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# [Proximal Maps](@id proximalMapFunctions)

For a function $\varphi\colon\mathcal M \to\mathbb R$ the proximal map is defined as

$\displaystyle\operatorname{prox}{\lambda\varphi}(x) = \operatorname*{argmin}{y\in\mathcal M} d_{\mathcal M}^2(x,y) + \varphi(y), \quad \lambda > 0,$

where $d_{\mathcal M}\colon \mathcal M \times \mathcal M \to \mathbb R$ denotes the geodesic distance on (\mathcal M). While it might still be difficult to compute the minimizer, there are several proximal maps known (locally) in closed form. Furthermore if $x^{\star} \in\mathcal M$ is a minimizer of $\varphi$, then

$\displaystyle\operatorname{prox}_{\lambda\varphi}(x^\star) = x^\star,$

i.e. a minimizer is a fixed point of the proximal map.

This page lists all proximal maps available within Manopt. To add you own, just extend the functions/proximalMaps.jl file.

Modules = [Manopt]
Pages   = ["proximalMaps.jl"]

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