This repository is meant as a collection of notebooks, introducing diverse concepts from category theory and algebraic topology using (and used by) the topos library.
The focus remains on introducing applications and showing how to build example models as promptly as possible, and this handbook might not substitute for precise texts on the subjects. We therefore tried to give appropriate references when relevant.
The concepts of categories and functors,
originating from algebraic topology (Maclane Eilenberg, 1945),
have since become ubiquitous in computer science and mathematics.
We begin with a quick introduction on the subject in (I).
The simplest kind of structure topos implements is that of
traditionnal 1-graphs, which we may
also call 1-dimensional simplicial complexes.
We show in (II) how graph convolutional networks (GCNs)
can be designed using the natural harmonic structure on Complex
instances.
An application of interest to us concerns
message-passing neural networks (MPNNs)
designed for us chemical materials and molecules.
We show in (III) how one can enforce invariance and equivariance
properties of such networks
using natural transformations of
Finally, we describe general statistical systems as they occur in thermodynamics. The laws of Boltzmann, although a couple centuries old, describe a great number of (almost) synonymous models used in machine learning and statistics: Boltzmann machines, graphical models, belief networks, energy based models, Markov random fiels...
The generalized belief propagation algorithm (GBP) introduced in (Yedidia Freeman Weiss, 2005), is a message-passing algorithm providing a powerful tool to estimate the statistics of such models and compute local approximations of global functionals. We introduce GBP and its variants in (IV).
- I-Sheaves and Functors
- II-Graph Convolutional Networks
- III-Equivariant Message-Passing for Materials and Molecules
- IV-Graphical Models, Boltzmann Machines and Belief Networks
- Maclane Eilenberg, 1945, General Theory of Natural Equivalences, Transactions of the AMS
- Yedidia Freeman Weiss, 2005, Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms, IEEE Transactions in Information Theory, 51 (7)