Josué Tonelli-Cueto

Luis Martínez, María Merino, Juan Manuel Montoya et al.

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. In this paper, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository.

Wilson G. Gregory, Josué Tonelli-Cueto, Nicholas F. Marshall et al.

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.