Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-Driven Matrix Recovery via Optimal Shrinkage and Spatially Resolved Singular Vector Denoising under High-Dimensional Separable Noise

Pei-Chun Su

Abstract

This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional and significantly larger than the signal rank, we derive exact leading-order variance formulas for the singular vector perturbation projected onto any spatial patch. The variance decomposes into a spatially non-uniform component governed by the local noise covariance and a spatially uniform component governed by the global noise level. These formulas provide the foundation for the \emph{extended optimal shrinkage and wavelet shrinkage} (e$\mathcal{OWS}$) algorithm, which recovers low-rank matrices satisfying a mixed Hölder condition. The pipeline begins with optimal shrinkage of singular values, then constructs coupled multiscale partition trees on the row and column spaces from the denoised estimate, generating a tensor Haar-Walsh wavelet basis. Spatially adaptive wavelet shrinkage is applied using data-driven, coefficient-level thresholds derived from the perturbation theory. We establish convergence rates that strictly improve upon both optimal shrinkage and wavelet shrinkage applied in isolation. Numerical simulations demonstrate reliable matrix recovery and accurate reconstruction of the underlying singular subspaces, including an application to fetal ECG extraction.

Data-Driven Matrix Recovery via Optimal Shrinkage and Spatially Resolved Singular Vector Denoising under High-Dimensional Separable Noise

Abstract

This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional and significantly larger than the signal rank, we derive exact leading-order variance formulas for the singular vector perturbation projected onto any spatial patch. The variance decomposes into a spatially non-uniform component governed by the local noise covariance and a spatially uniform component governed by the global noise level. These formulas provide the foundation for the \emph{extended optimal shrinkage and wavelet shrinkage} (e) algorithm, which recovers low-rank matrices satisfying a mixed Hölder condition. The pipeline begins with optimal shrinkage of singular values, then constructs coupled multiscale partition trees on the row and column spaces from the denoised estimate, generating a tensor Haar-Walsh wavelet basis. Spatially adaptive wavelet shrinkage is applied using data-driven, coefficient-level thresholds derived from the perturbation theory. We establish convergence rates that strictly improve upon both optimal shrinkage and wavelet shrinkage applied in isolation. Numerical simulations demonstrate reliable matrix recovery and accurate reconstruction of the underlying singular subspaces, including an application to fetal ECG extraction.

Paper Structure

This paper contains 38 sections, 13 theorems, 124 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spiked Model and Optimal Shrinkage of Singular Values
  3. Spiked Model
  4. Optimal Shrinkage of Singular Values
  5. Mixed Hölder Matrix Recovery via Wavelet Shrinkage
  6. Wavelet Analysis
  7. Hölder and Mixed Hölder Space
  8. Tree Structure
  9. Wavelet Shrinkage
  10. Motivations
  11. Contributions
  12. The e$\mathcal{OWS}$ Pipeline
  13. Applications and Experiments
  14. Preliminaries
  15. Required Assumptions
  16. ...and 23 more sections

Key Result

Proposition 2.7

When $d_i \geq \alpha$, the optimal shrinker is $\varphi^*_i=d_i \sqrt{a_{1,i} a_{2,i}}$, $\varphi^*_i=d_i\sqrt{\frac{a_{1,i}\wedge a_{2,i}}{a_{1,i}\vee a_{2,i}}}$ and $\varphi^*_i=d_i(\sqrt{a_{1,i} a_{2,i}}-\sqrt{(1-a_{1,i})(1-a_{2,i})}\ )$ when the Frobenius norm, operator norm, and nuclear norm a

Figures (6)

  • Figure 1: Helmholtz kernel matrix between a planar point cloud and a helix in $\mathbb{R}^3$. (a) The three-dimensional geometry: $p=512$ points sampled uniformly on a flat plane (blue) and $n=512$ points along a helix of radius $r=1$ (red). (b) The resulting kernel matrix $K_{ij} = C\cos(2\pi\nu\lVert x_i - y_j\rVert) / \lVert x_i - y_j\rVert$. (c) The same matrix with rows and columns organized by the row and column trees based on the affinity from Algorithm \ref{['alg:questionnaire']}, revealing the smooth oscillatory pattern induced by the pairwise distances between the two point sets.
  • Figure 2: Comparison of denoised Helmholtz kernels. Top-left: the noisy kernel $\widetilde{S}$ contaminated with TYPE3 noise. Top-right: denoised result using $\mathcal{WS}$. Middle-left: denoised result using eOptShrink. Middle-right: denoised result using e$\mathcal{OWS}$. Bottom-left: the clean ground-truth kernel $S$.
  • Figure 3: Comparison of the denoising performance of eOptShrink, $\mathcal{WS}$, and e$\mathcal{OWS}$ on the Helmholtz kernel. The top, middle, and bottom rows correspond to the contamination with TYPE1, TYPE2, and TYPE3 noise, respectively. The first column displays the MSE with the $y$-axis on a log scale, the second column shows the left inner product, and the third column presents the right inner product. The $x$-axis in all plots represents the value of $n$ on a log scale. The blue circles mark the results of eOptShrink, the red triangles represent $\mathcal{WS}$, and the green stars denote e$\mathcal{OWS}$.
  • Figure 4: Comparison of denoised Sinusoidal waves. Top-left: the noisy kernel $\widetilde{S}$ contaminated with TYPE2 noise. Top-right: denoised result using $\mathcal{WS}$. Middle-left: denoised result using eOptShrink. Middle-right: denoised result using e$\mathcal{OWS}$. Bottom-left: the clean ground-truth kernel $S$.
  • Figure 5: Comparison of the denoising performance of eOptShrink, $\mathcal{WS}$, and e$\mathcal{OWS}$ on the matrix of sinusoidal waves. The top, middle, and bottom rows correspond to the contamination with TYPE1, TYPE2, and TYPE3 noise, respectively. The first column displays the MSE with the $y$-axis on a log scale, the second column shows the left inner product, and the third column presents the right inner product. The $x$-axis in all plots represents the value of $n$ on a log scale. The blue circles mark the results of eOptShrink, the red triangles represent $\mathcal{WS}$, and the green stars denote e$\mathcal{OWS}$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 1.1: Multiscale Partition Tree Structure $\mathcal{T}_X$
  • Definition 1.2: Tree metric
  • Definition 2.1: Empirical spectral Distribution
  • Definition 2.2: Integral transforms
  • Definition 2.3: Stochastic domination
  • Definition 2.5: Asymptotic loss
  • Definition 2.6: Optimal shrinker
  • Proposition 2.7: GDleeb2020optimal
  • Theorem 2.8: Theorem 4.2 su2025data
  • Theorem 2.9: Theorem 4.4 of su2025data
  • ...and 20 more