HAL: hal-00556652, version 2

arXiv: 1101.3229

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v1 (2011-01-17)

v2 (2011-10-06)

Sparse single-index model

Pierre Alquier 1, 2, Gérard Biau 1, 3, 4, 5

(2011-10-05)

Let $(\bX, Y)$ be a random pair taking values in $\mathbb R^p \times \mathbb R$. In the so-called single-index model, one has $Y=f^{\star}(\theta^{\star T}\bX)+\bW$, where $f^{\star}$ is an unknown univariate measurable function, $\theta^{\star}$ is an unknown vector in $\mathbb R^d$, and $W$ denotes a random noise satisfying $\mathbb E[\bW|\bX]=0$. The single-index model is known to offer a flexible way to model a variety of high-dimensional real-world phenomena. However, despite its relative simplicity, this dimension reduction scheme is faced with severe complications as soon as the underlying dimension becomes larger than the number of observations (''$p$ larger than $n$'' paradigm). To circumvent this difficulty, we consider the single-index model estimation problem from a sparsity perspective using a PAC-Bayesian approach. On the theoretical side, we offer a sharp oracle inequality, which is more powerful than the best known oracle inequalities for other common procedures of single-index recovery. The proposed method is implemented by means of the reversible jump Markov chain Monte Carlo technique and its performance is compared with that of standard procedures.

1:	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
2:	Centre de Recherche en Économie et Statistique (CREST)
	INSEE – École Nationale de la Statistique et de l'Administration Économique
3:	Laboratoire de Statistique Théorique et Appliquée (LSTA)
	Université Pierre et Marie Curie [UPMC] - Paris VI
4:	Département de Mathématiques et Applications (DMA)
	CNRS : UMR8553 – Ecole normale supérieure de Paris - ENS Paris
5:	CLASSIC (INRIA Paris - Rocquencourt)
	Ecole normale supérieure de Paris - ENS Paris – INRIA

Subject

:

Mathematics/Statistics Statistics/Statistics Theory

Keyword(s): Nonparametric statistics – single-index model – sparsity – PAC-Bayesian inequalities – oracle inequalities – MCMC.

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From: Pierre Alquier <>

Submitted on: Wednesday, 5 October 2011 11:47:17

Updated on: Tuesday, 19 June 2012 15:36:07