提出旋转不变主成分分析，高效处理分子三维数据
SO(3)-invariant PCA with application to molecular data
- 基于SO(3)群不变性，隐式建模所有旋转，无需数据增强
- 计算复杂度降至原始方法的平方根级别，大幅降低开销
- 适用于大尺度分子结构重建，尤其适合高维三维数据
主成分分析（PCA）是降维与去噪的基础方法，但在结构生物学中常见的任意取向三维数据上应用面临重大挑战。传统方法需对每一样本生成大量旋转副本，导致计算成本过高。本文提出一种高效的SO(3)-不变PCA框架，无需显式数据增强即可隐式处理所有旋转。通过利用底层代数结构，计算仅需总协方差条目数的平方根，显著降低复杂度。在真实分子数据集上验证了该方法的有效性，为大规模高维重构问题开辟新路径。
原文摘要 · Abstract (English)
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.
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