Existence of best low rank tensor approximations

Eric Evert
Katholieke Universiteit Leuven, Campus Kortrijk

We give deterministic methods which guarantee that a given tensor has a best low rank approximation. The bounds are produced by viewing low rank canonical polyadic decompositions as a “joint generalized eigenvalue decomposition”. We show that low border rank tensors with rank strictly greater than border rank are defective in the sense of this joint generalized eigenvalue problem. Perturbation theoretic bounds for joint generalized eigenvalues are developed and used to produce a certificate which can guarantee that a tensor is nondefective in this sense, hence has rank equal to border rank.


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