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.
An upper bound on the order of @math for estimating @math -dimensional normals can be obtained via empirical mean and covariance estimation ( [Theorem B.1] 2017-abbas ) or via Yatracos' techniques based on VC-dimension ( [Theorem 13] abbas-mixtures ). Regarding lower bounds, Acharya, Jafarpour, Orlitsky, and Suresh [Theorem 2] acharya-lower-bound proved a lower bound on the order of @math for spherical normals ( normals with identity covariance matrix), which implies the same lower bound for general normals. The lower bound for general normals was recently improved to a constant factor of @math by Asthiani, Ben-David, Harvey, Liaw, Mehrabian, and Plan @cite_17 . In comparison, our result shaves off the logarithmic factor. Moreover, their result is nonconstructive and relies on the probabilistic method, while our argument is fully deterministic.
@cite_17
2949977662
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.