🤖 AI Summary
This paper addresses the problem of sparse signal detection for Gaussian random fields on Riemannian manifolds. Methodologically, it introduces an exact extremal testing framework based on the second maximum of the field—novelly employing the second maximum to characterize extremal behavior, deriving its exact distribution via the conditional Kac–Rice formula, and constructing the first rigorously calibrated (t-)spacing test in continuous domains. Theoretical contributions include: (i) a general testing framework applicable to arbitrary Riemannian submanifolds; (ii) perfect control of Type-I error under the null hypothesis; and (iii) optimal detection power in symmetric tensor testing, continuous sparse deconvolution, and two-layer smoothed ReLU networks. Numerical experiments confirm both theoretical calibration accuracy and superior statistical power.
📝 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.