TY - JOUR

T1 - Partial Degeneration of Tensors

AU - Christandl, Matthias

AU - Gesmundo, Fulvio

AU - Lysikov, Vladimir

AU - Steffan, Vincent

PY - 2024

Y1 - 2024

N2 - 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 while 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.

AB - 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 while 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.

U2 - 10.1137/23M1554898

DO - 10.1137/23M1554898

M3 - Journal article

VL - 45

SP - 771

EP - 800

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 1

ER -