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.
By Elizabeth Collins-Woodfin, Inbar Seroussi
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...
By Zeyong Zhang, Tung Sum Thomas Kwok, Tengfei Ma, Mengjia Xu
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.
By Shirin Panahi, Amirhossein Nazerian, Ali Pezeshki
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.
By Yun Young Choi, Asung Kil, Sun Woo Park, Minho Lee, Seokhwan Kim
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