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Information geometry of diffeomorphism groups

Boris Khesin, Gerard Misiołek, Klas Modin

Abstract

The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.

Information geometry of diffeomorphism groups

Abstract

The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.

Paper Structure

This paper contains 52 sections, 40 theorems, 249 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Infinite dimensional manifolds
  3. Differential calculus in Fréchet spaces
  4. Fréchet spaces
  5. The Gateaux derivative and its properties
  6. Manifolds modelled on Fréchet spaces
  7. Geometric tools: connections, curvature, geodesics and metrics
  8. The tame category of Sergeraert and Hamilton
  9. Tame Fréchet spaces
  10. Manifolds of maps
  11. Diffeomorphism groups and their quotients
  12. Fréchet Lie groups
  13. Diffeomorphism groups as Fréchet manifolds and Lie groups
  14. The quotient space of probability densities
  15. Riemannian structures on infinite dimensional Lie groups
  16. ...and 37 more sections

Key Result

Proposition 3.7

Let $f: \mathfrak{X}\, {\supset}\, \mathscr{U} \to \mathfrak{Y}$ be a function of class $\mathcal{C}^r$ between Fréchet spaces and let $\mathscr{U} \subset \mathfrak{X}$ be an open set.

Figures (11)

  • Figure 1: Moser's fibration of diffeomorphisms $\mathfrak{D}(M)$ over smooth probability densities $\mathfrak{Dens}(M)$. The identity fiber $\mathfrak{D}_\mu(M)$ is determined by a reference density $\mu \in \mathfrak{Dens}(M)$. The fiber structure provides an infinite dimensional principal bundle in the tame Fréchet category (cf. Proposition \ref{['prop:PB']}).
  • Figure 2: Euler--Arnold equations related to various Lie groups and metrics.
  • Figure 3: The geometry of Euler-Arnold equations. The vector $v$ in the Lie algebra $\mathfrak{g}$ traces the evolution of the velocity vector of a geodesic $g(t)$ on the group $\mathfrak{G}$. The inertia operator $A$ sends $v$ to an element $m$ in the dual space $\mathfrak{g}^*$.
  • Figure 4: Defining the Lie--Poisson structure: $df_m,\ dg_m\in \mathfrak{g}$, while $m\in \mathfrak{g}^*$.
  • Figure 5: The Riemannian geometry behind $L^2$ optimal mass transport. Shortest geodesics between fibers, with respect to the (non-invariant) $L^2$ metric on $\mathfrak{D}(M)$, project via push-forward to Wasserstein-Otto geodesics on $\mathfrak{Dens}(M)$ which give the $L^2$ Wasserstein distance between $\nu$ and $\lambda$.
  • ...and 6 more figures

Theorems & Definitions (111)

  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • Example 3.6
  • Proposition 3.7
  • Definition 3.8
  • Remark 3.9
  • Remark 3.10
  • ...and 101 more