  Artificial neural networks can achieve impressive performances, and even
outperform humans in some specific tasks. Nevertheless, unlike biological
brains, the artificial neural networks suffer from tiny perturbations in
sensory input, under various kinds of adversarial attacks. It is therefore
necessary to study the origin of the adversarial vulnerability. Here, we
establish a fundamental relationship between geometry of hidden representations
(manifold perspective) and the generalization capability of the deep networks.
For this purpose, we choose a deep neural network trained by local errors, and
then analyze emergent properties of trained networks through the manifold
dimensionality, manifold smoothness, and the generalization capability. To
explore effects of adversarial examples, we consider independent Gaussian noise
attacks and fast-gradient-sign-method (FGSM) attacks. Our study reveals that a
high generalization accuracy requires a relatively fast power-law decay of the
eigen-spectrum of hidden representations. Under Gaussian attacks, the
relationship between generalization accuracy and power-law exponent is
monotonic, while a non-monotonic behavior is observed for FGSM attacks. Our
empirical study provides a route towards a final mechanistic interpretation of
adversarial vulnerability under adversarial attacks.
