# Improved Conic Reformulations for K-means Clustering

## TL;DR

This paper introduces a new conic reformulation of K-means clustering that leads to a tighter SDP relaxation and a novel approximation algorithm, improving cluster identification and outperforming existing methods.

## Contribution

It presents a polynomial-sized conic reformulation of K-means, a tighter SDP relaxation, and a new approximation algorithm with empirical performance gains.

## Key findings

- Tighter SDP relaxation for K-means clustering.
- New approximation algorithm outperforms existing methods.
- Reformulation enables better cluster identification.

## Abstract

In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes.