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We empirically found (see
Section 6.5.2
below) that the quality of the above
approximation can be improved by adding a correction term of
−
r
3
/
(1
−
r
2
)
to it. Thus, we finally get
r
3
rd
−
κ
=
.
(6.17)
1
−
r
2
Recently Tanabe et al. (50) used some inequalities regarding the Bessel func-
tion ratio
A
d
(
κ
) (3) to bound the solution to
A
d
(
κ
)=
r
as
r
(
d
−
rd
2)
≤ κ ≤
r
2
.
1
−
r
2
1
−
Our solution (6.17) lies within these bounds, thus leading to a better theoret-
ical justification in retrospect.
The approximation in (6.17) could perhaps be made even more accurate
by adding other correction terms that are functions of
r
and
d
. However, we
remark that if one wants a more accurate approximation, it is easier to use
(6.17) as a starting point and then perform Newton-Raphson iterations for
solving
A
d
(
r
= 0, since it is easy to evaluate
A
d
(
κ
)=1
A
d
(
κ
)
2
κ
)
−
−
−
d−
1
κ
A
d
(
κ
). However, for high-dimensional data, accurately computing
A
d
(
κ
)
can be quite slow compared to eciently approximating
κ
using (6.17), and
a very high accuracy for
κ
is not that critical. For other approximations of
κ
and some related issues, the reader is referred to (21; 5).
We now show some numerical results to assess the quality of our approxima-
tion in comparison to (6.14) and (6.15). First note that a particular value of
r
may correspond to many different combinations of
κ
and
d
values. Then, one
needs to evaluate the accuracy of the approximations over the parts of the
d
-
κ
plane that are expected to be encountered in the target application domains.
Section 6.5.2 below provides such an assessment by comparing performances
over different slices of the
d
-
κ
plane and over a range of
r
values. Below we
simply compare the accuracies at a set of points on this plane via Table 6.1
which shows the actual numerical values of
κ
that the three approximations
(6.14), (6.15), and (6.17) yielded at these points. The
r
values shown in the
table were computed using (6.5).
TABLE 6.1:
κ
for a sampling of
κ
and
d
Approximations
values.
(
d, r, κ
)
κ
in (6.14)
κ
in (6.15)
κ
in (6.17)
(10
,
0
.
633668
,
10)
12
.
3
9
.
4
10
.
2
(100
,
0
.
46945
,
60)
93
.
3
59
.
4
60
.
1
(500
,
0
.
46859
,
300)
469
.
5
296
.
8
300
.
1
(1000
,
0
.
554386
,
800)
1120
.
9
776
.
8
800
.
1
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