Publications

2018

• Efficient semiparametric estimation and model selection for multidimensional mixtures
  
  • Gassiat Elisabeth
  • Rousseau Judith
  • Vernet Elodie
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.703-740. In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results. (10.1214/17-ejs1387)
  DOI : 10.1214/17-ejs1387
• Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
  
  • Akimoto Youhei
  • Auger Anne
  • Glasmachers Tobias
  , 2018. This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $\Theta(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.
• Volume Viscosity and Internal Energy Relaxation: Error Estimates
  
  • Giovangigli Vincent
  • Yong Wen An
  Nonlinear Analysis: Real World Applications, Elsevier, 2018. We investigate the fast relaxation of internal energy in nonequilibrium gas models derived from the kinetic theory of gases. We establish uniform a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources. We prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms.
• Monochromatic identities for the Green function and uniqueness results for passive imaging
  
  • Agaltsov Alexey
  • Hohage Thorsten
  • Novikov Roman
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (5), pp.2865–2890. For many wave propagation problems with random sources it has been demonstrated that cross correlations of wave fields are proportional to the imaginary part of the Green function of the underlying wave equation. This leads to the inverse problem to recover coefficients of a wave equation from the imaginary part of the Green function on some measurement manifold. In this paper we prove, in particular, local uniqueness results for the Schrödinger equation with one frequency and for the acoustic wave equation with unknown density and sound speed and two frequencies. As the main tool of our analysis, we establish new algebraic identities between the real and the imaginary part of Green's function, which in contrast to the well-known Kramers-Kronig relations involve only one frequency. (10.1137/18M1182218)
  DOI : 10.1137/18M1182218
• Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.
  
  • Durmus Alain
  • Moulines Éric
  • Pereyra Marcelo
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (1). In this paper, two new algorithms to sample from possibly non-smooth log-concave probability measures are introduced. These algorithms use Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a de-convolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic bounds for one of the proposed methods are derived. These bounds follow from non-asymptotic results for ULA applied to probability measures with a convex continuously differentiable log-density with respect to the Lebesgue measure. (10.1137/16M110834)
  DOI : 10.1137/16M110834