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Shape spaces in terms of Wasserstein geometry

Bernadette Lessel

TL;DR

The work defines Shape space as the Wasserstein space $W_p(X)$ modulo a subgroup of isometries, equipping it with a Shape distance $D_p$ that arises from pushing one measure by an isometry before comparing it with another via $W_p$. It proves existence of minimizers for the Shape distance under proper action, establishing when $D_p$ becomes a genuine metric and showing Polish and topological properties of the Shape space, including its completeness and geodesic relations to Wasserstein geodesics. It further develops a tangent-space framework for the Shape space, by quotienting out orbit directions generated by the isometry group (via fundamental vector fields), producing a well-defined tangent space $T_{[\mu]}\mathcal{S}(\mathbb{R}^n) = T_\mu W(\mathbb{R}^n)/U_\mu$, and extends these ideas to Shape spaces on Riemannian manifolds. The results provide a rigorous mathematical foundation for shape analysis of probability measures under symmetry, linking optimal transport geometry with quotient geometry and enabling principled, intrinsic comparisons and interpolations of shapes.

Abstract

For a Polish space $X$, we define the Shape space $\mathcal{S}_p(X)$ to be the Wasserstein space $W_p(X)$ modulo the action of a subgroup $G$ of the isometry group $ISO(X)$ of $X$, where the action is given by the pushforward of measures. The Wasserstein distance can then naturally be transformed into a \emph{Shape distance} on Shape space if $X$ and the action of $G$ are proper. This is shown for example to be the case for complete connected Riemannian manifolds with $G$ being equipped with the compact-open topology. Before finally proposing a notion for tangent spaces on the Shape space $\mathcal{S}_2(\mathbb{R}^n)$, it is shown that $\mathcal{S}_p(X)$ is Polish as well in case $X$ and the action of $G$ are indeed proper. Also, the metric geodesics in $\mathcal{S}_p(X)$ are put in relation to the ones in $W_p(X)$.

Shape spaces in terms of Wasserstein geometry

TL;DR

The work defines Shape space as the Wasserstein space modulo a subgroup of isometries, equipping it with a Shape distance that arises from pushing one measure by an isometry before comparing it with another via . It proves existence of minimizers for the Shape distance under proper action, establishing when becomes a genuine metric and showing Polish and topological properties of the Shape space, including its completeness and geodesic relations to Wasserstein geodesics. It further develops a tangent-space framework for the Shape space, by quotienting out orbit directions generated by the isometry group (via fundamental vector fields), producing a well-defined tangent space , and extends these ideas to Shape spaces on Riemannian manifolds. The results provide a rigorous mathematical foundation for shape analysis of probability measures under symmetry, linking optimal transport geometry with quotient geometry and enabling principled, intrinsic comparisons and interpolations of shapes.

Abstract

For a Polish space , we define the Shape space to be the Wasserstein space modulo the action of a subgroup of the isometry group of , where the action is given by the pushforward of measures. The Wasserstein distance can then naturally be transformed into a \emph{Shape distance} on Shape space if and the action of are proper. This is shown for example to be the case for complete connected Riemannian manifolds with being equipped with the compact-open topology. Before finally proposing a notion for tangent spaces on the Shape space , it is shown that is Polish as well in case and the action of are indeed proper. Also, the metric geodesics in are put in relation to the ones in .
Paper Structure (11 sections, 39 theorems, 56 equations)

This paper contains 11 sections, 39 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Wasserstein calculus
  3. Wasserstein spaces $W_p(X)$
  4. Tangent spaces on $W_2(M)$
  5. Shape spaces $\mathcal{S}_p(X)$
  6. Shape distances $D_p(\mu,\nu)$
  7. Existence of a minimizer for the Shape distance-problem
  8. Shape spaces $\mathcal{S}_p(M)$ on Riemannian manifolds
  9. Some topological properties of $\mathcal{S}_p(X)$
  10. Geodesic properties of $\mathcal{S}_p(X)$
  11. Tangent spaces on $\mathcal{S}_2(M)$

Key Result

Theorem 7

Let $(F_t)_{t\in[0,T)}$ be a family of maps on $M$ such that $F_{t}:M\rightarrow M$ is a bijection for every $t\in[0,T)$, $F_0=Id$ and both $(t,x)\mapsto F_t(x)$ and $(t,x)\mapsto F_t^{-1}(x)$ are locally Lipschitz on $[0,T)\times M$. Let further $v_t(x)$ be a family of velocity fields on $M$ such t

Theorems & Definitions (107)

  • Definition 1: Wasserstein distances and Wasserstein spaces
  • Definition 2: Weak convergence in $\mathcal{P}_p(X)$
  • Definition 3: Constant speed geodesic
  • Definition 4: Geodesic space
  • Definition 5: Non branching space
  • Definition 6: Continuity equation
  • Theorem 7
  • Definition 8: Absolutely continuous curve
  • Definition 9: Metric derivative
  • Theorem 10: Differential characterization of a.c. curves
  • ...and 97 more