Publications

2015

• Quadratic BSDEs with jumps: a fixed-point approach
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (66), pp.1-28. (10.1214/EJP.v20-3363)
  DOI : 10.1214/EJP.v20-3363
• Sample Complexity of Dictionary Learning and other Matrix Factorizations
  
  • Gribonval Rémi
  • Jenatton Rodolphe
  • Bach Francis
  • Kleinsteuber Martin
  • Seibert Matthias
  IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2015, 61 (6), pp.3469-3486. Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), $K$-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to $\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix factorization techniques. (10.1109/TIT.2015.2424238)
  DOI : 10.1109/TIT.2015.2424238
• Analysis and simulation of rare events for SPDEs
  
  • Bréhier Charles-Edouard
  • Gazeau Maxime
  • Goudenège Ludovic
  • Rousset Mathias
  ESAIM: Proceedings and Surveys, EDP Sciences, 2015, 48, pp.364-384. In this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling). (10.1051/proc/201448017)
  DOI : 10.1051/proc/201448017
• Zubov's equation for state-constrained perturbed nonlinear systems
  
  • Grüne Lars
  • Zidani Hasnaa
  Mathematical Control and Related Fields, AIMS, 2015, 5 (1), pp.55-71. The paper gives a characterization of the uniform robust domain of attraction for a nite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable in nite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations. (10.3934/mcrf.2015.5.55)
  DOI : 10.3934/mcrf.2015.5.55
• Derivation of nonlinear shell models combining shear and flexure: application to biological membranes
  
  • Pantz Olivier
  • Trabelsi Karim
  Mathematics and Mechanics of Complex Systems, International Research Center for Mathematics & Mechanics of Complex Systems (M&MoCS),University of L’Aquila in Italy, 2015, 3 (2), pp.101--138. Biological membranes are often idealized as incompressible elastic surfaces whose strain energy only depends on their mean curvature and pos-sibly on their shear. We show that this type of model can be derived using a formal asymptotic method by considering biological membranes to be thin, strongly anisotropic, elastic, locally homogeneous bodies.
• Sharp asymptotics of metastable transition times for one dimensional SPDEs
  
  • Barret Florent
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2015, 51 (1), pp.129-166. We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
• Accelerated Share Repurchase: pricing and execution strategy
  
  • Guéant Olivier
  • Pu Jiang
  • Guillaume Royer
  International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2015, 18 (3). In this article, we consider a specific optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank. (10.1142/S0219024915500193)
  DOI : 10.1142/S0219024915500193