# Approximation of Riemannian Distances and Applications to Distance-Based Learning on Manifolds

## TL;DR

This paper introduces a novel method to efficiently approximate Riemannian distances on manifolds, enabling scalable distance-based learning by solving a linear number of boundary value problems, demonstrated in shape analysis applications.

## Contribution

The paper presents a new approximation technique for Riemannian distances that reduces computational cost, suitable for large datasets in machine learning and data analysis.

## Key findings

- Significantly reduces computation time for Riemannian distances
- Effective in shape analysis on landmark spaces
- Applicable to large-scale distance-based learning

## Abstract

Several important algorithms for machine learning and data analysis use pairwise distances as input. On Riemannian manifolds these distances may be prohibitively costly to compute, in particular for large datasets. To tackle this problem, we propose a distance approximation which requires only a linear number of geodesic boundary value problems to be solved. The approximation is constructed by fitting a two-dimensional model space with constant curvature to each pair of samples. We demonstrate the usefulness of our approach in the context of shape analysis on landmarks spaces.