Classification of Tripartite Tensors with Small Geometric Ranks

Abstract

Geometric Rank of tensors was introduced by Kopparty et al. as a useful tool to study algebraic complexity theory, extremal combinatorics and quantum information theory. This dissertation studies the classification of tripartite tensors with small geometric ranks. We introduce primitive tensors and compression tensors, which reduces the classification problem to finding all primitive tensors.

There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces.

Using the results on linear determinantal varieties, we find all primitive tensors with geometric rank 1, 2 and 3 up to change of coordinates, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. Finally, we explicitly classify all tripartite tensors with geometric rank at most 1, 2 and 3.