The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known to be susceptible to local optima. In this paper, we first give a general characterization of the basin of attraction associated with any global optimum of the population likelihood. By exploiting this characterization, we provide non-asymptotic finite sample guarantees on the Baum-Welch updates, guaranteeing geometric convergence to a small ball of radius on the order of the minimax rate around a global optimum. As a concrete example, we prove a linear rate of convergence for a hidden Markov mixture of two isotropic Gaussians given a suitable mean separation and an initialization within a ball of large radius around (one of) the true parameters. To our knowledge, these are the first rigorous local convergence guarantees to global optima for the Baum-Welch algorithm in a setting where the likelihood function is nonconvex. We complement our theoretical results with thorough numerical simulations studying the convergence of the Baum-Welch algorithm and illustrating the accuracy of our predictions.
 We show that a simple spectral algorithm for learning a mixture of k spherical Gaussians in Rn works remarkably well--it succeeds in identifying the Gaussians assuming essentially the minimum possible separation between their centers that keeps them unique (solving an open problem of Arora and Kannan (Proceedings of the 33rd ACM STOC, 2001). The sample complexity and running time are polynomial in both n and k. The algorithm can be applied to the more general problem of learning a mixture of "weakly isotropic" distributions (e.g. a mixture of uniform distributions on cubes).  In recent years analysis of complexity of learning Gaussian mixture models from sampled data has received significant attention in computational machine learning and theory communities. In this paper we present the first result showing that polynomial time learning of multidimensional Gaussian Mixture distributions is possible when the separation between the component means is arbitrarily small. Specifically, we present an algorithm for learning the parameters of a mixture of k identical spherical Gaussians in n-dimensional space with an arbitrarily small separation between the components, which is polynomial in dimension, inverse component separation and other input parameters for a fixed number of components k. The algorithm uses a projection to k dimensions and then a reduction to the 1-dimensional case. It relies on a theoretical analysis showing that two 1-dimensional mixtures whose densities are close in the L norm must have similar means and mixing coefficients. To produce the necessary lower bound for the L norm in terms of the distances between the corresponding means, we analyze the behavior of the Fourier transform of a mixture of Gaussians in one dimension around the origin, which turns out to be closely related to the properties of the Vandermonde matrix obtained from the component means. Analysis of minors of the Vandermonde matrix together with basic function approximation results allows us to provide a lower bound for the norm of the mixture in the Fourier domain and hence a bound in the original space. Additionally, we present a separate argument for reconstructing variance. Mixtures of Gaussians are among the most fundamental and widely used statistical models. Current techniques for learning such mixtures from data are local search heuristics with weak performance guarantees. We present the first provably correct algorithm for learning a mixture of Gaussians. This algorithm is very simple and returns the true centers of the Gaussians to within the precision specified by the user with high probability. It runs in time only linear in the dimension of the data and polynomial in the number of Gaussians. Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has running time and data requirements polynomial in the dimension and the inverse of the desired accuracy, with provably minimal assumptions on the Gaussians. As a simple consequence of our learning algorithm, we we give the first polynomial time algorithm for proper density estimation for mixtures of k Gaussians that needs no assumptions on the mixture. It was open whether proper density estimation was even statistically possible (with no assumptions) given only polynomially many samples, let alone whether it could be computationally efficient. The building blocks of our algorithm are based on the work (Kalai , STOC 2010) that gives an efficient algorithm for learning mixtures of two Gaussians by considering a series of projections down to one dimension, and applying the method of moments to each univariate projection. A major technical hurdle in the previous work is showing that one can efficiently learn univariate mixtures of two Gaussians. In contrast, because pathological scenarios can arise when considering projections of mixtures of more than two Gaussians, the bulk of the work in this paper concerns how to leverage a weaker algorithm for learning univariate mixtures (of many Gaussians) to learn in high dimensions. Our algorithm employs hierarchical clustering and rescaling, together with methods for backtracking and recovering from the failures that can occur in our univariate algorithm. Finally, while the running time and data requirements of our algorithm depend exponentially on the number of Gaussians in the mixture, we prove that such a dependence is necessary. Hidden Markov Models (HMMs) are one of the most fundamental and widely used statistical tools for modeling discrete time series. In general, learning HMMs from data is computationally hard (under cryptographic assumptions), and practitioners typically resort to search heuristics which suffer from the usual local optima issues. We prove that under a natural separation condition (bounds on the smallest singular value of the HMM parameters), there is an efficient and provably correct algorithm for learning HMMs. The sample complexity of the algorithm does not explicitly depend on the number of distinct (discrete) observations-it implicitly depends on this quantity through spectral properties of the underlying HMM. This makes the algorithm particularly applicable to settings with a large number of observations, such as those in natural language processing where the space of observation is sometimes the words in a language. The algorithm is also simple, employing only a singular value decomposition and matrix multiplications.