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Section: New Results

Tensor decomposition and homotopy continuation

Participant : Bernard Mourrain.

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop in [3] computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,...,X_k\subset {\mathbb{P}}^N$ defined over $ℂ$. After computing ranks over $ℂ$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.

This is a joint work with Alessandra Bernardi, Noah S. Daleo, Jonathan D. Hauenstein.