The inverse problem for kernel means
Abstract: We discuss the inverse problem for the kernel embedding of measures. We identify which elements of a reproducing kernel Hilbert space which are in the cone generated by some set of kernel functions as polar dual of the Herglotz-type functions, the functions with positive imaginary part. Over certain spaces, such as Sobelev spaces, the duality to Herglotz functions reduces to a classical multivariate moment problem, and, over analytic spaces, we see more complex analytic type conditions. We give conditions for when Herglotz functions have representations in terms of kernel functions in terms of reflexive reproducing kernel Hilbert spaces. We identify the orbits of a dynamical system in terms of the Koopmanism philosophy: we give a way to decide when there is an orbit contained in some compact subset of the domain.
Submission Length: Regular submission (no more than 12 pages of main content)
Assigned Action Editor: ~Brian_Kulis1
Submission Number: 1221
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