Computing the proximal operator of the $\ell_1$ induced matrix norm

In this short article, for any matrix $X\in\mathbb{R}^{n\times m}$ the proximity operator of two induced norms $\|X\|_1$ and $\|X\|_{\infty}$ are derived. Although no close form expression is obtained, an algorithmic procedure is described which costs roughly $\mathcal{O}(nm)$. This algorithm relies on a bisection on a real parameter derived from the Karush-Kuhn-Tucker conditions, following the proof idea of the proximal operator of the $\max$ function found in Parikh(2014).