Currently CNRS visiting professor in the team DATA at Département d'Informatique, ENS Paris.
Email : wainrib [at] math.univ-paris13.fr
10-11 March 2014 Université Paris 13
Recent preprints & publications
Regular graphs maximize the variability of random neural networks
G. Wainrib and M. Galtier, submitted [arxiv]
Considering a model of dynamical units interconnected through a random weighted directed graph, we investigate how the behavior of the system is influenced by intricate relationships between the in-degree distribution, the variance of the weights and the non-linearity of the model. To this end, we develop an original theoretical approach based on a combination of a classical mean-field theory originally developed in the context of dynamical spin-glass models and the heterogeneous mean-field theory developed to study epidemic propagation on graphs. Our main result is that, surprisingly, increasing the variance of the in-degree distribution does not result in a more variable dynamical behavior, but on the contrary that the most variable behaviors are obtained in the regular graph setting. We further study how the dynamical complexity of the attractors is influenced by the statistical properties of the in-degree distribution.
A local Echo State Property through the largest Lyapunov exponent
M. Galtier and G.Wainrib, submitted [arxiv]
Echo State Networks are efficient time series predictors which highly depend on the value of the spectral radius of the reservoir connections. Based on recent results on the mean field theory of driven random recurrent neural networks, which allow the computation of the largest Lyapunov exponent of an ESN, we develop a cheap algorithm to establish a local and operational version of the Echo State Property which guarantees good prediction performances. The value of the spectral radius tuning the network to the edge of chaos is specific to the considered input and is larger than 1.
Relative entropy minimizing noisy non-linear neural network to approximate stochastic processes
M. Galtier, C. Marini, G. Wainrib, H. Jaeger, submitted. [arxiv]
A method is provided for designing and training noise-driven recurrent neural networks as models of stochastic processes. The method unifies and generalizes two known separate modeling approaches, Echo State Networks (ESN) and Linear Inverse Modeling (LIM), under the common principle of relative entropy minimization. The power of the new method is demonstrated on a stochastic approximation of the El Nino phenomenon studied in climate research.
Absorption properties of stochastic equations with Hölder diffusion coefficients
J.Touboul and G.Wainrib, submitted [arxiv]
In this article, we address the absorption properties of a class of stochastic differential equations around singular points where both the drift and diffusion functions vanish. According to the H\"older coefficient alpha of the diffusion function around the singular point, we identify different regimes. Stability of the absorbing state, large deviations for the absorption time, existence of stationary or quasi-stationary distributions are discussed. In particular, we show that quasi-stationary distributions only exist for alpha < 3/4, and for alpha in the interval (3/4, 1), no quasi-stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state (as is demonstrated to be true for alpha = 1). Applications of these results to stochastic bifurcations are discussed.
Index Distribution of the Ginibre Ensemble Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this Letter, we compute analytically the probability distribution of the number of eigenvalues NR with modulus greater than R (the index) of a large N×N random matrix in the real or complex Ginibre ensemble. We show that the fraction NR/N=p has a distribution scaling as exp(−βN2ψR(p)) with β=1 (respectively β=1/2) for the complex (resp. real) Ginibre ensemble. For any p∈[0,1], the equilibrium spectral densities as well as the rate function ψR(p) are explicitly derived. This function displays a third order phase transition at the critical (minimum) value p∗R=1−R2, associated to a phase transition of the Coulomb gas. We deduce that, in the central regime, the fluctuations of the index NR around its typical value p∗RN scale as N1/3.
R. Allez, J.Touboul and G.Wainrib, Journal of Physics A: Mathematical and Theoretical (2014) [arxiv]
Probability theory / Dynamical systems- limit theorems
- large deviations
- piecewise-deterministic Markov processes
- singular perturbations
- averaging principles
- stochastic bifurcations
- random matrix theory
- random walk on graphs
- partial differential equations
- random fields
Applications in theoretical biology / computer science- Hodgkin Huxley models with stochastic ion channels
- Action potential generation and propagation
- Information transmission
- Noise-induced phenomena
- Synaptic plasticity and learning
- Random neural networks
- Links with machine learning, reservoir computing
- Immune system modeling
- Gene regulatory networks
Laboratoire Analyse Géométrie et Applications (LAGA)
Université Paris 13
99, avenue Jean-Baptiste Clément
93430 - Villetaneuse
Bureau D 306
Tel : 01 49 40 35 83
Fax : 01 49 40 35 68
Email : wainrib [at] math.univ-paris13.fr