Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities
This provides a theoretical guarantee for a variant of EM in a broader class of mixture models, addressing a foundational problem in statistical estimation with potential applications in signal processing or machine learning, though it is an incremental extension from Gaussian to log-concave settings.
The paper tackles the location estimation problem for a mixture of two rotation invariant log-concave densities, demonstrating that Least Squares EM converges globally to the true location parameter from random initialization with explicit convergence rates and sample complexity bounds dependent on signal-to-noise ratio and tail properties.
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.