On the Sample Complexity of Learning for Blind Inverse Problems

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, and the random forward operators. We also prove convergence results of the reconstruction error under appropriate source condition assumptions. Furthermore, we derive 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.

💡 Research Summary

This paper addresses the challenging setting of blind inverse problems, where the forward operator A in the observation model y = A x + ε is unknown or only partially known. While classical non‑blind approaches assume a fixed A and rely on variational regularization (e.g., Tikhonov) or MAP estimation, these methods either perform poorly when A is random or lack theoretical guarantees when combined with modern data‑driven techniques. The authors therefore focus on linear minimum mean‑square error (LMMSE) estimators, which are the linear counterpart of the optimal MMSE estimator and enjoy a rich analytical framework (Wiener‑Kolmogorov filtering, Kalman filtering).

The first major contribution is the derivation of a closed‑form expression for the LMMSE estimator when A is a random matrix. Under standard independence assumptions (A, x, ε are mutually independent, have finite first and second moments, and ε is zero‑mean Gaussian with covariance β I), the authors compute the cross‑covariance C_xy = E