  We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS)
embeddings of conditional distributions and vector-valued regressors. This
connection introduces a natural regularized loss function which the RKHS
embeddings minimise, providing an intuitive understanding of the embeddings and
a justification for their use. Furthermore, the equivalence allows the
application of vector-valued regression methods and results to the problem of
learning conditional distributions. Using this link we derive a sparse version
of the embedding by considering alternative formulations. Further, by applying
convergence results for vector-valued regression to the embedding problem we
derive minimax convergence rates which are O(\log(n)/n) -- compared to current
state of the art rates of O(n^{-1/4}) -- and are valid under milder and more
intuitive assumptions. These minimax upper rates coincide with lower rates up
to a logarithmic factor, showing that the embedding method achieves nearly
optimal rates. We study our sparse embedding algorithm in a reinforcement
learning task where the algorithm shows significant improvement in sparsity
over an incomplete Cholesky decomposition.
