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来源：泰然健康网时间：2026年03月06日 14:04

Abstract

Utilizing the structure of a probabilistic model can significantly increase its learning speed. Motivated by several recent applications, in particular bigram models in language processing, we consider learning low-rank conditional probability matrices under expected KL-risk. This choice makes smoothing, that is the careful handling of low-probability elements, paramount. We derive an iterative algorithm that extends classical non-negative matrix factorization to naturally incorporate additive smoothing and prove that it converges to the stationary points of a penalized empirical risk. We then derive sample-complexity bounds for the global minimizer of the penalized risk and show that it is within a small factor of the optimal sample complexity. This framework generalizes to more sophisticated smoothing techniques, including absolute-discounting.

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