LIMEFLDL: A Local Interpretable Model-Agnostic Explanations Approach for Label Distribution Learning

Thank you for your positive review of our paper; we greatly appreciate your comments and questions.
[Comment 1] Steps 3, 4 in Algorithm 1 lack sufficient rigor.
A: 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.

[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?
A: 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.

[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)?
A: 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.

[Comment 4] There are some typos in this paper.
A: 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 $B$.

Thank you for your constructive review of our paper.
[Comment 1]About the structure of the proofs.
A:We will restructure proofs in the revised version. The proof of Theorem 3.2 is appeared in Appendix B.2 (lines 1025~1040).

[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.
A: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.

1. 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.
2. 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.
3. 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.

[Comment 3]LDL metrics may not directly assess interpretability quality.
A:We conducted supplementary experiments using R²-score aligned with LIME metric and expanded baselines to GLIME and DLIME, all evaluated under default parameters.

These experiments will be included in revision (space permitting).

[Comment 4]About the decisions are objective and influenced by the choice of participants.
A: 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.

[Comment 5]About how label dependencies are accounted for.
A: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).

[Comment 6]About some writing errors.
A:We will change "Explanations" to "Explanation" and correct the problem with the two periods in the revised version.

[Comment 7]About some inadequate or incorrect descriptions.
A: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.

[Comment 8]About the $\pi$ function.
A: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.

[Comment 9]About the purple color in Figure 3.
A: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.

Thank you for your positive review of our paper; we greatly appreciate your comments and questions.

[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.
A: We will follow up by scrutinizing the paper for writing issues, as well as discussing the experimental results in the appendix in more detail.

[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.
A: "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.

[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?
A: 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.

Thank you for your detailed review of our paper; we greatly appreciate your comments and questions.
[Comment 1]About the errors and unclear mathematical symbols.
A:

1. 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.
2. 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.
3. 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.
4. It should be a $v$-vector instead of $u$, we will correct it in the revised version.

[Comment 2]The captions for figures and tables in the paper are not sufficiently detailed.
A: 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.

[Comment 3]About the statistical information in the comparison results between LIMEFLDL and PULIME.
A: We will give ranking statistics in the revised version, as follows,

LIMEFLDL leads the rankings for the vast majority of measures, we will add this section in the revised version as space permits.

[Comment 4]About the abbreviation RBF-LDL-LRR, and the text in Figure 10.
A: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.

[Comment 5]About the introduction of the label dependency problem and the initial value for the feature attribution matrix A.
A: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.

[Comment 6]About the computational complexity of the LIMEFLDL algorithm compared to PULIME.
A: 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.

[Comment 7]About the Y_mean in Equation 8, and a class-related prior probability distribution.
A: 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.