Expressive Sign Equivariant Networks for Spectral Geometric Learning

In the paper titled "Expressive Sign Equivariant Networks for Spectral Geometric Learning," researchers introduce a novel approach to developing neural network architectures that respect the symmetries of eigenvectors associated with the structure of data. In many machine learning applications, such as processing data on manifolds or graphs, it is crucial to handle eigenvectors that are sign and basis symmetric; these vectors can flip sign or change basis without changing their intrinsic meaning. Traditionally, models invariant to such symmetries have been used to improve empirical performances across various tasks.

The primary contribution of this work is to argue that sign equivariance, as opposed to invariance, can be significantly more beneficial for tasks involving the processing of eigenvectors. Sign equivariance ensures that the output of a function changes in a structured way—a sign flip—when the input eigenvector is sign-flipped, maintaining positional information that is vital in certain applications, like link prediction in graphs. The paper proceeds to establish a formal connection between sign equivariance and the ability to learn more expressive representations, particularly for edge representations in graphs and orthogonally equivariant models for point cloud data.

To implement sign equivariant models practically, the authors present a theoretical analysis leading to the design of novel sign-equivariant neural network architectures. They encounter an initial challenge: popular approaches to constructing equivariant networks, which mingle equivariant linear maps with elementwise nonlinearities, fall short in providing adequate expressive power for sign equivariance. Therefore, the authors provide an analytical characterization of sign equivariant polynomial functions and demonstrate how their construction can inspire the development of equivariant neural network architectures that embody the same expressiveness.

Empirical validation confirms the theoretical benefits of sign equivariant models. The researchers conduct numerical experiments on synthetic datasets related to link prediction and node clustering problems, which reinforce the utility of the proposed sign equivariant approach. The networks designed not only approximate sign equivariant polynomials but also exhibit universality properties, suggesting that they are capable of learning any sign equivariant function under certain conditions. Lastly, the authors extend their findings by characterizing and discussing sign invariant polynomials, providing an alternative proof of the universality of SignNet, a previously presented neural network.

In conclusion, this paper advances the field of geometric deep learning by tackling the treatment of eigenvector symmetries with sign equivariant functions, enabling the learning of richer and more informative representations. The presented architectures offer an efficient and powerful tool for geometric learning tasks with significant potential to outperform existing methods that solely rely on sign invariance.