Dynamics to decision: A mathematical theory of Lyapunov spectra and decision boundaries in deep classifiers
Paper ties finite-time Lyapunov exponents of deep classifiers to decision-boundary geometry.
The paper treats a feed-forward classifier as a finite nonautonomous discrete dynamical system and studies the finite-time maximum Lyapunov exponent of sample trajectories through depth. For Gaussian classes, probability-level FTMLE aligns with the decision-boundary normal; at the logits the relationship depends on the loss, and in hidden layers it persists only under identifiable structural conditions. The authors propose geometry-aware fine-tuning and conditions that concentrate high hidden FTMLE near the boundary, with numerical results supporting the theory.
- Models classifier depth as discrete dynamical time
- Probability-level FTMLE aligns with the boundary normal
- Logit-level link depends on the training loss
- Hidden-layer signature needs structural conditions
- Proposes geometry-aware fine-tuning near the boundary
From the original article
1 min read at arxiv.orgA 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 across layers provides a natural framework for asking how decision geometry emerges through depth and how far back we can trace a boundary's dynamical signature.
By Shirin Panahi · arxiv.org
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