A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization

Louis Filstroff; Olivier Gouvert; Cedric Fevotte; Olivier Cappe

doi:10.1109/TSP.2021.3060000

A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization

Louis Filstroff^*, Olivier Gouvert, Cedric Fevotte, Olivier Cappe

^*Corresponding author for this work

Not published at Aalto University

Research output: Contribution to journal › Article › Scientific › peer-review

3 Citations (Scopus)

Abstract

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.

Original language	English
Article number	9359515
Pages (from-to)	1614-1626
Number of pages	13
Journal	IEEE Transactions on Signal Processing
Volume	69
DOIs	https://doi.org/10.1109/TSP.2021.3060000
Publication status	Published - 2021
MoE publication type	A1 Journal article-refereed

Funding

Manuscript received June 5, 2020; revised October 19, 2020 and January 21, 2021; accepted January 27, 2021. Date of publication February 19, 2021; date of current version March 19, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Athanasios A. Rontogiannis. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under grant agreement No 681839 (project FACTORY). (Corresponding author: Louis Filstroff.) Louis Filstroff was with IRIT, Univ. Toulouse, CNRS, 31062 Toulouse, France. He is now with the Department of Computer Science, School of Science, Aalto University, 02150 Espoo, Finland (e-mail: [email protected]).

Keywords

Gamma Markov chains
MAP estimation
Non-negative matrix factorization
time series data

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10.1109/TSP.2021.3060000

Cite this

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title = "A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization",

abstract = "Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood. ",

keywords = "Gamma Markov chains, MAP estimation, Non-negative matrix factorization, time series data",

author = "Louis Filstroff and Olivier Gouvert and Cedric Fevotte and Olivier Cappe",

note = "Funding Information: Manuscript received June 5, 2020; revised October 19, 2020 and January 21, 2021; accepted January 27, 2021. Date of publication February 19, 2021; date of current version March 19, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Athanasios A. Rontogiannis. This work was supported by the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation program under grant agreement No 681839 (project FACTORY). (Corresponding author: Louis Filstroff.) Louis Filstroff was with IRIT, Univ. Toulouse, CNRS, 31062 Toulouse, France. He is now with the Department of Computer Science, School of Science, Aalto University, 02150 Espoo, Finland (e-mail: [email protected]). Publisher Copyright: {\textcopyright} 1991-2012 IEEE. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

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N1 - Funding Information: Manuscript received June 5, 2020; revised October 19, 2020 and January 21, 2021; accepted January 27, 2021. Date of publication February 19, 2021; date of current version March 19, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Athanasios A. Rontogiannis. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under grant agreement No 681839 (project FACTORY). (Corresponding author: Louis Filstroff.) Louis Filstroff was with IRIT, Univ. Toulouse, CNRS, 31062 Toulouse, France. He is now with the Department of Computer Science, School of Science, Aalto University, 02150 Espoo, Finland (e-mail: [email protected]). Publisher Copyright: © 1991-2012 IEEE. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.

AB - Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.

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A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization

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