Some Theory on Non-negative Tucker Decomposition
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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.
KeywordsNon-negative Tucker Decomposition Non-negative Canonical Polyadic Decomposition Dimensionality reduction Non-negative Matrix Factorization
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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