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

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, $x-G(x)=\nabla {\Psi }_t(x)$, 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, ${\rho }_g^{\star }=1-e^{-h_t}<0.326$ throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on ${\Psi }_t$, unstable wherever ${\lambda }_{\max }({\nabla }^2{\Psi }_t)>2$, so oscillation certifies nothing; damping below $2/{\lambda }_{\max }$ cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth $w=r{\kappa }_{\max }$ the oscillation shell sits at $w=\frac{1}{2}$, schedule-free, and the divergence shell at $w=1/(1+{\rho }_g^{\star })$, with a measured finite-noise correction in $\Vert \mathrm{II}{\Vert }^2$. 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 ${\sigma }_t{\lambda }_{\max }(\mathrm{sym}J)\le 1$, from $\mathrm{Cov}(x_0\mid x_t)⪰0$ alone, holds for the exact score to $3\times 10^{-7}$ but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by $1.26$-$4.66\times$.

Revised follow-up of "Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step" (ICML 2026 Workshop SPIGM)