Introduce proxes.

src/proximal_maps.jl

@@ -0,0 +1,50 @@
+@doc raw"""
+    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.
+
+# Output
+* `y` – the result of the proximal map of ``φ``
+"""
+function prox_distance(M::AbstractManifold, λ, f, x, p::Int = 2)
+    d = distance(M, f, x)
+    if p == 2
+        t = λ / (1 + λ)
+    elseif p == 1
+        t = (λ < d) ? λ / d : 1.0
+    else
+        throw(
+            ErrorException(
+                "Proximal Map of distance(M,f,x) not implemented for p=$(p) (requires p=1 or 2)",
+            ),
+        )
+    end
+    return exp(M, x, log(M, x, f), t)
+end
+function prox_distance!(M::AbstractManifold, y, λ, f, x, p::Int = 2)
+    d = distance(M, f, x)
+    if p == 2
+        t = λ / (1 + λ)
+    elseif p == 1
+        t = (λ < d) ? λ / d : 1.0
+    else
+        throw(
+            ErrorException(
+                "Proximal Map of distance(M,f,x) not implemented for p=$(p) (requires p=1 or 2)",
+            ),
+        )
+    end
+    return exp!(M, y, x, log(M, x, f), t)
+end