Orbit recovery for spherical functions

TL;DR

This paper proves that degree three invariants (bispectrum) are sufficient to recover generic orbits of functions on spheres under rotations, with explicit bounds on sampling requirements, and demonstrates an efficient algorithm with practical applications in structural biology.

Contribution

It establishes that the bispectrum invariants suffice for orbit recovery of functions on spheres, providing explicit sampling bounds and an efficient recovery algorithm.

Findings

Degree three invariants suffice for generic orbit recovery.

Explicit bounds on radial samples needed for recovery.

Algorithm successfully applied to protein structure data.

Abstract

Orbit recovery is a central problem in both mathematics and applied sciences, with important applications to structural biology. This paper focuses on recovering generic orbits of functions on ${\mathbb R}^{n}$ and the sphere $S^{n-1}$ under the rotation action of $SO(n)$. Specifically, we demonstrate that invariants of degree three (called the bispectrum) suffice to recover generic orbits of functions in finite-dimensional approximations of $L^2({\mathbb R}^n)$ obtained by band-limiting the spherical component and discretizing the radial direction. In particular, our main result explicitly bounds the number of samples in the radial direction required for recovery from the degree three invariants. From an application perspective, the most important case is $SO(3)$, which arises in many scientific fields, and in particular, plays a central role in leading structural biology applications such as cryo-electron tomography and cryo-electron microscopy. Our result for $SO(3)$ states that considering three spherical shells (i.e., samples in the radial direction) is sufficient to recover generic orbits, which verifies an implicit conjecture made in a paper of Bandeira et al. Our proof technique provides an explicit, computationally efficient algorithm to recover the signal by successively solving systems of linear equations. We implemented this algorithm and demonstrated its effectiveness on two protein structures.