The Proximity Operator of the Log-Sum Penalty

The log-sum penalty is often adopted as a replacement for the ℓ pseudo-norm in compressive sensing and low-rank optimization. The proximity operator of the ℓ penalty, i.e., the hard-thresholding operator, 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 ℓ₁ 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 ℓ₁ solution disagrees with the true proximity operator in certain regions. As a by-product, the iteratively reweighted ℓ₁ 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. We present applications to solving compressive sensing problems and to mixed additive Gaussian white noise and impulse noise removal.