Sparse Kronecker Product Decomposition: A General Framework of Signal Region Detection in Image Regression

TL;DR

This paper introduces a scalable Frequentist framework called Sparse Kronecker Product Decomposition (SKPD) for detecting signal regions in high-resolution image regression, applicable to matrices and tensors, with theoretical guarantees and real data validation.

Contribution

The paper develops a novel, scalable SKPD framework for image region detection that works for high-dimensional matrices and tensors, with convergence guarantees and applicability to neural network models.

Findings

Effective in real brain imaging data from UK Biobank

Guarantees convergence of the nonconvex optimization algorithms

Achieves consistent region detection in high-resolution images

Abstract

This paper aims to present the first Frequentist framework on signal region detection in high-resolution and high-order image regression problems. Image data and scalar-on-image regression are intensively studied in recent years. However, most existing studies on such topics focused on outcome prediction, while the research on image region detection is rather limited, even though the latter is often more important. In this paper, we develop a general framework named Sparse Kronecker Product Decomposition (SKPD) to tackle this issue. The SKPD framework is general in the sense that it works for both matrices (e.g., 2D grayscale images) and (high-order) tensors (e.g., 2D colored images, brain MRI/fMRI data) represented image data. Moreover, unlike many Bayesian approaches, our framework is computationally scalable for high-resolution image problems. Specifically, our framework includes: 1) the one-term SKPD; 2) the multi-term SKPD; and 3) the nonlinear SKPD. We propose nonconvex optimization problems to estimate the one-term and multi-term SKPDs and develop path-following algorithms for the nonconvex optimization. The computed solutions of the path-following algorithm are guaranteed to converge to the truth with a particularly chosen initialization even though the optimization is nonconvex. Moreover, the region detection consistency could also be guaranteed by the one-term and multi-term SKPD. The nonlinear SKPD is highly connected to shallow convolutional neural networks (CNN), particular to CNN with one convolutional layer and one fully connected layer. Effectiveness of SKPDs is validated by real brain imaging data in the UK Biobank database.