The Ensemble Inverse Problem: Applications and Methods

2026-01-29

AI Summary

This work addresses the challenge of accurately reconstructing underlying physical quantity distributions in inverse problems—such as those in high-energy physics, full-waveform inversion, and inverse imaging—when the prior is unknown and the observation system is distorted. To overcome this, the authors propose the Ensemble Inverse Generative Model (EIGM), a non-iterative conditional generative framework trained on ensembles of ground-truth–observation pairs drawn from diverse priors. EIGM implicitly learns the forward process and leverages the collective information from an ensemble of observations to perform posterior sampling. The study formalizes, for the first time, the notion of “ensemble inverse problems,” enabling generalization to unseen priors. Extensive experiments on both synthetic and real-world datasets demonstrate that the method substantially improves reconstruction accuracy and generalization capability. The implementation is publicly released.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We introduce a new multivariate statistical problem that we refer to as the Ensemble 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 of quantities, such as momentum and angle, from measurements that are distorted by detector effects. In recent applications, 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 measurements. 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 demonstrate that this training procedure implicitly encodes the likelihood model. The use of ensemble information helps posterior inference and enables generalization to unseen priors. We benchmark the proposed method on several synthetic and real datasets in inverse imaging, HEP, and FWI. The codes are available at https://github.com/ZhengyanHuan/The-Ensemble-Inverse-Problem--Applications-and-Methods.

Problem

• Ensemble Inverse Problem
• Unfolding
• Inverse Imaging
• Full Waveform Inversion
• Prior Reconstruction

Innovation

• Ensemble Inverse Problem
• Conditional Generative Models
• Posterior Sampling
• Forward Model Avoidance
• Prior Generalization