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Journal Articles Applied and Computational Harmonic Analysis Year : 2018

## Testing Gaussian Process with Applications to Super-Resolution

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Jean-Marc Azaïs
Yohann de Castro

#### 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 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.

### Dates and versions

hal-01686434 , version 1 (02-07-2018)

### Identifiers

• HAL Id : hal-01686434 , version 1
• ARXIV :

### Cite

Jean-Marc Azaïs, Yohann de Castro, Stéphane Mourareau. Testing Gaussian Process with Applications to Super-Resolution. Applied and Computational Harmonic Analysis, 2018. ⟨hal-01686434⟩

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