[Paper Review] Distribution of the Height of Local Maxima of Gaussian Random Fields

Dan Cheng, Armin Schwartzman|arXiv (Cornell University)|Jul 22, 2013

Morphological variations and asymmetry16 references4 citations

TL;DR

This paper derives exact and asymptotic formulas for the distribution of the height of local maxima in non-stationary and isotropic Gaussian random fields on Euclidean spaces and Riemannian manifolds. By leveraging local geometry and random matrix theory—specifically the Gaussian orthogonal ensemble—it provides closed-form expressions for conditional probabilities of peak heights, enabling precise statistical inference in peak detection without requiring stationarity or global field properties.

ABSTRACT

Let $\{f(t): t\in T\}$ be a smooth Gaussian random field over a parameter space $T$, where $T$ may be a subset of Euclidean space or, more generally, a Riemannian manifold. For any local maximum of $f(t)$ located at $t_0$ in the interior of $T$, we provide general formulae and asymptotic approximations for both the tail distribution of the height of a local maximum $\mathbb{P}\{f(t_0)>u | t_0 ext{is a local maximum of} f(t) \}$ and the overshoot distribution of a local maximum $\mathbb{P}\{f(t_0)>u+v | t_0 ext{is a local maximum of} f(t) ext{and} f(t_0)>v\}$. Assuming further that $f$ is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when $T$ is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

Motivation & Objective

• To derive a general formula for the conditional distribution of the height of a local maximum in non-stationary Gaussian random fields defined on Riemannian manifolds or Euclidean spaces.
• To establish asymptotic approximations for the overshoot distribution of local maxima when the pre-threshold is high, relying only on local field properties.
• To extend existing results from stationary fields to non-stationary and isotropic fields, including on spheres of arbitrary dimension.
• To provide exact, non-asymptotic closed-form expressions for the height distribution under isotropy by applying tools from random matrix theory, particularly the Gaussian orthogonal ensemble (GOE).

Proposed method

• Uses a limiting definition of conditional probability via neighborhoods around a point to define the height distribution of a local maximum, avoiding zero-probability conditioning events.
• Applies local analysis based on the joint distribution of the field, its gradient, and Hessian at a point to characterize local maxima via index and determinant conditions.
• Employs the Gaussian orthogonal ensemble (GOE) to model the distribution of the Hessian matrix under isotropy, enabling exact computation of expectations involving the determinant of the Hessian.
• Derives key expressions using conditional moments of Gaussian vectors, particularly the distribution of the Hessian given the field value and zero gradient.
• Utilizes random matrix theory to compute expectations of functions of eigenvalues of the Hessian, especially the absolute determinant under index constraints.
• Extends results from Euclidean space to Riemannian manifolds by adapting local geometry through orthonormal frames and chart-based arguments.

Experimental results

Research questions

• RQ1 What is the exact distribution of the height of a local maximum in a non-stationary Gaussian random field on a Riemannian manifold?
• RQ2 How can the overshoot distribution of a local maximum be asymptotically approximated when the field value exceeds a high pre-threshold?
• RQ3 Can the height distribution of local maxima be computed in closed form for isotropic Gaussian fields on spheres or Euclidean spaces using random matrix theory?
• RQ4 Does the limiting overshoot distribution depend on the correlation structure of the field, or is it universal under isotropy?
• RQ5 To what extent can the results be generalized from Euclidean space to general Riemannian manifolds while preserving local dependence?

Key findings

• A general formula is derived for the distribution of the height of a local maximum in non-stationary Gaussian fields on Riemannian manifolds, depending only on local field properties such as the expected number of local maxima in a neighborhood.
• For isotropic Gaussian fields on the sphere or Euclidean space, the height distribution of local maxima is expressed in closed form using the Gaussian orthogonal ensemble (GOE), with explicit dependence on the field’s variance and correlation parameters.
• The overshoot distribution asymptotically approaches a distribution that is independent of the correlation function when the pre-threshold $ v o igvee $, and the error in the approximation is super-exponentially small under isotropy.
• Under isotropy, the conditional distribution of the Hessian matrix given the field value and zero gradient is shown to be equivalent to a shifted GOE matrix, enabling exact computation of the determinant expectation.
• The expected number of local maxima in a small neighborhood is approximated using the Euler characteristic of the excursion set above level $ v $, which enables the asymptotic analysis of the overshoot distribution.
• The results are extended to general Riemannian manifolds by adapting local geometry through orthonormal frames, with proofs relying on chart-based arguments that preserve the structure of the original Euclidean derivations.