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

2025-10-21

AI Summary

Conventional principal component analysis (PCA) on 3D molecular data suffers from high computational cost and reliance on explicit rotational data augmentation to achieve SO(3) invariance. Method: We propose the first SO(3)-invariant PCA framework, which avoids generating explicit rotated copies by leveraging group representation theory to construct an implicit covariance estimation mechanism—embedding rotational equivalence directly into the algebraic structure. This reduces computational complexity to the square root of the number of covariance matrix entries. An efficient numerical algorithm enables rotation-invariant dimensionality reduction and denoising for 3D volumetric data. Contribution/Results: Evaluated on real cryo-electron microscopy (cryo-EM) molecular datasets, our method significantly reduces computational overhead while enabling scalable, high-resolution 3D structural reconstruction. It establishes a novel, symmetry-aware, and computationally tractable paradigm for dimensionality reduction in structural biology.

Technology Category

• Computer Vision: 3D Computer Vision
• Machine Learning: Dimensionality Reduction/Feature Selection
• Data Mining & Knowledge Management: Data Compression

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

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.

Problem

• Developing SO(3)-invariant PCA for 3D volumetric datasets
• Addressing arbitrary orientation challenges in molecular data
• Enabling efficient rotation-invariant analysis without data augmentation

Innovation

• SO(3)-invariant PCA framework for 3D volumetric datasets
• Implicit rotation handling without explicit data augmentation
• Algebraic structure reduces covariance computation complexity