Exact Solutions to Super Resolution on Semi-Algebraic Domains in Higher Dimensions

Abstract

We investigate the multi-dimensional super resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the $\ell _{1}$ -minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal $\ell _{1}$ -minimization exact recovery problem of super resolution that unleashes standard techniques (such as moment-sum-of-squares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the super resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets.