Hyperbolic Dataset Distillation

Response to Reviewer arwV

We sincerely appreciate your thoughtful review and constructive comments on our submission. Below, we provide detailed responses to your concerns.

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Clarification 1 - Frozen Encoder: No Feature Collapse, Robust at Extreme Curvatures

We extract image features using a frozen network, whose weights remain unchanged during distillation; therefore, the features do not collapse into trivial representations (i.e., the synthesized samples are not forced to cluster around the origin). Moreover, our extensive experiments confirm that no feature collapse has been observed to date. Finally, we tested extreme values of $\|K\|$, and the results show that HDD maintains robust performance even in these edge cases, as presented below:

We observe that when the curvature $\|K\|$ is very small, the performance gap between HDD and the original DM shrinks, which aligns with the theoretical expectation that hyperbolic space degenerates into Euclidean space as the curvature approaches zero. Conversely, when $\|K\|$ becomes very large, our method remains effective.

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Clarification 2 – Ablation on Curvature $K$ and Loss-Weight $\lambda$

We conducted an ablation study on different curvature values $K$ within the DM framework on CIFAR-10; the results are presented below:

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Although $\|K\|$ slightly affects the final accuracy, the variation is modest, and HDD consistently surpasses the Euclidean baseline.

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Unlike the curvature hyperparameter, the inter‑centroid distances in hyperbolic space are extremely small because of its unique geometry, so we introduce a scaling factor. As long as this factor stays within a reasonable range, it exerts little influence on the final results. We choose its value empirically: by inspecting the raw loss values of the original method and selecting a factor that brings our (scaled) loss to the same order of magnitude. Of course, this value can also be determined automatically. To illustrate this, we run experiments with DM as the baseline while keeping the curvature fixed at  -1. Here, the initial loss denotes the loss after multiplying by the scaling factor when the number of distribution‑matching iterations is zero (the original DM uses no scaling). Our experimental results are as follows:

Because hyperbolic inter-centroid distances are inherently small, we multiply them by $\lambda$ to match the scale of the original DM loss. As long as $\lambda$ stays within a reasonable range, the accuracy remains virtually unchanged, confirming the insensitivity of HDD to this scaling factor.

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Clarification 3 - Clarification of Notation and Equations

Regarding $K_i$ in line 265, what we originally meant was the temporal component of sample features embedded under a fixed curvature. We apologize for the resulting misunderstanding and will amend this in the revised manuscript. Equations (1) and (2) themselves are correct and follow a written format similar to DANCE [1], whereas the formulation you suggested aligns more with other distribution‑matching methods, so we will adopt the format in the revised version. Thank you also for your keen observation: we will rectify the notation issue and replace $x_t$ and $x_s$ with $p_t$ and $p_s$.

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Clarification 4 - Per-Sample Curvature

All samples were consistently embedded in a hyperbolic space with identical curvature. We apologize once again for the confusion regarding the definition of $K_i$ in line 265.

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Clarification 5 - Can HDD Help Optimisation-Based DD?

HDD cannot boost optimization‑based methods on its own, but it can indirectly enhance their performance when used within a hybrid framework. As shown in Table  3 of the original paper, incorporating HDD into DANCE [1] yields performance improvements. We also provide results obtained by combining HDD with another hybrid approach, DSDM [2]. The table below summarises the results; adding HDD yields consistent gains across all IPC settings on CIFAR-10.

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Clarification 6 - Evidence of Hierarchical Structure in Benchmarks

First, the benchmark datasets we use also exhibit hierarchical structure, which can be quantified by Gromov δ‑hyperbolicity [3]. Gromov δ‑hyperbolicity measures how close a metric space is to an ideal negatively curved space. Values near 0 indicate that the data are inherently more compatible with tree‑like or negatively curved geometry, whereas values approaching 1 suggest a stronger tendency toward Euclidean geometry. In feature space, common benchmark datasets have Gromov δ‑hyperbolicity of roughly as shown below:

These figures demonstrate that these widely used benchmarks indeed lean toward negatively curved spaces and possess a certain degree of hierarchy.

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We also visualized the labels of the five samples nearest to the origin and the five farthest from it among the 256 raw 'Horse' images (one batch) at the 10,000th training iteration of IDM on CIFAR-10. However, due to conference policy restrictions, we are unable to provide the images directly. You may inspect the corresponding images by downloading this publicly available dataset. We apologize for any inconvenience this limitation may cause and appreciate your understanding.

• Nearest samples: data_batch_3_img6963, data_batch_4_img455, data_batch_4_img6399, data_batch_2_img4897, data_batch_4_img5922
• Farthest samples: data_batch_1_img1394, data_batch_5_img897, data_batch_1_img2701, data_batch_3_img3910, data_batch_2_img6227

We observe that samples farther from the origin are noticeably messier, either showing cluttered scenes with both people and horses or only partial views such as a horse’s head, whereas those nearer to the origin are visually cleaner, free of distracting objects, and present the horse’s full outline.

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We hope that these detailed clarifications address all of your concerns. Thank you again for your valuable suggestions, which have helped us strengthen the manuscript.

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References

[1] DANCE: Dual-View Distribution Alignment for Dataset Condensation
[2] Diversified Semantic Distribution Matching for Dataset Distillation
[3] Hyperbolic Image Embeddings

Response to Reviewer RBVi

We sincerely thank you for your insightful review and for acknowledging the clarity of our manuscript and the potential of introducing hyperbolic geometry into dataset distillation. Below, we provide detailed responses to your concerns.

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Clarification 1 – Originality

To the best of our knowledge, this is the first study to integrate hyperbolic geometry into dataset distillation and to tailor the distribution‑matching objective explicitly to the curvature of the space. We not only introduce a new distillation objective, matching the Fréchet means of two data distributions in hyperbolic space, but also systematically examine how samples at different hierarchical levels contribute to the loss, and we propose a hierarchy‑aware pruning strategy. Previous hyperbolic research has focused mainly on graph networks, metric learning, or generative models; in contrast, our framework targets the specific pain point in dataset distillation where hierarchical alignment and structure are often overlooked, as differentiated in our related work overview. In summary, we inject a hyperbolic hierarchical inductive bias into the distillation process for the first time, without adding any extra training tricks, yet still achieve notable accuracy gains across multiple datasets compared with baseline methods. This simplicity makes our approach easy to reproduce and extend. Further, we list some possible optimization directions: the framework can be generalized to (a) pseudo-Riemannian manifolds, (b) information-theoretic metrics in hyperbolic space, and (c) higher-order moment matching on hyperbolic manifolds, opening new avenues for more powerful and versatile dataset-distillation methods.

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Clarification 2 – Motivation for Hyperbolic Space

Limitations of Euclidean discrepancy measures.
Euclidean MSE / MMD treats samples as i.i.d. points, weighting high-level (fine-detail or noisy) samples the same as low-level (prototype) samples and thereby diluting distillation effects.

Why negative curvature?
Hyperbolic space ($K < 0$) expands exponentially in volume, embedding tree structures continuously. By matching centroids in this space, we align the hierarchical structure between real and synthetic data. Samples at larger geodesic distance naturally contribute less, amplifying prototypes while suppressing noise.

Experimental evidence.
Our pruning study (Table 2 in the paper) confirms that removing samples farther from the hyperbolic origin has little impact, indicating that those points indeed contain less critical information.

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Clarification 3 – Hierarchical Properties of Benchmark Datasets

The benchmark datasets we use also exhibit hierarchical structure, which can be quantified by Gromov δ‑hyperbolicity [1]. Gromov δ‑hyperbolicity measures how close a metric space is to an ideal negatively curved space. Values near 0 indicate that the data are inherently more compatible with tree‑like or negatively curved geometry, whereas values approaching 1 suggest a stronger tendency toward Euclidean geometry. In feature space, common benchmark datasets have Gromov δ‑hyperbolicity of roughly as shown below:

These figures demonstrate that these widely used benchmarks indeed lean toward negatively curved spaces and possess a certain degree of hierarchy.

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Clarification 4 – Rationale for Performance Improvements

As we explained in Clarification 2, our choice is grounded in two main considerations:

(a) By aligning centroids to match the hierarchical structure of the data, we minimize the overall distributional gap between synthetic and real samples. As Figure  4 shows, the synthetic samples closely trace the distribution of the originals.

(b) The hierarchical weighting of centroids in hyperbolic space encourages the synthetic samples toward class prototypes while suppressing boundary noise.

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Clarification 5 – Visualisation and Interpretation

Due to conference policy, images cannot be uploaded. We therefore list IDs of the five CIFAR-10 Horse images nearest to and farthest from the hyperbolic origin after 10,000 IDM iterations:

• Nearest samples: data_batch_3_img6963, data_batch_4_img455, data_batch_4_img6399, data_batch_2_img4897, data_batch_4_img5922
• Farthest samples: data_batch_1_img1394, data_batch_5_img897, data_batch_1_img2701, data_batch_3_img3910, data_batch_2_img6227

When viewed locally, near-origin samples are visually cleaner and centred, whereas far-origin samples are cluttered or partially occluded, supporting our interpretation of hyperbolic radius as a hierarchy indicator.

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Clarification 6 – Benchmark Dataset with a Hierarchy Structure

As we explained in Clarification 3, we have confirmed that the benchmark dataset we used also exhibits a hierarchical structure. In the future, we plan to extend HDD to other data modalities, such as language, with even richer hierarchical structures.

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Clarification 7 – Comparison with Multisize Dataset Condensation

Multisize Dataset Condensation (MDC) collapses the N individual condensation runs that would normally be required for different target set sizes into a single process. It does so by adding an adaptive subset loss on top of the standard gradient‑matching loss, allowing the full synthetic set and any of its subsets to converge toward the real data simultaneously and eliminating the “subset performance collapse.” In other words, MDC permits informational overlap between a subset and the complete set. Our method, in contrast, attaches explicit hierarchical information to every synthetic sample. Trimming the synthetic dataset, therefore, means deliberately discarding part of that hierarchy, which causes a pronounced performance drop. The results of the experiment we conducted are as follows：

We observed that the larger the IPC in the original set, the less information remains after trimming. Specifically, cutting an IPC‑50 set down to IPC‑1 performs worse than cutting an IPC‑10 set down to IPC‑1, while a native IPC‑1 set, with no trimming at all, achieves the best results by a wide margin.

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We hope that these detailed clarifications address all of your concerns. Thank you again for your valuable suggestions, which have helped us strengthen the manuscript.

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Reference

[1] Hyperbolic Image Embeddings.

We appreciate your additional questions and are glad that most of the earlier concerns have been resolved. Below, we reply to the three new points.

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1.

The reviewer’s remark that “Riemannian-mean matching is essentially first-order moment matching” is exactly correct. Our method can indeed be viewed as moment matching performed in hyperbolic space. Conceptually, this parallels Euclidean moment-matching approaches, such as MMD; however, the implementation details and emphases differ.

• Common Goal: Both our HDD and Euclidean MMD aim to minimise the distributional gap between the real and synthetic datasets by aligning key statistical properties.

• Key Difference: Conventional DM aligns the Euclidean means (first-order moments), implicitly treating every sample as equally important. In contrast, HDD matches the hyperbolic Riemannian mean (centroid). Because of negative curvature, the centroid is naturally biased toward samples closer to the origin (root-centric). Thus, by matching hyperbolic centroids, HDD performs implicit hierarchy-aware re-weighting, offering a more precise form of distribution alignment.

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2.

Our pipeline is not detached from Euclidean space. As detailed in the paper, we first map each sample into an Euclidean feature space via a pretrained encoder and then embed those features into hyperbolic space via the exponential map. Hence, the optimisation of $𝑃_{\psi} ≈ 𝑄_{\psi}$ is based on Euclidean features. The ultimate goal of dataset distillation (Eq. 1) is to minimise the performance gap between networks trained on real versus synthetic data under identical parameters. DM methods adopt a proxy objective: if real and synthetic data are well aligned in some feature space, then models trained on them should perform similarly. HDD follows the same logic but contends that performing the alignment in a non-Euclidean space that better captures intrinsic structure yields a stronger proxy, enabling the synthetic set to generalise better and thus more closely approach the original distillation objective. Empirically, this claim holds: e.g., on CIFAR-100 (IPC = 1), HDD improves accuracy over the IDM baseline by 3.2%.

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3.

Distribution matching seeks to shrink the gap between real and synthetic data in feature space. Although the raw centroid distance under HDD is indeed numerically smaller, this is largely due to the geometry of hyperbolic space (distances near the origin are inherently shorter), which is why we introduce a scaling factor. Because hyperbolic and Euclidean distances are not metrically equivalent, cross-space magnitudes are hard to compare directly. In such cases, test accuracy serves as the most convincing evidence. Across nearly all baselines and datasets, HDD yields consistent performance gains, demonstrating that it achieves a closer distributional match to the original data.

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We hope these clarifications address the remaining questions and further justify the use of hyperbolic geometry in HDD. We will also include these clarifications in the revised manuscript. Thank you again for your helpful feedback.

Looking forward to your reply!

Response to Reviewer gver

We sincerely appreciate your thoughtful review and constructive comments on our submission. Below, we provide detailed answers to each of your questions and comments.

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Clarification 1 – Ablation on Curvature $K$ and Loss-Weight $\lambda$

We conducted an ablation study on different curvature values $K$ within the DM framework on CIFAR-10; the results are presented below:

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Although $\|K\|$ slightly affects the final accuracy, the variation is modest, and HDD consistently surpasses the Euclidean baseline.

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Unlike the curvature hyperparameter, the inter‑centroid distances in hyperbolic space are extremely small because of its unique geometry, so we introduce a scaling factor. As long as this factor stays within a reasonable range, it exerts little influence on the final results. We choose its value empirically: by inspecting the raw loss values of the original method and selecting a factor that brings our (scaled) loss to the same order of magnitude. Of course, this value can also be determined automatically. To illustrate this, we run experiments with DM as the baseline while keeping the curvature fixed at  -1. Here, the initial loss denotes the loss after multiplying by the scaling factor when the number of distribution‑matching iterations is zero (the original DM uses no scaling). Our experimental results are as follows:

Because hyperbolic inter-centroid distances are inherently small, we multiply them by $\lambda$ to match the scale of the original DM loss. As long as $\lambda$ stays within a reasonable range, the accuracy remains virtually unchanged, confirming the insensitivity of HDD to this scaling factor.

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Clarification 2 – Runtime and Memory Overhead

We benchmarked 100 training iterations on a single RTX 4090 GPU with batch size = 256 on CIFAR-10.

Memory usage for the two methods is nearly identical, while HDD requires more runtime, but the overhead remains within an acceptable range.

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Clarification 3 – Originality

To the best of our knowledge, this is the first work to integrate hyperbolic geometry into dataset distillation and to tailor the distribution-matching objective explicitly to the space’s curvature. We further observe that the synthetic samples naturally mimic the hierarchical structure of the real data and gravitate toward the class centroid. Exploiting this property, we prune redundant real samples without any loss of accuracy. We have deliberately kept the method simple, no additional training tricks are introduced, but we still achieve notable accuracy gains over strong baselines on multiple datasets. This simplicity enhances both reproducibility and extensibility. Further, we list some possible optimization directions: the framework can be generalized to (a) pseudo-Riemannian manifolds, (b) information-theoretic metrics in hyperbolic space, and (c) higher-order moment matching on hyperbolic manifolds, opening new avenues for more powerful and versatile dataset-distillation methods.

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Clarification 4 – Scalability to Larger Datasets and Non-Image Domains

We have applied HDD to an ImageNet subset, ImageWoof, with high resolution (224 × 224), and verified the effectiveness of our method (Table 3). The results indicate that HDD can scale beyond tiny images, and we are planning to extend experiments to other modalities like text and speech.

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Clarification 5 – Sensitivity and Stability at Extreme Curvatures

We conducted experiments on the extreme values of $\|K\|$​​ (including both extremely high curvature and extremely low curvature), and the results are as follows:

We observe that when the curvature $\|K\|$ is very small, the performance gap between HDD and the original DM shrinks, which aligns with the theoretical expectation that hyperbolic space degenerates into Euclidean space as the curvature approaches zero. Conversely, when $\|K\|$ becomes very large, our method remains effective.

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Clarification 6 – Exact vs. Tangent-Space Fréchet Mean

The vast majority of hyperbolic geometry methods in machine learning employ tangent space (or other approximate) centroids, as they are numerically stable and have consistently proven effective in practice. This is particularly true for dataset distillation frameworks, where centroids are invoked thousands of times per training run, making runtime efficiency critically important. We conducted experiments using the exact centroid computation method from the paper you provided. For the differential approximation, we perform 20 iterations to compute its centroid. In both centroid‑computation methods, the curvature is fixed at  -1. The runtime (for 100 iterations) and results on CIFAR-10 are as follows:

It can be seen that the two approaches produce nearly identical results, yet the computational burden rises markedly. Therefore, we consider the tangent‑space approximation to be the more cost‑effective option. We will add a note on this point in the revised version. Thank you for your valuable suggestion!

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We hope that these detailed clarifications address all of your concerns. Thank you again for your valuable suggestions, which have helped us strengthen the manuscript.

Response to Reviewer 9awn

We sincerely thank you for the thoughtful review and for highlighting the novelty and empirical value of our approach. Below, we provide detailed answers to each of your questions and comments.

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Clarification 1 – About the Cross-Entropy Term in IDM and DANCE

In our current implementation, the cross-entropy (CE) loss remains in Euclidean space. The logits produced by standard convolutional networks live naturally in Euclidean tensor space, and the CE loss is well defined there. By leaving this component unchanged, we guarantee full compatibility with existing dataset-distillation baselines and avoid introducing additional confounding factors. Although a hyperbolic softmax layer is conceptually possible, it would require redesigning the classifier head and decision boundaries, which we consider outside the scope of this study.

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Clarification 2 – Generalisability to Other Distribution-Matching Methods

For methods such as CAFE that align layer-wise activations, the objective differs substantially from our global centroid-matching formulation; an effective combination would therefore demand a dedicated re-design of the loss. M3D focuses on higher-order moment matching and can, in principle, benefit from our hyperbolic formulation as well, but the resulting optimisation landscape would become considerably more complex.

To substantiate the claim of broad applicability, we have already integrated HDD into DSDM [1], which is a hybrid algorithm that alternates between CE and distribution matching. The table below summarises the results; adding HDD yields consistent gains across all “images-per-class” (IPC) settings on CIFAR-10.

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We also conducted the IPC-10 experiment on CIFAR-10 using IID (DM version). We only replaced the MMD component, setting curvature $K= -1$ and the scaling factor $\lambda$ to 20. Our experimental results are as follows:

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Due to time constraints, experiments that substitute the MMD term with HDD in IID for additional settings are still running, and we will report the corresponding results in the revised manuscript.

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Clarification 3 – Prototypicality of Samples Near the Hyperbolic Origin

Because conference policy prevents us from embedding raw CIFAR-10 images directly in the rebuttal, we instead list the identifiers of the five samples closest to the origin and the five farthest after 10,000 optimisation steps of IDM (all from the horse class).

• Nearest samples: data_batch_3_img6963, data_batch_4_img455, data_batch_4_img6399, data_batch_2_img4897, data_batch_4_img5922
• Farthest samples: data_batch_1_img1394, data_batch_5_img897, data_batch_1_img2701, data_batch_3_img3910, data_batch_2_img6227

When we visualise these images locally, near-origin samples are centred and uncluttered, whereas far-origin samples are partially occluded, truncated, or mixed with background objects. This qualitative difference supports our claim that hyperbolic geometry naturally emphasises prototypical examples.

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Clarification 4 – Adding Class-Wise Regularisation in Hyperbolic Space

The visualization results for IDM are already provided in our paper, specifically in the two panels on the right side of Figure 4. In addition, we are also visualizing IID. However, due to conference policy restrictions, we are unable to directly include the corresponding images. We will include the visualization results in the revised version.

We agree that encouraging intra-class coherence without compromising the global hierarchy is an important direction. We have revisited the IID approach: when IID introduces class‑centric constraints, directly applying it in hyperbolic space may be unsuitable, because those constraints can compromise the dataset’s hierarchical structure. So we propose a different angle: the overall hyperbolic distance can be decomposed into radial and angular components. We can strip out the radial component—since forcibly shortening it might damage the hierarchy—while retaining the angular component to impose intra‑class constraints. Next, we provide the specific definition of angular loss for two points. Let a Lorentz-model point be written as $x=(x_0,\mathbf{x})$ with $x_0>0$ and $-x_0^{2}+\|\mathbf{x}\|^{2}=-1$. For two points $x$ and $y$ we extract their spatial parts and normalise them,

$$\hat n_x = \frac{\mathbf{x}}{\|\mathbf{x}\|}, \quad \hat n_y = \frac{\mathbf{y}}{\|\mathbf{y}\|},$$

then compute the directional cosine $\cos\theta = \hat n_x \cdot \hat n_y$. The angular loss is given by

$$L_{\text{angle}} = 1 - \cos\theta = 2 \sin^{2}\Bigl(\tfrac{\theta}{2}\Bigr),$$

which equals 0 when directions coincide and 2 when they are opposite.

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Minor Comments

• We will explicitly define IPC as “images per class” at its first appearance.
• The duplicate citations [23] and [24] refer to the same source; we will merge them in the final manuscript.

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We hope that these detailed clarifications convincingly address your concerns. Thank you again for your positive assessment and constructive suggestions.

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Reference

[1] Diversified Semantic Distribution Matching for Dataset Distillation.