SO(3)-invariant PCA with application to molecular data
This addresses the challenge of handling arbitrarily oriented 3D data in structural biology, offering a more efficient solution for large-scale reconstruction problems.
The paper tackled the problem of applying PCA to 3D volumetric data with unknown orientations, common in structural biology, by developing an SO(3)-invariant PCA framework that avoids explicit data augmentation and reduces computational complexity, validated on real-world molecular datasets.
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.