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Laplace operators in finite energy and dissipation spaces | | Sergey Bezuglyi
; Palle E.T. Jorgensen
; | | Date: |
22 Mar 2019 | | Abstract: | Recent applications of large network models to machine learning, and to
neural network suggest a need for a systematic study of the general
correspondence, (i) discrete vs (ii) continuous. Even if the starting point is
(i), limit considerations lead to (ii), or, more precisely, to a measure
theoretic framework which we make precise. Our motivation derives from graph
analysis, e.g., studies of (infinite) electrical networks of resistors, but our
focus will be (ii), i.e., the measure theoretic setting. In electrical networks
of resistors, one considers pairs (of typically countably infinite), sets $V$
(vertices), $E$ (edges) a suitable subset of $V imes V$, and prescribed
positive symmetric functions $c$ on $E$ . A conductance function $c$ is defined
on $E$ (edges), or on $V imes V$, but with $E$ as its support. From an
initial triple $(V, E, c)$ , one gets graph-Laplacians, generalized Dirichlet
spaces (also called energy Hilbert spaces), dipoles, relative reproducing
kernel-theory, dissipation spaces, reversible Markov chains, and more.
Our main results include: spectral theory and Green’s functions for measure
theoretic graph-Laplace operators; the theory of reproducing kernel Hilbert
spaces related to Laplace operators; a rigorous analysis of the Laplacian on
Borel equivalence relations; a new decomposition theory; irreducibility
criteria; dynamical systems governed by endomorphisms and measurable fields;
orbit equivalence criteria; and path-space measures and induced dissipation
Hilbert spaces. We consider several applications of our results to other fields
such as machine learning problems, reproducing kernel Hilbert spaces, Gaussian
and determinantal processes, and joinings. | | Source: | arXiv, 1903.9572 | | Services: | Forum | Review | PDF | Favorites | |
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