  It has been shown that the maximum stable set problem in some infinite
graphs, and the kissing number problem in particular, reduces to a minimization
problem over the cone of copositive kernels. Optimizing over this infinite
dimensional cone is not tractable, and approximations of this cone have been
hardly considered in literature. We propose two convergent hierarchies of
subsets of copositive kernels, in terms of non-negative and positive definite
kernels. We use these hierarchies and representation theorems for invariant
positive definite kernels on the sphere to construct new SDP-based bounds on
the kissing number. This results in fast-to-compute upper bounds on the kissing
number that lie between the currently existing LP and SDP bounds.
