A geometric framework for understanding multi-category classification is introduced, through which many existing 'all-together' algorithms can be understood. The structure enables parsimonious optimisation, through a direct extension of the binary methodology. The focus is on Support Vector Classification, with parallels drawn to related methods. The ability of the framework to compare algorithms is illustrated by a brief discussion of Fisher consistency. Its utility in improving understanding of multi-category analysis is demonstrated through a derivation of improved generalisation bounds. It is also described how this architecture provides insights regarding how to further improve on the speed of existing multi-category classification algorithms. An initial example of how this might be achieved is developed in the formulation of a straightforward multi-category Sequential Minimal Optimisation algorithm. Proof-of-concept experimental results have shown that this, combined with the mapping of pairwise results, is comparable with benchmark optimisation speeds.

A geometric construction is presented which is shown to be an effective tool for understanding and implementing multi-category support vector classification. It is demonstrated how this construction can be used to extend many other existing two-class kernel-based classification methodologies in a straightforward way while still preserving attractive properties of individual algorithms. Reducing training times through incorporating the results of pairwise classification is also discussed and experimental results presented.