arXiv:2608. 02668v1 Announce Type: cross Abstract: Residual connections are the de facto mechanism for training deep neural networks stably.
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hyperbolix is an open‑source library for hyperbolic deep learning in JAX, built on Flax NNX. It provides six manifolds—including Euclidean, Poincaré ball, hyperboloid, κ‑stereographic, mixed‑curvature product, and proper velocity space—through a common interface, and implements a wide range of layer families (linear, convolution, attention, normalization, positional encoding, regression, vector quantization). The library also supplies Riemannian optimizers, wrapped distributions, dimensionality‑reduction techniques, and precision‑tested operations that replace numerically unstable formulas on the hyperboloid, ensuring accurate float32 computations at large distances.
Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direction is equally meaningful, but there is no reason...
arXiv:2608. 10416v1 Announce Type: cross Abstract: We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver).
The paper introduces Hyper^2, a dual‑space consistency framework that applies hyperbolic geometry consistently to both the loss and the encoder in point‑cloud completion tasks. By reusing the same arcosh(1+αd²) function as a positional bias in refinement attention and as the Chamfer loss, Hyper^2 achieves significant Chamfer error reductions—up to 22.9% on ShapeNet‑55 and 37.5% on unseen ShapeNet‑34—while adding only ~1.6% FLOPs. The authors demonstrate that geometric consistency across encoder and loss, rather than either component alone, is key to effective hyperbolic supervision, supported by two model‑agnostic indicators that peak only when both are hyperbolic.
arXiv:2609.27988v1 Announce Type: cross Abstract: Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direc...
The paper introduces the spherical Cauchy distribution as a new hyperspherical posterior for variational autoencoders, avoiding the complications of the von Mises–Fisher and Power Spherical alternatives. By using stereographic projection and a Möbius transformation, the authors obtain exact posterior samples and a closed‑form KL divergence that terminates in a finite polynomial for even dimensions and admits certified truncation for odd dimensions. Empirical results show that the spherical Cauchy yields faster inference and lower reconstruction loss on MNIST and improved negative log‑likelihood on smallNORB compared to existing methods.