Selecting Bases in Spectral learning of Predictive State Representations via Model Entropy

Predictive State Representations (PSRs) are powerful techniques for modelling dynamical systems, which represent a state as a vector of predictions about future observable events (tests). In PSRs, one of the fundamental problems is the learning of the PSR model of the underlying system. Recently, spectral methods have been successfully used to address this issue by treating the learning problem as the task of computing an singular value decomposition (SVD) over a submatrix of a special type of matrix called the Hankel matrix. Under the assumptions that the rows and columns of the submatrix of the Hankel Matrix are sufficient~(which usually means a very large number of rows and columns, and almost fails in practice) and the entries of the matrix can be estimated accurately, it has been proven that the spectral approach for learning PSRs is statistically consistent and the learned parameters can converge to the true parameters. However, in practice, due to the limit of the computation ability, only a finite set of rows or columns can be chosen to be used for the spectral learning. While different sets of columns usually lead to variant accuracy of the learned model, in this paper, we propose an approach for selecting the set of columns, namely basis selection, by adopting a concept of model entropy to measure the accuracy of the learned model. Experimental results are shown to demonstrate the effectiveness of the proposed approach.