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Canonical polyadic decomposition (CPD) of big tensors with low multilinear rank

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Abstract

Tensor decomposition methods have been widely applied to big data analysis as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the challenges posed by big data dilemma. This paper attempts to improve the scalability of tensor decompositions and makes two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms.

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Notes

  1. See www.mathworks.com/help/pdf_doc/mdce/mdce.pdf.

  2. We have assumed that each sub-tensor can be efficiently handled in by the associated worker, otherwise we increase the number of workers and decrease the size of each sub-tensor accordingly

  3. The performance index SIR is defined as \(\text {SIR}(\mathbf {a}_{r}^{(n)},{\hat {\mathbf {a}}}_{r}^{(n)})=20\log {\left \|\left \|{\mathbf {a}_{r}^{(n)}-{\hat {\mathbf {a}}}_{r}^{(n)}}\right \|\right \|}/{\left \|\left \|{\mathbf {a}_{r}^{(n)}}\right \|\right \|}\), where \(\mathbf {a}_{r}^{(n)}\) is an estimate of \(\mathbf {a}_{r}^{(n)}\), both \(\mathbf {a}_{r}^{(n)}\) and \(\mathbf {a}_{r}^{(n)}\) are normalized to be zero-mean and unit-variance.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (NSFC) Grant 61673124, Grant 61973090, and Grant 61727810.

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Authors and Affiliations

  1. School of Automation, Guangdong University of Technology, Guangzhou, China

    Yichun Qiu & Guoxu Zhou

  2. School of Automation, Guangdong-HongKong-Macao Joint Laboratory for Smart Discrete Manufacturing, Guangdong University of Technology, Guangzhou, 510006, China

    Guoxu Zhou

  3. Department of Psychiatry and Behavioral Sciences, Stanford University, Stanford, CA, USA

    Yu Zhang

  4. Skolkovo Institute of Science and Technology (SKOLTECH), 143026, Moscow, Russia

    Andrzej Cichocki

  5. System Research Institute, Polish Academy of Sciences, Warsaw, 00-901, Poland

    Andrzej Cichocki

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  1. Yichun Qiu
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Correspondence to Guoxu Zhou.

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Qiu, Y., Zhou, G., Zhang, Y. et al. Canonical polyadic decomposition (CPD) of big tensors with low multilinear rank. Multimed Tools Appl 80, 22987–23007 (2021). https://doi.org/10.1007/s11042-020-08711-1

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