A Canonical Form for Weighted Automata and Applications to Approximate Minimization

We study the problem of constructing approximations to a weighted automaton. Weighted finite automata (WFA) are closely related to the theory of rational series. A rational series is a function from strings to real numbers that can be computed by a WFA. Among others, this includes probability distributions generated by hidden Markov models and probabilistic automata. The relationship between rational series and WFA is analogous to the relationship between regular languages and ordinary automata. Associated with such rational series are infinite matrices called Hankel matrices which play a fundamental role in the theory of minimal WFA. Our contributions are: (1) an effective procedure for computing the singular value decomposition (SVD) of such infinite Hankel matrices based on their finite representation in terms of WFA, (2) a new canonical form for WFA based on this SVD decomposition, and, (3) an algorithm to construct approximate minimizations of a given WFA. The goal of our approximate minimization algorithm is to start from a minimal WFA and produce a smaller WFA that is close to the given one in a certain sense. The desired size of the approximating automaton is given as input. We give bounds describing how well the approximation emulates the behavior of the original WFA. The study of this problem is motivated by the analysis of machine learning algorithms that synthesize weighted automata from spectral decompositions of finite Hankel matrices. It is known that when the number of states of the target automaton is correctly guessed, these algorithms enjoy consistency and finite-sample guarantees in the probably approximately correct (PAC) learning model. It has also been suggested that asking the learning algorithm to produce a model smaller than the true one will still yield useful models with reduced complexity. Our results in this paper vindicate these ideas and confirm intuitions provided by empirical studies. Beyond learning problems, our techniques can also be used to reduce the complexity of any algorithm working with WFA, at the expense of incurring a small, controlled amount of error.