Broadcasted Nonparametric Tensor Regression

TL;DR

This paper introduces a nonparametric tensor regression method using broadcasting operations to model nonlinear relationships, with proven convergence rates and optimality, validated by numerical experiments.

Contribution

It presents a novel broadcasting-based approach for nonparametric tensor regression, including penalized estimation and theoretical analysis of convergence and optimality.

Findings

Estimation achieves desirable convergence rates.

Method outperforms linear models in experiments.

Provides minimax lower bounds for estimator optimality.

Abstract

We propose a novel use of a broadcasting operation, which distributes univariate functions to all entries of the tensor covariate, to model the nonlinearity in tensor regression nonparametrically. A penalized estimation and the corresponding algorithm are proposed. Our theoretical investigation, which allows the dimensions of the tensor covariate to diverge, indicates that the proposed estimation yields a desirable convergence rate. We also provide a minimax lower bound, which characterizes the optimality of the proposed estimator for a wide range of scenarios. Numerical experiments are conducted to confirm the theoretical findings, and they show that the proposed model has advantages over its existing linear counterparts.