  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.
