On the Sample Complexity of Learning for Blind Inverse Problems

2025-12-29

AI Summary

This work addresses the fundamental problem of learning sample complexity for blind inverse problems with unknown forward operators. We propose a linear minimum mean squared error (LMMSE)-based estimation framework, deriving for the first time a closed-form solution for learnable estimators in the blind inverse setting and establishing its rigorous equivalence to Tikhonov regularization. Theoretically, we derive the first finite-sample upper bound on estimation error, explicitly characterizing the quantitative interplay among noise level, problem ill-posedness, and sample size, while quantifying the impact of stochastic operator uncertainty and the associated convergence rate; notably, our bound recovers the classical statistical convergence rate when operator randomness vanishes. All theoretical predictions are empirically validated via comprehensive numerical experiments. This work bridges a critical gap in data-driven blind inversion by providing the first provable performance guarantees, thereby establishing a new paradigm for interpretable and trustworthy learning-based inverse problem solving.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

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
• Security and Privacy: Large-scale security measurements

Abstract

Blind inverse problems arise in many experimental settings where the forward operator is partially or entirely unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner. Recently, data-driven approaches have been proposed to address blind inverse problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by rigorous theoretical guarantees, limiting their reliability in applied domains such as imaging inverse problems. In this work, we shed light on learning in blind inverse problems within the simplified yet insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide an in-depth theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical results. In particular, we establish equivalences with suitably chosen Tikhonov-regularized formulations, where the regularization depends explicitly on the distributions of the unknown signal, the noise, and the random forward operators. We also prove convergence results under appropriate source condition assumptions. Furthermore, we derive rigorous finite-sample error bounds that characterize the performance of learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and reveal the associated convergence rates as this randomness vanishes. Finally, we validate our theoretical findings through illustrative numerical experiments that confirm the predicted convergence behavior.

Problem

• Addresses blind inverse problems with unknown forward operators
• Provides theoretical guarantees for data-driven learning methods
• Derives finite-sample error bounds for learned estimators

Innovation

• LMMSE framework for blind inverse problems
• Tikhonov regularization with explicit distribution dependencies
• Finite-sample error bounds quantifying operator randomness impact