arXiv subjects

Florent Bouchard

Publications and source records attributed to Florent Bouchard.

16 recordsLinked to original sources

arXiv2026

Benchmarking External Generalization of SPD Matrix Learning for Resting-State fMRI Connectome Prediction

Resting-state functional magnetic resonance imaging (rs-fMRI) functional connectivity (FC) matrices are widely used for individual-level prediction, but strong performance within one cohort may not generalize to a new cohort. We ask whether within-dataset performance remains when the test data come from an entirely held-out rs-fMRI dataset. Each scan is represented as a regularized symmetric positive definite (SPD) correlation connectome, which allows methods to use the geometry of the SPD manifold. We introduce a reproducible age-prediction benchmark across six rs-fMRI datasets: COBRE, ADNIDOD, Cam-CAN, ABIDE, OASIS-3, and ADNI. The benchmark compares a vectorized correlation baseline, Tangent-Space Ridge, SPDNet, and split-wise Riemannian harmonization under within-dataset GroupKFold, pooled GroupKFold, and leave-one-dataset-out (LODO) evaluation. Within-dataset and pooled GroupKFold results are substantially more favorable than LODO results. When an entire dataset is held out, prediction error increases, differences among methods narrow, and performance is strongly affected by age-range mismatch and cohort heterogeneity. The benchmark provides common inputs, model settings, data splits, and analysis scripts so that future SPD matrix learning methods can be evaluated under the same external-validation protocol.

eess.SP↗

arXiv2026

FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet

We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with Stiefel-constrained parameters. Unlike standard Euclidean averaging, which violates orthogonality, our approach preserves geometric structure through two efficient aggregation strategies: ProjAvg, projecting arithmetic means onto the Stiefel manifold, and RLAvg, approximating tangent-space averaging via retractions and liftings. Both methods are computationally efficient, independent of the optimizer, and enable scalable federated learning for signal processing applications whose features are SPD matrices. Simulations on EEG motor imagery benchmarks show that FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation, while using fewer parameters per communication round.

stat.ML↗

arXiv2026

SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization

Implementations of symmetric positive definite (SPD) matrix-based neural networks for neural decoding remain fragmented across research codebases and Python packages. Existing implementations often employ ad hoc handling of manifold constraints and non-unified training setups, which hinders reproducibility and integration into modern deep-learning workflows. To address this gap, we introduce SPD Learn, a unified and modular Python package for geometric deep learning with SPD matrices. SPD Learn provides core SPD operators and neural-network layers, including numerically stable spectral operators, and enforces Stiefel/SPD constraints via trivialization-based parameterizations. This design enables standard backpropagation and optimization in unconstrained Euclidean spaces while producing manifold-constrained parameters by construction. The package also offers reference implementations of representative SPDNet-based models and interfaces with widely used brain computer interface/neuroimaging toolkits and modern machine-learning libraries (e.g., MOABB, Braindecode, Nilearn, and SKADA), facilitating reproducible benchmarking and practical deployment.

q-bio.NC↗

arXiv2025

Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning

This paper addresses the problem of learning an undirected graph from data gathered at each nodes. Within the graph signal processing framework, the topology of such graph can be linked to the support of the conditional correlation matrix of the data. The corresponding graph learning problem then scales to the squares of the number of variables (nodes), which is usually problematic at large dimension. To tackle this issue, we propose a graph learning framework that leverages a low-rank factorization of the conditional correlation matrix. In order to solve for the resulting optimization problems, we derive tools required to apply Riemannian optimization techniques for this particular structure. The proposal is then particularized to a low-rank constrained counterpart of the GLasso algorithm, i.e., the penalized maximum likelihood estimation of a Gaussian graphical model. Experiments on synthetic and real data evidence that a very efficient dimension-versus-performance trade-off can be achieved with this approach.

cs.LG↗

arXiv2025

Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds

In this paper, the issue of averaging data on a manifold is addressed. While the Fr\'echet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

stat.ML↗

arXiv2024

Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning

This paper deals with Elliptical Wishart distributions - which generalize the Wishart distribution - in the context of signal processing and machine learning. Two algorithms to compute the maximum likelihood estimator (MLE) are proposed: a fixed point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated such as consistency, asymptotic normality and an intrinsic version of Fisher efficiency. On the statistical learning side, novel classification and clustering methods are designed. For the $t$-Wishart distribution, the performance of the MLE and statistical learning algorithms are evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.

stat.ML↗

arXiv2024

Random matrix theory improved Fr\'echet mean of symmetric positive definite matrices

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fr\'echet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine-learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory-based method that estimates Fr\'echet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.

stat.ML↗

arXiv2024

On Elliptical and Inverse Elliptical Wishart distributions

This paper deals with the Elliptical Wishart and Inverse Elliptical Wishart distributions, which play a major role when handling covariance matrices. Similarly to multivariate elliptical distributions, these form a large family of covariance distributions, encompassing, e.g., the Wishart or \textit{t}-Wishart ones. Our first major contribution is to derive a stochastic representation for Elliptical Wishart and Inverse Elliptical Wishart matrices. This later enables us to obtain various key statistical properties of Elliptical Wishart and Inverse Elliptical Wishart distributions such as expectations, variances, and Kronecker moments up to any orders. The stochastic representation also allows us to provide an efficient method to generate random matrices from Elliptical Wishart and Inverse Elliptical Wishart distributions. Finally, the practical interest of Elliptical Wishart distributions - in particular the \textit{t}-Wishart one - is demonstrated through a fitting experiment on real electroencephalographic data. This showcases their effectiveness in accurately modeling real covariance matrices.

math.ST↗

arXiv2023

Online Change Detection in SAR Time-Series with Kronecker Product Structured Scaled Gaussian Models

We develop the information geometry of scaled Gaussian distributions for which the covariance matrix exhibits a Kronecker product structure. This model and its geometry are then used to propose an online change detection (CD) algorithm for multivariate image times series (MITS). The proposed approach relies mainly on the online estimation of the structured covariance matrix under the null hypothesis, which is performed through a recursive (natural) Riemannian gradient descent. This approach exhibits a practical interest compared to the corresponding offline version, as its computational cost remains constant for each new image added in the time series. Simulations show that the proposed recursive estimators reach the Intrinsic Cram\'er-Rao bound. The interest of the proposed online CD approach is demonstrated on both simulated and real data.

stat.AP↗

arXiv2023

Intrinsic Bayesian Cram\'er-Rao Bound with an Application to Covariance Matrix Estimation

This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a Riemannian manifold (a smooth manifold endowed with a Riemannian metric) and follows a given prior distribution. In this setup, the chosen Riemannian metric induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure. Performance bound for such error measure were previously obtained in the non-Bayesian case (when the unknown parameter is assumed to deterministic), and referred to as \textit{intrinsic} Cram\'er-Rao bound. The presented result then appears either as: \textit{a}) an extension of the intrinsic Cram\'er-Rao bound to the Bayesian estimation framework; \textit{b}) a generalization of the Van-Trees inequality (Bayesian Cram\'er-Rao bound) that accounts for the aforementioned geometric structures. In a second part, we leverage this formalism to study the problem of covariance matrix estimation when the data follow a Gaussian distribution, and whose covariance matrix is drawn from an inverse Wishart distribution. Performance bounds for this problem are obtained for both the mean squared error (Euclidean metric) and the natural Riemannian distance for Hermitian positive definite matrices (affine invariant metric). Numerical simulation illustrate that assessing the error with the affine invariant metric is revealing of interesting properties of the maximum a posteriori and minimum mean square error estimator, which are not observed when using the Euclidean metric.

math.ST↗

arXiv2023

The Fisher-Rao geometry of CES distributions

When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher-Rao information geometry. Interestingly, this yields a point of view that allows for leveragingmany tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cram\'er-Rao bounds, and classification using Riemannian distances.

stat.ML↗

arXiv2022

Learning Graphical Factor Models with Riemannian Optimization

Graphical models and factor analysis are well-established tools in multivariate statistics. While these models can be both linked to structures exhibited by covariance and precision matrices, they are generally not jointly leveraged within graph learning processes. This paper therefore addresses this issue by proposing a flexible algorithmic framework for graph learning under low-rank structural constraints on the covariance matrix. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution (a generalization of Gaussian graphical models to possibly heavy-tailed distributions), where the covariance matrix is optionally constrained to be structured as low-rank plus diagonal (low-rank factor model). The resolution of this class of problems is then tackled with Riemannian optimization, where we leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models. Numerical experiments on real-world data sets illustrate the effectiveness of the proposed approach.

stat.ML↗

arXiv2021

Riemannian classification of EEG signals with missing values

This paper proposes a strategy to handle missing data for the classification of electroencephalograms using covariance matrices. It relies on the observed-data likelihood within an expectation-maximization algorithm. This approach is compared to two existing state-of-the-art methods: (i) covariance matrices computed with imputed data; (ii) Riemannian averages of partially observed covariance matrix. All approaches are combined with the minimum distance to Riemannian mean classifier and applied to a classification task of two widely known paradigms of brain-computer interfaces. In addition to be applicable for a wider range of missing data scenarios, the proposed strategy generally performs better than other methods on the considered real EEG data.

cs.HC↗

arXiv2020

Riemannian geometry for Compound Gaussian distributions: application to recursive change detection

A new Riemannian geometry for the Compound Gaussian distribution is proposed. In particular, the Fisher information metric is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series: a recursive approach based on Riemannian optimization is developed. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient.

stat.ML↗

arXiv2020

A Riemannian Framework for Low-Rank Structured Elliptical Models

This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contribution is to consider an original Riemannian metric, leading to new representations of tangent spaces and geodesics. From this geometry, we derive a new Riemannian optimization framework for robust covariance estimation, which is leveraged to minimize the popular Tyler's cost function on the considered quotient manifold. We also obtain a new divergence function, which is exploited to define a geometrical error measure on the quotient, and the corresponding intrinsic Cramér-Rao lower bound is derived. Thanks to the structure of the chosen parametrization, we further consider the subspace estimation error on the Grassmann manifold and provide its intrinsic Cramér-Rao lower bound. Our theoretical results are illustrated on some numerical experiments, showing the interest of the proposed optimization framework and that performance bounds can be reached.

math.DG↗

arXiv2019

Random Matrix Improved Covariance Estimation for a Large Class of Metrics

Relying on recent advances in statistical estimation of covariance distances based on random matrix theory, this article proposes an improved covariance and precision matrix estimation for a wide family of metrics. The method is shown to largely outperform the sample covariance matrix estimate and to compete with state-of-the-art methods, while at the same time being computationally simpler. Applications to linear and quadratic discriminant analyses also demonstrate significant gains, therefore suggesting practical interest to statistical machine learning.

stat.ML↗