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| HAL: hal-00556652, version 2 |
| arXiv: 1101.3229 |
| Detailed view |  Export this paper |
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| Available versions: | v1 (2011-01-17) | v2 (2011-10-06) |
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| Sparse single-index model |
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| Pierre Alquier 1, 2Gérard Biau 1, 3, 4, 5 |
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| (2011-10-05) |
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| 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. |
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| 1: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA) |
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CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot |
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| 2: | Centre de Recherche en Économie et Statistique (CREST) |
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INSEE – École Nationale de la Statistique et de l'Administration Économique |
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| 3: | Laboratoire de Statistique Théorique et Appliquée (LSTA) |
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Université Pierre et Marie Curie [UPMC] - Paris VI |
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| 4: | Département de Mathématiques et Applications (DMA) |
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CNRS : UMR8553 – Ecole normale supérieure de Paris - ENS Paris |
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| 5: | CLASSIC (INRIA Paris - Rocquencourt) |
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Ecole normale supérieure de Paris - ENS Paris – INRIA |
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| Subject | : | Mathematics/Statistics Statistics/Statistics Theory |
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| Nonparametric statistics – single-index model – sparsity – PAC-Bayesian inequalities – oracle inequalities – MCMC. |
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| Attached file list to this document: | |||||
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| hal-00556652, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00556652 | |
| oai:hal.archives-ouvertes.fr:hal-00556652 | |
| From: Pierre Alquier | |
| Submitted on: Wednesday, 5 October 2011 11:47:17 | |
| Updated on: Tuesday, 19 June 2012 15:36:07 | |
