Title: Proper Velocity Neural Networks
Primary area: learning on graphs and other geometries & topologies
Keywords: Hyperbolic geometry, Geometric deep learning, Manifold learning, Proper velocity, Representation learning, Riemannian geometry

Abstract: Hyperbolic neural networks (HNNs) have shown remarkable success in representing hierarchical and scale-free structures, yet most existing work relies on the Poincaré ball and hyperboloid models. While these models admit closed-form Riemannian operators, their constrained nature potentially leads to numerical instabilities, especially near model boundaries. In this work, we explore the Proper Velocity (PV) manifold, an unconstrained representation of hyperbolic space rooted in Einstein’s special relativity, as a stable alternative. We first establish the complete Riemannian toolkit of the PV space. Building on this foundation, we introduce Proper Velocity Neural Networks (PVNNs) with core layers including Multinomial Logistic Regression (MLR), Fully Connected (FC), convolutional, activation, and batch normalization layers. Extensive experiments across four domains, namely numerical stability, graph benchmarks, image classification, and genomic sequence prediction, demonstrate that PVNNs offer both improved stability and competitive accuracy compared with Poincaré and hyperboloid counterparts.

TL;DR: We establish the Riemannian toolkit of the Proper Velocity manifold, introduce core layers, and build PV neural networks as stable and competitive alternatives to Poincaré and Lorentz networks.