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Some Theory on Non-negative Tucker Decomposition

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10169)

Abstract

Some theoretical difficulties that arise from dimensionality reduction for tensors with non-negative coefficients is discussed in this paper. A necessary and sufficient condition is derived for a low non-negative rank tensor to admit a non-negative Tucker decomposition with a core of the same non-negative rank. Moreover, we provide evidence that the only algorithm operating mode-wise, minimizing the dimensions of the features spaces, and that can guarantee the non-negative core to have low non-negative rank requires identifying on each mode a cone with possibly a very large number of extreme rays. To illustrate our observations, some existing algorithms that compute the non-negative Tucker decomposition are described and tested on synthetic data.

Keywords

Non-negative Tucker Decomposition Non-negative Canonical Polyadic Decomposition Dimensionality reduction Non-negative Matrix Factorization 

Notes

Acknowledgements

The authors wish to thank the reviewers as well as the editor for very precious technical comments on a first version of this communication.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departement of Mathematics and Operational Research, Faculté PolytechniqueUniversité de MonsMonsBelgium
  2. 2.Gipsa-labSt Martin d’HèresFrance

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