Title: Robust Wasserstein  $k$-center Clustering: Algorithms and Acceleration
Venue: NeurIPS 2025
Accepted: no
Keywords: clustering; coreset; Wasserstein distance
Abstract: The classical metric $k$-center problem is widely used in data representation tasks. However, real-world datasets often contain noise and exhibit complex structures, making the traditional metric $k$-center problem insufficient for such scenarios. To address these challenges, we present the \textbf{R}obust \textbf{W}asserstein \textbf{C}enter clustering (RWC-clustering)  problem.
Compared to the classical setting, the main challenge in designing an algorithm for the RWC-clustering problem lies in effectively handling noise in the cluster centers. To this end, we introduce a dedicated purification step to eliminate noise, based on which we develop our clustering algorithm.
Furthermore, when dealing with large-scale datasets, both storage and computation become highly resource-intensive. To alleviate this, we adopt the \textit{coreset} technique to improve the computational and storage efficiency by compressing the dataset.  
Roughly speaking, this coreset method enables us to calculate the objective value on a small-size coreset, while ensuring a close approximation to the value on the original dataset in theory; thus, it substantially saves the storage and computation resources.  
Finally, experimental results show the effectiveness of our RWC-clustering  problem and the efficiency of the coreset method.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper formalises the Robust Wasserstein $k$-center clustering (RWC-clustering) problem, where each data point is a probability measure possibly contaminated by outliers. \nThe authors argue that comparing a noisy datum $\\mu$ to a clean centre $\\nu$ using a one-sided robust Wasserstein distance $\\mathcal{W}(\\mu,\\nu)$ preserves intra-cluster similarity, whereas the fully robust distance $\\tilde{\\mathcal{W}}$ can collapse dissimilar points into the same ball. \n\n---\n\n### Algorithms\nA two-stage algorithm is proposed.\n\nThe seeding algorithm iteratively picks the point farthest from the current centre set (Gonzalez rule) but purifies each candidate via a “$\\tau$-nearest neighbours under $\\tilde{\\mathcal{W}}$” procedure to strip noise before adding it to the centre set .\n\nThen, a local-search style refinement algorithm repeatedly samples a point proportional to its cost, purifies it, and swaps it with an existing centre when beneficial.\n\n---\n\n### Coreset\n\nIn order to improve practical performance, the authors develop some coreset theory for RWC.\nBecause WW is not a metric, standard coreset theory does not apply. The authors \n1. derive a 2-approximate lower bound $\\Gamma$ on the optimal radius via a modified Gonzalez routine using $\\tilde{\\mathcal{W}}$, and \n1. build a doubling-metric style coreset whose size is $O\\left( \\left( \\frac{R}{\\varepsilon \\Gamma} \\right)^{2{\\rm ddim}} \\right)$ and show that any k-centre cost computed on the coreset approximates that on the full set within $\\varepsilon$ relative error. Here $R$ is the radius of the dataset.\n\n---\n\n### Experiments\nExperiments on small datasets are performed to empirically verify the efficiency and performance of the algorithms.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. The main conceptual contribution seems to be a principled formulation for robust clustering in Wasserstein distance, which is an interesting and useful concept.\n1. The authors develop a novel coreset algorithm with provable guarantees despite the fact that the objective does not come from a metric, which may be of independent interest\n1. Empirical studies include ablation studies of seeding and coreset effectiveness\n\n---\n\n### Weaknesses\n\n1. There is no end-to-end theoretical guarantee of the proposed algorithms. This is not necessarily a weakness if there are extensive experiments. However,\n1. the empirical studies seem to be completed on small datasets (size 200), which may not be a good measure of the efficiency of the coreset technique for larger datasets\n\nQUESTIONS:\n1. Do the authors believe an end-to-end algorithm with provable approximation guarantees is possible with their technique?\n1. How is the performance of the proposed algorithm on larger instances?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the RWC-clustering method is proposed to address the shortcomings of the traditional metric k - center clustering in noisy and complex structured data. The approach employs the Robust Wasserstein Distance (RWD) to measure the difference between data points and cluster centers, and designs a dedicated \"purification step\" to eliminate noise from candidate centers, ensuring the purity of cluster centers. To tackle the computational and storage bottlenecks in large-scale data scenarios, the authors adopt the coreset technique to compress the dataset, theoretically proving that the compressed data can effectively approximate the objective values of the original data, thus enhancing algorithm efficiency. The experiments verify the effectiveness of the RWC clustering algorithm proposed in the paper and the efficiency of the coreset method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths：\n1.This paper is theoretically sound. The authors analyze the anti-noise mechanism of RWD and the approximation guarantee of coreset compression from the theoretical level, which provides theoretical support for the rationality of the method.\n2.This approach demonstrates certain innovation. To address the limitations of traditional k-center clustering in handling noisy and complex-structured data, the RWC-clustering method is proposed, which integrates the Robust Wasserstein Distance (RWD) and a \"purification step\" to enhance clustering robustness from both the metric design and noise filtering mechanisms. \nWeaknesses：\n1.The readability of the article is poor. The author provides a series of theoretical analysis, but lacks specific examples to explain the algorithmic process in detail, which would help readers understand the flow of the algorithm.\n2.The main text presents only a small number of experimental results, and a large amount of key data (e.g., more dataset comparisons, ablation experiments, etc.) are placed in the appendices, which leads to an incomplete logic of argumentation in the main text, and readers need to rely on the appendices to comprehensively assess the performance of the method, which affects the readability and rigor of the paper.\n\nQUESTIONS:\n1.In the steps of the RWC algorithm, both Wasserstein distance and robust Wasserstein distance (RWD) are mixture used. Could the authors elaborate on the specific roles of these two distances in the RWC algorithm respectively?\n2.Does the robust Wasserstein distance improve noise immunity by discarding ζ mass, potentially leading to the loss of critical information (e.g., small probability but important features)?\n3.Adopting only a single evaluation index and lacking a systematic evaluation of the core indicators of clustering accuracy (e.g., ACC, NMI, ARI, etc.), it is difficult to comprehensively verify the clustering effect of the algorithm.\n4.The experimental comparisons are limited to UOT-based and WD-based clustering algorithms, without covering mainstream other baseline methods, making it insufficient to demonstrate the superiority of the proposed algorithm.\n5.Only the parameters τ and ζ are ablated individually (as in Tables 11 and 12), without exploring the combined effects of the two on algorithm performance. This lack of systematic exploration of the parameter space may lead to deviations in the selection of optimal parameter combinations.\n\nLIMITATIONS:\nThe authors acknowledge that the robust Wasserstein distance used does not satisfy the triangle inequality of metric spaces, leading to the inapplicability of theoretical guarantees for traditional clustering algorithms. Although the paper enhances practicality through purification steps and coreset techniques, it lacks a rigorous approximation ratio proof for RWC clustering in non-metric spaces, which renders the theoretical framework incomplete.\nAdditionally, the construction of coreset relies on the \"low doubling dimension\" assumption of data, which may not be satisfied by actual complex datasets. The paper fails to fully validate the universality of this assumption in real-world scenarios, nor does it propose coping strategies for performance degradation of coreset in high-dimensional cases—for instance, it does not elaborate on how to adjust parameters to ensure approximation accuracy in high-dimensional contexts.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nRobust Wasserstein Center (RWC) clustering is proposed in this paper to address the issue of noise contamination in empirical datasets which standard Wasserstein distances are sensitive to. By adopting a specific variant of robust Wasserstein distance, the RWC-clustering problem is formulated and the algorithm built on the backbone of Gonzalez’s algorithm to select robust Wasserstein centers are proposed.   On the side, a technique that aims to find corsets of the original dataset is proposed to improve the computation efficiency while maintaining accuracy by both theoretically proving that the coreset serves as a set of Wasserstein centers that the RWC-objective is achieved by a negligible distance determined by the number of points in the corset and the optimal value of the RWC-objective.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper is very comprehensive and clearly written by organizing and sequencing the problem formulation, intuition, theories, algorithms, experiments, and vivid simple examples.\n2. The proof is rigorous and solid.\n3. The experiments are very detailed and well presented, serving as strong arguments for the proposed algorithms.\n4. The authors are very thorough in considering the whole pipeline from both theoretical and computational perspectives and have made valid points in each involved steps.\n\nWeakness:\n1. While there is innovation and good power of broad applications in the proposal of RWC clustering and the associated algorithms, a big portions of the underlying theories and the algorithms seem to stem from and be built on previous related work without much novel design and/or theoretical findings.\n\nQUESTIONS:\nIt is assumed in the paper that $\\mu$ and $\\nu$ both have $\\zeta$ mass of outliers: is there any specific reason for imposing this assumption and does it change the conclusion if two distributions are allowed to have different mass of outliers? And how is $\\zeta$ chosen in reality if it is unkown?\n\nIn terms of the computation of robust Wasserstein distance, how is it handled given that it is a minimum of Wasserstein distance over a set of distributions and the computation of Wasserstein distance (especially in high dimensions) are known to be inefficient?\n\nLIMITATIONS:\nThere is no major limitations but the concerns mentioned in the questions above may limit the effectiveness and broadness of the applications to real-world problems.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the k-center problem under the Wasserstein distance. The Wasserstein distance (also known as the Earth Mover's Distance) measures the cost of optimally transporting mass to transform one distribution into another. While it is widely used in machine learning and computational geometry, it is important to note that the Wasserstein distance is generally not a metric over the space of discrete distributions when the supports differ. This raises questions about how traditional metric-space clustering algorithms, such as k-center, can meaningfully be applied.\n\nI found the paper difficult to follow. I have worked on the k-center problem for almost 20 years, but the technical exposition here is quite difficult to understand. From what I understand, the authors propose adding a purification step before running the Gonzalez algorithm. Specifically, Algorithm 1 on page 5 simulates the Gonzalez procedure, and in each iteration it selects the tau closest points to the most recently chosen center to find a better candidate center among them. However, on page 6 the authors write “we choose tau that retain the smallest tau for Equation 5.” I do not understand what this means, why it is justified, or how it fits into the algorithm.\n\nMoreover, the paper does not mention the approximation factor achieved by the proposed algorithm in the “Our Contribution” section on page 2, which is a crucial omission. In fact, this section is generally unclear. For example, the authors write “For effectively representing datasets with complex structures and outliers, we introduce the RWC-clustering problem and provide the underlying intuition for its formulation.” However, the terms “complex structures” and “outliers” are not clearly defined, making it hard to understand what precise problem the authors are solving.\n\nLater, the paper states: “To solve this robust clustering problem, we first design a purification step to eliminate the noise in the candidate centers; then, we integrate it into existing methods [Gonzalez 1985; Lattanzi and Sohler 2019; Choo et al. 2020] to develop customized initialization and post-processing algorithms.” However, the purification step itself is not clearly explained or sketched in intuitive terms in the “Our Contribution” section. Furthermore, it seems the authors may not be aware that the result of Lattanzi and Sohler 2019 is for k-means clustering, not k-center clustering.\n\nThe paper also mentions: “Furthermore, to enhance scalability, we introduce the coreset technique to accelerate the computation by compressing the dataset. Additionally, we theoretically demonstrate that the coreset is a good proxy of the original dataset.” However, the construction of the coreset is not clearly described in the “Our Contribution” section, and the theoretical guarantees are not stated precisely. Importantly, the coreset construction appears to rely on bounded doubling dimension, which is a property defined for metric spaces. Since the Wasserstein distance here is generally not a metric, this assumption should be explicitly acknowledged and justified.\n\nIn summary, the paper needs significant polishing and clarification:\n- First, clearly define what is meant by “complex structures and outliers.”\n- Next, explicitly state the main theorems, approximation factors, and coreset guarantees achieved by the algorithm.\n- Finally, rewrite the “Our Contribution” section to clearly and concretely describe the proposed method and why it is novel.\n\nAddressing these issues would greatly help readers understand the motivation and technical content of the paper.\n\nSTRENGTHS AND WEAKNESSES:\n**Weaknesses**\n- The paper is difficult to follow: key ideas (e.g., purification step, coreset construction) are not clearly explained.\n- Lacks precise statements of approximation factors, theoretical guarantees, and assumptions.\n- The related work discussion is incomplete and occasionally inaccurate (e.g., citing k-means results for k-center).\n- Relies on bounded doubling dimension without explaining how this applies under the Wasserstein distance, which is generally not a metric.\n\nQUESTIONS:\nI have mentioned them in above.\n\nLIMITATIONS:\nMentioned in above.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-07-05"}