Data-Driven Matrix Recovery with High-Dimensional Noise via Optimal Shrinkage of Singular Values and Wavelet Shrinkage of Singular Vectors

This paper presents a novel data-driven algorithm designed to recover low-rank matrices whose entries satisfy a mixed Hölder condition in the presence of high-dimensional noise with a separable covariance structure. The algorithm, coined extended optimal shrinkage and wavelet shrinkage (e$\mathcal{OWS}$), emphasizes the asymptotic structure, where the matrix size is significantly larger than the rank of the signal matrix. The denoising process begins with the adaptation of the well-known optimal shrinkage of singular values. This is followed by an iterative procedure that organizes the matrix using a coupled metric on the rows and columns, constructed by building a tree structure for both dimensions. This hierarchical organization induces a tensor Haar-Walsh basis on the matrix. An adapted wavelet shrinkage technique is applied to further denoise the reconstructed matrix, modifying the Haar-Walsh coefficients based on the analysis of the first-order perturbation of singular vectors. We provide theoretical guarantees for these estimators, demonstrating a convergence rate that highlights the efficacy of our algorithm. Simulations show successful matrix recovery, with a small mean squared error between the estimate and the ground truth, and accurate reconstruction of the singular vector spaces.