" 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 …

We introduce giotto-tda, a Python library that integrates high-performance topological data
analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ …

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode
topological properties of data and have already proved pivotal in many different applications …

Many and varied methods currently exist for featurization, which is the process of map**
persistence diagrams to Euclidean space, with the goal of maximally preserving structure …

In topological data analysis and visualization, topological descriptors such as persistence
diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an …

Recently a new feature representation and data analysis methodology based on a
topological tool called persistent homology (and its persistence diagram summary) has …

Persistent homology (PH) provides topological descriptors for geometric data, such as
weighted graphs, which are interpretable, stable to perturbations, and invariant under, eg …

Graph neural networks (GNNs) have emerged as a powerful tool for graph classification and
representation learning. However, GNNs tend to suffer from over-smoothing problems and …

Embedding data in hyperbolic spaces has proven beneficial for many advanced machine
learning applications such as image classification and word embeddings. However, working …

Optimal transport (\OT) theory defines a powerful set of tools to compare probability
distributions.\OT~ suffers however from a few drawbacks, computational and statistical …