Title

Convergence and Energy Landscape for Cheeger Cut Clustering

Authors

X Bresson
T Laurent
David Uminsky, University of San FranciscoFollow
J von Brecht

Document Type

Conference Proceeding

Publication Date

2012

Abstract

This paper provides both theoretical and algorithmic results for the l ₁-relaxation of the Cheeger cut problem. The l₂- relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The l₁-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The l₁-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for algorithms that minimize the l₁-relaxation. The second challenge entails comprehending the l₁-energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that l ₁-algorithms can get trapped in local minima that are not globally optimal and we provide a classification theorem to interpret these local minima. This classification gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the l₁-relaxation provides the solution of the original Cheeger cut problem.

Comments

Presented at: 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012; Lake Tahoe, NV; United States; 3 December 2012 through 6 December 2012

Published by MIT Press

Recommended Citation

Bresson, X., Laurent, T., Uminsky, D., & von Brecht, J.H. (2012). Convergence and energy landscape for cheeger cut clustering. Advances in Neural Information Processing Systems 25 (NIPS 2012), 1394-1402.