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2023 Regression in tensor product spaces by the method of sieves
Tianyu Zhang, Noah Simon
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Electron. J. Statist. 17(2): 3660-3727 (2023). DOI: 10.1214/23-EJS2188


Estimation of a conditional mean (linking a set of features to an outcome of interest) is a fundamental statistical task. While there is an appeal to flexible nonparametric procedures, effective estimation in many classical nonparametric function spaces, e.g., multivariate Sobolev spaces, can be prohibitively difficult – both statistically and computationally – especially when the number of features is large. In this paper, we present some sieve estimators for regression in multivariate product spaces. We take Sobolev-type smoothness spaces as an example, though our general framework can be applied to many reproducing kernel Hilbert spaces. These spaces are more amenable to multivariate regression, and allow us to, in-part, avoid the curse of dimensionality. Our estimator can be easily applied to multivariate nonparametric problems and has appealing statistical and computational properties. Moreover, it can effectively leverage additional structure such as feature sparsity.

Funding Statement

The authors gratefully acknowledge NIH grant R01HL137808.


The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.


Download Citation

Tianyu Zhang. Noah Simon. "Regression in tensor product spaces by the method of sieves." Electron. J. Statist. 17 (2) 3660 - 3727, 2023.


Received: 1 January 2023; Published: 2023
First available in Project Euclid: 7 December 2023

arXiv: 2206.02994
Digital Object Identifier: 10.1214/23-EJS2188

Primary: 62G05 , 62G08

Keywords: multivariate regression , orthonormal basis , sparse nonparametric models

Vol.17 • No. 2 • 2023
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