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

Abstract

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