$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

The adjoint differential of the exponential map base point

This function evaluates for $F(x)=\exp_x\xi$ with fixed $\xi\in T_x\M$ the adjoint differenital $D^*_xF(x)[\eta].$

It is calculated a corresponding adjoint Jacobi field AdjJacoiField. Since the weights are the same as for the differential DxExp, we refer to that page for details.

Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
%   nu = AdjDxExp(x,xi,eta) - Adjoint of the Derivative of Exp with
%   respect to the basis point
%    INPUT
%      x   : base point of the exponential
%      xi  : direction of the exponential
%     eta  : (in TExp(x,xi)M) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     nu   : ( in TxM ) - the adjoint of DxExp with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04