Revisiting CUR Perturbation Analysis: A Local Tangent-Space Expansion

2026-05-13

AI Summary

This work addresses the limitations of classical perturbation analyses for CUR decomposition, which rely solely on global noise levels and fail to capture how sampling structures influence local reconstruction errors. By employing a local tangent space expansion, the study establishes, for the first time, a precise connection between the first-order perturbation error of rank-truncated CUR mappings with fixed index sets and the oblique projection operators induced by sampling. This reveals a mechanism whereby invisible perturbations are automatically eliminated under first-order approximation. Leveraging Fréchet derivatives and local Taylor expansions, the authors theoretically derive first- and second-order local convergence rates. Numerical experiments further validate the perturbation-removal effect across distinct subspaces, highlighting CUR’s unique advantage over truncated SVD in structural sensitivity.

Abstract

CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we develop a local perturbation expansion for a fixed-index rank-truncated CUR map near an admissible rank-\(r\) matrix. We show that the Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix is governed not by the full perturbation norm alone, but by the image of the perturbation under this sampling-induced tangent projector. In particular, perturbations that are invisible to the selected rows and columns are removed to first order. We compare this behavior with the classical local expansion of the rank-\(r\) SVD truncation. SVD removes orthogonal-normal perturbations to first order, whereas rank-truncated CUR removes perturbations in the kernel of the sampling-induced oblique tangent projector. Numerical experiments illustrate these regimes and confirm the predicted first- and second-order local rates.

Problem

• CUR decomposition
• perturbation analysis
• low-rank matrix
• tangent-space projector
• rank truncation

Innovation

• CUR decomposition
• perturbation analysis
• Fréchet derivative
• tangent-space projector
• low-rank approximation