On Learning Mixtures of Well-Separated Gaussians
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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 δ using poly(k,d, 1/δ) 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 {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(√( 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( k) dimensions. 2. We show that with separation Ω(√( k)), poly(k,d,1/δ) samples suffice. Note that the bound on the separation is independent of δ. 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/δ) outputs estimates of the means up to arbitrary accuracy δ assuming the separation between the means is Ω({√( k),√(d)}) (independently of δ). We also present a computationally efficient algorithm in d=O(1) dimensions with only Ω(√(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.
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