Approximating probabilistic models as weighted finite automata

Weighted finite automata (WFA) are often used to represent probabilistic models, such as $n$-gram language models, since they are efficient for recognition tasks in time and space. The probabilistic source to be represented as a WFA, however, may come in many forms. Given a generic probabilistic model over sequences, we propose an algorithm to approximate it as a weighted finite automaton such that the Kullback-Leiber divergence between the source model and the WFA target model is minimized. The proposed algorithm involves a counting step and a difference of convex optimization, both of which can be performed efficiently. We demonstrate the usefulness of our approach on various tasks, including distilling $n$-gram models from neural models, building compact language models, and building open-vocabulary character models.