On the geometry of geometric rank

TL;DR

This paper explores the geometric properties of tensor rank, providing classifications for tensors with specific geometric ranks and establishing relationships between geometric rank bounds and tensor rank.

Contribution

It offers a geometric perspective on tensor rank, classifies tensors with certain geometric ranks, and links geometric rank bounds to tensor rank lower bounds.

Findings

Classified tensors with degenerate geometric rank in C^3⊗C^3⊗C^3.

Classified tensors with geometric rank two.

Proved that upper bounds on geometric rank imply lower bounds on tensor rank.

Abstract

We make a geometric study of the Geometric Rank of tensors recently introduced by Kopparty et al. Results include classification of tensors with degenerate geometric rank in $C^3\otimes C^3\otimes C^3$, classification of tensors with geometric rank two, and showing that upper bounds on geometric rank imply lower bounds on tensor rank.