Section: New Results
Testing Gaussian Process with Applications to Super-Resolution
J.-M. Azaïs, Y. De Castro, S. Mourareau
In [13], we introduce exact testing procedures on the mean of a Gaussian process derived from the outcomes of -minimization over the space of complex valued measures. The process can be thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.