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

TL;DR

Problem: 3D volumes observed under unknown rotations make standard PCA ill-suited due to rotational variability. Approach: develop an $SO(3)$-invariant PCA by expanding volumes in ball harmonics, producing a block-diagonal covariance $\mathcal{C}=\bigoplus_{\ell=0}^L (I_{2\ell+1} \otimes C_\ell)$ and per-$\ell$ eigenproblems that yield an eigenvolume basis. Contributions: reduce computational complexity from the naive $O(N^9)$ to $O(N^4)$ under Nyquist sampling, enable accurate 3D reconstructions and generation of synthetic structures, and extend naturally to $\mathsf{O}(3)$-invariance. Significance: provides scalable, rotation-robust dimensionality reduction for large-volume molecular data and supports integration into cryo-EM workflows, including subspace EM and moment-method analyses.

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.

Figures (4)

• Figure 1: Eigenvolumes for $N = 64$. The notation is retained from Section \ref{['sec:pca']}: $j$ stands for the rank of the eigenvolume $\psi_j$ in the ordering of ${u_{\ell_j s_j m_j}}$, while $\mathbf{j}$ stands for the rank of the eigenvalue $\lambda_{\ell_\mathbf{j} s_\mathbf{j}}$.
• Figure 2: Volume reconstructions with $N=128$. In each row: (a) reference volume; (b) ball harmonics expansion (bandlimit $L=20$); (c--f) reconstructions using the top $d$ eigenvectors with $d = 10,20,100,200$, respectively. Top row: volume with PDB index 1avo; bottom row with 1dgb.
• Figure 3: Comparison of approximations with $\mathsf{SO}(3)$-invariant PCA and the ball harmonics basis (BH) for the sample volume 1fzf. The plots show $w^V_\phi (k)$ for $k = 1,...,d$ under three choices of the basis $V$: PCA (solid blue), sorted BH (dashed orange), BH sorted by $u_{ls}$ (dotted green).
• Figure 4: Examples of synthesis of random proteins using $d = 200$ eigenvectors with resolution $N = 64$ and bandlimit $L = 20$. The PDB indexes of the real proteins are 1dgb, 1cb5, 1fzf, respectively.