Publications

2015

• Innovation-based sparse estimation of functional connectivity from multivariate autoregressive models
  
  • Deloche François
  • de Vico Fallani Fabrizio
  • Allassonniere Stéphanie
  , 2015. One of the main limitations of functional connectivity estimators of brain networks is that they can suffer from statistical reliability when the number of areas is large and the available time series are short. To estimate directed functional connectivity with multivariate autoregressive (MVAR) model on sparse connectivity assumption, we propose a modified Group Lasso procedure with an adapted penalty. Our procedure includes the innovation estimates as explaining variables. This approach is inspired by two criteria that are used to interpret the coefficients of the MVAR model, the Directed Transfer Function (DTF) and the Partial Directed Coherence (PDC). A causality measure can be deduced from the output coefficients which can be understood as a synthesis of PDC and DTF. We demonstrate the potential of our method and compare our results with the standard Group Lasso on simulated data. © (2015) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only. (10.1117/12.2189640)
  DOI : 10.1117/12.2189640
• Microlocal analysis of the generalized Radon transform arising in a model of pure industry
  
  • Agaltsov Alexey
  , 2015. We show that the generalized Radon transform arising in a model of pure industry taking into account the substitution of production factors at the micro-level is an integral Fourier operator satisfying the condition of microlocal regularity. We describe a method for reconstruction of singularities of a function from the singularities of its generalized Radon transform.
• Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions
  
  • Gobet Emmanuel
  • Turkedjiev Plamen
  Mathematics of Computation, American Mathematical Society, 2015, 85 (299), pp.1359-1391. We design a numerical scheme for solving the Multi step-forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical least-squares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy. Numerical experiments are presented illustrating theoretical convergence estimates. (10.1090/mcom/3013)
  DOI : 10.1090/mcom/3013
• Identifying defects in an unknown background using differential measurements
  
  • Audibert Lorenzo
  • Alexandre Girard
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2015, 9 (3). We present a new qualitative imaging method capable of selecting defects in complex and unknown background from differential measurements of farfield operators: i.e. far measurements of scattered waves in the cases with and without defects. Indeed, the main difficulty is that the background physical properties are unknown. Our approach is based on a new exact characterization of a scatterer domain in terms of the far field operator range and the link with solutions to so-called interior transmission problems. We present the theoretical foundations of the method and some validating numerical experiments in a two dimensional setting. (10.3934/ipi.2015.9.625)
  DOI : 10.3934/ipi.2015.9.625
• On the convergence of the iterates of "FISTA
  
  • Chambolle Antonin
  • Dossal Charles H
  Journal of Optimization Theory and Applications, Springer Verlag, 2015, Volume 166 (Issue 3), pp.25. FISTA is a classical optimization algorithm to minimize convex functions. The article gives new results on the properties of the sequences generated by this algorithm for non classical choices of parameters. The main result is the proof of the convergence of the iterates of the algorithm.
• On the Fluid Limits of a Resource Sharing Algorithm with Logarithmic Weights
  
  • Robert Philippe
  • Veber Amandine
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, 25 (5), pp.45. The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has x requests to transmit, then it receives a fraction of the capacity proportional to log(1+x), the logarithm of its current load. A fluid scaling analysis of such a network is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system, in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors which give an interesting fairness property to this class of algorithms. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the log function is replaced by an increasing function. (10.1214/14-AAP1057)
  DOI : 10.1214/14-AAP1057
• Non-parametric Stochastic Approximation with Large Step sizes
  
  • Dieuleveut Aymeric
  • Bach Francis
  Annals of Statistics, Institute of Mathematical Statistics, 2015, 44 (4). We consider the random-design least-squares regression problem within the reproducing kernel Hilbert space (RKHS) framework. Given a stream of independent and identically distributed input/output data, we aim to learn a regression function within an RKHS $\mathcal{H}$, even if the optimal predictor (i.e., the conditional expectation) is not in $\mathcal{H}$. In a stochastic approximation framework where the estimator is updated after each observation, we show that the averaged unregularized least-mean-square algorithm (a form of stochastic gradient), given a sufficient large step-size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in $\mathcal{H}$. (10.1214/15-AOS1391)
  DOI : 10.1214/15-AOS1391
• Internal exponential stabilization to a nonstationary solution for 1D Burgers equations with piecewise constant controls
  
  • Kröner Axel
  • Rodrigues Sergio S.
  , 2015, pp.2676-2681. The feedback stabilization of the Burgers system to a nonstationary solution using a finite number of internal piecewise constant controls is considered. Estimates for the number of needed controls are derived. In the particular case of no constraint on the support of the control a better estimate is derived, so the possibility of getting an analogous estimate for the general case is discussed.That possibility is suggested by the results of some numerical simulations. (10.1109/ECC.2015.7330942)
  DOI : 10.1109/ECC.2015.7330942
• Ergodicity Condition for Zero-Sum Games
  
  • Akian Marianne
  • Gaubert Stephane
  • Hochart Antoine
  , 2015. For zero-sum repeated stochastic games, basic questions are whether the mean payoff per time unit is independent of the initial state, and whether this property is robust to perturbations of rewards. In the case of finite action spaces, we show that the answer to both questions is positive if and only if an ergodicity condition involving fixed points of the recession function of the Shapley operator or reachability in directed hypergraphs is satisfied.
• Long and Winding Central Paths
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stephane
  • Joswig Michael
  , 2015. We disprove a continuous analog of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with 3r+4 inequalities in dimension 2r+2 where the central path has a total curvature in Ω(2^r/r). Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of linear programs viewed through logarithmic glasses.
• Eigenvectors of Non-Linear Maps on the Cone of Positive Semidefinite Matrices Application to Stability Analysis
  
  • Stott Nikolas
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Goubault Eric
  • Putot Sylvie
  , 2015. We show that the problem of synthesis of a common Lyapunov function for some classes of switched linear systems can be approached by solving an eigenproblem involving a nonlinear map on the cone of positive semidefinite matrices. This map involves the selection of a maximal lower bound of a family of matrices in this cone. We present some variants of the power algorithm, allowing one to solve the nonlinear eigenproblem in a scalable way.
• Tropicalizing Semialgebraic Pivoting Rules, Or How to Solve Mean Payoff Games in Polynomial Time on Average
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stephane
  , 2015. We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring. The proof relies on the observation that certain semi-algebraic pivoting rules can be tropicalized.
• Generic Uniqueness of the Bias Vector of Mean-Payoff Zero-Sum Games
  
  • Hochart Antoine
  • Akian Marianne
  • Gaubert Stéphane
  , 2015. Under some ergodicity conditions, finite state space mean payoff zero-sum games can be solved using a nonlinear fixed point problem, involving a vector (bias or potential), which determines the optimal strategies. A basic issue is to check when the bias is unique. We show that this is always the case for generic values of the payments of the game. We also discuss the application of this result to the perturbation analysis of policy iteration.
• Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates
  
  • Guilloux Agathe
  • Lemler Sarah
  • Taupin Marie-Luce
  , 2015. We propose a novel kernel estimator of the baseline function in a general high-dimensional Cox model, for which we derive non-asymptotic rates of convergence. To construct our estimator, we first estimate the regression parameter in the Cox model via a LASSO procedure. We then plug this estimator into the classical kernel estimator of the baseline function, obtained by smoothing the so-called Breslow estimator of the cumulative baseline function. We propose and study an adaptive procedure for selecting the bandwidth, in the spirit of Goldenshluger and Lepski [14]. We state non-asymptotic oracle inequalities for the final estimator, which leads to a reduction in the rate of convergence when the dimension of the covariates grows.
• Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation
  
  • Kuhn Estelle
  • Allassonniere Stéphanie
  • Durrleman Stanley
  , 2015. Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate estimation algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data, which is particularly the case of medical images. The use of an optimized Monte Carlo Markov Chain method for a stochastic Expectation Maximization algorithm, is proposed to estimate the model parameters by maximizing the likelihood. A new Anisotropic Metropolis Adjusted Langevin Algorithm is used as transition in the MCMC method. First it is proven that this new sampler leads to a geometrically uniformly ergodic Markov chain. Furthermore, it is proven also that under mild conditions, the estimated parameters converge almost surely and are asymptotically Gaussian distributed. The methodology developed is then tested on handwritten digits and some 2D and 3D medical images for the deformable model estimation. More widely, the proposed algorithm can be used for a large range of models in many fields of applications such as pharmacology or genetic. The technical proofs are detailed in an appendix.
• Bayesian Mixed Effect Atlas Estimation with a Diffeomorphic Deformation Model
  
  • Allassonnière Stéphanie
  • Durrleman S
  • Kuhn E
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2015, 8 (3), pp.29. In this paper we introduce a diffeomorphic constraint on the deformations considered in the deformable Bayesian mixed effect template model. Our approach is built on a generic group of diffeo-morphisms, which is parameterized by an arbitrary set of control point positions and momentum vectors. This enables us to estimate the optimal positions of control points together with a template image and parameters of the deformation distribution which compose the atlas. We propose to use a stochastic version of the expectation-maximization algorithm where the simulation is performed using the anisotropic Metropolis adjusted Langevin algorithm. We propose also an extension of the model including a sparsity constraint to select an optimal number of control points with relevant positions. Experiments are carried out on the United States Postal Service database, on mandibles of mice, and on three-dimensional murine dendrite spine images. (10.1137/140971762)
  DOI : 10.1137/140971762
• A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes
  
  • Allaire Grégoire
  Ingénieurs de l'Automobile, SIA, 2015, 836, pp.33-36. The goal of this paper is to briefly recall the importance of the adjoint method in many problems of sensitivity analysis, uncertainty quantification and optimization when the model is a differential equation. We illustrate this notion with some recent examples. As is well known, from a computational point of view the adjoint method is intrusive, meaning that it requires some changes in the numerical codes. Therefore we advocate that any new software development must take into account this issue, right from its inception.
• Global dynamics of the buffered chemostat for a general class of response functions
  
  • Rapaport Alain
  • Haidar Ihab
  • Harmand Jérôme
  Journal of Mathematical Biology, Springer, 2015, 71 (1), pp.69-98. We study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model. We show that the occurrence of a buffer can allow a species to setup or on the opposite to go to extinction, depending on the characteristics of the buffer. For non-monotonic response function, we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable, while this is not possible with single, serial or parallel vessels of the same total volume and input flow. These results are illustrated with the Haldane kinetic function. (10.1007/s00285-014-0814-7)
  DOI : 10.1007/s00285-014-0814-7
• E. Bretin - About phase field method to approximate the willmore flow
  
  • Bretin Elie
  • Bastien Fanny
  • Magnien Jérémy
  , 2015. We discuss in this video phase-field approximations of the Willmore functional and the associated $${\mathrm L}^{2}$$L2-flow. After recollecting known results on the approximation of the Willmore energy and its $${\mathrm L}^{1}$$L1 relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence is proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.
• Further remarks on Markus-Yamabe instability for time-varying delay differential equations
  
  • Haidar Ihab
  • Mason Paolo
  • Niculescu Silviu-Iulian
  • Sigalotti Mario
  • Chaillet Antoine
  , 2015.
• A mixed-effects model with time reparametrization for longitudinal univariate manifold-valued data
  
  • Schiratti Jean-Baptiste
  • Allassonniere Stéphanie
  • Routier Alexandre
  • Colliot Olivier
  • Durrleman Stanley
  , 2015, Lecture Notes In Computer Science (9123), pp.564-575. Mixed-effects models provide a rich theoretical framework for the analysis of longitudinal data. However , when used to analyze or predict the progression of a neurodegenerative disease such as Alzheimer ' s disease , these models usually do not take into account the fact that subjects may be at different stages of disease progression and the interpretation of the model may depend on some implicit reference time. In this paper , we propose a generative statistical model for longitudinal data , described in a univariate Riemannian manifold setting , which estimates an average disease progression model , subject-specific time shifts and acceleration factors. The time shifts account for variability in age at disease-onset time. The acceleration factors account for variability in speed of disease progression. For a given individual , the estimated time shift and acceleration factor define an affine reparametrization of the average disease progression model. This statistical model has been used to analyze neuropsychological assessments scores and cortical thickness measurements from the Alzheimer ' s Disease Neuroimaging Initiative database. The numerical results showed that we can distinguish between slow versus fast progressing and early versus late-onset individuals .
• A quasi-backscattering problem for inverse acoustic scattering in the Born regime
  
  • Haddar Houssem
  • Rezac Jacob
  Inverse Problems, IOP Publishing, 2015, 31 (7), pp.075008. We consider the problem of detecting three-dimensional inclusions from quasi-backscattering far field data generated by an incident field of time-harmonic fixed frequency plane waves modeled with the Born approximation. We assume only partial far field data is known and use a sampling-type method to reconstruct small obstacles and extended spherical obstacles. In particular, at the location of a device transmitting an incident wave, we assume far field data is collected only along a line extending a short distance from the transmitting device. Several numerical examples are provided to demonstrate the effectiveness of the approach. (10.1088/0266-5611/31/7/075008)
  DOI : 10.1088/0266-5611/31/7/075008
• Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions
  
  • Barilari Davide
  • Boscain Ugo
  • Donne Enrico Le
  • Sigalotti Mario
  , 2015. In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.
• Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (65), pp.1-31. (10.1214/EJP.v20-3569)
  DOI : 10.1214/EJP.v20-3569
• A low-complexity 2D signal space diversity solution for future broadcasting systems
  
  • Yang Jianxiao
  • Kai Wan
  • Geller Benoit
  • Abdel Nour Charbel
  • Rioul Olivier
  • Douillard Catherine
  , 2015, 63 (1), pp.2762-2767. —DVB-T2 was the first industrial standard deploying rotated and cyclic Q delayed (RCQD)modulation to improve performance over fading channels. This enablesimportantgains compared toconventional quadrature amplitude modulations(QAM) under severe channel conditions.However, the corresponding demodulation complexitystill prevents its use forwider applications. This paper proposes several rotation angles for different QAM constellations anda corresponding low-complexity detection method. Results show that the proposed solution simplifies both the transmitter and the receiver with often betterperformancethan the proposed angles in DVB-T2. Compared with the lowest complexity demappers currently used in DVB-T2, the proposed solution achieves an additional reduction bymore than 60%. Index Terms— DVB-T2, Rotated and Cyclic Q Delayed (RCQD) Modulations, Signal Space Diversity (SSD), Fading Channel, Quadrature Amplitude Modulations (QAM), Max-Log, ComputationalComplexity. (10.1109/ICC.2015.7248744)
  DOI : 10.1109/ICC.2015.7248744