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Journal Articles Journal of Machine Learning Research Year : 2013

Sparse single-index model

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Abstract

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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Dates and versions

hal-00556652 , version 1 (17-01-2011)
hal-00556652 , version 2 (05-10-2011)

Identifiers

Cite

Pierre Alquier, Gérard Biau. Sparse single-index model. Journal of Machine Learning Research, 2013, 14, pp.243−280. ⟨hal-00556652v2⟩
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