Monday Febuary 5th 2018
Gabriel Peyré, CNRS and Ecole Normale Supérieure

Title: Off-the-grid Methods for Sparse Spikes Super-resolution

Abstract: In this talk, I study sparse spikes super-resolution over the space of measures (as initiated for instance in [2,3]), in order to solve a wide variety of inverse problems in imaging sciences. For non-degenerate sums of Diracs, we show that, when the signal-to-noise ratio is large enough, total variation regularization of measures (which the natural extension of the l1 norm of vectors to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. When the measure is positive, it is known that l1-type methods always succeed when there is no noise. We show that exact support recovery is still possible when there is noise. The signal-to-noise ratio should then scale like $1/t^{2N-1}$ where there are N spikes separated by a distance t. This reflects the intrinsic explosion of the ill-posedness of the problem [4]. Lastly, I will also discus computational methods to solve the corresponding grid-free infinite-dimensional optimization problem. I will advocate for the use of an hybridization between a convex Frank-Wolfe algorithm (which progressively adds points to the computation grid) and a non-convex update (that moves the grid points), and show that it performs surprisingly well on deconvolution and Laplace inversion. This is joint work with Vincent Duval and Quentin Denoyelle, see [1,4] for more details.

[1] V. Duval, G. Peyré, Exact Support Recovery for Sparse Spikes Deconvolution, Foundation of Computational Mathematics, 15(5), pp. 1315–1355, 2015.
[2] E. J. Candès and C. Fernandez-Granda. Towards a mathematical theory of super-resolution. Communications on Pure and Applied Mathematics, 67(6), 906-956, 2013.
[3] K. Bredies and H.K. Pikkarainen. Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations, 19:190-218, 2013.
[4] Q. Denoyelle, V. Duval, G. Peyré. to appear in Journal of Fourier Analysis and Applications, 2017.

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