On the discretization of the object space in inverse problems with application to cryo-electron microscopy
Abstract
In many inverse problems, the aim is to recover a probability distribution on a latent (or object state) space from indirect, noisy observations. When the observations can be modelled as noisy samples from the mixing latent distribution, the recovery problem is a deconvolution problem on the space of probability measures. A common strategy is to fix a finite set of candidate points and estimate a weight for each, turning the problem into a finite-dimensional concave maximum likelihood problem on the simplex. We study the combined effect of this discretization and of the noise in the context of cryo-electron microscopy (cryo-EM), where the candidate points are biomolecular conformations and the weights describe the relative frequency of each conformation. Our results pertain to both statistical and algorithmic aspects of the estimator. We analyze the weight-recovery problem in which the candidate states and their likelihoods are known. A nearby pair of candidates forces a near-null direction. More generally, the grid and the noise level impose a uniform lower bound on the achievable Kullback--Leibler divergence between observation densities, even with infinite data. The finite-grid estimator is asymptotically normal when its population target is in the interior of the simplex; at a boundary target, its limit is a cone-projected Gaussian. Finally, the exact proximal form of Expectation--Maximization leads to a global high-noise comparison between an early iterate and a KL-penalized likelihood, without a basin assumption or a linearization of the recursion. Tests on synthetic images of the Hsp90 molecule illustrate the theoretical findings and translate them into practical guidelines for interpreting reweighted ensembles.
Related
- Notes on the discretization of TV-NORM regularized inverse potential problems
- Linear and non-linear thermal system identification based on the integral of non-integer order — Application to solve inverse heat conduction linear and non-linear problems
- Right Bregman proximal gradient with application to Poisson inverse problems
- Reduced order model for parametric Boltzmann equation and its application to inverse problems
- Efficient C1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
BibTeX
@misc{mordant2026discretization,
title = {{On the discretization of the object space in inverse problems with application to cryo-electron microscopy}},
author = {Gilles Mordant and Luke Evans and David Silva-Sánchez and Pilar Cossio and Roy Lederman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01688},
eprint = {2609.01688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01688},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the discretization of the object space in inverse problems with application to cryo-electron microscopy
AU - Gilles Mordant
AU - Luke Evans
AU - David Silva-Sánchez
AU - Pilar Cossio
AU - Roy Lederman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01688
UR - https://arxiv.org/abs/2609.01688
AB - In many inverse problems, the aim is to recover a probability distribution on a latent (or object state) space from indirect, noisy observations. When the observations can be modelled as noisy samples from the mixing latent distribution, the recovery problem is a deconvolution problem on the space of probability measures. A common strategy is to fix a finite set of candidate points and estimate a weight for each, turning the problem into a finite-dimensional concave maximum likelihood problem on the simplex. We study the combined effect of this discretization and of the noise in the context of cryo-electron microscopy (cryo-EM), where the candidate points are biomolecular conformations and the weights describe the relative frequency of each conformation. Our results pertain to both statistical and algorithmic aspects of the estimator. We analyze the weight-recovery problem in which the candidate states and their likelihoods are known. A nearby pair of candidates forces a near-null direction. More generally, the grid and the noise level impose a uniform lower bound on the achievable Kullback--Leibler divergence between observation densities, even with infinite data. The finite-grid estimator is asymptotically normal when its population target is in the interior of the simplex; at a boundary target, its limit is a cone-projected Gaussian. Finally, the exact proximal form of Expectation--Maximization leads to a global high-noise comparison between an early iterate and a KL-penalized likelihood, without a basin assumption or a linearization of the recursion. Tests on synthetic images of the Hsp90 molecule illustrate the theoretical findings and translate them into practical guidelines for interpreting reweighted ensembles.
ER -