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Entrywise tensor-train approximation of large tensors via random embeddings

Stanislav Budzinskiy

TL;DR

This work delivers a new a priori bound on the entrywise TT approximation error for large tensors in terms of the TT factorization quasinorm $\gamma_{\mathrm{F}}^{\mathrm{TT}}(\mathcal{A})$, bridging tensor networks with random embedding techniques via a tensor-structured Hanson--Wright inequality. The main result shows that for any TT factorization there exists a rank-$\mathrm{TT}$ approximation $\mathcal{B}$ with $\|\mathcal{A}-\mathcal{B}\|_{\max} \le \varepsilon \cdot \gamma_{\mathrm{F}}^{\mathrm{TT}}(\mathcal{A})$, where the TT rank scales logarithmically with tensor size. The paper also introduces TT core coherences and demonstrates how they bound the TT-quasinorm, yielding corollaries for CP representations and for TT-SVD-based approximations, along with numerical experiments using alternating projections to illustrate practical behavior. This provides a foundational link between tensor-network representations and entrywise accuracy, with potential extensions to more general tensor networks and practical implications for high-dimensional data analysis. The results highlight how the intrinsic TT structure and coherence properties influence the feasibility of accurate entrywise compression in large-scale applications.

Abstract

The theory of low-rank tensor-train approximation is well understood when the approximation error is measured in the Frobenius norm. The entrywise maximum norm is equally important but is significantly weaker for large tensors, making the estimates obtained via the Frobenius norm and norm equivalence pessimistic or even meaningless. In this article, we derive a direct estimate of the entrywise approximation error that is applicable in some of these cases. The estimate is given in terms of the higher-order generalization of the matrix factorization norm, and its proof is based on the tensor-structured Hanson--Wright inequality. The theoretical results are accompanied by numerical experiments carried out with the method of alternating projections.

Entrywise tensor-train approximation of large tensors via random embeddings

TL;DR

This work delivers a new a priori bound on the entrywise TT approximation error for large tensors in terms of the TT factorization quasinorm , bridging tensor networks with random embedding techniques via a tensor-structured Hanson--Wright inequality. The main result shows that for any TT factorization there exists a rank- approximation with , where the TT rank scales logarithmically with tensor size. The paper also introduces TT core coherences and demonstrates how they bound the TT-quasinorm, yielding corollaries for CP representations and for TT-SVD-based approximations, along with numerical experiments using alternating projections to illustrate practical behavior. This provides a foundational link between tensor-network representations and entrywise accuracy, with potential extensions to more general tensor networks and practical implications for high-dimensional data analysis. The results highlight how the intrinsic TT structure and coherence properties influence the feasibility of accurate entrywise compression in large-scale applications.

Abstract

The theory of low-rank tensor-train approximation is well understood when the approximation error is measured in the Frobenius norm. The entrywise maximum norm is equally important but is significantly weaker for large tensors, making the estimates obtained via the Frobenius norm and norm equivalence pessimistic or even meaningless. In this article, we derive a direct estimate of the entrywise approximation error that is applicable in some of these cases. The estimate is given in terms of the higher-order generalization of the matrix factorization norm, and its proof is based on the tensor-structured Hanson--Wright inequality. The theoretical results are accompanied by numerical experiments carried out with the method of alternating projections.
Paper Structure (25 sections, 23 theorems, 96 equations, 2 figures, 1 table)

This paper contains 25 sections, 23 theorems, 96 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Matrix factorization norm and random embeddings
  3. Tensor-train factorization quasinorm
  4. Contributions
  5. Outline
  6. Notation
  7. Proof of Theorem \ref{['theorem:tt_approx']}
  8. Overview
  9. Arrays and indices
  10. Array transformations
  11. Hierarchy of norms
  12. Tensor-structured Hanson--Wright inequality
  13. Technical lemmas
  14. Analysis of the specific quadratic form
  15. Moment and tail bounds
  16. ...and 10 more sections

Key Result

Theorem 1.2

\newlabeltheorem:matrix_approx0 Let $\varepsilon \in (0,1)$ and $n_1, n_2 \in \mathbb{N}$. Consider For every $\bm{\mathrm{A}} \in \mathbb{R}^{n_1 \times n_2}$, there exists $\bm{\mathrm{B}} \in \mathbb{R}^{n_1 \times n_2}$ of $\mathrm{rank} ( \bm{\mathrm{B}} ) \leq r$ such that

Figures (2)

  • Figure 1: Entrywise TT approximation errors computed with a combination of quasioptimal alternating projections and binary search for the varying approximation rank $r$.
  • Figure 2: Entrywise TT approximation errors computed with a combination of quasioptimal alternating projections and binary search for the varying tensor size $n$.

Theorems & Definitions (53)

  • Remark 1.1
  • Theorem 1.2: srebro2005rank
  • Example 1.3
  • Remark 1.4
  • Lemma 1.5
  • Proof 1
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • Proof 2
  • ...and 43 more