Shape is data and data is shape. Biologists are accustomed to thinking about how the shape
of biomolecules, cells, tissues, and organisms arise from the effects of genetics …

Applications in data science, shape analysis, and object classification frequently require
comparison of probability distributions defined on different ambient spaces. To accomplish …

The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding
the size and prominence of topological features at varying scales, the PD provides important …

Abstract The Euler Curve Transform (ECT) of Turner et al. is a complete invariant of an
embedded simplicial complex, which is amenable to statistical analysis. We generalize the …

We study the inverse problem for persistent homology: For a fixed simplicial complex K, we
analyze the fiber of the continuous map PH on the space of filters that assigns to a filter f: K→ …

Topological data analysis and, more specifically, persistent homology have received
significant attention as a method of describing the shape of complex data. Persistent …

Let K be a finite simplicial, cubical, delta or CW complex. The persistence map PH takes a
filter f: K→ R as input and returns the barcodes of the sublevel set persistent homology of f in …

In this paper, we consider topological featurizations of data defined over simplicial
complexes, like images and labeled graphs, obtained by convolving this data with various …

Applications in data science, shape analysis and object classification frequently require
comparison of probability distributions defined on different ambient spaces. To accomplish …

This thesis is concerned with the following two naive questions about the widely used
Persistent Homology1 (PH) descriptor for data analysis: 1 Persistent Homology [EH08 …