Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

摘要

Consider L groups of point sources or spike trains, with the lth group represented by xl (t). For a function g : R. R, let gl (t) = g(t/mu l) denote a point spread function with scale mu l > 0, and with mu 1 < center dot center dot center dot < mu L. With y(t) = L l=1(gl xl)(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 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).