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

Jean-Marc Azaïs; Federico Dalmao; Yohann De Castro

math.STJun 26, 2024

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

Authors: Jean-Marc Azaïs, Federico Dalmao, Yohann De Castro

Organizations: Institut de Mathématiques de Toulouse Université Paul Sabatier, France · DMEL, CENUR Litoral Norte Universidad de la República, Salto, Uruguay · Institut Camille Jordan, CNRS UMR 5208 École Centrale Lyon, France · Institut Universitaire de France (IUF)

Abstract

In this article, we introduce the novel concept of the second maximum of a Gaussian random field defined on a Riemannian manifold: Once the field reaches its global maximum at a point $t_1$ , the second maximum is the largest value of the field away from $t_1$ , after the value and the gradient of the field at $t_1$ have been removed. This statistic serves as a powerful tool for characterizing the distribution of the global maximum. By employing an ad-hoc Kac-Rice formula, we derive the explicit distribution of the maximum, conditioned on the second maximum and the independent part of the Riemannian Hessian (the part of the Hessian at $t_1$ that is independent of the value and the gradient of the field at $t_1$ ). This approach yields an exact test based on the spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test for detecting sparse alternatives in Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Theoretical results are supported by numerical experiments that illustrate the calibration and power of the proposed procedures. More generally, we provide a framework for applying the spacing test to continuous sparse regression for any Gaussian random field on a Riemannian manifold. Furthermore, when the variance-covariance function is known only up to a scaling factor, we derive an exact Studentized version of the procedure, coined the t-spacing test. We show that this test is perfectly calibrated under the null hypothesis and exhibits high power for detecting sparse alternatives. As a corollary, we derive an exact, non-asymptotic test for the spiked tensor model without requiring prior knowledge of the noise level.

Figures & tables

Panel of Figure 1 — Figure 1: [Spiked tensor PCA, Section 5.2 , example 1/5] Visualisation of the random fields on the sphere $\mathds{S}^{2}$ for the Spiked Tensor PCA problem. Left: The Gaussian homogeneous polynomial $X(\cdot)$ , where the arrow indicates the global maximizer $t_{1}$ (first eigenvector). Middle: The conditional random field $X^{|t_{1}}(\cdot)$ defined in ( 4 ), where the arrow indicates the second maximizer $t_{2}$ . Right: A volumetric view of $X^{|t_{1}}(\cdot)$ where the radial height represents the field value.

Panel of Figure 2 — Figure 2: [Spiked tensor PCA, Section 5.2 , example 2/5] The CDFs of the $p$ -value of ( $t$ -)spacing tests over $250,000$ Monte-Carlo samples for each value of $\gamma$ . Note that these tests are perfectly calibrated under the null hypothesis, for which $\gamma=0$ . The parameter $\gamma$ is a scaling factor of eigenvalue of the rank-one tensor to be detected. The value $\gamma=1$ corresponds to the so-called phase transition in Spiked tensor PCA as presented in Perry et al., (2020, Theorem 1.3) . In the $t$ -spacing test, the variance $\sigma$ has been estimated on $X^{|t_{1}}(\cdot)$ .

Panel of Figure 3 — Figure 3: [Spiked tensor PCA, Section 5.2 , example 3/5] The PDFs of the maxima $\lambda_{1},\lambda_{2}$ and the CDF of the distance $\mathrm{d}(t_{0},t_{1})$ over $250,000$ Monte-Carlo samples for each value of the parameter $\gamma$ . The alternative is given by $t_{0}$ fixed and $\lambda_{0}=\gamma\times\sigma\sqrt{3\log 3+3\log\log 3}\simeq 0.684\gamma\sigma$ where $\gamma=1$ corresponds to the so-called phase transition in Spiked tensor PCA as presented in Perry et al., (2020, Theorem 1.3) . The distance $\mathrm{d}(t_{0},t_{1})$ is normalized so that it is uniformly distributed on $(0,1)$ if $t_{1}$ is uniformly distributed on the sphere (e.g., $\gamma=0$ ).

Figure 4: [Spiked tensor PCA, Section 5.2 , example 4/5] The variance estimator is not distributed according to $\chi^{2}$ -distribution with $\kappa=7$ degrees of freedom, it underestimates the variance. The dashed black line is the distribution of a $\chi(7)/\sqrt{7}$ , and $\sigma=1$ in these experiments. The probability distribution function is estimated over $250,000$ Monte-Carlo samples for each value of $\gamma$ .

Application	Manifold $\boldsymbol{M}$	Dim. $\boldsymbol{d}$	Section
Spiked Tensor PCA (Rank-1)	Sphere $\mathds{S}^{2}$	$2$	5.2
Two-Spiked Tensor Model	$\mathds{S}^{1}\times\mathcal{M}_{n,2}$ (Stiefel)	$2n-2$	5.3
Super-Resolution	Torus $\mathbb{T}^{2}=[0,2\pi)^{2}$	$2$	A.1
2-Layer Neural Networks	$\mathds{S}^{r-1}\times(\mathds{S}^{n-1})^{r}$	$nr-1$	A.2

Table 1: Summary of applications considered. We specify the underlying manifold

M

and its dimension

d

Panel of Figure 5 — Figure 5: [Spiked tensor PCA, Section 5.2 , example 5/5] Behaviour of the spacing test when the two maxima are close. Left: $p$ -value against the observed spacing $\lambda_{1}-\lambda_{2}$ (subsample of $3{,}000$ draws per value of $\gamma$ from the experiments of Figure 2 ). The $p$ -value goes to $1$ as the spacing goes to $0$ ; no rejection at level $5\%$ (dashed line) occurs below a spacing of $0.6$ , and $97\%$ of the rejections have a spacing larger than $1.5$ . Middle: $p$ -value as the spacing $\varepsilon=\lambda_{1}-\lambda_{2}$ goes to $0$ at fixed $(\lambda_{2},\Omega)$ ; the direct evaluation (circles) and the log-scale evaluation ( 26 ) (lines) agree to about $10^{-14}$ . Right: $p$ -value as $\lambda_{1}$ grows with the spacing fixed at $1$ ; the direct double-precision evaluation returns NaN beyond $\lambda_{1}\approx 39$ (shaded area) while the log-scale evaluation ( 26 ) follows the exact value.

Figure 6: [Super-Resolution, Section A.1 ] The super-resolution random field

Z(\cdot)

depicts the observation of a point source at location

x_{0}

on the interval

[0,2\pi)

(spatial domain on the

x

-axis) with phase

\theta_{0}

given on the

y

-axis. The

z

-axis corresponds to value of the profile likelihood

Z(x,\theta)

at point

x

with phase

\theta

. This figure is presented in Azaïs et al., (2020, Fig. 2) .

Panel of Figure 7 — Figure 7: a two-layer neural network with $c_{k}$ and $v_{i,j}$ real numbers.

General notation
$[a]$	Set of integers $\{1,\ldots,a\}$ ;
$A^{\top}$ $(\text{resp. }A_{i:})$	transpose of a matrix $A$ (resp. $i$ -th line);
$\mathrm{Id}$	Identity matrix of dimension $d\times d$ ;
$\mathcal{S}_{d}$	Space of $d\times d$ symmetric matrices;
$\mathds{S}^{d-1}$	Euclidean sphere of $\mathds{R}^{d}$ ;
$t$ (and $s$ )	Generic value for a vector on the manifold $M$ ;

Table 2: List of notation: Riemannian gradients and Hessians are defined using the Levi-Civita connection. Riemannian Hessians are represented in matrix form.

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