Ranks of tensors and a generalization of secant varieties

TL;DR

This paper introduces subspace rank to analyze tensor ranks, provides new bounds, and characterizes ranks of certain partially symmetric tensors, offering a geometric perspective on tensor rank theory.

Contribution

It presents a novel subspace rank concept, derives an improved upper bound for tensor rank, and determines ranks of specific partially symmetric tensors.

Findings

New upper bound for tensor rank

Ranks of partially symmetric tensors in C^2 ⊗ C^b ⊗ C^b determined

Geometric perspective on tensor rank literature

Abstract

We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We review the literature from a geometric perspective.