Learning Mixtures of Spherical Gaussians via Fourier Analysis
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Suppose that we are given independent, identically distributed samples x_l from a mixture μ of no more than k of d-dimensional spherical gaussian distributions μ_i with variance 1, such that the minimum ℓ_2 distance between two distinct centers y_l and y_j is greater than √(d)Δ for some c ≤Δ, where c∈ (0,1) is a small positive universal constant. We develop a randomized algorithm that learns the centers y_l of the gaussians, to within an ℓ_2 distance of δ < Δ√(d)/2 and the weights w_l to within cw_min with probability greater than 1 - (-k/c). The number of samples and the computational time is bounded above by poly(k, d, 1/δ). Such a bound on the sample and computational complexity was previously unknown when ω(1) ≤ d ≤ O(log k). When d = O(1), this follows from work of Regev and Vijayaraghavan. These authors also show that the sample complexity of learning a random mixture of gaussians in a ball of radius Θ(√(d)) in d dimensions, when d is Θ( log k) is at least poly(k, 1/δ), showing that our result is tight in this case.
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