The Hidden Convexity of Spectral Clustering

In recent years, spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain "contrast function" over a sphere. These algorithms are simple to implement, efficient and, unlike most of the existing algorithms for multiclass spectral clustering, are not initialization-dependent. Moreover, they are applicable without modification for normalized and un-normalized clustering, which are two common variants of spectral clustering. Geometrically, the proposed algorithms can be interpreted as recovering a discrete weighted simplex by means of function optimization. We give complete necessary and sufficient conditions on contrast functions for the optimization to guarantee recovery of clusters. We show how these conditions can be interpreted in terms of certain "hidden convexity" of optimization over a sphere.