Mathematics for AI: What Algebraic Geometry Brings to the Table?

In this talk, I will give an overview of (neuro)algebraic geometry, an emerging field analogous to algebraic statistics that uses algebraic geometry to study the theory of deep learning. I will focus on feedforward neural networks. The universal approximation theorem tells us that, with sufficiently large hidden width, a shallow neural network can approximate any continuous function on a compact set arbitrarily well. Rather than taking width to infinity, we fix a neural network architecture and ask about its expressivity. To study this question, we associate a geometric object called a neuromanifold to a fixed architecture: the image of its parameter space inside an ambient space of functions. When the activation function is algebraic, this ambient space can be chosen to be finite-dimensional. This allows us to take the Zariski closure of the neuromanifold over the real or complex numbers, thereby associating an algebraic variety to the architecture. I will explain how geometric invariants of this variety, such as dimension, degree, defining equations, and singularities, can reveal structural properties of neural networks and shed light on their black-box behavior. More broadly, the talk illustrates how algebro-geometric tools can contribute to the mathematical foundations of deep learning alongside more familiar statistical and probabilistic methods.