Coresets for Clustering with Noisy Data

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

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Keywords: clustering, noise, coreset

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TL;DR: We study the problem of data reduction for clustering in the presence of stochastic noise, propose a measure that better captures the quality of a coreset for this setting and show its effectiveness theoretically and empirically.

Abstract:

We study the problem of data reduction for clustering when the input dataset $\widehat{P}$ is a noisy version of the true dataset $P$.
Motivation for this problem derives from settings where data is obtained from inherently noisy measurements or noise is added to data for privacy or robustness reasons.
In the noise-free setting, coresets have been proposed as a solution to this data reduction problem -- a coreset is a subset $S$ of $P$ that comes with a guarantee that the maximum difference, over all center sets, in cost of the center set for $S$ versus that of $P$ is small.
We find that this well-studied measure which determines the quality of a coreset is too strong when the data is noisy because the change in the cost of the optimal center set in the case $S=\widehat{P}$ when compared to that of $P$ can be much smaller than other center sets.
To bypass this, we consider a modification of this measure by 1) restricting only to approximately optimal center sets and 2) considering the *ratio* of the cost of $S$ for a given center set to the minimum cost of $S$ over all approximately optimal center sets.
This new measure allows us to get refined estimates on the quality of the optimal center set of a coreset as a function of the noise level.
Our results apply to a wide class of noise models satisfying certain bounded-moment conditions that include Gaussian and Laplace distributions.
Our results are not algorithm-dependent and can be used to derive estimates on the quality of a coreset produced by any algorithm in the noisy setting.
Empirically, we present results on the performance of coresets obtained from noisy versions of real-world datasets, verifying our theoretical findings and implying that the variance of noise is the main characterization of the coreset performances.

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Submission Number: 5866