Applied Algebraic Geometry: Polynomial Neural Networks

This project aims to enhance the development and understanding of machine learning and artificial intelligence by employing techniques from applied algebraic geometry, in the context of polynomial neural networks. The analysis of polynomial neural networks has implications both for applications and timely machine learning approaches, including generative modelling, and algebraic techniques are well-adapted for providing global insight into these neural networks. These insights about polynomial neural networks can then be used to make informed a priori design choices and to improve the learning process for a given neural network. Graduate students will participate in this research, enhancing their training at the intersection of mathematics and artificial intelligence. This award deepens recent connections between algebraic geometry and machine learning made by considering neuromanifolds of polynomial neural networks--algebraic spaces consisting of functions representable by a neural network with a fixed architecture and an algebraic activation function. By leveraging classical results in algebraic geometry and number theory, this research determines algebraic invariants of these neuromanifolds such as their dimension, learning degree, and singular locus. These invariants are then understood from the view of machine learning to better understand properties of the original neural network such as expressivity, complexity of the learning process, and limitations on gradient descent based learning algorithms. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.