Deep Kronecker Network

We propose Deep Kronecker Network (DKN), a novel framework designed for analyzing medical imaging data, such as MRI, fMRI, CT, etc. Medical imaging data is different from general images in at least two aspects: i) sample size is usually much more limited, ii) model interpretation is more of a concern compared to outcome prediction. Due to its unique nature, general methods, such as convolutional neural network (CNN), are difficult to be directly applied. As such, we propose DKN, that is able to i) adapt to low sample size limitation, ii) provide desired model interpretation, and iii) achieve the prediction power as CNN. The DKN is general in the sense that it not only works for both matrix and (high-order) tensor represented image data, but also could be applied to both discrete and continuous outcomes. The DKN is built on a Kronecker product structure and implicitly imposes a piecewise smooth property on coefficients. Moreover, the Kronecker structure can be written into a convolutional form, so DKN also resembles a CNN, particularly, a fully convolutional network (FCN). Furthermore, we prove that with an alternating minimization algorithm, the solutions of DKN are guaranteed to converge to the truth geometrically even if the objective function is highly nonconvex. Interestingly, the DKN is also highly connected to the tensor regression framework proposed by Zhou et al. (2010), where a CANDECOMP/PARAFAC (CP) low-rank structure is imposed on tensor coefficients. Finally, we conduct both classification and regression analyses using real MRI data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) to demonstrate the effectiveness of DKN.