Adaptive Estimation and Statistical Inference for High-Dimensional Graph-Based Linear Models

TL;DR

This paper introduces adaptive estimation and inference methods for high-dimensional linear models with graph-structured coefficients, utilizing graph-based regularization and piecewise polynomial modeling to improve accuracy and interpretability.

Contribution

The paper proposes the Graph-Piecewise-Polynomial-Lasso and its one-step update for adaptive estimation and inference in graph-structured high-dimensional linear models, with theoretical guarantees.

Findings

The methods outperform existing approaches in simulations.

The approach effectively captures piecewise polynomial structures.

Application to microarray data demonstrates practical utility.

Abstract

We consider adaptive estimation and statistical inference for high-dimensional graph-based linear models. In our model, the coordinates of regression coefficients correspond to an underlying undirected graph. Furthermore, the given graph governs the piecewise polynomial structure of the regression vector. In the adaptive estimation part, we apply graph-based regularization techniques and propose a family of locally adaptive estimators called the Graph-Piecewise-Polynomial-Lasso. We further study a one-step update of the Graph-Piecewise-Polynomial-Lasso for the problem of statistical inference. We develop the corresponding theory, which includes the fixed design and the sub-Gaussian random design. Finally, we illustrate the superior performance of our approaches by extensive simulation studies and conclude with an application to an Arabidopsis thaliana microarray dataset.