The Guedon-Vershynin Semi-definite Programming Approach to Low Dimensional Embedding for Unsupervised Clustering

Abstract

This paper proposes a method for estimating the cluster matrix in the Gaussian mixture framework via Semi-Definite Programming. Theoretical error bounds are provided and a (non linear) low dimensional embedding of the data is deduced from the cluster matrix estimate. The method and its analysis is inspired by the work by Guédon and Vershynin on community detection in the stochastic block model. The adaptation is non trivial since the model is different and new Gaussian concentration arguments are needed. Our second contribution is a new Bregman-ADMM type algorithm for solving the semi-definite program and computing the embedding. This results in an efficient and scalable algorithm taking only the pairwise distances as input. The performance of the method is illustrated via Monte Carlo experiments and comparisons with other embeddings from the literature.