Optimal Clustering of Discrete Mixtures: Binomial, Poisson, Block Models, and Multi-layer Networks

TL;DR

This paper establishes the fundamental limits of clustering in multi-layer networks and discrete mixture models, proposing a novel two-stage algorithm that achieves minimax optimal error rates even in sparse networks.

Contribution

It introduces a new two-stage clustering method that is minimax optimal for multi-layer stochastic block models and discrete mixture models, extending theoretical understanding and practical algorithms.

Findings

The proposed algorithm achieves minimax optimal error rates.

It works effectively under extreme network sparsity.

Validated by simulations and real data experiments.

Abstract

In this paper, we first study the fundamental limit of clustering networks when a multi-layer network is present. Under the mixture multi-layer stochastic block model (MMSBM), we show that the minimax optimal network clustering error rate, which takes an exponential form and is characterized by the Renyi divergence between the edge probability distributions of the component networks. We propose a novel two-stage network clustering method including a tensor-based initialization algorithm involving both node and sample splitting and a refinement procedure by likelihood-based Lloyd algorithm. Network clustering must be accompanied by node community detection. Our proposed algorithm achieves the minimax optimal network clustering error rate and allows extreme network sparsity under MMSBM. Numerical simulations and real data experiments both validate that our method outperforms existing methods. Oftentimes, the edges of networks carry count-type weights. We then extend our methodology and analysis framework to study the minimax optimal clustering error rate for mixture of discrete distributions including Binomial, Poisson, and multi-layer Poisson networks. The minimax optimal clustering error rates in these discrete mixtures all take the same exponential form characterized by the Renyi divergences. These optimal clustering error rates in discrete mixtures can also be achieved by our proposed two-stage clustering algorithm.