Wednesday, 29 October 2025
Note: This program includes links that allow direct access to detailed sections of the website (keynotes). Clicking a link will automatically take you to the relevant section, even if it is located on a different page.
09:00 - Opening & Keynote Session
09:00
Conference Opening Session
09:15
Nina MIOLANE
(UC Santa Barbara, USA)
10:30 - Coffee Break + GSI'25 Posters Session
Auditorium Maupertuis
Geometric Statistics (Session 1)
Chairman : Xavier PENNEC, Stefan SOMMER, Benjamin ELTZNER
11:00 - Ridge Regression for Manifold-valued Time-Series with Application to Hurricane Forecasting (7) Esfandiar Nava‑Yazdani
We propose a natural intrinsic extension of the ridge regression from Euclidean spaces to general manifolds, which relies on Riemannian least-squares fitting, empirical covariance, and Mahalanobis distance. We utilize it for time-series prediction and apply the approach to forecast hurricane tracks and their intensities (maximum wind speeds).
11:20 - On the approximation of the Riemannian barycenter (63) Simon Mataigne
We present a method to compute an approximate Riemannian barycenter of a collection of points lying on a Riemannian manifold. Our approach relies on the use of theoretically proven under- and overapproximations of the Riemannian distance function. We compare it to the exact computation of the Riemannian barycenter and to an approach that approximates the Riemannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold.
11:40 - Accelerated Stein Variational Gradient Flow (33) Viktor Stein
Stein variational gradient descent (SVGD) is a kernel-based particle method for sampling from a target distribution, e.g., in generative modeling and Bayesian inference. SVGD does not require estimating the gradient of the log-density, which is called score estimation. In practice, SVGD can be slow compared to score-estimation based sampling algorithms. To design fast and efficient high-dimensional sampling algorithms, we introduce ASVGD, an accelerated SVGD, based on an accelerated gradient flow in a metric space of probability densities following Nesterov’s method. We then derive a momentum-based discrete-time sampling algorithm, which evolves a set of particles deterministically. To stabilize the particles’ momentum update, we also study a Wasserstein metric regularization. For the generalized bilinear kernel and the Gaussian kernel, toy numerical examples with varied target distributions demonstrate the effectiveness of ASVGD compared to SVGD and other popular sampling methods.
12:00 - Geodesic Non-completeness of the Truncated Normal Family (68) Baalu Ketema
Motivated by robustness studies under uncertainty of computer codes that simulate the behavior of a physical system, we are brought to inspect geodesic completeness of parametric families of truncated probability distributions. Specifically, we focus on the parametric family of truncated normal distributions with fixed truncation interval. Endowed with the Fisher information metric, this family can be seen as a Riemannian manifold. We prove that it is not geodesically complete and conjecture a potential candidate for the completion.
Room Vauban 1
A geometric approach to differential equations (Session 1)
Chairman : Javier de Lucas ARAUJO
11:00 - Reduction of hybrid Hamiltonian systems with non-equivariant momentum maps (18) Asier López‑Gordón
We develop a reduction scheme à la Marsden-Weinstein-Meyer for hybrid Hamiltonian systems. Our method does not require the momentum map to be equivariant, neither to be preserved by the impact map. We illustrate the applicability of our theory with an example.
11:20 - New Lie systems from Goursat distributions: reductions and reconstructions (121) Oscar Carballal
We show that types of bracket-generating distributions lead to new classes of Lie systems with compatible geometric structures. Specifically, the n-trailer system is analysed, showing that its associated distribution is related to a Lie system if n = 0 or n = 1. These systems allow symmetry reductions and the reconstruction of solutions of the original system from those of the reduced one. The reconstruction procedure is discussed and indicates potential extensions for studying broader classes of differential equations through Lie systems and new types of superposition rules.
11:40 - Symplectic approach to global stability (124) Jordi Gaset Rifà
We present a new approach to the problem of proving global stability, based on symplectic geometry and with a focus on systems with several conserved quantities. We also provide a proof of instability for integrable systems whose momentum map is everywhere regular. Our results take root in the recently proposed notion of a confining function and are motivated by ghost-ridden systems, for whom we put forward the first geometric definition.
12:00 - Reduction of exact symplectic manifolds and energy hypersurfaces (145) Bartosz Zawora
This article introduces two reduction schemes for Hamiltonian systems on an exact symplectic manifold admitting Lie group symmetries. It is demonstrated that these reduction procedures are equivalent, by employing a modified Marsden–Meyer–Weinstein reduction theorem for exact symplectic manifolds and contact manifolds given by energy hypersurfaces. Each approach is illustrated through an example.
Room Vauban 2
Lie Group in Learning Distributions & in Filters (Session 1)
Chairman : Eren M. KIRAL, Koichi TOJO, Ha Q. MINH
11:00 - F‑t Joint Distribution on a Real Siegel Domain and Simultaneous Hypothesis Test (61) Hiroto Inoue
This article introduces the $F$-$t$ distributions on the real Siegel domain associated to the cone of positive definite symmetric matrices.
These distributions arise from a random variable defined in a group theoretical way using the quadratic map to the cone.
Its density function is also provided based on analysis on the real Siegel domains.
As an application, we present a numerical experiment for an invariant simultaneous test for two-sample problem.
11:20 - Note on harmonic exponential families on homogeneous spaces (77) Koichi Tojo
In [5], we proposed a method to construct a G-invariant exponential family on a homogeneous space G/H by using a representation of G. In this paper, we prove that for any G-invariant exponential family P on G/H, we can construct a family P_0 by the method such that P ⊂ P_0.
11:40 - The Fisher metric and the Amari–Chentsov tensor of the family of Poincaré distributions (76) Koichi Tojo
We give a simple description of the Fisher metric and the Amari–Chentsov tensor of the family of Poincaré distributions on the upper half plane by using a coordinate compatible with SL(2, R)-action.
12:00 - Fast equivariant k-means on SPD matrices (90) Gabriel Trindade
In this paper, we propose an efficient alternative to the affine-invariant Riemannian k-means algorithm on symmetric positive definite matrices. Recently introduced log-extrinsic means are coupled with the Jensen-Bregman log-det divergence, as a replacement for the Riemannian Fréchet mean and the Riemannian distance. Performances and computation times are compared for several frameworks on point clouds sampled from Riemannian Gaussians. Results show that our algorithm matches the clustering accuracy of the affine-invariant Riemannian k-means, while achieving runtimes comparable to those of log-Euclidean k-means.
12:20-12:40 - Lunch break + GSI'25 Posters Session
14:00
Alice Le Brigant
(Université Paris 1 Panthéon-Sorbonne, France)
Auditorium Maupertuis
Geometric Statistics (Session 2)
Chairman : Xavier PENNEC, Stefan SOMMER, Benjamin ELTZNER
15:00 - Intrinsic LDA for 3D Shape Classification via Parallel Transport (94) Maria Victoria Ibáñez-Gual
In this paper we propose a novel methodology that extends Linear Discriminant Analysis (LDA) to Kendall’s shape space to classify 3D shapes and analyze which features most influence class differentiation. Our approach adapts LDA to the non-Euclidean geometry of the shape space, generalizing assumptions about the probability distribution of data in Euclidean spaces and incorporating parallel transport to improve the estimation of shape variability between clusters. A simulation study is performed to show the effectiveness of the proposed methodology.
15:20 - Eigengap Sparsity for Covariance Parsimony (99) Tom Szwagier
Covariance estimation is a central problem in statistics. An important issue is that there are rarely enough samples $n$ to accurately estimate the $p (p+1) / 2$ coefficients in dimension $p$. Parsimonious covariance models are therefore preferred, but the discrete nature of model selection makes inference computationally challenging. In this paper, we propose a relaxation of covariance parsimony termed « eigengap sparsity » and motivated by the good accuracy–parsimony tradeoff of eigenvalue-equalization in covariance matrices. This new penalty can be included in a penalized-likelihood framework that we propose to solve with a projected gradient descent on a monotone cone. The algorithm turns out to resemble an isotonic regression of mutually-attracted sample eigenvalues, drawing an interesting link between covariance parsimony and shrinkage.
15:40 - RNA Structure Correction – the Importance of Small Clusters (138) Benjamin Eltzner
RNA residues come in a plethora of geometric conformational shapes, some of which are very common and some of which are rather rare. In this contribution we extend the previously developed clustering algorithm MINT-AGE in order to find within a large database rare conformational clusters of very small sizes. To this end in the MINT step we replace the previous nonparametric circular mode hunting by a parametric version. In validation, this allows to identify conformational classes of sizes ≥ 2 via statistical unsupervised learning on the gold standard database, hand curated by the Richardson Laboratory.
Room Vauban 1
A geometric approach to differential equations (Session 2)
Chairman : Bartosz ZAWORA
15:00 - Novel pathways in $k$-contact geometry (142) Tomasz Sobczak, Tymon Frelik
Our study of Goursat distributions originates new types of $k$-contact distributions and Lie systems with applications. In particular, families of generators for Goursat distributions on $\mathbb{R}^4, \mathbb{R}^5$ and $\mathbb{R}^6$ give rise to Lie systems, and we characterise Goursat structures that are $k$-contact distributions. Our results are used to study zero-trailer and other systems via Lie systems and $k$-contact manifolds. New ideas for the development of superposition rules via geometric structures and the characterisation of $k$-contact distributions are given and applied. Some relations of $k$-contact geometry with Cartan theory are inspected.
15:20 - A relation between $k$-symplectic and $k$-contact Hamiltonian systems (129) Silvia Vilariño
Systems of partial differential equations which appear in classical field theories can be studied geometrically using different geometrical structures, for example, k-symplectic geometry, k-cosymplectic geometry, multisymplectic geometry, etc.
In recent years, there has been a notable increase in the study of k-contact Hamiltonian systems. These are based on the description of the dynamics of field theories using the so-called k-contact manifolds. Such structures are generalizations of contact structures and k-symplectic structures.
The relation between k-symplectic manifolds and k-contact manifolds was established in \cite{LRS24}. In light of the above relation, this work seeks to explore the relationship between k-symplectic Hamiltonian systems and k-contact Hamiltonian systems.
15:40 - Applications of standard and Hamiltonian stochastic Lie systems (139) Javier de Lucas Araujo
A stochastic Lie system on a manifold $M$ is a stochastic differential equation whose dynamics is described by a linear combination with functions depending on $\mathbb{R}^\ell$-valued semi-martigales of vector fields on $M$ spanning a finite-dimensional Lie algebra. We analyse new examples of stochastic Lie systems and Hamiltonian stochastic Lie systems, and review the coalgebra method for Hamiltonian stochastic Lie systems. We apply the theory to biological and epidemiological models, stochastic oscillators, stochastic Riccati equations, coronavirus models, etc.
Room Vauban 2
Lie Group in Learning Distributions & in Filters (Session 2)
Chairman : Eren M. KIRAL, Koichi TOJO, Ha Q. MINH
15:00 - A new geometric regression with inputs-outputs on matrix Lie groups (88) Serigne Daouda Pene
This paper investigates a new Lie group regression model for input-output data belonging to Lie groups. The originality of the model lies in the fact that the unknown weights also lie in Lie groups and are learned using an intrinsic optimization algorithm based on maximum likelihood estimation. The model is validated through numerical simulations conducted using synthetic data belonging to the Lie group SO(3), which is commonly used in robotics to represent rotational observations.
15:20 - Equivariant Filter: navigation on the rotating round-earth model using a left-error state (89) Alexandre Cellier-Devaux
For navigation problems based on Flat Earth equations with inertial sensor biases, the system’s equivariance principle on the Semi-Direct Group enables the design of filters that improves estimation error and covariance compared to legacy Extended or Invariant Kalman Filtering. We derived the equivariance principle for comprehensive navigation on a rotating, round Earth by choosing appropriate reference frames and filter architecture to preserve natural symmetries in the system equations. For the filter design, we chose a right-equivariance structure with a left error (in the local reference frame) of the estimation state and compared it to the usual academic choice of right error (in the global reference frame). This approach aims to take advantage of inertial sensor outputs during filter propagation and avoid making the observation matrix dependent on attitude error. In the end, we compare the performances of bias estimation, covariance dynamics, and estimated error accuracy of our L-EqF filter against both EKF (SO(3)xR¹²) and IEKF (SE₂(3)xR⁶) in a simulation representative of a GNSS-denied scenario and fast alignment.
15:40 - Sequential parallel Metropolis-Adjusted Langevin Algorithm on Matrix Lie Groups (131) Enzo Lopez
Langevin-based Monte Carlo Markov Chain methods provide a powerful framework for nonlinear state estimation. Using Langevin dynamics for efficient state transitions, these methods offer a robust alternative to traditional nonlinear filtering techniques. However, standard approaches suffer from high computational costs, the curse of dimensionality, and sensitivity to local maxima. To address these challenges, we extend sequential Metropolis Adjusted Langevin Algorithm (MALA) techniques to parallel chains on Lie Groups, leading to the Lie Group parallel Metropolis-Adjusted Langevin Algorithm (LG-pMALA) filter.
16:00-16:30 - Coffee Break + GSI'25 Posters Session
Auditorium Maupertuis
Neurogeometry
Chairman : Alessandro SARTI, Giovanna CITTI, Giovanni PETRI
16:30 - Geometric neural fields for cortical activity (36) Emre Baspinar
Neural fields refer to integro-differential equations which model the average neural activity of a neural population in a coarse-grained limit. In classical neural fields, which follow Wilson-Cowan-Amari formalism, the neural interactions are modeled based on a distance-based connectivity, without
taking into account the modulatory effects of functional properties of neurons on the connectivity. Such effects are observed in particular in the primary visual cortex (V1). In this work, we consider a neural field which takes into account these effects in the connectivity by focusing on the functional architecture of V1. This model was applied to a specific family of visual illusions, to reproduce the cortical activity generating the illusions. We will explain this model, and discuss its potential to an extension towards pathological cortical activity.
16:50 - Log-Euclidean Frameworks for Smooth Brain Connectivity Trajectories (48) Olivier Bisson
The brain is often studied from a network perspective, where functional activity is assessed using functional Magnetic Resonance Imaging (fMRI) to estimate connectivity between predefined neuronal regions. Functional connectivity can be represented by correlation matrices computed over time, where each matrix captures the Pearson correlation between the mean fMRI signals of different regions within a sliding window. We introduce several Riemannian Log-Euclidean framework for constructing smooth approximations of functional brain connectivity trajectories. Representing dynamic functional connectivity as time series of full-rank correlation matrices, we leverage recent theoretical Log-Euclidean diffeomorphisms to map these trajectories in practice into Euclidean spaces where polynomial interpolation becomes feasible. Pulling back the interpolated curve ensures that each estimated point remains a valid correlation matrix, enabling a smooth, interpretable, and geometrically consistent approximation of the original brain connectivity dynamics. Experiments on fMRI-derived connectivity trajectories demonstrate the geometric consistency and computational efficiency of our approach.
17:10 - A heterogeneous model of boundary and figure completion in V1 (91) Mattia Galeotti
We propose a neurally based model of joint boundary and figure completion, which takes into account the arrangements of simple cells in orientation maps. This map is modeled as a regular surface in a sub-Riemannian structure and the propagation process is studied on the surface. Application to Kanizsa triangle is considered, and results compared with high-resolution fMRI measurements of completion phenomena in V1.
17:30 - Geometry of Cells Sensible to Curvature and Their Receptive Profiles (97) Vasiliki Liontou
We propose a model of the functional architecture of curvature sensible cells in the visual cortex that associates curvature with scale. The feature space of orientation and position is naturally enhanced via its oriented prolongation, yielding a 4-dimensional manifold endowed with a canonical Engel structure. This structure encodes position, orientation, signed curvature, and scale. We associate an open submanifold of the prolongation with the quasi-regular representation of the similitude group $SIM(2)$, and find left-invariant generators for the Engel structure. Finally, we use the generators of the Engel structure to characterize curvature-sensitive receptive profiles .
17:50 - Orientation Scores should be a Piece of Cake (60) Finn Sherry
We axiomatically derive a family of wavelets for an orientation score, lifting from position space R^2 to position and orientation space R^2 x S^1, with fast reconstruction property, that minimise position-orientation uncertainty.
We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1.
Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets.
Finally, we show experimentally that in this way we can reduce the network complexity and improve the neurogeometric interpretability of (PDE-)G-CNNs, with only a slight impact on the model’s performance.
Room Vauban 1
New trends in Nonholonomic Systems
Chairman : Manuel de LEON, Leonardo COLOMBO
16:30 - Virtual nonlinear nonholonomic constraints from a symplectic point of view (43) Alexandre Simoes
In this paper, we provide a geometric characterization of virtual nonlinear nonholonomic constraints from a symplectic perspective. Under a transversality assumption, there is a unique control law making the trajectories of the associated closed-loop system satisfy the virtual nonlinear nonholonomic constraints. We characterize them in terms of the almost-tangent and a symplectic structure on $TQ$. In particular, we show that the closed-loop vector field satisfies a geometric equation of Chetaev type. Moreover, the closed-loop dynamics is obtained as the projection of the uncontrolled dynamics to the tangent bundle of the constraint submanifold defined by the virtual constraints.
16:50 - Geometric Stabilization of Virtual Nonlinear Nonholonomic Constraints (108) Leonardo Colombo
In this paper, we address the problem of stabilizing a system around a desired manifold determined by virtual nonlinear nonholonomic constraints. Virtual constraints are relationships imposed on a control system that are rendered invariant through feedback control. Virtual nonholonomic constraints represent a specific class of virtual constraints that depend on the system’s velocities in addition to its configurations. We derive a control law under which a mechanical control system achieves exponential convergence to the virtual constraint submanifold, and rendering it control-invariant. The proposed controller’s performance is validated through simulation results in an application to the control of an unmanned surface vehicle (USV) navigating a stream.
17:10 - Trajectory generation for nonholonomic control systems using reconstruction techniques on SE(2) (130) Nicola Sansonetto
In this note we investigate the trajectory generation problem for nonholonomic mechanical shape- control systems, focusing in the case in which the symmetry group is SE(2), by using techniques from reconstruction theory.
17:30 - Homogeneous bi-Hamiltonian structures and integrable contact systems (4) Asier López-Gordón
Bi-Hamiltonian structures can be utilised to compute a maximal set of functions in involution for certain integrable systems, given by the eigenvalues of the recursion operator relating both Poisson structures. We show that the recursion operator relating two compatible Jacobi structures cannot produce a maximal set of functions in involution. However, as we illustrate with an example, bi-Hamiltonian structures can still be used to obtain a maximal set of functions in involution on a contact manifold, at the cost of symplectisation.
17:50 - Deep Dirac Neural Networks for Holonomic Mechanical Systems (80) Kenshin Okuwaki
We propose a physics-informed machine learning method for mechanical systems using the framework of Dirac dynamical systems. Specifically, we focus on mechanical systems with holonomic constraints. Our approach enables the learning of the generalized energy, which is theoretically derived from a Lagrangian, using both training and target data. Notably, it does not require prior knowledge of constraint forces to separately learn the generalized energy and constraint forces. This is achieved by enforcing a nonenergic condition through a loss function that ensures the constraint forces perform no work. The proposed approach outperforms existing methods by eliminating the need for predefined holonomic constraints as prerequisites for learning. We demonstrate the effectiveness of our method using a double pendulum as an example and conduct a comparative analysis of both approaches.In this paper, we propose a physics-informed machine learning method for mechanical systems using the framework of Dirac dynamical systems. Specifically, we focus on mechanical systems with holonomic constraints. Our approach enables the learning of the generalized energy, which is theoretically derived from a Lagrangian, using both training and target data. Notably, it does not require prior knowledge of constraint forces to separately learn the generalized energy and constraint forces. This is achieved by enforcing a nonenergic condition through a loss function that ensures the constraint forces perform no work. The proposed approach outperforms existing methods by eliminating the need for predefined holonomic constraints as prerequisites for learning. We demonstrate the effectiveness of our method using a double pendulum as an example and conduct a comparative analysis of both approaches.
Room Vauban 2
Learning of Dynamic Processes
Chairman : Stéphane CHRETIEN
16:30 - Memory capacity of nonlinear recurrent networks: Is it informative? (128) Giovanni Ballarin
The total memory capacity (MC) of linear recurrent neural networks (RNNs) has been proven to be equal to the rank of the corresponding Kalman controllability matrix, and it is almost surely maximal for connectivity and input weight matrices drawn from regular distributions. This fact questions the usefulness of this metric in distinguishing the performance of linear RNNs in the processing of stochastic signals. This note shows that the MC of random nonlinear RNNs yields arbitrary values within established upper and lower bounds depending just on the input process scale. This confirms that the existing definition of MC in linear and nonlinear cases has no practical value.
16:50 - Lie-Adaptive Inversion of Signature via Pfeffer-Seigal-Sturmfels Algorithm (140) Remi Vaucher
Since rough path signatures were introduced into machine learning by Terry Lyons, the practical inversion of the Signature transform remains an open problem. Several approaches have been proposed, ranging from insertion methods to optimal transport techniques. Each of these methods is only an approximation of the inversion, based on optimization problems. Our work extends the framework of Pfeffer, Seigal, and Sturmfels to incorporate the Lie group structure of $G^N(\mathbb{R}^d)$, the signature space. The original framework use an expression of the $i$-th level of signature $S^{(i)}$ as a sequence of $k$-mode tensor product between a given functional base and the decomposition matrix of the aimed path in this base. In this paper, we aim to go beyond a mean square loss by constructing a sequence of tensors that adhere to the Lie group structure. Additionally, we propose a method to recover the exact-length path rather than the shortest one. Finally, we evaluate our approach on multi-dimensional Brownian paths.
17:10 - Hypergraphs on high dimensional time series set using signature transform (141) Remi Vaucher
Over the past decades, hypergraphs and their study with topological data analysis (TDA) have become first-rate tools. Accordingly to this phenomena, a significant amount of tools appeared to build hypergraphs (named simplicial complexes in TDA) on top of data. Such structures allow us to create edges between more than two vertices.
In this paper, we adress the problem of constructing an hypergraph on top of multiple multivariate time series. The case of an hypergraph over a single multivariate time series has been addressed multiple times these past years. We succed to this task by generalizing a pre-existing algorithm for multivariate time series in the case of multiple multivariate time series. In addition, we exploit the properties of the signature transform to propose the introduction of some randomness in the algorithm in order to robustify the construction. Finally, our method is tested on synthetic data, and gives promising results.
17:30 - Using Signatures and Koopman operators to learn non-linear dynamics (146) Stephane Chretien
We propose a novel framework for predicting the evolution of dynamical systems by learning the Koopman operator in the space of linear functionals on the Signature transform of trajectory data. The Signature, a central object in rough path theory, provides a universal and compact representation of paths through iterated integrals, enabling linear models to approximate a wide class of nonlinear functionals. By restricting observables to lie in the span of truncated Signatures, we construct a finite-dimensional approximation of the Koopman operator, which we estimate directly from data using regularized linear regression. This approach merges the expressiveness of operator-theoretic methods with the structural richness of Signature features.
17:50 - A kernel-based global method for the learning of elastic potentials on Lie groups (152) Jianyu Hu
We propose a structure-preserving kernel ridge regression method for learning elastic potentials on Lie groups from noisy observations of force and torque fields. The approach is demonstrated on the special Euclidean group $SE(3)$, where the elastic potential acts as an external control. A key advantage of our method is that the potential function estimator admits a globally defined closed-form solution, with provable convergence analysis. Numerical experiments confirm the effectiveness of the proposed scheme.