Equivariant neural networks for recovery of Hadamard matrices

Geometric deep learning is an emerging area of research in machine learning focusing on exploiting symmetries in problems to improve models. Its goal is to understand how transformations to the input should affect the output and design neural networks around the corresponding inductive bias. We present a message passing neural network architecture designed to be equivariant to column and row permutations of a matrix. We illustrate its advantages over traditional architectures like multi-layer perceptrons (MLPs), convolutional neural networks (CNNs) and even Transformers, on the combinatorial optimization task of recovering a set of deleted entries of a Hadamard matrix. We argue that this is a powerful application of the principles of Geometric Deep Learning to fundamental mathematics, and a potential stepping stone toward more insights on the Hadamard conjecture using Machine Learning techniques.