Functional Tensor Regression

TL;DR

This paper introduces a functional tensor regression framework that combines tensor and functional data analysis, employing low-rank decomposition and smooth regularization, with an efficient algorithm and demonstrated effectiveness in simulations and neuroimaging.

Contribution

It proposes a novel functional tensor regression model with a low Tucker rank and smooth regularization, along with a quadratic convergence algorithm.

Findings

Demonstrates finite sample performance through simulations.

Shows effectiveness in neuroimaging data analysis.

Provides theoretical convergence guarantees.

Abstract

Tensor regression has attracted significant attention in statistical research. This study tackles the challenge of handling covariates with smooth varying structures. We introduce a novel framework, termed functional tensor regression, which incorporates both the tensor and functional aspects of the covariate. To address the high dimensionality and functional continuity of the regression coefficient, we employ a low Tucker rank decomposition along with smooth regularization for the functional mode. We develop a functional Riemannian Gauss--Newton algorithm that demonstrates a provable quadratic convergence rate, while the estimation error bound is based on the tensor covariate dimension. Simulations and a neuroimaging analysis illustrate the finite sample performance of the proposed method.