Investigating on nonlinear relationship in high-dimensional setting

The high dimensional setting is a modern and dynamic research area in Statistics. It covers numerous situations where the number of explanatory variables is much larger than the sample size. This is the case in genomics when one observes (dozens of) thousands genes expression ; typically one has at hand a small sample of high dimensioned vectors derived from a large set of covariates. Such datasets will be abbreviated to HDD-I for High Dimensional Data of type I. A particular setting may correspond to the observation of a collection of curves, surfaces, ... sampled at high frequencies (design points) ; these sets of data are gathered under the terminology functional data (or functional variables) and will be abbreviated to HDD-II (High Dimensional Data of type II). The main feature of HDD-II (and difference with HDD-I) is due to the existence of high colinearities between explanatory variables which reduces the overall dimensionality of the data. Last twenty years have been devoted to develop successful methodologies able to manage such high dimensional data. Essentially sparse linear modelling involving variable selection techniques has been proposed to investigate on HDD-I whereas non selective functional linear approaches have been introduced to handle HDD-II mainly. However, as in the standard multivariate setting, linear assumption may too much restrictive by hiding relevant nonlinear aspects. This is why in the last decade flexible methodologies taking into account nonlinear relationship have been developed to better understand the structure of such high dimensional data. So, the aim of this talk is to present and illustrate on various examples recent approaches connecting nonparametric, selective and functional techniques in order to handle nonlinear relationship in HDD-I or HDD-II settings which allow us to tackle various challenging issues.