Two-layer neural networks provably converge almost surely to irreducible representations of finite groups when trained on the group composition task, with the dynamics governed by Riemannian gradient ascent on a representation-theoretic energy functional.
arXiv preprint arXiv:2506.06489 , year =
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Derives exact Frobenius norm imbalance identity for deep nonlinear networks, classifies activations into four classes, and obtains critical-depth escape time law τ★ = Θ(ε^{-(r-2)}) from reduction to scalar ODE on permutation-symmetric submanifold.
Manifold steering along activation geometry induces behavioral trajectories matching the natural manifold of outputs, while linear steering produces off-manifold unnatural behaviors.
Llama-3.1-8B computes sums for cyclic concepts using base-10 addition via task-agnostic Fourier features with periods 2, 5, and 10 rather than modular arithmetic in the concept period.
A mechanics of the learning process is emerging in deep learning theory, characterized by dynamics, coarse statistics, and falsifiable predictions across idealized settings, limits, laws, hyperparameters, and universal behaviors.
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A Theory of Saddle Escape in Deep Nonlinear Networks
Derives exact Frobenius norm imbalance identity for deep nonlinear networks, classifies activations into four classes, and obtains critical-depth escape time law τ★ = Θ(ε^{-(r-2)}) from reduction to scalar ODE on permutation-symmetric submanifold.
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Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior
Manifold steering along activation geometry induces behavioral trajectories matching the natural manifold of outputs, while linear steering produces off-manifold unnatural behaviors.