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

TL;DR

This paper derives an algorithmic procedure to compute the proximity operators of the  and  induced matrix norms, enabling efficient optimization involving these norms.

Contribution

It introduces a novel algorithmic approach for computing the proximity operators of  and  induced matrix norms with (nm) complexity.

Findings

Provides a bisection-based algorithm for proximity operator computation.

No closed-form solution but efficient approximation method.

Applicable to optimization problems involving these norms.

Abstract

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).