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.

翻译：我们引入了一个新的多元统计问题，称之为集成逆问题。EIP的目标是反演一个按照先验分布在前向过程推演下形成的集成分布。在高能物理中，这与一个广为人知的称为"解卷积"的问题相关，其目标是从受探测器效应扭曲的测量数据中重建物理量的真实分布（如动量与角度）。在近期应用中，EIP同样出现在全波形反演以及先验未知的逆成像问题中。我们提出了非迭代的推理时方法，该方法基于一类新的条件生成模型构建后验采样器，我们称之为集成逆生成模型。对于后验建模，这些模型除了利用单次测量信息外，还额外利用了观测数据集中蕴含的集成信息。与现有方法不同，我们提出的方法通过在多组符合相同前向模型但源自不同先验分布的真实-观测数据对上进行训练，避免了推理时显式且迭代地使用前向模型。我们证明该训练过程隐式编码了似然模型。集成信息的使用有助于后验推断，并能泛化到未见过的先验分布。我们在逆成像、HEP和FWI的多个合成与真实数据集上对所提方法进行了基准测试。代码发布于https://github.com/ZhengyanHuan/The-Ensemble-Inverse-Problem--Applications-and-Methods。