Papers › Testing Gaussian Process with Applications to Super-Resolution
Testing Gaussian Process with Applications to Super-Resolution
Jean-Marc Azaïs, Yohann De Castro, Stéphane Mourareau
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This article introduces exact testing procedures on the mean of a Gaussian process X derived from the outcomes of ℓ₁-minimization over the space of complex valued measures. The process X 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~X 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 X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X 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.
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