Partial Degeneration of Tensors
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Partial Degeneration of Tensors. / Christandl, Matthias; Gesmundo, Fulvio; Lysikov, Vladimir; Steffan, Vincent.
In: SIAM Journal on Matrix Analysis and Applications, Vol. 45, No. 1, 2024, p. 771-800.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 385690111