Persistent homology, algorithms, and stochastic geometry

The field of topological data analysis immerged some two decades ago from a combination of computational geometry and algebraic topology. It is primarily a subject pursued by mathematicians, but its products are directly applicable and a natural complement to timely work in machine learning, which is a domain within computer science/statistics. This project studies questions that further develop the field of topological data analysis but are also instrumental in forming the bridge to applications, and in particular in machine learning. The main directions are A. the establishment of persistent homology as an important method within stochastic geometry, B. the extension of discrete Morse theoretic ideas beyond the current framework. With the expected results, we will develop a powerful connection between yet separate mathematical disciplines and lead them toward applications outside of mathematics.

The inter-related topics studied under this project are persistent homology, algorithms for computing its various aspects, and stochastic geometry questions aimed at casting light on the difference between topological noise and features in data. Two major insights gained during the course of these studies deserve to be mentioned: - The stochastic analysis of Voronoi paths and scapes of geometric shapes, which can be used to measure shapes that are otherwise difficult to measure. For example, a compact smooth surface in three-dimensional space is shown to have Voronoi scapes (piecewise linear surfaces dual to the intersection of the given surface with the Voronoi tessellations of Poisson point processes) whose expected areas are one-and-a-half times the area of the given surface. Because of the piecewise linear nature, the areas of the Voronoi scapes are easy to compute, and by multiplying with two-over-three we get a stochastic approximation of the area of the surface. - The depth poset of a filtered complex records the dependencies between cancellations aimed at simplifying the complex. Focusing on the cancellation of shallow pairs that preserve the homology groups, this partial order can be computed efficiently by running two particular matrix reduction algorithms, one based on column and the other on row operations. The latter result is motivated by the desire to extend the theory of persistent homology to discrete dynamical systems. This is indeed a vast area of research with far-reaching applications in the sciences and industry.