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

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

Technology and economic inequality effects on international trade

The notion that technology plays a key role in explaining trade performed was supported in the last decades by many empirical studies. In this paper we use a different perspective trying to analyse the effects of technology and economic inequalities on international trade inequalities. The theoretical framework where we built our empirical analysis is the technological gap aproach. We considered eight European countries and 13 manufacture industries in the time period 1995-2002. We made a panel data model with a cross-sectional unit of analysis: the Euclidean distance among countries in each industry. We considered the Euclidian distance as a proxy of inequality among countries in each industry. We observed that technology and economic inequalities affect trade inequality and that the effect depends on the technological contend of each industry.Technology inequality, trade inequality, Euclidean distance, technological contend.