Sketch-and-Lift: Scalable Subsampled Semidefinite Program for $K$-means Clustering

TL;DR

This paper introduces a scalable, linear-time algorithm for approximate $K$-means clustering using a subsampled SDP approach, achieving high accuracy and similar recovery thresholds as full SDP methods.

Contribution

The paper presents the sketch-and-lift (SL) method, a novel scalable approach that approximates SDP relaxation for $K$-means clustering with theoretical guarantees and practical efficiency.

Findings

SL achieves similar exact recovery thresholds as full SDP.

SL outperforms state-of-the-art fast clustering algorithms in accuracy.

SL significantly reduces runtime while maintaining high statistical accuracy.

Abstract

Semidefinite programming (SDP) is a powerful tool for tackling a wide range of computationally hard problems such as clustering. Despite the high accuracy, semidefinite programs are often too slow in practice with poor scalability on large (or even moderate) datasets. In this paper, we introduce a linear time complexity algorithm for approximating an SDP relaxed $K$-means clustering. The proposed sketch-and-lift (SL) approach solves an SDP on a subsampled dataset and then propagates the solution to all data points by a nearest-centroid rounding procedure. It is shown that the SL approach enjoys a similar exact recovery threshold as the $K$-means SDP on the full dataset, which is known to be information-theoretically tight under the Gaussian mixture model. The SL method can be made adaptive with enhanced theoretic properties when the cluster sizes are unbalanced. Our simulation experiments demonstrate that the statistical accuracy of the proposed method outperforms state-of-the-art fast clustering algorithms without sacrificing too much computational efficiency, and is comparable to the original $K$-means SDP with substantially reduced runtime.