A note on spike localization for line spectrum estimation

This note considers the problem of approximating the locations of dominant spikes for a probability measure from noisy spectrum measurements under the condition of residue signal, significant noise level, and no minimum spectrum separation. We show that the simple procedure of thresholding the smoothed inverse Fourier transform allows for approximating the spike locations rather accurately

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)