Speaker: Krishnakumar Balasubramanian (https://sites.google.com/view/kriznakumar/)

Title: Unified RKHS Methodology and Analysis for Functional Linear and Single-Index Models

Abstract: Functional linear and single-index models are elementary methods in the functional data analysis toolkit and are widely used methods for performing regression when the covariates are observed as random functions in various applications. In the existing literature, however, constructing the associated estimators and studying their theoretical properties are invariably carried out on a case-by-case basis for the individual model under consideration. In this work, we provide a unified methodological and theoretical framework for estimating the index in the functional linear and single-index models; in the latter case the proposed approach is agnostic to the specification of the link function. On the methodological side, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. On the theoretical side, we characterize the convergence rates of the estimator for both the linear and single-index model. Our analysis has several advantages: (i) we do not require restrictive commutativity assumptions on the covariance operator of the random covariates and the integral operator associated with the reproducing kernel, and (ii) we also allow for the true index parameter to lie outside of the chosen RKHS thereby allowing for and quantifying the degree of index misspecification in the models. We recover several existing results as special case of our analysis.

Short bio: Krishna Balasubramanian is an assistant professor in the Department of Statistics, University of California, Davis. His recent research interests include stochastic optimization and sampling, reproducing kernel Hilbert space methods, and geometric and topological statistics. His research was/is supported by a Facebook PhD fellowship, and CeDAR and NSF grants.