Sparse Kronecker product decomposition: a general framework of signal region detection in image regression

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 focussed on outcome prediction, while the research on 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 and tensors represented image data. Our framework includes one-term, multi-term, and nonlinear SKPDs. We propose nonconvex optimization problems for one-term and multi-term SKPDs and develop path-following algorithms for the nonconvex optimization. Under a Restricted Isometric Property, 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. The nonlinear SKPD is highly connected to shallow convolutional neural networks (CNN), particularly to CNN with one convolutional layer and one fully-connected layer. Effectiveness of SKPD is validated by real brain imaging data in the UK Biobank database.