Proximal Operator of the $ {L}_{2} $ Norm Cubed (Proximal Operator of Norm Composition - Cubic Euclidean Norm) What is the proximal operator of $ {\left\| x \right\|}_{2}^{3} $ where $ {\left\| x \right\|}_{2} $ is the $ {L}_{2} $ norm?
Using Moreau Decomposition (Someone needs to create a Wikipedia page for it) one could solve it as following:
$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} (v) = v - \Pi_B \left( v \right) $$
Where $ \Pi_B \left( \cdot \right) $ is the projection of onto the unit ball of the dual norm.
Yet I'm not sure how to derive for the case mentioned above.
 A: The solution isn't trivial.
The problem is given by:
$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda f \left( x \right) = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2}^{3} $$
I succeeded using identity of Euclidean Norm Composition (Calculus of the Prox Operator, From Amir Beck - First Order Methods in Optimization):
Norm Composition: Let $ f : E \to R $ be given by $ f \left( x \right) = g \left( \left\| x \right\| \right) $, where $ g : R \to \left(−\infty, \infty \right] $ is a proper closed and convex function satisfying $ \operatorname{dom} \left( g \right) \subseteq \left[0, \infty \right) $. Then:
$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} \left( y \right) = \begin{cases}
\operatorname{Prox}_{ \lambda g \left( \cdot \right) } \left( \left\| y \right\| \right) \frac{y}{\left\| y \right\|} & \text{ if } y \neq 0 \\ 
\left\{ x \in \mathbb{E} : \left\| x \right\| = \operatorname{Prox}_{ \lambda g \left( \cdot \right)} \left( 0 \right) \right\} & \text{ if } y = 0 
\end{cases} $$
So in our case:
$$ g \left( t \right) = \begin{cases}
{t}^{3} & \text{ if } t \geq 0 \\ 
\infty & \text{ if } t < 0 
\end{cases} $$
It is easy to derive that
$$ \operatorname{Prox}_{\lambda g \left( \cdot \right)} \left( t \right) = \frac{-1 + \sqrt{ 1 + 12 \lambda {\left[ t \right]}_{+} }}{6 \lambda} $$
So for $ \operatorname{Prox}_{\lambda g \left( \cdot \right)} \left( 0 \right) = 0 $.
Hence the composition becomes:
$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} \left( y \right) = \begin{cases}
\frac{-1 + \sqrt{ 1 + 12 \lambda \left\| y \right\| }}{6 \lambda} \frac{y}{\left\| y \right\|} & \text{ if } y \neq 0 \\ 
0 & \text{ if } y = 0 
\end{cases} = \frac{2}{1 + \sqrt{1 + 12 \lambda \left\| y \right\|}} y $$
I validated the result comparing it to CVX solution. The MATLAB code is available at my Mathematics StackExchange Question 2071774 Repository.
Remark
Your idea about using the Moreau Decomposition is valid (Though I don't think it will yield solution in this case).
Yet you must pay attention that your function $ {\left\| \cdot \right\|}_{2}^{3} $ isn't a norm (Hence it is not the support of a norm etc...). You can't use the projection in this case but you need to use the Dual Function which as I wrote, I don't think will create a simpler solution.
