Let (X, Y ) be a random pair taking values in R p × R. In the socalled single-index model, one has Y = f (θ T X) + W , where f is an unknown univariate measurable function, θ is an unknown vector in R d , and W denotes a random noise satisfying E[W |X] = 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 1 Corresponding author. 2 Research partially supported by the French "Agence Nationale pour la Recherche" under grant ANR-09-BLAN-0128 "PARCIMONIE". 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.
Alquier and Biau REF consider learning high dimensional single index models.
Sparse single-index model
Journal of Machine Learning Research 14 (2013) 243-280
Mathematics Computer Science