Abstract
The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as for some unknown index vector v and link function f. Conditional methods provide a simple and effective approach to estimate v by averaging moments of X conditioned on Y, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on f. In this paper we propose a new conditional method converging at rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Hölder functions when combined to any -convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time.
Funding Statement
This research was partially supported by AFOSR FA9550-17-1-0280, NSF-DMS-1821211, NSF-ATD-1737984.
Acknowledgements
S.V. thanks Timo Klock for the discussion and the useful exchange of views about this and related problems.
Citation
Alessandro Lanteri. Mauro Maggioni. Stefano Vigogna. "Conditional regression for single-index models." Bernoulli 28 (4) 3051 - 3078, November 2022. https://doi.org/10.3150/22-BEJ1482