Discrete Gaussian distributions via theta functions

TL;DR

This paper introduces a discrete analogue of the multivariate Gaussian distribution using Riemann theta functions, exploring its properties and connections to algebraic geometry, and highlighting its entropy-maximizing nature.

Contribution

It presents a novel discrete Gaussian distribution parametrized by theta functions and analyzes its statistical properties and geometric connections.

Findings

Discrete Gaussian supported on integer lattice

Characterized by entropy maximization

Connections to abelian varieties in algebraic geometry

Abstract

We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the property of maximizing entropy, just as its continuous counterpart. We capitalize on the theta function representation to derive statistical properties. Throughout, we exhibit strong connections to the study of abelian varieties in algebraic geometry.