Open Access
2023 Optimal function-on-scalar regression over complex domains
Matthew Reimherr, Bharath Sriperumbudur, Hyun Bin Kang
Author Affiliations +
Electron. J. Statist. 17(1): 156-197 (2023). DOI: 10.1214/22-EJS2096


In this work we consider the problem of estimating function-on-scalar regression models when the functions are observed over multi-dimensional or manifold domains and with potentially multivariate output. We establish the minimax rates of convergence and present an estimator based on reproducing kernel Hilbert spaces that achieves the minimax rate. To better interpret the derived rates, we extend well-known links between RKHS and Sobolev spaces to the case where the domain is a compact Riemannian manifold. This is accomplished using an interesting connection to Weyl’s Law from partial differential equations. We conclude with a numerical study and an application to 3D facial imaging.

Funding Statement

M. Reimherr was supported by NSF grant SES 1853209. B. Sriperumbudur was supported by NSF grant DMS 1945396.


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Matthew Reimherr. Bharath Sriperumbudur. Hyun Bin Kang. "Optimal function-on-scalar regression over complex domains." Electron. J. Statist. 17 (1) 156 - 197, 2023.


Received: 1 November 2021; Published: 2023
First available in Project Euclid: 16 January 2023

MathSciNet: MR4533744
zbMATH: 07649360
Digital Object Identifier: 10.1214/22-EJS2096

Keywords: Functional data analysis , functional regression , optimal regression , ‎reproducing kernel Hilbert ‎space , Weyl’s law

Vol.17 • No. 1 • 2023
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