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Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation

Nguyen, Minh-Lien; Lacour, Claire; Rivoirard, Vincent (2022), Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation, Journal of Machine Learning Research, 23, 254, p. 1−74

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Type
Article accepté pour publication ou publié
External document link
https://www.jmlr.org/papers/volume23/21-0582/21-0582.pdf
Date
2022
Journal name
Journal of Machine Learning Research
Volume
23
Number
254
Publisher
Microtome Publishing
Published in
Paris
Pages
1−74
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Author(s)
Nguyen, Minh-Lien
Laboratoire de Mathématiques d'Orsay [LMO]
Lacour, Claire
Laboratoire d'Analyse et de Mathématiques Appliquées [LAMA]
Rivoirard, Vincent
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
This paper studies the estimation of the conditional density f (x, ·) of Y i given X i = x, from the observation of an i.i.d. sample (X i , Y i) ∈ 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. Its computational complexity is only O(dn log n).
Subjects / Keywords
sparsity; greedy algorithm; kernel density estima- tors; minimax rates; high dimension; conditional density; nonparametric inference