Recovery of a mixture of Gaussians by sum-of-norms clustering

TL;DR

This paper demonstrates that sum-of-norms clustering can reliably recover Gaussian mixture components even with infinitely many samples, extending previous results by removing sample size restrictions.

Contribution

It extends the theoretical guarantees of sum-of-norms clustering to infinite sample sizes, using a novel characterization of clusters related to the agglomeration conjecture.

Findings

Sum-of-norms clustering recovers Gaussian mixtures with unlimited samples.

A new characterization of clusters was developed and proved.

The proof leverages and restates a key result from the agglomeration conjecture.

Abstract

Sum-of-norms clustering is a method for assigning $n$ points in $\mathbb{R}^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to lift this restriction, i.e., show that sum-of-norms clustering with equal weights can recover a mixture of Gaussians even as the number of samples tends to infinity. Our proof relies on an interesting characterization of clusters computed by sum-of-norms clustering that was developed inside a proof of the agglomeration conjecture by Chiquet et al. Because we believe this theorem has independent interest, we restate and reprove the Chiquet et al.\ result herein.