Tensor learning with orthogonal, Lorentz, and symplectic symmetries

TL;DR

This paper develops machine learning architectures that leverage orthogonal, Lorentz, and symplectic symmetries in tensor functions, improving performance in scientific applications like material science and time series analysis.

Contribution

It introduces universally expressive equivariant models for tensors exploiting specific group symmetries, with applications across multiple scientific domains.

Findings

Equivariant models outperform non-equivariant baselines.

Models effectively handle symmetries in tensor functions.

Applications include material science, computer science, and time series analysis.

Abstract

Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.