Random points on an algebraic manifold

TL;DR

This paper introduces a novel, simple method for sampling and integrating over algebraic manifolds by intersecting them with random linear spaces, with applications in physics and data analysis.

Contribution

It presents a new approach for sampling on algebraic manifolds using random linear intersections, handling multiple components and ensuring i.i.d. samples.

Findings

Method produces independent samples efficiently.

Applicable to manifolds with multiple connected components.

Demonstrated in physics and topological data analysis.

Abstract

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic manifolds. This method is based on intersecting with random linear spaces. It produces i.i.d. samples, works in the presence of multiple connected components, and is simple to implement. We present applications to computational statistical physics and topological data analysis.