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A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization
IEEE Transactions on Signal Processing ( IF 5.4 ) Pub Date : 2021-02-19 , DOI: 10.1109/tsp.2021.3060000
Louis Filstroff , Olivier Gouvert , Cedric Fevotte , Olivier Cappe

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.

中文翻译:

时间非负矩阵分解的Gamma马尔可夫链的比较研究

非负矩阵分解(NMF)已成为一类公认的非负数据分析方法。特别地,已经对概率NMF进行了大量工作,即基于泊松或指数似然来描述数据的概率模型中的估计或推理任务。当处理时间序列数据时,已经提出了许多工作来将激活系数的演化建模为非负马尔可夫链,大多数时间与Gamma分布有关,从而产生了所谓的时间NMF模型。在本文中,我们回顾了NMF文献中的四个Gamma Markov链,并显示它们都具有相同的缺点:缺乏明确定义的平稳分布。然后,我们引入第五个过程,克服了这一局限性的名为BGAR(1)的时间序列文献模型。然后,在泊松似然的情况下,在预测任务的MAP框架中比较这些时间NMF模型。
更新日期:2021-03-23
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