A Scale Invariance Property of PCA
Abstract
The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's k-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's k-th largest principal component, for each k. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.
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BibTeX
@misc{landy2026scale,
title = {{A Scale Invariance Property of PCA}},
author = {Jonathan Landy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11016},
eprint = {2609.11016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11016},
}
RIS
TY - UNPB
TI - A Scale Invariance Property of PCA
AU - Jonathan Landy
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11016
UR - https://arxiv.org/abs/2609.11016
AB - The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's k-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's k-th largest principal component, for each k. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.
ER -