# Hermite interpolation and data processing errors on Riemannian matrix manifolds

## TL;DR

This paper develops a general Hermite interpolation framework on Riemannian manifolds and links data processing errors to manifold curvature, with applications to matrix factorizations like SVD and QR.

## Contribution

It introduces a versatile interpolation method on Riemannian manifolds and relates data errors to sectional curvature, providing new error bounds for manifold data processing.

## Key findings

- Interpolation framework applicable to matrix manifolds in data analysis and signal processing.
- Error bounds depend on the sectional curvature of the manifold.
- Numerical experiments demonstrate the method on orthogonal matrix factorizations.

## Abstract

The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable, if algorithms for the associated Riemannian exponential and logarithm mappings are available. This includes many of the matrix manifolds that arise in practical Riemannian computing application such as data analysis and signal processing, computer vision and image processing, structured matrix optimization problems and model reduction. On the other hand, we expose a natural relation between data processing errors and the sectional curvature of the manifold in question. This provides general error bounds for manifold data processing methods that rely on Riemannian normal coordinates. Numerical experiments are conducted for the compact Stiefel manifold of rectangular column-orthogonal matrices. As use cases, we compute Hermite interpolation curves for orthogonal matrix factorizations such as the singular value decomposition and the QR-decomposition.