  This paper provides a general framework to study the effect of sampling
properties of training data on the generalization error of the learned machine
learning (ML) models. Specifically, we propose a new spectral analysis of the
generalization error, expressed in terms of the power spectra of the sampling
pattern and the function involved. The framework is build in the Euclidean
space using Fourier analysis and establishes a connection between some high
dimensional geometric objects and optimal spectral form of different
state-of-the-art sampling patterns. Subsequently, we estimate the expected
error bounds and convergence rate of different state-of-the-art sampling
patterns, as the number of samples and dimensions increase. We make several
observations about generalization error which are valid irrespective of the
approximation scheme (or learning architecture) and training (or optimization)
algorithms. Our result also sheds light on ways to formulate design principles
for constructing optimal sampling methods for particular problems.
