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

TL;DR

It is concluded that successful recovery of posterior maximization depends on strategic initialization within the basin of favorable local minima, and this is validated with numerical experiments on both synthetic and real-world data.

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.

Figures (15)

• Figure 1: Critical points obtained by a gradient descent starting from real images in FFHQ-256 using the FFHQ-256 potential. The critical points behave as denoised versions of the initial image. The potential $q=- \log p_{\boldsymbol{x}}$ displayed on the top of each image is about twice lower for critical points.
• Figure 2: Illustration of the convergence of a gradient descent on the potential $q$ using different models. The potential stabilizes after a few thousands iterations.
• Figure 3: Distance and difference of potential between natural images and the corresponding critical points. The dataset used to train the model is highlighted in blue. Observe that the relative distance between the critical points and the initial images is less than 10% even when starting from pure noise. Yet, the potential is much lower.
• Figure 4: Spectra of the Hessian of the potential $\nabla^2 q(\bar{x}_i)$ for 20 different critical points $\bar{x}_i$. By thresholding all values below $10^{-5}$, the local image manifold dimension is estimated at 36 for FFHQ-64 and 16 for AFHQ-64.
• Figure 5: Different 1D blur families used in the forthcoming experiments. The parameter $\theta$ roughly accounts for the diameter of the PSF with $h_{0} = \delta$ and the higher $\theta$, the more blur.

Theorems & Definitions (18)

• Definition 2.1: Morse-Bott Potential
• Remark 2.1
• Remark 2.2: On the choice of diffusion priors
• Remark 2.3: Conservative fields
• Definition 3.1: Second order critical point
• Theorem 3.1: Recovery guarantees -- non-blind case
• Theorem 3.2: Stable recovery conditions -- blind case
• Corollary 3.1: Stability bounds
• Remark 3.1: Dependence on noise
• Remark 3.2: On the size of the basin