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Coresets for Clustering Under Stochastic Noise

Lingxiao Huang, Zhize Li, Nisheeth K. Vishnoi, Runkai Yang, Haoyu Zhao

TL;DR

This work addresses coreset construction for $(k,z)$-clustering when the input data is corrupted by stochastic noise, a setting in which the true dataset $P$ is unobservable. It introduces two surrogate metrics, the classical $\mathsf{Err}$ and a new, noise-aware $\mathsf{Err}_{\alpha}$, and demonstrates that $\mathsf{Err}$ can yield overly pessimistic guarantees under noise. The authors propose two algorithms, $\mathbf{CN}$ and $\mathbf{CN}_{\alpha}$, with the latter leveraging cluster-wise sampling and structural assumptions to produce significantly smaller coresets and tighter bounds on the true clustering cost, validated by theory and experiments on real datasets. This work advances robust clustering in noisy environments and provides practical coreset mechanisms that preserve near-optimal clustering performance despite stochastic perturbations.

Abstract

We study the problem of constructing coresets for $(k, z)$-clustering when the input dataset is corrupted by stochastic noise drawn from a known distribution. In this setting, evaluating the quality of a coreset is inherently challenging, as the true underlying dataset is unobserved. To address this, we investigate coreset construction using surrogate error metrics that are tractable and provably related to the true clustering cost. We analyze a traditional metric from prior work and introduce a new error metric that more closely aligns with the true cost. Although our metric is defined independently of the noise distribution, it enables approximation guarantees that scale with the noise level. We design a coreset construction algorithm based on this metric and show that, under mild assumptions on the data and noise, enforcing an $\varepsilon$-bound under our metric yields smaller coresets and tighter guarantees on the true clustering cost than those obtained via classical metrics. In particular, we prove that the coreset size can improve by a factor of up to $\mathrm{poly}(k)$, where $n$ is the dataset size. Experiments on real-world datasets support our theoretical findings and demonstrate the practical advantages of our approach.

Coresets for Clustering Under Stochastic Noise

TL;DR

This work addresses coreset construction for -clustering when the input data is corrupted by stochastic noise, a setting in which the true dataset is unobservable. It introduces two surrogate metrics, the classical and a new, noise-aware , and demonstrates that can yield overly pessimistic guarantees under noise. The authors propose two algorithms, and , with the latter leveraging cluster-wise sampling and structural assumptions to produce significantly smaller coresets and tighter bounds on the true clustering cost, validated by theory and experiments on real datasets. This work advances robust clustering in noisy environments and provides practical coreset mechanisms that preserve near-optimal clustering performance despite stochastic perturbations.

Abstract

We study the problem of constructing coresets for -clustering when the input dataset is corrupted by stochastic noise drawn from a known distribution. In this setting, evaluating the quality of a coreset is inherently challenging, as the true underlying dataset is unobserved. To address this, we investigate coreset construction using surrogate error metrics that are tractable and provably related to the true clustering cost. We analyze a traditional metric from prior work and introduce a new error metric that more closely aligns with the true cost. Although our metric is defined independently of the noise distribution, it enables approximation guarantees that scale with the noise level. We design a coreset construction algorithm based on this metric and show that, under mild assumptions on the data and noise, enforcing an -bound under our metric yields smaller coresets and tighter guarantees on the true clustering cost than those obtained via classical metrics. In particular, we prove that the coreset size can improve by a factor of up to , where is the dataset size. Experiments on real-world datasets support our theoretical findings and demonstrate the practical advantages of our approach.

Paper Structure

This paper contains 36 sections, 21 theorems, 149 equations, 1 figure, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Noise models and metrics for coreset quality
  5. Theoretical results
  6. Proof of Theorem \ref{['thm:err']}: Using $\mathsf{Err}$
  7. Proof of Lemma \ref{['lm:err_bound']}: Bounding $\mathsf{Err}(\widehat{P}, P)$
  8. Lower bound of Err(P-hat, P)
  9. Proof of Theorem \ref{['thm:err_alpha']}: Using $\mathsf{Err}_{\alpha}$
  10. Proof of Lemma \ref{["lemma:structural property of C(P')"]}: structural properties of $\mathcal{C}_{\alpha}(S)$
  11. Useful properties of $\widehat{P}$.
  12. Proof of Lemma \ref{['lemma:err_alpha_step1']}: properties of $S$
  13. Proof of Lemma \ref{['lemma:err_alpha_step2']}: movement of centers of $P'$
  14. Justifying Assumption \ref{['assumption:additional']}: Finding center sets within local balls
  15. Empirical results
  16. ...and 21 more sections

Key Result

Theorem 3.1

Let $\widehat{P}$ be drawn from $P$ via noise model i@ with known $\theta \geq 0$. Let $\varepsilon \in (0,1)$ and fix $\alpha \geq 1$. Let $\mathcal{A}$ be an algorithm that constructs a weighted subset $S\subset \widehat{P}$ for $k$-Means of size $\mathcal{A}(\varepsilon)$ and with guarantee $\mat

Figures (1)

  • Figure 1: Empirical comparison of error metrics $\mathsf{Err}_1(\widehat{P}, P)$ and $\mathsf{Err}(\widehat{P}, P)$ on synthetic data as the separation level $\beta$ varies. While $\mathsf{Err}_1$ responds to changing cluster geometry, $\mathsf{Err}$ remains nearly constant, illustrating its insensitivity to structural properties in noisy settings.

Theorems & Definitions (47)

  • Definition 2.1: Noise model i@
  • Theorem 3.1: Coreset using $\mathsf{Err}$
  • Theorem 3.3: Coreset using the $\mathsf{Err}_\alpha$ metric
  • Theorem 4.1: Restatement of Theorem \ref{['thm:err']}
  • Lemma 4.2: Bounding $\mathsf{Err}(\widehat{P}, P)$
  • Lemma 4.3: Composition property
  • proof
  • proof : Proof of Theorem \ref{['thm:err']}
  • Claim 4.4: Bounding $\sum_{p\in P} \|\xi_p\|_2^2$
  • proof
  • ...and 37 more