The Risk Shadow of Principal Component Analysis: When 99.9999% Variance Preservation Causes Catastrophic Decision Errors

Title:The Risk Shadow of Principal Component Analysis: When 99.9999% Variance Preservation Causes Catastrophic Decision Errors

Abstract: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.

5 tables, 1 figure. all references fully checked manually

Machine Learning (cs.LG); Computer Science and Game Theory (cs.GT)

arXiv:2606.14533 [cs.LG]

(or arXiv:2606.14533v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.14533 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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