Testing relevant difference in high-dimensional linear regression with applications to detect transferability

TL;DR

This paper introduces a new hypothesis test for high-dimensional linear regression that assesses the relevance of coefficient differences, with applications to transfer learning, using eigenvalue estimation and random matrix theory.

Contribution

It proposes a novel test procedure for relevant differences in coefficients, incorporating eigenvalue estimation and addressing high-dimensional nuisance parameters.

Findings

The test accurately detects relevant coefficient differences in simulations.

Application to GTEx data demonstrates improved transferability detection.

The method achieves lower estimation and prediction errors compared to existing approaches.

Abstract

Most of researchers on testing a significance of coefficient $\ubeta$ in high-dimensional linear regression models consider the classical hypothesis testing problem $H_0^{c}: \ubeta=\uzero \mbox{ versus } H_1^{c}: \ubeta \neq \uzero$. We take a different perspective and study the testing problem with the null hypothesis of no relevant difference between $\ubeta$ and $\uzero$, that is, $H_0: \|\ubeta\|\leq \delta_0 \mbox{ versus } H_1: \|\ubeta\|> \delta_0$, where $\delta_0$ is a prespecified small constant. This testing problem is motivated by the urgent requirement to detect the transferability of source data in the transfer learning framework. We propose a novel test procedure incorporating the estimation of the largest eigenvalue of a high-dimensional covariance matrix with the assistance of the random matrix theory. In the more challenging setting in the presence of high-dimensional nuisance parameters, we establish the asymptotic normality for the proposed test statistics under both the null and alternative hypotheses. By applying the proposed test approaches to detect the transferability of source data, the unified transfer learning models simultaneously achieve lower estimation and prediction errors with comparison to existing methods. We study the finite-sample properties of the new test by means of simulation studies and illustrate its performance by analyzing the GTEx data.