SVM can is very effective for binary classification. In order to handle classification with many categories, there are also some extensions for multi-class SVM. Anyway, a more frequently used approach is to adopt a binary classifcation with a one-against all approach (each category is considered against the others and n categories are obtained by levaging n binary classifiers).
Large Scale Multi-Label Classification via MetaLaber suggests extending the one-against all approach by adopting an auxiliar classifier which learns the number of relevant top-k scores.
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