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We consider the problem of learning low-dimensional representations forlarge-scale Markov chains. We formulate the task of representation learning asthat of mapping the state space of the model to a low-dimensional state space,called the kernel space. The kernel space contains a set of meta states whichare desired to be representative of only a small subset of original states. Topromote this structural property, we constrain the number of nonzero entries ofthe mappings between the state space and the kernel space. By imposing thedesired characteristics of the representation, we cast the problem as aconstrained nonnegative matrix factorization. To compute the solution, wepropose an efficient block coordinate gradient descent and theoreticallyanalyze its convergence properties.

Identifying Sparse Low-Dimensional Structures in Markov Chains: A Nonnegative Matrix Factorization Approach