The paper investigates the implicit bias of Riemannian gradient flow in hyperbolic multiclass classification with fixed class prototypes. It introduces a decomposition of distances into radial and direction-dependent terms via the Busemann function, leading to two key results: a radial dichotomy that predicts whether trajectories move toward the boundary or return inward, and convergence of boundary directions to critical points of the Busemann risk. These findings offer a rigorous asymptotic explanation for boundary saturation and near-boundary clustering observed in hyperbolic representation learning.
arXiv:2510. 14074v2 Announce Type: replace-cross Abstract: We develop a framework for analyzing the learning dynamics of high-dimensional problems trained using one-pass stochastic gradient descent (SGD) with data from multiple anisotropic classes.
arXiv:2610.02391v1 Announce Type: new Abstract: When a large language model solves a mathematical problem, its reasoning is largely hierarchical, and the solution often branches at a few tokens where...
The paper presents a mathematical framework that treats a deep classifier as a finite, nonautonomous discrete dynamical system, with each layer acting as a time step. It introduces the Finite-Time Maximum Lyapunov Exponent (FTMLE) to quantify how nearby data trajectories converge or diverge as they propagate through the network, and shows that at the probability level the FTMLE’s dominant direction aligns with the decision boundary normal for Gaussian classes. The authors further analyze how this relationship changes when moving backward to logits and hidden representations, identify conditions under which boundary-related FTMLE persists, and propose geometry-aware fine‑tuning to concentrate high hidden FTMLE near the decision boundary.
A deep classifier is defined not only by the decision it produces, but also by the sequence of transformations through which that decision is formed. Treating this evolution as a dynamical system acro...
The paper introduces the Graph Signature Index to quantify the mismatch between positive-definite geometric priors in many Graph Neural Networks and the negative distances often needed for heterophilic graphs. Using this signature, the authors propose the Pseudo‑Riemannian Attention Network (PRAT), which learns a signed squared pseudo‑distance metric that can capture both attractive and repulsive relationships. Experiments on fifteen node‑classification benchmarks show that PRAT matches or surpasses state‑of‑the‑art heterophily‑adapted models, and ablation studies confirm that the sign structure provided by the signature is essential for its performance.
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.
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.
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.
arXiv:2607. 06497v1 Announce Type: new Abstract: We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths.
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. 09589v1 Announce Type: new Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces.
arXiv:2607. 19335v1 Announce Type: cross Abstract: Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability.