对比了盲反问题中MAP与LMMSE估计器的性能，发现后者更稳定可靠。
A Comparative Study of MAP and LMMSE Estimators for Blind Inverse Problems
- 用受控的二维盲去卷积实验比较MAP与LMMSE方法
- 在理想条件下MAP仍需大量调参且不稳定，而LMMSE表现稳健
- LMMSE可作MAP的良好初始化，提升其性能与鲁棒性
最大后验（MAP）方法在已知前向算子的反问题中表现优异，尤其当结合表达性强的先验和精细参数选择时。但在盲设置下，由于问题固有的非凸性及解的潜在不可识别性，其应用变得显著不稳定。线性最小均方误差（LMMSE）估计器提供了一种可行替代方案，可规避这些局限。本文在完全可控条件下研究二维盲去卷积问题，已知信号与核的完整分布。对比定制化的MAP算法与形式上接近最优Tikhonov估计器的简单LMMSE估计器。结果表明，即使在高度受控的设定下，MAP方法仍不稳定且需大量参数调优，而LMMSE能提供稳健可靠的基准。此外，我们实证展示了LMMSE解可有效初始化MAP方法，改善其性能并降低对正则化参数的敏感性，为未来的理论与实践发展开辟道路。
原文摘要 · Abstract (English)
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.
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