Perturbation Bounds for Low-Rank Inverse Approximations under Noise

2025-10-29

AI Summary

This work addresses the spectral-norm robustness of low-rank pseudoinverse approximations to matrix inversion under observational noise. To handle noisy perturbations, we introduce, for the first time, a compact non-asymptotic perturbation bound based on contour integral techniques applied to the non-holomorphic function $f(z) = 1/z$, overcoming the asymptotic and loose nature of classical bounds—achieving up to $sqrt{n}$-fold improvement in theoretical accuracy. The bound explicitly characterizes error dependence on eigenvalue gaps, spectral decay, and alignment between noise and low-curvature directions. By integrating matrix perturbation theory with spectral analysis, our framework enables quantifiable robustness modeling. Experiments demonstrate that the derived bound tightly tracks empirical errors and significantly outperforms existing estimates across diverse synthetic and real-world datasets. This provides a new theoretical guarantee for efficient, spectrum-aware matrix computation in noisy environments.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and quantization. The spectral-norm robustness of low-rank inverse approximations remains poorly understood. We systematically study the spectral-norm error $| ( ilde{A}^{-1})_p - A_p^{-1} |$ for an $n imes n$ symmetric matrix $A$, where $A_p^{-1}$ denotes the best rank-(p) approximation of $A^{-1}$, and $ ilde{A} = A + E$ is a noisy observation. Under mild assumptions on the noise, we derive sharp non-asymptotic perturbation bounds that reveal how the error scales with the eigengap, spectral decay, and noise alignment with low-curvature directions of $A$. Our analysis introduces a novel application of contour integral techniques to the emph{non-entire} function $f(z) = 1/z$, yielding bounds that improve over naive adaptations of classical full-inverse bounds by up to a factor of $sqrt{n}$. Empirically, our bounds closely track the true perturbation error across a variety of real-world and synthetic matrices, while estimates based on classical results tend to significantly overpredict. These findings offer practical, spectrum-aware guarantees for low-rank inverse approximations in noisy computational environments.

Problem

• Analyzing spectral-norm robustness of low-rank pseudoinverses under noise perturbations
• Deriving sharp perturbation bounds for low-rank inverse approximations with noise
• Establishing spectrum-aware guarantees for noisy low-rank matrix inversions

Innovation

• Novel contour integral technique for non-entire function
• Sharp non-asymptotic bounds for spectral-norm error
• Improved perturbation bounds by factor of sqrt(n)