EM Converges for a Mixture of Many Linear Regressions

We study the convergence of the Expectation-Maximization (EM) algorithm for mixtures of linear regressions with an arbitrary number $k$ of components. We show that as long as signal-to-noise ratio (SNR) is $\tilde{\Omega}(k)$, well-initialized EM converges to the true regression parameters. Previous results for $k \geq 3$ have only established local convergence for the noiseless setting, i.e., where SNR is infinitely large. Our results enlarge the scope to the environment with noises, and notably, we establish a statistical error rate that is independent of the norm (or pairwise distance) of the regression parameters. In particular, our results imply exact recovery as $\sigma \rightarrow 0$, in contrast to most previous local convergence results for EM, where the statistical error scaled with the norm of parameters. Standard moment-method approaches may be applied to guarantee we are in the region where our local convergence guarantees apply.