Landscape $k$-complexity of isotropic centered Gaussian fields

TL;DR

The paper analyzes the high-dimensional behavior of the mean number of critical points of index $k$ below level $u$ for isotropic centered Gaussian fields defined on sets $T_d$ in $\mathbb{R}^d$ whose volume grows like a ball with parameter $V$. It identifies three distinct regimes, characterized by two critical volumes $V_c^1$ and $V_c^2$, including a layered regime where the availability of critical points depends on both level and index; in the Bargmann-Fock special case only two regimes remain. The main tool is a combination of the Kac–Rice formula and Large Deviations for the $(k+1)$-th GOE eigenvalue, yielding explicit logarithmic asymptotics $\Theta^{\star}(V)$ and $\Theta^{\star}_k(V,v)$ that govern the exponential growth/decay rates. The paper also provides concrete computations for the Matérn class and the Bargmann-Fock field, illustrating how spectral moments $\lambda_2$, $\lambda_4$ drive regime boundaries and layered structures, with implications for the geometry of high-dimensional Gaussian landscapes.

Abstract

In large dimension, we study the asymptotic behavior of the mean number of critical points with index $k$ below a level $u$ for an isotropic centered Gaussian random field defined on a family of subsets of $\mathbb{R}^d$ depending on $d$. We prove the existence of three regimes depending on the speed of growth of the volume the parameter set. In the first regime the mean number of critical points decreases exponentially with the dimension. For the second regime, there exists a critical level $u_c$ such that the mean number of critical points with index $k$ below a level $u$ with $u>u_c$ increases exponentially with the dimension $d$ independently of the index $k$ and decreases exponentially with $d$ when $u<u_c$. In the third regime, there exists a layered structure depending on the level $u$ considered and on the index $k$ of the critical points. This behavior is similar to the one encountered on the sphere by Auffinger et al. [5]. In the particular case of the Bargmann-Fock field, only two regimes coexist.

Figures (2)

• Figure 1: Function $\Theta^{\star}_k(V,v)$ for $V=3/4$ (left) and $V=3$ (right), ${\bf{r}}(\|s-t\|^2)$ being the Matérn covariance function with $\nu=4$ and $\ell=1$, $k=0$ in black, $k=1$ in green and $k=100$ in red.
• Figure 2: Function ${\tilde{\Theta}}_k(V,v)$ for ${\bf{r}}(\|s-t\|^2)=\exp \left(-5\|s-t\|^2\right)$, $V=0$, $k=0$ in black, $k=1$ in green and $k=100$ in red.

Theorems & Definitions (12)

• Proposition 2.1: schoenberg1938
• Proposition 2.2
• Remark
• Lemma 3.1
• Theorem 3.2
• Remark
• Lemma 3.3
• Remark
• Theorem 3.4
• Remark