Maximum-a-posteriori (MAP) approaches are an effective framework for inverse problems with known forward operators, particularly when combined with expressive priors and careful parameter selection. In blind settings, however, their use becomes significantly less stable due to the inherent non-convexity of the problem and the potential non-identifiability of the solutions. (Linear) minimum mean square error (MMSE) estimators provide a compelling alternative that can circumvent these limitations. In this work, we study synthetic two-dimensional blind deconvolution problems under fully controlled conditions, with complete prior knowledge of both the signal and kernel distributions. We compare tailored MAP algorithms with simple LMMSE estimators whose functional form is closely related to that of an optimal Tikhonov estimator. Our results show that, even in these highly controlled settings, MAP methods remain unstable and require extensive parameter tuning, whereas the LMMSE estimator yields a robust and reliable baseline. Moreover, we demonstrate empirically that the LMMSE solution can serve as an effective initialization for MAP approaches, improving their performance and reducing sensitivity to regularization parameters, thereby opening the door to future theoretical and practical developments.

翻译：最大后验概率方法对于前向算子已知的逆问题是一个有效的框架，尤其当结合了表达能力强的先验分布和细致的参数选择时。然而，在盲设置下，由于问题固有的非凸性以及解可能存在的不可辨识性，这类方法的使用变得极不稳定。（线性）最小均方误差估计器提供了一种有吸引力的替代方案，能够规避这些限制。在本工作中，我们在完全受控条件下研究合成的二维盲反卷积问题，并对信号与核的分布拥有完整的先验知识。我们将定制的MAP算法与简单的LMMSE估计器进行比较，后者的函数形式与最优Tikhonov估计器密切相关。我们的结果表明，即使在这些高度受控的设置下，MAP方法仍不稳定且需要大量的参数调优，而LMMSE估计器则能提供一个稳健可靠的基线。此外，我们通过实验证明，LMMSE解可以作为MAP方法的有效初始化，从而提升其性能并降低对正则化参数的敏感性，这为未来的理论与应用发展开辟了道路。