Modeling Inverse Covariance Matricesby Basis Expansion

Abstract

This paper proposes a new covariance modeling technique for Gaussian mixture models. Specifically the inverse covariance (precision) matrix of each Gaussian is expanded in a rank-1 basis i.e., /spl Sigma//sub j//sup -1/=P/sub j/=/spl Sigma//sub k=1//sup D//spl lambda//sub k//sup j/a/sub k/a/sub k//sup T/, /spl lambda//sub k//sup j//spl isin//spl Ropf/,a/sub k//spl isin//spl Ropf//sup d/. A generalized EM algorithm is proposed to obtain maximum likelihood parameter estimates for the basis set {a/sub k/a/sub k//sup T/}/sub k=1//sup D/ and the expansion coefficients {/spl lambda//sub k//sup j/}. This model, called the extended maximum likelihood linear transform (EMLLT) model, is extremely flexible: by varying the number of basis elements from D=d to D=d(d+1)/2 one gradually moves from a maximum likelihood linear transform (MLLT) model to a full-covariance model. Experimental results on two speech recognition tasks show that the EMLLT model can give relative gains of up to 35% in the word error rate over a standard diagonal covariance model, 30% over a standard MLLT model.