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

2025-08-27

AI Summary

In scalar-on-image regression, conventional smoothness assumptions often compromise the interpretability of coefficient functions. To address this, we propose a two-dimensional coefficient function estimation method that jointly enforces sparsity and smoothness. Our core innovation is a generalized Dantzig selector framework incorporating both an ℓ₁-norm penalty for sparsity and a second-order difference penalty for smoothness, enabling precise identification of image regions with no contribution to the response. Theoretically, we derive non-asymptotic error bounds, ensuring estimation stability and optimal convergence rates. Computationally, we design an efficient optimization algorithm. Simulation studies and real-data analyses demonstrate that our method maintains high estimation accuracy while substantially improving spatial interpretability—outperforming existing approaches dominated either by smoothness or by sparsity alone.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Identifies sparse image regions affecting scalar response
• Enhances interpretability by combining sparsity and smoothness
• Addresses overfitting while maintaining estimation stability

Innovation

• Generalized Dantzig Selector for sparse estimation
• Jointly enforces sparsity and smoothness constraints
• Identifies non-contributing image regions with stability