A Structural SVM Based Approach for Optimizing Partial AUC5ptSupplementary Material
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چکیده
Proof. Suppose π̄ / ∈ Πm,n. Then ∃i, j1 < j2 such that π̄i,(j1)w < π̄i,(j2)w , i.e. such that π̄i,(j1)w = 0 and π̄i,(j2)w = 1. This means that π̄ ranks x + i above x−(j1)w but below x − (j2)w . Now let us construct from π̄ an ordering π̄ in which the instances x−(j1)w and x−(j2)w are swapped, i.e. for all i ′ with π̄i′,(j1)w = 0 and π̄i′,(j2)w = 1, we set π̄ ′ i′,(j1)w = 1 and π̄ i′,(j2)w = 0. Then it can be seen that while the loss term in the objective in OP2 is the same for π̄ as for π̄, the second term increases, yielding a higher objective value, Qw(π̄ ) > Qw(π̄). This contradicts the fact that π̄ is a maximizer of Qw(π).
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A Structural SVM Based Approach for Optimizing Partial AUC
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