A Morse-Bott Framework for Blind Inverse Problems: Local Recovery Guarantees and the Failure of the MAP
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 applications such as blind deblurring, this principle is used to recover sharp images from degraded observations. In this paper, we analyze the recovery guarantees of MAP-based methods by adopting a \emph{Morse--Bott framework}. We model the image 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 descents converge to the same minimizer) and to small perturbations of the data (solutions vary smoothly with the observations). This local stability potentially provides a theoretical justification for the empirical success of well designed gradient-based optimization in these settings. However, we also demonstrate that this local stability is a \textbf{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 depends on strategic initialization around favorable local minima.