Semiring Activation in Neural Networks

We introduce a class of trainable nonlinear operators based on semirings that are suitable for use in neural networks. These operators generalize the traditional alternation of linear operators with activation functions in neural networks. Semirings are algebraic structures that describe a generalised notation of linearity, greatly expanding the range of trainable operators that can be included in neural networks. In fact, max- or min-pooling operations are convolutions in the tropical semiring with a fixed kernel. We perform experiments where we replace the activation functions for trainable semiring-based operators to show that these are viable operations to include in fully connected as well as convolutional neural networks (ConvNeXt). We discuss some of the challenges of replacing traditional activation functions with trainable semiring activations and the trade-offs of doing so.

PDE-CNNs: Axiomatic Derivations and Applications

PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) utilize solvers of geometrically meaningful evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs offer several key benefits all at once: fewer parameters, inherent equivariance, better performance, data efficiency, and geometric interpretability. In this article we focus on Euclidean equivariant PDE-G-CNNs where the feature maps are two dimensional throughout. We call this variant of the framework a PDE-CNN. We list several practically desirable axioms and derive from these which PDEs should be used in a PDE-CNN. Here our approach to geometric learning via PDEs is inspired by the axioms of classical linear and morphological scale-space theory, which we generalize by introducing semifield-valued signals. Furthermore, we experimentally confirm for small networks that PDE-CNNs offer fewer parameters, better performance, and data efficiency in comparison to CNNs. We also investigate what effect the use of different semifields has on the performance of the models.

Mathematics of Neural Networks (Lecture Notes Graduate Course)

These are the lecture notes that accompanied the course of the same name that I taught at the Eindhoven University of Technology from 2021 to 2023. The course is intended as an introduction to neural networks for mathematics students at the graduate level and aims to make mathematics students interested in further researching neural networks. It consists of two parts: first a general introduction to deep learning that focuses on introducing the field in a formal mathematical way. The second part provides an introduction to the theory of Lie groups and homogeneous spaces and how it can be applied to design neural networks with desirable geometric equivariances. The lecture notes were made to be as self-contained as possible so as to accessible for any student with a moderate mathematics background. The course also included coding tutorials and assignments in the form of a set of Jupyter notebooks that are publicly available at https://gitlab.com/bsmetsjr/mathematics_of_neural_networks.