  Motivated by applications, we consider here new operator theoretic approaches
to Conditional mean embeddings (CME). Our present results combine a spectral
analysis-based optimization scheme with the use of kernels, stochastic
processes, and constructive learning algorithms. For initially given non-linear
data, we consider optimization-based feature selections. This entails the use
of convex sets of positive definite (p.d.) kernels in a construction of optimal
feature selection via regression algorithms from learning models. Thus, with
initial inputs of training data (for a suitable learning algorithm,) each
choice of p.d. kernel $K$ in turn yields a variety of Hilbert spaces and
realizations of features. A novel idea here is that we shall allow an
optimization over selected sets of kernels $K$ from a convex set $C$ of
positive definite kernels $K$. Hence our \textquotedblleft
optimal\textquotedblright{} choices of feature representations will depend on a
secondary optimization over p.d. kernels $K$ within a specified convex set $C$.
