A Morse-Bott Framework for Blind Inverse Problems: Local Recovery Guarantees and the Failure of the MAP

TL;DR

This paper introduces a Morse-Bott framework to analyze the local recovery guarantees of MAP-based methods in blind inverse problems, revealing intrinsic landscape limitations and emphasizing the importance of initialization.

Contribution

It models the image prior as a Morse-Bott function, aligning theoretical analysis with learned priors, and demonstrates the intrinsic landscape challenges in blind deconvolution.

Findings

Local minimizers are stable near the ground truth.

Gradient-based methods converge to the same minimizer from similar initializations.

MAP failure in blind deconvolution is due to landscape properties, not prior quality.

Abstract

Maximum A Posteriori (MAP) estimation is a cornerstone framework for blind inverse problems, where an image and a forward operator are jointly estimated as the maximizers of a posterior distribution. In this paper, we analyze the recovery guarantees of MAP-based methods by adopting a Morse-Bott framework. We model the image prior potential as a Morse-Bott function, where natural images are modeled as residing locally on a critical submanifold. This means that while the potential is locally flat along the natural directions of the image manifold, it is strictly convex in the directions normal to it. We demonstrate that this Morse-Bott hypothesis aligns with the structural properties of state-of-the-art learned priors, a finding we validate through an experimental analysis of the potential landscape and its Hessian spectrum. Our theoretical results show that, in a neighborhood of the ground-truth image and operator, the posterior admits local minimizers that are stable both with respect to initialization (gradient steps converge to the same minimizer) and to small noise perturbations (solutions vary smoothly). This local stability explains the empirical success of well-designed gradient-based optimization in these settings. However, we also demonstrate that this stability is a local property: the blurry trap, well-known for sparse priors in blind deconvolution, persists even with state-of-the-art learned priors. Our findings demonstrate that the failure of MAP in blind deconvolution is not a limitation of prior quality, but an intrinsic characteristic of the landscape. We conclude that successful recovery of posterior maximization depends on strategic initialization within the basin of favorable local minima, and we validate this with numerical experiments on both synthetic and real-world data.