Publications

2022

• Asymptotic Analysis of a Matrix Latent Decomposition Model
  
  • Mantoux Clément
  • Durrleman​ Stanley
  • Allassonnière Stéphanie
  ESAIM: Probability and Statistics, EDP Sciences, 2022, 26, pp.208-242. Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model. (10.1051/ps/2022004)
  DOI : 10.1051/ps/2022004
• Firm non-expansive mappings in weak metric spaces
  
  • Gutiérrez Armando W.
  • Walsh Cormac
  Archiv der Mathematik, Springer Verlag, 2022,  119, pp.389-400. We introduce the notion of firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.
• Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions
  
  • Isaev Mikhail
  • Novikov Roman
  • Sabinin Grigory
  Inverse Problems, IOP Publishing, 2022. We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882, hal-03289374] for finding a compactly supported function v on R^d , d ≥ 1, from its Fourier transform F[v] given within the ball B_r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit. (10.1088/1361-6420/ac87cb)
  DOI : 10.1088/1361-6420/ac87cb
• DAMPING OPTIMIZATION OF VISCOELASTIC CANTILEVER BEAMS AND PLATES UNDER FREE VIBRATION
  
  • Joubert A
  • Allaire G
  • Amstutz S
  • Diani J
  Computers & Structures, Elsevier, 2022. The goal of this work is to significantly enhance the damping of linear viscoelastic structures under free vibration by relying on optimal design. Homogeneous cantilever slender beams and plates satisfying, respectively, the Euler-Bernoulli and Kirchhoff-Love assumptions are considered. A sizing optimization of the beam or plate thickness is proposed, as well as a coupled optimization of the thickness and geometry of the plate applying Hadamard's boundary variation method. The isotropic linear viscoelastic material is modeled by a classical generalized Maxwell model, well suited for polymers. Gradients of the objective functions are computed by an adjoint approach. Optimization is performed by a projected gradient algorithm and the mechanical models are evaluated by the finite element method. Numerical tests indicate that the optimal designs, as well as their damping properties, strongly depend on the material parameters.
• Multiply Accelerated Value Iteration for Non-Symmetric Affine Fixed Point Problems and application to Markov Decision Processes
  
  • Akian Marianne
  • Gaubert Stéphane
  • Qu Zheng
  • Saadi Omar
  SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2022, 43 (1). We analyze a modified version of Nesterov accelerated gradient algorithm, which applies to affine fixed point problems with non self-adjoint matrices, such as the ones appearing in the theory of Markov decision processes with discounted or mean payoff criteria. We characterize the spectra of matrices for which this algorithm does converge with an accelerated asymptotic rate. We also introduce a $d$th-order algorithm, and show that it yields a multiply accelerated rate under more demanding conditions on the spectrum. We subsequently apply these methods to develop accelerated schemes for non-linear fixed point problems arising from Markov decision processes. This is illustrated by numerical experiments. (10.1137/20M1367192)
  DOI : 10.1137/20M1367192
• On scaling limits of random trees and maps with a prescribed degree sequence
  
  • Marzouk Cyril
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2022, 5, pp.317-386. (10.5802/ahl.125)
  DOI : 10.5802/ahl.125
• Enriched nonconforming multiscale finite element method for Stokes flows in heterogeneous media based on high-order weighting functions
  
  • Feng Qingqing
  • Allaire Grégoire
  • Omnes Pascal
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2022, 20 (1). This paper addresses an enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling \& Simulation 2015 13:4, 1146-1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389 [math.NA], 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials. (10.1137/21M141926X)
  DOI : 10.1137/21M141926X
• Linear-sized independent sets in random cographs and increasing subsequences in separable permutations
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Drmota Michael
  • Feray Valentin
  • Gerin Lucas
  • Maazoun Mickaël
  • Pierrot Adeline
  Combinatorial Theory, eScholarship, 2022, 2 (3), pp.23340676. This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions. (10.5070/C62359179)
  DOI : 10.5070/C62359179
• Generating natural adversarial Remote Sensing Images
  
  • Burnel Jean-Christophe
  • Fatras Kilian
  • Flamary Rémi
  • Courty Nicolas
  IEEE Transactions on Geoscience and Remote Sensing, Institute of Electrical and Electronics Engineers, 2022, 60, pp.1-14. Over the last years, Remote Sensing Images (RSI) analysis have started resorting to using deep neural networks to solve most of the commonly faced problems, such as detection, land cover classification or segmentation. As far as critical decision making can be based upon the results of RSI analysis, it is important to clearly identify and understand potential security threats occurring in those machine learning algorithms. Notably, it has recently been found that neural networks are particularly sensitive to carefully designed attacks, generally crafted given the full knowledge of the considered deep network. In this paper, we consider the more realistic but challenging case where one wants to generate such attacks in the case of a black-box neural network. In this case, only the prediction score of the network is accessible, given a specific input. Examples that lure away the network's prediction, while being perceptually similar to real images, are called natural or unrestricted adversarial examples. We present an original method to generate such examples, based on a variant of the Wasserstein Generative Adversarial Network. We demonstrate its effectiveness on natural adversarial hyper-spectral image generation and image modification for fooling a state-of-the-art detector. Among others, we also conduct a perceptual evaluation with human annotators to better assess the effectiveness of the proposed method. (10.1109/TGRS.2021.3110601)
  DOI : 10.1109/TGRS.2021.3110601