Connecting Weighted Automata, Tensor Networks and Recurrent Neural Networks through Spectral Learning

• URL: http://arxiv.org/abs/2010.10029v2
• Date: Thu, 6 Jan 2022 19:06:45 GMT
• Title: Connecting Weighted Automata, Tensor Networks and Recurrent Neural Networks through Spectral Learning
• Authors: Tianyu Li, Doina Precup, Guillaume Rabusseau
• Abstract summary: 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. We introduce the first provable learning algorithm for linear 2-RNN defined over sequences of continuous vectors input.
• Score: 58.14930566993063
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 high-order 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 second-orderrecurrent neural networks~(2-RNN): in the case of sequences of discrete symbols, WFA and 2-RNN 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 2-RNN defined over sequences of continuous input vectors.This algorithm relies on estimating low rank sub-blocks of the Hankel tensor, from which the parameters of a linear 2-RNN can be provably recovered. The performances of the proposed learning algorithm are assessed in a simulation study on both synthetic and real-world data.

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