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.
Principal component analysis (PCA) is one of the most widely used unsupervised dimension reduction techniques. We study PCA for data from multiple related domains. Since principal components generally differ across domains, one way to obtain a shared low-rank embedding is to perform PCA on the pooled data. However, this approach can focus on spurious directions that exhibit high variation in only a few domains. To find a robust embedding that still explains most variance in unseen but similar domains, we propose instead to focus on shared directions of variation. To this end, we introduce Anchor PCA which trades off overall explained variance with agreement between the shared and domain-specific low-rank embeddings. Anchor PCA amounts to PCA on a modified target matrix and thus can be solved efficiently. Moreover, we show that Anchor PCA recovers a maximal invariant subspace and admits a minimax reconstruction interpretation under bounded domain-specific covariance inflations. On simulated and real-world gas sensor data with temporal drift, we demonstrate, respectively, that Anchor PCA recovers the maximally invariant subspace and yields embeddings that explain more variance on unseen domains than the pooling baseline and a worst-case alternative. Taken together, these findings establish Anchor PCA as a promising approach to robust unsupervised dimension reduction from multi-domain data.
Benedikt Seiter, Anya Fries, Julius von Kügelgen +1
Principal Component Analysis (PCA) preserves variance, not the information needed to detect rare catastrophic events. This paper proves the existence of a {\it Risk Shadow}: PCA can retain over 99.9999 percent of total variance while completely erasing all signal about rare, high-impact failures. When this happens, even the best possible classifier operating on the PCA representation reduces to a constant predictor. The root cause is a fundamental mismatch between variance maximization and tail risk awareness. To break the shadow, we introduce Expectile PCA (ExPCA) and Tail-Preserving PCA (TP-PCA), two methods that reweight the data covariance toward high-impact events. We prove theoretically that ExPCA strictly outperforms PCA in retaining rare-event information, and we validate our claims on synthetic data and a real-world credit card fraud detection benchmark. Our results call for a fundamental rethinking of variance-based dimensionality reduction in high-stakes decisions.
This methodological study analyzes the effects of collinearity, effective dimensionality, and cluster stability in a 2023 study of US airline profit cycles from 1995 to 2020 by Renold et al., which uses k-means clustering, principal component analysis, and system dynamic modelling.We replicate their clustering experiment in three spaces -- the original 7-dim. raw-variable space, a 3-dim. PC score space, and a 4-dim. PC score space using their dataset. We show that the six-cluster taxonomy is geometrically robust: k-means in 3-PC space produces bit-for-bit identical cluster assignments relative to 7D raw space. As a nonlinearity check we apply kernel PCA under six kernels spanning three families plus a linear baseline. The kernels confirm an intrinsically linear manifold with no detectable curvature. The silhouette criterion reveals that the dataset structurally supports only three clusters, not six. Collinearity in the raw 7D space suppresses the silhouette signal. A kernel ridge regression check confirms no nonlinear accuracy gain over linear ridge once the COVID19 year is excluded. Together, these results argue for clustering on PC scores rather than raw variables in collinearity-prone panel data.