Tight Continuous Relaxation of the Balanced $k$-Cut Problem

• Main Authors: Rangapuram, Syama Sundar, Mudrakarta, Pramod Kaushik, Hein, Matthias
• Format: Journal Article
• Language: English
• Published: 24.05.2015
• Subjects: Computer Science - Learning; Statistics - Machine Learning
• DOI: 10.48550/arxiv.1505.06478

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.