arXiv subjects

Stéphane Puechmorel

Publications and source records attributed to Stéphane Puechmorel.

8 recordsLinked to original sources

arXiv2026

Ricci Curvature and Betti Numbers of Hessian Manifolds

We study Ricci curvature properties of Hessian metrics on the leaves of the codimension-one foliation $\mathcal{F}_ω= \ker\,ω$ generated by the first Koszul form $ω$ of a closed oriented Hessian manifold. Our main result reveals a striking rigidity phenomenon: non-negative Ricci curvature on a single leaf of $\mathcal{F}_ω$ compels the Hessian metric to be flat, yields sharp bounds on the first Betti number in terms of the dimension of the Hessian manifold and the topology of the leaves. This rigidity also shows that Koszul-type and radiant affine manifolds admit no leaf carrying non-negative Ricci curvature, reflecting a fundamental incompatibility between affine hyperbolicity and leafwise curvature positivity. In dimension three, we obtain a complete classification of the underlying manifold, extended to the non-orientable setting via the orientation double cover.

math.DG↗

arXiv2024

Cartan moving frames and the data manifolds

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and through experiments, explanations on the response of a neural network are given by pointing out the output classes that are easily reachable from a given input. This emphasizes how the proposed mathematical relationship between the output of the network and the geometry of its inputs can be exploited as an explainable artificial intelligence tool.

stat.ML↗

arXiv2024

Adversarial attacks on neural networks through canonical Riemannian foliations

Deep learning models are known to be vulnerable to adversarial attacks. Adversarial learning is therefore becoming a crucial task. We propose a new vision on neural network robustness using Riemannian geometry and foliation theory. The idea is illustrated by creating a new adversarial attack that takes into account the curvature of the data space. This new adversarial attack, called the two-step spectral attack is a piece-wise linear approximation of a geodesic in the data space. The data space is treated as a (degenerate) Riemannian manifold equipped with the pullback of the Fisher Information Metric (FIM) of the neural network. In most cases, this metric is only semi-definite and its kernel becomes a central object to study. A canonical foliation is derived from this kernel. The curvature of transverse leaves gives the appropriate correction to get a two-step approximation of the geodesic and hence a new efficient adversarial attack. The method is first illustrated on a 2D toy example in order to visualize the neural network foliation and the corresponding attacks. Next, we report numerical results on the MNIST and CIFAR10 datasets with the proposed technique and state of the art attacks presented in Zhao et al. (2019) (OSSA) and Croce et al. (2020) (AutoAttack). The result show that the proposed attack is more efficient at all levels of available budget for the attack (norm of the attack), confirming that the curvature of the transverse neural network FIM foliation plays an important role in the robustness of neural networks. The main objective and interest of this study is to provide a mathematical understanding of the geometrical issues at play in the data space when constructing efficient attacks on neural networks.

stat.ML↗

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

Convergence of the Hesse-Koszul flow on compact Hessian manifolds

We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse-Einstein metric by Cheng-Yau and Caffarelli-Viaclovsky as well as the real Calabi theorem by Cheng-Yau, Delanoë and Caffarelli-Viaclovsky.

math.DG↗

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

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↗