1. 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 :
d 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
2a a a1n
2a21 2a a2n : '
2an ann 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.