Related papers: Gaussian likelihood geometry of projective varieti…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

Let $\lambda(n)$ be the Liouville function. We study the distribution of \[ \frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n)) \] over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular…

Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express…

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros…

For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x})…

In this article, we discuss two specific classes of models - Gaussian Mixture Copula models and Mixture of Factor Analyzers - and the advantages of doing inference with gradient descent using automatic differentiation. Gaussian mixture…

Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree…

Let $\phi: X \to \mathbb P^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \leq m \leq n-1$. We study the varieties of minimal rational tangents $\mathcal C_x \subset \mathbb P T_x(X)$ at a general point $x$ of…

This paper primarily concerns the variance estimate of zeros of systems of random holomorphic sections associated with a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X. The probability measures taken…

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued…

We define a Gaussian measure on the space $H^0_J(M, L^N)$ of almost holomorphic sections of powers of an ample line bundle $L$ over a symplectic manifold $(M, \omega)$, and calculate the joint probability densities of sections taking…

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two…

We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given…

We consider a latent space model for random graphs where a node $i$ is associated to a random latent point $X_i$ on the Euclidean unit ball. The probability that an edge exists between two nodes is determined by a ``link'' function, which…

We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically…

We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$ variables attains a…

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the…

This paper studies Gaussian random fields with Mat\'ern covariance functions with smooth parameter $\nu>2$. Two cases of parameter spaces, the Euclidean space and $N$-dimensional sphere, are considered. For such smooth Gaussian fields, we…