One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion

Gordei Verbii

stat.MLAug 24, 2026

One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion

Authors: Gordei Verbii

Organizations: Independent Researcher

Abstract

One implicit DDIM inversion step is the cheapest probe of whether a pretrained diffusion model encodes local manifold geometry. It is the stationarity condition of an explicit potential, $x-G(x)=\nablaΨ_t(x)$ , strongly convex at the Bayes limit with modulus exactly $e^{-h_t}$ for the step's log-SNR gap $h_t$ $-$ for every data law, schedule and point, with no manifold, reach or unimodality hypothesis. Three consequences must be kept apart. (i) The solution is unique at the Bayes limit; a second one requires the trained score to violate the posterior-covariance bound by $1/(1-e^{-h_t})$ , a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, $ρ_g^{\star}=1-e^{-h_t}<0.326$ throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on $Ψ_t$ , unstable wherever $λ_{\max}(\nabla^2Ψ_t)>2$ , so oscillation certifies nothing; damping below $2/λ_{\max}$ cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth $w=rκ_{\max}$ the oscillation shell sits at $w=\tfrac12$ , schedule-free, and the divergence shell at $w=1/(1+ρ_g^{\star})$ , with a measured finite-noise correction in $\|\mathrm{II}\|^2$ . Exact scores reproduce both to within $0.54\%$ on three classes; no trained score we probe shows a shell $-$ a derived limitation, not a null result: the Fermi window conflicts with the model's own training support by $3.6$ - $5.6\times$ , and the trained Hessian-Lipschitz constant is $2$ - $12\%$ of the curvature the law reads, $0$ on a ReLU net. Finally the unconditional ceiling $σ_tλ_{\max}(\mathrm{sym}\,J)\le1$ , from $\mathrm{Cov}(x_0\mid x_t)\succeq0$ alone, holds for the exact score to $3\times10^{-7}$ but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by $1.26$ - $4.66\times$ .

Figures & tables

$\mathcal{M}$	$\sigma_{t}$	$\mathsf{S}_{t}/\mathsf{S}^{\star}_{t}$	$\dfrac{2\sqrt{\bar{\alpha}_{t}}\\|\varepsilon^{\star}(x^{\circ})\\|}{\sigma_{t}\\|H\\|}$	$\dfrac{\hat{d}(x^{\circ})}{\hat{d}^{\star}(x^{\circ})}$	$\nu_{\perp}/\nu_{\parallel}$
$S^{1}\!\subset\!\mathbb{R}^{2}$ , $d{=}1$	0.10	1.0000	1.003	1.0000	$+1.02/-1.06$
	0.02	1.0000	1.000	1.0000	$+1.00/-1.00$
$S^{2}\!\subset\!\mathbb{R}^{3}$ , $d{=}2$	0.10	1.0000	1.000	1.0000	$+1.05/-1.04$
	0.02	1.0000	1.000	1.0000	$+1.00/-1.00$
$S^{1}\!\subset\!\mathbb{R}^{3}$ , $d{=}1$	0.10	1.0000	1.003	1.0000	$+1.01/-1.06$
	0.02	1.0000	1.000	1.0000	$+1.00/-1.00$

Table 1: Bayes-limit calibration; no trained model enters this table. Above the rule every entry is a dimensionless ratio with Bayes value

1

(or

\pm 1

for

\nu

). Below it are the identities that make the table a calibration rather than a list, and the sweep that shows it resolves.

Figure 1: The convergence domain, and why the trained shell is unmeasurable rather than absent. (a) Dominant Picard multiplier along an inward normal ray (sphere, $\sigma_{t+1}{=}0.0585$ ): on the exact score the negative branch (dashed) takes over at $w_{\rm osc}=0.4999$ and reaches $1$ at $w_{\rm div}=0.7891$ , against the closed forms $\tfrac{1}{2}$ and $0.7856$ ; the trained field is flat and crosses neither. (b) The Fermi window (vertical) caps the reachable depth at $1.4\,\sigma_{t+k}\kappa_{\max}$ (diagonal, one point per convergence setting), $3.6\times$ short of $w_{\rm osc}$ : the hatched box where both hold is empty by arithmetic (§ 6 ).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

family	stride $k$	$\rho_{g}^{\star}$	$1/\rho_{g}^{\star}$	$w_{P}$	$\|B_{t}\|/\sigma_{t}$ (old)
DDPM-linear $T{=}1000$	1	0.325699	3.070	0.7543	0.4830
	10	0.786704	1.271	0.5597	3.6845
	20	0.874046	1.144	0.5336	6.9180
	50	0.943182	1.060	0.5146	16.3496
	100	0.970781	1.030	0.5074	31.4357
cosine $T{=}1000$ (clip $.999$ )	1	0.968377	1.033	0.5080	0.9684

Appendix

Table 2: Seven schedule families at five strides.

\rho_{g}^{\star}=1-e^{-h_{t}}

is the Bayes contraction rate at the family’s worst step,

w_{P}=1/(1+\rho_{g}^{\star})

the Picard divergence boundary of Thm. 5 , and the last column is the superseded

|B_{t}|/\sigma_{t}

normalisation, shown to make the size of the correction visible. Only a zero-terminal-SNR top step attains

\rho_{g}^{\star}=1

Figure 2: Bayes-limit calibration on analytically integrable manifolds; no trained model is involved. (a) The two dimensionless certificates $\mathsf{S}_{t}$ (solid) and $\sigma_{t}^{2}\hat{\lambda}$ (dashed). The offset of $\mathsf{S}_{t}$ from $1$ is not error: it is the closed-form curvature deficit of ( 8 ), matched to all printed digits in codimension one, with an $O(\sigma_{t}^{4})$ residual. (b) The scaling exponents ( 14 ) converge to $+1$ (normal) and $-1$ (tangential); the value $0$ often assumed for the tangential family is the dotted threshold, not a limit. (c) The same functional evaluated at the sampled point returns the ambient $D$ at every noise level (Prop. 6 ) and at the noiseless point returns the intrinsic $d$ (Prop. 7 ); error bars are $95\%$ intervals over $2\times 10^{4}$ draws.

Figure 3: The descent-based Łojasiewicz estimator is uninformative by construction (Remark 11 ). (a) For a quadratic energy the log–log locus is the line of slope $\tfrac{1}{2}$ over ten decades whether the iteration converges or diverges; the two traces coincide with the theorem line. (b) What the estimator returns on analytic energies of known exponent: $\tfrac{1}{2}$ on quadratic and Morse–Bott energies including under a divergent step, a different value under a stable step on the same energy, and $0.077$ on a quartic energy whose true exponent is $\tfrac{3}{4}$ (dotted arrow to the truth). Open diamonds: the field estimator of ( 13 ) on the same energies, which returns the true exponent in all four cases.

setting	$\mathcal{M}$	$\sigma_{t+k}\kappa_{\max}$	Fermi	$r_{\rm osc}/\sigma$	$r_{\rm div}/\sigma$	support	both
$t{=}2$ , $k{=}1$	sphere	0.0588	yes	8.51	13.37	no	no
	torus	0.2265	no	2.21	3.47	no	no
	swiss roll	0.1887	no	2.65	4.16	no	no
	trefoil	0.1167	no	4.28	6.73	no	no
	two spheres	0.1628	no	3.07	4.82	no	no
$t{=}2$ , $k{=}5$	sphere	0.1230	no	4.06	4.92	no	no

Appendix

Table 3: The diagnostic grid. Fermi:

\sigma_{t+k}\kappa_{\max}\leq 0.10

. Support: the predicted oscillation shell within

2\sigma_{t+k}

of the base point. Radii are in units of

\sigma_{t+k}

$\mathcal{M}$	score	$t$	$a/e^{h_{t}}$	$b/b^{\star}$	$L$	$L/(\rho_{g}^{\star}\kappa_{\max})$	$L_{\rm rand}/L$	$\ell=abL$
sphere	exact	2	0.9987	0.9987	0.2723	0.9950	0.122	0.00545
sphere	exact	3	0.9985	0.9985	0.2132	0.9920	0.122	0.00464
torus	exact	2	0.9895	0.9898	1.0161	0.9630	0.092	0.03950
torus	exact	3	0.9875	0.9878	0.7773	0.9381	0.092	0.03300
swiss roll	exact	2	0.9979	0.8528	1.3401	1.5591	0.048	0.08537
swiss roll	exact	3	0.9982	0.8898	0.8117	1.2025	0.057	0.05419

Appendix

Table 4: The Kantorovich triple against its Bayes references. Columns

a/e^{h_{t}}

b/b^{\star}

and

L/(\rho_{g}^{\star}\kappa_{\max})

each have Bayes value

1

;

\ell=abL

certifies at

\leq\tfrac{1}{2}

L_{\rm rand}/L

is the same constant read along an unprojected direction.

readout	Bayes value	exact score	trained	verdict
$\hat{d}(x^{\circ})$ , flattened	$256$ / $128$	$255.99999$ / $128.16$	$382$ – $383$	model
trefoil $\div$ 2-D ratio	$0.500$	$0.5049$	$0.9989$	model
rank gap, $K=4D$	index $128$	$12.31$ at $128$	$1.16$ at $373$ , abstains	model
rank gap, $K=1.08D$	index $128$	not run	$5.06$ at $128$	budget
$\nu$ split	$128$ / $256$	$128$ / $256$ (4 of 5)	$n_{+}=0$ – $46$	model
$\nu$ , swiss roll	$128$ / $256$	$174$ / $210$	—	window (free boundary)

Appendix

Table 5: Attribution. Verdicts are estimator , model , budget , window or untested . “Exact” is the analytic or oracle score under the identical harness, probe cloud, flattening and budget. The three hypothesis-free rows assume no manifold, reach or unimodality, so the windows of § 5.2 do not apply to them and none may be discounted as out of regime. Image entries are certified lower bounds. A dash is “not run”, never “returned nothing”.

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FB-GDM: Fully-Bayesian Guided Diffusion Models for High-Dimensional Linear Inverse Problems via Unsupervised Variational Inference

Diffusion models are powerful priors for linear inverse problems, but the reference guidance methods, Diffusion Posterior Sampling (DPS) and Pseudoinverse-Guided Diffusion Models (

Π

GDM), rely on scalar hyperparameters tuned per task, usually against the ground truth. We introduce FB-GDM, a fully-Bayesian guided diffusion method that removes this calibration step. Starting from the Gaussian approximation of

Π

GDM, we derive a closed-form conditional score that depends on two precision parameters (inverse variances), one associated with the denoising approximation and one with the observation likelihood, and treat them as latent variables inferred by variational inference at each reverse step. A separable factorization makes each update scale linearly with the number of pixels, so the inference stays tractable at full image resolution, at a cost comparable to one

Π

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Π

GDM at its nominal setting, even when the latter is given the true noise level, by up to 14 dB depending on the operator, and to match the ground-truth-calibrated

Π

GDM oracle within 0.1 dB. (ii) FB-GDM is robust when the forward operator, the noise level, or the image distribution changes: it stays close to a per-problem

Π

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Π

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Pretrained score-based diffusion models provide strong unconditional priors, yet enforcing measurement or physics consistency in inverse problems is often handled by heuristic guidance, intermittent projections, or task-specific conditional training, with limited guarantees of feasibility at the end of inference. We propose terminal-conditioned inversion for score-based SDE priors. Given a frozen Score-SDE prior and a task-defined terminal feasibility specification, we construct an associated backward stochastic differential equation whose adapted solution defines a principled inverse map from the terminal requirement to a prior state at a chosen noise level. Under standard regularity conditions, we establish existence and uniqueness of the adapted solution and obtain terminal consistency by construction. We further develop a practical neural BSDE solver that composes arbitrary pretrained diffusion priors with domain constraints without modifying the score-defined coefficients, producing an anchored prior state that enables neighborhood sampling for uncertainty characterization. Experiments on toy datasets validate stable terminal-conditioned inversion and distributionally consistent neighborhood sampling. As a real-world case study, we apply the framework to sparse-view CT reconstruction and achieve improved reconstruction quality over representative training-free baselines while satisfying strict measurement feasibility under the prescribed terminal specification. Project is available in: https://laplace.center/icmlbsdeI/

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May 7, 2026

Why DDIM Hallucinates More Than DDPM: A Theoretical Analysis of Reverse Dynamics

We theoretically study the hallucination phenomena in two canonical diffusion samplers: the stochastic Denoising Diffusion Probabilistic Model (DDPM) and the deterministic Denoising Diffusion Implicit Model (DDIM). We analyze the reverse ODE (DDIM) and SDE (DDPM) for a Gaussian mixture target, proving that after a critical time

τ

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Muhammad H. Ashiq, Samanyu Arora, Abhinav N. Harish +3

University of Wisconsin-Madison, Madison, WI, USA.

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