In this paper, we utilize hyperspheres and regular n-The orthogonal group O(n) fully encapsulates the symmetry simplexes and propose an approach to learning deep features structure of an nD sphere, including both rotational equivariant under the transformations of nD reflections and reflection symmetries. Integrating these symmetries and rotations, encompassed by the powerful group into a model as an inductive bias is often a crucial requirement of O(n). Namely, we propose O(n)-equivariant neurons for problems in natural sciences and the respective with spherical decision surfaces that generalize to applications, e.g., molecular analysis, protein design and any dimension n, which we call Deep Equivariant assessment, or catalyst design (Rupp et al., 2012; Ramakrishnan Hyperspheres. We demonstrate how to combine them et al., 2014; Townshend et al., 2021; Jing et al., 2021; in a network that directly operates on the basis of the input Lan et al., 2022).

Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of neurons with spherical decision surfaces and operates on point clouds. Such spherical neurons are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Focusing on 3D geometry, we exploit the isometry property of spherical neurons and derive a 3D steerability constraint. After training spherical neurons to classify point clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate the neurons and construct rotation-equivariant spherical filter banks. We then apply the derived constraint to interpolate the filter bank outputs and, thus, obtain a rotation-invariant network. Finally, we use a synthetic point set and real-world 3D skeleton data to verify our theoretical findings. The code is available at https://github.com/pavlo-melnyk/steerable-3d-neurons.