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Decomposing tensors via rank-one approximations

Alvaro Ribot, Emil Horobet, Anna Seigal, Ettore Teixeira Turatti

TL;DR

The paper develops a tensor analogue of the matrix SVD via successive rank-one approximations, defining two-orthogonal decompositions as the order-independent case. It proves a maximal-length bound N for such decompositions, establishes a dimension lower bound for the corresponding two-orthogonal variety that exceeds the odeco dimension, and provides both combinatorial (Latin hypercubes, graphs) and algebraic descriptions (for binary and 2×2×2 tensors) of the variety. The work also characterizes identifiability in small cases and discusses optimal truncations, border rank considerations, and extensions to partially symmetric tensors. Overall, it deepens understanding of tensor decompositions beyond odeco, linking algebraic geometry, combinatorics, and tensor approximation theory with implications for rank notions and identifiability.

Abstract

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors.

Decomposing tensors via rank-one approximations

TL;DR

The paper develops a tensor analogue of the matrix SVD via successive rank-one approximations, defining two-orthogonal decompositions as the order-independent case. It proves a maximal-length bound N for such decompositions, establishes a dimension lower bound for the corresponding two-orthogonal variety that exceeds the odeco dimension, and provides both combinatorial (Latin hypercubes, graphs) and algebraic descriptions (for binary and 2×2×2 tensors) of the variety. The work also characterizes identifiability in small cases and discusses optimal truncations, border rank considerations, and extensions to partially symmetric tensors. Overall, it deepens understanding of tensor decompositions beyond odeco, linking algebraic geometry, combinatorics, and tensor approximation theory with implications for rank notions and identifiability.

Abstract

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors.

Paper Structure

This paper contains 13 sections, 29 theorems, 61 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two-orthogonal decompositions
  3. Maximal length of two-orthogonal decompositions.
  4. Two-orthogonal rank
  5. Optimal truncations.
  6. Two-orthogonality for partially symmetric tensors
  7. Dimension of the two-orthogonal variety
  8. Parametric description of the two-orthogonal variety
  9. Graphical descriptions.
  10. Binary tensors
  11. $2 \times 2 \times 2$ tensors
  12. Stratification and equations of the two-orthogonal variety.
  13. Generic identifiability of two-orthogonal decompositions.

Key Result

Theorem 1.2

The set of two-orthogonal tensors in $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ coincides with the set of tensors with a decomposition $\mathcal{T} = \sum_{i=1}^r x_i$ such that each $x_j$ is a critical rank-one approximation of $\mathcal{T} - \sum_{i \in \mathcal{I}} x_i$ for all $\

Figures (3)

  • Figure 1: The affine slice of the two-orthogonal variety in $\mathbb{R}^2 \otimes \mathbb{R}^2 \otimes \mathbb{R}^2$ when $t_{010}=1, t_{100}=2t_{000}, t_{110}=1,t_{101}=2, t_{011}=3$. The image on the right is a zoomed-in and rescaled version of the image on the left. The surface is $W_4 \setminus W_3$. The black curve is $W_3 \setminus W_2$. The red, green, and blue curves are the three components of $W_2$. The intersection of these three curves is the odeco variety. These plots were made with MathematicaMathematica.
  • Figure 2: $\mathcal{G} = \mathcal{G}_1 \cup \mathcal{G}_2 \cup \mathcal{G}_3$
  • Figure :

Theorems & Definitions (88)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.3
  • Remark 2.4
  • ...and 78 more