Point Source Super-resolution Via Non-convex $L_1$ based methods

Yifei Lou University of Texas at Dallas Penghang Yin University of California at Irvine Jack Xin University of California at Irvine

Information Theory mathscidoc:1608.19001

Journal of Scientific Computing, 68, (3), 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), L1 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 $L_2$ norms $(L_{1-2})$ and capped $L_1 (CL_1)$, subject to the measurement constraints. In one and two dimensional numerical SR examples,the local optimal solutions from di erence 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 nding a sparse solution satisfying the constraints more accurately.
Rayleigh Length  $L_{1-2}$, Capped $L_1$, Difference of Convex Algorithm (DCA)
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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. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818145539722003239.
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