Partial Degeneration of Tensors

TL;DR

This paper introduces the concept of partial degeneration of tensors, exploring its properties, obstructions, and implications for tensor rank and complexity, with applications to algebraic complexity, quantum entanglement, and tensor networks.

Contribution

It defines and studies partial degeneration, connecting it to aided rank and providing new bounds and obstructions, advancing understanding of tensor transformations.

Findings

Partial degeneration is a new tensor transformation concept.

Upper bounds on aided rank are derived from partial degenerations.

Examples include W-tensor and Coppersmith-Winograd tensors showing obstructions.

Abstract

Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local linear maps may vary along a curve, and the resulting tensor is expressed as a limit along this curve. In this work we introduce and study partial degeneration, a special version of degeneration where one of the local linear maps is constant whereas the others vary along a curve. Motivated by algebraic complexity, quantum entanglement and tensor networks, we present constructions based on matrix multiplication tensors and find examples by making a connection to the theory of prehomogeneous tensor spaces. We highlight the subtleties of this new notion by showing obstruction and classification results for the unit tensor. To this end, we study the notion of aided rank, a natural generalization of tensor rank. The existence of partial degenerations gives strong upper bounds on the aided rank of a tensor, which allows one to turn degenerations into restrictions. In particular, we present several examples, based on the W-tensor and the Coppersmith-Winograd tensors, where lower bounds on aided rank provide obstructions to the existence of certain partial degenerations.