# Intrinsic Riemannian Functional Data Analysis

## TL;DR

This paper introduces a new geometric framework for analyzing Riemannian functional data, enabling principal component analysis and linear regression directly on manifolds using intrinsic methods.

## Contribution

It develops tensor Hilbert spaces along curves on manifolds and derives Karhunen-Loeve expansion for Riemannian processes, advancing intrinsic geometric analysis tools.

## Key findings

- Framework applies to Euclidean and non-Euclidean manifolds.
- Develops intrinsic PCA and regression methods with asymptotic analysis.
- Demonstrates effectiveness through simulations and real data examples.

## Abstract

In this work we develop a novel and foundational framework for analyzing general Riemannian functional data, in particular a new development of tensor Hilbert spaces along curves on a manifold. Such spaces enable us to derive Karhunen-Loeve expansion for Riemannian random processes. This framework also features an approach to compare objects from different tensor Hilbert spaces, which paves the way for asymptotic analysis in Riemannian functional data analysis. Built upon intrinsic geometric concepts such as vector field, Levi-Civita connection and parallel transport on Riemannian manifolds, the developed framework applies to not only Euclidean submanifolds but also manifolds without a natural ambient space. As applications of this framework, we develop intrinsic Riemannian functional principal component analysis (iRFPCA) and intrinsic Riemannian functional linear regression (iRFLR) that are distinct from their traditional and ambient counterparts. We also provide estimation procedures for iRFPCA and iRFLR, and investigate their asymptotic properties within the intrinsic geometry. Numerical performance is illustrated by simulated and real examples.