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Sufficient recovery conditions for noise-buried low rank tensors

Sergey Petrov, Nikolai Zamarashkin

TL;DR

This work addresses the problem of recovering low-rank tensor structure from noisy measurements by deriving a probabilistic bound on the recovery error that decays with tensor dimensionality. It introduces a Gaussian width framework with a sufficient approximation condition and analyzes the rank-2 canonical tensor case, obtaining a bound of the form $\mathbb{P}\{ \sup_{\mathrm{rank}(\mathbf V) \le 2, \|\mathbf V\|_F = 1} ( {\bf N}, \mathbf V )_F > \mu \sqrt{m d^2 \ln m} + t \} \le e^{-t^2/4} + 2 e^{-m^{d/2}/8}$, modulo constants. The results indicate that high-dimensional low-rank tensor representations can outperform matrices with the same rank and element count in filtering noise, supported by numerical experiments for canonical, Tensor-Train, and Tucker formats that exhibit consistent scaling with the number of representation parameters. These findings have practical impact for wireless communications and other high-dimensional data settings, where tensorization and quantization can substantially enhance denoising performance.

Abstract

Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured approximations provide superior noise-filtering properties compared to matrices with the same rank and total element count.

Sufficient recovery conditions for noise-buried low rank tensors

TL;DR

This work addresses the problem of recovering low-rank tensor structure from noisy measurements by deriving a probabilistic bound on the recovery error that decays with tensor dimensionality. It introduces a Gaussian width framework with a sufficient approximation condition and analyzes the rank-2 canonical tensor case, obtaining a bound of the form , modulo constants. The results indicate that high-dimensional low-rank tensor representations can outperform matrices with the same rank and element count in filtering noise, supported by numerical experiments for canonical, Tensor-Train, and Tucker formats that exhibit consistent scaling with the number of representation parameters. These findings have practical impact for wireless communications and other high-dimensional data settings, where tensorization and quantization can substantially enhance denoising performance.

Abstract

Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured approximations provide superior noise-filtering properties compared to matrices with the same rank and total element count.
Paper Structure (6 sections, 7 theorems, 81 equations, 5 figures)

This paper contains 6 sections, 7 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Noise perturbation functional definition
  3. Gaussian width framework
  4. Bounds for the set of rank-two tensors
  5. Numerical experiments
  6. Discussion

Key Result

Lemma 4.1

Figures (5)

  • Figure 1: Scheme of the proposed analysis
  • Figure 2: Rank-one approximation perturbations in dependence of dimensionality.
  • Figure 3: Canonical ALS approximation perturbation dependence on the tensor rank
  • Figure 4: Tensor-Train TT-SVD approximation perturbation dependence on the tensor rank
  • Figure 5: Tucker HOSVD approximation perturbation dependence on the tensor rank

Theorems & Definitions (14)

  • Lemma 4.1: Structured orthogonal projection
  • proof
  • Theorem 4.1: Gaussian width bound for the set of unit rank-two tensors
  • Lemma 4.2: Approximate cosine
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Theorem 4.2
  • ...and 4 more