Publications

2015

• 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
• 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.
• 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
• 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.
• Robust utility maximization in nondominated models with 2BSDE: the uncertain volatility model
  
  • Matoussi Anis
  • Possamaï Dylan
  • Zhou Chao
  Mathematical Finance, Wiley, 2015, 25 (2), pp.258-287. The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2]. (10.1111/mafi.12031)
  DOI : 10.1111/mafi.12031
• 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
• An iterative approach to non-overdetermined inverse scattering at fixed energy
  
  • Novikov Roman
  Sbornik: Mathematics, Turpion, 2015, 206 (1), pp.120-134. We propose an iterative approximate reconstruction algorithm for non-overdetermined inverse scattering at fixed energy E with incomplete data in dimension d >= 2. In particular, we obtain rapidly converging approximate reconstructions for this inverse scattering for E --> +infinity.