(9)Proof ��We first rewrite the f(��(1), ��(2)) as follows:f(��(1),��(2))=[b?(?��(1),x(1)???��(2),x(2)?)]+=max?����[0,1]?��(b?(?��(1),x(1)???��(2),x(2)?)).(10)Based on the definition of Fenchel conjugate for multivariables functions, we can +max?��(2)?��(2)?��x(2),��(2)?).(11)Since?????=min?����[0,1]?(?��b+max?��(1)?��(1)+��x(1),��(1)??+?��(2)?��x(2),��(2)?)????????=min?����[0,1]max?(��(1),��(2))?(?��b+?��(1)+��x(1),��(1)???��(b?(?��(1),x(1)???��(2),x(2)?)))????????=max?(��(1),��(2))?min?����[0,1]?(?��(1),��(1)?+?��(2),��(2)???max?����[0,1]?��(b?(?��(1),x(1)???��(2),x(2)?)))??????=max?(��(1),��(2))?(?��(1),��(1)?+?��(2),��(2)??=max?(��(1),��(2))?(?��(1),��(1)?+?��(2),��(2)??f(��(1),��(2)))?obtain selleck chem U0126 thatf?(��(1),��(2)) the previous third equality follows from the strong max-min property, it can be transferred into a min-max problem.

If ��(1) + �� x(1) �� 0, max ��(1)��(1) + �� x(1), ��(1) is ��, and if ��(2) ? �� x(2) �� 0, max ��(2)��(2) ? �� x(2), ��(2) is ��; otherwise, if ��(1) + �� x(1) = 0 and ��(2) ? �� x(2) = 0, we have f*(��(1), ��(2)) = ?��b.Back to the primal problem, we want to get a sequence of boundary ��0(1,2), ��1(1,2),��, ��T(1,2)(��t(1,2) = (��t(1), ��t(2)), t 0,1,��, T) which makes J(��0(1,2)) �� J(��1(1,2)) �� ��J(��T(1,2)). Unfortunately, decreasing the objective function J(��(1), ��(2)) directly is impossible, while the training examples arrive in a steam. In order to avoid this contradiction, we propose a Fenchel conjugate transform for multi-view S2L problems based on coregularization.An equivalent ??i��1,2,��,T,��i(1)=��0(1),��i(2)=��0(2).

(12)Using????s.t.???12��1||��0(1)||2+12��2||��0(2)||2+��t=1Tgt(��t(1,2)),?problem of (6) ismin?��0(1,2),��1(1,2),��,��T(1,2) the Lagrange dual function, we can rewrite (12) by introducing a vector group (��1(1,2), ��2(1,2),��, +��t=1T?��1(2),��0(2)?��t(2)?).(13)Consider????????????+��t=1T?��1(1),��0(1)?��t(1)?????????????+��t=1Tgt(��t(1,2))????????????��T(1,2)):max?��1(1,2),��2(1,2),��,��T(1,2)??min?��0(1,2),��1(1,2),��,��T(1,2)?(12��1||��0(1)||2+12��2||��0(2)||2? Batimastat the dual ?12��2(?��t=1T��t(2))2?��t=1Tgt?(��t(1,2)),(14)where???=?12��1(?��t=1T��t(1))2???��t=1Tmax?��t(1,2)(?��t(1),��t(1)?+?��t(2),��t(2)??gt(��t(1,2)))????max?��0(2)?(��t=1T??��t(2),��0(2)??12��2||��0(2)||2)???=?max?��0(1)?(��t=1T??��t(1),��0(1)??12��1||��0(1)||2)??+��t=1T?��t(2),��0(2)?��t(2)?)??????????+��t=1T?��t(1),��0(1)?��t(1)???????????+��t=1Tgt(��t(1,2))??????????=min?��0(1,2),��1(1,2),��,��T(1,2)?(12��1||��0(1)||2+12��2||��0(2)||2??functionD(��1(1,2),��2(1,2),��,��T(1,2)) gt*(��t(1,2)) is the Fenchel conjugate of gt(��t(1,2)). The primal problem can be described as maximizing the dual function as in the followingmin?��(1,2)??J(��(1,2))=max?��1(1,2),��2(1,2),��,��T(1,2)D(��1(1,2),��2(1,2),��,��T(1,2)).