Boundary volume tracks generalization for CNNs but not FC nets

measured in 1 paper

- Decision-boundary volume is estimated via Weyl's tube formula (small-neighbourhood volume proportional to boundary volume) using Monte-Carlo sampling with FGSM as a fast distance-to-boundary oracle, in three variants (Bvol, TrainBvol, LAdvBvol). [burfitt-brodzki-dlotko-2026-decision-boundary-volume] - For CNNs, optimal test accuracy consistently coincides with a local minimum of boundary volume (smaller/smoother boundaries track better generalization). [burfitt-brodzki-dlotko-2026-decision-boundary-volume] - For fully-connected networks the volume-generalization relationship is unstable, varying with dataset and even training algorithm, so the link is architecture-dependent. [burfitt-brodzki-dlotko-2026-decision-boundary-volume] - Custom CNN and FC classifiers on MNIST, Fashion-MNIST and CIFAR-10; observational (results are graphs, no correlation statistics). [burfitt-brodzki-dlotko-2026-decision-boundary-volume]

Structure

Decision boundary (as a codimension-1 hypersurface)

Context

Weyl tube formula (small-epsilon tubular-neighborhood volume proportional to hypersurface volume), Monte Carlo estimation via FGSM as a fast distance-to-boundary oracle, three region-specific volume measures (Bvol, TrainBvol, LAdvBvol), architecture-dependent (CNN vs. fully-connected) stability of the volume-generalization relationship, contrast with persistent homology's high-dimensional curse-of-dimensionality failure mode

Confirmed in models

Custom small CNN classifier (MNIST/Fashion-MNIST/CIFAR-10) · Custom CNN/FC image classifiers (Burfitt, Brodzki & Dłotko 2026), Custom fully-connected classifier (MNIST/Fashion-MNIST/CIFAR-10) · Custom CNN/FC image classifiers (Burfitt, Brodzki & Dłotko 2026)

Method

Decision-boundary volume estimation (Weyl tube formula + Monte Carlo)

Papers

Understanding the Geometry of Deep Learning with Decision Boundary Volume — Burfitt, Matthew, Brodzki, Jacek, Dłotko, Pawel2026 · arXiv:2603.14768