Coresets for Clustering Under Stochastic Noise

TL;DR

This paper develops coreset construction methods for clustering in noisy data environments, introducing new error metrics that improve coreset size and accuracy guarantees under stochastic noise.

Contribution

It introduces a novel error metric for coreset construction under stochastic noise and provides algorithms with provable guarantees that outperform classical metrics.

Findings

Coreset size can be reduced by up to a polynomial factor in k.

The new metric yields tighter approximation guarantees for true clustering cost.

Experimental results validate theoretical improvements and practical effectiveness.

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.