Dynamics to decision: A mathematical theory of Lyapunov spectra and decision boundaries in deep classifiers

arXiv:2609.39190 · cs.LG, cs.SY, eess.SY, math.DS · Submitted 2026-09-30 · Read on arXiv

cs.LG, cs.SY, eess.SY, math.DS

Submitted: 2026-09-30

Updated: 2026-09-30

Terminology

• Robustness — 23× in this paper · explained in 5 episodes
• Fine-Tuning — 22× in this paper · explained in 5 episodes
• Regularization — 13× in this paper · explained in 3 episodes
• Accuracy — 7× in this paper · explained in 1 episode
• Alignment — 7× in this paper · explained in 1 episode
• Adversarial Robustness — 5× in this paper · explained in 3 episodes
• Adam — 4× in this paper · explained in 2 episodes
• Fisher Information — 4× in this paper · explained in 3 episodes
• Generalization — 4× in this paper · explained in 13 episodes
• Adversarial Attacks — 3× in this paper · explained in 1 episode
• Finite-Time Maximum Lyapunov Exponent — 3× in this paper
• Adversarial Examples — 1× in this paper · explained in 2 episodes
• Adversarial Training — 1× in this paper · explained in 1 episode
• Feature Extraction — 1× in this paper · explained in 2 episodes
• Generalizability — 1× in this paper · explained in 6 episodes
• Interpretability — 1× in this paper · explained in 15 episodes
• Inverse Problem — 1× in this paper · explained in 2 episodes
• Loss Function — 1× in this paper · explained in 2 episodes
• Representation Learning — 1× in this paper · explained in 2 episodes
• Semi-supervised Learning — 1× in this paper · explained in 4 episodes
• Theoretical Guarantees — 1× in this paper · explained in 2 episodes
• Trace — 1× in this paper · explained in 2 episodes

Sources

• Robustness of classifiers: from adversarial to random noise
• Stable Architectures for Deep Neural Networks
• Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation
• Robust Learning with Jacobian Regularization
• SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries
• Exact Spectral Norm Regularization for Neural Networks
• Art of singular vectors and universal adversarial perturbations
• Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness
• Robustness of classifiers to universal perturbations: a geometric perspective
• A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks
• The Limitations of Deep Learning in Adversarial Settings
• Exponential expressivity in deep neural networks through transient chaos
• Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients
• Deep Information Propagation
• Gradient-Regularized Out-of-Distribution Detection
• Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks
• Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach
• Adversarially robust generalization theory via Jacobian regularization for deep neural networks
• Spectral Norm Regularization for Improving the Generalizability of Deep Learning
• How does training shape the Riemannian geometry of neural network representations?

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