$ \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}} $

Notation

This is a list of mathematical notations used throughout the MVIRT pages sorted alphabetically

norm in the tangent space , , induced by the inner product.
inner product in the tangent space of , where the index might be ommited if it is clear from the context.
covariant derivative
differential of the function with respect to its variable , e.g. if then and
differential of the function evaluated at , e.g. if then and . One might also use , i.e. after derivation with respect to we again use as the variable of the differential
directional derivative of the function with respect to evaluated at with respect to the direction , e.g. if then .
distance on the manifold , i.e. for two points the value is the length of a shortest geodesic connecting and .
functions defined on the manifold and mapping into (a possibily different) manifold. Capital letters might be used to indicate the main objective, while for general theory or atoms of the main objetive, small letters are used.
, ,
A geodesic is a function with vanishing covariant derivative with either fixed start point and end point or given start point and initial velocity , where denotes the covariant derivative. Sometimes geodesic only refers to a length minimizing geodesic.
the gradient of defined by the property
A complete -dimensional Riemannian manifold.
Matrices are denoted in bold capital letters. The only exception is, when we speak of points on a manifold or in the tangent plane, where we empahsize the character of being a point on the manifold or tangent plane, respectively, while they are (also) represented as matrices.
proximal map of a function is given by
the tangent bundle, , see also tangent space
the tangent space at
tangent vectors from the tangent space at . The index is ommited, if the point is clear from the context.
points on the manifold
tangent vector fields, i.e. ,
Vectors are denotes in bold letters. The only exception is, when we speak of points on a manifold or in the tangent plane, where we empahsize the character of being a point on the manifold or tangent plane, respectively, while they are also represented as vectors.