  This work provides a computable, direct, and mathematically rigorous
approximation to the differential geometry of class manifolds for
high-dimensional data, along with nonlinear projections from input space onto
these class manifolds. The tools are applied to the setting of neural network
image classifiers, where we generate novel, on-manifold data samples, and
implement a projected gradient descent algorithm for on-manifold adversarial
training. The susceptibility of neural networks (NNs) to adversarial attack
highlights the brittle nature of NN decision boundaries in input space.
Introducing adversarial examples during training has been shown to reduce the
susceptibility of NNs to adversarial attack; however, it has also been shown to
reduce the accuracy of the classifier if the examples are not valid examples
for that class. Realistic "on-manifold" examples have been previously generated
from class manifolds in the latent of an autoencoder. Our work explores these
phenomena in a geometric and computational setting that is much closer to the
raw, high-dimensional input space than can be provided by VAE or other black
box dimensionality reductions. We employ conformally invariant diffusion maps
(CIDM) to approximate class manifolds in diffusion coordinates, and develop the
Nystr\"{o}m projection to project novel points onto class manifolds in this
setting. On top of the manifold approximation, we leverage the spectral
exterior calculus (SEC) to determine geometric quantities such as tangent
vectors of the manifold. We use these tools to obtain adversarial examples that
reside on a class manifold, yet fool a classifier. These misclassifications
then become explainable in terms of human-understandable manipulations within
the data, by expressing the on-manifold adversary in the semantic basis on the
manifold.
