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.

翻译：盲逆问题广泛存在于实验科学中，其中前向算子部分或完全未知。在此背景下，为已知前向算子的非盲问题开发的方法无法直接适用。近年来，基于数据驱动的方法被提出以解决盲逆问题，展现出优异的实证性能与适应性。然而，这些方法通常缺乏可解释性，且缺乏严格的理论保证，限制了其在成像逆问题等应用领域的可靠性。本文在线性最小均方误差估计器（LMMSE）这一简化但富有启发性的框架下，对盲逆问题的学习机制进行理论阐释。我们通过理论分析推导了最优估计器的闭式解，并推广了经典结论。特别地，我们建立了与适当选择的Tikhonov正则化表述的等价性，其中正则化项显式依赖于未知信号、噪声及随机前向算子的分布。在适当的源条件假设下，我们证明了重构误差的收敛性结果。此外，我们推导了有限样本误差界，该误差界将学习估计器的性能表征为噪声水平、问题条件数及可用样本数量的函数。这些误差界明确量化了算子随机性的影响，并揭示了当随机性消失时的收敛速率。最后，我们通过数值实验验证了理论结论，实验结果证实了预测的收敛行为。