Does SLOPE outperform bridge regression?

A recently proposed SLOPE estimator (arXiv:1407.3824) has been shown to adaptively achieve the minimax ℓ2 estimation rate under high-dimensional sparse linear regression models (arXiv:1503.08393). Such minimax optimality holds in the regime where the sparsity level k, sample size n, and dimension p satisfy k/p rightarrow 0, klog p/n rightarrow 0. In this paper, we characterize the estimation error of SLOPE under the complementary regime where both k and n scale linearly with p, and provide new insights into the performance of SLOPE estimators. We first derive a concentration inequality for the finite sample mean square error (MSE) of SLOPE. The quantity that MSE concentrates around takes a complicated and implicit form. With delicate analysis of the quantity, we prove that among all SLOPE estimators, LASSO is optimal for estimating k-sparse parameter vectors that do not have tied non-zero components in the low noise scenario. On the other hand, in the large noise scenario, the family of SLOPE estimators are sub-optimal compared with bridge regression such as the Ridge estimator.