arXiv Machine Learning
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.
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.
By Guantian Zheng, Haiyang Xu, Tianyu Gao
arXiv:2609.37817v1 Announce Type: new
Abstract: Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori ($\mathbb{T}^K$). Howev...
By Zhongping Ji
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).
By Liangchen Ge
The paper introduces Riemannian–Lorentz Parameter Fusion (RLPF), a method for merging a Vision Transformer and a state‑space model without gradient descent. RLPF aligns parameter groups by semantic role, projects them onto a common coordinate system, lifts selected coordinates to the Lorentz hyperboloid, computes a regularized geodesic barycenter, and decodes the result back into the two branches, with a learned gate combining their logits. The resulting fine‑tuned system achieves 82.37 % on CIFAR‑10, 75.04 % on Oxford‑IIIT Pet, and 78.58 % top‑1 accuracy on ImageNet‑1K, surpassing the best‑parent accuracies of 76.54 %, 71.42 %, and 76.42 % respectively.
By Badri N. Patro, Vijay S. Agneeswaran
arXiv:2609.35436v2 Announce Type: replace
Abstract: Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications....
By Ziheng Chen
arXiv:2607. 19305v1 Announce Type: cross Abstract: Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations.
By Chen Ziheng
arXiv:2608. 02668v1 Announce Type: cross Abstract: Residual connections are the de facto mechanism for training deep neural networks stably.
By Jie Zhang, Cheng-Fang Su, Yi-Jui Huang, Min-Te Sun
arXiv:2607. 19305v2 Announce Type: replace-cross Abstract: Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations.
By Chen Ziheng
arXiv:2609.10305v1 Announce Type: new
Abstract: Language models under one million parameters matter for edge deployment, domain adaptation, and reproducible research, yet a two-layer LSTM or Transfor...
By Fang Li
arXiv:2609.21039v1 Announce Type: new
Abstract: A pervasive structural pattern in modern deep learning is the linear factorization block: a submodule of the form $W = BA$ in which two parameter matri...
By Emanuele Zangrando, Marco Sutti, Francesco Tudisco
arXiv:2607. 17513v1 Announce Type: cross Abstract: Expert domains are trees; the Euclidean transformer is not, diluting parent-child structure exponentially at depth.
By Kwan Soo Shin, In Seok Kang, Munho Lee
While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals.