  The log-sum penalty is often adopted as a replacement for the $\ell_0$
pseudo-norm in compressive sensing and low-rank optimization. The
hard-thresholding operator, i.e., the proximity operator of the $\ell_0$
penalty, plays an essential role in applications; similarly, we require an
efficient method for evaluating the proximity operator of the log-sum penalty.
Due to the nonconvexity of this function, its proximity operator is commonly
computed through the iteratively reweighted $\ell_1$ method, which replaces the
log-sum term with its first-order approximation. This paper reports that the
proximity operator of the log-sum penalty actually has an explicit expression.
With it, we show that the iteratively reweighted $\ell_1$ solution disagrees
with the true proximity operator of the log-sum penalty in certain regions. As
a by-product, the iteratively reweighted $\ell_1$ solution is precisely
characterized in terms of the chosen initialization. We also give the explicit
form of the proximity operator for the composition of the log-sum penalty with
the singular value function, as seen in low-rank applications. These results
should be useful in the development of efficient and accurate algorithms for
optimization problems involving the log-sum penalty.
