Some improvements of the spectral learning approach for probabilistic grammatical inference

International audienceSpectral methods propose new and elegant solutions in probabilistic grammatical inference. We propose two ways to improve them. We show how a linear representation, or equivalently a weighted automata, output by the spectral learning algorithm can be taken as an initial point for the Baum Welch algorithm, in order to increase the likelihood of the observation data. Secondly, we show how the inference problem can naturally be expressed in the framework of Structured Low-Rank Approximation. Both ideas are tested on a benchmark extracted from the PAutomaC challenge

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

Unsupervised spectral learning of WCFG as low-rank matrix completion

We derive a spectral method for unsupervised learning ofWeighted Context Free Grammars. We frame WCFG induction as finding a Hankel matrix that has low rank and is linearly constrained to represent a function computed by inside-outside recursions. The proposed algorithm picks the grammar that agrees with a sample and is the simplest with respect to the nuclear norm of the Hankel matrix.Peer ReviewedPreprin

Sp2Learn: A Toolbox for the spectral learning of weighted automata *

Abstract Sp2Learn is a Python toolbox for the spectral learning of weighted automata from a set of strings, licensed under Free BSD. This paper gives the main formal ideas behind the spectral learning algorithm and details the content of the toolbox. Use cases and an experimental section are also provided