Square Root {LASSO}: well-posedness, Lipschitz stability and the tuning trade off

This paper studies well-posedness and parameter sensitivity of the Square Root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over the standard LASSO) is that the optimal tuning of the regularization parameter is robust with respect to measurement noise. This paper provides three point-based regularity conditions at a solution of the SR-LASSO: the weak, intermediate, and strong assumptions. It is shown that the weak assumption implies uniqueness of the solution in question. The intermediate assumption yields a directionally differentiable and locally Lipschitz solution map (with explicit Lipschitz bounds), whereas the strong assumption gives continuous differentiability of said map around the point in question. Our analysis leads to new theoretical insights on the comparison between SR-LASSO and LASSO from the viewpoint of tuning parameter sensitivity: noise-robust optimal parameter choice for SR-LASSO comes at the "price" of elevated tuning parameter sensitivity. Numerical results support and showcase the theoretical findings.