Convex Optimization in Infinite Sums of Banach Spaces Using Besov Regularization
Benoît Sauty De Chalon

Meeting • 13 July 2020

Abstract
We caracterize the solutions of a broad class of convex optimization problems for the recovery of a function from a finite number of linear measurements. We take interest in the case where the solution is decomposable as $f=\sum_{n\in\mathbb{N}} f_n$ in an infinite amount of subcomponents, where each component belongs to a prescribed Banach space $\mathcal{B}_n$, while ensuring the problem is well posed by penalizing some composite norm of the solution. We begin by deriving a general representer theorem that states conditions for existence of solutions and gives the parametric representation of the solution components. Namely, they can be decomposed as a sparse sum of extremal points of the unit balls of the $\mathcal{B}_n$ spaces. Then, we apply this framework by studying functions that can be decomposed in a multi-resolution wavelet decomposition using the well known Shannon wavelet and a fitting regularization norm inspired by Besov spaces to obtain a generalized sparse dictionnary learning technique.