**Shape Analysis, Stochastic Mechanics and Optimal Transport**

**OFFICIAL WEBSITE**

**Boris Khesin**, University of Toronto: Beyond Arnold’s geodesic framework of an ideal hydrodynamics

We discuss a ramification of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. We show such problems of mathematical physics as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries, and describe the corresponding geometry and equations. (This is a joint work with Anton Izosimov.)

Watch video | Download video**Gerard Misiolek**, University of Notre Dame: The L2 exponential map in 2D and 3D hydrodynamics

In the 1960's V. Arnold showed how solutions of the incompressible Euler equations can be viewed as geodesics on the group of diffeomorphisms of the fluid domain equipped with a metric given by fluid's kinetic energy. The study of the exponential map of this metric is of particular interest and I will describe recent results concerning its properties as well as some necessary background.

Watch video | Download video**Klas Modin**, Chalmers University of Technology / University of Gothenburg: Semi-invariant metrics on diffeos

We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy---a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity---under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the Γ-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.

Watch video | Download video**Ana Cruzeiro**, University of Lisbon : On some relations between Optimal Transport and Stochastic Geometric Mechanics

We formulate the so-called Schrodinger problem in Optimal Transport on lie group and derive the corresponding Euler-Poincaré equations.

Watch video | Download video | PDF presentation**Christian Léonard**, Universite Paris Nanterre: Some ideas and results about gradient flows and large deviations

In several situations, the empirical measure of a large number of random particles evolving in a heat bath is an approximation of the solution of a dissipative PDE. The evaluation of the probabilities of large deviations of this empirical measure suggests a way of defining a natural ``large deviation cost'' for these fluctuations, very much in the spirit of optimal transport. Some standard Wasserstein gradient flow evolutions are revisited in this perspective, both in terms of heuristic results and a few rigorous ones. This talk gathers several joint works with Julio Backhoff, Giovanni Conforti, Ivan Gentil, Luigia Ripani and Johannes Zimmer.

Watch video | Download video | PDF presentation**Marc Arnaudon**, Université de Bordeaux: A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces

"Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. I will present a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula will allow to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein in the context of discrete generation models to continuous time Feynman-Kac models and their interacting jump particle interpretations. I will also provide new propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. These results allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers. "

Watch video | Download video | PDF presentation**Alexis Arnaudon**, Imperial College London : Geometric modelling of uncertainties

In mechanics, and in particular in shape analysis, taking into account the underlying geometric properties of a problem to model it is often crucial to understand and solve it. This approach has mostly been applied for isolated systems, or for systems interacting with a well-defined, deterministic environment. In this talk, I want to discuss how to go beyond this deterministic description of isolated systems to include random interactions with an environment, while retaining as much as possible the geometric properties of the isolated systems. I will discuss examples from geometric mechanics to shape analysis, ranging from interacting rigid bodies with a heath bath to uncertainties quantification in computational anatomy.

Watch video | Download video**Bernhard Schmitzer**, University of Münster: Semi-discrete unbalanced optimal transport and quantization

"Semi-discrete optimal transport between a discrete source and a continuous target has intriguing geometric properties and applications in modelling and numerical methods. Unbalanced transport, which allows the comparison of measures with unequal mass, has recently been studied in great detail by various authors. In this talk we consider the combination of both concepts. The tessellation structure of semi-discrete transport survives and there is an interplay between the length scales of the discrete source and unbalanced transport which leads to qualitatively new regimes in the crystallization limit."

Watch video | Download video | PDF presentation**Carola-Bibiane Schönlieb**, University of Cambridge : Wasserstein for learning image regularisers

In this talk we will discuss the use of a Wasserstein loss function for learning regularisers in an adversarial manner. This talk is based on joint work with Sebastian Lunz and Ozan Öktem, see https://arxiv.org/abs/1805.11572

Watch video | Download video | PDF presentation**Tryphon Georgiou**, University of California, Irvine : Interpolation of Gaussian mixture models and other directions in Optimal Mass Transport

Watch video | Download video**Laurent Younes**, John Hopkins University : Normal coordinates and equivolumic layers estimation in the cortex (tentative)

Watch video | Download video**Barbara Gris**, Université Pierre-et-Marie-Curie: Analyze shape variability via deformations

I will present how shape registration via constrained deformations can help understanding the variability within a population of shapes.

Watch video | Download video**Dongyang Kuang**, University of Ottawa : Convnets, a different view of approximating diffeomorphisms in medical image registration

As with the heat of artificial intelligence, there are more and more researches starting to investigate the possible geometric transformations using data-driven methods such as convolutional neural networks. In this talk, I will start by introducing some existing work that learn 2D linear transformations in an unsupervised way. This then will be followed by an overview of some recent works focusing on nonlinear transformations in 3D volumetric data. Finally, I will present results from the joint work with my supervisor using our network architecture called FAIM.

Watch video | Download video**Stephen Preston**, Brooklyn College : Solar models for Euler-Arnold equations

Many one-dimensional Euler-Arnold equations can be recast in the form of a central-force problem Γtt(t,x)=−F(t,x)Γ(t,x), where Γ is a vector in ℝ2 and F is a nonlocal function possibly depending on Γ and Γt. Angular momentum of this system is precisely the conserved momentum for the Euler-Arnold equation. In particular this picture works for the Camassa-Holm equation, the Hunter-Saxton equation, and the Okamoto-Sakajo-Wunsch family of equations. In the solar model, breakdown comes from a particle hitting the origin in finite time, which is only possible with zero angular momentum. Results due to McKean (for Camassa-Holm), Lenells (for Hunter-Saxton), and Bauer-Kolev-Preston/Washabaugh (for the Wunsch equation) show that breakdown of smooth solutions occurs exactly when momentum changes from positive to negative. I will discuss some conjectures and numerical evidence for the generalization of this picture to other equations such as the μ-Camassa-Holm equation or the DeGregorio equation.

Watch video | Download video**Cy Maor**, University of Toronto : Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

Since the seminal work of Arnold on the Euler equations, many important PDEs were shown to be geodesic equations of diffeomorphism groups of manifolds, with respect to various Sobolev norms. But what about the geodesic distance induced by these norms? Is it positive between different diffeomorphisms, or not? In this talk I will show that the geodesic distance on the diffeomorphism group of an n-dimensional manifold, induced by the Ws,p norm, does not vanish if and only if s≥1 or sp>n. The first condition detects changes of volume, while the second one detects transport of arbitrary small sets. I will focus on the case where both conditions fail, and how this enables the construction of arbitrary short paths between diffeomorphisms. Based on a joint work with Robert Jerrard, following works of Michor-Mumford, Bauer-Bruveris-Harms-Michor and Bauer-Harms-Preston.

Watch video | Download video**Philipp Harms**, University of Freiburg : Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics.

We show that the functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. Using this result we are able to prove that fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics. (Joint work with Martins Bruveris, Martin Bauer, and Peter W. Michor).

Watch video | Download video**Eric Klassen**, Florida State University : Comparing Shapes of Curves, Surfaces, and Higher Dimensional Immersions in Euclidean Space.

Comparing shapes and treating them as data for statistical analyses has many applications in biology and elsewhere. Certain elastic metrics on spaces of immersions have proved very effective for comparing curves and surfaces. The elastic metrics which have proved most useful for computation have been first order metrics, i.e., they compare tangent vectors on the shapes rather than points on the shapes. In this talk I will present a unifying view of these metrics, shedding new light on old methods and, I hope, suggesting new methods for analyzing surfaces and higher dimensional shapes.

Watch video | Download video**Facundo Memoli**, The Ohio State University : Metrics on the collection of dynamic shapes.

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of groups of animals. In a similar vein, when attempting to classify motion capture data according to action one is confronted with having to match/compare shapes that evolve with time. Motivated by these applications, we study the question of suitably metrizing the collection of all dynamic metric spaces (DMSs). We construct a suitable metric on this collection and prove the stability of several natural invariants of DMSs under this metric. In particular, we prove that certain zigzag persistent homology invariants related to dynamic clustering are stable w.r.t. this distance. These lower bounds permit the efficient classification of dynamic shape data in applications. We will show computational experiments on dynamic data generated via distributed behavioral models. This is joint work with Woojin Kim and Zane Smith https://research.math.osu.edu/networks/formigrams/

Watch video | Download video**Tom Needham**, Ohio State University : Gromov-Monge Quasimetrics and Distance Distributions.

In applications in computer graphics and computational anatomy, one seeks a measure-preserving map from one shape to another which preserves geometry as much as possible. Inspired by this, we consider a notion of distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces. This distance has convenient lower bounds defined in terms of distance distributions; these are functions associated to mm-spaces which have been used frequently as summaries in data and shape analysis applications. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of mm-spaces such as metric graphs or plane curves.This is joint work with Facundo Mémoli.

Watch video | Download video**Jean-David Benamou**, INRIA Rocquencourt : Dynamic formulations of optimal transportation and variational relaxation of Euler equations.

We will briefly recall the classical Optimal Transportation Framework and its Dynamic relaxations. We will show the link between these Dynamic formulation and the so-called MultiMarginal extension of Optimal Transportation. We will then describe the so-called Iterative Proportional Fitting Procedure (aka Sinkhorn method) which can be efficiently applied to the multi-marginal OT setting. Finally we will show how this can be used to compute generalized Euler geodesics due to Brenier. This problem can be considered as the oldest instance of Multi-Marginal Optimal Transportation problem. Joint work with Guillaume Carlier (Ceremade, Universite Paris Dauphine, France) and Luca Nenna (U. Paris Sud, France).

Watch video | Download video**Tudor Ratiu**, Shanghai Jiao Tong University: Group valued momentum maps

Watch video | Download video**Andrea Natale**, Inria : Generalized H(div) geodesics and solutions of the Camassa-Holm equation

Watch video | Download video**Jean Feydy**, Ecole Normale Supérieure : Robust shape matching with optimal transport

Watch video | Download video | PDF presentation**Alice Le Brigant***, ENAC - Ecole Nationale de l'Aviation Civile : Quantization on a Riemannian manifold with application to air traffic control

Watch video | Download video

**Shape Analysis, Stochastic Mechanics and Optimal Transport**

**OFFICIAL WEBSITE**

**Boris Khesin**, University of Toronto: Beyond Arnold’s geodesic framework of an ideal hydrodynamics

We discuss a ramification of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. We show such problems of mathematical physics as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries, and describe the corresponding geometry and equations. (This is a joint work with Anton Izosimov.)

Watch video | Download video**Gerard Misiolek**, University of Notre Dame: The L2 exponential map in 2D and 3D hydrodynamics

In the 1960's V. Arnold showed how solutions of the incompressible Euler equations can be viewed as geodesics on the group of diffeomorphisms of the fluid domain equipped with a metric given by fluid's kinetic energy. The study of the exponential map of this metric is of particular interest and I will describe recent results concerning its properties as well as some necessary background.

Watch video | Download video**Klas Modin**, Chalmers University of Technology / University of Gothenburg: Semi-invariant metrics on diffeos

We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy---a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity---under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the Γ-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.

Watch video | Download video**Ana Cruzeiro**, University of Lisbon : On some relations between Optimal Transport and Stochastic Geometric Mechanics

We formulate the so-called Schrodinger problem in Optimal Transport on lie group and derive the corresponding Euler-Poincaré equations.

Watch video | Download video | PDF presentation**Christian Léonard**, Universite Paris Nanterre: Some ideas and results about gradient flows and large deviations

In several situations, the empirical measure of a large number of random particles evolving in a heat bath is an approximation of the solution of a dissipative PDE. The evaluation of the probabilities of large deviations of this empirical measure suggests a way of defining a natural ``large deviation cost'' for these fluctuations, very much in the spirit of optimal transport. Some standard Wasserstein gradient flow evolutions are revisited in this perspective, both in terms of heuristic results and a few rigorous ones. This talk gathers several joint works with Julio Backhoff, Giovanni Conforti, Ivan Gentil, Luigia Ripani and Johannes Zimmer.

Watch video | Download video | PDF presentation**Marc Arnaudon**, Université de Bordeaux: A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces

"Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. I will present a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula will allow to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein in the context of discrete generation models to continuous time Feynman-Kac models and their interacting jump particle interpretations. I will also provide new propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. These results allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers. "

Watch video | Download video | PDF presentation**Alexis Arnaudon**, Imperial College London : Geometric modelling of uncertainties

In mechanics, and in particular in shape analysis, taking into account the underlying geometric properties of a problem to model it is often crucial to understand and solve it. This approach has mostly been applied for isolated systems, or for systems interacting with a well-defined, deterministic environment. In this talk, I want to discuss how to go beyond this deterministic description of isolated systems to include random interactions with an environment, while retaining as much as possible the geometric properties of the isolated systems. I will discuss examples from geometric mechanics to shape analysis, ranging from interacting rigid bodies with a heath bath to uncertainties quantification in computational anatomy.

Watch video | Download video**Bernhard Schmitzer**, University of Münster: Semi-discrete unbalanced optimal transport and quantization

"Semi-discrete optimal transport between a discrete source and a continuous target has intriguing geometric properties and applications in modelling and numerical methods. Unbalanced transport, which allows the comparison of measures with unequal mass, has recently been studied in great detail by various authors. In this talk we consider the combination of both concepts. The tessellation structure of semi-discrete transport survives and there is an interplay between the length scales of the discrete source and unbalanced transport which leads to qualitatively new regimes in the crystallization limit."

Watch video | Download video | PDF presentation**Carola-Bibiane Schönlieb**, University of Cambridge : Wasserstein for learning image regularisers

In this talk we will discuss the use of a Wasserstein loss function for learning regularisers in an adversarial manner. This talk is based on joint work with Sebastian Lunz and Ozan Öktem, see https://arxiv.org/abs/1805.11572

Watch video | Download video | PDF presentation**Tryphon Georgiou**, University of California, Irvine : Interpolation of Gaussian mixture models and other directions in Optimal Mass Transport

Watch video | Download video**Laurent Younes**, John Hopkins University : Normal coordinates and equivolumic layers estimation in the cortex (tentative)

Watch video | Download video**Barbara Gris**, Université Pierre-et-Marie-Curie: Analyze shape variability via deformations

I will present how shape registration via constrained deformations can help understanding the variability within a population of shapes.

Watch video | Download video**Dongyang Kuang**, University of Ottawa : Convnets, a different view of approximating diffeomorphisms in medical image registration

As with the heat of artificial intelligence, there are more and more researches starting to investigate the possible geometric transformations using data-driven methods such as convolutional neural networks. In this talk, I will start by introducing some existing work that learn 2D linear transformations in an unsupervised way. This then will be followed by an overview of some recent works focusing on nonlinear transformations in 3D volumetric data. Finally, I will present results from the joint work with my supervisor using our network architecture called FAIM.

Watch video | Download video**Stephen Preston**, Brooklyn College : Solar models for Euler-Arnold equations

Many one-dimensional Euler-Arnold equations can be recast in the form of a central-force problem Γtt(t,x)=−F(t,x)Γ(t,x), where Γ is a vector in ℝ2 and F is a nonlocal function possibly depending on Γ and Γt. Angular momentum of this system is precisely the conserved momentum for the Euler-Arnold equation. In particular this picture works for the Camassa-Holm equation, the Hunter-Saxton equation, and the Okamoto-Sakajo-Wunsch family of equations. In the solar model, breakdown comes from a particle hitting the origin in finite time, which is only possible with zero angular momentum. Results due to McKean (for Camassa-Holm), Lenells (for Hunter-Saxton), and Bauer-Kolev-Preston/Washabaugh (for the Wunsch equation) show that breakdown of smooth solutions occurs exactly when momentum changes from positive to negative. I will discuss some conjectures and numerical evidence for the generalization of this picture to other equations such as the μ-Camassa-Holm equation or the DeGregorio equation.

Watch video | Download video**Cy Maor**, University of Toronto : Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

Since the seminal work of Arnold on the Euler equations, many important PDEs were shown to be geodesic equations of diffeomorphism groups of manifolds, with respect to various Sobolev norms. But what about the geodesic distance induced by these norms? Is it positive between different diffeomorphisms, or not? In this talk I will show that the geodesic distance on the diffeomorphism group of an n-dimensional manifold, induced by the Ws,p norm, does not vanish if and only if s≥1 or sp>n. The first condition detects changes of volume, while the second one detects transport of arbitrary small sets. I will focus on the case where both conditions fail, and how this enables the construction of arbitrary short paths between diffeomorphisms. Based on a joint work with Robert Jerrard, following works of Michor-Mumford, Bauer-Bruveris-Harms-Michor and Bauer-Harms-Preston.

Watch video | Download video**Philipp Harms**, University of Freiburg : Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics.

We show that the functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. Using this result we are able to prove that fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics. (Joint work with Martins Bruveris, Martin Bauer, and Peter W. Michor).

Watch video | Download video**Eric Klassen**, Florida State University : Comparing Shapes of Curves, Surfaces, and Higher Dimensional Immersions in Euclidean Space.

Comparing shapes and treating them as data for statistical analyses has many applications in biology and elsewhere. Certain elastic metrics on spaces of immersions have proved very effective for comparing curves and surfaces. The elastic metrics which have proved most useful for computation have been first order metrics, i.e., they compare tangent vectors on the shapes rather than points on the shapes. In this talk I will present a unifying view of these metrics, shedding new light on old methods and, I hope, suggesting new methods for analyzing surfaces and higher dimensional shapes.

Watch video | Download video**Facundo Memoli**, The Ohio State University : Metrics on the collection of dynamic shapes.

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of groups of animals. In a similar vein, when attempting to classify motion capture data according to action one is confronted with having to match/compare shapes that evolve with time. Motivated by these applications, we study the question of suitably metrizing the collection of all dynamic metric spaces (DMSs). We construct a suitable metric on this collection and prove the stability of several natural invariants of DMSs under this metric. In particular, we prove that certain zigzag persistent homology invariants related to dynamic clustering are stable w.r.t. this distance. These lower bounds permit the efficient classification of dynamic shape data in applications. We will show computational experiments on dynamic data generated via distributed behavioral models. This is joint work with Woojin Kim and Zane Smith https://research.math.osu.edu/networks/formigrams/

Watch video | Download video**Tom Needham**, Ohio State University : Gromov-Monge Quasimetrics and Distance Distributions.

In applications in computer graphics and computational anatomy, one seeks a measure-preserving map from one shape to another which preserves geometry as much as possible. Inspired by this, we consider a notion of distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces. This distance has convenient lower bounds defined in terms of distance distributions; these are functions associated to mm-spaces which have been used frequently as summaries in data and shape analysis applications. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of mm-spaces such as metric graphs or plane curves.This is joint work with Facundo Mémoli.

Watch video | Download video**Jean-David Benamou**, INRIA Rocquencourt : Dynamic formulations of optimal transportation and variational relaxation of Euler equations.

We will briefly recall the classical Optimal Transportation Framework and its Dynamic relaxations. We will show the link between these Dynamic formulation and the so-called MultiMarginal extension of Optimal Transportation. We will then describe the so-called Iterative Proportional Fitting Procedure (aka Sinkhorn method) which can be efficiently applied to the multi-marginal OT setting. Finally we will show how this can be used to compute generalized Euler geodesics due to Brenier. This problem can be considered as the oldest instance of Multi-Marginal Optimal Transportation problem. Joint work with Guillaume Carlier (Ceremade, Universite Paris Dauphine, France) and Luca Nenna (U. Paris Sud, France).

Watch video | Download video**Tudor Ratiu**, Shanghai Jiao Tong University: Group valued momentum maps

Watch video | Download video**Andrea Natale**, Inria : Generalized H(div) geodesics and solutions of the Camassa-Holm equation

Watch video | Download video**Jean Feydy**, Ecole Normale Supérieure : Robust shape matching with optimal transport

Watch video | Download video | PDF presentation**Alice Le Brigant***, ENAC - Ecole Nationale de l'Aviation Civile : Quantization on a Riemannian manifold with application to air traffic control

Watch video | Download video