Title: Coreset for Robust Geometric Median: Eliminating Size Dependency on Outliers
Venue: NeurIPS 2025
Accepted: yes
Keywords: coreset

robust geometric median

Euclidean space

outlier

robust clustering
Abstract: We study the robust geometric median problem in Euclidean space $\mathbb{R}^d$, with a focus on coreset construction. A coreset is a compact summary of a dataset $P$ of size $n$ that approximates the robust cost for all centers $c$ within a multiplicative error $\varepsilon$. Given an outlier count $m$, we construct a coreset of size $\tilde{O}(\varepsilon^{-2} \cdot \min \\{ \varepsilon^{-2}, d \\})$ when $n \geq 4m$, eliminating the $O(m)$ dependency present in prior work [Huang et al., 2022 & 2023]. For the special case of $d = 1$, we achieve an optimal coreset size of $\tilde{\Theta}(\varepsilon^{-1/2} + \frac{m}{n} \varepsilon^{-1})$, revealing a clear separation from the vanilla case studied in [Huang et al., 2023; Afshani and Chris, 2024]. Our results further extend to robust $(k,z)$-clustering in various metric spaces, eliminating the $m$-dependence under mild data assumptions. The key technical contribution is a novel non-component-wise error analysis, enabling substantial reduction of outlier influence, unlike prior methods that retain them. Empirically, our algorithms consistently outperform existing baselines in terms of size-accuracy tradeoffs and runtime, even when data assumptions are violated across a wide range of datasets.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes improved coreset constructions for the robust geometric median problem that remove the dependency on the number of outliers. It also extends to robust $(k, z)$-clustering and shows strong empirical performance.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality:**\nThe paper appears technically sound, with well-defined theoretical contributions and empirical validation across multiple datasets. The improvements over prior work, particularly in terms of coreset size and runtime efficiency, are clearly articulated and seem well-supported by both analysis and experiments.\n\n**Clarity:**\nThe manuscript is generally clear and well-organized. However, there are some presentation issues—specifically, the placement of figures, tables, and algorithm blocks could be more closely aligned with the corresponding narrative. Additionally, the structure of the paper could be improved to highlight key experimental results more prominently in the main body, rather than deferring them to the appendix or supplementary materials.\n\n**Significance:**\nThe problem of robust coreset construction for geometric median and $(k, z)$-clustering is important for scalable learning. The elimination of outlier dependency and the extension to general metric spaces indicate potential for meaningful impact in both theory and practice.\n\n**Originality:**\nThe work introduces a novel non-component-wise error analysis technique and achieves theoretical and empirical improvements over existing methods. These contributions appear to be a non-trivial and original step forward in the field.\n\nQUESTIONS:\n1. **Presentation Alignment:** Several figures, tables, and algorithm blocks are not placed near the corresponding explanatory text. Reorganizing the layout so that each element appears closer to the relevant discussion would significantly improve readability and reader comprehension.\n\n2. **Structure of Experimental Results:** Key experimental results are currently not sufficiently highlighted in the main text. Could the authors restructure the paper to better emphasize these results in the main body, rather than relegating them to the appendix or supplementary material?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

comments: SUMMARY:
This paper studies the robust geometric median problem in the d=1 and general d case, and provides improved results for coresets in these settings.  For d=1 their algorithm polynomially reduces the dependence on the error term eps, removes the dependence on the number of outliers m, and is tight up to poly-logarithmic factors.  For the general d case, they use the insight from the d=1 case to make a simple and clever adaptation of previous results, and this also improves the previous results by removing the dependence on the number of outliers m.  
These results extend to provide improved results for robust (k,z)-clustering.  
Experiments are provided to show on real data the improvement in size/error trade-off on previous robust-median coresets algorithms.

STRENGTHS AND WEAKNESSES:
I found the new methods and analysis relatively simple, but it improves the results which appeared in STOC, SODA, ICALP, FOCS in the last few years -- in that it removes the dependence on the number of outliers.  I appreciate the simple analysis, it is a positive.  And removing the dependence on the number of outliers is an important improvement.  

The topic of coresets, and clustering, and robust estimators are all very relevant for NeurIPS, so since this is an important improvement, I think the paper should be accepted.  

I found the writing to be formal, and no errors or concerned, but a bit dense.  I appreciate the Figure 1, but I did not find it very well labeled.  

I was glad to see the experiments, even though it is a theory paper.  They convinced me that the improvements were meaningful.

QUESTIONS:
N/A

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of constructing coresets for the robust geometric median in Euclidean space $\\mathbb{R}^d$. The key innovation is a method that removes the dependence on the number of outliers $m$ in the coreset size, which was a central limitation in prior works. The authors achieve this through a novel non-component-wise error analysis, leading to a coreset of size $\\tilde{O}(\\varepsilon^{-2} \\cdot \\min\\{\\varepsilon^{-2}, d\\})$ when $n \\geq 4m$. For $d = 1$, they obtain an optimal bound of $\\Theta(\\varepsilon^{-1/2} + \\frac{m}{n} \\varepsilon^{-1})$. The paper extends the results to robust $(k,z)$-clustering in various metric spaces. In terms of theoretical novelty, the paper introduces a novel non-component-wise error analysis, which enables a substantial reduction in the influence of outliers. The authors also provide empirical results on real-world datasets showing improved size-accuracy trade-offs and runtime.\n\nSTRENGTHS AND WEAKNESSES:\n$\\mathbf{Strengths}$: \n1. The paper is technically sound and presents significant theoretical advancements. The proposed non-component-wise error analysis is an elegant departure from prior component-wise methods, and the theoretical claims are backed with detailed proofs and matching lower bounds.\n\n2. The paper is well-organized and offers a detailed exposition of the techniques. However, clarity suffers due to occasional overuse of notation and a lack of high-level intuition in the main text. The manuscript would benefit from more concrete examples or illustrations, especially to help interpret the new error analysis approach.\n\n3. The elimination of $O(m)$ dependency is a notable contribution to the literature on robust coreset construction. However, the improvement is restricted to the regime where $n \\geq 4m$. Moreover, the practical impact is limited by the modest empirical gains over strong baselines. \n\n\n$\\mathbf{Weakness}$:\n\n1. The paper lacks a comparison with recent state-of-the-art literature such as Ding, H. and Wang, Z., 2020; Paul, D. et. al., 2021; Cohen-Addad, V. et. al., 2022 and related coreset literature. These should be included to better contextualize the contribution.\n\n2. The empirical section is limited in scope. While results are presented across 6 datasets, there is insufficient analysis on the robustness of the proposed methods under adversarial or heavy-tailed distributions.\n\n3. The paper makes some strong assumptions (e.g., $n \\geq 4m$, existence of VC-dimension bounds) but does not clearly discuss their practical validity or limitations.\n\n4. While the paper claims that the proposed methodology generalizes to broader robust clustering applications, this generalization is not rigorously substantiated—neither through theoretical guarantees nor through empirical demonstrations on such tasks.\n\n\n$\\mathbf{References:}$\n\nDing, H. and Wang, Z., 2020, November. Layered sampling for robust optimization problems. In International Conference on Machine Learning (pp. 2556-2566). PMLR.\n\nPaul, D., Chakraborty, S., Das, S. and Xu, J., 2021. Uniform concentration bounds toward a unified framework for robust clustering. Advances in Neural Information Processing Systems, 34, pp.8307-8319.\n\nCohen-Addad, V., Larsen, K.G., Saulpic, D. and Schwiegelshohn, C., 2022, June. Towards optimal lower bounds for k-median and k-means coresets. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (pp. 1038-1051).\n\nQUESTIONS:\n1. Does the proposed non-component-wise error analysis can be generalized to other robust learning tasks, such as robust regression or PCA, or is it fundamentally specific to clustering-like cost functions?\n\n2. How does the proposed method perform when the assumption $n \\geq 4m$ is violated in a controlled way?\n\n3. How does the coreset behave under adversarial or heavy-tailed conditions for the datasets? \n\n4. Is it possible to construct an $\\varepsilon$-coreset for the robust geometric median (or more generally, robust $(k,z)$-clustering) in high-dimensional Euclidean space $\\mathbb{R}^d$ with optimal or near-optimal size, without relying on uniform sampling of outliers or the ball range space framework—perhaps via alternative geometric or combinatorial constructions?\n\nLIMITATIONS:\nThe main theoretical results rely on the assumption $n \\geq 4m$, which may not hold in many practical scenarios, and the coreset bounds for $d > 1$ are not tight. The analysis depends on structural properties such as bounded VC-dimension, but the sensitivity of the method to violations of these assumptions is not studied. Computational aspects, including the overhead of constructing the coreset and its scalability to large-scale or streaming settings, are not addressed. The generalization to robust $(k, z)$-clustering is stated but not formally supported. Overall, the novelty of this paper is incremental and lacks thorough and extensive theoretical comparison, empirical validation, and contextualization within the broader literature.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nLet P be a set of n points in R^d, m>1 and epsilon>0. Given a center c in R^d, we denote by cost^m(P,c) the sum of Euclidean distances over the points of P to c, excluding the m farthest points.\n\nThe main result is a randomized algorithm that gets P, m>1 and epsilon>0, and return a subset of size ~d/epsilon^2 that approximates cost^m(P,c) to every given c in R^d, up to a multiplicative factor of 1+epsilon. It is independent of $m$.\nFor d=1, the size is ~1/sqrt(eps)+m/(epsilon * n).\nThere are also some generalization to k-means with more assumptions,\n\nSTRENGTHS AND WEAKNESSES:\nStrong:\n- This is a very fundamental problem that would probably affect many other types of related coresets in many areas.\n- Unlike previous coresets, the size is independent of m.\n- The theory seems correct and interesting.\n- The technique is novel, at least to me.\n- Code in C++.\n\nWeak:\n- \\tilde{O} is undefined, although it is used everywhere, including the main theorem.\n- Assumption F6 is critical but appears only in the appendix. It is not clear, since most of the terms there (i,r,Pi) are undefined.\n- A single center seems easy and less interesting then k-median/mean.\n- Paper [1] is not cited, although it is one of the few papers that handle this problem.\n- It was very hard to understand the pseudo-code. I Some figures would have helped.\n- Put points on the curves, when you actually made the experiment.\n- Datasets: Does it make sense to run your algorithm on these datasets? For what purposes?\n\n[1] article{DBLP:journals/algorithms/StatmanRF20,\n  author       = {Adiel Statman and\n                  Liat Rozenberg and\n                  Dan Feldman},\n  title        = {k-Means: Outliers-Resistant Clustering+++},\n  journal      = {Algorithms},\n  volume       = {13},\n  number       = {12},\n  pages        = {311},\n  year         = {2020},\n  url          = {https://doi.org/10.3390/a13120311},\n  doi          = {10.3390/A13120311},\n  timestamp    = {Sat, 09 Jan 2021 14:14:06 +0100},\n  biburl       = {https://dblp.org/rec/journals/algorithms/StatmanRF20.bib},\n  bibsource    = {dblp computer science bibliography, https://dblp.org}\n}\n\nQUESTIONS:\n- Can you commit to an open code upon acceptance?\n- What is the definition of \\tilde{O}?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}