Tight Continuous Relaxation of the Balanced k-Cut Problem
Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced k-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced k-cut criteria.
The paper presents a novel algorithm to solve the balanced k-cut problem. The k-cut criterion is defined by using the Lovasz extension of a set function. A relaxation of balanced k-cut functionals can be formulated using Lovasz extensions of cuts and of normalizing constraints. The appealing feature of this relaxation is the connection between convex Lovasz extensions and submodular set functions. The authors propose an algorithm that minimizes a sum of auxiliary variables that are lower-bounded by ratios. The minimization is further constraint by simplex and unique membership constraints. Clarity: The paper is very well written and it appropriately refers to prior art. Originality: There exists significant prior art and the novel idea modifies the optimization strategy. I consider the originality and in particular the novelty as limited but its significance due to the success in the experiments might justify publication of this paper at NIPS. Significance: Due to the surprisingly clear superiority of the new method, this paper might show high impact in the future despite its high similarity with prior art. Minor comments:Figure 1 has five subfigures but only four are explained in the caption. What is the fifth subfigure about?