Multi-kernel unmixing and super-resolution using the Modified Matrix Pencil method
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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=1^L (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 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).
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