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A chiseling algorithm for low-rank Grassmann decomposition of skew-symmetric tensors

Nick Vannieuwenhoven

TL;DR

This work develops a numerically efficient algorithm for exact Grassmann decompositions of low-Gr-rank skew-symmetric tensors by leveraging a sparsification framework built on the kernel of a multilinear map attached to the tensor. The method reduces to a concise tensor space, uses an eigen-decomposition in the kernel to identify elementary Grassmann components, and reconstructs the rank-$r$ Grassmann decomposition up to scale. The authors provide a detailed, scalable numerical implementation and demonstrate strong performance and robustness on synthetic data, with clear guidance on parameter choices and refinement procedures. The approach offers a principled alternative to traditional pencil-based methods and connects Grassmann decompositions to broader sparsification and identifiability results in algebraic geometry. Overall, the work advances practical computation of Grassmann decompositions and opens avenues for initializing optimization-based Grassmann-rank methods in noisy or larger-scale settings.

Abstract

A numerical algorithm to decompose an exact low-rank skew-symmetric tensor into a sum of elementary (rank-$1$) skew-symmetric tensors is introduced. The algorithm uncovers this Grassmann decomposition based on linear relations that are encoded by the kernel of the differential of the natural action of the general linear group on the tensor, following the ideas of [Brooksbank, Kassabov, and Wilson, Detecting null patterns in tensor data, arXiv:2408.17425v2, 2025]. The Grassmann decomposition can be recovered, up to scale, from a diagonalization of a generic element in this kernel. Numerical experiments illustrate that the algorithm is computationally efficient and quite accurate for mathematically low-rank tensors.

A chiseling algorithm for low-rank Grassmann decomposition of skew-symmetric tensors

TL;DR

This work develops a numerically efficient algorithm for exact Grassmann decompositions of low-Gr-rank skew-symmetric tensors by leveraging a sparsification framework built on the kernel of a multilinear map attached to the tensor. The method reduces to a concise tensor space, uses an eigen-decomposition in the kernel to identify elementary Grassmann components, and reconstructs the rank- Grassmann decomposition up to scale. The authors provide a detailed, scalable numerical implementation and demonstrate strong performance and robustness on synthetic data, with clear guidance on parameter choices and refinement procedures. The approach offers a principled alternative to traditional pencil-based methods and connects Grassmann decompositions to broader sparsification and identifiability results in algebraic geometry. Overall, the work advances practical computation of Grassmann decompositions and opens avenues for initializing optimization-based Grassmann-rank methods in noisy or larger-scale settings.

Abstract

A numerical algorithm to decompose an exact low-rank skew-symmetric tensor into a sum of elementary (rank-) skew-symmetric tensors is introduced. The algorithm uncovers this Grassmann decomposition based on linear relations that are encoded by the kernel of the differential of the natural action of the general linear group on the tensor, following the ideas of [Brooksbank, Kassabov, and Wilson, Detecting null patterns in tensor data, arXiv:2408.17425v2, 2025]. The Grassmann decomposition can be recovered, up to scale, from a diagonalization of a generic element in this kernel. Numerical experiments illustrate that the algorithm is computationally efficient and quite accurate for mathematically low-rank tensors.

Paper Structure

This paper contains 31 sections, 17 theorems, 75 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Tensor rank decomposition
  3. Block term decomposition
  4. Preliminaries
  5. Linear and multilinear algebra concepts
  6. Algebraic geometry concepts
  7. Identifiability of rank decompositions
  8. Chiseling algorithms to detect sparsity patterns in tensor data
  9. An algorithm for Grassmann decomposition
  10. Reduction to concise spaces
  11. The key ingredients
  12. The high-level algorithm
  13. An efficient numerical implementation
  14. S0: Computing a basis of the concise tensor space
  15. Flattenings
  16. ...and 16 more sections

Key Result

proposition 1

Let $\mathcal{G}_n^d \subset \mathrm{V}$ be a Grassmannian with $3 \le d \le n$. If then there exists a Zariski open subset $\mathcal{V} \subset \sigma_r(\mathcal{G}_n^d)$ such that $\Sigma_r^{-1} : \mathcal{V} \to (\mathcal{G}_n^d)^{\times r}/\mathfrak{S}([r])$ is a continuous inverse map of $\Sigma_r$.

Figures (4)

  • Figure 1: The relative backward error of decomposing $100$ random Gr-rank-$15$ tensors in $\wedge^3 \mathbb{R}^{50}$ for a varying number of iterations $p$ in \ref{['alg_fixed_point_improv']}.
  • Figure 2: Relative breakdown of the execution times of the steps in \ref{['alg_main_computation']} for a random Grassmann tensor in $\wedge^d \mathbb{R}^{65}$ of rank $r = 60/d$ for $d=3,4,5,6$. The absolute total execution times were $4.1$s, $7.6$s, $60.5$s, and $1208.3$s for, respectively, $d=3,4,5,$ and $6$.
  • Figure 3: The execution time in seconds and the base-$10$ logarithm of the relative backward and forward errors for decomposing noiseless random Gr-rank-$r$ tensors in $\wedge^d \mathbb{F}^{dr}$ for feasible combinations of $3 \le d \le 10$ and $1 \le r \le 33$.
  • Figure 4: The maximum relative forward error under relative perturbations for $100$ random noisy Gr-rank-$10$ tensors in $\wedge^d \mathbb{C}^{50}$ for $d=3,4,5$.

Theorems & Definitions (17)

  • proposition 1
  • lemma 1
  • lemma 2: Flattening
  • lemma 3: Multilinear rank
  • corollary 1
  • lemma 4: Compression
  • lemma 5: Decomposing elementary tensors
  • lemma 6: Differential
  • theorem 1: Kernel structure theorem
  • lemma 7: Generic diagonalizability
  • ...and 7 more