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

# List of proximal maps

### List of proximal maps for atoms

The proximal map of an absolute difference
1
% proxAbsoluteDifference(M,f1,f2,lambda) prox of d_M(f1,f2) in both args


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = d_{\M}(x,y)$.

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The proximal map of an absolute difference with Huber relaxation
1
% proxAbsoluteDifferenceHuber(M,f1,f2,lambda,tau,omega) prox d(x,y) relaxed


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = h(d_{\M}(x,y))$, where $h$ relaxes the nondifferential point at $d=0$.

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The proximal map of an absolute difference squared
1
% proxAbsoluteDifference(M,f1,f2,lambda) prox of d^2_M(f1,f2) in both args


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = d^2_{\M}(x,y)$.

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The proximal map of an absolute second order difference
1
% proxAbsoluteSecondOrderDifference(M,f1,f2,f3,lambda) prox d2(f1,f2,f3)


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^3\to\R,\quad f(x,y,z) = d_2(x,y,z) = d_{\M}(\geo{x}{z}(\tfrac{1}{2}),y)$, where the geodesic is the one closest to $y$.

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The proximal map of an absolute second order mixed difference
1
% proxAbsoluteSecondOrderMixedDifference(M,f1,f2,f3,f4,lambda)


Proximal map $\prox_{\lambda f}$ for the function

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The proximal map of the distance
1
% proxDistanceSquared(M,x,f,lambda) prox of f(x) = d(x,y), y fixed,


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M\to\R,\quad f(x) = d_{\M}(x,y)$ for some fixed value $y\in\M$.

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The proximal map of the distance squared
1
% proxDistanceSquared(M,x,f,lambda) prox of f(x) = d^2(x,y), y fixed,


Proximal map $\prox_{\lambda f}$ for the function $f\colon\M\to\R,\quad f(x) = d^2_{\M}(x,y)$ for some fixed value $y\in\M$.

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The proximal map of total variation parallel
1
% proxParallelTV(M,x,lambda) compute all proxima the tv prior in parallel


For given data $x\in\M^{\vect{n}}$ of dimension $\vect{n}=(n_1,\ldots,n_m)$, $m\in\mathbb N$, (usually $m=1,2$) this function computes the proximal map of the TV prior in parallel, returning all results in one array.

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The proximal map of total variation
1
% proxTV(M,x,lambda) compute all proxima of TV in a cyclic manner.


For given data $x\in\M^{\vect{n}}$ of dimension $\vect{n}=(n_1,\ldots,n_m)$, $m\in\mathbb N$, (usually $m=1,2$) this function computes the proximal map of the TV prior in a cyclic manner.

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The proximal map of second order total variation
1
% proxTV(M,x,lambda) compute all proxima of second order finite differences


For given data $x\in\M^{\vect{n}}$ of dimension $\vect{n}=(n_1,\ldots,n_m)$, $m\in\mathbb N$, (usually $m=1,2$) this function computes the proximal map of the second order TV prior in a cyclic manner.

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