Universal Collection of Euclidean Invariants between Pairs of Position-Orientations
This provides a foundational improvement for researchers and practitioners in fields like molecular dynamics prediction by enabling more accurate equivariant neural networks.
The paper tackles the problem of designing Euclidean invariant kernels for equivariant neural networks by rigorously deriving an optimal collection of 4 smooth scalar invariants on position-orientation space, and experiments show that using this universal collection significantly improves the accuracy of the PONITA architecture.
Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties. To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3). In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels. We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3). With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them. We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture. Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly.