Multi-kernel unmixing and super-resolution using the Modified Matrix Pencil method

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 μ₁ <⋯<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).