Convergence and Energy Landscape for Cheeger Cut Clustering
CALIFORNIA UNIV LOS ANGELES DEPT OF MATHEMATICS
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This paper provides both theoretical and algorithmic results for the l1-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 1-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 l1-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.
- Numerical Mathematics