Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX 201

Hypothesis testing near singularities and boundaries

The likelihood ratio statistic, with its asymptotic $\chi^2$ distribution at regular model points, is often used for hypothesis testing. At model singularities and boundaries, however, the asymptotic distribution may not be $\chi^2$, as highlighted by recent work of Drton. Indeed, poor behavior of a $\chi^2$ for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood ratio statistic. For two example trinomial models, arising in the context of inference of evolutionary trees, we show the new distributions outperform a $\chi^2$.Comment: 32 pages, 12 figure

The Medusa of Spatial Sorting: Topological Construction

We consider the simultaneous movement of finitely many colored points in space, calling it a spatial sorting process. The name suggests a purpose that drives the collection to a configuration of increased or decreased order. Mapping such a process to a subset of space-time, we use persistent homology measurements of the time function to characterize the process topologically

Auto-completion of contours in sketches, maps and sparse 2D images based on topological persistence.

We design a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane. The only input can be a scanned map with almost closed curves, a hand-drawn artistic sketch or any sparse dotted image in 2D without any extra parameters. The output is a hierarchy of closed contours that have a long enough life span (persistence) in a sequence of nested neighborhoods of the input points. We prove theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contours in the unknown graph

ON INTERVAL UNCERTAINTIES OF CARDINAL NUMBERS OF SUBSETS OF FINITE SPACES WITH TOPOLOGIES WEAKER THAN T 1

In the work using interval mathematics, we develop knowledge for cardinal numbers from the viewpoint of uncertainty analysis. In the finite non-T 1 topological spaces, the inclusion-exclusion formula provide interval estimations for the closure and interior of given sets. This paper introduces a novel approach that combines combinatorial and point-set topology, which leads to a number of results. Among these is the cardinality estimation for the intersection of two open sets that cover a hyperconnected topo-logical space