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.
J.E. Cohen—Research funded by ERC advanced grant “DECODA” no. 320594, ERC starting grant “COLORAMAP” no. 679515, and F.R.S.-FNRS incentive grant for scientific research n\(^\text {o}\) F.4501.16.
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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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Cohen, J.E., Comon, P., Gillis, N. (2017). Some Theory on Non-negative Tucker Decomposition. In: Tichavský, P., Babaie-Zadeh, M., Michel, O., Thirion-Moreau, N. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2017. Lecture Notes in Computer Science(), vol 10169. Springer, Cham. https://doi.org/10.1007/978-3-319-53547-0_15
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