On Comon's and Strassen's conjectures

TL;DR

This paper surveys key conjectures in tensor decomposition, proves them under certain bounds, and improves bounds for Comon's conjecture by developing new equations for secant varieties.

Contribution

It provides new proofs of Comon's and Strassen's conjectures under specific conditions and advances the bounds for Comon's conjecture using novel equations.

Findings

Proved Comon's and Strassen's conjectures under certain rank bounds.

Developed new equations for secant varieties of Veronese and Segre varieties.

Improved the known bounds for Comon's conjecture.

Abstract

Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.