Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any k d-dimensional spherical Gaussians, we derive an intuitive spectral-estimator samples and runs in time O k,ǫ (d 3 log 5 d), both significantly lower than previously known. The constant factor O k is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that Ω k d ǫ 2 samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for k-component one-dimensional mixtures that uses O k log k ǫ ǫ 2 samples and runs in timeÕ k ǫ 3k+1 . Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the ℓ 1 distance to an unknown underlying distribution.
For mixtures of spherical Gaussians, a polynomial time algorithm for the realizable setting with sample complexity O(dk 9 log 2 (d)/ε 4 ) was proposed in REF Theorem 11] .
Near-optimal-sample estimators for spherical Gaussian mixtures
ArXiv
ArXiv
Mathematics Computer Science