Wasserstein distances are metrics on probability distributions inspired by the problem of
optimal mass transportation. Roughly speaking, they measure the minimal effort required to …

Topological Data Analysis (TDA) is a recent and fast growing field providing a set of new
topological and geometric tools to infer relevant features for possibly complex data. This …

" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

Many data sets can be viewed as a noisy sampling of an underlying space, and tools from
topological data analysis can characterize this structure for the purpose of knowledge …

Persistent homology (PH) is a method used in topological data analysis (TDA) to study
qualitative features of data that persist across multiple scales. It is robust to perturbations of …

We define a new topological summary for data that we call the persistence landscape. Since
this summary lies in a vector space, it is easy to combine with tools from statistics and …

A suitable feature representation that can both preserve the data intrinsic information and
reduce data complexity and dimensionality is key to the performance of machine learning …

New representations of tree-structured data objects, using ideas from topological data
analysis, enable improved statistical analyses of a population of brain artery trees. A number …

We develop in this paper a theoretical framework for the topological study of time series
data. Broadly speaking, we describe geometrical and topological properties of sliding …

In this paper we discuss the adaptation of the methods of homology from algebraic topology
to the problem of pattern recognition in point cloud data sets. The method is referred to as …