On the Sample Complexity of Learning for Blind Inverse Problems

TL;DR

This work analyzes blind inverse problems where the forward operator is random and unknown, proposing Linear Minimum Mean Square Estimators (LMMSEs) as a principled, tractable framework. It derives closed-form LMMSE estimators for both recovering the unknown signal $\mathbf{x}$ and the forward operator $\mathbf{A}$, showing their equivalence to Tikhonov-regularized problems and establishing finite-sample guarantees under a Hölder source condition. The authors quantify how operator randomness and sampling affect reconstruction via a bias-variance decomposition, providing rigorous bounds that separate approximation and sampling errors and validating them with one-dimensional numerical experiments. The results illuminate the role of operator randomness in blind settings and yield practical, provable performance guarantees for learning in blind inverse problems with linear estimators. This work lays a theoretical foundation for reliable, data-driven approaches to blind inverse problems and motivates extension to nonlinear or bilevel formulations and learned synthesis operators.

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.

Figures (6)

• Figure 1: Illustrative synthetic dataset: statistics of a sinusoidal signal $\mathbf{x}$ (Figure \ref{['fig:data_generation1']}), observations obtained by convolution with Gaussian kernel (Figure \ref{['fig:data_generation2']}) and convolution kernels (Figure \ref{['fig:data_generation3']}) with uncertainties. For all quantities, empirical means are denoted by solid lines, together with $\pm$ one empirical standard deviation (shaded area) computed from $N=1000$ training samples.
• Figure 2: Samples of reference and observed (blurred + noisy) signals.
• Figure 3: Reconstruction of a test signal using the theoretical LMMSE estimator. Ground truth (solid blue), sinusoidal mean (dotted red), measurement (dotted green), and theoretical-LMMSE reconstruction (dashed orange).
• Figure 4: Numerical verification of the error bound \ref{['eq:approx_bound']} as a function of the source condition parameter $\alpha$.
• Figure 5: Numerical verification of the approximation error bound \ref{['eq:approx_bound']} as a function of kernel variability, parameterized via the standard deviation $\sigma_{\mathrm{std}}$ (or equivalently $\mathrm{CV}^2$) of the kernel width.

Theorems & Definitions (32)

• Lemma 2.1: \ref{['eq:LMMSE-B']} as Tikhonov solution
• proof
• Lemma 2.2
• proof
• Definition 3.1: Hölder source condition with random singular values
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4: Sampling bound of LMMSE