Convolutional density estimation in hidden Markov models for speech recognition

In continuous density Hidden Markov Models (HMMs) for speech recognition, the probability density function (pdf) for each state is usually expressed as a mixture of Gaussians. In this paper, we present a model in which the pdf is expressed as the convolution of two densities. We focus on the special case where one of the convolved densities is a M-Gaussian mixture, and the other is a mixture of N impulses. We present the reestimation formulae for the parameters of the M x N convolutional model, and suggest two ways for initializing them, the residual K-Means approach, and the deconvolution from a standard HMM with MN Gaussians per state using a genetic algorithm to search for the optimal assignment of Gaussians. Both methods result in a compact representation that requires only O(M + N) storage space for the model parameters, and O(MN) time for training and decoding. We explain how the decoding time can be reduced to O(M + kN), where k < M. Finally results are shown on the 1996 Hub-rf Development test, demonstrating that a 32 x 2 convolutional model can achieve performance comparable to that of a standard 64-Gaussian per state model.