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.

在本文中，我们介绍了高斯随机场在黎曼子流形上的第二最大值的新概念，并利用一个特定的Kac Rice公式，推导了在第二最大值和黎曼海森矩阵的一些回归分量条件下的最大值分布的显式形式。我们将其作为间距测试的基础，通过评估这些最大值之间的间距实施精确检验。我们研究了这个测试方法在检测高斯对称张量、连续稀疏反卷积和具有平滑整流器的两层神经网络中的稀疏替代方面的适用性。我们的理论结果得到了数值实验的支持，展示了提出的测试方法的校准性和功效。此外，当高斯随机场的方差协方差函数已知的比例不变时，我们推导出了我们测试的精确标准化版本，称为$t$-间距测试。该检验在零假设下完全校准，并具有检测稀疏替代方案的高功效。

高斯随机场的第二最大值和精确(t-)间距检验