arXiv subjects

Alice Le Brigant

Publications and source records attributed to Alice Le Brigant.

18 recordsLinked to original sources

arXiv2026

On the Wasserstein Geodesic Principal Component Analysis of probability measures

This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

stat.ML↗

arXiv2025

Conjugate points on Lie groups with left-invariant metrics

We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an arbitrary left-invariant metric, we show that all geodesics must have a conjugate point, and we give upper and lower bounds on conjugate times. In particular this applies to the left-invariant metrics on $SU(n)$ and $SO(n)$ which are of importance in fluid dynamics and rigid body motion, and yields estimates for the diameter and injectivity radius. We also establish criteria in the noncompact case: we show that every closed nonhomogeneous geodesic has a conjugate point, and determine explicit conditions for them in the three-dimensional unimodular case. For homogeneous geodesics, we relate conjugate points to Lagrangian stability, and Eulerian stability of the corresponding steady velocity. Finally, we obtain as by-products criteria for conjugate points in general homogeneous spaces, by lifting the problem to the total Lie group of the quotient and using a result of O'Neill. Through several examples, we show that our theorems apply when well-known criteria relying on positive Ricci curvature or other curvature bounds fail, and in some cases even when Ricci curvature is negative in all directions.

math.DG↗

arXiv2025

A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations

We give a new geometric interpretation of the Amari-Cencov $α$-connections $\nabla^{(α)}$ from information geometry: On the space of densities $\operatorname{Dens}_+(M)$, we show that there exist Riemannian metrics $G^α$, which we call $α$-Fisher-Rao metrics, whose Levi-Civita connections are $\nabla^{(α)}$. With the exception of $α=0$ (the Fisher-Rao metric), these metrics are non-invariant to the action of the diffeomorphism group $\operatorname{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\nabla^{(α)}$ as energy-minimizing curves. On the space of probability densities $\operatorname{Prob}(M)$, we show that the same phenomenon holds for $α\in \{-1,0,1\}$ and that the $α$-connections are not metric otherwise. We show that $\nabla^{(α)}$-geodesics on this space can be interpreted as radial projections of straight lines on appropriate hyper-surfaces, and use this geometric picture to obtain geodesic convexity for any $α\in \mathbb{R}$. In addition, we prove analogous results for appropriate metrics and connections on $\operatorname{Diff}(M)$, which, for the case $M=\mathbb{R}$, imply that the generalized Proudman-Johnson equations on the real line are the Euler-Arnold equations of non-right invariant metrics. Finally, in the finite-dimensional case, we show that $\nabla^{(α)}$ can be metric or non-metric depending on the considered statistical model.

math.DG↗

arXiv2024

Learning from landmarks, curves, surfaces, and shapes in Geomstats

We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape spaces, such as the Kendall shape space, as well as elastic spaces of discrete curves and surfaces. The shape module further implements the abstract mathematical structures of group actions, fiber bundles, quotient spaces and associated Riemannian metrics which allow users to build their own shape spaces. The Riemannian geometry tools enable users to compare, average, interpolate between shapes inside a given shape space. These essential operations can then be leveraged to perform statistics and machine learning on shape data. We present the object-oriented implementation of the shape module along with illustrative examples and show how it can be used to perform statistics and machine learning on shape spaces.

cs.GR↗

arXiv2024

Optimal matching between curves in a manifold

This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the action of the reparameterization group. This group induces a quotient structure classically interpreted as the ''shape space''. We introduce a simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle. We consider the particular case of elastic metrics and show simulations for open curves in the plane, the hyperbolic plane and the sphere.

math.DG↗

arXiv2023

Conjugate points along Kolmogorov flows on the torus

The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold $M$, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of conjugate points along such geodesics reveal that these cease to be infinitesimally length-minimizing between their endpoints. In this work, we focus on the case of the torus $M=\T^2$ and on geodesics corresponding to steady solutions of the Euler equation generated by stream functions $ψ=-\cos(mx)\cos(ny)$ for integers $m$ and $n$, called Kolmogorov flows. We show the existence of conjugate points along these geodesics for all pairs of strictly positive integers $(m,n)$, thereby completing the characterization of all pairs $(m,n)$ such that the associated Kolmogorov flow generates a geodesic with conjugate points.

math.DG↗

arXiv2023

The $L^p$-Fisher-Rao metric and Amari-Cencov $α$-connections

We introduce a family of Finsler metrics, called the $L^p$-Fisher-Rao metrics $F_p$, for $p\in (1,\infty)$, which generalizes the classical Fisher-Rao metric $F_2$, both on the space of densities Dens$_+(M)$ and probability densities Prob$(M)$. We then study their relations to the Amari-uCencov $α$-connections $\nabla^{(α)}$ from information geometry: on Dens$_+(M)$, the geodesic equations of $F_p$ and $\nabla^{(α)}$ coincide, for $p = 2/(1-α)$. Both are pullbacks of canonical constructions on $L^p(M)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of $α$-geodesics as being energy minimizing curves. On Prob$(M)$, the $F_p$ and $\nabla^{(α)}$ geodesics can still be thought as pullbacks of natural operations on the unit sphere in $L^p(M)$, but in this case they no longer coincide unless $p=2$. Using this transformation, we solve the geodesic equation of the $α$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of $F_p$, and study their relation to $\nabla^{(α)}$.

math.DG↗

arXiv2022

Parametric information geometry with the package Geomstats

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher-Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher-Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

cs.LG↗

arXiv2021

Classifying histograms of medical data using information geometry of beta distributions

In this paper, we use tools of information geometry to compare, average and classify histograms. Beta distributions are fitted to the histograms and the corresponding Fisher information geometry is used for comparison. We show that this geometry is negatively curved, which guarantees uniqueness of the notion of mean, and makes it suitable to classify histograms through the popular K-means algorithm. We illustrate the use of these geometric tools in supervised and unsupervised classification procedures of two medical data-sets, cardiac shape deformations for the detection of pulmonary hypertension and brain cortical thickness for the diagnosis of Alzheimer's disease.

stat.ML↗

arXiv2020

Fisher-Rao geometry of Dirichlet distributions

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fr{é}chet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

math.DG↗

arXiv2020

Intrinsic Riemannian metrics on spaces of curves: theory and computation

This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine different numerical approaches that have been proposed to estimate such distances in practice and in particular to quotient out curve reparametrization in the resulting minimization problems.

math.DG↗

arXiv2020

Geomstats: A Python Package for Riemannian Geometry in Machine Learning

We introduce Geomstats, an open-source Python toolbox for computations and statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, and many more. We provide object-oriented and extensively unit-tested implementations. Among others, manifolds come equipped with families of Riemannian metrics, with associated exponential and logarithmic maps, geodesics and parallel transport. Statistics and learning algorithms provide methods for estimation, clustering and dimension reduction on manifolds. All associated operations are vectorized for batch computation and provide support for different execution backends, namely NumPy, PyTorch and TensorFlow, enabling GPU acceleration. This paper presents the package, compares it with related libraries and provides relevant code examples. We show that Geomstats provides reliable building blocks to foster research in differential geometry and statistics, and to democratize the use of Riemannian geometry in machine learning applications. The source code is freely available under the MIT license at \url{geomstats.ai}.

cs.LG↗

arXiv2019

The Fisher-Rao geometry of beta distributions applied to the study of canonical moments

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrinsic representation of the moments of a probability distribution. Drawing on the fact that a uniform distribution in the regular moment space corresponds to a product of beta distributions in the canonical moment space, we propose a mapping from the space of canonical moments to the product beta manifold, allowing us to use the Fisher-Rao geometry of beta distributions to compare and analyze canonical moments.

math.ST↗

arXiv2018

Math in the Black Forest: Workshop on New Directions in Shape Analysis

These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical foundations in infinite-dimensional Riemannian geometry, geometric statistics, and geometric stochastics, and with applications in medical imaging, evolutionary development, and fluid dynamics. The workshop is the 6th instance of a series of workshops on the same topic.

math.DG↗

arXiv2018

Optimal Riemannian quantization with an application to air traffic analysis

The goal of optimal quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a K-means-type clustering. In this paper, we introduce Competitive Learning Riemannian Quantization (CLRQ), an online algorithm that computes the optimal summary when the data does not belong to a vector space, but rather a Riemannian manifold. We prove its convergence and show simulated examples on the sphere and the hyperbolic plane. We also provide an application to real data by using CLRQ to create summaries of images of covariance matrices estimated from air traffic images. These summaries are representative of the air traffic complexity and yield clusterings of the airspaces into zones that are homogeneous with respect to that criterion. They can then be compared using discrete optimal transport and be further used as inputs of a machine learning algorithm or as indexes in a traffic database.

stat.AP↗

arXiv2018

A discrete framework to find the optimal matching between manifold-valued curves

The aim of this paper is to find an optimal matching between manifold-valued curves, and thereby adequately compare their shapes, seen as equivalent classes with respect to the action of reparameterization. Using a canonical decomposition of a path in a principal bundle, we introduce a simple algorithm that finds an optimal matching between two curves by computing the geodesic of the infinite-dimensional manifold of curves that is at all time horizontal to the fibers of the shape bundle. We focus on the elastic metric studied in the so-called square root velocity framework. The quotient structure of the shape bundle is examined, and in particular horizontality with respect to the fibers. These results are more generally given for any elastic metric. We then introduce a comprehensive discrete framework which correctly approximates the smooth setting when the base manifold has constant sectional curvature. It is itself a Riemannian structure on the product manifold of "discrete curves" given by a finite number of points, and we show its convergence to the continuous model as the size of the discretization goes to infinity. Illustrations of optimal matching between discrete curves are given in the hyperbolic plane, the plane and the sphere, for synthetic and real data, and comparison with dynamic programming is established.

math.DG↗

arXiv2017

Computing distances and geodesics between manifold-valued curves in the SRV framework

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing.

math.DG↗

arXiv2015

Reparameterization invariant metric on the space of curves

This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M' = Imm([0,1], M) by pullback of a metric on the tangent bundle TM' derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM' induces a first-order Sobolev metric on M' with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .

math.DG↗