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[2512.23405] On the Sample Complexity of Learning for Blind Inverse Problems
Summary
This article explores the sample complexity of learning in blind inverse problems, providing theoretical insights and empirical validation for data-driven approaches in this challenging domain.
Why It Matters
Blind inverse problems are prevalent in various fields, including imaging and signal processing, where the forward operator is unknown. Understanding the sample complexity and developing robust learning methods is crucial for improving the reliability and interpretability of solutions in practical applications.
Key Takeaways
- The paper provides a theoretical framework for understanding learning in blind inverse problems using Linear Minimum Mean Square Estimators (LMMSEs).
- It establishes closed-form expressions for optimal estimators and derives finite-sample error bounds that quantify performance based on noise and sample size.
- The findings highlight the impact of operator randomness on convergence rates, offering insights into the reliability of learned estimators.
- Numerical experiments validate the theoretical predictions, confirming the expected convergence behavior.
- This research contributes to bridging the gap between empirical performance and theoretical guarantees in machine learning applications.
Computer Science > Machine Learning arXiv:2512.23405 (cs) [Submitted on 29 Dec 2025 (v1), last revised 19 Feb 2026 (this version, v3)] Title:On the Sample Complexity of Learning for Blind Inverse Problems Authors:Nathan Buskulic, Luca Calatroni, Lorenzo Rosasco, Silvia Villa View a PDF of the paper titled On the Sample Complexity of Learning for Blind Inverse Problems, by Nathan Buskulic and 3 other authors View PDF HTML (experimental) 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 a 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, an...
