A Decoupled Approach for Composite Sparse-plus-Smooth Penalized Optimization

We consider a linear inverse problem whose solution is expressed as a sum of two components: one smooth and the other sparse. This problem is addressed by minimizing an objective function with a least squares data-fidelity term and a different regularization term applied to each of the components. Sparsity is promoted with an $\ell_1$ norm, while the smooth component is penalized with an $\ell_2$ norm. We characterize the solution set of this composite optimization problem by stating a Representer Theorem. Consequently, we identify that solving the optimization problem can be decoupled by first identifying the sparse solution as a solution of a modified single-variable problem and then deducing the smooth component. We illustrate that this decoupled solving method can lead to significant computational speedups in applications, considering the problem of Dirac recovery over a smooth background with two-dimensional partial Fourier measurements.