TITLE: The Sub Gradient and the Proximal Operator of the of $ {L}_{2, 1} $ Norm (Mixed Norm)
QUESTION [5 upvotes]: What would be the the Sub Gradient of
$$ f \left( X \right) = {\left\| A X \right\|}_{2, 1} $$
Where $ X \in \mathbb{R}^{m \times n} $, $ {A} \in \mathbb{R}^{k \times m} $ and $ {\left\| Y \right\|}_{2, 1} = \sum_{j} \sqrt{ \sum_{i} {Y}_{i,j}^{2} } $.
What would be the Prox of:
$$ \operatorname{Prox}_{\lambda {\left\| \cdot \right\|}_{2,1}} \left( Y \right) = \arg \min_{X} \frac{1}{2} {\left\| X - Y \right\|}_{F}^{2}  + \lambda {\left\| X \right\|}_{2, 1}, \; X, Y \in {\mathbb{R}}^{m \times n} $$
Can either be generalized for $ {\left\| \cdot \right\|}_{q, p} $?

REPLY [3 votes]: Let $x_j$ be the $j$th column of $X$. (So $\|X\|_{2,1} = \sum_{j=1}^n \| x_j \|_2$.) Then the separable sum rule for proximal operators tells us that $\text{prox}_{\lambda\| \cdot \|_{2,1}}(X)$ is the $m \times n$ matrix whose $j$th column is $\text{prox}_{\lambda\| \cdot \|_2}(x_j)$.