Modeling Inverse Covariance Matrices by 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., ∑j-1 = Pj = ∑k=1D λkjakakT, λkj ∈ ℝ, ak ∈ ℝd. A generalized EM algorithm is proposed to obtain maximum likelihood parameter estimates for the basis set {akakT} k=1D and the expansion coefficients {λ kj}. 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.