In the response, we provde three theoretical guarantees for the prosoed method. We will add them to the final version of our paper. First, we prove the existence and uniqueness of the optimal solution for our entropy-regularized optimal transport (OT) problem. Second, we demonstrate that the Sinkhorn-Knopp algorithm, which we employ, is guaranteed to converge to this unique solution. Finally, we explain how our innovative LLP-Proportion Penalty, incorporated into the cost function, guides the pseudo-label distribution to align with the true LLP prior.

1. Existence and Uniqueness of the Optimal Solution We first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The problem is defined as

$$min_{T in U(a, b)} langle C, T rangle - gamma H(T),$$

where $langle C, T rangle = sum_{k,i} C_{k,i} T_{k,i}$ is the transport cost, $H(T) = -sum_{k,i} T_{k,i} (log T_{k,i} - 1)$ is the entropy term, $gamma 0$ is the regularization strength, and $U(a, b)$ is the transport polytope defined by marginal constraints. Proof Let $F(T) = langle C, T rangle - gamma H(T)$. Then we will prove the strict convexity of the object function. For the term $langle C, T rangle$, it is linear in $T$, hence convex. For the entropy term $-H(T) = sum_{k,i} T_{k,i} (log T_{k,i} - 1)$, we can compute out its Hessian matrix

$$frac{partial^2 (-H)}{partial T_{k,i}^2} = frac{1}{T_{k,i}} 0 quad text{since } T_{k,i} 0,$$

implying strict convexity of $-H(T)$. Therefore, $F(T) = langle C, T rangle - gamma H(T)$ is the sum of a convex and strictly convex function, and thus strictly convex. Meanwhile, for $U(a,b)= left{ T in mathbb{R}_{geq 0}^{K times N} , middle , sum_i T_{k,i} = a_k, sum_k T_{k,i} = b_i right }$, we will prove it to be convex, closed and bounded. As $U(a,b)$ is defined by linear equalities and nonnegativity constraints, it is a convex polytope. Because all constraints are equation or non-strict non-equation, $U(a,b)$ is closed. Since $sum a_k = sum b_i = 1$, we have$0 le T_{k,i} le 1$, and $U(a,b)$ is a compact convex set. Finally, by standard results in convex optimization, a strictly convex function over a non-empty compact convex set has a unique global minimizer T^.

2. Convergence of the Sinkhorn-Knopp Algorithm Next, we prove that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution T^. The optimal transport plan T∗ is known to have a specific structure

T^_{k,i} = u_k cdot K_{k,i} cdot v_i, quad text{where } K_{k,i} = expleft(-frac{C_{k,i}}{gamma}right)

The Sinkhorn-Knopp algorithm is an iterative procedure to find the scaling vectors $u in mathbb{R}^K$ and $v in mathbb{R}^N$ that satisfy the marginal constraints. Proof： The convergence can be shown by interpreting the algorithm as an alternating projection procedure using the Kullback-Leibler (KL) divergence. Let the sets of matrices satisfying the row and column constraints be

• $mathcal{C}_1 = { T in mathbb{R}_{geq 0}^{K times N} mid sum_i T_{k,i} = a_k }$ (Row constraints)
• mathcal{C}_2 = { T in mathbb{R}_{geq 0}^{K times N} mid sum_k T_{k,i} = b_i (Column constraints) Each iteration of the Sinkhorn-Knopp algorithm can be viewed as performing the following alternating KL projections

begin{aligned} T^{(l+frac{1}{2})} &= argmin_{T in mathcal{C}_1} mathrm{KL}(T parallel T^{(l)}), T^{(l+1)} &= argmin_{T in mathcal{C}_2} mathrm{KL}(T parallel T^{(l+frac{1}{2})}), end{aligned}

where the KL divergence is a specific type of Bregman divergence. Since $mathcal{C}_1$ and $mathcal{C}_2$ are convex sets, the theory of alternating Bregman projections guarantees that this iterative process converges [1]. The intersection $mathcal{C}_1 cap mathcal{C}_2 =U(a,b)$ is non-empty and contains the unique optimal solution T∗. Therefore, the sequence of iterates T(l) produced by the Sinkhorn-Knopp algorithm is guaranteed to converge to the unique optimal solution T^.

3. Role of the LLP-Proportion Penalty Finally, we provide the intuition for why our innovative LLP-Proportion Penalty helps align the pseudo-label distribution with the true LLP prior. Our effective cost function is

$$C_{text{eff}}(c_k, z_i) = d(c_k, z_i) + lambda_{text{LLP}}(1 - p(k mid B_j(i)))$$

The key component is the penalty term. To understand its effect, we analyze the part of the objective function corresponding to this penalty, which we denote $mathcal{L}_{text{LLP}}(T)$

$$mathcal{L}_{text{LLP}}(T) = sum_j sum_{i in B_j} sum_k T_{k,i} (1 - p(k mid B_j)).$$

Minimizing this term is equivalent to maximizing the alignment between the transport plan's implied label proportions and the true bag proportions. By defining the transport plan's label distribution in bag $B_j$ as $P_T(k mid B_j)$, minimizing $mathcal{L}_{text{LLP}}(T)$ becomes equivalent to maximizing the sum of dot products between the predicted and true label distributions within each bag

$$max_{T} sum_j langle P_T(cdot mid B_j), p(cdot mid B_j) rangle.$$

Maximizing this dot product encourages the distribution $P_T$ to align with the true prior $p$, which is analogous to minimizing statistical divergences like the KL divergence. Therefore, by introducing the LLP-Proportion Penalty into the cost matrix, we directly enforce that the optimal transport plan T^—the unique solution to the optimization problem—favors assignments that are consistent with the known bag-level supervision.

Reference [1] Byrne, Charles L. Alternating minimization and alternating projection algorithms A tutorial.

Limited experimental scope on modality (W2 & Q1)

This is a very valid point. To address this limitation and demonstrate the broader applicability of our method, we have conducted new experiments on tabular data. Specifically, we evaluated RLPL on the UCI Adult dataset and the LLP-Bench [1] dataset as suggested. Furthermore, to explore bag generation beyond random sampling, we constructed feature-based bags on the UCI Adult dataset, where bags are formed by clustering instances in the feature space. Our method consistently achieves strong performance in these new settings, showcasing its versatility.

We performed experiments on the UCI Adult dataset, a commonly used benchmark in tabular data, and compared our method with two existing methods, LLP-AHIL and ROT. The experimental results are summarized in the table below:

From the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods.

Furthermore, we also tested our method on the LLP-Bench [1] dataset, where we achieved an accuracy of 66.85% on the Criteo dataset. This result further confirms that our method is capable of handling tabular data effectively, demonstrating its broad applicability beyond image-based modalities.

These results reinforce the robustness of our method across different non-vision modalities. We believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method.

Performance on large vs. small bag sizes (Q2)

This is a very insightful question. Our hypothesis is that the performance gap correlates with the degree of label ambiguity inherent in different bag sizes.

• With smaller bags, the label proportions provide a relatively strong and less ambiguous supervisory signal. In this setting, even simpler methods can perform reasonably well, leading to a smaller performance gap between different approaches.
• With larger bags, the label ambiguity increases significantly, as a single proportion vector can correspond to a vast number of instance-level label configurations. This is where the robustness of our pipeline, particularly the LLP-OTD refinement process, becomes critical. Our method's ability to effectively denoise pseudo-labels from a highly ambiguous signal allows it to excel and outperform baselines more significantly in these challenging scenarios. We will include this analysis in the final version of the paper to further explain the relationship between bag size and performance in different settings.

Number of OT passes in LLP-OTD (Q3)

Thank you for this excellent question. We investigated the effect of the number of OT passes, and the results on CIFAR-10 (bag size=256) are as follows:

Concerns about the computational complexity aof the Optimal Transport module (W1)

We thank the reviewer for raising this important point. We provide a detailed complexity analysis and empirical runtime measurements to demonstrate the practicality of our approach.

Complexity Analysis:
In our implementation, we solve the entropy-regularized OT problem using the highly efficient Sinkhorn-Knopp algorithm. The complexity is determined by two main steps:

1. Constructing the $K \times N$ cost matrix, which takes $O(KNd)$ time, where $K$ is the number of classes, $N$ is the number of instances, and $d$ is the feature dimension.
2. Running the Sinkhorn-Knopp algorithm for $L$ iterations, which takes $O(LKN)$ time.

This results in a total complexity of $O(KN(d + L))$. As this is linear with respect to the number of instances $N$, our method is scalable and efficient for large-scale LLP problems, especially compared to traditional OT solvers with polynomial complexity (e.g., $O(N^3)$). For extreme-scale scenarios, our framework can also incorporate more advanced solvers like Greedy Sinkhorn, and we will add a discussion of these options in the appendix of our revised paper.

Empirical Training Time:
To empirically verify the cost, we profiled our method on CIFAR-10 with a bag size of 256. The LLP-OTD module accounts for only 9.6% of the total training time in the second stage, demonstrating that the OT refinement does not pose a computational bottleneck in practice.

Impact of Sinkhorn Iterations (L):
We also evaluated the impact of the number of Sinkhorn iterations ($L$) on accuracy. The results from the CIFAR-10 dataset (bag size=256) at epoch 50 are shown in the table below:

As shown, the accuracy quickly saturates with a small number of iterations (e.g., $L = 5$ to $10$). This confirms that a small, constant $L$ is sufficient, keeping the practical computational cost low.

We will include this detailed analysis and runtime measurement in the appendix of our revised paper.

On the novelty of using established components like Optimal Transport (W2)

We agree that OT and consistency loss are established tools. Our core novelty lies not in using these tools, but in how we fundamentally adapt OT for the LLP problem. While prior works typically modify the overall OT objective function, our key innovation is to redesign the internal cost matrix for each sample-prototype pair. By embedding the LLP-Proportion Penalty directly into the cost function, $C_{k,i} = \|f_{\text{main}}(x_i) - \mu_k\|_2^2 + \lambda_{\text{OTD}}(1-p_{j(i),k})$, we reshape the cost landscape (refer to response to reviewer z6fq). This modification forces the optimal transport plan to a new equilibrium that is inherently more consistent with the true label proportions. This micro-level intervention, which alters the matching cost for each instance, is the central contribution that distinguishes our method from previous OT-based approaches.

Request for comparison against other baselines like Count Loss and GLWS (W4, Q2)

Following the reviewer's suggestion, we have broadened our experimental comparison to include Count Loss [1] and GLWS [2]. We have run new experiments on CIFAR-10 to compare with these baselines. Our method consistently outperforms them across various bag sizes. We will add the following table into section 5.3 in the revised paper:

On Count Loss replacement(Q1)

Thank you for your valuable suggestion to explore potential improvements for our work. Following your advice, we implemented Count Loss, referencing the official implementation provided in [1], to replace the KL divergence loss used for averaging predicted instance probabilities in our two-stage method. However, our experiments revealed that this approach introduces a prohibitive computational overhead. The Count Loss algorithm requires a dynamic programming step to compute the loss for each instance within a bag, leading to a complexity of O(N²), where N is the bag size. This computational cost proved to be impractically expensive for our setting. To substantiate this, we ran a comparative experiment on an RTX 4090 GPU to measure the average training time per epoch. The results are presented below:

As the table demonstrates, the training time per epoch for Count Loss increases substantially with the bag size, exceeding 5 hours for a bag size of 128. This computational demand made it infeasible for us to obtain complete training results within a reasonable timeframe. In contrast, the epoch time for our method using KL divergence loss remains highly efficient. It even slightly decreases with larger bag sizes due to the corresponding reduction in the total number of bags, demonstrating that it introduces no extra computational overhead. This highlights a significant efficiency advantage of our proposed approach. Given this substantial computational bottleneck, we maintain that our original choice of KL divergence loss represents a more practical and scalable solution.

On Observing Adherence to Bag Proportions (Q3)

To directly address your question, we conducted comprehensive bag proportions prediction experiments that quantitatively measure how well our final classifier adheres to the original bag-level constraints. We evaluate this through Mean Absolute Error (MAE) between predicted and true bag proportions, where MAE represents the average absolute deviation of predicted class proportions from ground-truth proportions across all bags. Our method demonstrates superior adherence:

Experimental Results (MAE $\downarrow$):

Key Observations:

1. Superior Bag Constraint Adherence: Our RLPL method consistently achieves lower MAE across all bag sizes, demonstrating that our final classifier better respects the original bag proportion constraints compared to the strong baseline LLP-AHIL.
2. Robust Scale Performance: The performance gap widens as bag size decreases (more challenging scenarios), with RLPL achieving 30.6% lower MAE at bag size 16 compared to LLP-AHIL, indicating our method's robustness in preserving bag-level supervision signals.
3. Practical Case Study: For bag size 256, we present a detailed example of one bag below, which demonstrates remarkable precision in proportion prediction:

This demonstrates that our LLP-OTD mechanism successfully preserves bag-level constraints while enabling accurate instance-level classification. The integration of LLP consistency loss in our LLPMix framework (Eq. 9 in our paper) ensures that the final classifier's predictions, when aggregated at the bag level, closely match the ground-truth proportions, directly answering your concern about bag proportion adherence.

On Training Time Details(Q4)

Thank you for raising the training-time concern. Here are the single-epoch numbers on CIFAR-10 (bag size = 256):

As the table shows, RLPL is a touch slower than some baselines. The extra cost comes almost entirely from the LLP-OTD module—the Sinkhorn iterations and the on-the-fly cost-matrix updates. Three quick points to put this in perspective:

1. The jump in accuracy is well worth it. RLPL outperforms LLP-VAT by 47.02 %, ROT by 30.85 %, and even beats LLP-AHIL by 1.07 pp—a large margin in the LLP world.
2. Overhead is modest and tunable. The OT piece accounts for only 9.6 % of a training step (see our reply to Reviewer z6fq). Dialing the Sinkhorn iterations down to L ≤ 5 cuts that slice further, with < 0.5 % accuracy loss.
3. Real-world trade-offs favor accuracy. In domains like healthcare or finance, a single-point absolute gain often outweighs a few extra seconds per epoch. The complexity is O(KN(d + L)), so scaling is linear; on UCI Adult the added cost is only 2.1 s per epoch.

We’ll fold this discussion into the camera-ready version and release the code so users can freely trade speed for precision.

Hello! Thank you very much for your hard work. I can imagine running all those experiments must have been difficult in such a short time so thank you very much. I believe you have answered all of my concerns so I will raise my score by 1.

I am curious about why your new framework does better than Count Loss and GLWS in LLP. Do you have any intuition for it?

Also in some of the other rebuttals there is some issue with the formatting of the text. just a heads up. :)

Theoretical Analysis of LLP-OTD (W1 & Q3)

Thank you for your valuable suggestion. We provide a three-step theoretical proof for our method.

1. Existence and Uniqueness of the Optimal Solution: We first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The optimization problem is defined as:

   $$\min_{T \in U(a, b)} \langle C, T \rangle - \gamma H(T),$$

   where $\langle C, T \rangle$ is the transport cost, $H(T)$ is the entropy term, and $U(a, b)$ is the transport polytope. The proof of existence and uniqueness is based on the strict convexity of the objective function and the compactness of the feasible set $U(a, b)$, guaranteeing a unique global minimizer.

2. Convergence of the Sinkhorn-Knopp Algorithm: We show that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution. The optimal transport plan $T^*$ is known to have a specific structure, and the convergence follows from interpreting the algorithm as an alternating projection procedure with Kullback-Leibler (KL) divergence. The process converges to the unique optimal solution $T^*$ due to the convexity of the feasible sets.

3. Role of the LLP-Proportion Penalty: Finally, we explain the intuition behind our innovative LLP-Proportion Penalty, which helps align the pseudo-label distribution with the true LLP prior. This penalty term effectively shifts the position of the optimal solution, ensuring that the resulting label assignments are more consistent with the given bag-level proportions.

Note: Due to space constraints, we refer to the detailed theoretical analysis provided in our response to Reviewer oaZQ. We will include the full discussion in the final version of the paper.

Performance on Noisy and Imbalanced Data (W2)

This is a very valid point. To evaluate our model's robustness, we conducted new experiments under more challenging conditions, including datasets with imbalanced class distributions and noisy label proportions. Evaluations on Imbalanced Datasets:
For the imbalanced data experiments, we used the CIFAR-10 dataset and constructed a long-tailed distribution by setting the imbalance ratio as the ratio of the maximum number of samples in one class to the minimum number of samples in another class. The imbalance ratios tested are 5, 10, 15, 50, and 100. The accuracy results are as follows:

As shown in the table, our method maintains strong performance even under high levels of imbalance. The accuracy decreases as the imbalance ratio increases, which is expected due to the growing difficulty of learning from such skewed data. However, our method outperforms the baseline, demonstrating its robustness in handling class imbalance. Evaluations on Noisy Datasets:
For noisy label experiments, we used the CIFAR-10 dataset and generated LLP data with clean labels, then introduced Gaussian noise and uniform noise with different intensities. The noisy labels were generated by the following formulas:

• Gaussian noise: $p' = \text{clip}(p + N(0, \sigma^2))$
• Uniform noise: $p' = \text{clip}(p + U(-r, r))$ The evaluation results are as follows:

As can be seen from the table, our method remains resilient under both moderate and heavy noise conditions. Even with significant label noise, the accuracy only slightly decreases, which highlights the robustness of our LLP-OTD mechanism in handling imperfect and ambiguous supervision signals. The results from both the imbalanced and noisy datasets demonstrate that our method consistently maintains high accuracy and robustness, making it highly effective in settings with imperfect supervision. This analysis will be included in the final version of the paper.

Computational Cost and Scalability of OT (W3 & Q1)

We thank the reviewer for raising this critical concern. We have analyzed the computational cost of OT from three perspectives. 1. Computational Complexity: Our LLP-OTD module, solved via the Sinkhorn-Knopp algorithm, has a time complexity of $O(KN(d + L))$, where $K$ is the number of classes, $N$ is the number of instances, $d$ is the dimensionality of each sample, and $L$ is the number of iterations required by the Sinkhorn-Knopp algorithm. This complexity is linear with respect to the number of instances $N$, which provides a significant advantage over classical Optimal Transport (OT) solvers that typically have a time complexity of $O(N^3)$. As a result, our approach is highly scalable and efficient for large datasets. 2. Empirical Runtime: To empirically verify the cost, we profiled our model's runtime on CIFAR-10 (bag size=256). The results show that the entire LLP-OTD module accounts for only 9.6% of the total training time in the second stage. This directly demonstrates that the OT refinement is not a computational bottleneck in practice. 3. Impact of Sinkhorn Iterations (L): The complexity depends on the number of Sinkhorn iterations, $L$. We tested its impact on accuracy. Table: Accuracy(at epoch 50, on CIFAR10, bag size=256) vs. Sinkhorn Iterations

As shown, the accuracy quickly saturates with a small number of iterations (e.g., L=5-10). This confirms that a small, constant L is sufficient, keeping the practical computational cost low.

Hyperparameter Tuning (Q4)

This is an important aspect. We have performed a detailed sensitivity analysis for the two weighting coefficients, namely $\lambda_{\text{OTD}}$ and $\omega_{\text{LLP}}$, on the CIFAR-10 dataset with a bag size of 256. The results, summarized in the table below, show that our model is robust across a reasonable range of hyperparameter settings. The accuracy at 100 epochs for different combinations of $\lambda_{\text{OTD}}$ and $\omega_{\text{LLP}}$ is as follows: Table: Accuracy (at 100 epochs) on different combinations of $\lambda_{\text{OTD}}$ and $\omega_{\text{LLP}}$

As shown in the table, the model achieves high and stable accuracy across a range of values for both $\lambda_{\text{OTD}}$ and $\omega_{\text{LLP}}$, with accuracy values ranging from 90.82% to 91.44%. The results demonstrate that our model is robust to variations in these hyperparameters, with no significant degradation in performance even for extreme values. This confirms that our method is not overly sensitive to specific hyperparameter choices, allowing for flexibility in tuning. We will include this analysis in Section 5.3 in the final version of the paper.

Evaluation on Other Modalities

We agree that demonstrating broader applicability is important. We have conducted new experiments on tabular data, including the UCI Adult dataset, and observed strong performance. Specifically, we performed experiments on the UCI Adult dataset, a widely used benchmark in tabular data. We also reproduced the results of two existing methods, LLP-AHIL and ROT, on this dataset for comparison. The experimental results are summarized in the table below:

From the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods. These results reinforce the robustness of our method across different non-vision modalities. We believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method.

Dear Reviewer z6fq,

We appreciate your valuable comments, which are important to improve the quality of this paper. We also appreciate your positive rating. As the deadline for the author-reviewer discussion period is approaching, please check whether your previous concerns have been fully addressed.

Looking forward to your reply. Thanks.

Regards.

On Theoretical Justification

In the response, we provide three theoretical guarantees for the proposed method. We will add them to the final version of our paper. First, we prove the existence and uniqueness of the optimal solution for our entropy-regularized optimal transport (OT) problem. Second, we demonstrate that the Sinkhorn-Knopp algorithm, which we employ, is guaranteed to converge to this unique solution. Finally, we explain how our innovative LLP-Proportion Penalty, incorporated into the cost function, guides the pseudo-label distribution to align with the true LLP prior.

1. Existence and Uniqueness of the Optimal Solution

We first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The problem is defined as:

$$\min_{T \in U(a, b)} \langle C, T \rangle - \gamma H(T),$$

where $\langle C, T \rangle = \sum_{k,i} C_{k,i} T_{k,i}$ is the transport cost, $H(T) = -\sum_{k,i} T_{k,i} (\log T_{k,i} - 1)$ is the entropy term, $\gamma > 0$ is the regularization strength, and $U(a, b)$ is the transport polytope defined by marginal constraints.

Proof:

Let $F(T) = \langle C, T \rangle - \gamma H(T)$. Then we will prove the strict convexity of the objective function. For the term $\langle C, T \rangle$, it is linear in $T$, hence convex. For the entropy term $-H(T) = \sum_{k,i} T_{k,i} (\log T_{k,i} - 1)$, we can compute its Hessian matrix:

$$\frac{\partial^2 (-H)}{\partial T_{k,i}^2} = \frac{1}{T_{k,i}} > 0 \quad \text{since } T_{k,i} > 0,$$

implying strict convexity of $-H(T)$. Therefore, $F(T) = \langle C, T \rangle - \gamma H(T)$ is the sum of a convex and strictly convex function, and thus strictly convex. Meanwhile, for $U(a,b)= \left\{ T \in \mathbb{R}_{\geq 0}^{K \times N} \, \middle| \, \sum_i T_{k,i} = a_k, \sum_k T_{k,i} = b_i \right\}$, we will prove it to be convex, closed and bounded. As $U(a,b)$ is defined by linear equalities and nonnegativity constraints, it is a convex polytope. Because all constraints are equation or non-strict inequality, $U(a,b)$ is closed. Since $\sum a_k = \sum b_i = 1$, we have $0 \le T_{k,i} \le 1$, and $U(a,b)$ is a compact convex set. Finally, by standard results in convex optimization, a strictly convex function over a non-empty compact convex set has a unique global minimizer $T^*$.

2. Convergence of the Sinkhorn-Knopp Algorithm

Next, we prove that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution $T^*$. The optimal transport plan $T^*$ is known to have a specific structure:

$$T^*_{k,i} = u_k \cdot K_{k,i} \cdot v_i, \quad \text{where } K_{k,i} = \exp\left(-\frac{C_{k,i}}{\gamma}\right)$$

The Sinkhorn-Knopp algorithm is an iterative procedure to find the scaling vectors $u \in \mathbb{R}^K$ and $v \in \mathbb{R}^N$ that satisfy the marginal constraints.

Proof:

The convergence can be shown by interpreting the algorithm as an alternating projection procedure using the Kullback-Leibler (KL) divergence. Let the sets of matrices satisfying the row and column constraints be:

• $\mathcal{C}_1 = \{ T \in \mathbb{R}_{\geq 0}^{K \times N} \mid \sum_i T_{k,i} = a_k \}$ (Row constraints)
• $\mathcal{C}_2 = \{ T \in \mathbb{R}_{\geq 0}^{K \times N} \mid \sum_k T_{k,i} = b_i \}$ (Column constraints) Each iteration of the Sinkhorn-Knopp algorithm can be viewed as performing the following alternating KL projections:

$$\begin{aligned} T^{(l+\frac{1}{2})} &= \arg\min_{T \in \mathcal{C}_1} \mathrm{KL}(T \parallel T^{(l)}), \\ T^{(l+1)} &= \arg\min_{T \in \mathcal{C}_2} \mathrm{KL}(T \parallel T^{(l+\frac{1}{2})}), \end{aligned}$$

where the KL divergence is a specific type of Bregman divergence. Since $\mathcal{C}_1$ and $\mathcal{C}_2$ are convex sets, the theory of alternating Bregman projections guarantees that this iterative process converges [1]. The intersection $\mathcal{C}_1 \cap \mathcal{C}_2 = U(a,b)$ is non-empty and contains the unique optimal solution $T^*$. Therefore, the sequence of iterates $T^{(l)}$ produced by the Sinkhorn-Knopp algorithm is guaranteed to converge to the unique optimal solution $T^*$.

3. Role of the LLP-Proportion Penalty

Finally, we provide the intuition for why our innovative LLP-Proportion Penalty helps align the pseudo-label distribution with the true LLP prior. Our effective cost function is:

$$C_{\text{eff}}(c_k, z_i) = d(c_k, z_i) + \lambda_{\text{LLP}}(1 - p(k \mid B_j(i)))$$

The key component is the penalty term. To understand its effect, we analyze the part of the objective function corresponding to this penalty, which we denote $\mathcal{L}_{\text{LLP}}(T)$:

$$\mathcal{L}_{\text{LLP}}(T) := \sum_j \sum_{i \in B_j} \sum_k T_{k,i} (1 - p(k \mid B_j)).$$

Minimizing this term is equivalent to maximizing the alignment between the transport plan's implied label proportions and the true bag proportions. By defining the transport plan's label distribution in bag $B_j$ as $P_T(k \mid B_j)$, minimizing $\mathcal{L}_{\text{LLP}}(T)$ becomes equivalent to maximizing the sum of dot products between the predicted and true label distributions within each bag:

$$\max_{T} \sum_j \langle P_T(\cdot \mid B_j), p(\cdot \mid B_j) \rangle.$$

Maximizing this dot product encourages the distribution $P_T$ to align with the true prior $p$, which is analogous to minimizing statistical divergences like the KL divergence. Therefore, by introducing the LLP-Proportion Penalty into the cost matrix, we directly enforce that the optimal transport plan $T^*$—the unique solution to the optimization problem—favors assignments that are consistent with the known bag-level supervision.

Reference:

[1] Byrne, Charles L. Alternating minimization and alternating projection algorithms: A tutorial.

Weakness：more experiments are suggested.

We appreciate the feedback on the experimental scope. To address this, we have significantly expanded our evaluation:

• Beyond image datasets: we have conducted additional experiments on tabular datasets to demonstrate the applicability of our method beyond image-based modalities. Specifically, we performed experiments on the UCI Adult dataset, a commonly used benchmark in tabular data. We also reproduced the results of two existing methods, LLP-AHIL and ROT, on this dataset for comparison. The experimental results are summarized in the table below: | Model | RLPL(ours) | LLP-AHIL | ROT | |:-----:|:----------:|:--------:|:---:| | UCI Adult | 77.57 | 75.99 | 72.82 |

From the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods. These results reinforce the robustness of our method across different non-vision modalities. We believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method.

Question： Multi-class vs. Multi-label Setting

Thank you for your comments. We give the following explanations.

1. Clarification of the Problem Setting: Our work addresses the standard multi-class Learning from Label Proportions (LLP) problem, where each instance belongs to exactly one class from a set of $K$ mutually exclusive classes. Multiple samples form a bag, and we have the label distribution of the bag as supervision information. The problem is formally defined in Section 3, where the proportion for class $k$ in a bag $B_j$ is given by $p_{j,k} = \frac{1}{n_j} \sum_{i \in B_j} I(y_i = c_k)$, with $\sum_{k=1}^K p_{j,k} = 1$. This formulation inherently assumes a multi-class ground truth. Consequently, our entire framework, including the pseudo-labeling step, is designed for this multi-class setting. The use of argmax to generate a single pseudo-label per instance is the standard and appropriate procedure for deriving hard labels from a multi-class classifier's probability distribution.

2. Potential Source of Misunderstanding: We believe the term "label proportions" itself, while standard in the LLP literature, might be interpreted in other machine learning contexts as relating to multi-label problems. We will ensure this distinction is made clearer in the final version of our paper to prevent future ambiguity.

3. On the Extension to a "Truly Multi-label" Setting: We agree that extending this work to the multi-label LLP setting is a fascinating and important future research direction. Such an extension would require some significant modifications to our framework:

• The classifier head would need to be changed from a softmax to multiple independent sigmoid activations.
• The bag proportion loss would need to be redefined, as the sum of proportions would no longer be 1.
• The OT formulation would be more complex, as the goal is no longer to assign an instance to a single class prototype. One might need to explore alternative matching frameworks.

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