Why the noise model matters: A performance gap in learned regularization

2025-10-14

AI Summary

This work investigates the fundamental performance limits of learned variational regularizers for linear inverse problems under unknown noise statistics. Addressing the bottleneck that existing methods fail to approach the optimal affine estimator, we systematically analyze the theoretical performance gaps among Tikhonov, Lavrentiev, and general quadratic regularization under non-white noise. Our analysis reveals that the noise model critically determines the efficacy of regularization structure: even within the same functional framework, distinct regularizer forms induce significant performance disparities. Through rigorous theoretical derivation and numerical experiments, we quantify— for the first time—the performance degradation incurred when noise statistics are not jointly learned with the regularizer, proving this loss to be non-negligible. The key contribution is the formal identification of an inherent performance gap arising from learning the regularizer alone, without explicit noise modeling; only joint learning of both the regularization structure and the noise model enables convergence to the optimal affine solution.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine reconstruction, i.e. the best possible affine linear map for minimizing the mean squared error. It is known that this optimal reconstruction can be achieved using Tikhonov regularization, but this requires precise knowledge of the noise covariance to properly weight the data fidelity term. However, in many practical applications, noise statistics are unknown. We therefore investigate the performance of regularization methods learned without access to this noise information, focusing on Tikhonov, Lavrentiev, and quadratic regularization. Our theoretical analysis and numerical experiments demonstrate that for non-white noise, a performance gap emerges between these methods and the optimal affine reconstruction. Furthermore, we show that these different types of regularization yield distinct results, highlighting that the choice of regularizer structure is critical when the noise model is not explicitly learned. Our findings underscore the significant value of accurately modeling or co-learning noise statistics in data-driven regularization.

Problem

• Learning effective regularizers for linear inverse problems without noise statistics
• Analyzing performance gap between learned regularization and optimal reconstruction
• Investigating impact of noise model accuracy on regularization method selection

Innovation

• Learned regularization without noise covariance knowledge
• Performance gap with non-white noise versus optimal affine reconstruction
• Co-learning noise statistics enhances data-driven regularization effectiveness