## Welcome

This workshop has the goal of bringing together researchers in the fields of stochastic analysis on manifolds, stochastic shape analysis, and stochastic geometric mechanics. These fields are closely interrelated: Stochastic shape analysis draws on stochastic variational principles, which have been developed in stochastic geometric mechanics, building on foundations in stochastics on manifolds. Conversely, stochastic shape analysis with its diverse real-world applications continues to inspire new theoretical questions, in particular regarding extensions to infinite dimensions and connections to statistical estimation.

We believe that the three communities will benefit from increased sharing of problems, methodology and solution strategies, and it is the goal of this workshop to initiate new collaborations across the three fields. The workshop will consist of scientific talks by the participants as well as time for collaboration. A detailed program will be announced later on.

## Workshop program

## Participants

The workshop is by invitation only. If you are interested, please contact one of the organizers. Click here to reach the registration site. If you are registered, submit your title and abstract using this form.

Current list of attendees:

- Ana Bela Cruzeiro, Dep. Matemática IST, Univ. Lisbon
- Wilfrid Kendall, University of Warwick
- Huiling Le, University of Nottingham
- Anton Thalmaier, University of Luxembourg
- Marc Arnaudon, Universite of Bordeaux
- Stephan Huckemann, University of Göttingen
- Benjamin Eltzner, University of Göttingen
- Sarang Joshi, University of Utah
- Xavier Pennec, INRIA Sophia-Antipolis
- François-Xavier Vialard, University Paris-Est Marne la Vallée
- Frank van der Meulen, Delft University of Technology
- Moritz Schauer, University of Leiden
- Jesper Møller, Aalborg University
- Mathias Jensen, University of Copenhagen
- Pernille Hansen, University of Copenhagen
- Erwin Luesink, Imperial College
- Alessandro Barp, Imperial College
- Do Tran, Duke University
- James Thompson, University of Luxembourg
- Jean-Claude Zambrini, University of Lisbon
- Marc Corstanje, Delft University of Technology
- Yann Thanwerdas, INRIA Sophia-Antipolis
- Nicolas Guigui, INRIA Sophia-Antipolis
- Helene Hauschultz, Aarhus University
- Morten Akhøj Pedersen, UCPH
- Philipp Harms, University of Freiburg
- Alexis Arnaudon, Imperial College
- Stefan Sommer, University of Copenhagen
- Eva Vedel Jensen, Aarhus University

## Practical information

*In case you need to contact us close to the start of the workshop, please email or call Stefan Sommer: sommer@di.ku.dk / +45 21179125*

The workshop starts on February 23rd in the afternoon/evening and ends on Thursday February 27th after lunch. Please plan your travel accordingly.

### Getting to Sandbjerg

The workshop will take place at the Sandbjerg Estate near Sønderborg in the southeastern part of Jutland, Denmark. The conference centre, which is affiliated with Aarhus University, is housed in manor buildings dating from the late 18th century, and is scenically located on an arm of the Baltic Sea. Sandbjerg is located in South Jutland, close to the German border.

### Address

Sandbjerg Estate. Sandbjergvej 102, DK-6400 Sønderborg

Telephone (+45) 73 46 52 00, E-mail: im@sandbjerg.dk, Web: www.sandbjerg.dk/en/

### Travel

Sandbjerg can be reached by train, bus or car from Denmark and Germany. When arriving by plane, it is most common to fly to Copenhagen Airport (Kastrup Lufthavn) and then taking the train or flying to Sønderborg. Alternatively, the venue can be reached from Hamburg airport.

### By train

By train from Copenhagen Airport to Sønderborg: Train connection every second hour, travel time about 4 hours. Travel information can be found at www.rejseplanen.dk. Train tickets can be bought in Terminal 3 at Copenhagen Airport. Distance between Sønderborg railway station and Sandbjerg Estate approx. 6 km.

### By airplane

There are multiple daily flights on weekdays between Copenhagen and Sønderborg, one flight on Saturdays and Sundays, flight time 45 minutes. See more information at alsieexpress.dk/en/. Distance between Soenderborg airport and Sandbjerg Estate 10 km.

### By bus from Flensburg

Bus Flensburg-Sønderborg: Travel information can be found at www.rejseplanen.dk. Travel time approx. 1 hour and 30 minutes.

### By car

See information at www.sandbjerg.dk/en/practical-information

### Taxi

Sønderborg Taxi: +45 7442 1818.

## Organization

The workshop is organized and funded by the Centre for Stochastic Geometry and Advanced Bioimaging.

The organizing committee is:

- Alexis Arnaudon (Imperial College)
- Philipp Harms (University of Freiburg)
- Stefan Sommer (University of Copenhagen)
- Eva Vedel Jensen (Aarhus University)

The venue Sandbjerg Estate serves as Aarhus University's conference center.

## Talks

### Wilfrid Kendall

Title: Poisson processes and Spatial Transportation Networks: network geodesics and Rayleigh random flights.

Abstract: Scale-invariant random spatial networks (SIRSN) are remarkable random structures which provide patterns of random routes that are scale-invariant, thus modelling apparent scale-invariance in online maps. My talk will review the rather non-trivial theory establishing the existence of the Poisson line SIRSN and the known properties of network geodesics. In order to develop a good intuition about the behaviour of these geodesics, attention has turned to random scattering processes on the Poisson line SIRSN. This in turn leads to an axiomatization of abstract scattering processes (Markov chains which algebraically look like scattering processes), perhaps of wider interest in reliability theory. Ergodic theory (in particular a continuum version of the famous range theorem of Kesten, Spitzer and Whitman) can then be applied to produce insight into the behaviour of a randomly broken geodesic on the SIRSN.

### Francois-Xavier Vialard

Title: Sinkhorn divergences for unbalanced optimal transport

In this talk, we will present a new family of divergences on positive Radon measures which are called Sinkhorn divergences. We show how they interpolate between unbalanced optimal transport and MMD distances and the necessary conditions to make them well-posed. Passing by, we prove linear rate of convergence of the Sinkhorn algorithm in an unbalanced setting. Last, we will discuss sample complexity of these new divergences.

### Huiling Le

Title: Stein's method for probability measures on manifolds

Abstract: Motivated by recent interest in applications of Stein's method to non-Euclidean data analysis in statistics, this talk discusses our recent investigation into how the diffusion method and coupling techniques can be used to generalise Stein's equation to probability measures on Riemannian manifolds and the study of properties of the corresponding Stein operator.

### Sarang Joshi

Title: Riemannian Brownian Bridges and Metric Estimation on Landmark Manifolds

Abstract: We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional non-linear PDE with no closed-form solution. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood.

### Benjamin Eltzner

Title: Topological and Geometrical Smeariness

Abstract: Surprisingly, for some probability measures on the circle and higher dimensional spheres, the fluctuations of the sample mean around the population mean are non-normally distributed and decrease more slowly than with the square root of sample size. This property of a probability measure is called smeariness. On the circle, this is dependent on a specific value of the probability density at the cut locus of the mean and thus on the topology of the circle, therefore we call it topological smeariness. On spheres, smeariness can occur even if a neighborhood of the cut locus of the mean is excluded from the support of the probability measure and the sphere is deformed near the cut locus of the mean to become diffeomorphic to Euclidean space. We call this phenomenon geometrical smeariness. We discuss a generalized central limit theorem and some data examples.

### Frank van der Meulen

Title: Conditional simulation of diffusions

Abstract: In this talk I will review some recently proposed methods for conditional simulation of diffusions. Focussing on the approach based on 'guided proposals' I will introduce efficient algorithms for inference in discretely observed diffusions. Applications include chemical reactions networks and landmark matching.

### Jean-Claude Zambrini

Title: Integration of Bernstein diffusions on manifolds

Abstract: We give a survey of the construction of Schroedinger-Bernstein diffusions on Riemannian manifolds and a glimpse at an associated notion of integrability.

### Ana Bela Cruzeiro

Title: Navier-Stokes via variational principles with Lagrange multipliers

Abstract: We derive and study Navier-Stokes Lagrangian flows obtained through stochastic variational principles with Lagrange multipliers. We consider several types of domains, with and without boundaries.

### Alessandro Barp

Title: Hamiltonian Monte Carlo on Manifolds

Abstract: We discuss the implementation of the Hamiltonian Monte Carlo algorithm to sample from distributions on manifolds, and its applications to molecular dynamics, gauge theory, and statistics.

### Do Tran

Title: Behavior of Frechet Means and Central Limit Theorems on Spheres

Abstract: Jacobi fields are used to compute higher derivatives of the Frechet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Frechet mean; (ii) a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.

### Marc Arnaudon

Title: Stochastic mean curvature flow and intertwined Brownian motion

Abstract: The evolution of a set by deformation of its boundary along mean curvature flow is involved in many physical phenomena. We are interested here in this evolution, to which we add a noise that acts uniformly on its boundary, and a renormalization term. It is shown that this evolution can be coupled to a Brownian motion which remains within the set, and which at all times is uniformly distributed inside the set. This is the phenomenon of duality and intertwining established by Diaconis and Fill in the context of Markov chains in finite state spaces. Different couplings are proposed, some of which involving the local time of the Brownian motion, either on the skeleton of the set or on its boundary. These couplings also differ by more or less strong correlation between the Brownian motion inside and the vibrating boundary. When the set is a symmetric real interval, the boundary evolves as a Bessel process of dimension 3, and we recover the 2M-X Pitman theorem as a special case, with one particular coupling. This is a common work with Koléhè Coulibaly (Institut Elie Cartan de Lorraine (IECL), Nancy), and Laurent Miclo (Institut de Mathématiques de Toulouse (IMT), Toulouse)

### Xavier Pennec

Title: Effect of curvature on the Empirical Fréchet mean estimation in manifolds

Abstract: Statistical inference in manifolds most often rely on the Fréchet mean in the Riemannian case, or on exponential barycenters in affine connection spaces. The uncertainty of the empirical mean estimation with a fixed number of samples is a key question. In sufficient concentration conditions, a central limit theorem was established in Riemannian manifolds by Bhattacharya & Patrangenaru in 2005. We present in this talk an asymptotic development valid in Riemannian and affine cases which better explain the role of the curvature in the modulation of the speed of convergence of the empirical mean. We also establish a non-asymptotic development in high concentration which shows a statistical bias on the empirical mean in the direction of the average gradient of the curvature. These curvature effects become important with large curvature and can drastically modify the estimation of the mean. They could partly explain the phenomenon of sticky means recently put into evidence in stratified spaces, notably in the case of negative curvature.

Reference: Xavier Pennec. Curvature effects on the empirical mean in Riemannian and affine Manifolds: a non-asymptotic high concentration expansion in the small-sample regime. ARXIV preprint 1906.07418, June 2019.

### Stephan Huckemann

Title: Asymptotic inference for PCA generalizations to non-Euclidean spaces

Abstract: We review the concept of generalized Fréchet means and what is known about their asymptotics. We put such generalized Fréchet means in the context of classical PCA and PCA extensions to non-Euclidean spaces. In contrast to the asymptotic of classical PCA which is that of a Gaussian scaled with the square root of sample size, faster and slower rates with non-Gaussian limiting distributions may occur on non-Euclidean spaces. In particular, the rate can be infinitely fast, resulting in stickiness. Both, stickiness and slower rates, called smeariness, may also affect a wide range of distributions up to rather large sample sizes.

### Pernille E.H. Hansen

Title: Asymptotic behavior of diffusion means on manifolds

Abstract: Defining a mean values for probability measures on manifolds is non-trivial because of the lack of vector space structure, and manifold means can behave quite different compared to their Euclidean counterparts due to the influence of curvature. In this talk, we investigate the asymptotic behavior of sample estimators of the diffusion mean. We extend results by Benjamin Eltzner and Stephan Huckemann on the law of large numbers and central limit theorem for the Fréchet mean to the diffusion mean. For specific distributions on the 2-sphere, we compare the asymptotic behavior of the diffusion mean and the Fréchet mean, in particular, we identify cases in which the diffusion mean is smeary.

### Mathias Højgaard Jensen

Title: Simulation of Conditioned Diffusions on Riemannian Manifolds

Abstract: We develop an approach to simulating stochastic diffusion processes on Riemannian manifolds that does not rely on their transition densities. Instead, we propose a method which involves the radial process.

### James Thompson

Title: Submanifold Brownian bridges

Abstract: In this talk, we will introduce and study submanifold Brownian bridges. By this we mean Brownian motions conditioned to arrive in a fixed submanifold at a fixed positive time. Our study of these processes involves deriving a path integral formula for the heat kernel and, more generally, its integral over the submanifold. We use this formula to derive lower bounds, an asymptotic relation and derivative estimates, yielding connections to hypersurface local time and mean curvature flow.

### Jesper Møller

Title: Statistics for point processes on the d-dimensional unit sphere

Abstract: We discuss statistical models and methods for the analysis of point patterns on the d-dimensional unit sphere, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case d=2. Two models are studied in detail: determinantal point processes (based on reference [2]) and log Gaussian Cox processes (based on [1]), which are models for regular and aggregated point patterns, respectively. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. We also study reduced Palm distributions and functional summary statistics for general point process models on the unit sphere (see [3]). The results are applied for the description of sky positions of galaxies (see [3]).

[1] F. Cuevas-Pacheco and J. Møller (2018). Log Gaussian Cox processes on the sphere. Spatial Statistics, 26, 69-82.

[2] J. Møller, M. Nielsen, E. Porcu and E. Rubak (2018). Determinantal point process models on the sphere. Bernoulli, 24, 1171-1201.

[3] J. Møller and E. Rubak (2016). Functional summary statistics on the sphere with an application to determinantal point processes. Spatial Statistics, 18, 4-23.

### Anton Thalmaier

Title: Brownian motion, Ricci curvature and geometric flows

Abstract: We describe recent developments on the problem of characterizing Ricci curvature and Ricci flow in terms of functional inequalities for diffusion semigroups, and if time permits, we discuss possible extensions to sub-Riemannian geometry. We show in particular that certain functional inequalities and gradient estimates on the path space over the manifold are equivalent to boundedness of the Ricci curvature tensor.

### Moritz Schauer

Title: Geometric statistics with shape deformation models based on stochastic differential equations

Abstract: Stochastic differential equation models for shapes represented by landmark configurations connect shape analysis to statistical boundary value problems for stochastic differential equations. The landmarks identify corresponding points on each shape, and their joint stochastic dynamics, derived from a hypo-elliptic Hamiltonian, define a likelihood giving weight to plausible deformations or configurations given the observational data.

Here we make use of novel Bayesian methods for inference and statistical boundary value problems for hypo-elliptic stochastic differential equations.

### Erwin Luesink

Abstract: In [Holm2015] a stochastic variational principle was introduced. This variational principle is of the Clebsch type. Recently in [BdLHLT2020], by deriving a stochastic chain rule for k-forms, it was shown that the other variational principles (Euler-Poincare and Hamilton-Pontryagin) are also valid for the derivation of equations in continuum mechanics. In this talk I will briefly speak about why this equivalence is now established and how it can be used to derive the equations for ocean modelling.

[Holm 2015] Holm, D.D., 2015. Variational principles for stochastic fluid dynamics. *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, *471*(2176), p.20140963.

[BdLHLT2020] de Leon, A.B., Holm, D., L, E. and Takao, S., 2019. Implications of Kunita-Ito-Wentzell formula for k-forms in stochastic fluid dynamics. *to appear in Journal of Nonlinear Science.*

### Yann Thanwerdas

Title: Principled Families of Riemannian Metrics on the Manifold of Symmetric Positive Definite Matrices

Abstract: Many Riemannian metrics have been proposed on the manifold of SPD matrices. There is an interest of finding parametric families of Riemannian metrics to characterize the existing ones and to be able to optimize the metric given data. In this talk, we present some recently introduced families of rotation-invariant Riemannian metrics that rely on two principles. The first principle consists in deforming existing metrics to create families indexed by diffeomorphisms: this allows to define the families of power-Euclidean, power-affine and alpha-Procrustes metrics. The second principle is the principle of balanced metrics which is closely related to the concept of dually-flat manifolds: this allows to define the family of mixed-power-Euclidean metrics which correspond in information geometry to AB-divergences.

### Nicolas Guigui

Title: Parallel transport with pole ladder and `geomstats`

Abstract:

Part I: Parallel transport is a fundamental tool to perform statistics on manifolds, yet it is not trivial to compute in general. Stemming from symmetric spaces, the pole ladder scheme turns out to be a simple and efficient method to perform parallel transport by using only exponential and logarithm maps. We present results on its performance in the general case and explore a simple way to improve it in the case of the landmark manifold with the LDDMM metric by using registration errors. Through basic properties of symmetries, we assess the accuracy of our method and compare it to other methods on a real-world data set of heart shapes.

Part II: We introduce `geomstats`, an open-source Python package for computations and statistics for data on non-linear manifolds such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, etc. We provide object-oriented and extensively unit-tested implementations. The manifolds come with families of Riemannian metrics, with associated Exponential/Logarithm maps, geodesics, and parallel transport. The learning algorithms follow scikit-learn API and provide methods for estimation, clustering and dimension reduction on manifolds. This talk will present the package, compare and show some examples. Code and documentation are available at: www.geomstats.ai.