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Clustering is a widely deployed unsupervised learning tool. Model-basedclustering is a flexible framework to tackle data heterogeneity when theclusters have different shapes. Likelihood-based inference for mixturedistributions often involves non-convex and high-dimensional objectivefunctions, imposing difficult computational and statistical challenges. Theclassic expectation-maximization (EM) algorithm is a computationally thriftyiterative method that maximizes a surrogate function minorizing thelog-likelihood of observed data in each iteration, which however suffers frombad local maxima even in the special case of the standard Gaussian mixturemodel with common isotropic covariance matrices. On the other hand, recentstudies reveal that the unique global solution of a semidefinite programming(SDP) relaxed $K$-means achieves the information-theoretically sharp thresholdfor perfectly recovering the cluster labels under the standard Gaussian mixturemodel. In this paper, we extend the SDP approach to a general setting byintegrating cluster labels as model parameters and propose an iterativelikelihood adjusted SDP (iLA-SDP) method that directly maximizes the exactobserved likelihood in the presence of data heterogeneity. By lifting thecluster assignment to group-specific membership matrices, iLA-SDP avoidscentroids estimation -- a key feature that allows exact recovery underwell-separateness of centroids without being trapped by their adversarialconfigurations. Thus iLA-SDP is less sensitive than EM to initialization andmore stable on high-dimensional data. Our numeric experiments demonstrate thatiLA-SDP can achieve lower mis-clustering errors over several widely usedclustering methods including $K$-means, SDP and EM algorithms.

Likelihood Adjusted Semidefinite Programs for Clustering Heterogeneous Data