Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test

TL;DR

This paper introduces the second maximum of a Gaussian random field and develops an exact spacing test for detecting sparse alternatives, with applications to neural networks and kernel regression, supported by theoretical and numerical validation.

Contribution

It proposes the novel concept of the second maximum and derives an explicit distribution for the maximum conditioned on it, enabling an exact spacing test for Gaussian fields.

Findings

The spacing test is well-calibrated and powerful in simulations.

The $t$-spacing test is exact under known covariance up to a scale.

Applications include neural networks and sparse kernel regression.

Abstract

In this article, we introduce the novel concept of the second maximum of a Gaussian random field on a Riemannian submanifold. This second maximum serves as a powerful tool for characterizing the distribution of the maximum. By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. This approach results in an exact test, based on the evaluation of spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test in detecting sparse alternatives within Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Our theoretical results are supported by numerical experiments, which illustrate the calibration and power of the proposed tests. More generally, this test can be applied to any Gaussian random field on a Riemannian manifold, and we provide a general framework for the application of the spacing test in continuous sparse kernel regression. Furthermore, when the variance-covariance function of the Gaussian random field is known up to a scaling factor, we derive an exact Studentized version of our test, coined the $t$-spacing test. This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives.