The Ensemble Inverse Problem: Applications and Methods
Zhengyan Huan, Camila Pazos, Martin Klassen, Vincent Croft, Pierre-Hugues Beauchemin, Shuchin Aeron
TL;DR
This paper introduces the Ensemble Inverse Problem, a new statistical challenge, and proposes non-iterative, generative-model-based methods that leverage ensemble data for improved posterior inference across various scientific fields.
Contribution
It presents a novel class of ensemble inverse generative models that encode likelihoods implicitly and generalize to unseen priors, avoiding iterative forward model evaluations during inference.
Findings
Effective on synthetic and real datasets in inverse imaging.
Improves posterior inference by leveraging ensemble information.
Generalizes well to unseen priors in diverse applications.
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…
Peer Reviews
Decision·Submitted to ICLR 2026
1. The formalization of the "Ensemble Inverse Problem" (EIP) is a significant contribution. This is essentially a new "in-context learning" or "meta-learning" framework for scientific inverse problems, which is potentially impactful. The proposed solution—conditioning on a learned, permutation-invariant embedding of the observation set $\mathcal{Y}$—is an elegant and general way to solve the problem. Instead of using hard-coded statistics (like moments), it allows the model to learn the most rel
1. Ambiguous Experimental Setups: A few points in the paper would need to be clarified, specifically about the experimental setups. - For the HEP experiment: My understanding was that the paper's premise is that the forward model and the noise model that allow building $p(y|x)$ need to be fixed. However, the data description mentions using "various parton distribution functions and parton shower models". This sounds like the underlying physics, and thus $p(y|x)$, is changing across the dif
The paper provides a clean formalization of the ensemble inverse problem, emphasizing inference across multiple priors with a shared but unknown forward operator.
- While the EIP formulation is interesting conceptually, its practical relevance to real-world inverse problems is not clear. The authors mention applications in high-energy physics and inverse imaging, but for readers unfamiliar with high-energy physics, the motivation in that domain is difficult to assess. For inverse imaging problems, it is unclear to me how the EIP problem setting arises in practice. - Despite the new terminology, Algorithms 1 and 2 follow standard conditional diffusion tra
For the most part, the paper is clearly written and easy to follow. The experiments cover a diversity of settings. The synthetic Gaussian experiment is easy to visualize and provides some good intuition, and also shows explicitly how the method can generalize beyond priors not seen during training. The particle physics experiments are helpful as the authors use problems from this field to motivate EIP. The MNIST experiment shows that the proposed method may also have applicability in other real
The methodological novelty is somewhat limited. As described in the summary, the method amounts to providing two additional inputs to the denoising network for a standard diffusion model setup. While it is intuitive that the inclusion of these two quantities can encode information about the prior to be reconstructed in a form which is usable by the denoising network, the paper also does not attempt to place this intuition on more rigorous footing, e.g. with theoretical results or even informal m
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGenerative Adversarial Networks and Image Synthesis · Gaussian Processes and Bayesian Inference · Statistical Mechanics and Entropy