Robust Multicategory Support Matrix Machines

We consider the classification problem when the input features are represented as matrices rather than vectors. To preserve the intrinsic structures for classification, a successful method is the Support Matrix Machine (SMM) in [19], which optimizes an objective function with a hinge loss plus a so-called spectral elastic net penalty. However, the issues of extending SMM to multicategory classification still remain. Moreover, in practice, it is common to see the training data contaminated by outlying observations, which can affect the robustness of existing matrix classification methods. In this paper, we address these issues by introducing a robust angle-based classifier, which boils down binary and multicategory problems to a unified framework. Benefitting from the use of truncated hinge loss functions, the proposed classifier achieves certain robustness to outliers. The underlying optimization model becomes nonconvex, but admits a natural DC (difference of two convex functions) representation. We develop a new and efficient algorithm by incorporating the DC algorithm and primal-dual first-order methods together. The proposed DC algorithm adaptively chooses the accuracy of the subproblem at each iteration while guaranteeing the overall convergence of the algorithm. The use of primal-dual methods removes a natural complexity of the linear operator in the subproblems and enables us to use the proximal operator of the objective functions, and matrix-vector operations. This advantage allows us to solve large-scale problems efficiently. Theoretical and numerical results indicate that for problems with potential outliers, our method can be highly competitive among existing methods.