{"ID":5438796,"CreatedAt":"2026-07-01T01:17:58.482524686Z","UpdatedAt":"2026-07-03T10:35:20.036867845Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31465","arxiv_id":"2606.31465","title":"Functional Principal Component Analysis for Manifold-Indexed Data","abstract":"Functional principal component analysis (FPCA) is a central tool for dimension reduction and covariance analysis in functional data analysis. We study FPCA for discretely observed scalar-valued functional data indexed by a compact d-dimensional Riemannian manifold M; that is, each subject is modeled as a random function from M to R. This setting is distinct from manifold-valued functional data, where the function values themselves lie on a manifold. We develop intrinsic kernel estimators for the mean and covariance functions using geodesic distances and a Riemannian volume-density correction. The proposed framework accommodates general subject-specific sampling frequencies and includes both equal-weight-per-observation and equal-weight-per-subject schemes. The uniform stochastic analysis uses VC-type empirical-process conditions for intrinsic kernel classes, together with clustered empirical-process compatibility conditions, allowing non-Lipschitz kernels under the stated assumptions. We establish uniform convergence rates for the mean and covariance estimators, Hilbert-Schmidt and operator-norm error bounds for the estimated covariance operator, and convergence rates for eigenvalues and eigenfunctions via spectral perturbation. The rates show that the sparse-to-dense transition is governed by the intrinsic dimension of the indexing manifold, reducing to the classical one-dimensional boundary when d=1. Simulations on S^1 and S^2 and a SONICOM head-related transfer function analysis illustrate the method and show modest but consistent improvements over a coordinate-based baseline when intrinsic geometry is ignored.","short_abstract":"Functional principal component analysis (FPCA) is a central tool for dimension reduction and covariance analysis in functional data analysis. We study FPCA for discretely observed scalar-valued functional data indexed by a compact d-dimensional Riemannian manifold M; that is, each subject is modeled as a random functio...","url_abs":"https://arxiv.org/abs/2606.31465","url_pdf":"https://arxiv.org/pdf/2606.31465v1","authors":"[\"Chang Jun Im\",\"Jeong Min Jeon\"]","published":"2026-06-30T10:44:16Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"math.ST\"]","methods":"[]","has_code":false}