Connecting Weighted Automata, Tensor Networks and Recurrent Neural Networks through Spectral Learning
Abstract
In this paper, we present connections between three models used in different research fields: weighted finite automata~(WFA) from formal languages and linguistics, recurrent neural networks used in machine learning, and tensor networks which encompasses a set of optimization techniques for highorder tensors used in quantum physics and numerical analysis. We first present an intrinsic relation between WFA and the tensor train decomposition, a particular form of tensor network. This relation allows us to exhibit a novel low rank structure of the Hankel matrix of a function computed by a WFA and to design an efficient spectral learning algorithm leveraging this structure to scale the algorithm up to very large Hankel matrices.We then unravel a fundamental connection between WFA and secondorderrecurrent neural networks~(2RNN): in the case of sequences of discrete symbols, WFA and 2RNN with linear activationfunctions are expressively equivalent. Leveraging this equivalence result combined with the classical spectral learning algorithm for weighted automata, we introduce the first provable learning algorithm for linear 2RNN defined over sequences of continuous input vectors.This algorithm relies on estimating low rank subblocks of the Hankel tensor, from which the parameters of a linear 2RNN can be provably recovered. The performances of the proposed learning algorithm are assessed in a simulation study on both synthetic and realworld data.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10029
 arXiv:
 arXiv:2010.10029
 Bibcode:
 2020arXiv201010029L
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Formal Languages and Automata Theory
 EPrint:
 Accepted as a journal paper in Machine Learning Journal. arXiv admin note: text overlap with arXiv:1807.01406