Clinical risk prediction is one important problem in medical informatics, and logistic regression is one of the most widely used approaches for clinical risk prediction. In many cases, the number of potential risk factors is fairly large and the actual set of factors that contribute to the risk is small. Therefore sparse logistic regression is proposed, which can not only predict the clinical risk but also identify the set of relevant risk factors. The inputs of logistic regression and sparse logistic regression are required to be in vector form. This limits the applicability of these models in the problems when the data cannot be naturally represented vectors (e.g., medical images are two-dimensional matrices). To handle the cases when the data are in the form of multi-dimensional arrays, we propose HOSLR: High-Order Sparse Logistic Regression, which can be viewed as a high order extension of sparse logistic regression. Instead of solving one classification vector as in conventional logistic regression, we solve for K classification vectors in HOSLR (K is the number of modes in the data). A block proximal descent approach is proposed to solve the problem and its convergence is guaranteed. Finally we validate the effectiveness of HOSLR on predicting the onset risk of patients with Alzheimer's disease and heart failure.