Matrix Factorization and Stochastic State Representations

Given a two-point finite valued process, we consider the problem of finding an underlying two-point state process such that the output at a certain time instant is a probabilistic function of the state at the same time instant. This problem is related to the hidden Markov realization problem for finite valued processes. It is shown that the problem is equivalent to the algebraic problem of decomposing a square nonnegative matrix P as VAT^T with A and V nonnegative. Both multiplicative update formulas and a heuristic approach, are proposed for the solution of this decomposition problem. A simulation example shows the effectiveness of the proposed methods