2017/06/02 by Jean‐Marc Azäis, Yohann de Castro, Azaïs, Jean-Marc +2
Computer Science · Engineering · Mathematics · #60G10 #60G15 #62E15 #62F03 #62H10 #62H15 (Primary) 60E05 #62J05 #94A08 (secondary) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Information Theory (cs.IT) #Probability (math.PR) #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1706.00679
openalex publication_date 2017/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.