  We investigate adversarial-sample generation methods from a frequency domain
perspective and extend standard $l_{\infty}$ Projected Gradient Descent (PGD)
to the frequency domain. The resulting method, which we call Spectral Projected
Gradient Descent (SPGD), has better success rate compared to PGD during early
steps of the method. Adversarially training models using SPGD achieves greater
adversarial accuracy compared to PGD when holding the number of attack steps
constant. The use of SPGD can, therefore, reduce the overhead of adversarial
training when utilizing adversarial generation with a smaller number of steps.
However, we also prove that SPGD is equivalent to a variant of the PGD
ordinarily used for the $l_{\infty}$ threat model. This PGD variant omits the
sign function which is ordinarily applied to the gradient. SPGD can, therefore,
be performed without explicitly transforming into the frequency domain.
Finally, we visualize the perturbations SPGD generates and find they use both
high and low-frequency components, which suggests that removing either
high-frequency components or low-frequency components is not an effective
defense.
