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Peer-reviewedCC BY-NC-NDJournal of Big Data

Principal component analysis in Φ α-linear spaces

  • Cassandra Mussard
  • Walter Briec
  • Charles Condevaux
  • Stéphane Mussard
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Abstract

This paper introduces a principal component analysis ( Φ α -PCA) in a topological structure called Φ α -linear spaces. The Φ α -PCA consists in an isomorphic deformation of the usual PCA through a hyperparameter α . It is shown that the statistics employed in standard PCA (cosines and correlations) exist in Φ α -linear spaces, these are U -statistics. As the regular PCA is a special case of Φ α -PCA when α = 1 , simulations and applications on images are provided to outline the relevance of the Φ α -PCA in various settings and specifically in the presence of noise and outliers.

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Cite

BibTeXRIS
BibTeX
@article{mussard2026principal,
  title = {{Principal component analysis in $$\Phi _\alpha$$-linear spaces}},
  author = {Cassandra Mussard and Walter Briec and Charles Condevaux and Stéphane Mussard},
  journal = {Journal of Big Data},
  year = {2026},
  month = may,
  doi = {10.1186/s40537-026-01473-x},
  url = {https://hal.science/hal-05780128v1},
  note = {CC BY-NC-ND},
}
RIS
TY  - JOUR
TI  - Principal component analysis in Φ _α-linear spaces
AU  - Cassandra Mussard
AU  - Walter Briec
AU  - Charles Condevaux
AU  - Stéphane Mussard
PY  - 2026
DA  - 2026/05/24
JO  - Journal of Big Data
DO  - 10.1186/s40537-026-01473-x
UR  - https://hal.science/hal-05780128v1
AB  - This paper introduces a principal component analysis ( Φ _α -PCA) in a topological structure called Φ _α -linear spaces. The Φ _α -PCA consists in an isomorphic deformation of the usual PCA through a hyperparameter α . It is shown that the statistics employed in standard PCA (cosines and correlations) exist in Φ _α -linear spaces, these are U -statistics. As the regular PCA is a special case of Φ _α -PCA when α = 1 , simulations and applications on images are provided to outline the relevance of the Φ _α -PCA in various settings and specifically in the presence of noise and outliers.
SN  - 2196-1115
ER  -