Point Source Super-resolution Via Non-convex L₁ Based Methods

We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), L₁ minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L₁ certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L₁ based nonconvex penalties, the difference of L₁ and L₂ norms (L_{1 - 2}) and capped L₁ (CL₁), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L₁ solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.