Computing geometric feature sizes for algebraic manifolds

TL;DR

This paper develops numerical algebraic geometry methods to compute and estimate geometric feature sizes like reach and local feature size of algebraic varieties, with applications to homology inference.

Contribution

It introduces algorithms for computing lower bounds and exact values of feature sizes directly from defining polynomials, including for special cases like non-quadratic complete intersections.

Findings

Algorithms successfully compute lower bounds on feature sizes.

Homology inference experiments demonstrate practical effectiveness.

Methods handle both dense and adaptively dense sampling scenarios.

Abstract

We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that non-quadratic complete intersections generically have finitely many geometric bottlenecks, and describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds. We also present homology inference experiments that combine persistent homology computations with implemented versions of our feature size algorithms, both with globally dense samples and samples that are adaptively dense with respect to the local feature size.