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XRAY Algorithm for Separable Nonnegative Tensor Factorization

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Advances in Computational Intelligence (IWANN 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10306))

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Abstract

Many computational problems in machine learning can be represented by separable matrix factorization models. In a geometric approach, linear separability means that the whole set of data points can be modeled by a convex combination of a few data points, referred to as the extreme rays. The aim of the XRAY algorithm is to find the extreme rays of the conic hull, generated by observed nonnegative vectors. In this paper, we extend the concept of this algorithm to a multi-linear data representation. Instead of searching into a vector space, we attempt to find the equivalent extreme rays in a space of tensors, under the linear separability assumption of subtensors, ordered along the selected mode. The proposed multi-way XRAY algorithm has been applied to Blind Source Separation (BSS) of natural images. The experiments demonstrate that if multi-way observations are at least one-mode linearly separable, the proposed algorithms can estimate the latent factors with high Signal-to-Interference (SIR) performance. The discussed methods may also be useful for analyzing video sequences.

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Notes

  1. 1.

    The inner product \(<{\mathcal {A}}, \; {\mathcal {B}}>\) between the tensors \({\mathcal {A}} = [a_{i_1,\ldots ,i_N}] \in {\mathbb {R}}^{I_1 \times \ldots \times I_N}\) and \({\mathcal {B}} = [b_{i_1,\ldots ,i_N}] \in {\mathbb {R}}^{I_1 \times \ldots \times I_N}\) is defined as follows: \(<{\mathcal {A}}, \; {\mathcal {B}}> = \sum _{i_1 = 1}^{I_1} \cdots \sum _{i_N = 1}^{I_N} a_{i_1,\ldots ,i_N} b_{i_1,\ldots ,i_N}\).

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Acknowledgment

This work was partially supported by the grant 2015/17/B/ST6/01865 funded by National Science Center (NCN) in Poland.

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Authors and Affiliations

  1. Faculty of Electronics, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego 27, 50-370, Wroclaw, Poland

    Rafał Zdunek & Tomasz Sadowski

Authors
  1. Rafał Zdunek
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  2. Tomasz Sadowski
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Correspondence to Rafał Zdunek .

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Editors and Affiliations

  1. Universidad de Granada, Granada, Spain

    Ignacio Rojas

  2. University of Malaga, Malaga, Spain

    Gonzalo Joya

  3. Polytechnic University of Catalonia, Vilanova i la Geltrú, Barcelona, Spain

    Andreu Catala

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Zdunek, R., Sadowski, T. (2017). XRAY Algorithm for Separable Nonnegative Tensor Factorization. In: Rojas, I., Joya, G., Catala, A. (eds) Advances in Computational Intelligence. IWANN 2017. Lecture Notes in Computer Science(), vol 10306. Springer, Cham. https://doi.org/10.1007/978-3-319-59147-6_22

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  • DOI: https://doi.org/10.1007/978-3-319-59147-6_22

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  • Online ISBN: 978-3-319-59147-6

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