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Three decompositions of symmetric tensors have similar condition numbers

Nick Dewaele, Paul Breiding, Nick Vannieuwenhoven

TL;DR

This work investigates how the numerical sensitivity of three symmetric-tensor decompositions—CPD, WD, and Q-WD—are interconnected, providing tight bounds that relate their condition numbers and revealing compression-invariance properties. A key contribution is a practical Terracini-matrix-based approach that speeds up the computation of WD-related condition numbers by orders of magnitude, enabling large-scale stability analyses. The authors prove a rank-2 equivalence between WD and CPD condition numbers and present extensive numerical evidence suggesting a close relationship beyond rank-2, with broader implications for partially symmetric decompositions. Collectively, the results deepen understanding of how symmetry and subspace constraints affect the stability of tensor decompositions and offer efficient tools for condition-number estimation on these manifolds.

Abstract

We relate the condition numbers of computing three decompositions of symmetric tensors: the canonical polyadic decomposition, the Waring decomposition, and a Tucker-compressed Waring decomposition. Based on this relation we can speed up the computation of these condition numbers by orders of magnitude

Three decompositions of symmetric tensors have similar condition numbers

TL;DR

This work investigates how the numerical sensitivity of three symmetric-tensor decompositions—CPD, WD, and Q-WD—are interconnected, providing tight bounds that relate their condition numbers and revealing compression-invariance properties. A key contribution is a practical Terracini-matrix-based approach that speeds up the computation of WD-related condition numbers by orders of magnitude, enabling large-scale stability analyses. The authors prove a rank-2 equivalence between WD and CPD condition numbers and present extensive numerical evidence suggesting a close relationship beyond rank-2, with broader implications for partially symmetric decompositions. Collectively, the results deepen understanding of how symmetry and subspace constraints affect the stability of tensor decompositions and offer efficient tools for condition-number estimation on these manifolds.

Abstract

We relate the condition numbers of computing three decompositions of symmetric tensors: the canonical polyadic decomposition, the Waring decomposition, and a Tucker-compressed Waring decomposition. Based on this relation we can speed up the computation of these condition numbers by orders of magnitude

Paper Structure

This paper contains 5 sections, 3 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Condition number of a $Q$-WD
  3. Equivalence between the CPD and WD
  4. Numerical experiments
  5. The condition number of the partially symmetric decomposition

Key Result

Theorem 1.1

Let $\mathpzc{G} = \mathpzc{G}_1 + \dots + \mathpzc{G}_R \in \mathbb{R}^{m \times \dots \times m}$ be a WD of an order-$D$ tensor.

Figures (1)

  • Figure 1: Ratio between the condition numbers of the CPD and WD of an $n \times n \times n$ symmetric tensor of rank $R$. The displayed value is the maximum over 500 test cases.

Theorems & Definitions (8)

  • Theorem 1.1
  • Conjecture 1.2
  • Proposition 1.3
  • proof : Proof of \ref{['prop:WaringCondInvariance']}
  • proof : Proof of \ref{['prop:waringCondSpecialCase']}
  • Theorem A.1
  • Remark A.2
  • proof