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Front. Appl. Math. Stat. | doi: 10.3389/fams.2019.00041

The Guedon-Vershynin Semi-Definite Programming approach to low dimensional embedding for unsupervised clustering

 Stephane Chretien1*,  Clement Dombry2 and Adrien Faivre2
  • 1National Physical Laboratory, United Kingdom
  • 2Université Bourgogne Franche-Comté, France

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 Gu\'edon 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. We propose furthermore an eigenvalue optimization approach 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.

Keywords: Semi-Definite Program (SDP), Clustering (un-supervised) algorithms, Gaussian mixture (GM) model, embedding, Convex relation

Received: 19 Sep 2018; Accepted: 18 Jul 2019.

Edited by:

Daniel Potts, Technische Universität Chemnitz, Germany

Reviewed by:

Junhong Lin, École Polytechnique Fédérale de Lausanne, Switzerland
Uwe Schwerdtfeger, Technische Universität Chemnitz, Germany  

Copyright: © 2019 Chretien, Dombry and Faivre. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Prof. Stephane Chretien, National Physical Laboratory, Teddington, United Kingdom,