Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities

Wei Qian; Yuqian Zhang; Yudong Chen

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arXiv · 1906.06776

Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities

Wei Qian·Yuqian Zhang·Yudong Chen

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Abstract

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the convergence results for Gaussian mixtures. In particular, we make use of an angle-decreasing property for establishing global convergence of Least Squares EM beyond Gaussian settings, as $\ell_2$ distance contraction no longer holds globally for general log-concave mixtures.

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BibTeX RIS

Wei Qian, Yuqian Zhang, Yudong Chen. 2019-06-16. Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities. https://arxiv.org/abs/1906.06776

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