Abstract

Lecture 1: Persistence homology and the space of natural images. A data set can typically be visualized as a point cloud in a higher dimensional euclidean space, and thought of as a sample obtained from a probability distribution function defined on it. In real world data, the support of these functions typically concentrate along lower dimensional subspaces of Rn whose geometry one would like to understand. Persistence homology constructs scale independent approximations to this underlying space and produces robust geometric invariants. In this lecture, I will review this technique illustrating it with an example constructed using natural images.

Lecture 2: The topology of neural networks. Deep learning has revolutionized aspects of science and technology ranging from image recognition tasks to the prediction of protein folded structures. These systems required a large amount of data to train and their performance, in many cases superhuman, is guided by a weighted network of relations that is hard to interpret. In this lecture, I will de- scribe the basics of neural networks and apply topological methods to visualize and understand geometrically examples of neural networks trained as image classifiers. We will see how the topology of natural images studied in the previous lecture reemerges in this context inside the internal parameters of these systems.

Lecture 3: Computational aspects and software demonstration. In this lecture we will focus on the computational and practical aspects of topological data analysis. We will overview one of the main tools for the incorporation of this ideas into machine learning pipelines, the open source package giotto-tda. This lecture will be presented jointly with the maintainer of giotto-tda Umberto Lupo (EPFL), and will be a live demonstration of how to use the basic functionalities of this software, as well as an overview of its capabilities.

Lecture 4: Symmetry and persistence homology refinements. In this final lecture, I will describe how to see further beyond the signals constructed by persistence homology, using symmetric groups to construct finer topological information on data sets. We will see how to geometrically interpret the signals produced by this persistence Steenrod theory in terms of the self-intersection properties of cycles constructed by traditional persistence homology.