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Computational Methods in Applied Mathematics

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Volume 19, Issue 1


Tensor Train Spectral Method for Learning of Hidden Markov Models (HMM)

Maxim A. Kuznetsov
  • Corresponding author
  • Skolkovo Institute of Science and Technology, Skolkovo Innovation Center Moscow, 143025, Moscow, Russia
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/ Ivan V. Oseledets
Published Online: 2018-08-11 | DOI: https://doi.org/10.1515/cmam-2018-0027


We propose a new algorithm for spectral learning of Hidden Markov Models (HMM). In contrast to the standard approach, we do not estimate the parameters of the HMM directly, but construct an estimate for the joint probability distribution. The idea is based on the representation of a joint probability distribution as an N-th-order tensor with low ranks represented in the tensor train (TT) format. Using TT-format, we get an approximation by minimizing the Frobenius distance between the empirical joint probability distribution and tensors with low TT-ranks with core tensors normalization constraints. We propose an algorithm for the solution of the optimization problem that is based on the alternating least squares (ALS) approach and develop its fast version for sparse tensors. The order of the tensor d is a parameter of our algorithm. We have compared the performance of our algorithm with the existing algorithm by Hsu, Kakade and Zhang proposed in 2009 and found that it is much more robust if the number of hidden states is overestimated.

Keywords: Multilinear Algebra; Tensor Train Decomposition; Alternating Least Squares (ALS); Hidden Markov Models (HMM); Spectral Algorithms

MSC 2010: 15A69; 65C40; 60J20


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About the article

Received: 2017-10-15

Revised: 2018-02-26

Accepted: 2018-05-02

Published Online: 2018-08-11

Published in Print: 2019-01-01

The authors gratefully acknowledge the financial support from Ministry of Education and Science of the Russian Federation under grant 14.756.31.0001.

Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 93–99, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0027.

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