高维半代数域上超归结的精确解
Exact solutions to Super Resolution on semi-algebraic domains in higher dimensions
摘要：
我们研究了多维超分辨率问题的封闭半代数域上的各种采样方案，如傅立叶或矩。在半代数集上的多维框架下，给出了Radon测度空间中1 -极小化问题的一个新的半定规划（SDP）公式。虽然标准的方法都集中在SDP松弛的对偶程序（一种流行的方法是基于革兰氏矩阵表示），本文介绍了一个确切的制定的原始1 -最小化精确恢复问题的超分辨率，释放标准技术（如矩和平方层次结构），以克服固有的局限性，以前的作品在文献中。值得注意的是，我们表明，可以精确地解决超分辨率问题的尺寸大于2，并为一个大家庭的域所描述的半代数集。
英文摘要：
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 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 1 -minimization exact recovery problem of Super Resolution that unleashes standard techniques (such as moment-sum-of-squares hier-archies) 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.
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