Performance of Johnson-Lindenstrauss Transform for k-Means and k-Medians Clustering

TL;DR

This paper proves that Johnson-Lindenstrauss projections approximately preserve the cost of optimal and all clusterings in Euclidean k-means and k-medians, with nearly optimal dimension bounds, resolving open problems in the field.

Contribution

It establishes nearly optimal dimension bounds for Johnson-Lindenstrauss transforms preserving clustering costs, solving open problems for k-means and k-medians.

Findings

Optimal cost preservation within (1+ε) for k-means and k-medians.

Dimension bound of O(log(k/ε)/ε^2) is nearly optimal.

Results extend to p-th power Euclidean distances for any constant p.

Abstract

Consider an instance of Euclidean $k$-means or $k$-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\varepsilon)$ under a projection onto a random $O(\log(k / \varepsilon) / \varepsilon^2)$-dimensional subspace. Further, the cost of every clustering is preserved within $(1+\varepsilon)$. More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean $k$-clustering with the distances raised to the $p$-th power for any constant $p$. For $k$-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for $k$-medians, it answers a question raised by Kannan.