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Deep neural network classifiers partition input space into high confidenceregions for each class. The geometry of these class manifolds (CMs) is widelystudied and intimately related to model performance; for example, the margindepends on CM boundaries. We exploit the notions of Gaussian width and Gordon'sescape theorem to tractably estimate the effective dimension of CMs and theirboundaries through tomographic intersections with random affine subspaces ofvarying dimension. We show several connections between the dimension of CMs,generalization, and robustness. In particular we investigate how CM dimensiondepends on 1) the dataset, 2) architecture (including ResNet, WideResNet \&Vision Transformer), 3) initialization, 4) stage of training, 5) class, 6)network width, 7) ensemble size, 8) label randomization, 9) training set size,and 10) robustness to data corruption. Together a picture emerges that higherperforming and more robust models have higher dimensional CMs. Moreover, weoffer a new perspective on ensembling via intersections of CMs. Our code is athttps://github.com/stanislavfort/slice-dice-optimize/

What does a deep neural network confidently perceive? The effective dimension of high certainty class manifolds and their low confidence boundaries