Functional Principal Component Analysis for Manifold-Indexed Data

Abstract

Functional principal component analysis has traditionally focused on processes indexed by Euclidean domains, although functional observations increasingly arise on curved domains. We develop FPCA for real-valued processes observed with noise at discrete locations on a general Riemannian manifold. We estimate the mean and covariance functions using intrinsic kernel smoothing based on geodesic distance and a normal-coordinate volume correction, with a general weighting scheme accommodating unequal numbers of observations across subjects. We establish L² and uniform convergence rates for these estimators and derive corresponding error bounds for the estimated eigenvalues and eigenfunctions. These results quantify how manifold dimension, sampling frequency and weighting affect estimation accuracy and characterize the phase transition between sparse and dense sampling regimes. Simulation studies on the circle and sphere, together with real-data applications, illustrate the finite-sample performance of the proposed methodology.

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BibTeX RIS

BibTeX

@misc{im2026functional,
  title = {{Functional Principal Component Analysis for Manifold-Indexed Data}},
  author = {Chang Jun Im and Jeong Min Jeon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.31465},
  eprint = {2606.31465},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.31465},
}

RIS

TY  - UNPB
TI  - Functional Principal Component Analysis for Manifold-Indexed Data
AU  - Chang Jun Im
AU  - Jeong Min Jeon
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2606.31465
UR  - https://arxiv.org/abs/2606.31465
AB  - Functional principal component analysis has traditionally focused on processes indexed by Euclidean domains, although functional observations increasingly arise on curved domains. We develop FPCA for real-valued processes observed with noise at discrete locations on a general Riemannian manifold. We estimate the mean and covariance functions using intrinsic kernel smoothing based on geodesic distance and a normal-coordinate volume correction, with a general weighting scheme accommodating unequal numbers of observations across subjects. We establish L^2 and uniform convergence rates for these estimators and derive corresponding error bounds for the estimated eigenvalues and eigenfunctions. These results quantify how manifold dimension, sampling frequency and weighting affect estimation accuracy and characterize the phase transition between sparse and dense sampling regimes. Simulation studies on the circle and sphere, together with real-data applications, illustrate the finite-sample performance of the proposed methodology.
ER  -

Abstract

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