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.

本文介绍了一种新的自适应算法，可以在给定定义多项式集的情况下，在实数代数多项式上找到证明的点样本。该算法利用了数值代数几何的方法，可以形式化保证采样的密度，并利用几何启发式方法减少采样的大小。结果表明，该算法可以使得使用 TDA 方法更加可行。