Uniqueness of Nonnegative Tensor Approximations
Résumé
We show that the best nonnegative rank-r approximation of a nonnegative tensor is almost always unique and that nonnegative tensors with nonunique best nonnegative rank-r approximation form a semialgebraic set contained in an algebraic hypersurface. We then establish a singular vector variant of the Perron-Frobenius Theorem for positive tensors that may be used for answering various questions involving nonnegative tensor approximations. In particular, we apply it to show that the best nonnegative rank-1 approximation of a positive tensor is always unique and that a best nonnegative rank-r approximation of a positive tensor can almost never be obtained by deflation.
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