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The condition number of many tensor decompositions is invariant under Tucker compression

Nick Dewaele, Paul Breiding, Nick Vannieuwenhoven

TL;DR

This work analyzes the sensitivity of additive tensor decompositions, including CPD, BTD, and sums of tree tensor networks, to perturbations. It defines the SBTD framework and develops the geometry of structured Tucker manifolds to enable a Terracini-matrix-based computation of the condition number $\kappa^{\mathrm{SBTD}}$. The central contribution proves that $\kappa^{\mathrm{SBTD}}$ is invariant under Tucker compression: projecting to a Tucker core $\mathpzc{G}$ via orthonormal $Q_d$ preserves the conditioning of the decomposition, across several perturbation domains. This invariance allows substantial computational speedups by performing condition-number calculations on the compressed core, as demonstrated on large CPD/BTD examples (e.g., a $265 \times 371 \times 7$ tensor reduced to a $3 \times 3 \times 3$ core yields millisecond timings). The results have practical impact for enabling efficient stability analysis and guiding optimization strategies in structured tensor decompositions.

Abstract

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an $265\times 371\times 7$ tensor of rank $3$ from a food science application whose condition number was computed in $6.9$ milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art.

The condition number of many tensor decompositions is invariant under Tucker compression

TL;DR

This work analyzes the sensitivity of additive tensor decompositions, including CPD, BTD, and sums of tree tensor networks, to perturbations. It defines the SBTD framework and develops the geometry of structured Tucker manifolds to enable a Terracini-matrix-based computation of the condition number . The central contribution proves that is invariant under Tucker compression: projecting to a Tucker core via orthonormal preserves the conditioning of the decomposition, across several perturbation domains. This invariance allows substantial computational speedups by performing condition-number calculations on the compressed core, as demonstrated on large CPD/BTD examples (e.g., a tensor reduced to a core yields millisecond timings). The results have practical impact for enabling efficient stability analysis and guiding optimization strategies in structured tensor decompositions.

Abstract

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an tensor of rank from a food science application whose condition number was computed in milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art.

Paper Structure

This paper contains 17 sections, 12 theorems, 50 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. The structured block term decomposition
  5. The geometry of the structured Tucker manifold
  6. Computing the condition number
  7. A direct algorithm
  8. Example 1 (BTD)
  9. Example 2 (CPD)
  10. Examples of well and ill-conditioned SBTDs
  11. Invariance of the condition number under Tucker compression
  12. Compress
  13. Decompose
  14. Expand
  15. Proof of the main result
  16. ...and 2 more sections

Key Result

Theorem 1.1

\newlabelthm:informalSBTDcondInvariance0 Let $\mathpzc{A} = \mathpzc{A}_1 + \dots + \mathpzc{A}_R$ be an SBTD. The condition number $\kappa^{\mathrm{SBTD}}(\mathpzc{A}_1,\dots,\mathpzc{A}_R)$ is the same for all four domains outlined above.

Figures (3)

  • Figure 1: Condition number of the BTD of $\mathpzc{G}_N \in \mathbb{R}^{4 \times 4 \times 2}$ and that of $\mathpzc{A}_N \in \mathbb{R}^{60 \times 40 \times 40}$ from the experiments in \ref{['sec:experiments']}
  • Figure 2: Ratio between the estimated forward error based on \ref{['eq:fstOrderErrBound']} and the true forward error for $\mathpzc{G}_N$ in the experiments in \ref{['sec:experiments']}. Only cases with a residual $\left\|\hat{\mathpzc{G}} - \mathpzc{G}\right\| \le 10^{-8}$ were considered.
  • Figure 3: Number of iterations of btd_nls applied to $\mathpzc{G}_N \in \mathbb{R}^{4 \times 4 \times 2}$ and $\mathpzc{A}_N \in \mathbb{R}^{60 \times 40 \times 40}$ from the experiments in \ref{['sec:experiments']}.

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1: Tucker core structure
  • Definition 2.2: Structured Tucker decomposition
  • Proposition 2.3
  • Definition 2.4: Structured block term decomposition
  • Proposition 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Proposition 3.3
  • ...and 15 more