Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner due to identifiability issues and symmetric solutions inherent to the blind setting. Recently, data-driven approaches have been proposed to address such problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by theoretical guarantees, limiting their reliability in domains such as applied imaging where a blind approach often relates to a calibration of the acquisition device. In this work, we shed light on learning in blind inverse problems within the insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical recovery results to the blind setting. In particular, we establish equivalences with tailored Tikhonov-regularized formulations, where the regularization structure depends explicitly on the distributions of the unknown signal, of the noise, and of the random forward operator. We also show how the reconstruction error converges as the noise and the randomness of the operator diminish when we use a source condition assumption. Furthermore, we derive finite-sample error bounds that characterize the performance of the 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 show explicitly the dependence of the associated convergence rates to this randomness factors. Finally, we validate our theoretical findings through illustrative exemplar numerical experiments that confirm the predicted convergence behavior.

翻译：盲反问题广泛存在于诸多实验场景中，其中感兴趣信号与前向算子均（部分）未知。在此背景下，由于盲设置固有的可辨识性问题和解对称性，为非盲情况设计的方法无法直接适用。近年来，数据驱动方法被提出用于解决此类问题，展现出强大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理论支撑，限制了其在应用成像等领域的可靠性——盲方法常与采集装置的校准密切相关。本研究在线性最小均方估计器（LMMSE）这一富有洞察力的框架内，揭示了盲反问题中的学习机制。我们提供理论分析，推导出最优估计器的闭式解，并将经典恢复结果拓展至盲设置。特别地，我们建立了与定制化吉洪诺夫正则化公式的等价性，其中正则化结构显式依赖于未知信号、噪声及随机前向算子的分布。我们还证明了当采用源条件假设时，重建误差如何随噪声与算子随机性的减弱而收敛。此外，我们推导了有限样本误差界，刻画了所学习估计器性能随噪声水平、问题条件数及可用样本量的变化规律。这些界显式量化了算子随机性的影响，并清晰展示了相关收敛速率对该随机性因子的依赖性。最后，通过说明性数值实验验证了理论发现，确认了预测的收敛行为。