Equivariant neural networks for recovery of Hadamard matrices

TL;DR

This paper introduces an equivariant neural network architecture tailored for recovering missing entries in Hadamard matrices, demonstrating advantages over traditional models and highlighting its potential for mathematical insights.

Contribution

The paper presents a novel equivariant message passing neural network architecture specifically designed for Hadamard matrix recovery, advancing geometric deep learning applications in mathematics.

Findings

Outperforms MLPs, CNNs, and Transformers in matrix recovery tasks

Demonstrates the effectiveness of equivariant architectures for combinatorial optimization

Provides a new approach to exploring the Hadamard conjecture using machine learning

Abstract

We propose 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.