Semisupervised Learning of Hidden Markov Models via a Homotopy Method

Hidden Markov model (HMM) classifier design is considered for the analysis of sequential data, incorporating both labeled and unlabeled data for training; the balance between the use of labeled and unlabeled data is controlled by an allocation parameter lambda isin (0, 1), where lambda = 0 corresponds to purely supervised HMM learning (based only on the labeled data) and lambda = 1 corresponds to unsupervised HMM-based clustering (based only on the unlabeled data). The associated estimation problem can typically be reduced to solving a set of fixed-point equations in the form of a "natural-parameter homotopy." This paper applies a homotopy method to track a continuous path of solutions, starting from a local supervised solution (lambda = 0) to a local unsupervised solution (lambda = 1). The homotopy method is guaranteed to track with probability one from lambda = 0 to lambda = 1 if the lambda = 0 solution is unique; this condition is not satisfied for the HMM since the maximum likelihood supervised solution (lambda = 0) is characterized by many local optima. A modified form of the homotopy map for HMMs assures a track from lambda = 0 to lambda = 1. Following this track leads to a formulation for selecting lambda isin (0, 1) for a semisupervised solution and it also provides a tool for selection from among multiple local-optimal supervised solutions. The results of applying the proposed method to measured and synthetic sequential data verify its robustness and feasibility compared to the conventional EM approach for semisupervised HMM training.