Sampling real algebraic varieties for topological data analysis
For researchers applying TDA to algebraic varieties, this algorithm reduces computational burden while maintaining formal guarantees on sample density.
The paper presents a new adaptive algorithm for generating provably dense samples of points on real algebraic varieties, minimizing sample size to make topological data analysis more computationally feasible.
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.