Community detection is a fundamental statistical problem in network data analysis. Many algorithms have been proposed to tackle this problem. Most of these algorithms are not guaranteed to achieve the statistical optimality of the problem, while procedures that achieve information theoretic limits for general parameter spaces are not computationally tractable. In this paper, we present a computationally feasible two-stage method that achieves optimal statistical performance in misclassification proportion for stochastic block model under weak regularity conditions. Our two-stage procedure consists of a refinement stage motivated by penalized local maximum likelihood estimation. This stage can take a wide range of weakly consistent community detection procedures as initializer, to which it applies and outputs a community assignment that achieves optimal misclassification proportion with high probability. The practical effectiveness of the new algorithm is demonstrated by competitive numerical results.

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Spectral clustering is one of the most popular algorithms to group high dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been fully understood. The spectral clustering algorithm is often used as a consistent initializer for more sophisticated clustering algorithms. However, in this paper, we show that spectral clustering is actually already optimal in the Gaussian Mixture Model, when the number of clusters of is fixed and consistent clustering is possible. Contrary to that spectral gap conditions are widely assumed in literature to analyze spectral clustering, these conditions are not needed in this paper to establish its optimality.

We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, and signs of regression coefficients from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd's algorithm and the iterative feature matching algorithm. A linear convergence result for the proposed algorithm is established in this paper under appropriate abstract conditions on stochastic errors and initialization. We illustrate our general theory by applying it on three representative problems: clustering in Gaussian mixture model, approximate ranking, and sign recovery in compressed sensing, and show that minimax rate is achieved in each case.

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