Statistical Foundations for the Analysis of Non-Euclidean Data

Hypothesis testing is one of the core tools in statistical inference and is also relevant for so-called random objects, i.e., data that reside in general (non-Euclidian) metric spaces. Computations must be based on the distance function only, not on algebraic operations applied to the data objects themselves. I am going to talk about hypothesis tests in two specific situations. In the first situation we consider an auto-regressive model for time series of random objects and test the hypothesis about independence over time. The test can be carried out as a permutation test. In the second situation we consider a sample of iid random objects and the hypothesis that the common Frechet mean is identical to a hypothesized value. The test is carried out as a randomization test and requires symmetry structures on the metric space. The talk is based on joint work with Matthieu Bulte.

Correlation matrices encode dependencies between variables, but imposing a rank constraint leads to a space with singularities where standard Riemannian methods do not apply directly. In this talk, we study this space through its representation as a quotient of a product of spheres by an orthogonal group. We show how reducing the factor dimension preserves the geometry of each fixed-rank stratum and provides a principal-bundle model with explicit horizontal lifts. This reduction also characterizes shortest paths constrained by a rank bound, connecting the geometry of the singular space with smooth Riemannian geometry. We conclude with consequences for curvature and holonomy.

Joint work with David Loiseaux and Xavier Pennec.

Objects are everywhere - from the smallest scale of atoms to the complex structure of human brains. Comparing different geometrical objects has been a challenging task over the centuries. It is almost 50 years since David Kendall developed his seminal work on quotient spaces for shapes, where geometrical transformations such as translation, rotation and scale are removed when comparing objects. Since then the quotient space idea has expanded into numerous object-oriented data analysis settings, including images, functions, covariance matrices, networks, compositional and other manifold-valued data, where correspondence, reparameterization and alignment are accounted for. We illustrate these ideas through challenges arising in the analysis of 3D volumetric data, such as forensic assessments of postmortem brains and kidneys where correspondence of volumes is not straightforward. Registration-based principal component analysis and principal nested cones yield interpretable decomposition orderings for the brain data, but not the kidney data. We therefore construct a registration-free pipeline using persistent homology and a fusion of complementary topological distances selected through simulations of cavity formation and growth. Comparison of the registration-based and registration-free methods demonstrates that the utility of both approaches depends on the application. This is joint work with Yanyan Zhan, Yuexuan Wu, Zeloso Tiago and Ed Schwalbe. 

Manifolds are appropriate representation spaces for many complex data objects, including directional data, shape data, data from diffusion tensor imaging. Distributions on manifolds are typically difficult to specify in closed form, and efficient strategies to sample from unnormalised densities must navigate the twofold challenge of high-dimensionality and curvature of the manifold. Starting with an intrinsic Langevin diffusion on a Riemannian manifold with a target distribution as its invariant distribution, I will discuss an intrinsic discrete-time Markov chain for sampling from the target, and identify conditions under which error bounds on approximations of linear functionals of the target distribution match optimal bounds in the Euclidean case. I will present some illustrations on positively and negatively curved manifolds, and comment on extensions. 

Modern omics technologies increasingly produce data that are not naturally represented as vectors. Examples include distributions of cells, geometric shapes, spatially resolved molecular measurements, and multimodal observations in which geometry is coupled with high-dimensional features. These settings raise statistical questions about how to compare, summarize, and align structured observations while respecting their non-Euclidean geometry.

In this talk, I will discuss several approaches motivated by these settings. I will consider problems ranging from comparing large collections of biological shapes, to analyzing shapes equipped with features supported on their geometry, to summarizing spatial transcriptomic tissue sections in which molecular measurements are represented as distributions over geometric domains. Most of the approaches I will discuss are based on optimal transport. Along the way, I will introduce the relevant biological settings and highlight practical challenges that arise in these applications, including heterogeneity, incomplete observations, and geometric distortions such as slicing bias. I hope the talk will also provide a starting point for discussion with researchers working on related non-Euclidean data problems, and I would be very happy to explore potential connections and collaborations.

While the manifold hypothesis is widely adopted in modern machine learning, complex data is often better modeled as stratified spaces---unions of manifolds (strata) of varying dimensions. Stratified learning is challenging due to varying dimensionality, intersection singularities, and lack of efficient models in learning the underlying distributions. We provide a deep generative approach to stratified learning by developing two generative frameworks for learning distributions on stratified spaces. The first is a sieve maximum likelihood approach realized via a dimension-aware mixture of variational autoencoders. The second is a diffusion-based framework that explores the score field structure of a mixture. We establish the convergence rates for learning both the ambient and  intrinsic distributions, which are shown to be dependent on the intrinsic dimensions and smoothness of the underlying strata.  Utilizing the spectral geometry of the score field, we propose a provably consistent algorithm for local intrinsic dimension estimation, enabling the unsupervised partitioning of data by intrinsic dimensionality.  Theoretical results for both frameworks provide fundamental insights into the interplay of the underlying geometry, the ambient noise level, and deep generative models. Extensive simulations and real dataset applications, such as molecular dynamics, demonstrate the effectiveness of our methods.