Sketch-and-solve approaches to k-means clustering by semidefinite programming

TL;DR

This paper presents a sketch-and-solve method that accelerates semidefinite programming for k-means clustering, providing optimality certification or bounds without assumptions on data distribution.

Contribution

It introduces a data-driven approach that efficiently approximates k-means solutions and certifies their optimality or provides bounds, improving computational speed and reliability.

Findings

Accelerates semidefinite relaxation computations.

Certifies approximate optimality of k-means solutions.

Provides high-confidence lower bounds on k-means objective.

Abstract

We introduce a sketch-and-solve approach to speed up the Peng-Wei semidefinite relaxation of k-means clustering. When the data is appropriately separated we identify the k-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal k-means value. This lower bound is data-driven; it does not make any assumption on the data nor how it is generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.