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

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

We propose the generative adversarial neural operator (GANO), a generative
model paradigm for learning probabilities on infinite-dimensional function
spaces. The natural sciences and engineering are known to have many types of
data that are sampled from infinite-dimensional function spaces, where
classical finite-dimensional deep generative adversarial networks (GANs) may
not be directly applicable. GANO generalizes the GAN framework and allows for
the sampling of functions by learning push-forward operator maps in
infinite-dimensional spaces. GANO consists of two main components, a generator
neural operator and a discriminator neural functional. The inputs to the
generator are samples of functions from a user-specified probability measure,
e.g., Gaussian random field (GRF), and the generator outputs are synthetic data
functions. The input to the discriminator is either a real or synthetic data
function. In this work, we instantiate GANO using the Wasserstein criterion and
show how the Wasserstein loss can be computed in infinite-dimensional spaces.
We empirically study GANO in controlled cases where both input and output
functions are samples from GRFs and compare its performance to the
finite-dimensional counterpart GAN. We empirically study the efficacy of GANO
on real-world function data of volcanic activities and show its superior
performance over GAN.

本研究提出了一种名为 GANO 的生成对抗神经操作器模型，可用于学习无限维函数空间上的概率。通过使用 Wasserstein 标准对其进行实例化，证明了其在真实世界函数数据上的优越性能。

生成对抗神经算子

Generative Adversarial Neural Operators

We give a probabilistic interpretation of sampling theory of graph signals.
To do this, we first define a generative model for the data using a pairwise
Gaussian random field (GRF) which depends on the graph. We show that, under
certain conditions, reconstructing a graph signal from a subset of its samples
by least squares is equivalent to performing MAP inference on an approximation
of this GRF which has a low rank covariance matrix. We then show that a
sampling set of given size with the largest associated cut-off frequency, which
is optimal from a sampling theoretic point of view, minimizes the worst case
predictive covariance of the MAP estimate on the GRF. This interpretation also
gives an intuitive explanation for the superior performance of the sampling
theoretic approach to active semi-supervised classification.

本研究给出了图信号采样理论的概率解释，通过概率高斯场与贝叶斯推断等方法，阐述了该理论在最小方差准则下，对最优采样及半监督分类等方面的应用。

图信号采样理论的概率解释

A Probabilistic Interpretation of Sampling Theory of Graph Signals

The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions
has been used for high-dimensional model representation and global sensitivity
analysis. When the objective function f has no simple analytic form and is
costly to evaluate, a practical limitation is that computing FANOVA terms may
be unaffordable due to numerical integration costs. Several approximate
approaches relying on random field models have been proposed to alleviate these
costs, where f is substituted by a (kriging) predictor or by conditional
simulations. In the present work, we focus on FANOVA decompositions of Gaussian
random field sample paths, and we notably introduce an associated kernel
decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of
tensor product projections is obtained, and it is shown that projected kernels
control both the sparsity of Gaussian random field sample paths and the
dependence structure between FANOVA effects. Applications on simulated data
show the relevance of the approach for designing new classes of covariance
kernels dedicated to high-dimensional kriging.

本文研究高维函数的表征和全局敏感性分析的 FANOVA 分解，阐述了一种基于高斯随机场样本路径 KANOVA 的核分解方法，以及相关联的张量积投影和协方差核的设计。