Tensor decomposition and homotopy continuation

TL;DR

This paper introduces a numerical algebraic geometric approach using tensor decomposition and homotopy continuation to compute tensor ranks, border ranks, and decompositions, overcoming classical elimination theory challenges.

Contribution

It develops computational methods for tensor rank and border rank determination using pseudowitness sets and homotopy continuation, applicable to complex and real tensors.

Findings

Successfully computes tensor ranks and border ranks numerically.

Demonstrates effectiveness on various tensor examples.

Extends methods to join varieties and real tensor ranks.

Abstract

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 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 $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over $\mathbb{C}$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.