Support detection in super-resolution

We study the problem of super-resolving a superposition of point sources from noisy low-pass data with a cut-off frequency f. Solving a tractable convex program is shown to locate the elements of the support with high precision as long as they are separated by 2/f and the noise level is small with respect to the amplitude of the signal

From seismography to compressed sensing and back: a brief history of optimization-based signal processing

In this talk we provide an overview of the history of l1-norm minimization applied to underdetermined inverse problems. In the 70s and 80s geophysicists proposed using l1-norm minimization for deconvolution from bandpass data in reflection seismography. In the 2000s, inspired by this approach and by magnetic resonance imaging, a method to provably recover sparse signals from random projections, known as compressed sensing, was developed. Theoretical insights used to analyze compressed sensing have recently been adapted to understand the potential and limitations of l1-norm minimization for deterministic problems. These include super-resolution from low-pass data and the deconvolution problem that originally motivated the geophysicists.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Power-law expansion and Higgs-type potential in a scalar-tensor model

In the scalar-tensor model with Gauss-Bonnet and kinetic couplings, the power-law dark energy solution may be described by Higgs-type potential. It was found that in the solution describing early time epoch of matter dominance, the potential presents symmetry breaking phase, and the power law solution leading to accelerated expansion corresponds to Higgs-type potential in its symmetric shape.Comment: 16 pages, 1 figure; accepted in EP

Towards a Mathematical Theory of Super-Resolution

This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary.Comment: 48 pages, 12 figure