  Many supervised learning problems involve high-dimensional data such as
images, text, or graphs. In order to make efficient use of data, it is often
useful to leverage certain geometric priors in the problem at hand, such as
invariance to translations, permutation subgroups, or stability to small
deformations. We study the sample complexity of learning problems where the
target function presents such invariance and stability properties, by
considering spherical harmonic decompositions of such functions on the sphere.
We provide non-parametric rates of convergence for kernel methods, and show
improvements in sample complexity by a factor equal to the size of the group
when using an invariant kernel over the group, compared to the corresponding
non-invariant kernel. These improvements are valid when the sample size is
large enough, with an asymptotic behavior that depends on spectral properties
of the group. Finally, these gains are extended beyond invariance groups to
also cover geometric stability to small deformations, modeled here as subsets
(not necessarily subgroups) of permutations.
