Grammatical Inference as a Principal Component Analysis Problem
Raphael Bailly, François Denis and Liva Ralaivola
In: ICML 09, 14-18 June 2009, Montreal, Canada.
One of the main problems in probabilistic grammatical inference consists in inferring a stochastic language, i.e. a probability distribution, in some class of probabilistic models, from a sample of words independently drawn according to a fixed unknown target distribution p. Here we consider the class of rational stochastic languages composed of stochastic languages that can be computed by muliplicity automata, which can be viewed as a generalization of probabilistic automata. Rational stochastic languages p have a useful algebraic characterization: all the mappings up:v->p(uv) lie in a finite dimensional vector subspace Vp of the vector space R(E) composed of all real-valued functions defined over E. Hence, a first step in the grammatial inference process can consist in identifying the subspace Vp. In this paper, we study the possibility of using principal component analysis to achieve this task. We provide an inference algorithm which computes an estimate of the target distribution. We prove some theoretical properties of this algorithm and we provide results from numerical simulations that confirm the relevance of our approach.