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 (eOWS), 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.
Pei-Chun Su (Fri,) studied this question.