Directional Nonlinear Principal Components: a measure transportation approach
Abstract
One of the earliest techniques in multivariate analysis, Principal Component Analysis (PCA) remains among the most popular ones in the area. PCA, however, is a linear and bidirectional technique (principal directions ±u are defined up to a sign), which is fine under Gaussian or elliptical assumptions but constitutes a major limitation outside that restrictive context. This has prompted the development of numerous nonlinear approaches. We show here how recent results in measure transportation allow for a nonlinear and strictly directional (disentangling u and −u) concept preserving the intuitive interpretation of the classical linear concept by maximizing expected squared arc lengths along data-driven principal curves rather than principal directions.
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BibTeX
@misc{hallin2026directional,
title = {{Directional Nonlinear Principal Components: a measure transportation approach}},
author = {Marc Hallin and Klaus Nordhausen and Una Radojicic},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02158},
eprint = {2611.02158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02158},
}
RIS
TY - UNPB
TI - Directional Nonlinear Principal Components: a measure transportation approach
AU - Marc Hallin
AU - Klaus Nordhausen
AU - Una Radojicic
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02158
UR - https://arxiv.org/abs/2611.02158
AB - One of the earliest techniques in multivariate analysis, Principal Component Analysis (PCA) remains among the most popular ones in the area. PCA, however, is a linear and bidirectional technique (principal directions ±u are defined up to a sign), which is fine under Gaussian or elliptical assumptions but constitutes a major limitation outside that restrictive context. This has prompted the development of numerous nonlinear approaches. We show here how recent results in measure transportation allow for a nonlinear and strictly directional (disentangling u and −u) concept preserving the intuitive interpretation of the classical linear concept by maximizing expected squared arc lengths along data-driven principal curves rather than principal directions.
ER -