The Ensemble Inverse Problem: Applications and Methods

TL;DR

This work defines the Ensemble Inverse Problem (EIP) as inverting for an ensemble distributed as the forward-pushed prior under a common forward model, with applications in unfolding, inverse imaging, and seismic FWI. It introduces ensemble inverse generative models (EI-DDPM and EI-FM) that perform non-iterative posterior sampling conditioned on both a single measurement and a permutation-invariant encoding of an observation set, enabling generalization to unseen priors. The method leverages training over diverse truth-observation pairs to implicitly encode the likelihood and uses a learned ensemble representation $\\phi_w(\\mathcal{Y})$ to extract information from the observation set. Across synthetic tasks and real data (HEP unfolding, FWI, image inversion), EI-DDPM / EI-FM demonstrate superior posterior inference and generalization to priors not seen during training, often outperforming established baselines. The work offers a practical, scalable approach to complex inverse problems where explicit forward-model access is costly or unavailable at inference time, with potential for theoretical guarantees and optimized ensemble-information encoders in future research.

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.

Figures (11)

• Figure 1: Consider a forward process in Fig. \ref{['fig:sub1']}, Fig. \ref{['fig:sub2']} shows EIP-II's solution, with its integration corresponding to EIP-I's solution. Fig. \ref{['fig:sub3']} shows an incorrect posterior; however, the integration of this incorrect posterior can lead to the correct prior.
• Figure 2: Visualization of $40000$ samples in the prior ($\gamma = 0.9$) and recovered distributions via various methods.
• Figure 3: Average sample-wise SWD($\downarrow$) between the truth and the recovery vs. $\gamma$, evaluated over $40000$ samples. Grey areas denote the priors contained in the training data.
• Figure 4: Unfolding results of jet kinematics from a $t\bar{t}$ process (modeled with the CT14lo PDF and Vincia parton showers) from the data-driven detector smearing using EI-FM.
• Figure 5: Upper: the recovered images via different methods, the truth ($t=0.5$), and the blurred images. Lower: the transformation process from digit "$9$" to "$6$".