The Annals of Statistics

CHIME: Clustering of high-dimensional Gaussian mixtures with EM algorithm and its optimality

T. Tony Cai, Jing Ma, and Linjun Zhang

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Unsupervised learning is an important problem in statistics and machine learning with a wide range of applications. In this paper, we study clustering of high-dimensional Gaussian mixtures and propose a procedure, called CHIME, that is based on the EM algorithm and a direct estimation method for the sparse discriminant vector. Both theoretical and numerical properties of CHIME are investigated. We establish the optimal rate of convergence for the excess misclustering error and show that CHIME is minimax rate optimal. In addition, the optimality of the proposed estimator of the discriminant vector is also established. Simulation studies show that CHIME outperforms the existing methods under a variety of settings. The proposed CHIME procedure is also illustrated in an analysis of a glioblastoma gene expression data set and shown to have superior performance.

Clustering of Gaussian mixtures in the conventional low-dimensional setting is also considered. The technical tools developed for the high-dimensional setting are used to establish the optimality of the clustering procedure that is based on the classical EM algorithm.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1234-1267.

Received: October 2017
First available in Project Euclid: 13 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62C20: Minimax procedures 62H35: Image analysis

High-dimensional data unsupervised learning Gaussian mixture model EM algorithm misclustering error Minimax optimality


Cai, T. Tony; Ma, Jing; Zhang, Linjun. CHIME: Clustering of high-dimensional Gaussian mixtures with EM algorithm and its optimality. Ann. Statist. 47 (2019), no. 3, 1234--1267. doi:10.1214/18-AOS1711.

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Supplemental materials

  • Supplement to “CHIME: Clustering of high-dimensional Gaussian mixtures with EM algorithm and its optimality”. This supplement provides detailed proofs of the Theorem 3.1 and 3.3, which are respectively the upper and lower bounds of the estimation error for $\boldsymbol{\beta}^{*}$. All technical lemmas used throughout the paper are also proved. In addition, extra simulation results are provided in the supplement.