MIAT-RPCA: Training-Free Robust PCA via Manifold-Inspired Interpolation and Adaptive Truncation

Thresholding-based robust principal component analysis (RPCA) separates sparse corruption from low-rank structure. In basic CUR updates, this process does not explicitly exploit local coefficient neighborhoods to refine corrupted observations. Yet observations affected by different corruption levels may retain shared structure that can support mutual refinement. Using this information requires considering both reconstruction inconsistency and neighborhood agreement. Motivated by this observation, we propose Manifold-Inspired Interpolation and Adaptive Truncation for RPCA (MIAT-RPCA), a training-free method that incorporates neighborhood information into sampled CUR recovery. Cosine similarities define local neighborhoods in coefficient space. A quantile-normalized residual score controls interpolation strength, allowing columns with greater reconstruction inconsistency to draw more heavily on neighboring coefficients. A complementary attenuation gate combines this score with neighborhood confidence, applying stronger suppression when inconsistency is high and neighborhood agreement is low. Both operations precede reconstruction and keep the corrected coefficients within their original span. We establish that the correction preserves the target rank bound and satisfies a neighborhood norm bound, characterize the gates' monotonic responses, and analyze the computational cost per iteration. All neighborhood weights and gates are computed from the current observation, with a prescribed threshold schedule and no training data or parameter learning.