High-dimensional linear classifiers, such as support vector machines (SVM) and distance weighted discrimination (DWD), are used to distinguish groups based on a large number of features. However, their use is limited to applications where a vector of features is measured per subject. In practice, data may be multi-way: measured over multiple dimensions, for example, metabolite abundance over multiple tissues, or gene expression over multiple time points. We propose a framework for linear classification of high-dimensional multi-way data, in which coefficients can be factorized into weights for each dimension. More generally, the coefficients for each measurement in a multi-way dataset have low-rank structure. This work extends existing classification techniques, and we have implemented and compared them to competing classifiers. We describe simulation results, and apply multi-way DWD to a neuroimaging study using magnetic resonance spectroscopy data over multiple brain regions to compare patients with and without spinocerebellar ataxia. Our method improves performance and simplifies interpretation over naive applications of full rank linear classification to multi-way data. This is joint work with Eric Lock and Tianmeng Lyu.