SCI. black-hox model-
(b) Top20-jaccard metrics for AA-KNN black-box modeling
ing under locally sampled exponential-level growth.
under locally sampled exponential-level growth.
Figure 4. Visualization the convergence explanation algorithms.
'Top-30 Jeccard Indlox
Tap-20 Jaccuard Tndlex
Top- 20 Jarrard Index
Top 20 Jaccard Index
odi
0.10
8.000
6.10
A.30
0000 Ca0o 1.000
1300
avg. abs. diff. per element
abs, diff. elemen: (2 models)
eng. abs. diff. per element
abs. diff. clement
(2 bb models)
(2 models)
(2 bb models)
(a) Top-20 jaccard metrics on the Yeast_alpha tabular
(b) Top-20 jaccard metrics on "east_cold tabular dataset.
(c) Top-20 jaccard metrics on
Top-20 jaccard metries on Emotionó labular dataset.
the Emotion6 image dataset.
dataset.
1bp-20 Jaccard Index
Top-20 Jaccard Tndex
Top 20 Jaccard Index
Top 20 Juccull Inlex
Seraa
SCFFE
003
0.12
0.13
8160
6.3
0.14
8100
507
Lic
one
sos sidi sic
110
6.00 sac
abss ciff. per cleacnnt (2 bb models)
abs, diff. element
ang. abs. diff. per element
abs. diff. element
(2 bb models)
(2 models)
(2 bb models)
(e) Top-20 jaccard metrics on the Natural Scone tabular dataset.
(I) Top-20 jaccard metrics on the SBU_3DFE tabular datasct.
(g) Top-20 jaccard metrics on the JAFFE image datasct.
(h) Top-20 jaccard metrics on the JAFFE tabular dataset.
Figure 5. Stability evaluation with AA-KNN black-box models.
15

---

LIMEFLDL: A Local Interpretable Model-Agnosti Explanations Approach for Label Distribution Learning
La
(a) Original image: 3.
(b) Original image: 29.
(c) Original image: 90.
(d) Original image: 201,
x
(c) Processed image: 3.
(f) Processed image: 29.
(g) Processed image: 90.
(h) Processed image: 201.
Figure 6. Dummy feature processing in image data.
F50alit
Fidelity
Fidelity
Fidelity
Number Added Features
Number of Added Features
Number Added Features
Number of Added Features
(a) Changes in fidelity met- rics under the JAFFE image dataset.
(b) Changes in fidelity met- rics under the Yeast.cold tab- ular dataset.
(c) Changes in fidelity met- under the Emotionó age dataset.
(d) Changes in fidelity met- rics under the JAFFE tabular dataset.
Figure Additive feature attrihution using AA-KNN black-box models.
Top- 20 Jacrard Indlxx
Top- 20 Jaceard Indlex
Jaccard Indlx
Top 20 Juceard Indls
a.
Roc
8
Test Set Index
Test Set. Index
Test Set Index
Test Set Index
(a) Interpretation consistency scatter distribution under the Yeasi.cold tabular data test
(b) Interpretation consistency scatter distribution under the Yeast.alpha tabular data test set.
(c) Interpretation consistency scatter distribution under the JAFFE tabular data test set.
(d) Interpretation consistency scatter distribution under the
Emotion6 tabular data test set.
set.
Figure 8. Dummy feature attribution of the LDL-SCLL black-box model.
16

---

LIMEFLDL: A Local Interpretable Model-Agnotic Explanations Approach for Label Distribution Learning
TTpp22 Jareard Index
Top 20y-20 Jancard Index
Top-20 Jarcard Ludex
Top- 20 Jaceard Incke
no
Datn Pazna
Data Pernts
Data Pents
Data Potzs
200
130
200 Test Set Index
200 Test Set Indlx
500
Test Set Index
Test Set Index
(a) Interpretation consistency scatter distribution under the Yeast_cole tabular data test
(b) Interpretation consistency scatter distribution under the Yeast alpha tabular data test set.
(c) Interpretation consistency scatter distribution under the JAFFE image data test set.
(d) Interpretation consistency
scatter distribution under the Emotion6 tabular data test set.
set.
Figure 9. Dummy feature attribution of the AA-KNN black-box model.
1.0
Values
Values
2.0
sj
sj
0.8
cold
cold
Motric size
Metric size
alpha
alpha
0.6
e6
1.0
0.4
0.5
0.2
0.0
0.0
che
cla
inter
che
cla
inter
Fidelity
Fidelity
(a) Modifying the lowest-ranked feature using the LDL-SCL black-box box model.
(b) Modifying the lowest-ranked feature using the AA-KNN black-box model.
Figure 10. Dummy feature attrihution under fidelity metrics.
B. Proofs
B.1. The range of B -1 ||C|| p. and the specific value of [E|]
First, the specific form of the B matrix is presented,
1 Σ n (an) (zz) ai
Σ (2)) 20 Σ. (zi) (22)
1 r (zi) 21,14
π (zi) 2222s411
B
(22)
(Z) 21441
1. Σ, (zi) 2224:411
Σ, n (zs) (z:f+1)²
17

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Table The effectiveness of two interpretation algorithms is evaluated on different datasets, using fidelity and consistency metrics under black-bor box modeling with AA-KNN, Results are marked with for better outcomes and o for anomalous data.
Index
Algorithms
Chebyshev
Clark
Canberra
4
Cosine ?
Intersection f
Jaccard
PULIME
.0055 20000
04000 = 0008
1319 +.
0001
=00001
9999 =0000
9927 = 0000
4206 + 2.462
LIMEFLDL
.0227 = 0000
.047
0002
.1485 + .000s
.002
0000
.998
2000
9918 0000
.4743
2
PULIME 0100 0000
0278
=0000
0477 2000
0003
00001
.999
-0000
9881 2.0002
.4088 #.07s
2
LIMEFLDL
0107 1000
.091
0000
0504 1. 0004
.004
.0000
.995
0000
.9755 1. 0001
.188 1561
PULIME
0350 00017
.1466
= .0588
.2841 .00sss
--3300
.0048
99400=.00s
.9533 .000
A261 +.107
LIMEFI DI.
0742 20001
0996
10066::00 .0522
1977
2.0104
.0038
=0004
99994
1 2,000
.6755
.0017
44222
PULIME 0576 0007
2080
#.011
.3004 20022
-0662
0000
.9886
2.0001
2.0005
.116 4.55:
LIMEFLDL 0322 20005
1256
.2493 20087
0062
=,0001
9939
=00001
.9881
-.0000
-,902
PULIME
0794 2012
6438
20000
1.225 1. 1.015
.0054
.0000
.923 2,000
8835
..007
.3742 1: 0006
LIMEFLDL
.055554 00011
5206
.0087
.971 2.0600
.0389
.0057
9839 =,000
9173
20013
42299 0,000
6
PULIME
.505 2,000
1666
.0883
3.777 +. .121
0088
.0126
.9220
2 #.012
.7791 +: 20000
.650 1.81
LIMEFLDL 1004 00055
1.610
1::10000m 0061
3.573 2.1422
.1494
00053
9523
84408
0018
.1555
7
PULIME .0000 00007
0249:
= =.007
0467 2,043
.0006
20000
.996=_0000 -.001
.9244 + .0007
..533
7
LIMEFLDL 0049 - 0002
01899 0001
0366 +. 0023
00003
.000
.9997
= 2.000
9937 + 0003
.3516
PULIME .0021
0000
1626 -0028
22166..002
0065
=0000
9966
=.000
9735 _0003
.1833
LIMEFLDI.
.0115 1. 0003
2030 1:, 0034
.3423 1. 1003
0083
2019
.9958 1: 0001
.707 1. 0004
.284 1, .0299
The expected value of each element in the matrix B is:
(t-f1 k a²
f!
E π (zx) (z1))
[ k
k=0
( 21 - ([)(k(((
(1-1)!
(K-1) aبت
(f 11)
0 e
(23)
2 - kk)((k- 1)!
k=0
(1-53/02 (1+
f-1
2
5-1
(1+e- x2
2
similarly,
(N-1) kikk: - 1) ft od
Σ
f(f -1) 1) 21 2f KIRR!
k=0
(k-9)
(2-
2f - kk)((k 2)1
k=0
(k-2)
((- ot
( - 2)
(24)
2f k)). 2)
k=0
f-2
1 1 22-1//0
1-2
(1+8 1
2
18

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
Table The effectiveness of two interpretation algorithms is evaluated on different datasets, using fidelity and consistency metrics under black-bor box modeling with MEM. Results are marked with for better outcomes and for anomalous data.
Index
Algoritims
Chebyshev 4
Clark J
Canberra
KL 4
Cosine
Intersection
Jaccard
PULIME 0008 + cono
0127 + 0001
1.147 + 0on4
o 0001 + cono
9999 + con0
9976 0000
.612 11112..1.0
LIMEFLDL
0009 I: 0000
0157 +: 0002
0523 + 0006
0000 +: 0000
9999 + 0000
9971 + 0000
.7029 4: 1506
2
PULIME
0034 co01
0087 + 0001
0149 + nons
o 0000 0000
9999 #999t.0mm 0on0
9962 + 0001
.7324 1984
2
LIMEFLDL
0036 0036 0001
0097 1. 0001
0168 1. 0002
.000 1: 000o
9999 1. 0o00
9958 1, 0001
77883 1001
PULIME
0721 coss
3030
5723 + 0188
.6122..0149
9810
9084 0031
.2218 d. 0752
LIMERLDI. DI
0470 0032
2198 0130
4296 I: 0241
0205 0.046
9874 0016
9349 117.2.o00 0087
6663 08884
PULIME 1158
3675 0004
6878 è. 005e
9651 oous
.712 0013
.213 1396
LIMEFLDL
.000
2096...00s ooas
4074 0012
0188 4: 0022
982811.000 oo0e
9323 100000000 0013
6115 H. 1428
PULIME 1529 1. 0026
8101 o.00
1,703 1. 0206
0078 1: 0160
9344 1. 0017
7758 1. 0026
.5412 1. 0022
LIMEFLDL
1051 0017
7260 0057
1.468 + 0121
.2555 + 02289
9420 0018
8442 0019
6344 # 1243
PULIME 4383 E. 0038
2,113 4: 0034
5.783 & 0139
66579 d: 0160
7737 1: 0023
o 1340 + 0083
0699 08622
LIMEFLDL
2022 t. 0027
1,868 + 0134
4,692 + 0451
1.855 0742
8339 0013
6,6454 + 2012
.2000 + 0053
7
PULIME
0086
0358 + 0012
.715 + 0025
0005 + 0o00
9995 -,00 0000
9883 + 0004
8755 #6639
LIMEFLDL
0050 +. 0001
0192 +: 0003
.384 1: 0007
0002 -: 0002
9999 + p000
9936 1
90000 =:
PULIME
0079 0001
0485 + 0007
1008 + 0018
0006 0ono
9095 00000 0000
9882 - 0001
8580 H. 0as0
LIMEFLDL
.007 1: 0001
(400 1. 0004
.0288 1. 0009
0007 1. 0001
9997 1. 0000
.900 1. 0001
.7670 1051
1051
Let,
(1 1+ e - J-1
1 1++e - f-2
M1
2
H2
2/
(25)
ignoring the added bias term, the expected value of the B matrix can be expressed as:
B = (M - p2) Ixx1 + #2 (15x1 x 11x)).
(26)
Then, the inverse of matrix B can be expressed as:
B ((11 - p2) +/2 (1xx1 x l1xx))"*
(27)
at the same time, we can obtain:
/-1
ltc
oz
25
2
21
2
2'
(28)
1-2
Ite oz
21-2
2
By applying the Sherman- formula, we obtain:
B B = (111 - P2) Ixxt + 12(1 x lixs)
1
Iixs+ +
112
1 1xx T
-1
ixs
11 - 1/2
111
1--2 P2
(29)
1
Ag
1xx 1xf
111 - 1/2
1 P2
Let,
#1 + (J 2)112
μ22
V1 (p) - n22 (wi + (f-1)pp2 1)p2))
1/2
(MI p2) (w1 + (s 1)
(30)
then, B- (v1 122 Lxxs + 12 (1xx1 1ixs) in that case, we get:
B- 1| = fuj + (
(31)
19

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
We get the ranges of of and vz.
11 + (f (f)22p
1
v
)25
(11 - 112) (111
+ + -1)112)
(11 122
25
S 221 2/0?
(1 (1+c c 22 2--2-- 0²
(32)
1/2 (-
42
)2
(11 (f 1) 22/02
(F1 (22 + (d 1)a2)
20
2 2222200
(1+ 1e5- ने 1)²
Thus, the upper bound of B-1 1| is:
B 1| S v fe2 + ( f) f-2 < (f
1f 2/02
(33)
The second step presents the specific form of the C matrix.
1 p
1 Σ, P2:1222 212
Ex p2 22f
T
m Ee P (212)²
71 1
1 E R221
m E, Rif
C =
m
(34)
1 m Σ, 25
Σ, 2:
1 E, E.p(Zer) (zap)
m
each element the C matrix less than or equal eL, then we have ||C|| E pm 4
Eventually, for the matrix E, which a X matrix where each element is L have E
B.2. Iterative optimal solution A
By the banach fixed point theorem, there exists self-mapping of A'T denoted as T(ATT) or T(ATT), ), which satisfies the conditions of the theorem.
| (ATT) - T (A2
|| - B
C
B
AT
-AT l|
|||||
0, pf
| T
(35)
T(-) E [0, 2 E (0, 1pmxf
| 1 (f 2
S
1, the condition on the condition on p is: p < 2 + 3 b This inequality guarantees that the product of the Frobenius norms 1| p. [[ and E is less than 1.
To ensure that |B- ||C|||||||
So, Assumption 3.1 holds and Z follows the (0.1) distribution, ignoring the regularization constraints, the condition for convergence is given by: P 2 (f + 1 fm² - d there exists an iterative fixed point A" that minimizes the loss.
Let Z II I F(h(Z))) + TT u1, u1,xx = 0, then,
AT Z ITZ") (Z (IF((2))) ZT (Tuix)), s.t. Z'A'T 1,x1 = 1mx11
(36)
B.3. Proof of convergence
Matrix Hoeffding's 's Inequality: Let M, be independent, random symmetric matrix of dimension क. Let V; represent sequence of fixed symmetric matrices, assume that each random matrix satisfies the following conditions:
E [&] and 2 x V? almost surely.
(37)
20

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
For any / 2 0,
-t2
P
Amax
Σ
Mi
2
S
+
exp
(38)
80
where o² o=:== -> V?2 Vollop Let 12 such that |/a/| we have
(Min)} E (Mi)j. un 1 [Mii) ||||| s V
(39)
This leads ||aaa|| S Φ, and for any we have Millop S ④. Therefore, if choose V; -LI,, we obtain
m
12
Amax
Σ
M2
2 < exp
(40)
8m
by exchanging V and M,, we have M, < Vi, which implies M2 & V?? Assuming that each element of M; satisfies (Mi),a E {-11, 1J. we can then conclude that for 2 O,
1
mt?
P
Mi
2t
S 24 exp
(41)
8d2
op
Then, for each element IIZZ"
belongs [0, 1], and similarly, for each element of the weight matrix [0, 1]. the same holds for each element of A. We define,
M: 11zZ A
(42)
in this case, the assumption is satisfied, and since $ = + + 1, we have m 2i=1
Mi
Â A. Therefore, 17 2 0,
-mt2 320²
P P(1
A llop 2 I 4Φ cxp
(43)
Since each element of Am A is within the range
[- 1 1]. for Vt 0, we obtain
mll2
A m
2 2f exp
(44)
8
similarly, since Im I is also in matrix form, with each element the rangel
u, 1 ul [-11,, 1], for Vt 2 0, we have
P P([T - F|| 21) s 2f exp
mt?
(45)
8f
let - f 7e m 25+2 + 1)5z 3 -
In
|f
we obtain
PP || A-1 r r
8
(46)
2
similarly, 1/ = 22=1--1-2r- de, m2 = firre
(4)
we obtain
P -
T
>
c) 2
5
8
(47)
F
therefore, when m = max(m), m2), we have
(1 A T
T m
AT
se)
2
(48)
21

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
It is worth noting that the theorem provides default statement for the theoretical realization of LIMEFLDI. The key distinction is that regularization terms are not considered. Additionally, Theorem 3.3 holds under fairly mild assumptions about the model F' to be explained. Essentially, we only require that F is bounded on the bins. If these bins are computed from a finite training set X, then Z is compact. We essentially require that F is well-defined on Z. This condition is almost always satisfied for most machine learning models.
Limitations. The strongest limitation of Theorem 3.3 is the poor concentration for small bandwidths. In fact, for small o (i.e., a < 1). Theorem 3.3 requires the use of n-degree polynomials in F to achieve convergence. This often means that, once the dimensionality exceeds 10, applying Theorem 3.3 in practice becomes very challenging for 1. In such cases, the effects of regularization start play key role. However, as long as f > 10 and the default bandwidth, the performance of Theorem 3.3 in LIMEFLDL remains highly satisfactory. On the other hand, as o - 0 it can be directly shown that A + 0 for any 1 S < f. In the extremes, the behavior of the interpretable coefficients is not straightforward. In particular, the sign of the interpretable coefficients can change as the bandwidth varies. This is a concerning phenomenon: for certain values of the bandwidth o, the explanations provided by LIMEFLDL can become zero.
B.4. Proof of stability
For any two similar black-box models P and Q, define (2)) = (P element (h( < E. To compare two similar black-box models, we first obtain their explanatory feature attribution matrices, A'p, A Then, we calculate the distance between these matrices,
- Q (Z)) and T (M()) of each
1A/ A/ F
(z (Z II') (Z" IT((()))
(49)
A || T IIT(A(())III
where ه| -1, S <f +ff 1) and that each element of [I(2)( is mosse, where the expected value of π= is
less than 2. we can conclude that when A [0. (0.m211(((111 / fr J. the following holds:|| Alp A F S
B.5. Proof of property
From a more practical perspective, considering the properties of additive feature attribution, 1et us assume that our knowledge of F is imperfect. More precisely, we can decompose the function into parts for explanation, the part originating from the black-box model F itself, and the part arising from perturbations.
Let AT represent the obtained interpretation, Exr be the feature originally located at the k-th position. By replacing it with Euj and ensuring that the output of the interpreted model remains consistent, we obtain the following:
(51, Sa ... . AT =E ..3...-6 Eug Eg)) AT
..j..-Ef)
(50)
based on this, we can obtain the following:
EKTaRI EkgaRs, Vt E (1,2,...,T).
(51)
where age is the explained value of the t-th label for the k-th feature in the explained feature attribution matrix. As seen above, the k-th row of A" all zeros, This means that the feature attribution for all labels of the k-th feature are This indicates the process of feature selection. For unselected features, which were not involved in training the local linear model, their feature attributions are all 0. We randomly select two features, &, and Ej. from the interpretable representation &, ensuring that & + Ej. We then exchange these two features to create new interpretable representation, ç'. Meanwhile, we ensure that the outputs of the interpretation models remain consistent across the different interpretable representations. This results in:
Eo.... (G.. EG.... AT =(Ei,E2,.. Ey. Eo. E.....E)) EE)) AT
(52)
Based on this, we can obtain
Eua + Ejaja Eaazk + Ejoas, Vk E (1,2,...,
(53)
based on this random exchange, we can deduce that the i-th and j-th rows of the feature attribution matrix A' are identical.
22

---

LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning
B.6. Probabilistic proof of the possibility of negative KL scatter
The KL divergence between two discrete probability distributions and θ, defined over the same probability space S2 given by:
((r)
DKL ( = Σ (x) log
(54)
(x))
for continuous probability distributions with density functions (x) and ((z), the KL divergence is defined as:
(KL(V = -f (x)x) log
(x))
dx,
(55)
our goal is to show that DKL (V] 2 constant when the sum of the probabilities of the predicted distribution is unity, 1.
However, (RI(())))) can less than 0 if the sum the probabilities the predicted distribution exceeds
Consider the function (t) = log t, which is convex for > 0, this follows from its second derivative:
6 (t)
(56)
for a random variable Y and convex function 3, Jensen's 's inequality states that:
B[B(Y)] 2 ((E())),
(57)
where denote the expected value. We define Y(x)= = VØz) points where (z) = 0 do not contribute to the expected value, we can disregard them. Now, we proceed by computing Ev [3(Y))
(()) When V(()) = 0, Y(x) is undefined. However, since the
((x)
Ey [3()] = Σ ((x)
(x)
xEQ
(xr)
Σ (x)
- log
(58)
xEG
Vxx))
(x) log
O()))
sEf
DRL, (O))
To compute 8 [Y))))) [Y]),
((xz)
Ey Y = Σ (x)
Σ (z)),
(59)
(Tr
xE
ZE
therefore,
8 (Ev 3( > (())) = log(
> (())).
(60)
zg
According to Jensen's 's inequality:
(V/1)) =FooB(() (E+(V]) = log Σ (x)))
(61)
xE8
When Even (xx) = 1, we have,
log( ((x)) log((1 0.
(62)
However, when Exen O(x) 1, - logoooooceeooo (z)) < 0. Consequently, the KL divergence DKL (KL (V/ or equal to a number less than O. Therefore, there is probability where the KL divergence can be negative. By dividing according to the interval, we find that this probability is given by,
is greater than
F(O(z)) Σ, O(x) - Ex0(z)
(63)
23

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LIMEFLDL: A Local Interpretable Model-Agostic Explanations Approach for Label Distribution Learning
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a Original image from the Emotionó dataset.
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Figure II. Processing of additive attribution of image data.
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Reviews:
[{"review_id": "S7bPfPoJx6", "reviewer": "Reviewer_3Lj7", "summary": "To address the local interpretability issues of label distribution learning (LDL), this paper proposes an improved LIME algorithm, namely LIMEFLDL. The algorithm is mainly manifested in three aspects: first, by introducing a feature attribution matrix to address the label dependency issue in LDL tasks; second, by minimizing the output differences between the black-box model and the explanation model within the generated local region to reduce computational complexity; third, by incorporating linear constraints and penalty functions to ensure that the predictions of the explanation model align with the distribution of the labels. In addition to the above, the article provides extensive theoretical proofs regarding the stability and convergence of the algorithm, as well as its analytical solution form. The effectiveness of the algorithm is demonstrated through multiple experiments and human experiments.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed review of our paper; we greatly appreciate your comments and questions.   \n[Comment 1]About the errors and unclear mathematical symbols.     \nA: \n1. Equation 23 should be modified: $\\sum_{k=0}^{f} e^{\\frac{(k-f)}{\\sigma^{2}}}\\frac{k}{f} \\frac{f!}{2^{f}(f-k)!k!}$, this equation is to represent the probability of picking $k$ non-zero elements from $f$ elements, $\\frac{f!}{(f-k)!k!}$ is the number of combinations to select $k$ from $f$ elements, $2^{f}$ is the total number of combinations. The same error is found in Eq.24. We will correct them in the revised version.\n2. What we want to describe is that if $\\sum_{x \\in \\Omega} \\Theta(x)>1$, Jensen's inequality (Eq.61) implies $E(KL)\\ge-\\log (\\sum_{x \\in \\Omega} \\Theta(x))<0$. Thus, if $KL\\in (-\\log (\\sum_{x \\in \\Omega} \\Theta(x)),0)$ when distributions violate normalization, the bound's numerator ($\\sum_{x}\\Theta(x)-1$) and denominator ($\\sum_{x } \\Theta(x)$) directly quantify this violation.\n3. In Eq.49, we replaced the original symbol $\\Sigma$ with $\\Delta$ to avoid ambiguity with the concatenation operator $\\Sigma$ used earlier. We'll correct it in the revised version.\n4. It should be a $v$-vector instead of $u$, we will correct it in the revised version.\n   \n[Comment 2]The captions for figures and tables in the paper are not sufficiently detailed.  \nA: We will add more details to describe the figures and tables in the revised version. Fig1a: x-axis: exponentially increasing samples (log scale); Fig1b: x-axis: decreasing divergence between two black-box models, both graphs of y-axis is the Top 20 Jaccard Index. Fig1a demonstrates convergence as Jaccard Index $\\uparrow$ with exponentially increasing samples (log-scale x-axis), thus illustrating the convergence of the interpretation algorithm. Fig1b shows Jaccard Index $\\uparrow$ when predictive distribution divergence between black-box models $\\downarrow$ (x-axis), thus illustrating the stability of the interpretation algorithm.    \n\n[Comment 3]About the statistical information in the comparison results between LIMEFLDL and PULIME.  \nA: We will give ranking statistics in the revised version, as follows,  \n|Model|Algorithm|Cheb.|Clark|Canb.|KL|Cos.|Inter.|Jac.|\n|-|-|-|-|-|-|-|-|-|\n|RBF-LDL-LRR|LIME|1.75|1.5|1.625|2|1.625|1.625|1.875|\n||LIMEFLDL|1.25|1.5|1.375|1|1.375|1.375|1.125|\n|LDL-SCL|LIME|1.625|1.5|1.5|1.875|1.5|1.5|1.625|\n||LIMEFLDL|1.375|1.5|1.5|1.125|1.5|1.5|1.375|\n|AA-KNN|LIME|1.625|1.625|1.625|1.5|1.625|1.625|1.75|\n||LIMEFLDL|1.375|1.375|1.375|1.5|1.375|1.375|1.25|\n|MEM|LIME|1.75|1.75|1.75|1.875|1.75|1.75|1.75|\n||LIMEFLDL|1.25|1.25|1.25|1.125|1.25|1.25|1.25|\n\nLIMEFLDL leads the rankings for the vast majority of measures, we will add this section in the revised version as space permits.  \n\n[Comment 4]About the abbreviation RBF-LDL-LRR, and the text in Figure 10.   \nA:We enhanced the LDL-LRR model's feature extraction via Gaussian kernel (renamed RBF-LDL-LRR). Figure 10 tests robustness by masking weakest features, measuring fidelity before/after masking. We will add the name of the modified model and the description to the revised version to make it more accessible.  \n\n[Comment 5]About the introduction of the label dependency problem and the initial value for the feature attribution matrix A.  \nA:We will clarify label dependence fundamentals in the revised version: LDL handles multi-label samples where labels exhibit co-occurrence patterns (simultaneous changes) and dependency propagation (one label's presence affects others' distributions). Matrix $A $ uses uniform initialization as initialization.  \n\n[Comment 6]About the computational complexity of the LIMEFLDL algorithm compared to PULIME.  \nA: We analyze the computational complexity of LIMEFLDL and PULIME. For LIMEFLDL, sampling and forming weight matrix $\\Pi$ requires $O(m)$. Matrix multiplication contributes $O(mfr)$, and black-box model inference accounts for $O(mrk)$. Element-wise matrix subtraction and F-paradigm operations each add $O(mr)$, with regularization terms contributing $O(fr)$. The total complexity is $O(mfr+mrk+mr+mr+fr+m)$. PULIME's per-label workflow involves sampling ($O(m)$), vector operations ($O(m)$ for subtraction/squaring and $O(mf)$ for multiplication), black-box inference ($O(mrk)$), and regularization ($O(f)$). With parallel computation, its total complexity is $O(mrf+mkr^2+mr+mr+mr+rf)$. PULIME's ridge regression solver has $O(f^3+mf^2)$ complexity, LIMEFLDL uses L-BFGS optimization with $O(T(zr+mf))$ cost, where $z≈5–20$ (stored gradient pairs) and $T$ denotes iterations.  \n\n[Comment 7]About the Y_mean in Equation 8, and a class-related prior probability distribution.  \nA: The idea of using a uniform distribution is to allow the current feature selection to go beyond the effect of a uniform distribution, but of course it is possible to make different prior distributions for specific datasets so that the selected features are more representative.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "1.For Equation 23 and 24, in the third equation, where does the 2 to the power of f come from?\n\n2.For Equation 64, does this expression satisfy the probability constraint, i.e., the sum of probabilities equals 1? Please provide a more detailed explanation.\n\n3.For Equation 49, the meaning of the sigma symbol has not been explained. Please check if it is written correctly.\n\n4.For Equation 39, there is a symbol error: the symbol u seems to be incorrect.", "experimental_designs_or_analyses": "Yes. I check the soundness and validity of all experimental designs and analyses.", "relation_to_broader_scientific_literature": "This paper proposes an improved LIME[1] algorithm, namely LIMEFLDL, to address the local interpretability issues of label distribution learning. The LIME algorithm is designed for single-label learning tasks. However, due to the label dependency in label distribution learning (LDL), LIME is unsuitable for direct application in LDL tasks. This paper introduces the feature attribution distribution matrix to address this issue.\n\n[1] Ribeiro M T, Singh S, Guestrin C. \" Why should i trust you?\" Explaining the predictions of any classifier[C]//Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. 2016: 1135-1144.", "essential_references_not_discussed": "There is no any essential references not discussed.", "other_strengths_and_weaknesses": "Strengths:\n\n1.The authors provide proofs regarding the stability and convergence of the explanation algorithm, thereby increasing the credibility and correctness of the algorithm.\n\n2.The authors compare the LIMEFLDL and PULIME algorithms on 8 datasets across 7 metrics, fully demonstrating the effectiveness of the proposed method.\n\nWeakness:\n\n1.There are some errors and unclear elements regarding mathematical symbols in the paper, which are detailed in the Theoretical Claims section.\n\n2.The captions for figures and tables in the paper are not sufficiently detailed. For example, in Figure 1, the legends are not explained in caption.", "other_comments_or_suggestions": "1.It is recommended to include statistical information in the comparison results between LIMEFLDL and PULIME, such as how many datasets showed better performance of LIMEFLDL compared to PULIME under the KL metric.\n\n2.In Table 1, what does the abbreviation RBF-LDL-LRR stand for? It is not found in the original text. Additionally, the text in Figure 10 is not clear.\n\n3.In the abstract, on the 22nd line, \"To address the label dependency problem,\" the introduction of the label dependency problem is not sufficiently clear and can be confusing. Adding some rationale would improve this.\nRegarding the initial value for the feature attribution matrix A, these are not found in the paper. This information could be added for completeness.", "questions_for_authors": "1.The Introduction mentions the computational complexity issues of the LIMEFLDL algorithm compared to parallel use of the traditional LIME algorithm (PULIME). Is there any theoretical or experimental proof to support this?\n\n2.Regarding the Y_mean in Equation 8, could it be influenced by the class distribution? It might be worth trying to adopt a class-related prior probability distribution.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kxCT2OKw3a", "reviewer": "Reviewer_UXYg", "summary": "In order to mitigate the interpretability challenge inherent in most label distribution learning (LDL) algorithms when applied to risk-sensitive decision-making scenarios, this paper introduces a novel local interpretable model-agnostic explanation framework specifically tailored for LDL. This approach takes into account the label distribution within the local region and constructs local linear models to effectively approximate the global behavior of the black-box LDL model. Furthermore, the paper conducts a thorough theoretical analysis of the proposed methodology, offering a theoretical assurance that the interpretations it yields in the context of LDL tasks are closely aligned with the actual decision-making process.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive review of our paper; we greatly appreciate your comments and questions.   \n\n[Comment 1] This paper also has some limitations. For example, the writing of this paper needs improvement, and the experimental results in appendix need more discussions.  \nA: We will follow up by scrutinizing the paper for writing issues, as well as discussing the experimental results in the appendix in more detail.\n\n[Comment 2] First, the word “Explanations” in the title “A Local Interpretable Model-Agnostic Explanations Approach” should be amended for accuracy. Second, the quotation marks on line 566 should be used correctly.     \nA: \"Explanations\" will be changed to \"Explanation\", the quotes used in line 566 are incorrect and that these essay writing will be corrected in the revised version.\n\n[Comment 3] In Figure 6, what are the differences between the original image and the processed image? Visually, they appear almost same. Besides, are there detailed discussions for the visualized cases to demonstrate the performance of LIMEFLDL?             \nA: The processed image in Fig.6 was changed by smudging and adding a green border change the lowest ranked hyperpixel block, which is not conspicuous enough because of its low rank and is often located in the edge region of the image. For both the image dataset and the table dataset we have done a visual comparison of the LIMEFLDL and the original LDL interpretation results, which we will show in the revised version.", "claims_and_evidence": "The claims made in the submission  are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The integration of label distribution constraints into LIME’s framework is novel. The feature attribution matrix and ADMM-based optimization are appropriate.\nEvaluation: Fidelity metrics and consistency are well-chosen.", "theoretical_claims": "I have checked the correctness of Theorem 3.2, Theorem 3.3, Theorem 3.4, Property 3.5, Property 3.6, and Property 3.7.", "experimental_designs_or_analyses": "I have checked the experiments. Specifically, I have examined the adequacy of the dataset in Section 4.1, the completeness of the evaluation metrics in Section 4.2, and the rationality of the experimental procedure in Section 4.3.", "relation_to_broader_scientific_literature": "The related literature is interpretable machine learning. The main contributions of this paper can be treated as an extension of traditional LIME (local interpretable model-agnostic explanations) method. Specifically, it adapts LIME to the $r$-dimensional label space with $r-1$ degrees of freedom.", "essential_references_not_discussed": "I did not find the essential missing references in this paper.", "other_strengths_and_weaknesses": "Strengths：\nThe problem addressed in this paper holds substantial significance, as label distribution learning algorithms are often complex and their decision-making correctness is challenging to validate in practical applications. This paper endeavors to resolve the interpretability issues inherent in label distribution learning, thereby significantly contributing to the expansion of the applicability of the label distribution learning paradigm. Furthermore, the adaptation of traditional machine learning explanation methods to the label distribution learning framework is non-trivial owing to the high-dimensional and interdependent label space. \n\nWeaknesses：\nThis paper also has some limitations. For example, the writing of this paper needs improvement, and the experimental results in appendix need more discussions.", "other_comments_or_suggestions": "First, the word “Explanations” in the title “A Local Interpretable Model-Agnostic Explanations Approach” should be amended for accuracy. Second, the quotation marks on line 566 should be used correctly.", "questions_for_authors": "In Figure 6, what are the differences between the original image and the processed image? Visually, they appear almost same. Besides, are there detailed discussions for the visualized cases to demonstrate the performance of LIMEFLDL?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nKHnAIoVDj", "reviewer": "Reviewer_EsHf", "summary": "This paper proposes an interpretability model for label distribution learning (LDL). The classification LIME approach is adapted to handle LDL. An optimization objective is proposed to estimate the parameters of the interpretability model. Theoretical analysis is performed, including convergence, stability, and some theoretical properties. Experimental results show that the proposed method can outperform PULIME, a parallel application of LIME.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive review of our paper.  \n[Comment 1]About the structure of the proofs.  \nA:We will restructure proofs in the revised version. The proof of Theorem 3.2 is appeared in Appendix B.2 (lines 1025~1040).\n \n[Comment 2]The method's novelty appears limited as it resembles a multi-label LIME extension. While targeting LDL (multi-label), Eq.4 decomposes into per-label subproblems akin to LIME's single-label optimization.  \nA:Equation 4 can be decomposed into each row for single-label LIME, which means that our method is a generalized version of LIME. However, combining multiple independent LIME cannot directly give rise to our method. The reasons can be summarized as follows. \n1. Label dependency. Interpretation of LDL requires handling label dependency, whereas the direct combination of multiple LIMEs cannot handle label dependency since per-label training produces disconnected weight vectors, and inability to consider the simultaneous impact of features on multiple labels. Our approach evaluates features’ impacts simultaneously across all labels from the weight matrices, revealing dependencies. \n2. Theoretical guarantees. We provide the following theoretical guarantees for our method, as the number of sampled examples increases, our interpretation results provably converge under the label distribution. When the black box model turbulence is smaller, the more stable the interpretation results are, however, these guarantees do not hold for applying multiple single-label LIMEs to approximate label distribution. \n3. Feature selection. Interpretation of LDL requires consideration of global representational features, and multi-LIME fails to achieve due to its single-label focus (label-specific features). Our method uses distribution information to prioritize features globally relevant features.\n\n[Comment 3]LDL metrics may not directly assess interpretability quality.  \nA:We conducted supplementary experiments using R²-score aligned with LIME metric and expanded baselines to GLIME and DLIME, all evaluated under default parameters.\n|Model|Datasets|LIME|DLIME|GLIME|LIMEFLDL|\n|-|-|-|-|-|-|\n||1|.16|.15|.16|.67|\n||2|.15|.15|.16|.67|\n||3|.48|.35|.34|.58|\n||4|.39|.33|.34|.40|\n|LDL-SCL|5|.34|.25|.25|.30|\n||6|.37|.34|.32|.38|\n||7|.94|-|.95|.99|\n||8|.89|-|.89|.98|\n||1|.17|.17|.16|.99|\n||2|.16|.15|.17|.99|\n||3|.34|.33|.33|.36|\n||4|.25|.31|.29|.36|\n|RBF-LDL-LRR|5|.26|.47|.36|1|\n||6|.52|.49|.47|.99|\n||7|.93|-|.96|.99|\n||8|.67|-|.94 |.99|\n\nThese experiments will be included in revision (space permitting).\n\n[Comment 4]About the decisions are objective and influenced by the choice of participants.  \nA: While single-study generalizability is limited, our design mitigated bias via: random recruitment (ML/non-ML backgrounds), independent evaluation of 20 randomized interpretation results, and high inter-rater agreement despite participant diversity.\n\n[Comment 5]About how label dependencies are accounted for.  \nA:Label dependencies directly influence the equilibrium states of the parameters $u$ and $\\rho$. These parameters jointly govern the weight matrix $A$, adjusting its entries to reflect label correlations, this ensures global consistency, features impacting correlated labels exhibit coordinated weight changes in $A$, perturbing a feature’s weight reveals its systemic effect on the entire label distribution (e.g., simultaneous probability shifts for co-dependent labels).\n\n[Comment 6]About some writing errors.  \nA:We will change \"Explanations\" to \"Explanation\" and correct the problem with the two periods in the revised version.\n\n[Comment 7]About some inadequate or incorrect descriptions.  \nA:We will add intuition before Eq.9 to clarify its role in streamlining Eq.10 and simplifying proofs. Clarify perturbation analysis (lines 257-258): black-box model (BM) perturbations (models $P$ vs. $Q$) impact interpretations. $T(h(Z))$ quantifies output distribution divergence between BM (models $P$ vs. $Q$), while $t(h(Z))$ describes the difference in the degree of descriptiveness of single label of the model outputs. Interpretation stability $\\propto (\\text{BM perturbation magnitude})^{-1} $. We will describe this part in more detail in the revised version. Property 3.6 uses superscripts ($\\xi_{k }^i$, $\\xi_{k }^j$) for distinct values of feature $k$. Property 3.7 uses ($\\xi_{i}$, $\\xi_{j}$) to denote different features. We will correct this part in the revised version.\n\n[Comment 8]About the $\\pi$ function.  \nA:For interpretation, the example to be interpreted $x$ is mapped to an all-ones vector in the interpretable space. Weights are computed via similarity between $x$'s and sampled instances' representations, aligning with prior work.\n\n[Comment 9]About the purple color in Figure 3.  \nA:Fig.3 uses a color scheme: Purple (blue+pink blend) indicates LIME$\\approx $LIMEFLDL; blue for LIME$>$LIMEFLDL; pink for LIMEFLDL$>$LIME. Detailed analysis will be discussed in the revised version.", "claims_and_evidence": "The claim that the interpretability model for LDL is important is good. However, it is not clear how the proposed method differs from the classical LIME approach. Although multiple labels are considered for LDL and a single label is considered for LIME, I think they are very similar, since the optimization problem in Eq. (4) can be decomposed for each row of $\\boldsymbol{A}\\'$, which corresponds to an optimization problem for each label in LIME. In this way, it seems that the proposed method, which mainly works by solving the optimization problem in Eq. (4), is very similar to LIME. The differences are mainly in the different optimization techniques (L-BFGS) and the label distribution constraints (non-zero and normalization).", "methods_and_evaluation_criteria": "The proposed method is valid.", "theoretical_claims": "The theoretical claims are good and support the proposed method. However, the structure of the proofs is unclear, and the proof of Theorem 3.2 is missing.", "experimental_designs_or_analyses": "- In the first part of the experiments, the fidelity is calculated as several LDL metrics using model outputs from black-box models and the interpretability model. However, the main purpose of the interpretability model is to determine the importance of different features, as described in the previous LIME paper. Therefore, it is uncertain whether a better metric value indicates better interpretability model performance. Therefore, it is recommended that the previous evaluation procedure in the LIME paper can be adopted to validate the effectiveness of the proposed method.\n\n- The second part of the experiments includes human experiments. Human participants are asked to decide whether the explanations are good or bad. It is unclear whether the decisions are objective and influenced by the choice of participants, since only one experiment is conducted.\n\n- Only LIME is chosen as a baseline, which may be inadequate and outdated. More recent interpretability models should be considered for experimental comparisons.", "relation_to_broader_scientific_literature": "N/A.", "essential_references_not_discussed": "There is no essential reference that needs to be discussed.", "other_strengths_and_weaknesses": "Strengths:\n- The interpretability of LDL is a good but under explored topic in the literature. Therefore, the research problem is beneficial and interesting.\n\n- The theoretical analysis is good and validate the effectivness of the proposed method.\n\nWeaknesses:\n- The novelty of the proposed method seems limited, since the optimization problem is quite similar to a multiple-label version of LIME. Therefore, the novelty and contributions should be clarified in detail.\n\n- It is unclear whether the experiments in the paper can help validate the effectiveness of the proposed method. It can be observed that the interpretability model is more accurate in fitting the black-box model locally. However, it does not mean that feature importance is well described by the proposed method.\n\n- It is unclear how label dependencies are accounted for in the proposed method, as noted in the Introduction section.", "other_comments_or_suggestions": "The writing of the paper can be improved. Here are some minor points:\n- The title should read \"Explanation\" instead of \"Explanations\".\n\n- There are two periods in line 18-19.\n\n- It is quite confusing with the introduction of Eq. (9) without any descriptions.\n\n- The wording in lines 257-258 is confusing.\n\n- In Property 3.6 and 3.7, the notation of the $k$-th function is inconsistent.", "questions_for_authors": "- The $\\pi$ function is different from the previous LIME method. In LIME, the anchor is the selected example $z$. In this paper, however, it is an all-one vector instead. I wonder why the function is changed in the paper.\n\n- I am confused by Figure 3. I do not know the meaning of the purple color that should be specified in the paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Responsible Research Practice (e.g., IRB, documentation, research ethics, participant consent)"], "ethical_review_concerns": "Since an experiment involves human participants, the ethical statement should describe in detail how the privacy of the participants will be protected and whether the evaluation will be unbiased."}}, {"review_id": "qTRGu8LR4w", "reviewer": "Reviewer_9QRm", "summary": "Existing interpretability models are designed for single-label paradigm and struggle to directly interpret label distribution learning (LDL) models. To solve this, the paper proposes an improved LIME algorithm capable of effectively interpreting black-box models in LDL. The authors also provide analysis on analytical solution, convergence, stability, and algorithm properties.\n## update after rebuttal\nI have carefully read the rebuttal. The rebuttal answer the question that how the scoring function in Equation (8) is used to select features and revised some typos. Considering the novelty and the advantage of the proposed method. I keep my socre.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive review of our paper; we greatly appreciate your comments and questions.   \n[Comment 1] Steps 3, 4 in Algorithm 1 lack sufficient rigor.   \nA: We will introduce Steps 3, 4 of Algorithm 1 in more detail in the revised version. Step 3 of Algorithm 1 generates $m$ samples based on the examples to be interpreted $x$. For image data, the input image is segmented into superpixels, we then generate $m$ samples via binary masking: 0 replaces a superpixel with mean values (per channel), while 1 retains the original pixels, this process preserves local structure while enabling efficient sampling. For tabular data, we first bin features based on $x$’s values (1 if in the same bin as $x$, 0 otherwise), samples are generated by sampling from Gaussian distributions parameterized by training-set statistics (mean/variance per feature). Steps 4 is to calculate pairwise weights between these sampled examples to the example to be interpreted using Eq. 3 and Appendix A.2, ensuring locality-awareness.    \n\n[Comment 2] Steps 8 in Algorithm 1 lack sufficient rigor. Could the authors briefly explain how the scoring function in Equation (8) is used to select features, i.e., in Step 8 of Algorithm 1, and whether the selected features are related to the subsequent steps?        \nA: In Step 8, we perform iterative feature selection by progressively adding features and evaluating their impact using the scoring function in Eq.8. At each iteration, we train model using only the currently selected features and compare the new score with the current maximum. If the score exceeds the maximum, the feature is retained in the selected set; otherwise, it is discarded. Unselected features are excluded from subsequent training iterations. For validation, we analyze the top 10 ranked features by incrementally adding them and monitoring their influence on explanation fidelity in our local accuracy experiments. In the dummy feature experiment, we search for the current lowest ranked feature and perform masking operation to observe the effect on the interpretation results before and after the change. We will introduce this section in more detail in the revised version to make it easier to understand.       \n\n[Comment 3] The authors introduce some notations in Equation (9) for convenience, but why are these notations not consistently used in the subsequent paragraphs, such as in Equations (11) & (12)?      \nA: The introduction of symbols in Eq.12, specifically the reformulation of $Z^{\\prime \\top}{\\Pi}F(h(Z))-Z^{\\prime \\top} u 1_{1 \\times r}=\\Gamma$, was designed to simplify the algebraic representation of subsequent theorems and proofs. While the original notation $Z^{ \\prime \\top} \\Pi Z^{ \\prime}=\\Delta$ remains mathematically rigorous, retaining $B$ (instead of $\\Delta$) streamlines derivations by reducing nested terms. We acknowledge that this substitution introduces minor notational complexity in the proof flow, and in the revised manuscript, we will add a footnote linking the notations.           \n       \n[Comment 4] There are some typos in this paper.        \nA: In the revised manuscript, we will correct the following issues, Line 74 should be \"adapt to\" instead of \"adopt to\", Lines 117-118 should be $a_{i}\\in \\mathbb{R} ^{r\\times 1}$, and Lines 159-160 should be \"entropy\" instead of \"entory\". The $B$ matrix of Eq.10 is just a more detailed description of Eq.9, which we will consolidate these equations to explicitly show that matrix $B$.", "claims_and_evidence": "The authors claim that existing interpretability models face challenges in explaining LDL from three aspects: label dependency, computational complexity, and label distribution constraints. These claims are clear and well-supported by evidence.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense for the problem at hand.", "theoretical_claims": "I did not check the correctness of any proofs for the theoretical claims, but they appear to be reasonable.", "experimental_designs_or_analyses": "Yes, the experiments follow a classic comparison protocol, making them sound and valid.", "relation_to_broader_scientific_literature": "Currently, there are no other work on interpretable LDL, so I find this paper well-positioned. This work significantly advances the understanding and application of existing LDL models.", "essential_references_not_discussed": "The paper has thoroughly discussed appropriate related work.", "other_strengths_and_weaknesses": "Strengths:\n1. The proposed algorithm is able to interpret any black-box model in LDL.\n2. The authors provide an in-depth analysis of the proposed algorithm, including discussions on its analytical solutions, convergence, stability, and overall properties.\n3. The paper is clear and has intuitive presentation of the analysis and experiments. For example, the visualization of dummy variable analysis helps in understanding the model’s behavior. Additionally, the introduction of the Jaccard index effectively demonstrates the superiority of the proposed algorithm.\n\nWeakness:\n1.\tSteps 3, 4 & 8 in Algorithm 1 lack sufficient rigor.\n2.\tThe authors introduce some notations in Equation (9) for convenience, but why are these notations not consistently used in the subsequent paragraphs, such as in Equations (11) & (12)?\n3.\tThere are some typos in this paper, see the next part.", "other_comments_or_suggestions": "Steps 3, 4 & 8 in Algorithm 1 lack sufficient rigor.\n\nTypos:\n1. Line 74: adapt to;\n2. Line 117-118: $\\mathbb{R}^{r \\times 1}$?\n3.  Line 159-160: entropy?\n4.  Duplicate definition of $\\boldsymbol{B}$ in Equations (9) & (10).", "questions_for_authors": "1. Could the authors briefly explain how the scoring function in Equation (8) is used to select features, i.e., in Step 8 of Algorithm 1, and whether the selected features are related to the subsequent steps?\n2.  The authors introduce some notations in Equation (9) for convenience, but why are these notations not consistently used in the subsequent paragraphs, such as in Equations (11) & (12)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission worked on interpretability of label distribution learning and proposed an extension of Local Interpretable Model-agnostic Explanations that can effectively interpret any black-box model in label distribution learning. All the four reviewers consistently agreed it is a good paper and should be accepted. There were some concerns about the human experiments, because human experiment results were reported but essential details were missing (including participant recruitment process, participant population description, experimental protocol details, and participant protections). I have checked the ethics review and rebuttal by myself, and I think the authors have addressed the issues. Therefore, we should accept this submission to be published at ICML 2025.

PS: The authors should give the full name of LIME the first time it appears in the abstract, since the abstract is non-technical and targets at a very broad readership.
Decision: Accept (poster)