  Historically, the machine learning community has derived spectral
decompositions from graph-based approaches. We break with this approach and
prove the statistical and computational superiority of the Galerkin method,
which consists in restricting the study to a small set of test functions. In
particular, we introduce implementation tricks to deal with differential
operators in large dimensions with structured kernels. Finally, we extend on
the core principles beyond our approach to apply them to non-linear spaces of
functions, such as the ones parameterized by deep neural networks, through
loss-based optimization procedures.
