  There are a number of hypotheses underlying the existence of adversarial
examples for classification problems. These include the high-dimensionality of
the data, high codimension in the ambient space of the data manifolds of
interest, and that the structure of machine learning models may encourage
classifiers to develop decision boundaries close to data points. This article
proposes a new framework for studying adversarial examples that does not depend
directly on the distance to the decision boundary. Similarly to the smoothed
classifier literature, we define a (natural or adversarial) data point to be
$(\gamma,\sigma)$-stable if the probability of the same classification is at
least $\gamma$ for points sampled in a Gaussian neighborhood of the point with
a given standard deviation $\sigma$. We focus on studying the differences
between persistence metrics along interpolants of natural and adversarial
points. We show that adversarial examples have significantly lower persistence
than natural examples for large neural networks in the context of the MNIST and
ImageNet datasets. We connect this lack of persistence with decision boundary
geometry by measuring angles of interpolants with respect to decision
boundaries. Finally, we connect this approach with robustness by developing a
manifold alignment gradient metric and demonstrating the increase in robustness
that can be achieved when training with the addition of this metric.
