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

# List of gradients

gradient of the distance functions first argument
1
% gradDistance(M,x,y,p) gradient of f(x) = 1/p d^p(x,y)for fixed y in M.


The gradient of $f(x) = \tfrac{1}{p} d^p(x,y)$ for a fixed value $y\in\M$ and $p\geq1$, with standard $p=2$.

more
gradient of the second order TV mid point model
1
%  gradientSecondOrderLog(M,f,alpha,beta) --- compute the gradient of


For data $x\colon\Grid\to\M$ on a $m$-dimensional grid $\Grid$ this function returns the gradient $\gradM F$ of the first and second order total variation additive mid point model.

more
gradient of the TV prior
1
% gradTV(M,x) compute the gradient of the manifold total variation


For data $x\colon\Grid\to\M$ on a $m$-dimensional grid $\Grid$ with $N_{\vect{k}} = \bigl\{\vect{l} : \vect{l}+\vect{e}_j,\ j\in\{1,\ldots,m\}\bigr\}\cap G$ this function comutes the gradient of the TV prior

with respect to all entries of $x$, i.e. $\gradM F\colon\Grid\to T\M$.

more
gradient of the second order TV mid point model prior
1
% gradTV2Midpoint(M,x) compute gradient of the second order mid point model


For data $x\colon\Grid\to\M$ on a $m$-dimensional grid $\Grid$ this function returns the gradient $\gradM F$ of the absolute second order total variation prior.

more