5.5 Further Issues
377
Fig. 5.116 A simple neural work with a single hidden layer
(C) if any sphere P k in S n is not in contact with others including P i and P j , a train
data u m = [P i , P j , S n ] can be obtained; Otherwise go to (A) and (B);
(D) Repeating (A)–(C). U = {u m }, m = 1, 2, . . . , M and delete the elements that
are close to other ones, i.e., |u m − u n | < γ , where γ is a small threshold.
The dimensionality reduction is performed by the transforms of the translation,
scaling, and rotation. The coordinate origin is positioned at P i , and all points are
scaled by the sphere radius, i.e.,
P
i =
P i − P i
r p
= 0,
P
j =
P j − P i
r p
,
P
1 =
P 1 − P i
r p
,
P
2 =
P 2 − P i
r p
,
. . .
P
n =
P n − P i
r p
,
(5.253)
The new sphere set is defined as S
n = {P
1 , P
2 , . . . , P
n }. The rotation matrix is
computed by the points P
i , P
j , and P
1 , which are not on a same line. The two vectors
are defined as ω
1 = P
j − P
i and ω
2 = P
1 − P
i . The orthogonal transformation