# Proximal Maps

For a function $\varphi:\mathcal M →ℝ$ the proximal map is defined as

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

where $d_{\mathcal M}: \mathcal M \times \mathcal M → ℝ$ 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 $x^{\star} ∈ \mathcal M$ is a minimizer of $\varphi$, then

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

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/proximal_maps.jl file.

Manopt.project_collaborative_TVFunction
project_collaborative_TV(M, λ, x, Ξ[, p=2,q=1])
project_collaborative_TV!(M, Θ, λ, x, Ξ[, p=2,q=1])

compute the projection onto collaborative Norm unit (or α-) ball, i.e. of the function

$$$F^q(x) = \sum_{i∈\mathcal G} \Bigl( \sum_{j∈\mathcal I_i} \sum_{k=1}^d \lVert X_{i,j}\rVert_x^p\Bigr)^\frac{q}{p},$$$

where $\mathcal G$ is the set of indices for $x∈\mathcal M$ and $\mathcal I_i$ is the set of its forward neighbors. The computation can also be done in place of Θ.

This is adopted from the paper Duran, Möller, Sbert, Cremers, SIAM J Imag Sci, 2016, see their Example 3 for details.

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Manopt.prox_TVFunction
ξ = prox_TV(M,λ,x [,p=1])

compute the proximal maps $\operatorname{prox}_{λ\varphi}$ of all forward differences occurring in the power manifold array, i.e. $\varphi(xi,xj) = d_{\mathcal M}^p(xi,xj)$ with xi and xj are array elemets of x and j = i+e_k, where e_k is the $k$th unit vector. The parameter λ is the prox parameter.

Input

• M – a manifold M
• λ – a real value, parameter of the proximal map
• x – a point.

Optional

(default is given in brackets)

• p – (1) exponent of the distance of the TV term

Ouput

• y – resulting point containing with all mentioned proximal points evaluated (in a cyclic order). The computation can also be done in place
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Manopt.prox_TVMethod
[y1,y2] = prox_TV(M, λ, [x1,x2] [,p=1])
prox_TV!(M, [y1,y2] λ, [x1,x2] [,p=1])

Compute the proximal map $\operatorname{prox}_{λ\varphi}$ of $φ(x,y) = d_{\mathcal M}^p(x,y)$ with parameter λ.

Input

• M – a manifold M
• λ – a real value, parameter of the proximal map
• (x1,x2) – a tuple of two points,

Optional

(default is given in brackets)

• p – (1) exponent of the distance of the TV term

Ouput

• (y1,y2) – resulting tuple of points of the $\operatorname{prox}_{λφ}($(x1,x2)$)$. The result can also be computed in place.
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Manopt.prox_TV2Method
(y1,y2,y3) = prox_TV2(M,λ,(x1,x2,x3),[p=1], kwargs...)
prox_TV2!(M, y, λ,(x1,x2,x3),[p=1], kwargs...)

Compute the proximal map $\operatorname{prox}_{λ\varphi}$ of $\varphi(x_1,x_2,x_3) = d_{\mathcal M}^p(c(x_1,x_3),x_2)$ with parameter λ>0, where $c(x,z)$ denotes the mid point of a shortest geodesic from x1 to x3 that is closest to x2. The result can be computed in place of y.

Input

• M – a manifold

• λ – a real value, parameter of the proximal map

• (x1,x2,x3) – a tuple of three points

• p – (1) exponent of the distance of the TV term

Optional

kwargs... – parameters for the internal subgradient_method (if M is neither Euclidean nor Circle, since for these a closed form is given)

Output

• (y1,y2,y3) – resulting tuple of points of the proximal map. The computation can also be done in place.
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Manopt.prox_TV2Method
y = prox_TV2(M, λ, x[, p=1])
prox_TV2!(M, y, λ, x[, p=1])

compute the proximal maps $\operatorname{prox}_{λ\varphi}$ of all centered second order differences occuring in the power manifold array, i.e. $\varphi(x_k,x_i,x_j) = d_2(x_k,x_i.x_j)$, where $k,j$ are backward and forward neighbors (along any dimension in the array of x). The parameter λ is the prox parameter.

Input

• M – a manifold M
• λ – a real value, parameter of the proximal map
• x – a points.

Optional

(default is given in brackets)

• p – (1) exponent of the distance of the TV term

Output

• y – resulting point with all mentioned proximal points evaluated (in a cylic order). The computation can also be done in place.
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Manopt.prox_distanceFunction
y = prox_distance(M,λ,f,x [, p=2])
prox_distance!(M, y, λ, f, x [, p=2])

compute the proximal map $\operatorname{prox}_{λ\varphi}$ with parameter λ of $φ(x) = \frac{1}{p}d_{\mathcal M}^p(f,x)$. For the mutating variant the computation is done in place of y.

Input

• M – a manifold M
• λ – the prox parameter
• f – a point $f ∈ \mathcal M$ (the data)
• x – the argument of the proximal map

Optional argument

• p – (2) exponent of the distance.

Ouput

• y – the result of the proximal map of $φ$
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Manopt.prox_parallel_TVFunction
y = prox_parallel_TV(M, λ, x [,p=1])
prox_parallel_TV!(M, y, λ, x [,p=1])

compute the proximal maps $\operatorname{prox}_{λφ}$ of all forward differences occurring in the power manifold array, i.e. $φ(x_i,x_j) = d_{\mathcal M}^p(x_i,x_j)$ with xi and xj are array elements of x and j = i+e_k, where e_k is the $k$th unit vector. The parameter λ is the prox parameter.

Input

• M – a PowerManifold manifold
• λ – a real value, parameter of the proximal map
• x – a point

Optional

(default is given in brackets)

• p – (1) exponent of the distance of the TV term

Ouput

• y – resulting Array of points with all mentioned proximal points evaluated (in a parallel within the arrays elements). The computation can also be done in place.

See also prox_TV

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# Literature

[DMSC16]
J. Duran, M. Moeller, C. Sbert and D. Cremers. Collaborative Total Variation: A General Framework for Vectorial TV Models. SIAM Journal on Imaging Sciences 9, 116-151 (2016), arxiv: [1508.01308](https://arxiv.org/abs/1508.01308).