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Detecting null patterns in tensor data

Peter A. Brooksbank, Martin D. Kassabov, James B. Wilson

TL;DR

The paper tackles detecting sparsity patterns in tensor data by introducing chisels, a tunable mechanism to parameterize and search for structured decompositions in modes. It connects observed patterns to derivations Der(π’ž,Ξ“), proving that sparsity patterns imply derivations and vice versa, and develops practical algorithms (including SVD-based heuristics) implemented in Julia to recover these patterns. It shows that many classical tensor decompositions, such as Tucker and HOSVD, arise as special cases of chiseling, while also enabling discovery of new continuous curve- and surface-like patterns. A central result is the universality and, under appropriate conditions, uniqueness of the pattern recovered using the universal chisel π’ž_{uni}, yielding reproducible decompositions across different derivations. The framework offers a principled, algebraic approach to identifying and ordering structured blocks in tensors, with practical impact for blind source separation, factorization, and high-dimensional data analysis.

Abstract

This article introduces a class of efficiently computable null patterns for tensor data. The class includes familiar patterns such as block-diagonal decompositions explored in statistics and signal processing, low-rank tensor decompositions, and Tucker decompositions. It also includes a new family of null patterns -- not known to be detectable by current methods -- that can be thought of as continuous decompositions approximating curves and surfaces. We present a general algorithm to detect null patterns in each class using a parameter we call a \textit{chisel} that tunes the search to patterns of a prescribed shape. We also show that the patterns output by the algorithm are essentially unique.

Detecting null patterns in tensor data

TL;DR

The paper tackles detecting sparsity patterns in tensor data by introducing chisels, a tunable mechanism to parameterize and search for structured decompositions in modes. It connects observed patterns to derivations Der(π’ž,Ξ“), proving that sparsity patterns imply derivations and vice versa, and develops practical algorithms (including SVD-based heuristics) implemented in Julia to recover these patterns. It shows that many classical tensor decompositions, such as Tucker and HOSVD, arise as special cases of chiseling, while also enabling discovery of new continuous curve- and surface-like patterns. A central result is the universality and, under appropriate conditions, uniqueness of the pattern recovered using the universal chisel π’ž_{uni}, yielding reproducible decompositions across different derivations. The framework offers a principled, algebraic approach to identifying and ordering structured blocks in tensors, with practical impact for blind source separation, factorization, and high-dimensional data analysis.

Abstract

This article introduces a class of efficiently computable null patterns for tensor data. The class includes familiar patterns such as block-diagonal decompositions explored in statistics and signal processing, low-rank tensor decompositions, and Tucker decompositions. It also includes a new family of null patterns -- not known to be detectable by current methods -- that can be thought of as continuous decompositions approximating curves and surfaces. We present a general algorithm to detect null patterns in each class using a parameter we call a \textit{chisel} that tunes the search to patterns of a prescribed shape. We also show that the patterns output by the algorithm are essentially unique.
Paper Structure (33 sections, 7 theorems, 55 equations, 11 figures, 2 algorithms)

This paper contains 33 sections, 7 theorems, 55 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multiarrays
  3. Partial evaluations of tensors
  4. Notational convention
  5. Changing bases
  6. Sparsity patterns
  7. Chisels
  8. Derivations
  9. Derivations associated with chisels
  10. Sparsity patterns produce derivations
  11. Derivations reveal sparsity patterns
  12. Scalar derivations reveal nothing
  13. Chiseling
  14. Basic chiseling
  15. Chiseling with SVDs
  16. ...and 18 more sections

Key Result

Theorem 5.4

Let $\Gamma$ be an $\ell$-tensor, let $\mathcal{C}$ be an $\ell$-chisel, and let $\delta\in \mathbb{K}^{k_1}\times \cdots \times \mathbb{K}^{k_{\ell}}$. Suppose we have decompositions $U_a=X_{a 1}\oplus \cdots \oplus X_{a k_a}$ of the modes that exhibit the sparsity pattern $\Delta(\mathcal{C},\delt

Figures (11)

  • Figure 1: Point clouds of tensor arrays recovered by our algorithms that exhibit various types of sparsity patterns.
  • Figure 2: Tensor interpretations visualized as sequences of partial evaluations. On the top is the interpretation in equation \ref{['def:tensor-product-interpretation']} and on the bottom is the interpretation in equation \ref{['def:multiarray-interpretation']}.
  • Figure 3: Sparsity patterns exhibited by decompositions of arrays. In (A), the pattern is $\Delta=\{(1,1,1)\}$; in (B), $\Delta=\{(1,1,1),(2,2,2)\}$; and in (C), $\Delta=\{(1,1,2),(1,2,1), (2,2,2)\}$.
  • Figure 4: Curve and surface sparsity patterns.
  • Figure 5: Sparsity pattern $\Delta(\mathcal{C},\delta)$ for the $(\mathcal{C},\delta)$ in equation \ref{['eq:sparsity-pattern-0']}.
  • ...and 6 more figures

Theorems & Definitions (22)

  • Example 4.1
  • Example 4.2
  • Definition 5.1
  • Remark 5.2
  • Example 5.3
  • Theorem 5.4
  • proof
  • Theorem 5.5
  • proof
  • Remark 6.1
  • ...and 12 more