A two session lecture on December 5, 2025 (10:00–10:50 and 11:10-12:00) at P3.10@Técnico and Online with
António Leitão, Scuola Normale Superiore di Pisa
Topological Expressive Power of Neural Networks
How many different problems can a neural network solve? What makes two machine learning problems different? In this talk, we'll show how Topological Data Analysis (TDA) can be used to partition classification problems into equivalence classes, and how the complexity of decision boundaries can be quantified using persistent homology. Then we will look at a network's learning process from a manifold disentanglement perspective. We'll demonstrate why analyzing decision boundaries from a topological standpoint provides clearer insights than previous approaches. We use the topology of the decision boundaries realized by a neural network as a measure of a neural network's expressive power. We show how such a measure of expressive power depends on the properties of the neural networks' architectures, like depth, width and other related quantities.
Mathematics for Artificial Intelligence is a series of seminars and lectures aimed primarily at mathematicians willing to contribute to mathematical challenges in the area of Artificial Intelligence.
