PCA可能误导对非线性数据结构的判断，真实结构是环状而非聚类。
Beyond Explained Variance: A Cautionary Tale of PCA
- 用t-SNE和持久同调揭示数据为环形结构
- 真实维度为1，与PCA显示的二维聚类矛盾
- 提出概率几何模型验证距离分布一致性
我们通过早期哺乳动物食虫类Kuehneotherium的化石牙齿数据集，揭示主成分分析（PCA）在可视化高维非线性流形数据时的局限性。尽管Jolliffe和Cadima（2016）的PCA散点图显示PC2 < 0区域存在聚类，但基于t-SNE和持久同调（PH）的分析发现，数据呈现环状结构，无明显聚类，且内在维度为1。我们进一步提出一个生成式概率-几何模型：数据均匀采样自单位圆。在此模型下，成对余弦距离服从反正弦分布，与观测到的U型分布定性一致，独立支持t-SNE与持久同调的分析结果。
原文摘要 · Abstract (English)
We address shortcomings of principal component analysis (PCA) for visualizing high-dimensional data lying on a nonlinear low-dimensional manifold via two-dimensional scatterplots, focusing on a fossil teeth dataset from the early mammalian insectivore Kuehneotherium. While the PCA scatterplot reported by Jolliffe and Cadima (Philosophical Transactions of the Royal Society A, 2016) shows clustering in the region where PC2 < 0, our analysis based on t-SNE and persistent homology (PH) reveals a ring-like structure with no evident clustering and intrinsic dimensionality equal to one. We further propose a generative probabilistic-geometric model in which the data are sampled uniformly from a unit circle. Under this model, pairwise cosine distances follow an arcsine distribution, in qualitative agreement with the observed U-shaped distribution, thereby independently supporting the analysis based on t-SNE and persistent homology.
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