In the last decade, there has been increasing interest in topological data analysis, a new
methodology for using geometric structures in data for inference and learning. A central …

Persistence landscapes map persistence diagrams into a function space, which may often
be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map …

This article explores the applications of Topological Data Analysis (TDA) in the finance field,
especially addressing the primordial problem of asset allocation. Firstly, we build a rationale …

Since persistence diagrams do not admit an inner product structure, a map into a Hilbert
space is needed in order to use kernel methods. It is natural to ask if such maps necessarily …

The output of standard persistent homology is represented in two ways, via persistence
barcodes and persistence diagrams. Initially persistent homology was used, as homology is …

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 …

The demands of science and industry for methods for understanding and utilizing large and
complex data sets have been growing very rapidly, driven in part by our ability to collect ever …

We propose a general technique for extracting a larger set of stable information from
persistent homology computations than is currently done. The persistent homology algorithm …

Topological data analysis (TDA) allows one to extract rich information from structured data
(such as graphs or time series) that occurs in modern machine learning problems. This …

We introduce a new notion of equivalence of discrete Morse functions on graphs called
persistence equivalence. Two functions are considered persistence equivalent if and only if …