On Learning Mixtures of Well-Separated Gaussians

We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of $k$ standard spherical Gaussians, and the goal is to estimate the means up to accuracy $\delta$ using $poly(k,d, 1/\delta)$ samples. In this work, we study the following question: what is the minimum separation needed between the means for solving this task? The best known algorithm due to Vempala and Wang [JCSS 2004] requires a separation of roughly $\min\{k,d\}^{1/4}$. On the other hand, Moitra and Valiant [FOCS 2010] showed that with separation $o(1)$, exponentially many samples are required. We address the significant gap between these two bounds, by showing the following results. 1. We show that with separation $o(\sqrt{\log k})$, super-polynomially many samples are required. In fact, this holds even when the $k$ means of the Gaussians are picked at random in $d=O(\log k)$ dimensions. 2. We show that with separation $\Omega(\sqrt{\log k})$, $poly(k,d,1/\delta)$ samples suffice. Note that the bound on the separation is independent of $\delta$. This result is based on a new and efficient "accuracy boosting" algorithm that takes as input coarse estimates of the true means and in time $poly(k,d, 1/\delta)$ outputs estimates of the means up to arbitrary accuracy $\delta$ assuming the separation between the means is $\Omega(\min\{\sqrt{\log k},\sqrt{d}\})$ (independently of $\delta$). We also present a computationally efficient algorithm in $d=O(1)$ dimensions with only $\Omega(\sqrt{d})$ separation. These results together essentially characterize the optimal order of separation between components that is needed to learn a mixture of $k$ spherical Gaussians with polynomial samples.Comment: Appeared in FOCS 2017. 55 pages, 1 figur

A Probabilistic Analysis of EM for Mixtures of Separated, Spherical Gaussians

We show that, given data from a mixture of k well-separated spherical Gaussians in ℜ^d, a simple two-round variant of EM will, with high probability, learn the parameters of the Gaussians to near-optimal precision, if the dimension is high (d >> ln k). We relate this to previous theoretical and empirical work on the EM algorithm

Learning mixtures of separated nonspherical Gaussians

Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a local-search heuristic from Dempster, Laird and Rubin [J. Roy. Statist. Soc. Ser. B 39 (1977) 1-38]. These do not provably run in polynomial time. We present the first algorithm that provably learns the component Gaussians in time that is polynomial in the dimension. The Gaussians may have arbitrary shape, but they must satisfy a ``separation condition'' which places a lower bound on the distance between the centers of any two component Gaussians. The mathematical results at the heart of our proof are ``distance concentration'' results--proved using isoperimetric inequalities--which establish bounds on the probability distribution of the distance between a pair of points generated according to the mixture. We also formalize the more general problem of max-likelihood fit of a Gaussian mixture to unstructured data.Comment: Published at http://dx.doi.org/10.1214/105051604000000512 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org