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 $µ_1 < · · · < µ_L$. With $y(t) = \sum_{l=1}^{L} (g_l * x_l)(t)$, 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)