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Optimally Sparse SVMs

We here prove a lower bound on the number of support vectors to achieve generalization bounds of the form which we consider. Importantly, this result holds not only for linear classifiers, but also for randomized classifiers of a more general form than those by our sparsification procedure (Section 4). Lemma 3.1. Let R,L∗, ≥ 0 be given, with L∗ + ≤ 1/4 and with R being an integer. There exists a data distribution D and a reference vector u such that ‖u‖ = R, Lhinge (gu) = L∗, and any w which satisfies: L0/1 (gw) ≤ L∗ +