Transfer Denoising of Low-Rank Plus Sparse Matrices under Ambient and Rank Growth

We study transfer learning for low-rank plus sparse matrix denoising when both ambient and latent dimensions grow. A data-rich source task provides subspace information for a higher-dimensional, data-scarce target with a small rank increment and localized sparse edits. We develop an anchored alternating-projection estimator that preserves transferred subspaces while learning target-specific coefficients, additional latent directions, and sparse modifications. We establish deterministic error bounds that separate target noise, representation growth, and source-estimation error, identifying improved rates through small structural increments and reduced noise within inherited subspaces. We further extend it to approximate anchoring quantifies robustness to subspace mismatch. We establish application-specific guarantees for Markov transition matrix estimation from single trajectories with dependent noise and for structured covariance estimation, with the latter analysis provided in the appendix. Experiments demonstrate consistent improvements over non-transfer baselines on synthetic data and an approximately 45\% reduction in principal-angle error on a NYC taxi transition-estimation task with expanding state spaces.