Coresets for Weight-Constrained Anisotropic Assignment and Clustering

TL;DR

This paper develops smaller coresets for weight-constrained anisotropic clustering, improving previous bounds and revealing limitations of importance sampling techniques due to high sensitivity in the constrained setting.

Contribution

It generalizes and improves coreset constructions for weight-constrained anisotropic clustering, reducing coreset size and demonstrating the limitations of importance sampling methods.

Findings

Coreset size reduced to O(k^2/ε^{d+1}) for constrained clustering.

Total sensitivity can be as large as the dataset size in the constrained case.

Importance sampling techniques are often inapplicable due to high sensitivity.

Abstract

The present paper constructs coresets for weight-constrained anisotropic assignment and clustering. In contrast to the well-studied unconstrained least-squares clustering problem, approximating the centroids of the clusters no longer suffices in the weight-constrained anisotropic case, as even the assignment of the points to best sites is involved. This assignment step is often the limiting factor in materials science, a problem that partially motivates our work. We build on a paper by Har-Peled and Kushal, who constructed coresets of size $\mathcal{O}\bigl(\frac{k^3}{\epsilon^{d+1}}\bigr)$ for unconstrained least-squares clustering. We generalize and improve on their results in various ways, leading to even smaller coresets with a size of only $\mathcal{O}\bigl(\frac{k^2}{\epsilon^{d+1}}\bigr)$ for weight-constrained anisotropic clustering. Moreover, we answer an open question on coreset designs in the negative, by showing that the total sensitivity can become as large as the cardinality of the original data set in the constrained case. Consequently, many techniques based on importance sampling do not apply to weight-constrained clustering.