Related papers: Density of critical points for a Gaussian random f…

We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate…

This paper focuses on stochastic saddle point problems with decision-dependent distributions. These are problems whose objective is the expected value of a stochastic payoff function and whose data distribution drifts in response to…

Let $f$ be a Gaussian random field on $\mathbb{R}^d$ and let $X$ be the number of critical points of $f$ contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on $f$, the…

We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three…

In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function…

Gaussian processes are the leading class of distributions on random functions, but they suffer from well known issues including difficulty scaling and inflexibility with respect to certain shape constraints (such as nonnegativity). Here we…

The definition and the properties of a Gaussian point distribution, in contrast to the well-known properties of a Gaussian random field are discussed. Constraints for the number density and the two-point correlation function arise. A simple…

We consider the conditional galaxy density around each galaxy, and study its fluctuations in the newest samples of the Sloan Digital Sky Survey Data Release 7. Over a large range of scales, both the average conditional density and its…

Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point.…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

In this note, we will get the estimate of the expected distribution of critical values of Gaussian SU(2) random polynomials as the degree large enough. The result is a direct application of the Kac-Rice formula. The critical values will…

We revisit the landscape of the simple matrix factorization problem. For low-rank matrix factorization, prior work has shown that there exist infinitely many critical points all of which are either global minima or strict saddles. At a…

Using the dynamical system approach, we describe the general dynamics of cosmological scalar fields in terms of critical points and heteroclinic lines. It is found that critical points describe the initial and final states of the scalar…

For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach…

We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail…

Interacting physical systems in the neighborhood of criticality (and massive continuum field theories) can often be characterized by just two physical scales: a (macroscopic) correlation length and a (microscopic) interaction range, related…

This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study…

The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We…

In this paper we show that the closure of a random sample for a k-dimensional random vector is almost surely a deterministic set of all heavy points of the distribution. A heavy point is defined to be a point for which all its neighborhoods…

Spectral properties of Gram matrices are central to high dimensional asymptotic analyses of statistical estimators in regression and covariance estimation. These properties, in turn, depend critically on the extreme singular values and…