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Model Selection for Non-Negative Tensor Factorization with Minimum Description Length

1
The Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku 113-8656, Japan
2
The Department of General Systems Studies, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku 153-8902, Japan
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(7), 632; https://doi.org/10.3390/e21070632
Received: 24 April 2019 / Revised: 14 June 2019 / Accepted: 23 June 2019 / Published: 27 June 2019
(This article belongs to the Special Issue Information-Theoretical Methods in Data Mining)
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Abstract

Non-negative tensor factorization (NTF) is a widely used multi-way analysis approach that factorizes a high-order non-negative data tensor into several non-negative factor matrices. In NTF, the non-negative rank has to be predetermined to specify the model and it greatly influences the factorized matrices. However, its value is conventionally determined by specialists’ insights or trial and error. This paper proposes a novel rank selection criterion for NTF on the basis of the minimum description length (MDL) principle. Our methodology is unique in that (1) we apply the MDL principle on tensor slices to overcome a problem caused by the imbalance between the number of elements in a data tensor and that in factor matrices, and (2) we employ the normalized maximum likelihood (NML) code-length for histogram densities. We employ synthetic and real data to empirically demonstrate that our method outperforms other criteria in terms of accuracies for estimating true ranks and for completing missing values. We further show that our method can produce ranks suitable for knowledge discovery. View Full-Text
Keywords: minimum description length; non-negative tensor factorization; model selection; normalized maximum likelihood code length minimum description length; non-negative tensor factorization; model selection; normalized maximum likelihood code length
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Fu, Y.; Matsushima, S.; Yamanishi, K. Model Selection for Non-Negative Tensor Factorization with Minimum Description Length. Entropy 2019, 21, 632.

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