1,110 research outputs found
Diffusion Variational Autoencoders
A standard Variational Autoencoder, with a Euclidean latent space, is
structurally incapable of capturing topological properties of certain datasets.
To remove topological obstructions, we introduce Diffusion Variational
Autoencoders with arbitrary manifolds as a latent space. A Diffusion
Variational Autoencoder uses transition kernels of Brownian motion on the
manifold. In particular, it uses properties of the Brownian motion to implement
the reparametrization trick and fast approximations to the KL divergence. We
show that the Diffusion Variational Autoencoder is capable of capturing
topological properties of synthetic datasets. Additionally, we train MNIST on
spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a
natural dataset like MNIST does not have latent variables with a clear-cut
topological structure, training it on a manifold can still highlight
topological and geometrical properties.Comment: 10 pages, 8 figures Added an appendix with derivation of asymptotic
expansion of KL divergence for heat kernel on arbitrary Riemannian manifolds,
and an appendix with new experiments on binarized MNIST. Added a previously
missing factor in the asymptotic expansion of the heat kernel and corrected a
coefficient in asymptotic expansion KL divergence; further minor edit
The Entropy of Lagrange-Finsler Spaces and Ricci Flows
We formulate a statistical analogy of regular Lagrange mechanics and Finsler
geometry derived from Grisha Perelman's functionals generalized for
nonholonomic Ricci flows. There are elaborated explicit constructions when
nonholonomically constrained flows of Riemann metrics result in Finsler like
configurations, and inversely, and geometric mechanics is modelled on Riemann
spaces with preferred nonholonomic frame structure.Comment: latex2e, 20 pages, v3, the variant accepted to Rep. Math. Phy
On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we
extend the concept of W-entropy for gravitational fields in the general
relativity, GR, theory. Such F- and W-functionals were introduced in the Ricci
flow theory of three dimensional, 3-d, Riemannian metrics by G. Perelman,
arXiv: math.DG/0211159. Nonrelativistic 3-d Ricci flows are characterized by
associated statistical thermodynamical values determined by W--entropy.
Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are
considered for models with local thermodynamical equilibrium and separation of
dissipative and non-dissipative processes in relativistic hydrodynamics. The
approach is elaborated in the framework of classical filed theories
(relativistic continuum and hydrodynamic models) without an underlying kinetic
description which will be elaborated in other works. The 3+1 splitting allows
us to provide a general relativistic definition of gravitational entropy in the
Lyapunov-Perelman sense. It increases monotonically as structure forms in the
Universe. We can formulate a thermodynamic description of exact solutions in GR
depending, in general, on all spacetime coordinates. A corresponding 2+2
splitting with nonholonomic deformation of linear connection and frame
structures is necessary for generating in very general form various classes of
exact solutions of the Einstein and general relativistic geometric flow
equations. Finally, we speculate on physical macrostates and microstate
interpretations of the W-entropy in GR, geometric flow theories and possible
connections to string theory (a second unsolved problem also contained in
Perelman's works) in the Polyakov's approach.Comment: latex2e, v4 is an accepted to EPJC substantial extension of a former
letter type paper on 10 pages to a research article on 41 pages; a new author
added, the paper's title and permanent and visiting affiliations were
correspondingly modified; and new results, conclusions and references are
provide
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