Sparse single-index model

TL;DR

This paper introduces a sparsity-based estimation method for high-dimensional single-index models using a PAC-Bayesian approach, providing theoretical guarantees and demonstrating competitive performance through MCMC implementation.

Contribution

It develops a novel PAC-Bayesian estimation framework for sparse single-index models, with a sharp oracle inequality and practical MCMC-based implementation.

Findings

The method achieves a sharper oracle inequality than existing procedures.

The MCMC implementation performs competitively with standard methods.

The approach effectively handles high-dimensional settings where p > n.

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.