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Partial Degeneration of Tensors

Matthias Christandl, Fulvio Gesmundo, Vladimir Lysikov, Vincent Steffan

TL;DR

The paper introduces partial degeneration as a natural intermediate notion between restriction and degeneration for tensors, establishing its inequivalence from both. It develops a framework around aided restriction and aided rank, using an aiding matrix ⟨1,1,p⟩ and interpolation tools to convert degenerations into restrictions, and introduces a substitution-based method to bound aided rank. Key results include Strassen-type partial degenerations, no-go theorems for the unit tensor, and exact aided-rank formulas for powers of the W-tensor, with applications to matrix pencils and prehomogeneous tensor spaces. The work clarifies the hierarchy among degeneration concepts, provides concrete constructions and obstructions, and offers techniques with potential impact on algebraic complexity, quantum entanglement, and tensor-network representations.

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.

Partial Degeneration of Tensors

TL;DR

The paper introduces partial degeneration as a natural intermediate notion between restriction and degeneration for tensors, establishing its inequivalence from both. It develops a framework around aided restriction and aided rank, using an aiding matrix ⟨1,1,p⟩ and interpolation tools to convert degenerations into restrictions, and introduces a substitution-based method to bound aided rank. Key results include Strassen-type partial degenerations, no-go theorems for the unit tensor, and exact aided-rank formulas for powers of the W-tensor, with applications to matrix pencils and prehomogeneous tensor spaces. The work clarifies the hierarchy among degeneration concepts, provides concrete constructions and obstructions, and offers techniques with potential impact on algebraic complexity, quantum entanglement, and tensor-network representations.

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.
Paper Structure (20 sections, 23 theorems, 111 equations, 2 figures)

This paper contains 20 sections, 23 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Partial degeneration
  4. Aided restriction and aided rank
  5. Conclusion and outlook
  6. Acknowledgements
  7. Preliminaries
  8. Partial degeneration
  9. Definition and motivation
  10. Strassen's tensor
  11. Compressibility of tensors and matrix multiplication
  12. Prehomogeneous spaces
  13. A no-go result for the unit tensor
  14. Aided restriction and aided rank
  15. Aided restriction and interpolation
  16. ...and 5 more sections

Key Result

Lemma 3.4

Let $T \in U_1 \otimes U_2 \otimes U_3$ and $S \in V_1 \otimes V_2 \otimes V_3$. Let $T \geq S$ and let $S$ be concise. If $S$ is $(a_1, a_2, a_3)$-compressible then $T$ is $(a_1, a_2, a_3)$-compressible.

Figures (2)

  • Figure 1: Visualizing the direct sum of two tensors $T$ and $S$: The tensor $T \oplus S$ is block-diagonal where one block is the tensor $T$ and the other block the tensor $S$.
  • Figure 2: Visualization of an $(a_1, a_2, a_3)$-compressible tensor: This large $u_1 \times u_2 \times u_3$-cube is the tensor $T \in U_1 \otimes U_2 \otimes U_3$. The entries of $T$ -- specified in some fixed basis -- can be written in the cells of this cube. The smaller, red $a_1 \times a_2 \times a_3$-cube depicts a block of size $a_1\times a_2\times a_3$ where each entry of $T$ equals zero. By choosing the linear maps as projectors onto the last $u_1 - a_1$ resp. $u_2 - a_2$ resp. $u_3 - a_3$ coordinates, we see that each such tensor is $(a_1,a_2,a_3)$-compressible.

Theorems & Definitions (49)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • Proposition 3.7
  • proof
  • ...and 39 more