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 topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

" 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 …

Most graph kernels are an instance of the class of R-Convolution kernels, which measure
the similarity of objects by comparing their substructures. Despite their empirical success …

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 …

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they
are routinely used to describe succinctly complex topological properties of complicated …

Attempts to incorporate topological information in supervised learning tasks have resulted in
the creation of several techniques for vectorizing persistent homology barcodes. In this …