Persistent homology (PH) is one of the most popular methods in Topological Data Analysis.
Even though PH has been used in many different types of applications, the reasons behind …

The network reconstruction task aims to estimate a complex system's structure from various
data sources such as time series, snapshots, or interaction counts. Recent work has …

For a closed Riemannian manifold M and a metric space S with a small Gromov–Hausdorff
distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale β> 0 …

Accurately and efficiently characterizing the decision boundary of classifiers is important for
problems related to model selection and meta-learning. Inspired by topological data …

In this paper we introduce a pruning of the medial axis called the (λ, α)-medial axis (axλα).
We prove that the (λ, α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under …

The convexity of a set can be generalized to the two weaker notions of positive reach and r-
convexity; both describe the regularity of a set's boundary. For any compact subset of R d …

This thesis presents a uniform treatment of different distances used in the applied topology
literature. We introduce the notion of a locally persistent category, which is a category with a …

In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger
on the learning of the homotopy type from a sample of an underlying space. In their work …

We introduce numerical algebraic geometry methods for computing lower bounds on the
reach, local feature size, and weak feature size of the real part of an equidimensional and …

We address the problem of estimating topological features from data in high dimensional
Euclidean spaces under the manifold assumption. Our approach is based on the …