Adaptive Greedy Algorithm for Moderately Large Dimensions in Kernel Conditional Density Estimation

Minh-Lien Jeanne Nguyen (1, 2) , Claire Lacour (3, 2) , Vincent Rivoirard (4)

1 Universiteit Leiden = Leiden University
2 CELESTE - Statistique mathématique et apprentissage
3 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper studies the estimation of the conditional density f(x,⋅) of Yi given Xi=x, from the observation of an i.i.d. sample (Xi,Yi)∈ℝ^d, i∈{1,…,n}. We assume that f depends only on r unknown components with typically r≪d.We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate f. To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty, 2006) to detect the sparsity structure of f. More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Our results also hold for (unconditional) density estimation. The computational complexity of our method is only O(dnlogn). A deep numerical study shows nice performances of our approach.

Mots clés: Sparsity; Greedy algorithm; Kernel density estimators; Minimax rates; Conditional density

Domaines: Statistiques [math.ST]

Minh-Lien Jeanne Nguyen, Claire Lacour, Vincent Rivoirard. Adaptive Greedy Algorithm for Moderately Large Dimensions in Kernel Conditional Density Estimation. Journal of Machine Learning Research, 2022, 23 (254). ⟨hal-02085677v3⟩