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.
These latter two results are abstract, applicable to a broad class of HMMs. We then specialize them to the case of a hidden Markov mixture consisting of two isotropic components, with means separated by a constant distance, and obtain concrete guarantees for this model. It is worth comparing these results to past work in the i.i.d. setting, for which the problem of Gaussian mixture estimation under various separation assumptions has been extensively studied (e.g., @cite_23 @cite_4 @cite_36 @cite_31 ). The constant distance separation required in our work is much weaker than the separation assumptions imposed in papers that focus on correctly labeling samples in a mixture model. Our separation condition is related to, but in general incomparable with the non-degeneracy requirements in other work @cite_12 @cite_30 @cite_7 .
@cite_30 @cite_4 @cite_7 @cite_36 @cite_23 @cite_31 @cite_12
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Hidden Markov model speech recognition systems typically use Gaussian mixture models to estimate the distributions of decorrelated acoustic feature vectors that correspond to individual subword units. By contrast, hybrid connectionist-HMM systems use discriminatively-trained neural networks to estimate the probability distribution among subword units given the acoustic observations. In this work we show a large improvement in word recognition performance by combining neural-net discriminative feature processing with Gaussian-mixture distribution modeling. By training the network to generate the subword probability posteriors, then using transformations of these estimates as the base features for a conventionally-trained Gaussian-mixture based system, we achieve relative error rate reductions of 35 or more on the multicondition Aurora noisy continuous digits task. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. 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. We consider the estimation of the number of hidden states (the order) of a discrete-time finite-alphabet hidden Markov model (HMM). The estimators we investigate are related to code-based order estimators: penalized maximum-likelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan (1994). We prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works. We prove a version of Stein's lemma for HMM order estimation and derive an upper bound on underestimation exponents. Then we prove that this upper bound can be achieved by the penalized ML estimator and by the penalized mixture estimator. The proof of the latter result gets around the elusive nature of the ML in HMM by resorting to large-deviation techniques for empirical processes. Finally, we prove that for any consistent HMM order estimator, for most HMM, the overestimation exponent is null.