Computing the proximal operator of the ${\mathrm{\ell }}_1$ induced matrix norm

In this short article, for any matrix $X\in {\mathbb{R}}^{n\times m}$ the proximity operator of two induced norms $\mathrm{\parallel }X{\mathrm{\parallel }}_1$ and $\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{\infty }}$ 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).