Sparse hidden Markov models for purer clusters

Sparse Hidden Markov Models for Speech Enhancement in Non-Stationary Noise Environments

“…The sparsity for the transition probabilities of the ARHMM can be encouraged by introducing the norm [26] to the second term of equation (14), and then maximizing (15) where is the previous parameters estimation of SARHMM, is the matrix of transition probabilities, is the regularization norm. Here, although the norm encourages sparsity, we cannot directly use the norm, because the transition probability and the observation probability are stochastic matrices, that is, their entries are non-negative and the summation of each row must be 1; the norm of each row is also 1, thus regularization is meaningless.…”

“…Here, although the norm encourages sparsity, we cannot directly use the norm, because the transition probability and the observation probability are stochastic matrices, that is, their entries are non-negative and the summation of each row must be 1; the norm of each row is also 1, thus regularization is meaningless. The in the norm is a regularization parameter, which encourages sparsity for [26]. By setting the derivative of equation (15) to zero and satisfying the constraints and for each state , we can obtain the update equation of transition probabilities of SARHMM: (16) where the maximization operation is added to make sure that all transition probabilities are greater than zero, and is the regularization term for the transition probability, which is defined as (17) where is the differential operator.…”

“…The model parameters of the ARHMM of [24] are estimated by maximizing their likelihood. Based on the approach for sparse HMMs described by [26], we introduce a prior distribution for the parameters that encourages the SARHMM to provide an approximation of the observed signal in terms of relatively few states. The introduction of a prior distribution can also be seen as the augmentation of the likelihood criterion with a regularization term.…”

“…The introduction of a prior distribution can also be seen as the augmentation of the likelihood criterion with a regularization term. From this perspective, we enforce the sparsity for the transition probabilities by means of an regularization [26] that drives low transition probability to zero and strengthens strong transition probabilities. Thus, the SARHMM presents fewer significant transition probabilities than the ARHMM.…”

“…Thus, the SARHMM presents fewer significant transition probabilities than the ARHMM. Sparsity can also be induced to the observation probability by an regularization [26], driving poorly supported posterior state probabilities to extinction while enforcing strong posterior state probabilities. As a result the new SARHMM model has more discriminative ability and encourages that only a few states have a significant contribution for any observation speech segment.…”