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such that
w
is a maximum (and, as discussed above, directions where
H
S
remains constant). So,
w
is a saddle point.
The above analysis again shows that SEE has a different behavior depending
on the distance between the classes. A graphical insight of this behavior can
be found in [212]. The following Example shows how to determine the
t
value
for Gaussian classes.
Example 4.6.
The eigenvalues of the Hessian can be used to compute the
minimum-to-saddle turn-about value for bivariate Gaussian classes. As
2
H
S
(
w
) is always a diagonal matrix with one zero entry, one positive entry, and
a third entry that changes sign as the classes get closer, corresponding to the
eigenvalues of the matrix, one can determine the minimum distance yielding
a minimum of
H
S
at [
w
1
∇
00
T
=0, by inspecting when the sign-
changing eigenvalue changes of sign. Setting
Σ
1
=
Σ
2
=
σ
2
I
and the class
means symmetrical about the origin and at the horizontal axis, it is possible
to derive that the third eigenvalue is positive if the following expression is
positive:
for
w
1
Φ
(
d
)) ln
2
Φ
(
d
)
1
√
2
πd
(1
e
−
d
2
,
−
−
(4.66)
−
Φ
(
d
)
where
d
is a normalized half distance between the classes. The turn-about
value is
d
=0
.
7026, which corresponds to a normalized distance between the
classes of 1
.
4052. This is precisely the same value found for the Stoller split
problem in Example 4.3.
