A Comparative Study of MAP and LMMSE Estimators for Blind Inverse Problems

2026-02-12

AI Summary

This work addresses the instability of maximum a posteriori (MAP) estimation in two-dimensional blind deconvolution—a canonical blind inverse problem—where non-convexity of the objective function and non-identifiability of solutions often lead to unreliable performance. Under fully controlled experimental conditions, the study systematically compares MAP algorithms employing exact priors against a structurally near-optimal Tikhonov-regularized linear minimum mean square error (LMMSE) estimator. The results demonstrate that LMMSE not only significantly outperforms MAP methods, which require meticulous hyperparameter tuning, but also serves as a robust initialization strategy for MAP optimization, effectively reducing its sensitivity to regularization parameters and enhancing both stability and reconstruction quality. These findings establish LMMSE as playing a dual role in blind deconvolution: a high-performance baseline and a reliable starting point for iterative refinement.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• blind inverse problems
• MAP estimation
• LMMSE estimation
• blind deconvolution
• non-convexity

Innovation

• blind deconvolution
• LMMSE estimator
• MAP estimation
• inverse problems
• initialization strategy