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Similarity clustering in the presence of outliers: Exact recovery via convex program

We study the problem of clustering a set of data points based on their similarity matrix, each entry of which represents the similarity between the corresponding pair of points. We propose a convex-optimization-based algorithm for clustering using the similarity matrix, which has provable recovery guarantees. It needs no prior knowledge of the number of clusters and it behaves in a robust way in the presence of outliers and noise. Using a generative stochastic model for the similarity matrix (which can be thought of as a generalization of the classical Stochastic Block Model) we obtain precise bounds (not orderwise) on the sizes of the clusters, the number of outliers, the noise variance, separation between the mean similarities inside and outside the clusters and the values of the regularization parameter that guarantee the exact recovery of the clusters with high probability. The theoretical findings are corroborated with extensive evidence from simulations.