# Inverse Problems Conditioned on Observation Ensembles: Applications and Methods

arXiv: https://arxiv.org/abs/2601.22029 · HTML: https://mlsanity.com/paper/2601.22029/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 32 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Sampling-Based Accuracy Testing of Posterior Estimators for General Inference](https://mlsanity.com/paper/2302.03026/) (2023)
- [OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion](https://mlsanity.com/paper/2111.02926/) (2021)
- [OmniFold: A Method to Simultaneously Unfold All Observables](https://mlsanity.com/paper/1911.09107/) (2019)
- [Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization](https://mlsanity.com/paper/1809.10262/) (2018)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [How to GAN away Detector Effects](https://mlsanity.com/paper/1912.00477/) (2019)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [An unfolding method based on conditional Invertible Neural Networks (cINN) using iterative training](https://mlsanity.com/paper/2212.08674/) (2022)
- [Improving Generative Model-based Unfolding with Schr"odinger Bridges](https://mlsanity.com/paper/2308.12351/) (2023)
- [A Restoration Network as an Implicit Prior](https://mlsanity.com/paper/2310.01391/) (2023)
- [Solving Empirical Bayes via Transformers](https://mlsanity.com/paper/2502.09844/) (2025)
- [Sourcerer: Sample-based Maximum Entropy Source Distribution Estimation](https://mlsanity.com/paper/2402.07808/) (2024)
- [Generative Unfolding with Distribution Mapping](https://mlsanity.com/paper/2411.02495/) (2024)
- [Provably Efficient Posterior Sampling for Sparse Linear Regression via Measure Decomposition](https://mlsanity.com/paper/2406.19550/) (2024)

## Abstract (arXiv)

We introduce a new multivariate statistical problem that we refer to as the Ensemble-conditioned Inverse Problem (EIP). The aim of EIP is to invert for an ensemble that is distributed according to the pushforward of a prior under a forward process. In high energy physics (HEP), this is related to a widely known problem called unfolding, which aims to reconstruct the true physics distribution from observations that are distorted by detector effects. The EIP also arises in full waveform inversion (FWI) and inverse imaging with unknown priors. We propose non-iterative inference-time methods that construct posterior samplers based on a new class of conditional generative models, which we call ensemble inverse generative models. For the posterior modeling, these models additionally use the ensemble information contained in the observation set on top of single observations. Unlike existing methods, our proposed methods avoid explicit and iterative use of the forward model at inference time via training across several sets of truth-observation pairs that are consistent with the same forward model, but originate from a wide range of priors. We empirically demonstrate that this training procedure can implicitly encode the likelihood model, enabling direct posterior inference for unseen priors to some degree. We benchmark the proposed method on several synthetic and real datasets in inverse imaging, HEP, and FWI. Our code is available at https://github.com/ZhengyanHuan/EIP.