# Testing Gaussian Process with Applications to Super-Resolution

## TL;DR

This paper develops exact Gaussian process testing methods for super-resolution applications, including grid-based and grid-less procedures, demonstrating their effectiveness in deconvolution and sparse signal detection.

## Contribution

It introduces novel grid-based and grid-less testing procedures for Gaussian processes, applicable to super-resolution and deconvolution, with proven convergence and power advantages.

## Key findings

- Grid-less test is more powerful in detecting sparse signals.
- Both tests work with unknown variance and known correlation.
- Numerical results show the grid-less method's superior performance.

## Abstract

This article introduces exact testing procedures on the mean of a Gaussian process $X$ derived from the outcomes of $\ell_1$-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.