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

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Pith's one-line read This paper establishes that with modern diffusion-based priors, the global MAP solution of blind deconvolution is still the no-blur estimate, while true recovery is confined to stable local minima near second-order critical points of the…

desk verdict A genuinely new local recovery theorem for blind deconvolution with learned priors, wrapped in a global MAP-failure narrative that outruns its own assumption (9). read the letter →

arxiv 2508.02923 v3 pith:FOKM6HV4 submitted 2025-08-04 cs.CV

Minh-Hai Nguyen , Edouard Pauwels , Pierre Weiss This is my paper
classification cs.CV
keywords blinddeconvolutiondiffusionmodelsmaximumaposterioriestimationMorse-Botttheorylearnedpriorsimagemanifoldslocalrecoveryguaranteesno-blursolution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether modern diffusion-based learned priors can rescue MAP estimation for blind deconvolution, and answers no. It shows that, on pretrained diffusion models, blurry images consistently receive lower prior cost than sharp ones, and natural images lie near flat critical submanifolds of the learned potential. Under the sufficient condition $q(h\star x)\le q(x)$, the global MAP solution is the no-blur pair: a Dirac kernel with a blurry image. At the same time, if the true image is a second-order critical point of the prior and three identifiability conditions hold, the true image-kernel pair is a strict local minimizer of the noiseless posterior and remains a stable minimizer under small noise. The practical moral is that recovery is decided by initialization and local minima, not by the global MAP estimate.

What carries the argument

The load-bearing object is the Morse-Bott structure of the prior potential $q=-\log p_x$: rather than being strictly convex, $q$ is flat along a low-dimensional manifold of natural images and strictly convex in normal directions, so its critical points form a submanifold. The paper's main theorem (Theorem 8 in the appendix, specialised as Theorems 4 and 5) reduces stable recovery to three kernel-intersection conditions—$\ker\nabla^2 q(\bar x)\cap\ker H_{\bar\theta}=\{0\}$, $\ker J(\bar\theta)=\{0\}$, and $(H_{\bar\theta}\,\ker\nabla^2 q(\bar x))\cap\operatorname{Im}J(\bar\theta)=\{0\}$—and proves via a block-matrix lemma that these make the posterior Hessian positive-definite. The companion mechanism for MAP failure is the prior-decay condition $q(h\star x)\le q(x)$, which makes the Dirac kernel and the MAP-denoised image global minimizers.

What would settle it

Compute the prior-decay gap $q(h_\theta\star x)-q(x)$ on a large corpus with diverse kernel families; one natural image and one kernel in the family with a positive gap refutes the sufficient condition behind Theorem 1. A second falsifier would be a second-order critical point $\bar x$ satisfying conditions (10)-(12) whose joint posterior Hessian has a negative eigenvalue, contradicting Theorem 4.

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Extended reading notes

Core claim

The central discovery is that the failure of MAP in blind deconvolution is intrinsic to the posterior landscape rather than a defect of the prior. The paper models the learned prior potential $q$ as a Morse-Bott function: natural images sit on a low-dimensional critical submanifold where $q$ is flat, while $q$ is strictly convex in normal directions. Empirically, blurring decreases $q$, so Theorem 1 shows that whenever the Dirac kernel lies in the kernel family and $q(h\star x)\le q(x)$ holds for every $x$ and $h$, every global minimizer of the negative log-posterior can be replaced by the no-blur solution (Dirac kernel plus a MAP-denoised image). Theorem 4 then shows that if $\bar x$ is a second-order critical point of $q$ and $\ker\nabla^2 q(\bar x)\cap\ker H_{\bar\theta}=\{0\}$, $\ker J(\bar\theta)=\{0\}$, and $(H_{\bar\theta}\,\ker\nabla^2 q(\bar x))\cap\operatorname{Im}J(\bar\theta)=\{0\}$, then $(\bar x,\bar\theta)$ is a strict local minimizer of the noiseless cost with positive-definite Hessian; Corollary 5 makes this stable under bounded noise. Together, these results say that the recoverable points are exactly the second-order critical points of the prior, while the global MAP estimator remains the blurry degenerate solution.

Load-bearing premise

The paper's claim that MAP is intrinsically blurry for diffusion priors rests on the condition $q(h\star x)\le q(x)$ holding for every image and every kernel in the family, a condition verified only on 100 training images and a few one-dimensional blur families.

Editorial extensions

If this is right

  • Global MAP blind deconvolution with diffusion priors will continue to return the no-blur solution, so methods that stop at the global minimizer are not a route to sharp images.
  • Images that are second-order critical points of a learned prior are the ones recoverable by local optimization, giving a concrete characterization of what a prior must encode for blind deblurring to succeed.
  • Gradient-descent algorithms that start near a favorable basin can recover the true image and kernel, and the recovered pair moves smoothly with the noise level.
  • Low-dimensional kernel parameterizations make the identifiability conditions easier to satisfy, so kernel families with fewer parameters should be preferred in blind deconvolution.
  • The stability is local: at higher noise the local minimizer near the truth can vanish, reverting the posterior to favoring no blur.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Morse-Bott analysis likely applies to other bilinear inverse problems such as blind super-resolution or joint calibration, where a similar global degeneracy and local recoverability structure should appear; the paper's Theorem 8 is already stated for general linear operators.
  • The empirical support for condition (9) is narrow—100 training images and one-dimensional blur families—so calling MAP failure 'intrinsic' to diffusion priors is only as strong as that sample; broader blur families or out-of-distribution images might violate it.
  • A practical testable extension would be to audit a learned prior before use by measuring the dimension of $\ker\nabla^2 q$ at representative images and checking the three intersection conditions, thereby predicting which images will be recoverable by local descent.
  • The initialization heuristic 'start with the largest kernel' could be formalized as a basin-size prior: the paper's experiments show larger initial kernels land in the recovery basin, which suggests a quantitative measure of basin width could guide initialization in other inverse problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies maximum-a-posteriori (MAP) estimation for blind deconvolution when the image prior is a learned diffusion prior. It proposes a "Morse-Bott" view of the prior landscape: natural images are modeled as lying near critical submanifolds where the negative log-prior q is locally flat in tangential directions and strictly convex in normal directions. Empirically, the paper reports two landscape properties on several diffusion models (FFHQ-256, ImageNet-64, FFHQ-64, AFHQ-64): blurry images have lower potential (higher likelihood) than sharp images, and gradient descent on q converges to critical points close to natural images whose Hessian spectra have a low-dimensional near-null space. The main theoretical results are: (i) Theorem 1, proving that under the global condition q(h*x) <= q(x) for all images x and all kernels h in the blur family, the no-blur pair is a global minimizer of the joint negative log-posterior; (ii) Theorems 4, 5, and 8, proving that under identifiability conditions (10)-(12), a second-order critical point of the prior is a strict local minimizer of the noiseless posterior and a stable local minimizer under bounded noise; and (iii) an alternating proximal-gradient algorithm with a deliberately "large" kernel initialization that empirically avoids the no-blur trap. The appendix contains the proofs, including a general linear blind-inverse-problem theorem (Theorem 8) proved via a block-matrix lemma and a strong-convexity argument.

Significance. The local recovery theorem is the most valuable part of the paper. If it stands, it gives a rigorous, non-convex recovery guarantee for blind deconvolution with learned priors, without requiring sparsity or convexity, and it identifies second-order critical points of the prior as the recoverable points. This is a meaningful extension of earlier blind-deconvolution theory. The empirical Hessian-spectrum analysis is also a useful contribution, and the authors provide code for reproducibility. I find the proof of Theorem 8 and its corollaries sound: the block-matrix argument in Lemma 7 correctly characterizes positive definiteness, and the strong-convexity/Brouwer step correctly yields a unique stable minimizer. However, the headline claim that MAP failure is an intrinsic property of diffusion priors is not supported by the stated assumptions. Theorem 1 relies on a global condition (Eq. (9)) that the paper itself concedes is false in some regimes (Remark 2) and that is tested only on a limited empirical sample. The local results are unaffected, but the global narrative needs to be reworded as conditional, or substantially supported by new evidence.

major comments (3)
  1. [Sec. 3.1, Theorem 1, Eq. (9)] The paper's central conclusion that MAP blind deconvolution fails intrinsically for diffusion priors (Abstract and Section 6) is not logically supported by Theorem 1. The theorem assumes q(h*x) <= q(x) for every image x and every kernel h in H, but Remark 2 explicitly concedes that this inequality fails at strict local minimizers of q when h is close to the Dirac delta. Section 2.3 checks the condition only on 100 training images over four 1D blur families, and it does not test the second-order critical points that Theorem 4 identifies as the recoverable points. The theorem is therefore a conditional statement, not a demonstration that diffusion priors intrinsically favor the no-blur MAP solution. The wording should be weakened to "under condition (9), which is empirically observed on natural images in the tested families," or the missing verification on the relevant region should be supplied.
  2. [Sec. 3.2.1, Theorem 4 and Corollary 5] The local recovery theorem is correct as a statement about an abstract C^2 potential q, but its application to diffusion priors uses q = -log p_{t_epsilon} with t_epsilon = 10^-3, computed through a score network and an ODE trace estimator. The identifiability conditions (10)-(12) are neither verified numerically nor connected to the score-network approximation error. The experiments in Section 4.1 show posterior profiles consistent with a local minimum around the true kernel when starting from a critical point, but they do not check the Jacobian and Hessian-kernel conditions. The paper should either verify these conditions for the models used, or explicitly present the numerical section as an illustration of the theorem's mechanism rather than a validation of its hypotheses.
  3. [Sec. 4.1, Eq. (13)-(14)] The empirical identification of the global no-blur minimizer relies on replacing the original convolution operator H_theta by the invertible regularized operator H_tilde_theta = H_theta + lambda I, with lambda tuned per family to satisfy (14). Since the original kernels may be non-invertible, the statement that the no-blur solution is the global minimizer of the true posterior is only demonstrated for this regularized surrogate. The main text should state this limitation explicitly and avoid presenting the profile as direct evidence for Theorem 1 in the non-invertible case.
minor comments (6)
  1. [Fig. 5 caption] The word "forthcomig" should be "forthcoming."
  2. [Table 1] The entry "AFQH" should be "AFHQ."
  3. [Sec. 2.3, MNIST experiment] The conclusion that blur preference is "a specificity of natural images and not a bias of the network architecture" is stronger than a single architecture comparison supports; the wording should be tempered.
  4. [Sec. 2.2, Hessian spectra] The reported intrinsic dimensions (36 for FFHQ-64, 16 for AFHQ-64) depend on the arbitrary eigenvalue threshold of 10^-5 and on only 20 critical points; the text should describe these as indicative estimates.
  5. [Sec. 5.2, Köhler dataset] The DRUNet denoiser used in the real-data experiment is treated as a proximal operator of a prior, but it is not tied to the diffusion potentials analyzed in Sections 2-4; this difference should be flagged more prominently in the main text.
  6. [Title and Sec. 2.2] The phrase "Morse-Bott framework" suggests a smooth critical submanifold with nondegenerate normal Hessian, but the theoretical results only use the weaker conditions grad q(bar x)=0 and grad^2 q(bar x)>=0. The terminology should be introduced as an analogy, or the manifold structure should be formalized and verified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's theorems are conditional on explicit assumptions and the empirical measurements are independent of the theorem conclusions.

full rationale

The paper's central theoretical claims are stated as conditional results with explicit assumptions, and the proofs in Appendix A are direct derivations from those assumptions. Theorem 1 is exactly the statement that the sufficient condition q(h*x) <= q(x) forces the Dirac-kernel/no-blur point to be a global minimizer; the proof is immediate and the paper does not disguise the condition as a prediction. The empirical checks in Section 2.3 measure potential values on images and blur families, and are not fitted parameters used to manufacture the conclusion; the gap between the sample-based validation of condition (9) on 100 images and the paper's broader 'intrinsic failure' narrative is a strength-of-evidence or overreach issue, not circularity. Similarly, Theorems 4, 5 and 8 are proved from the second-order critical-point assumptions and conditions (10)-(12), and the numerical posterior profiles in Section 4 illustrate rather than define the theoretical content. The self-citations to DeepInverse [Tachella et al., 2025] and to Nguyen and Weiss [2024] concern software implementation and an interpretation of DRUNet as a proximal operator; they are not load-bearing for the main derivation. No equation is defined in terms of a target conclusion, no fitted input is renamed as a prediction, and no uniqueness or ansatz is imported solely through a self-citation. Accordingly, no significant circularity is present and the honest finding is a score of 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central theorems are conditional and do not fit parameters to data. The empirical claims rely on several numerical approximations and arbitrary thresholds, and the global MAP-failure narrative adds an unverified universal condition. No new physical entities are introduced.

free parameters (4)
  • t_epsilon = 1e-3 = 1e-3
    Small time used to approximate -log p_0; all potential and Hessian estimates are for the smoothed marginal p_t_epsilon.
  • Hessian eigenvalue threshold = 1e-5
    Eigenvalues below 1e-5 are treated as zero, yielding manifold dimensions 36 for FFHQ-64 and 16 for AFHQ-64.
  • lambda inversion regularizer = grid-searched per blur family
    Chosen in Section 4.1 to make H_theta invertible and satisfy condition (14) for the one-dimensional posterior profiles.
  • ODE tolerances atol/rtol = 1e-5
    Used in the Skilling-Hutchinson likelihood estimate; affects all measured potentials.
assumptions (6)
  • domain assumption q_t_epsilon approximates -log p_0
    Section 2.1 sets t_epsilon=1e-3 for numerical stability and uses it as the prior for all experiments.
  • domain assumption The score network provides the exact gradient and Hessian of a C2 potential q
    Section 2.1 identifies -s_theta with nabla q; Hessians are obtained by automatic differentiation of the network.
  • domain assumption Kernel map theta -> h_theta is C2 and contains the Dirac delta
    Section 3 assumes twice differentiability; Theorems 1, 4, and 8 rely on the parameterized family, and h_0=delta for the tested families.
  • ad hoc to paper Condition (9): q(h*x) <= q(x) for all images and kernels
    This is the sufficient condition for Theorem 1; empirically checked only on a subset and contradicted by Remark 2 at strict local minima.
  • ad hoc to paper Identifiability conditions (10)-(12)
    Theorem 4 and 8 are conditional on these; the paper interprets them but does not verify them for the actual pretrained diffusion models.
  • ad hoc to paper Natural images lie near second-order critical points of q
    Used to connect the abstract theorem to image recovery; supported by gradient descent experiments but not by formal proof.

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Pith. "Pith review of A Morse-Bott Framework for Blind Inverse Problems: Local Recovery Guarantees and the Failure of the MAP." pith.science (2026). https://pith.science/paper/FOKM6HV4

@misc{pith2026250802923,
  author       = {Pith},
  title        = {Pith review of: A Morse-Bott Framework for Blind Inverse Problems: Local Recovery Guarantees and the Failure of the MAP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOKM6HV4}},
  note         = {Machine review of arXiv:2508.02923}
}
read the original 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 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.

Figures

Figures reproduced from arXiv: 2508.02923 by the authors.

Figure 1
Figure 1. Critical points obtained by a gradient descent starting from real images in FFHQ-256 using the FFHQ-256 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the convergence of a gradient descent on the potential [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Distance and difference of potential between natural images and the corresponding critical points. The dataset [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Spectra of the Hessian of the potential ∇2 q(¯xi) for 20 different critical points 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. 2.3 Blurry images are more likely A key property that …
Figure 5
Figure 5. Figure 5: Different 1D blur families used in the forthcomig experiments. The parameter [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Variation of the potential q with Gaussian blur on the FFHQ-256 model. A smaller potential means a more likely image. Observe that the potential is decreasing strictly with the blur level θ. −2.0 −1.5 −1.0 ×106 Gaussian motion Airy defocus −8 −6 −4 ×104 Dirac 1.0 2.0 3…
Figure 7
Figure 7. Figure 7: The evolution of the potential q(hθ ⋆ x) on various models and 100 different images. From top-left to bottom-right: FFHQ-256, ImageNet-64, FFHQ-64 and AFHQ-64. The initial images are taken from the same training dataset as the pre-trained model. The potential is consis…
Figure 8
Figure 8. Figure 8: variation of the potential q with the blur level, for a model trained on a binarized MNIST dataset. The potential increases, suggesting that the property “more blurry=more likely” is a specificity of natural images and not a bias of the neural network architecture. cas…
Figure 9
Figure 9. Figure 9: The function ℓ opt y¯ in (8) with noise-free measurements, using an FFHQ-256 model. Left: x¯ is from the dataset. Right: x¯ is a critical point of the prior. Observe that for critical points, the posterior has a local minimum around the true kernel ¯θ, validating theor…
Figure 10
Figure 10. Figure 10: The function ℓ opt y (θ) in (8) with noisy measurements for different noise levels. Left: x¯ is from the dataset. Right: x¯ is a critical point of the prior. Here, we used the FFHQ-256 model. This experiment supports theorem 5. Algorithm 1 Blind deconvolution with alt…
Figure 11
Figure 11. Figure 11: Example of joint posterior minimization for a diffraction-limited blur kernel. The algorithm 1 is initialized [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Illustration of the algorithm 1 and the influence of the kernel initilization h( ˆθ0). A “large” initialization (uniform kernel, top two rows) of the kernel results in a satisfactory recovery of the image and the kernel. A “small” initialization (a Gaussian kernel wit…
Figure 13
Figure 13. Figure 13: Results on the Köhler dataset [Köhler et al., 2012]. Top: real blurry images and true kernels. Middle: [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: More examples on the influence of initialization for blind deconvolution, with diffraction-limited blur kernels [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: More numerical results with a simplex constraint using different pairs [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]

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Forward citations

Cited by 1 Pith paper

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  1. On the Sample Complexity of Learning for Blind Inverse Problems

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