Factoring Gaussian Precision Matrices for Linear Dynamic Models

The linear dynamic model (LDM), also known as the Kalman filter model, has been the subject of research in the engineering, control, and more recently, machine learning and speech technology communities. The Gaussian noise processes are usually assumed to have diagonal, or occasionally full, covariance matrices. A number of recent papers have considered modelling the precision rather than covariance matrix of a Gaussian distribution, and this work applies such ideas to the LDM. A Gaussian precision matrix P can be factored into the form P = UTSU where U is a transform and S a diagonal matrix. By varying the form of U, the covariance can be specified as being diagonal or full, or used to model a given set of spatial dependencies. Furthermore, the transform and scaling components can be shared between models, allowing richer distributions with only marginally more parameters than required to specify diagonal covariances. The method described in this paper allows the construction of models with an appropriate number of parameters for the amount of available training data. We provide illustrative experimental results on synthetic and real speech data in which models with factored precision matrices and automatically-selected numbers of parameters are as good as or better than models with diagonal covariances on small data sets and as good as models with full covariance matrices on larger data sets.