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Cluster Analysis is Convex

Madhushini Narayana Prasad (madhushini***at***utexas.edu)
Grani A. Hanasusanto (grani.hanasusanto***at***utexas.edu)

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.

Keywords: K-means, cluster analysis, semidefinite programming, copositive programming, conic programming

Category 1: Convex and Nonsmooth Optimization

Category 2: Linear, Cone and Semidefinite Programming

Category 3: Combinatorial Optimization


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Entry Submitted: 06/19/2017
Entry Accepted: 06/19/2017
Entry Last Modified: 06/21/2017

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