Flag spaces for Geometric statistics
Flags are sequences of properly embedded linear subspaces. They appear in multiscale dimension reduction methods such as principal or independent component analysis. Non-linear flags also appear to be the right geometric objects to work with for generalizations of PCA to manifolds such as principal nested spheres or barycentric subspace analysis. One can actually show that extracting order principal components actually optimizes a criterion on the flag space and not on Grassmannians as usually thought. Flag spaces are Riemannian homogeneous spaces that generalize Grassmmann and Steifel manifolds. However, they are usually not symmetric. In this talk, I will present an extension of PCA based on flag spaces called Principal subspace analysis that may turn out to be much more stable that the classical PCA decomposition into unidimensional modes. I will also expose a method to obtain confidence regions on the resulting subspaces based on a geometric formulation of the central limit theorem directly in the space of flags. This is joint work with Tom Szwagier for the first part and Dimbihery Rabenoro for the second part.
Geodesic and stochastic completeness for landmark space
In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in R^d and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of (Q,g).
Symplectic approaches to Langevin diffusions
Langevin diffusions are ubiquitous in physics and statistics where they are used for modelling molecular dynamics and for sampling from distributions on Riemannian manifolds. In this talk, I show that kinetic Langevin diffusions are a special case of symplectic Langevin diffusions, which means that they can be tackled with symplectic numerical methods. Via bi-invariant metrics on reductive Lie groups, we can combine the analytical benefits of the Riemannian theory with algebraic benefits for numerics that Lie group theory provides. Combining this also with symplectic tools, which allow for backward error analysis, one obtains a framework in which both theory and computation can be done pathwise. This talk is based on joint work with Oliver Street from Imperial College London.
Stochastic frozen-flow methods for sampling stochastic dynamics on Riemannian manifolds with high accuracy
In stochastic optimization, molecular dynamics, quantum physics, or in the training of neural networks, one is interested in sampling from the law of a stochastic process. These dynamics are often subject to geometric constraints (fixed distance between particles, Physics Informed Neural Networks, ...). In this context, it is crucial to develop numerical approaches that take into account both the geometric and random features of the dynamics. The literature relies mostly on penalty formulations or extrinsic numerical integrators. These solutions are costly, subject to significant time-step restrictions, and require formulation of the dynamics in much higher-dimensional spaces. In this talk, we present a new class of methods that rely on intrinsic geometric operations (in the spirit of Crouch-Grossman methods), a new robust convergence analysis, and an algebraic formalism of planar exotic Butcher series for the foundation of the weak order theory of intrinsic stochastic methods. A second order integrator is presented and tested numerically for a handful of manifolds. This is joint work with Eugen Bronasco and Baptiste Huguet.
Unbalanced Riemannian Metric Transport and the Wasserstein-Ebin Metric
Work with M. Bauer and FX Viallard: We present weak Riemannian metrics on 5 different spaces related to the space of all Riemannian metrics which fit together via Riemannian submersions. Its geodesics are like the unbalanced version of the Brenier-Otto optimal transport with the most interesting version on the space of Riemannian metrics.
Conditioning Diffusions Using Malliavin Calculus
In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques.
Non-Euclidean Statistics for Learning Structural Biology
We address two problems of predicting ribonucleic acid (RNA) structure geometry on suite level (from one base to one of its adjacent bases) based on experimental maps of electron density. From such electron densities, at high resolution all atomic centers can be detected, except for hydrogen atom positions - as they give away their electron. For this and other reasons, reconstructed RNA geometry can feature clashes, i.e. placing atoms at positions that are chemically not possible. Further, at low resolution, location of the phosphate groups and the glyocosidic bonds can be determined from electron density maps but all other backbone atom positions cannot. The first problem addresses clash removal, the second prediction of high detail structure (all atomic locations).
We propose two learning methods, CLEAN for clash removal and RNAprecis for high detail prediction. This requires creating ground truth gold standard data bases of manually corrected suites. From these, structure clusters are identified by MINT-AGE, an adaptive iterative linkage preclustering method on metric spaces (AGE), mapping each precluster to a (stratified) sphere. Then dimension is reduced by a variant of principal nested spheres (PNS) analysis combined with mode hunting (using the variant WiZer of the SiZer) on the resulting one-dimensional torus (MINT). We illustrate the power of both methods and observe, in particular, that MINT-AGE clusters are in high agreement with curated gold standard suite conformers, sometimes even finer.
This is joint work with Vincent B. Chen, Benjamin Eltzner, Franziska Hoppe, Kanti V. Mardia, Ezra Miller, Michael G. Prisant, Jane S. Richardson, Henrik Wiechers and Christopher J. Williams.
Sampling from conditioned stochastic differential equations
In this talk I will discuss sampling from conditional stochastic differential equations, in particular a method of learning the score term that arises through Doob’s h-transform. We will start with some motivation arising from conditioning stochastic processes of shapes. Then we talk about finding a tractable loss function for the score term via the Kullback-Leibler divergence. In order to learn this term, we will use another type of process called the adjoint process which somehow encodes the reverse dynamics of the unconditioned process, and from which we can sample from. We will use these adjoint processes to get a tractable loss function that we minimise to learn the score term, resulting in a way to sample from the conditioned processes.
On robust learning of surface deformations
Computing physically plausible deformations of surface meshes is critical for applications ranging from animation to medical imaging. Traditional optimization techniques provide a valuable theoretical framework, for example, for statistical analysis, but often suffer from poor computational efficiency and extensive parameter tuning, limiting their practical applicability for unprocessed data. Recent deep-learning methods improve efficiency, yet they typically require registered meshes with fixed resolution and vertex-wise correspondence, which reduces their robustness and utility when such data is scarce.
In this talk, we present several approaches for robust deformation computation, from regularized latent-space models to geometry-informed neural deformation fields. We will also highlight key open research directions and challenges in developing more generalizable
and effective deformation models.
Score-Optimal Diffusion Schedules
Denoising diffusion models (DDMs) define a stochastic flow from data to a Gaussian prior, tracing a path of probability distributions. The quality of generated samples depends critically on how this path is discretised; however, current approaches largely rely on heuristics.
We introduce an adaptive discretisation scheme grounded in basic tools from information geometry and simple energy minimisation principles. Our method minimises a cost derived from the work required to transport samples along the diffusion path, using only the estimated Stein score. The optimal solution corresponds to a geometric invariant specific to the data distribution on which the model is trained. The approach is hyperparameter-free, structurally aligned with the diffusion process, and integrates with pre-trained models. Empirically, it recovers performant hand-tuned schedules and achieves competitive FID scores on standard image datasets.
In this talk, I will derive the cost function, identify the associated geometric invariant, and demonstrate how this framework improves sampling performance on both image data and mollified fractal distributions.
Score-based pullback Riemannian geometry: Extracting the Data Manifold Geometry using Anisotropic Flows
Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold
On the interplay between Mutual Information and Diffusion Processes
Mutual Information (MI) is essential for quantifying dependencies in complex systems, yet accurately estimating it in high-dimensional settings is challenging. In this talk, we introduce novel methods for MI estimation using diffusion processes. First, we present MINDE, an approach that leverages score-based diffusion models and an interpretation of the Girsanov theorem to estimate the Kullback-Leibler divergence between probability densities. This enhances MI and entropy estimation, outperforming existing techniques. Additionally, we introduce SΩI, a method for computing O-information (a generalization of mutual information to more than two variables) without restrictive assumptions, effectively capturing higher-order dependencies and revealing the synergy-redundancy balance in multivariate systems.
We then investigate how MI affects the dynamics of generative models based on SDEs. By extending Nonlinear Filtering (NLF), we develop a theoretical framework that shows how MI quantifies the influence of unobservable latent abstractions on generative pathways. Empirical studies validate our theory, demonstrating how MI guides the evolution of latent variables that steer the generative process.
SO(3)-Equivariant Neural Networks for Learning Vector Fields on Spheres
Analyzing vector fields on the sphere, such as wind speed and direction on Earth, is a difficult task. Models should respect both the rotational symmetries of the sphere and the inherent symmetries of the vector fields. We introduce a Fourier-type deep learning architecture that respects both symmetry types of vector fields on the sphere using novel techniques based on group convolutions in the 3-dimensional rotation group SO(3). This architecture is suitable for both scalar and vector fields on the sphere as they can be described as equivariant signals on the 3-dimensional rotation group. Our experiments show that this architecture achieves lower prediction and reconstruction error when tested on rotated data compared to a standard convolution neural network, and higher expressivity when compared to other equivariant architectures.
Inference of Factor Graphs under Topological Transformations
Graphical models and factor graphs are probabilistic models that incorporate prior knowledge of dependencies between variables; celebrated examples include hidden Markov models. Computing the posterior distribution for a given collection of observations is called inference and is, in general, computationally very costly. In practice, one often resorts to variational inference, which consists in optimizing a weighted mean free energy over subcollections of variables, under the constraint that their probability distributions are compatible by marginalization. This compatibility condition defines the space of sections of specific presheaves. The General Belief Propagation algorithm is used to find the critical points of the weighted free energy. We will first explain how one can extend factor graphs to account for a broader class of relations between subcollections of variables, by generalizing results from those specific presheaves to arbitrary presheaves over a poset. Given this broader framework, we ask how transformations on those presheaves affect the optimization problem and the associated algorithms. In particular, we show that natural transformations induce transformations between algorithms in a functorial manner. We then demonstrate that inference on minimal deformation retracts of a poset of rank 2 is sufficient for inference on the entire poset, yielding a « topological classification » of inference on factor graphs.
Metric geometry of spaces of persistence diagrams
Persistence diagrams are central objects in topological data analysis. They are pictorial representations of persistence homology modules and describe topological features of a data set at different scales. In this talk, I will discuss the geometry of spaces of persistence diagrams and connections with the theory of Alexandrov spaces, which are metric generalizations of complete Riemannian manifolds with sectional curvature bounded below. In particular, I will discuss how one can assign to a metric pair $(X, A)$ a one-parameter family of pointed metric spaces of (generalized) persistence diagrams $D_p(X, A)$ with points in $(X, A)$ via a family of functors $D_p$ with $p \in [1, \infty]$. These spaces are equipped with the $p$-Wasserstein distance when $p \geq 1$ and the bottleneck distance when $p = \infty$. The functors $D_p$ preserve natural metric properties of the space $X$, including non-negative curvature in the triangle comparison sense when $p = 2$. When $p = \infty$, the functor $D_\infty$ is continuous with respect to a suitable notion of Gromov–Hausdorff distance of metric pairs. When $(X, A) = (\mathbb{R}^2,\Delta)$, where $\Delta$ is the diagonal of $\mathbb{R}^2$, one recovers previously known properties of the usual spaces of persistence diagrams. I will also discuss some connections of these results with optimal partial transport. The talk is based on joint work with Mauricio Che, Luis Guijarro, Ingrid Membrillo Solis, and Motiejus Valiunas.
Iterative Importance Fine-Tuning of Diffusion Models
Diffusion models are an important tool for generative modelling, serving as effective priors in applications such as imaging and protein design. A key challenge in applying diffusion models to downstream tasks is efficiently sampling from resulting posterior distributions, which can be addressed using the h-transform. This talk presents a self-supervised algorithm for fine-tuning diffusion models by estimating the h-transform, enabling amortised conditional sampling. Our method iteratively refines the h-transform using a synthetic dataset resampled with path-based importance weights. The resampling step follows prior work on rejection-resampling steps for sampling from unnormalized probability density functions. We demonstrate the effectiveness of this framework on class-conditional sampling and reward fine-tuning for text-to-image diffusion models.
Debiasing Guidance for Discrete Diffusion with Sequential Monte Carlo
Discrete diffusion models are a class of generative models that produce samples from an approximated data distribution within a discrete state space. Often, there is a need to target specific regions of the data distribution. Current guidance methods aim to sample from a distribution with mass proportional to p(x0)p(ζ|x0)^α but fail to achieve this in practice. We introduce a Sequential Monte Carlo algorithm that generates unbiasedly from this target distribution, utilising the learnt unconditional and guided process. We validate our approach on low-dimensional distributions, controlled images and text generations. For text generation, our method provides strong control while maintaining low perplexity compared to guidance-based approaches.