  This paper discusses the theory and application of learning Boolean functions
that are concentrated in the Fourier domain. We first estimate the VC dimension
of this function class in order to establish a small sample complexity of
learning in this case. Next, we propose a computationally efficient method of
empirical risk minimization, and we apply this method to the MNIST database of
handwritten digits. These results demonstrate the effectiveness of our model
for modern classification tasks. We conclude with a list of open problems for
future investigation.
