A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space

TL;DR

This paper constructs non-parametric infinite-dimensional manifolds of finite measures modeled on Sobolev spaces, supporting the Fisher-Rao metric, with applications to nonlinear filtering and Fokker-Planck equations.

Contribution

It introduces a new class of Sobolev space-based manifolds with a nonlinear superposition operator supporting the Fisher-Rao metric, extending the geometric framework of statistical models.

Findings

Supports the Fisher-Rao metric on Sobolev manifolds.

The deformed exponential function acts continuously on mixed-norm Sobolev spaces.

Contains a submanifold of probability measures with applications to SPDEs.

Abstract

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $R^d$. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space $W^{2,1}$; i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined.