Friday, 31 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.
08:30
Rita FIORESI
(University of Bologna, Italy)
Auditorium Maupertuis
Geometric Mechanics (Session 1)
Chairman : Gery de SAXCE, Zdravko TERZE, François DUBOIS
In a previous paper, we proposed a symplectic version of the Brezis-Ekeland-Nayroles principle. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non associated laws. For this aim, we introduce the concept of symplectic bipotential which extends that of bipotential to dynamical systems. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis-Ekeland-Nayroles principle to the non associated dissipative laws. As example, we apply it to the unilateral contact law with Coulomb’s dry friction.
We shall elucidate the foliation structures, namely, the 3-web and the bi-Lagrangian structure, that were jointly employed by the physicist and mathematician Jean-Marie Souriau in his Lie Groups Thermodynamics, extended to include dissipation models, and by Gérard Debreu, the Nobel Laureate in Economics, within the context of preferences and utility theory. Debreu examined the conditions under which a web is considered trivial, that is, whether there exists a change of coordinates rendering the web equivalent to a standard orthogonal grid, integrable into a potential function. He employed the Frobenius theorem and Pfaffian forms to investigate whether certain distributions, in the sense of foliations, satisfy the necessary conditions for integrability. Souriau developed a symplectic model of thermodynamics based on symplectic foliation, wherein dissipation dynamics are defined on a transverse Riemannian foliation, thereby inducing a web structure linked to a bi-Lagrangian manifold. A bi-Lagrangian manifold may be endowed with a metric 3-web structure via a Riemannian metric. For every 3-web, one can associate a canonical Chern connection, whose flatness guarantees additive separability. This connection, originally introduced by Hess for Lagrangian 2-webs, also known as bipolarised symplectic manifolds, is particularly employed in the study of bi-Lagrangian manifolds.
We construct conformal Carroll geometry by gauging the conformal Carroll algebra. In doing so, we pay special attention to the way the so-called intrinsic torsion tensor components enter into the transformation rules of the geometric fields. As an application of our results, we couple a single electric/magnetic massless scalar to conformal Carroll gravity and show how, upon gauge-fixing the dilatations, we obtain a non-conformal version of electric/magnetic Carroll gravity.
We present a variational integrator based on the Lobatto quadrature for the time integration of dynamical systems issued from the least action principle. This numerical method uses a cubic interpolation of the states and the action is approximated at each time step by Lobatto’s formula. Numerical analysis is performed on a harmonic oscillator. The scheme is conditionally stable, sixth-order accurate, and symplectic. It preserves an approximate energy quantity. Simulation results illustrate the performance of the proposed method.
[GSI 2025, 28 March 2025.]
A triplet $(\omega, \mathcal{F}_{1},\mathcal{F}_{2})$ is a bilagrangian structure on a manifold $M$, if $\omega$ is a 2-form, closed and non-degenerate (called symplectic form) on $M$, and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations on the symplectic manifold $(M,\omega)$. The quadruplet $(M, \omega, \mathcal{F}_{1},\mathcal{F}_{2})$ is called a bilagrangian manifold.
We prolong a bi-Lagrangian structure on $M$ on its tangent bundle $TM$, and its cotangent bundle $T^{*}M$ in different ways. As a consequence, some dynamics on the set of bi-Lagrangian structures of $M$ can be prolonged as dynamics on the set of the bi-Lagrangian structures of $TM$ and $T^{*}M$.
Room Vauban 1
Computational Information Geometry and Divergences (session 1)
Chairman : Frank NIELSEN and Olivier RIOUL
This work studies the Geometric Jensen-Shannon divergence, based on the notion of geometric mean of probability measures, in the setting of Gaussian measures on an infinite-dimensional Hilbert space. On the set of all Gaussian measures equivalent to a fixed one, we present a closed form expression for this divergence that directly generalizes the finite-dimensional version. By utilizing the notion of Log-Determinant divergences between positive definite unitized trace class operators, we then define a Regularized Geometric Jensen-Shannon divergence that is valid for any pair of Gaussian measures and that recovers the exact Geometric Jensen-Shannon divergence between two equivalent Gaussian measures when the regularization parameter approaches zero.
Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data analysis in the presence of noise. Similar to its classical single-parameter counterpart, however, it is challenging to compute and use in practice due to its complex algebraic construction. In this paper, we study a popular and tractable invariant for multiparameter persistent homology in a statistical setting: the multiparameter persistence landscape. We derive a functional central limit theorem for multiparameter persistence landscapes, from which we compute confidence bands, giving rise to one of the first statistical inference methodologies for multiparameter persistence landscapes. We provide an implementation of confidence bands and demonstrate their application in a machine learning task on synthetic data.
We introduce a new version of the KL-divergence for Gaussian distributions which is based on Wasserstein geometry and referred to as WKL-divergence. We show that this version is consistent with the geometry of the sample space ${\Bbb R}^n$. In particular, we can evaluate the WKL-divergence of the Dirac measures concentrated in two points which turns out to be proportional to the squared distance between these points.
Gaussian Mixture Models (GMMs) are important tool for modeling complex data in many tasks such as image recognition and retrieval, pattern recognition, speaker recognition and varification etc. Various GMM similarity measures are in place but most of them consume large computing resources and high computation time. This is mainly due to the lack of a closed form expression for divergence on GMM. We address this by using the embedding of the manifold of K-component GMMs into the manifold of symmetric positive definite (SPD) matrices. The manifold of SPD matrices is identified with the manifold of centered multivariate normal distribution which provides a computationally efficient formula for the divergence. First, we prove that the f -divergence between any two GMMs is greater than or equal to the f -divergence
computed between their corresponding centered multivariate normal representations. A local second-order analysis via Taylor series expansion shows that, under small perturbations of the GMM parameters, the difference between the f -divergences is quadratic. This enables to have a closed form formula for divergence on GMMs. To demonstrate the computational efficiency of this divergence we conducted an audio classification experiment, where Mel Frequency Cepstral Coefficient (MFCC) features extracted from audio signals are modeled by GMMs and classify
them using closed-form Symmetric KL divergence. The analysis indicate that the proposed method shows competative accuracy and significantly reduced the computational time compared to the existing methods.
Analyzing relationships between objects is a pivotal problem within data science. In this context, dimensionality reduction (DR) techniques are employed to generate smaller and more manageable data representations. This paper proposes a new method for dimensionality reduction, based on optimal transportation theory and the Gromov Wasserstein (GW) distance. We offer a new probabilistic view of the classical multidimensional scaling (MDS) algorithm and the nonlinear dimensionality reduction algorithm, Isomap (Isometric mapping or Isometric feature mapping) that extends the classical MDS, in which we use the GW distance between the probability measure of high-dimensional data, and its low-dimensional representation. Through gradient descent, our method embeds high-dimensional data into a lower-dimensional space, providing a robust and efficient solution for analyzing complex highdimensional datasets.
Room Vauban 2
Statistical Manifolds and Hessian information geometry (Session 1)
Chairman : Michel NGUIFFO BOYOM, Stéphane PUECHMOREL
By the results of Furuhata–Inoguchi–Kobayashi [Inf. Geom. (2021)] and Kobayashi–Ohno [Osaka Math. J. (2025)], the Amari–Chentsov $\alpha$-connections on the space $\mathcal{N}$ of all $n$-variate normal distributions are uniquely characterized by the invariance under the transitive action of the affine transformation group among all conjugate symmetric statistical connections with respect to the Fisher metric. In this paper, we investigate the Amari–Chentsov $\alpha$-connections on the submanifold $\mathcal{N}_0$ consisting of zero-mean $n$-variate normal distributions. It is known that $\mathcal{N}_0$ admits a natural transitive action of the general linear group $GL(n,\mathbb{R})$. We establish a one-to-one correspondence between the set of $GL(n,\mathbb{R})$-invariant conjugate symmetric statistical connections on $\mathcal{N}_0$ with respect to the Fisher metric and the space of homogeneous cubic real symmetric polynomials in $n$ variables. As a consequence, if $n \geq 2$, we show that the Amari–Chentsov $\alpha$-connections on $\mathcal{N}_0$ are not uniquely characterized by the invariance under the $GL(n,\mathbb{R})$-action among all conjugate symmetric statistical connections with respect to the Fisher metric. Furthermore, we show that any invariant statistical structure on a Riemannian symmetric space is necessarily conjugate symmetric.
Contrast functions play a fundamental role in information geometry, providing a means for generating the geometric structures of a statistical manifold: a pseudo-Riemannian metric and a pair of torsion-free affine connections. However, conventional contrast-based approaches become insufficient in settings where torsion is naturally present, such as quantum information geometry. This work introduces contrast bi-forms, a generalisation of contrast functions that systematically encode metric and connection data, allowing arbitrary affine connections regardless of torsion. It will be shown that they provide a unified framework for statistical potentials, offering new insights into the inverse problem in information geometry. As an application, we explore teleparallel manifolds, where torsion is intrinsic to the geometry, demonstrating how bi-forms naturally accommodate these structures.
We investigate a foliation by deformed escort distributions for the transition of $q$-parameters, not for a fixed $q$-parameter.
In particular,.this study considers a natural foliation of dualistic structures of escort distributions of exponential families from the information geometrical point of view.
We then propose a decomposition of an extended divergence on the foliation, which is an analogue of the previously proposed one for discrete escort distributions.
This paper explores hyperboloid models as statistical manifolds through the framework of flat F-manifolds. We show that these models admit a flat F-manifold structure, offering an alternative to the Fisher information metric. This new perspective deepens the geomet- ric understanding of probabilistic models and opens pathways to more efficient and interpretable methods in machine learning and statistical inference.
The λ-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d. sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the q-Gaussian distribution and the Dirichlet perturbation.
11:10-11:30 - Coffee Break + GSI'25 Posters Session
Auditorium Maupertuis
Geometric Mechanics (Session 2)
Chairman : Gery de SAXCE, Zdravko TERZE, François DUBOIS
In this paper we discuss nonholonomic contact Lagrangian and Hamiltonian systems, that is, systems with a kind of dissipation that are also subject to nonholonomic constraints. We introduce the so-called contact Eden bracket that allows us to simplify the calculation of the evolution of any observable. Finally, we present a particular vector subspace of observables where the dynamics remain unconstrained.
In this work, we revisit the Kaluza-Klein theory from the perspective of the classification of elementary particles based on the coadjoint orbit method. We propose a symmetry group for which the electric charge is invariant and, on this basis, a cosmological scenario in which the three former spatial dimensions inflate quickly while the fifth one shrinks, leading to a 4D era where the particles correspond to the coadjoint orbits of this group. By this mechanism, the elementary particles can acquire electric charge as a by-product of the 4 + 1 symmetry breaking of the Universe. By pullback over the space-time, we construct the non-Riemannian connection corresponding to this symmetry group, allowing to recover conservation of the charge and the equation of motion with the Lorentz force. On this ground, we develop a five dimensional extension of the variational relativity allowing to deduce in the classical limit Maxwell’s equation.
In a previous work of the first authors, a non-holonomic model, generalising the micromorphic models and allowing for curvature (disclinations) to arise from the kinematic values, was presented. In the present paper, a generalisation of the classical models of Euler-Bernoulli and Timoshenko bending beams based on the mentioned work is proposed. The former is still composed of only one unidimensional scalar field, while the latter introduces a third unidimensional scalar field, correcting the second order terms. The generalised Euler-Bernoulli beam is then shown to exhibit curvature (i.e. disclinations) linked to a third order derivative of the displacement, but no torsion (dislocations). Parallelly, the generalised Timoshenko beam is shown to exhibit both curvature and torsion, where the former is linked to the non-holonomy introduced in the generalisation. Lastly, using variational calculus, asymptotic values for the value taken by the curvature in static equilibrium are obtained when the second order contribution becomes negligible; along with an equation for the torsion in the generalised Timoshenko beam.
Morphoelastic bodies are elastic materials that undergo complex shape changes due to intrinsic growth, remodelling, or active internal processes. In a theoretical context, these materials are typically represented by non-Euclidean material manifolds characterized by an evolving metric structure. In this work, we formulate remodelling on such manifolds through a gradient system, where the dynamics of the system are driven by the steepest descent of an energy functional within an appropriate metric space. We obtain a gradient flow equation by combining isotropic non-linear elasticity with growth-induced dissipative mechanisms and illustrate the formalism through numerical simulations of stress relaxation at fixed strain.
Because of the strict separation of mass and energy in Galilei physics, a Galilei-invariant tensor formalism is most at home in a 5-dimensional extended spacetime associated with the Bargmann-Galilei (traditionally `Bargmann’) group, a central extension of the Galilei group that explicitly exhibits the transformation properties of kinetic energy. While not necessary for a tensor formalism fully embodying Poincar\’e physics, a similar central extension of the Poincar\’e group to the Bargmann-Poincar\’e group may illuminate a path towards a strong-field `Galilei general relativity’. Here the Bargmann metric is generalized to curved spacetime by extending the usual 1+3 (traditionally `3+1′) formalism of general relativity on 4-dimensional spacetime to a 1+3+1 formalism, whose spacetime kinematics is shown to be consistent with that of the usual 1+3 formalism. On Bargmann spacetime, tensor laws governing the motion of an elementary classical material particle and the dynamics of a simple fluid reference the foliation of spacetime in a manner that partially reverts the Einstein perspective (accelerated fiducial observers, and geodesic material particles and fluid elements) to a Newton-like perspective (geodesic fiducial observers, and accelerated material particles and fluid elements subject to a gravitational force).
Room Vauban 1
Computational Information Geometry and Divergences (session 2)
Chairman : Frank NIELSEN and Olivier RIOUL
Knowledge Distillation (KD) seeks to transfer the knowledge of a teacher, towards a student neural net. This process is often done by matching the networks’ predictions (i.e., their output), but, recently several works have proposed to match the distributions of neural nets’ activations (i.e., their features), a process known as \emph{distribution matching}. In this paper, we propose an unifying framework, Knowledge Distillation through Distribution Matching (KD$^{2}$M), which formalizes this strategy. Our contributions are threefold. We i) provide an overview of distribution metrics used in distribution matching, ii) benchmark on computer vision datasets, and iii) derive new theoretical results for KD.
By analogy to curved exponential families, we define curved Bregman divergences as restrictions of Bregman divergences to sub-dimensional parameter subspaces,
and prove that the barycenter of a finite weighted parameter set with respect to a curved Bregman
divergence amounts to the Bregman projection onto the subspace induced by the constraint of the barycenter with respect to the unconstrained full Bregman divergence.
We demonstrate the significance of curved Bregman divergences with two examples: (1) symmetrized Bregman divergences and (2) the Kullback-Leibler divergence between circular complex normal distributions.
We then consider monotonic embeddings to define representational curved Bregman divergences and show that the $\alpha$-divergences are representational curved Bregman divergences with respect to $\alpha$-embeddings of the probability simplex into the positive measure cone.
As an application, we report an efficient method to calculate the intersection of a finite set of $\alpha$-divergence spheres.
For a smooth manifold $M$, the tangent groupoid « glues » the set $M \times M$ with $TM$ as two underlying pieces in smooth transition from one to the other. We show that any contrast function defined on $M \times M$ naturally leads to a Riemannian metric and a pair of conjugate connections that are objects (so-called « statistical structure ») defined for sections of $TM$. This is achieved through smooth « extension » of the contrast function and its anti-symmetrized version on $M \times M$ to, respectively, a quadratic and a cubic function on $TM$. We recovered the standard formulae \cite{Eguchi1983,Eguchi1985,Eguchi1992,Blaesild1991} linking contrast functions to statistical structure through differentiation of the former by two (to obtain the metric) and three (to obtain the connections) vector fields.
We investigate the hyperbolic decomposition of the Dirichlet norm and distance between autoregressive moving average (ARMA) models. Beginning with the K\ »ahler information geometry of linear systems in the Hardy space and weighted Hardy spaces, we demonstrate that the Dirichlet norm and distance of ARMA models, corresponding to the mutual information between the past and future, are decomposed into functions of the hyperbolic distance between the poles and zeros of the ARMA models.
Room Vauban 2
Applied Geometry-Informed Machine Learning (Session 2)
Chairman : Pierre-Yves LAGRAVE, Santiago VALASCO-FORERO and Teodora PETRISOR
Transformers excel when dealing with sequential data.
Generalizing transformer models to geometric domains, such as manifolds, we encounter the problem of not having a well-defined global order.
We propose a solution with attention heads following a space-filling curve.
As a first experimental example, we present the Spiroformer, a transformer that follows a polar spiral on the $2$-sphere.
We introduce a new approach to the reconstruction of hidden structures from incomplete data, unifying techniques from geometric integration and topological analysis within the pioneering frameworks of Vaisman and Neifeld. Our method transcends traditional iterative schemes by employing a refined geometric decomposition of configuration spaces into invariant foliations and moment maps, thereby resolving the intrinsic ambiguities of underdetermined inverse problems. By synergistically combining Vaisman’s deep insights into symmetry and Neifeld’s analytic methodologies, we establish a robust, noise-resistant paradigm that not only ensures computational tractability but also fundamentally redefines the landscape of reconstruction in imaging and structural analysis. This framework paves the way for transformative applications across diverse scientific domains, heralding a new era in the synthesis of geometry and topology for inverse problem solving.
Unlike conventional telemetry systems, this solution delivers unparalleled flexibility and scalability, as the number of nodes in the network can be expanded as needed. The system’s physical layer capabilities enable long-range, high-data-rate communication, while the unique network layer algorithm ensures reliable relaying capabilities.
This paper details the practical implementation of the telemetry system within the D328 Deutsche Aircraft flight test campaign, highlighting its advantages over traditional solutions. Key use cases include data acquisition and relay between multiple airborne systems and ground stations, demonstrating the system’s potential to significantly enhance range and reliability in demanding environments. By reducing dependency on direct line-of-sight, this approach paves the way for more robust and efficient telemetry operations in future aerospace applications.
The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph G can be realized in the Euclidean space so that the distances between embedded vertices correspond to the edge weights. The DGP is a rich and active research field, with many important applications. Several approaches to the DGP are based on the idea of directly placing the vertices of G in space. Our model uses a completely new approach: we focus our attention on the edges, and not on the vertices, and we attempt placing in space the “sticks” that can be associated to each edge of the graph. Sticks have fixed length (hence they always satisfy all distance constraints), and they admit three total degrees of freedom (position of one vertex, plus the stick orientation) in 2D. The automatic satisfaction of all DGP constraints comes at the cost of possibly having several distinct positions associated to the same vertex, potentially a different one for every stick where each vertex is involved. Therefore, we formulate a problem consisting in finding stick configurations where all vertices involved in multiple sticks can find a unique position in space, implying in turn the definition of a valid realization for the original DGP. We initially focus the attention on DGPs where the information on the stick orientations is a priori given, so that to formulate a convex quadratic optimization problem with linear constraints. For the general case, we propose a heuristic which solves, at each iteration, an instance of the quadratic problem.
This paper focuses on the challenge of interactively modeling street networks. In order to provide usable datasets for artificial intelligence applications, it is often necessary to generate random cities with adjustable parameters, such as the spatial extent of the city and its traffic distribution. Several models have been developed for this purpose, including the hyperfractal model introduced by Philippe Jacquet. This model offers a significant advantage as it accounts not only for the fractal geometry of urban structures but also for the statistical distribution of traffic within a city.
In this work, we extend the simple fractal model, which is particularly useful for describing small cities or individual districts, by constructing random cities based on a tiling structure over which hyperfractals are distributed. This approach enables the connection of multiple hyperfractal districts, providing a more comprehensive urban representation.
Furthermore, we demonstrate how this decomposition can be used to segment a city into distinct districts through fractal analysis. Finally, we present tools for the numerical generation of random cities following this model.
Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data? In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah–Molino framework. In stark contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid—an intrinsic invariant that encapsulates the averaged geometry of a data set—to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering an innovative perspective for Topological Data Analysis that eschews persistent homology in favor of a unified geometric paradigm. The subsequent paper will extend these ideas to a full reconstruction scheme via the Atiyah–Molino framework. Our method not only provides a robust and computationally tractable framework for the recovery of hidden structures but also opens new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.
13:20-14:50 - Lunch Break + GSI'25 Posters Session
14:50
Mário A.T. FIGUEIREDO
(Universidade de Lisboa, Portugal)
Auditorium Maupertuis
Divergences in Statistics and Machine Learning
Chairman : Michel BRONIATOWSKI and Wolfgang STUMMER
Generalized moment problems –called feature moments in the area of machine learning– are here considered with and without relaxation. The solution is the minimum of phi-divergences, according to an extended Maximum Entropy Principle. Inference from sampled data constraints leads to balance the divergence by a relaxation term.
In the literature, the form of the minimizing solution is obtained by resorting to Fenchel’s duality theorem or the method of Lagrange multipliers. An alternative method, resorting to Hilbert spaces, presented here, yields a necessary and sufficient condition under second order assumptions. It is based on a decomposition via a nested procedure of the relaxed problem into two successive problems, one of which without relaxation.
In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom—the H\ »{o}lder divergence and the functional density power divergence (FDPD)—based on their intersection and generalization.
These divergence families include the density power divergence and the $\gamma$-divergence as special cases.
First, we prove that the intersection of the H\ »{o}lder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001).
Subsequently, motivated by the fact that H\ »{o}lder’s inequality is used in the proofs of nonnegativity for both the H\ »{o}lder divergence and the FDPD, we define a generalized divergence family, referred to as the $\xi$-H\ »{o}lder divergence.
The nonnegativity of the $\xi$-H\ »{o}lder divergence is established through a combination of the inequalities used to prove the nonnegativity of the H\ »{o}lder divergence and the FDPD.
Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the $\xi$-H\ »{o}lder divergence.
Finally, we prove that imposing the mathematical structure of the H\ »{o}lder score on a composite scoring rule results in the $\xi$-H\ »{o}lder divergence.
Parameter estimation of probabilistic models for discrete variables is often infeasible due to the calculation of the normalization constant required to ensure the model represents a valid probability distribution, and various approaches have been developed to resolve this problem.
In this paper, we consider a computationally feasible estimator for discrete probabilistic models based on a concept of empirical localization. Furthermore, we propose a computationally feasible estimator similar to the MAP estimator in Bayesian estimation by extending the above estimator.
For some smooth special case of generalized phi-divergences
as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) ell1-distance and (weighted) ell1-norm.
Learning to reject provide a learning paradigm which allows for our models to abstain from making predictions. One way to learn the rejector is to learn an ideal marginal distribution (w.r.t. the input domain) — which characterizes a hypothetical best marginal distribution — and compares it to the true marginal distribution via a density ratio. In this paper, we consider learning a joint ideal distribution over both inputs and labels; and develop a link between rejection and thresholding different statistical divergences. We further find that when one considers a variant of the log-loss, the rejector obtained by considering the joint ideal distribution corresponds to the thresholding of the skewed Bhattacharyya divergence between class-probabilities. This is in contrast to the marginal case — that is equivalent to a typical characterization of optimal rejection, Chow’s Rule — which corresponds to a thresholding of the Kullback-Leibler divergence. In general, we find that rejecting via a Bhattacharyya divergence is less aggressive than Chow’s Rule.
Room Vauban 1
Statistical Manifolds and Hessian information geometry (Session 2)
Chairman : Michel NGUIFFO BOYOM, Stéphane PUECHMOREL
A fundamental question in information theory is to quantify the loss of information under a noisy channel. Partial orders are typical tools to that end, however, they are often also challenging to evaluate. For the special class of binary input symmetric output (BISO) channels, Geng et al. showed that among channels with the same capacity, the binary symmetric channel (BSC) and binary erasure channel (BEC) are extremal with respect to the more capable order. Here we extend on this result by considering partial orders based on Renyi mutual information. We establish the extremality of the BSC and BEC in this setting with respect to the generalized Renyi capacity. In the process, we also generalize the needed tools and introduce alpha-Lorenz curves.
This paper addresses the observation of finite energy signals in noise and the estimation of their parameters, based on a geometric science of information approach. The parameters define the coordinate system of a statistical manifold. On this manifold, the Fisher-Rao distance characterizes the statistical dissimilarity between observations of two signals represented by their respective parameters. This work proposes a representation of finite energy signal observations and investigates the possibility of obtaining closed-form expressions for the Fisher-Rao distance. We derive the expressions for the Christoffel symbols and the tensorial equations of the geodesics. This leads to geodesic equations expressed as second-order differential equations. We show that the tensor differential equations can be transformed into matrix equations. These equations depend on the parametric model but simplify to only two vectorial equations, which combine the magnitude and phase of the signal and their gradients with respect to the parameters. These equations lead to closed-form expressions of the Fisher-Rao distance in certain cases. We study the example of observing an attenuated signal with a known magnitude spectrum and unknown phase spectrum and calculate the Fisher-Rao distance. We demonstrate that the finite energy signal manifold corresponds to the manifold of the Gaussian distribution with a known covariance matrix, and that the manifold of known magnitude spectrum signals is a submanifold. We compute closed-form expressions of the Fisher-Rao distances and show that the submanifold is non-geodesic, indicating that the Fisher-Rao distance measured within the submanifold is greater than in the full manifold. The results show that prior knowledge of the magnitude spectrum provides an advantage for signal phase parameter estimation when the difference in phase spectrum between the signals varies significantly throughout the bandwidth,
Results on nonparametric exponential models are presented by exploiting the notion of sub-exponential random variable. Applications of these models to exponential utility maximization problems are also highlighted.
Production of labelled foliations
Statistical states are introduced as coherent states seen as probability amplitudes in the Koopman representation. In Hamiltonian dynamical systems they can be studied as Hilbert bundles over a symplectic manifolds. The duality in Bochner’s theorem: probability measures ↔ functions of positive type is interpreted a statistical states and measurements. We present Hamiltonian dynamics on the statistical manifold by representations of the symplectic algebra acting on coherent time series.
Room Vauban 2
Geometric Learning and Differential Invariants on Homogeneous Spaces
Chairman : Remco DUITS, Erik BEKKERS
Contrary to popular belief, the global positioning problem on earth may have more than one solutions even if the user position is restricted to a sphere. With 3 satellites, we show that there can be up to 4 solutions on a sphere. With 4 or more satellites, we show that, for any pair of points on a sphere, there is a family of hyperboloids of revolution such that if the satellites are placed on one sheet of one of these hyperboloid, then the global positioning problem has both points as solutions. We give solution methods that yield the correct number of solutions on/near a sphere.
Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based Group Convolutional Neural Networks (PDE-G-CNNs), where the Riemannian metric enters in multilayer perceptrons, message passing, and max-pooling over Riemannian balls.
In PDE-G-CNNs it is proposed to take the minimum left-invariant Riemannian distance over the fiber in SE(3)/SO(2), whereas in G-CNNs and in many geometric image processing methods an efficient SO(2)-conjugation invariant section is advocated.
The conjecture rises whether that computationally much more efficient section indeed always selects distance minimizers over the fibers. We show that this conjecture does NOT hold in general, and in the logarithmic norm approximation setting used in practice we analyze the small (and sometimes vanishing) differences. We first prove that the minimal distance section is reached by minimal horizontal geodesics with constant momentum and zero acceleration along the fibers, and we generalize this result to (reductive) homogeneous spaces with legal metrics and commutative structure groups.
Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties.
To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3).
In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels.
We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3).
With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them.
We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture.
Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly.
Riemannian metrics on the position-orientation space M(3) that are roto-translation group SE(3) invariant play a key role in image analysis tasks like enhancement, denoising, and segmentation.
These metrics enable roto-translation equivariant algorithms, with the associated Riemannian distance often used in implementation.
However, computing the Riemannian distance is costly, which makes it unsuitable in situations where constant recomputation is needed.
We propose the mav (minimal angular velocity) distance, defined as the Riemannian length of a geometrically meaningful curve, as a practical alternative.
We see an application of the mav distance in geometric deep learning.
Namely, neural networks architectures such as PONITA, relies on geometric invariants to create their roto-translation equivariant model.
The mav distance offers a trainable invariant, with the parameters that determine the Riemannian metric acting as learnable weights.
In this paper we:
1) classify and parametrize all SE(3) invariant metrics on M(3),
2) describes how to efficiently calculate the mav distance,
and 3) investigate if including the mav distance within PONITA can positively impact its accuracy in predicting molecular properties.
This paper explores the theoretical integration of mathematical morphology with deep learning, specifically focusing on creating neural network layers that inherently produce fixed points through iterative application of operators. It investigates how principles from mathematical morphology, such as idempotence and convergence of operators in complete lattices, can be leveraged to design efficient and stable deep learning architectures. The work extends these concepts to group-equivariant operators on homogeneous spaces, aiming to build nonlinear iterative layers in deep convolutional neural networks that respect symmetries present in the data. By examining group convolutions, dilations and erosions, the paper lays theoretical groundwork for constructing equivariant fixed-point layers using either maxplus group operations or group convolutions.
Flow Matching (FM) is a recent generative modelling technique by Lipman et al. (2022): we aim to learn how to sample from distribution X1 by flowing samples from some distribution X0 that is easy to sample from.
The key trick is that this flow field can be trained while conditioning on the end point in X1: given an end point, simply move along a straight line segment to the end point.
However, straight line segments are only well-defined on Euclidean space.
Consequently, Chen et al. (2023) generalised the method to FM on Riemannian manifolds, replacing line segments with geodesics or their spectral approximations.
We take an alternative point of view: we generalise to FM on Lie groups by instead substituting exponential curves for line segments. This leads to a simple, intrinsic, and fast implementation for many matrix Lie groups, since the required Lie group operations (products, inverses, exponentials, logarithms) are simply given by the corresponding matrix operations.
FM on Lie groups could then be used for generative modelling with data consisting of sets of features (in R^n) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields Wessels et al. (2025).