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Towards a condition number theorem for the tensor rank decomposition
We show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions (i.e., decompositions with unbounded geometric condition number, derived in [P. Breiding and N. Vannieuwenhoven, The condition number of join decompositions, SIAM J. Matrix Anal. Appl. (2018)]) is bounded from below by the inverse of this condition number. That is, we prove one inequality towards a condition number theorem for the tensor rank decomposition. Numerical experiments suggest that the other inequality could also hold (at least locally).