Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with L0-``norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on L1-norm relaxations. In this paper, we propose new iterative conic quadratic relaxations that exploit not only the L0-``norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the L0-``norm" and its L1 surrogate. Experiments using an off-the-shelf conic quadratic solver on synthetic as well as real datasets indicate that the proposed iterative convex relaxations lead to significantly better estimators than L1-norm while preserving the computational efficiency. In addition, the parameters of the model and the resulting estimators are easily interpretable.
BCOL Research Report 18.05, IEOR, University of California-Berkeley.