Computing the proximal operator of the ℓ1 induced matrix norm

In this short article, for any matrix X∈ℝn× m the proximity operator of two induced norms |X|1 and |X|∞ are derived. Although no close form expression is obtained, an algorithmic procedure is described which costs roughly 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).