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

arXiv:2608.23094v1 Announce Type: cross 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)=nablaPsi_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}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=rkappa_{max} the oscillation shell sits at w=frac12, schedule-free, and the divergence shell at w=1/(1+rho_g^{star}), with a measured finite-noise correction in |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 3.6-5.6imes, 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_tlambda_{max}(sym,J)le1, from Cov(x_0mid x_t)succeq0 alone, holds for the exact score to 3imes10^{-7} but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by 1.26-4.66imes.

Source: arXiv cs.LG | 2026-08-25