Optimal Function-on-Scalar Regression over Complex Domains

Abstract

In this work we consider the problem of estimating function-onscalar regression models when the functions are observed over multidimensional 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.1. Introduction. Functional data analysis has seen a precipitous development in recent decades, in terms of methodology, theory, and applications. As with classical statistics, functional linear regression models are used extensively in practice. In recent years, there has also been a surge in the development of so-called next generation functional data analysis, which involves functional data with highly complex structures. Much of this development has been spurred by advances in biomedical imaging, where dense measurements are taken over various tissues, including the brain,

Functional Linear Regression with Mixed Predictors

“…To establish Theorem 2, we develop new peeling techniques to obtain new exponential tail bounds for our estimators and this is crucial in dealing with the potentially exponentially growing dimension. We remark that our finite sample guarantees are not consequences of existing results in the literature such as Sun et al (2018) or Reimherr et al (2019). To the best of our knowledge, all the existing literature that are able to achieve minimax optimality only provide asymptotic guarantees and require the functional noise satisfying the second moment condition:…”

“…Regarding the statistical inference tasks, in the regression context, prediction and estimation are two indispensable pillars. In the functional linear regression literature, Valencia and Yuan (2013), Cai and Yuan (2011), Yuan and Cai (2010), Lin and Yuan (2006), Park et al (2018), Fan et al (2014), Fan et al (2015), and Reimherr et al (2019),, among many others, have studied different aspects of the estimation problem. As for prediction, the existing literature includes Cai and Yuan (2012), Cai and Hall (2006), Ferraty and Vieu (2009), Sun et al (2018), Reimherr et al (2018) to name but a few.…”