Modeling inverse covariance matrices by basis expansion

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., Σ_j^-1=P_j=Σ_k=1^Dλ_k^ja_ka_k^T, λ_k^j∈R,a_k∈R^d. A generalized EM algorithm is proposed to obtain maximum likelihood parameter estimates for the basis set {a_ka_k^T}_k=1^D and the expansion coefficients {λ_k^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.