We show in this article that there exists no H-fractional Brownian field indexed by the cylinder $1×]0,ε[ endowed with its product distance d for any ε > 0 and H > 0. This is equivalent to say that d2H is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels-or valid covariances - on the cylinder. We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle. As a consequence of our result, we show that the set of H such that d2H is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces. These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modeling and kernel method learning on cylinders.

(2019). Nonexistence of fractional Brownian fields indexed by cylinders [journal article - articolo]. In ELECTRONIC JOURNAL OF PROBABILITY. Retrieved from http://hdl.handle.net/10446/145100

Nonexistence of fractional Brownian fields indexed by cylinders

Venet, Nil
2019-01-01

Abstract

We show in this article that there exists no H-fractional Brownian field indexed by the cylinder $1×]0,ε[ endowed with its product distance d for any ε > 0 and H > 0. This is equivalent to say that d2H is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels-or valid covariances - on the cylinder. We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle. As a consequence of our result, we show that the set of H such that d2H is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces. These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modeling and kernel method learning on cylinders.
articolo
2019
Venet, Nil
(2019). Nonexistence of fractional Brownian fields indexed by cylinders [journal article - articolo]. In ELECTRONIC JOURNAL OF PROBABILITY. Retrieved from http://hdl.handle.net/10446/145100
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10446/145100
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