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

TL;DR

This paper introduces a novel spatially resolved perturbation theory for singular vectors under high-dimensional separable noise, enabling an advanced data-driven matrix recovery method that combines optimal and wavelet shrinkage techniques.

Contribution

It develops a new perturbation theory for singular vectors in high-dimensional settings and introduces the extended optimal shrinkage and wavelet shrinkage (eOWS) algorithm for improved low-rank matrix recovery.

Findings

Theoretical variance formulas for singular vector perturbations are derived.

The eOWS algorithm achieves convergence rates better than traditional methods.

Numerical simulations confirm effective matrix recovery and subspace reconstruction.

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 H\"older 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.