Sampling real algebraic varieties for topological data analysis

TL;DR

This paper introduces an adaptive algorithm for sampling points densely on real algebraic varieties, improving the efficiency of topological data analysis by reducing computational costs.

Contribution

The paper presents a novel sampling algorithm that guarantees density on algebraic varieties and integrates geometric heuristics to minimize sample size, enhancing TDA applications.

Findings

Algorithm provides provably dense samples on algebraic varieties.

Software implementation demonstrates practical effectiveness.

Sampling reduces computational resources needed for TDA.

Abstract

Topological data analysis (TDA) provides a growing body of tools for computing geometric and topological information about spaces from a finite sample of points. We present a new adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to reduce the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We provide a software package that implements the algorithm and also demonstrate the implementation with several examples.