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## Abstract

Consider L groups of point sources or spike trains, with the l

^{th}group represented by x_{l }(t). For a function g:R→R , let g_{l}(t)=g(t/μ_{l}) denote a point spread function with scale μ_{l}>0 , and with μ_{1}<⋯<μ_{L}. With y(t)=∑ L l=1 (g_{l}⋆x_{l})(t), our goal is to recover the source parameters given samples of y , or given the Fourier samples of y . This problem is a generalization of the usual super-resolution setup wherein L=1 ; we call this the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples of y , we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L . In particular, the estimation process at stage 1≤l≤L involves (i) carefully sampling the tail of the Fourier transform of y , (ii) a \emph{deflation} step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).Original language | English |
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Publisher | ArXiv |

Number of pages | 49 |

Publication status | Published - 8 Jul 2018 |

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