Sparse Markov source estimation via transformed Lasso

Teemu Roos, Bin Yu

Abstract: We establish a connection between Lasso-type ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> regularization and learning variable length Markov chains (VLMCs). This is achieved by a parameterization of discrete-valued finite-memory Markov sources in which setting a parameter value equal to zero is equivalent to eliminating a node in the corresponding context tree model. The parameterization involves a Haar wavelet transformation on a set of indicator functions, the output of which is mapped to symbol probabilities via logistic regression. The optimization problem is convex and can be solved efficiently using existing tools. We present preliminary results, comparing the method to an earlier algorithm for learning VLMCs in terms of model selection and prediction performance. We also discuss other transformations which lead to a flexible family of sparse representations of Markov sources.