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We provide a novel interpretation of the dual of support vector machines (SVMs) in terms of scatter with respect to class prototypes and their mean. As a key contribution, we extend this framework to multiple classes, providing a new joint Scatter SVM algorithm, at the level of its binary counterpart in the number of optimization variables. We identify the associated primal problem and develop a fast chunking-based optimizer. Promising results are reported, also compared to the state-of-the-art, at lower computational complexity.