Author Archives: Jérémie Bigot

Axis 9: Nonparametric estimation and statistical process

In this axis, one of the research direction is focused on nonparametric and semi-parametric statistics for the design of optimal estimators (in the minimax sense, or from Oracle inequalities) for statistical inference problems in large dimension (deformable models in signal processing, covariance matrix estimation, inverse problems).

In this context, the first part focuses on the minimization of the Stein unbiased risk estimator (SURE) for variational models. A first theoretical difficulty is to design such estimators when the targeted functionals are non-smooth, non-convex or discontinuous. A second difficulty relates to the development of efficient algorithms for the calculation and minimization of the SURE when solutions of these models are themselves derived from an optimization algorithm. Finally, a last difficulty concerns the extension of the SURE to complex inference problems (ill-posed problems, non white Gaussian noise, etc.).

Another part of this axis concerns semi-parametric regression models when the regression function is estimated by a recursive Nadaraya-Watson type estimator. In this context, a “région Aquitaine” contract was obtained in 2014 for 3 years. It covers the development of new non-parametric estimation methods with applications in valvometry and environmental sciences.

Axis 5: Stochastic calculus, probabilities and statistics on manifolds

This axis relates to the use of all methods of stochastic calculus, particularly the detailed analysis of process trajectories, their probabilities, their variation, their couplings, in order to:

  • analyse the diffusion semigroups and evolution equations in manifolds (heat equation, mean curvature equation, Ricci flow), and their use in signal and image processing,
  • get functional inequalities,
  • study the Poisson boundaries,
  • computing price sensitivity in financial models,
  • get transport inequality,
  • design search and optimization algorithms in manifolds for images and signals.

Existence and uniqueness problems are also studied for martingales with given terminal value in manifolds. Several contributions also cover the notion of Fréchet mean which is an extension of the usual Euclidean barycenter to spaces equiped with non-Euclidean distances. In this context, many statistical properties of the Fréchet mean were established for deformable models of signals.