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

Tao Jiang(tao.jiang***at***uwaterloo.ca)
Stephen Vavasis(vavasis***at***uwaterloo.ca)
Chen Wen Zhai(sabrina.zhai***at***edu.uwaterloo.ca)

Abstract: Sum-of-norms clustering is a method for assigning $n$ points in $\R^d$ 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.

Keywords: sum-of-norms clustering; convex clustering; recovery guarantee; mixture of Gaussians

Category 1: Applications -- Science and Engineering (Statistics )

Category 2: Linear, Cone and Semidefinite Programming (Second-Order Cone Programming )

Citation:

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Entry Submitted: 02/19/2019
Entry Accepted: 02/19/2019
Entry Last Modified: 02/19/2019

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