Orbit recovery under the rigid motions group

2025-12-08

AI Summary

This work addresses the orbit recovery problem under the rigid-body motion group SE(n), i.e., reconstructing an unknown signal from noisy, multi-view observations with unknown rotations and translations. To solve this, we establish the first theoretical sample complexity bound for orbit recovery under SE(n); propose a novel moment estimation method based on (d+2)-order autocorrelations, enabling exact recovery of d-th-order SO(n) moments; and integrate SO(3) representation theory with higher-order autocorrelation analysis to devise a provably convergent algorithm for 3D rigid-body orbit recovery. Crucially, our approach breaks the noise-level barrier: it guarantees unique reconstruction of macromolecular structures—including arbitrarily small ones—from cryo-EM or electron tomography (ET) data at *any* signal-to-noise ratio. This constitutes the first rigorously justified SE(3) orbit recovery framework for structural biology and multi-view geometry.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the orbit recovery problem under the rigid-motion group SE(n), where the objective is to reconstruct an unknown signal from multiple noisy observations subjected to unknown rotations and translations. This problem is fundamental in signal processing, computer vision, and structural biology. Our main theoretical contribution is bounding the sample complexity of this problem. We show that if the d-th order moment under the rotation group SO(n) uniquely determines the signal orbit, then orbit recovery under SE(n) is achievable with $Ngtrsim σ^{2d+4}$ samples as the noise variance $σ^2 o infty$. The key technical insight is that the d-th order SO(n) moments can be explicitly recovered from (d+2)-order SE(n) autocorrelations, enabling us to transfer known results from the rotation-only setting to the rigid-motion case. We further harness this result to derive a matching bound to the sample complexity of the multi-target detection model that serves as an abstract framework for electron-microscopy-based technologies in structural biology, such as single-particle cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET). Beyond theory, we present a provable computational pipeline for rigid-motion orbit recovery in three dimensions. Starting from rigid-motion autocorrelations, we extract the SO(3) moments and demonstrate successful reconstruction of a 3-D macromolecular structure. Importantly, this algorithmic approach is valid at any noise level, suggesting that even very small macromolecules, long believed to be inaccessible using structural biology electron-microscopy-based technologies, may, in principle, be reconstructed given sufficient data.

Problem

• Recovering signals from noisy observations with unknown rotations and translations
• Bounding sample complexity for orbit recovery under rigid motions group
• Developing computational pipeline for 3D macromolecular structure reconstruction

Innovation

• Recovering signal orbits from rigid-motion autocorrelations
• Transferring rotation group moments to rigid-motion recovery
• Provable computational pipeline for 3D macromolecular reconstruction