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

arXiv: https://arxiv.org/abs/2608.23094 · HTML: https://mlsanity.com/paper/2608.23094/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 25 of its 70 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 25 of 25)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [ReNoise: Real Image Inversion Through Iterative Noising](https://mlsanity.com/paper/2403.14602/) (2024)
- [Effective Real Image Editing with Accelerated Iterative Diffusion Inversion](https://mlsanity.com/paper/2309.04907/) (2023)
- [Stabilizing Equilibrium Models by Jacobian Regularization](https://mlsanity.com/paper/2106.14342/) (2021)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [On Exact Inversion of DPM-Solvers](https://mlsanity.com/paper/2311.18387/) (2023)
- [Deep Equilibrium Approaches to Diffusion Models](https://mlsanity.com/paper/2210.12867/) (2022)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [Lightning-Fast Image Inversion and Editing for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2312.12540/) (2023)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models](https://mlsanity.com/paper/2402.03845/) (2024)
- [Mathematical analysis of singularities in the diffusion model under the submanifold assumption](https://mlsanity.com/paper/2301.07882/) (2023)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Score-based generative models learn manifold-like structures with constrained mixing](https://mlsanity.com/paper/2311.09952/) (2023)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)
- [A Connection Between Score Matching and Local Intrinsic Dimension](https://mlsanity.com/paper/2510.12975/) (2025)

## Abstract (arXiv)

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)=nablaPsit(x), strongly convex at the Bayes limit with modulus exactly e-ht for the step's log-SNR gap ht - 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-ht), a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, ρgstar=1-e-ht<0.326 throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on Psit, unstable wherever λmax(nabla2Psit)>2, so oscillation certifies nothing; damping below 2/λmax cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth w=rkappamax the oscillation shell sits at w=tfrac12, schedule-free, and the divergence shell at w=1/(1+ρgstar), 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 3.6-5.6×, and the trained Hessian-Lipschitz constant is 2-12% of the curvature the law reads, 0 on a ReLU net. Finally the unconditional ceiling σtλmax(sym,J)≤1, from Cov(x0mid xt)succeq0 alone, holds for the exact score to 3×10-7 but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by 1.26-4.66×.