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A Robust and Non-Iterative Tensor Decomposition Method with Automatic Thresholding

Hiroki Hasegawa, Yukihiko Okada

TL;DR

This work addresses the challenge of extracting reliable, low-rank structure from high-dimensional, noisy tensor data without predefining ranks or relying on iterative optimization. It introduces TARST, a non-iterative tensor decomposition method that applies mode-wise Singular Value Hard Thresholding (SVHT) with thresholds derived from the Marčenko–Pastur distribution, enabling automatic rank estimation. Theoretical contributions include a polynomial-time upper bound on computational cost and conditions guaranteeing tractable scaling, along with empirical evidence showing TARST achieves superior reconstruction accuracy and robustness to noise and outliers compared with HOSVD, HOOI, and Tucker-L2E. The method offers reproducible, parameter-free tensor denoising suitable for large-scale, real-time analytics in IoT, biometrics, and related data-driven decision contexts.

Abstract

Recent advances in IoT and biometric sensing technologies have led to the generation of massive and high-dimensional tensor data, yet achieving accurate and efficient low-rank approximation remains a major challenge. Most existing tensor decomposition methods require predefined ranks and iterative optimization, resulting in high computational costs and dependence on analyst expertise. This study proposes a novel tensor low-rank approximation method that eliminates both prior rank specification and iterative optimization. The method applies statistical singular value hard thresholding to each mode-wise unfolded matrix to automatically extract statistically significant components, effectively reducing noise while preserving the intrinsic structure. Theoretically, the optimal thresholds for each mode are derived from the asymptotic properties of the Marcenko-Pastur distribution. Simulation experiments demonstrate that the proposed method outperforms conventional approaches (HOSVD, HOOI, and Tucker-L2E) in both estimation accuracy and computational efficiency. These results indicate that the proposed approach provides a theoretically grounded, fully automatic, and non-iterative framework for tensor decomposition.

A Robust and Non-Iterative Tensor Decomposition Method with Automatic Thresholding

TL;DR

This work addresses the challenge of extracting reliable, low-rank structure from high-dimensional, noisy tensor data without predefining ranks or relying on iterative optimization. It introduces TARST, a non-iterative tensor decomposition method that applies mode-wise Singular Value Hard Thresholding (SVHT) with thresholds derived from the Marčenko–Pastur distribution, enabling automatic rank estimation. Theoretical contributions include a polynomial-time upper bound on computational cost and conditions guaranteeing tractable scaling, along with empirical evidence showing TARST achieves superior reconstruction accuracy and robustness to noise and outliers compared with HOSVD, HOOI, and Tucker-L2E. The method offers reproducible, parameter-free tensor denoising suitable for large-scale, real-time analytics in IoT, biometrics, and related data-driven decision contexts.

Abstract

Recent advances in IoT and biometric sensing technologies have led to the generation of massive and high-dimensional tensor data, yet achieving accurate and efficient low-rank approximation remains a major challenge. Most existing tensor decomposition methods require predefined ranks and iterative optimization, resulting in high computational costs and dependence on analyst expertise. This study proposes a novel tensor low-rank approximation method that eliminates both prior rank specification and iterative optimization. The method applies statistical singular value hard thresholding to each mode-wise unfolded matrix to automatically extract statistically significant components, effectively reducing noise while preserving the intrinsic structure. Theoretically, the optimal thresholds for each mode are derived from the asymptotic properties of the Marcenko-Pastur distribution. Simulation experiments demonstrate that the proposed method outperforms conventional approaches (HOSVD, HOOI, and Tucker-L2E) in both estimation accuracy and computational efficiency. These results indicate that the proposed approach provides a theoretically grounded, fully automatic, and non-iterative framework for tensor decomposition.
Paper Structure (25 sections, 2 theorems, 51 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 25 sections, 2 theorems, 51 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notation
  4. HOSVD
  5. Singular Value Hard Thresholding(SVHT)
  6. Proposed Method
  7. Observation Model
  8. Mode-wise Denoising via SVHT
  9. Tensor Reconstruction
  10. Key Advantages and Theoretical Basis
  11. Simulations
  12. Computational Cost Analysis
  13. Reconstruction Accuracy Analysis
  14. Results
  15. Computational Cost Analysis
  16. ...and 10 more sections

Key Result

Proposition 4.1

Under the above assumptions, the total computational cost of TARST satisfies where $\varepsilon = \max\{1/2,\, \alpha_{\max}\}$ and $\alpha_{\max} = \max_k (\alpha_k) < 1$. Hence, TARST operates within polynomial time with respect to the input size $P$.

Figures (3)

  • Figure 1: Overview of the proposed TARST method. (A) The observed tensor is decomposed mode-wise via SVD. (B) Statistically optimal thresholds are applied to remove noise, and the tensor is reconstructed to estimate the low-rank signal. Abbreviation: SVD, Singular Value Decomposition.
  • Figure 2: Results of Pattern 1 (Gaussian Noise). From left to right, the figures correspond to tensors of size $10\times10\times10$ with true tensor mean $10$ and standard deviation $2$, $10\times10\times10$ with mean $10$ and standard deviation $0.25$, $50\times50\times50$ with mean $10$ and standard deviation $2$, and $50\times50\times50$ with mean $10$ and standard deviation $0.25$. Five methods are compared: Baseline (no denoising), HOSVD, HOOI, Tucker-L2E, and TARST. Both axes are shown on a logarithmic scale: the horizontal axis represents the noise level $\sigma$ (standard deviation of additive Gaussian noise), and the vertical axis shows the RRSE, where smaller values indicate better performance.
  • Figure 3: Results of Pattern 2 (Outlier Robustness).This figure illustrates how each method (Baseline, HOSVD, HOOI, Tucker-L2E, TARST) responds to changes in outlier ratio and noise dispersion, visualized as a heatmap of RRSE values.The vertical axis represents the outlier ratio $r$. For example, $r=10\%, 50$ indicates that 10% of true values are replaced with outliers scaled by a factor of 50. The horizontal axis denotes the noise level, corresponding to the standard deviation of Gaussian noise, ranging from $10^{-1}$ to $10^{1}$. The color scale shows the RRSE (Relative Reconstruction Squared Error): smaller values (blue) indicate better performance, while larger values (yellow) denote higher error.

Theorems & Definitions (2)

  • Proposition 4.1: Polynomial-Time Upper Bound under General Anisotropic Conditions
  • Corollary 4.1.1: Boundedness of the Worst-Case Time Ratio