  Kernel methods are powerful tools in machine learning. Classical kernel
methods are based on positive-definite kernels, which map data spaces into
reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces,
positive-definite kernels are difficult to come by. In this case, we propose
the use of reproducing kernel Krein space (RKKS) based methods, which require
only kernels that admit a positive decomposition. We show that one does not
need to access this decomposition in order to learn in RKKS. We then
investigate the conditions under which a kernel is positively decomposable. We
show that invariant kernels admit a positive decomposition on homogeneous
spaces under tractable regularity assumptions. This makes them much easier to
construct than positive-definite kernels, providing a route for learning with
kernels for non-Euclidean data. By the same token, this provides theoretical
foundations for RKKS-based methods in general.
