2018/07/08 by Stéphane Chrétien, Chrétien, Stéphane, Hemant Tyagi +1 · 1 citation
Decision Sciences · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1807.02862
openalex publication_date 2018/07/08 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Consider L groups of point sources or spike trains, with the\nl\th group represented by xl(t). For a function g:\ℝ\n\→ \ℝ, let gl(t) = g(t/\μl) denote a point spread\nfunction with scale \μl > 0, and with \μ1 < \⋯ < \μL. With y(t)\n= \∑l=1L (gl \⋆ xl)(t), our goal is to recover the source\nparameters given samples of y, or given the Fourier samples of y. This\nproblem is a generalization of the usual super-resolution setup wherein L =\n1; we call this the multi-kernel unmixing super-resolution problem. Assuming\naccess to Fourier samples of y, we derive an algorithm for this problem for\nestimating the source parameters of each group, along with precise\nnon-asymptotic guarantees. Our approach involves estimating the group\nparameters sequentially in the order of increasing scale parameters, i.e., from\ngroup 1 to L. In particular, the estimation process at stage 1 \≤ l \≤\nL involves (i) carefully sampling the tail of the Fourier transform of y,\n(ii) a \deflation step wherein we subtract the contribution of the groups\nprocessed thus far from the obtained Fourier samples, and (iii) applying\nMoitra's modified Matrix Pencil method on a deconvolved version of the samples\nin (ii).\n