Klein Neural Networks++

Hyperbolic neural networks (HNNs) have shown significant advantages in representing hierarchical and tree-structured data through their exponential volume growth. However, existing methods rely almost exclusively on the Poincare ball and Lorentz hyperboloid, while the Klein model, another classical representation, has long been unexplored. The Klein model has Euclidean line-segment geodesics and a closed-form Einstein midpoint, where the latter avoids the iterative Frechet mean required for batch normalization. In this work, we systematically explore its modeling potential and build a Klein neural network framework of closed-form operators, namely Klein MLR, Klein Busemann MLR, Klein fully connected layer, Klein Busemann fully connected layer, and Klein GyroBN. Klein demonstrates strong training stability under feature clipping, curvature mis-specification, and gradient scaling, where the Poincare and Lorentz references diverge. The closed-form Einstein midpoint in Klein GyroBN offers a favorable trade-off between computational efficiency and performance against iterative Frechet means. Across numerical stability, image classification, graph link prediction, graph node classification, and EEG decoding, the proposed Klein layers match or improve upon their Poincare, Lorentz, and PV counterparts. Klein MLR heads further achieve the fastest runtime among hyperbolic counterparts.