Injective and coarse embeddings of persistence diagrams and Wasserstein space

In this dissertation we will examine questions related to two fields of mathematics, topological data analysis (TDA) and optimal transport (OT). Both of these fields center on complex data types to which one often needs to apply standard machine learning or statistical methods. Such application will typically mandate that these data types are embedded into a vector space. It has been shown that for many natural metrics such embeddings necessarily have high distortion, i.e. are not even coarse embeddings. Whether coarse embeddings exist with respect to the p-Wasserstein distance for 1 = p = 2 remains an open question, however, both for persistence diagrams (from TDA) and planar distributions (from OT). In this first part of this dissertation, we use coarse geometric techniques to show that the TDA and OT sides of this open question are equivalent for p > 1. In the second, we study an embedding of persistence diagrams, and show that under mild conditions it is injective, i.e. distinguishes between distinct diagrams

Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two parametric algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a data set of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, statistical inference, and geometric morphometrics.Comment: 99 page

Metric Geometry of Spaces of Persistence Diagrams

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $\mathcal D_p$, $1\leq p \leq\infty$, that assign, to each metric pair $(X,A)$, a pointed metric space $\mathcal D_p(X,A)$. Moreover, we show that $\mathcal D_{\infty}$ is sequentially continuous with respect to the Gromov--Hausdorff convergence of metric pairs, and we prove that $\mathcal D_p$ preserves several useful metric properties, such as completeness and separability, for $p \in [1,\infty)$, and geodesicity and non-negative curvature in the sense of Alexandrov, for $p=2$. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $\mathcal D_2(\mathbb R^{2n},\Delta_n)$, has infinite covering, Hausdorff, and asymptotic dimensions.Comment: Improved version; minor errors corrected; 30 page