Adaptive test of independence based on HSIC measures

Nom de l'orateur
Anouar Meynaoui
Etablissement de l'orateur
IRMAR
Date et heure de l'exposé
Lieu de l'exposé
Salle des Séminaires

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level $\alpha \in (0,1)$. From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikol'skii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.

References:

-> This presentation will be based on the paper :

Adaptive test of independence based on HSIC measures (https://arxiv.org/abs/1902.06441)

-> To be prepared :

  • First part of the talk deals with optimality of non parametric independence statistical tests, see Sections 0.1 and 0.2 of the PhD thesis of Melisande Albert

https://perso.math.univ-toulouse.fr/albert/files/2016/11/These-malbert.pdf

  • Second part of the talk deals with kernel based measures of independence, see

Measuring Statistical Dependence with Hilbert-Schmidt Norms (https://is.mpg.de/publications/3774)