  We address the problem of sufficient dimension reduction for feature
matrices, which arises often in sensor network localization, brain
neuroimaging, and electroencephalography analysis. In general, feature matrices
have both row- and column-wise interpretations and contain structural
information that can be lost with naive vectorization approaches. To address
this, we propose a method called principal support matrix machine (PSMM) for
the matrix sufficient dimension reduction. The PSMM converts the sufficient
dimension reduction problem into a series of classification problems by
dividing the response variables into slices. It effectively utilizes the matrix
structure by finding hyperplanes with rank-1 normal matrix that optimally
separate the sliced responses. Additionally, we extend our approach to the
higher-order tensor case. Our numerical analysis demonstrates that the PSMM
outperforms existing methods and has strong interpretability in real data
applications.
