Section: New Results

Axis 2: Multi-kernel unmixing and super-resolution using the Modified Matrix Pencil method

Participant: Hemant Tyagi.

Consider $L$ groups of point sources or spike trains, with the $l^{th}$ group represented by $x_l\left(t\right)$. For a function $g:R\to R$, let $g_l\left(t\right)=g(t/{\mu }_l)$ denote a point spread function with scale ${\mu }_l>0$, and with ${\mu }_1<\cdots <{\mu }_L$. With $y\left(t\right)={\sum }_{l=1}^L(g_l☆x_l)\left(t\right)$, 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 $1\le l\le L$ 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).

This is joint work with Stephane Chretien (National Physical Laboratory, UK & Alan Turing Institute, London) and was mostly done while Hemant Tyagi was affiliated to the Alan Turing Institute. It is currently under revision in an international journal and is available as a preprint [56].