Approximating the Derivative of Manifold-valued Functions

TL;DR

This paper develops explicit error bounds for approximating derivatives of manifold-valued functions by embedding the manifold, applying vector approximation, and projecting back, highlighting the role of the manifold's reach.

Contribution

It provides the first pre-asymptotic error bounds for derivatives of manifold-valued functions with explicit constants depending on the manifold's reach.

Findings

Derived explicit constants for error bounds based on manifold reach

Showed approximation error for derivatives differs from that of functions

Provided theoretical guarantees for derivative approximation accuracy

Abstract

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold.