On the recovery of measures without separation conditions

Applied and Computational Mathematics Seminar
Monday, October 2, 2017 - 13:55
1 hour (actually 50 minutes)
Skiles 005
University of Maryland, College Park
We formulate super-resolution as an inverse problem in the space of measures, and introduce a discrete and a continuous model. For the discrete model, the problem is to accurately recover a sparse high dimensional vector from its noisy low frequency Fourier coefficients. We determine a sharp bound on the min-max recovery error, and this is an immediate consequence of a sharp bound on the smallest singular value of restricted Fourier matrices. For the continuous model, we study the total variation minimization method. We borrow ideas from Beurling in order to determine general conditions for the recovery of singular measures, even those that do not satisfy a minimum separation condition. This presentation includes joint work with John Benedetto and Wenjing Liao.