Statistics of shapes has a long tradition in medical image analysis and biology. This spans from classical landmark representations and Kendall's shape space to infinite dimensional shape spaces with rich geometric structures. Recent work has added stochastics and statistical inference for stochastic processes to the picture. Shape analysis, shape statistics, and shape stochastics give rise to new research directions in both pure and applied mathematics, and new developments have direct impact in applied fields such as evolutionary biology. The aim of the conference is to establish connections between the communities of shape analysis, differential geometry and statistics for stochastic processes. Focus will be on intersections between geometry including sub-Riemannian geometry, shape analysis, stochastic analysis, and applications in biology, including phylogenetic inference.
A lot of unexpected mathematical developments around shape spaces and statistics of shapes has been fueled by the need to analyze biological shape variations in neuroimaging leading to the new field of computational anatomy. Nevertheless, current descriptions of brain diseases usually need to put together several orders of magnitude ranging from the millimeter scale for tissues in standard imaging devices to the micron or even nano scale for neural cells and molecules and to go across modalities. I will present some of the new challenges for mathematicians and our recent attempts to make a step in that direction.
The impact of manifold curvature on the statistical estimation or on the accuracy of algorithms to compute on manifolds is not evident to establish. We show in this talk that Gavrilov's Taylor expansions of the double exponential can be completed by a companion neighboring log expansion which measures how the Riemannian logarithm changes when its foot-point is moving. These two tensorial and coordinate free Taylor expansions constitute a complete toolbox for the polynomial approximations of problems related to infinitesimal geodesic triangles. Moreover they are valid in general affine connection manifolds and computations can surprisingly easily be pushed to higher orders if needed.
We exemplify the use of this toolbox for two of fundamental tools for geometric statistics: the estimation of the Fréchet mean and parallel transport. We first present a new non-asymptotic (small sample) expansion in high concentration conditions which quantifies the concentration of the empirical Fréchet mean towards the population mean. This shows a statistical bias on the empirical mean in the direction of the average gradient of the curvature and a modulation of the convergence speed which could partly explain smeary means in positive curvature spaces and sticky means in stratified spaces of negative curvature.
Parallel transport is a second major tool to compare local tangent information at different points of the manifold. We previously proposed a modification of Schild's ladder called pole ladder that is surprisingly exact in only one step on symmetric spaces. Iterating geodesic parallelograms was thought to be of first order but a real convergence analysis was lacking. We show that pole and Schild's ladder naturally converges with quadratic speed even when geodesics are approximated by numerical schemes. This contrasts with Jacobi fields approximations that are bound to linear convergence. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy.
For a stochastic process evolving on a (directed) tree I will show that sampling from the smoothing distribution can essentially be done by defining forward and backward maps, together with composition rules for the pairing of these maps. If transitions over edges can be composed, then it is natural to ask whether the corresponding composition rule of the resulting pairing is equivalent to composition of the separate pairings. The answer is affirmative.
Based on joint work with Moritz Schauer
Finding meaningful ways to determine the dependency between two random variables ξ and ζ is a timeless statistical endeavor with vast practical relevance. In recent years, several concepts that aim to extend classical means (such as the Pearson correlation or rank-based coefficients like Spearman's ρ) to more general spaces have been introduced and popularized, a well-known example being the distance correlation. In this article, we propose and study an alternative framework for measuring statistical dependency, the transport dependency τ≥0, which relies on the notion of optimal transport and is applicable in general Polish spaces. It can be estimated consistently via the corresponding empirical measure, is versatile and adaptable to various scenarios by proper choices of the cost function. Notably, statistical independence is characterized by τ=0, while large values of τ indicate highly regular relations between ξ and ζ. Indeed, for suitable base costs, τ is maximized if and only if ζ can be expressed as 1-Lipschitz function of ξ or vice versa. Based on sharp upper bounds, we exploit this characterization and define three distinct dependency coefficients (a-c) with values in [0,1], each of which emphasizes different functional relations. These transport correlations attain the value 1 if and only if ζ=φ(ξ), where φ is a) a Lipschitz function, b) a measurable function, c) a multiple of an isometry. The properties of coefficient c) make it comparable to the distance correlation, while coefficient b) is a limit case of a) that was recently studied independently by Wiesel (2021). We address several practical issues, such as fast computational schemes for transport dependency and permutation based independence testing. Numerical results suggest that the transport dependency is a robust quantity that efficiently discerns structure from noise in simple settings, often out-performing other commonly applied coefficients of dependency. Finally, we reanalyze a data set from cancer genetics based on hierarchical trees and by means of its gene expression correlation. This is joint work with Giacomo Nies and Thomas Staudt.
Statistical shape models have been established as one of the most successful methods for understanding the geometric variability in shape populations. There is increasing evidence that intrinsic approaches, which account for the non-Euclidean structure inherent to shapes, are essential to faithfully capture the large variability. This talk presents a nonlinear, rigid motion invariant approach for shape analysis based on concepts from differential geometry of smooth surfaces. Performing geodesic calculus on this representation allows for fast computations opening up the potential in large shape databases accessible through longitudinal and multi-site imaging studies. The rich structure of the derived shape space yields highly differentiating shape descriptors providing a compact representation that is amenable to learning algorithms. The advantages over alternative approaches will be demonstrated on the example of shape-based assessment and classification of morphological disorders.
This talk brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Signed distance fields are fundamental to shape representation and analysis. Topological data analysis uses algebraic topology to characterize geometric and topological patterns in shapes. The most well-known tool from this approach is persistent homology that tracks the evolution of topological features in a dynamic manner as a barcode; here we study the persistent homology of shape textures using sublevel set filtration induced by the signed distance. Morse theory is a framework from differential topology that studies critical points of functions on manifolds and has been used to characterize the lifetimes of persistent homology features. However, a significant limitation here is that it cannot be readily applied to distance functions because of their lack of smoothness. In this paper, we generalize Morse theory to Euclidean distance functions of bounded sets with smooth boundaries. We use transversality theory to prove that for generic embeddings of a smooth compact surface in R3, signed distance functions admit finitely-many non-degenerate critical points. Thus, signed distance persistence modules of generic shapes admit a finite barcode decomposition whose birth and death points can be classified and described geometrically, providing a rigorous characterization of shape textures. We use this approach to quantify shape textures on both simulated data and real vascular data from biology. This is joint work with Anna Song (Imperial College London, The Francis Crick Institute) and Ka Man (Ambrose) Yim (Oxford).
Many methods of shape analysis, such as the LDDMM framework, use sub-Riemannian geometry. Moreover, one often needs to consider infinite dimensions. However, in this case, many complications can arise. I will give a brief overview of what can and cannot be done in infinite dimensional sub-riemannian manifolds, and give an in-depth study of the geodesics of a non-trivial example: the free 2-step nilpotent group modeled on a Hilbert space.
Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack pluss to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds.
Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications.
The results are part of joint work with Stefan Sommer (Copenhagen, Denmark)
In this paper, we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles, denoted principal subbundles. In particular, we compute local approximations by local PCAs and collect them into a rank \textit{k} tangent subbundle on $\mathbb{R}^d$, $k
Complex object data such as networks and shapes are becoming increasingly available, and so there is a need to develop suitable methodology for statistical analysis. Networks can be represented as graph Laplacian matrices, which are a type of manifold-valued data. Shapes of 3D objects are also a type ofmanifold-valued data, invariant to translation, rotation and scale. Our main objective is to estimate a regression curve from a sample of graph Laplacian matrices or 3D shapes conditional on a set of Euclidean covariates, for example in dynamic objects where the covariate is time. We develop an adapted Nadaraya-Watson estimator which has uniform weak consistency for estimation using Euclidean and power Euclidean metrics, and we also explore splines on shape spaces.
In statistics on manifolds, we introduce a new family of location statistics describing centers of isotropic diffusion for different diffusion times. In contrast to the situation in Euclidean data, these diffusion means on manifolds do not generally coincide for different diffusion times. In the limit of vanishing diffusion time, diffusion means can be shown to converge to the intrinsic mean in general. For the limit of infinite diffusion time, we show for the circle and spheres of arbitrary dimension that diffusion means converge to the extrinsic mean in the canonical embedding. This yields an appealing interpretation of the extrinsic mean and a definition without reference to an embedding. Furthermore, we show that diffusion means with simultaneously estimated diffusion time have appealing regularity properties which can make them preferable to the intrinsic mean.
In this talk I will present a sheaf theoretic construction of shape space (or the space of all shapes). We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision. This is joint work with Justin Curry and Sayan Mukherjee.
In this talk, we will explore how geometric objects such as manifolds and fiber bundles can aid in studying the statistics of shape space. We will begin by discussing the concept of shape space as a manifold and review the use of Diffusion Maps (DM) in analyzing the underlying structure of the space. We will also present a new parameter tuning method for DM that takes advantage of the semigroup properties of the diffusion kernel. Next, we will consider the shape space as approximately a fiber bundle and review Horizontal Diffusion Maps (HDM). We will introduce a probabilistic model for learning the generative process of data on fiber bundles, and define Gaussian processes on fiber bundles. Throughout the talk, we will use examples from geometric morphometrics to illustrate the use of these methods.
Most existing metrics between phylogenetic trees directly measure differences in topology and edge weights, and are unrelated to the models of evolution used to infer trees. The wald space is a newly developed geodesic metric space which arose from a shift of viewpoint, in which trees are identified with the probability models on genetic sequence data they induce. We describe a family of metrics between trees which are pull-backs of metrics between probability distributions of discrete characters induced by trees. These behave very differently from existing metrics and we illustrate this using some simple examples. In order to do statistics on data sets of trees, it is highly desirable to construct metric spaces which are lengths spaces, or even better, geodesic spaces, and we describe how to construct a length space using probabilistic metrics. Locally, this space consists of a Riemannian manifold for each collection of trees with a fixed branching topology, in which the Riemannian metric is the Fisher information metric. Calculations in this space are highly computationally intensive due to the use of discrete characters. By developing a related Gaussian Markov process model for a continuous trait, we are able to identify trees with certain Gaussian distributions and this enables much faster computation. The same geometry is obtained by embedding trees in the space of positive definite matrices with the affine invariant metric, by considering covariance matrices of Gaussians. This embedded space is the wald space.
We further explore the wald space for phylogenetic trees introduced in Tom Nye's talk. As a point set, it sits between the BHV space (Billera, Holmes and Vogtmann, 2001) and the edge-product space (Moulton and Steel 2004). It has a natural embedding in the space of positive definite matrices, equipped with the information geometry. Thus, singularities such as overlapping leaves are infinitely far away, proper forests, however, comprising the "BHV-boundary at infinity", are part of the wald space, adding boundary correspondences to groves (corresponding to orthants in the BHV space). In fact the wald space contracts to the complete disconnected forest. Further, it is a geodesic space, exhibiting the structure of a Whitney stratified space of type (A) where strata carry compatible Riemannian metrics. We explore some more geometric properties, but the full picture remains open. We conclude by identifying interesting and open problems, we deem interesting.
Co-authors Tom Nye, Jonas Lueg, Maryam Garba
The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think "curvature") is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural enhancement, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran.
Empirical shapes, for instance those realized in biological organisms, constitute a tiny subset of the mathematically possible ones. They are constrained by coherence between their features. And their classification should incorporate the invariance under suitable classes of transformations. I shall sketch some mathematical framework to account for that.
RNA secondary structures are a discrete, graphical model of the 3D shapes of RNA molecules that capture the dominating part of the energy of structure formation and constitute a key intermediate in the formation of the final 3D structure. The relative simplicity of secondary structures makes it possible to efficiently compute ground states and thermodyanmic parameters for given sequences. This sets the stage for exploring the evolution of RNA shapes in response to variations of the underling sequences. At the same time, the same model can be used to study the dynamical process of folding, i.e., structure formation, for a given sequence. The sequence-shape map of RNA has a number of unusual properties, in particular a high degree of neutrality combined with a the proximity of of very different shapes. These properties together explain key features of RNA evolution.
Scientific Organizers
Karen Habermann
University of Warwick
Sayan Mukherjee
Max Planck Institute for Mathematics in the Sciences, Leipzig
Max von Renesse
Leipzig University
Stefan Horst Sommer
University of Copenhagen
Administrative Contact
Katharina Matschke
Max Planck Institute for Mathematics in the Sciences, Leipzig
Contact by email
