  It is a commonly held belief that enforcing invariance improves
generalisation. Although this approach enjoys widespread popularity, it is only
very recently that a rigorous theoretical demonstration of this benefit has
been established. In this work we build on the function space perspective of
Elesedy and Zaidi arXiv:2102.10333 to derive a strictly non-zero generalisation
benefit of incorporating invariance in kernel ridge regression when the target
is invariant to the action of a compact group. We study invariance enforced by
feature averaging and find that generalisation is governed by a notion of
effective dimension that arises from the interplay between the kernel and the
group. In building towards this result, we find that the action of the group
induces an orthogonal decomposition of both the reproducing kernel Hilbert
space and its kernel, which may be of interest in its own right.
