Let @math be an undirected graph with @math edges and @math vertices. We show that @math -dimensional Ising models on @math can be learned from @math i.i.d. samples within expected total variation distance some constant factor of @math , and that this rate is optimal. We show that the same rate holds for the class of @math -dimensional multivariate normal undirected graphical models with respect to @math . We also identify the optimal rate of @math for Ising models with no external magnetic field.
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 @math @math -dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses @math samples and runs in time @math , both significantly lower than previously known. The constant factor @math is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that @math 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 one-dimensional mixtures that uses @math samples and runs in time @math . Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the @math distance to an unknown underlying distribution.