The second maximum of a gaussian random field and statistical application to sparse models.

Arising from continuous sparse kernel regression, we introduce the concept of the so-called second maximum of a Gaussian random field. This second maximum proves to be a valuable means for characterizing the distribution of the maximum. In this paper, we use an ad-hoc Kac Rice formula to derive the explicit form of the distribution of the maximum of a Gaussian random field, conditional to the second maximum and the so-called random part of the Riemannian Hessian. This method leads to an exact testing procedure based on the evaluation of spacing between these maxima, referred to as the spacing test.

In addition, if the variance-covariance function of the Gaussian random field is known up to some scaling factor, we derive an exact studentized version of our testing procedure called $t$-spacing test. We illustrate that the ($t$-)spacing tests can be used to detect sparse alternatives in Gaussian symmetric tensor, continuous sparse deconvolution, and two-layers neural networks with smooth rectifier.