Learning hidden Markov sparse models

This paper considers the problem of separating streams of unknown non-stationary signals from under-determined mixtures of sources. The source signals are modeled as a hidden Markov model (HMM) where each state in the Markov chain is determined by a set of on (i.e., active) or off (i.e., inactive) states of the sources, with some unknown probability density functions (pdfs) in the on-state. Under the assumption that the number of active sources is small compared to the total number of sources (thus the sources are sparse), the goal is to recursively estimate the HMM state and the over-complete mixing matrix (subsequently the source signals) for signal recovery. The proposed approach combines the techniques of HMM-based filtering and manifold-based dictionary learning for estimating both the state and the mixing matrix. Specifically, we model the on/off state of the source signals as a hidden Markov model. In particular, we consider only a sparse set of simultaneously active sources. Thus, this setting generalizes the typical scenario considered in dictionary learning in which there is a sparse number of temporally independent active signals. To extract the activity profile of the sources from the observations, a technique known as change-of-measure is used to decouple the observations from the sources by introducing a new probability measure over the set of observations. Under this new measure, the un-normalized conditional densities of the state and the transition matrix of the Markov chain can be computed recursively. Due to the scaling ambiguity of the mixing matrix, we introduce an equivalence relation, which partitions the set of mixing matrices into a set of equivalence classes. Rather than estimating the mixing matrix by imposing the unit-norm constraint, the proposed algorithm searches directly for an equivalence class that contains the true mixing matrix. In our simulations, the proposed recursive algorithm with manifold-based dictionary learning, compared to algorithms with unit-norm constraint, estimates the mixing matrix more efficiently while maintaining high accuracy.