Interpretable Scalar-on-Image Linear Regression Models via the Generalized Dantzig Selector

TL;DR

This paper introduces a new sparse and smooth scalar-on-image regression method called the Generalized Dantzig Selector, improving interpretability and estimation stability in image response modeling.

Contribution

It proposes the Generalized Dantzig Selector that enforces sparsity and smoothness simultaneously, with theoretical error bounds and superior interpretability.

Findings

Enhanced interpretability by identifying non-contributing image regions

Improved estimation stability and accuracy over existing methods

Theoretical non-asymptotic error bounds established

Abstract

The scalar-on-image regression model examines the association between a scalar response and a bivariate function (e.g., images) through the estimation of a bivariate coefficient function. Existing approaches often impose smoothness constraints to control the bias-variance trade-off, and thus prevent overfitting. However, such assumptions can hinder interpretability, especially when only certain regions of an image influence changes in the response. In such a scenario, interpretability can be better captured by imposing sparsity assumptions on the coefficient function. To address this challenge, we propose the Generalized Dantzig Selector, a novel method that jointly enforces sparsity and smoothness on the coefficient function. The proposed approach enhances interpretability by accurately identifying regions with no contribution to the changes of response, while preserving stability in estimation. Extensive simulation studies and real data applications demonstrate that the new method is highly interpretable and achieves notable improvements over existing approaches. Moreover, we rigorously establish non-asymptotic bounds for the estimation error, providing strong theoretical guarantees for the proposed framework.