SO(3)-invariant PCA with application to molecular data

TL;DR

This paper introduces an SO(3)-invariant PCA method for 3D molecular data that efficiently accounts for arbitrary orientations without extensive data augmentation, significantly reducing computational costs.

Contribution

The authors develop a novel SO(3)-invariant PCA framework that leverages algebraic structures to efficiently handle 3D data with unknown orientations.

Findings

Effective on real-world molecular datasets

Reduces computational complexity compared to naive methods

Enables large-scale 3D data analysis

Abstract

Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and denoising; however, its application to three-dimensional data with arbitrary orientations -- common in structural biology -- presents significant challenges. A naive approach requires augmenting the dataset with many rotated copies of each sample, incurring prohibitive computational costs. In this paper, we extend PCA to 3D volumetric datasets with unknown orientations by developing an efficient and principled framework for SO(3)-invariant PCA that implicitly accounts for all rotations without explicit data augmentation. By exploiting underlying algebraic structure, we demonstrate that the computation involves only the square root of the total number of covariance entries, resulting in a substantial reduction in complexity. We validate the method on real-world molecular datasets, demonstrating its effectiveness and opening up new possibilities for large-scale, high-dimensional reconstruction problems.