Sparse and Smooth Signal Estimation: Convexification of L0 Formulations
This addresses the challenge of efficiently solving L0-based signal estimation problems for applications requiring sparse and smooth signals, representing a strong specific gain rather than a broad breakthrough.
The paper tackled the non-convex problem of signal estimation with sparsity and smoothness priors by proposing iterative convex conic quadratic relaxations, which yield solutions within 1% of the exact L0 approach and handle up to 100,000 variables in under a minute.
Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper, we propose a new iterative (convex) conic quadratic relaxations that exploit not only the $\ell_0$-"norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the $\ell_0$-"norm" and its $\ell_1$ surrogate. These stronger relaxations lead to significantly better estimators than $\ell_1$-norm approaches and also allow one to utilize affine sparsity priors. In addition, the parameters of the model and the resulting estimators are easily interpretable. Experiments with a tailored Lagrangian decomposition method indicate that the proposed iterative convex relaxations \rev{yield solutions within 1\% of the exact $\ell_0$ approach, and can tackle instances with up to 100,000 variables under one minute.