# Measurement Geometry and Design for Trustworthy Generative Inverse Problems

arXiv: https://arxiv.org/abs/2606.02309 · HTML: https://mlsanity.com/paper/2606.02309/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 38 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Solving 3D Inverse Problems using Pre-trained 2D Diffusion Models](https://mlsanity.com/paper/2211.10655/) (2022)
- [Robust Compressed Sensing using Generative Models](https://mlsanity.com/paper/2006.09461/) (2020)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [Learning-Based Compressive MRI](https://mlsanity.com/paper/1805.01266/) (2018)
- [Deep Learning-Based Solvability of Underdetermined Inverse Problems in Medical Imaging](https://mlsanity.com/paper/2001.01432/) (2020)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Tackling Hallucination from Conditional Models for Medical Image Reconstruction with DynamicDPS](https://mlsanity.com/paper/2503.01075/) (2025)
- [Constant-Expansion Suffices for Compressed Sensing with Generative Priors](https://mlsanity.com/paper/2006.04237/) (2020)
- [Fast Variable Density Poisson-Disc Sample Generation with Directional Variation](https://mlsanity.com/paper/2004.06789/) (2020)
- [RODS: Robust Optimization Inspired Diffusion Sampling for Detecting and Reducing Hallucination in Generative Models](https://mlsanity.com/paper/2507.12201/) (2025)
- [Compressed Sensing: From Research to Clinical Practice with Data-Driven Learning](https://mlsanity.com/paper/1903.07824/) (2019)

## Abstract (arXiv)

Generative models are increasingly used as priors for inverse problems, but their ability to produce realistic images creates a basic trust problem: a plausible reconstruction may be supported by the measurements, or it may be filled in by the prior along unobserved directions. This distinction is especially important in medical imaging, where acquisition operators are designed under scan-time, dose, and calibration constraints. We study generative inverse problems from a measurement-geometry perspective. The central question is whether a fixed measurement operator can distinguish nearby images that are plausible under the generative prior, and whether this relationship can guide better measurements. We introduce a local measurement-manifold compatibility measure that quantifies how well the operator observes prior-relevant tangent directions. Under local regularity assumptions, we prove that this quantity controls the stable part of the reconstruction error, while the generative prior controls off-manifold drift. This worst-direction certificate motivates practical fixed and sequential acquisition rules based on overall local volume preservation, including a posterior-cloud design that adapts measurements at test time without training a sampling policy. Across row-sampling, tomographic, and MR acquisition settings, the proposed scores predict failure modes, explain measurement-induced hallucinations, and guide better sampling. In fastMRI Cartesian sampling, posterior-cloud measurement design improves over strong non-learned ACS-preserving baselines, including variable-density and Poisson-like masks.