Point Source Super-resolution Via Non-convex L_1 Based Methods

Yifei Lou Penghang Yin Jack Xin

Numerical Analysis and Scientific Computing mathscidoc:1912.43864

Journal of Scientific Computing, 68, (3), 1082-1100, 2016.9
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), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 1 norms ( L 1 ) and capped L 1 (C L 1 ), 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 L 1 solutions near or below Rayleigh length scales either in
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  title={Point Source Super-resolution Via Non-convex L_1 Based Methods},
  author={Yifei Lou, Penghang Yin, and Jack Xin},
  booktitle={Journal of Scientific Computing},
Yifei Lou, Penghang Yin, and Jack Xin. Point Source Super-resolution Via Non-convex L_1 Based Methods. 2016. Vol. 68. In Journal of Scientific Computing. pp.1082-1100. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210305527901428.
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