arXiv subjects

Arnaud Breloy

Publications and source records attributed to Arnaud Breloy.

At least 19 recordsLinked to original sources

arXiv2026

Including Node Textual Metadata in Laplacian-constrained Gaussian Graphical Models

This paper addresses graph learning in Gaussian Graphical Models (GGMs). In this context, data matrices often come with auxiliary metadata (e.g., textual descriptions associated with each node) that is usually ignored in traditional graph estimation processes. To fill this gap, we propose a graph learning approach based on Laplacian-constrained GGMs that jointly leverages the node signals and such metadata. The resulting formulation yields an optimization problem, for which we develop an efficient majorization-minimization (MM) algorithm with closed-form updates at each iteration. Experimental results on a real-world financial dataset demonstrate that the proposed method significantly improves graph clustering performance compared to state-of-the-art approaches that use either signals or metadata alone, thus illustrating the interest of fusing both sources of information.

stat.ML↗

arXiv2025

Riemannian Change Point Detection on Manifolds with Robust Centroid Estimation

Non-parametric change-point detection in streaming time series data is a long-standing challenge in signal processing. Recent advancements in statistics and machine learning have increasingly addressed this problem for data residing on Riemannian manifolds. One prominent strategy involves monitoring abrupt changes in the center of mass of the time series. Implemented in a streaming fashion, this strategy, however, requires careful step size tuning when computing the updates of the center of mass. In this paper, we propose to leverage robust centroid on manifolds from M-estimation theory to address this issue. Our proposal consists of comparing two centroid estimates: the classical Karcher mean (sensitive to change) versus one defined from Huber's function (robust to change). This comparison leads to the definition of a test statistic whose performance is less sensitive to the underlying estimation method. We propose a stochastic Riemannian optimization algorithm to estimate both robust centroids efficiently. Experiments conducted on both simulated and real-world data across two representative manifolds demonstrate the superior performance of our proposed method.

cs.LG↗

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↗

arXiv2024

Intrinsic Bayesian Cramér-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} Cramér-Rao bound. The presented result then appears either as: \textit{a}) an extension of the intrinsic Cramér-Rao bound to the Bayesian estimation framework; \textit{b}) a generalization of the Van-Trees inequality (Bayesian Cramér-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↗

arXiv2024

Covariance Fitting Interferometric Phase Linking: Modular Framework and Optimization Algorithms

Interferometric phase linking (IPL) has become a prominent technique for processing images of areas containing distributed scaterrers in SAR interferometry. Traditionally, IPL consists in estimating consistent phase differences between all pairs of SAR images in a time series from the sample covariance matrix of pixel patches on a sliding window. This paper reformulates this task as a covariance fitting problem: in this setup, IPL appears as a form of projection of an input covariance matrix so that it satisfies the phase closure property. Given this modular formulation, we propose an overview of covariance matrix estimates, regularization options, and matrix distances, that can be of interest when processing multi-temporal SAR data. In particular, we will observe that most of the existing IPL algorithms appear as special instances of this framework. We then present tools to efficiently solve related optimization problems on the torus of phase-only complex vectors: majorization-minimization and Riemannian optimization. We conclude by illustrating the merits of different options on a real-world case study.

stat.AP↗

arXiv2024

Sparse PCA with False Discovery Rate Controlled Variable Selection

Sparse principal component analysis (PCA) aims at mapping large dimensional data to a linear subspace of lower dimension. By imposing loading vectors to be sparse, it performs the double duty of dimension reduction and variable selection. Sparse PCA algorithms are usually expressed as a trade-off between explained variance and sparsity of the loading vectors (i.e., number of selected variables). As a high explained variance is not necessarily synonymous with relevant information, these methods are prone to select irrelevant variables. To overcome this issue, we propose an alternative formulation of sparse PCA driven by the false discovery rate (FDR). We then leverage the Terminating-Random Experiments (T-Rex) selector to automatically determine an FDR-controlled support of the loading vectors. A major advantage of the resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical experiments and a stock market data example demonstrate a significant performance improvement.

stat.ML↗

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 Cramér-Rao bound. The interest of the proposed online CD approach is demonstrated on both simulated and real data.

stat.AP↗

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 Cramér-Rao bounds, and classification using Riemannian distances.

stat.ML↗

arXiv2023

Multifrequency Array Calibration in Presence of Radio Frequency Interferences

Radio interferometers are phased arrays producing high-resolution images from the covariance matrix of measurements. Calibration of such instruments is necessary and is a critical task. This is how the estimation of instrumental errors is usually done thanks to the knowledge of referenced celestial sources. However, the use of high sensitive antennas in modern radio interferometers (LOFAR, SKA) brings a new challenge in radio astronomy because there are more sensitive to Radio Frequency Interferences (RFI). The presence of RFI during the calibration process generally induces biases in state-of-the-art solutions. The purpose of this paper is to propose an alternative to alleviate the effects of RFI. For that, we first propose a model to take into account the presence of RFI in the data across multiple frequency channels thanks to a low-rank structured noise. We then achieve maximum likelihood estimation of the calibration parameters with a Space Alternating Generalized Expectation-Maximization (SAGE) algorithm for which we derive originally two sets of complete data allowing close form expressions for the updates. Numerical simulations show a significant gain in performance for RFI corrupted data in comparison with some more classical methods.

astro-ph.IM↗

arXiv2023

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↗

arXiv2023

Through the Wall Radar Imaging via Kronecker-structured Huber-type RPCA

The detection of multiple targets in an enclosed scene, from its outside, is a challenging topic of research addressed by Through-the-Wall Radar Imaging (TWRI). Traditionally, TWRI methods operate in two steps: first the removal of wall clutter then followed by the recovery of targets positions. Recent approaches manage in parallel the processing of the wall and targets via low rank plus sparse matrix decomposition and obtain better performances. In this paper, we reformulate this precisely via a RPCA-type problem, where the sparse vector appears in a Kronecker product. We extend this approach by adding a robust distance with flexible structure to handle heterogeneous noise and outliers, which may appear in TWRI measurements. The resolution is achieved via the Alternating Direction Method of Multipliers (ADMM) and variable splitting to decouple the constraints. The removal of the front wall is achieved via a closed-form proximal evaluation and the recovery of targets is possible via a tailored Majorization-Minimization (MM) step. The analysis and validation of our method is carried out using Finite-Difference Time-Domain (FDTD) simulated data, which show the advantage of our method in detection performance over complex scenarios.

stat.AP↗

arXiv2023

Riemannian optimization for non-centered mixture of scaled Gaussian distributions

This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log-likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set.

cs.LG↗

arXiv2023

Entropic Wasserstein Component Analysis

Dimension reduction (DR) methods provide systematic approaches for analyzing high-dimensional data. A key requirement for DR is to incorporate global dependencies among original and embedded samples while preserving clusters in the embedding space. To achieve this, we combine the principles of optimal transport (OT) and principal component analysis (PCA). Our method seeks the best linear subspace that minimizes reconstruction error using entropic OT, which naturally encodes the neighborhood information of the samples. From an algorithmic standpoint, we propose an efficient block-majorization-minimization solver over the Stiefel manifold. Our experimental results demonstrate that our approach can effectively preserve high-dimensional clusters, leading to more interpretable and effective embeddings. Python code of the algorithms and experiments is available online.

stat.ML↗

arXiv2022

Robust Geometric Metric Learning

This paper proposes new algorithms for the metric learning problem. We start by noticing that several classical metric learning formulations from the literature can be viewed as modified covariance matrix estimation problems. Leveraging this point of view, a general approach, called Robust Geometric Metric Learning (RGML), is then studied. This method aims at simultaneously estimating the covariance matrix of each class while shrinking them towards their (unknown) barycenter. We focus on two specific costs functions: one associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M -estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian distance, which enjoys nice properties of geodesic convexity and affine invariance. The optimization is performed using the Riemannian geometry of symmetric positive definite matrices and its submanifold of unit determinant. Finally, the performance of RGML is asserted on real datasets. Strong performance is exhibited while being robust to mislabeled data.

stat.ML↗

arXiv2021

Robust low-rank covariance matrix estimation with a general pattern of missing values

This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume unstructured signal models. The former can be inaccurate in real-world data sets in which heterogeneity causes heavy-tail distributions, while the latter does not profit from the usual low-rank structure of the signal. Taking advantage of both worlds, a covariance matrix estimation procedure is designed on a robust (mixture of scaled Gaussian) low-rank model by leveraging the observed-data likelihood function within an expectation-maximization algorithm. It is also designed to handle general pattern of missing values. The proposed procedure is first validated on simulated data sets. Then, its interest for classification and clustering applications is assessed on two real data sets with missing values, which include multispectral and hyperspectral time series.

stat.ME↗

arXiv2021

Regularized tapered sample covariance matrix

Covariance matrix tapers have a long history in signal processing and related fields. Examples of applications include autoregressive models (promoting a banded structure) or beamforming (widening the spectral null width associated with an interferer). In this paper, the focus is on high-dimensional setting where the dimension $p$ is high, while the data aspect ratio $n/p$ is low. We propose an estimator called Tabasco (TApered or BAnded Shrinkage COvariance matrix) that shrinks the tapered sample covariance matrix towards a scaled identity matrix. We derive optimal and estimated (data adaptive) regularization parameters that are designed to minimize the mean squared error (MSE) between the proposed shrinkage estimator and the true covariance matrix. These parameters are derived under the general assumption that the data is sampled from an unspecified elliptically symmetric distribution with finite 4th order moments (both real- and complex-valued cases are addressed). Simulation studies show that the proposed Tabasco outperforms all competing tapering covariance matrix estimators in diverse setups. A space-time adaptive processing (STAP) application also illustrates the benefit of the proposed estimator in a practical signal processing setup.

stat.ME↗

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

Asymptotic Performance of Complex M-estimators for Multivariate Location and Scatter Estimation

The joint estimation of means and scatter matrices is often a core problem in multivariate analysis. In order to overcome robustness issues, such as outliers from Gaussian assumption, M-estimators are now preferred to the traditional sample mean and sample covariance matrix. These estimators are well established and studied in the real case since the seventies. Their extension to the complex case has drawn recent interest. In this letter, we derive the asymptotic performance of complex M-estimators for multivariate location and scatter matrix estimation.

eess.SP↗