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

The adjoint differential of the logarithmic map base point

This function evaluates for $F(x)=\log_xy$ with fixed $y\in\M$ 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 DxLog, we refer to that page for details.

Matlab Documentation

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%       with respect to the basis point x.
%   INPUT
%      x   : base point of the logarithm
%      y   : argument of the logarithm
%     eta  : (in TxM) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     nu   : ( in TxM ) - the adjoint of DxLog with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04