Maximizing theVariance = : xN x1od x2od xNod = Xod=Fd(X)=DPPd(X) d1 dn How do we use this theory? For DPP-gap based Clustering, we can hill-climb akk below to a d that gives us the global maximum variance. Heuristically higher variance means more prominent gaps. Given any table, X(X1, ..., Xn), and any unit vector, d, in n-space, let V(d)≡VarDPPd(X)= (Xod)2 - (Xod)2 = i=1..N(j=1..n xi,jdj)2 - ( j=1..nXj dj )2 N 1 For DPP-gap based Classification, we can start with the table of the C Training Set Class Means: X≡(X1...Xn) where Mk≡MeanVectorOfClass=k . M1 M2 : MC - (jXj dj) (kXk dk) = i(j xi,jdj) (k xi,kdk) N 1 = ijxi,j2dj2 + j<k xi,jxi,kdjdk - jXj2dj2 +2j<k XjXkdjdk N 1 2 Then Xi = Mean(X)i and and XiXj = Mean Mi1 Mj1 . : MiC MjC = jXj2 dj2 +2j<kXjXkdjdk - " = j=1..n(Xj2 - Xj2)dj2 + +(2j=1..n<k=1..n(XjXk - XjXk)djdk ) subject to i=1..ndi2=1 dT o A o d=VarDPPdX≡V(d) V i XiXj-XiX,j : d1 ... dn d1 dn We can write this separating out the diagonal elements or not: These computations are O(C) (the number of classes) and are therefore instantaneous. Once we have the matrix A, we apply UT-2 and hill-climb to obtain a d that maximizes the variance of the class means. + jkajkdjdk V(d)=jajjdj2 ijaijdidj V(d) =  d0, one can hill-climb it to locally maximize the variance, V as follows: d1≡(V(d0)); d2≡(V(d1)):... where 2a11 2a12 ... 2a1n 2a21 2a22 ... 2a2n : ' 2an1 ... 2ann d1 di dn V(d)≡Gradient(V)=2Aod or V(d)= 2a11d1 +j1a1jdj 2a22d2 +j2a2jdj : 2anndn +jnanjdj Ubhaya Theorem1:  k{1,...,n} s.t. d=ek will hill-climb V to its globally maximum. Let d=ek s.t. akk is a maximal diagonal element of A, Ubhaya Theorem2: d=ek will hill-climb V to its globally maximum.