🤖 AI Summary
This study addresses dimensionality reduction and covariance analysis for scalar-valued functional data indexed over a compact d-dimensional Riemannian manifold, where the indexing domain—not the function values—exhibits geometric structure. The work extends functional principal component analysis (FPCA) to manifold-indexed settings by introducing an intrinsic kernel estimation framework that incorporates geodesic distance and Riemannian volume density correction, accommodating heterogeneous sampling frequencies and weighted estimation strategies across individuals. Theoretically, by integrating intrinsic kernel methods, VC-type empirical process theory, and spectral perturbation analysis, the paper establishes uniform convergence rates for the mean, covariance, and eigen-objects under non-Lipschitz kernels. It further reveals that the transition from sparse to dense observation regimes is governed by the intrinsic manifold dimension, reducing to classical results when d = 1. Experiments on S¹, S², and real-world SONICOM head-related transfer function data demonstrate consistent and substantial performance gains over baseline methods that ignore geometric structure.
📝 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.