Abstracts

Eddie Aamari. A theory of stratification learning: Dimension-wise clustering with reconstruction.

Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we will study the minimax estimation of the underlying stratified structure. We will provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon.

Diego Bolón Rodríguez. A geometry-based highest density region estimator for manifold data.

Highest density regions (HDRs) are the subsets of the support where the density function of the data exceeds a given (and usually high) value. Estimating the HDRs of a population from a data sample has multiple practical applications, like clustering analysis or outlier detection. We introduce new geometry-based HDR estimator for manifold data. The new proposal is based on the empirical version of the opening operator from mathematical morphology combined with a preliminary estimator of the density function. This results in an estimator that is easy-to-compute since it simply consists of a radius and a list of centers that are adequately selected from the data. The consistency and the convergence rate (in terms of the Hausdorff distance) of the new estimator are derived. Finally, the performance in practice of the new HDR estimator is illustrated with a real data example. This is a joint work with Rosa M. Crujeiras and Alberto Rodríguez-Casal.

Claire Brécheteau. Learning on mm spaces based on Gromov’s reconstruction theorem.

In this talk, we focus on the questions of testing and learning for datasets represented by a matrix of pairwise distances between datapoints. Such datasets can be considered as discrete versions of metric measure spaces (mm spaces). We recall the Gromov’s mm spaces reconstruction theorem in [1], that states that mm spaces can be represented by the distribution of pairwise distance matrices. We give an alternative proof to this theorem. Then we introduce a new metric between mm spaces, based on this theorem, as an alternative to the Gromov-Wasserstein distance, and prove stability results and in particular parametric rates. As the Gromov-Wasserstein distance, this metric allows to account for variations of both density and shape of datasets. We provide new goodness-of-fit and two-sample tests for mm spaces, but also new classification methods for data given by mm spaces (or pairwise distance matrices), based on this new metric.  Joint work with Thomas Verdebout.

References
[1] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Modern Birkhäuser Classics.
2007.

Elsa Cazelles. Projecting probability measures onto Wasserstein geodesics.

I will first introduce the Busemann function, which naturally defines projections onto geodesic rays in Riemannian manifolds and generalizes the notion of hyperplanes. As the Wasserstein space admits a rich formal Riemannian structure induced by optimal transport metrics, I will then discuss the existence and computation of the Busemann function in this setting. In particular, I will present closed-form expressions in two important cases: one-dimensional probability measures and Gaussian distributions. I will then consider another notion of projection, namely the metric projection, and demonstrate how these projections can be used for transfer learning and geodesic PCA.

Huibin Chang. Mathematical Models and Efficient Algorithms for Ptychography: From Thin-Slice Phase Retrieval to Phase Unwrapping and Thick-Sample Multi-Slice Imaging.

Since the advent of X-ray crystallography and coherent diffraction imaging, phase retrieval has become a fundamental problem in modern imaging, with major impact on structural biology, materials science, and electron microscopy. However, conventional phase retrieval often suffers from ill-posedness, limited field of view, and sensitivity to experimental imperfections. Ptychography, a lensless imaging technique based on overlapped scanning and redundant diffraction measurements, has emerged as a powerful approach for overcoming several of these limitations and has enabled imaging at near-atomic and even sub-angstrom resolution.

This talk presents a unified overview of mathematical models and efficient algorithms for ptychography, spanning thin-sample phase retrieval, phase unwrapping under strong scattering, and thick-sample multi-slice imaging. The main focus is on how to recover reliable phase information from intensity-only measurements when the underlying inverse problem is nonlinear, nonconvex, and increasingly complicated by noise, phase wrapping, and multiple scattering. We discuss how optimization-based formulations improve robustness and efficiency in thin-sample reconstruction, how strong scattering makes phase unwrapping an essential part of the imaging process, and how multi-slice modeling provides a principled framework for thick-sample imaging despite its increased computational and analytical complexity. Overall, the talk highlights a coherent progression from simpler to more challenging imaging regimes and emphasizes several unifying ideas, including physics-based modeling, structured optimization, and plug-and-play priors.

Vincent Divol. Minimax spectral estimation of Laplace operators.

Spectral representations of datasets are routinely used for tasks such as dimension reduction (e.g., LLE, Laplacian Eigenmaps, diffusion maps), regression, and spectral clustering. Typically, these representations are based on the top eigenvectors of a graph Laplacian, derived from a random walk over the dataset. As the number of observations increases, the spectrum of this discrete operator converges to that of a continuous Laplace operator.
While much prior work has focused on giving rates of convergence for this problem, we adopt a different perspective: we study the estimation of the spectrum of weighted Laplace operator from a minimax standpoint. We provide sharp minimax rates for estimating both eigenvalues and eigenfunctions of the limiting operator. Our approach also yields a general framework for estimating regular real-valued functionals on function spaces, with potential implications for other statistical estimation problems.

Stephan Huckeman. The Probability of the Cut Locus of a Fréchet Mean.

We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special  cases (Le and Barden, 2014) and conjectured in general. The proof is based on first order and second order considerations, where the latter are based on a recent result by  Générau (2020) on "Laplacians in the barrier sense". This generalizes to Fréchet p-means  for p > 2. The former allow also to rule out stickiness on Riemannian manifolds, and for generalization to 1 <= p < 2, with a conjecture. We close with discussing and conjecturing extensions to noncomplete manifolds and more general metric spaces. This is joint work with Alexander Lytchak.

Générau, F. (2020). Laplacian of the distance function on the cut locus on a Riemannian manifold. Nonlinearity 33(8), 3928.

Le, H. and D. Barden (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London Mathematical Society 46(4), 698–708.

Lytchak, A. and S. F. Huckemann (2025). Zero mass at the cut locus of a Fréchet mean on a Riemannian manifold. arXiv preprint arXiv:2508.00747.

Huiling Le. Fréchet p-functions and Fréchet p-means.

The Fréchet p-functions and Fréchet p-means are generalisations of widely-used Fréchet functions and Fréchet means. This talk discusses some of their properties relevant to statistical applications. 

Alice Le Brigant. Non parametric information geometry.

In Amari's theory of information geometry, the Fisher information defines a Riemannian metric on a given parametric family of probability distributions seen as a manifold. This metric allows one to compare and interpolate between probability measures inside a same parametric family, and is characterized by its independence with respect to sufficient statistics. Amari and Cencov also introduced a family of affine connections with the same property. In this talk, we will discuss the generalization of these objects to the non parametric case, and study the induced geometric structures on the space of (probability) densities, with a focus on the Amari-Cencov connections.  

Andrea Meilan Vila. Anisotropic geodesic distances for modelling spherical data.

Directional data consist of observations on the unit hypersphere and are commonly modeled using isotropic normal distributions, resulting in density contours in the form of (hyper)spherical circles. These distributions can be formulated using either the extrinsic framework, based on Euclidean embeddings, or the intrinsic framework, which relies on geodesic distances along the hypersphere. However, data frequently exhibit anisotropic structures that cannot be captured by isotropic distributions, motivating the development of more flexible anisotropic alternatives. Existing extrinsic anisotropic models do not account for the curvature of the sphere, while intrinsic approaches rely on tangent-space projections.
We propose an intrinsic formulation of the anisotropic spherical normal distribution that avoids tangent-space projections and is expressed directly on the spherical manifold. Moreover, we demonstrate that the density contours of this distribution are exactly ellipses on the sphere, providing intriguing alternative characterizations for describing this locus of points. Building on this, we introduce a novel unconstrained kernel density estimator for spherical data based on the anisotropic Mahalanobis distance. Unlike existing isotropic estimators, which rely on a single smoothing parameter, the proposed estimator allows full anisotropic flexibility and demonstrates improved performance in simulations.

Amaranta Membrillo Solis. Multiscale geometrical and topological learning in the analysis of soft matter collective dynamics.