  Proximal operators are of particular interest in optimization problems
dealing with non-smooth objectives because in many practical cases they lead to
optimization algorithms whose updates can be computed in closed form or very
efficiently. A well-known example is the proximal operator of the vector
$\ell_1$ norm, which is given by the soft-thresholding operator. In this paper
we study the proximal operator of the mixed $\ell_{1,\infty}$ matrix norm and
show that it can be computed in closed form by applying the well-known
soft-thresholding operator to each column of the matrix. However, unlike the
vector $\ell_1$ norm case where the threshold is constant, in the mixed
$\ell_{1,\infty}$ norm case each column of the matrix might require a different
threshold and all thresholds depend on the given matrix. We propose a general
iterative algorithm for computing these thresholds, as well as two efficient
implementations that further exploit easy to compute lower bounds for the mixed
norm of the optimal solution. Experiments on large-scale synthetic and real
data indicate that the proposed methods can be orders of magnitude faster than
state-of-the-art methods.
