Text Clustering with Mixture of von Mises-Fisher Distributions - Text Mining: Classification, Clustering, and Applications - page 136

Comparisions of approximations for κ =500, d varying

1200

κ =500
(1.14)
(1.15)
(1.17)

1000

800

600

400

200

0

0 500 1000 1500

Dimension d

FIGURE 6.2 : Comparison of approximations for varying d , κ = 500.

We begin by holding d fixed at 1000, and allow κ to vary from 10 to 5010.
Figure 6.1 shows the values of computed

κ (estimation of κ ) using the three
approximations. From this figure one can see that (6.14) overestimates the
true κ , while (6.15) underestimates it. However, our approximation (6.17) is
very close to the true κ values.
Next we illustrate the quality of approximation when κ is held fixed and d is
allowed to vary. Figure 6.2 illustrates how the various approximations behave
as the dimensionality d is varied from d = 4 till d = 1454. The concentration
parameter κ was set at 500 for this experiment. We see that (6.15) catches up
with the true value of κ after approximately d

2 κ (because the associated
r values become small), whereas (6.17) remains accurate throughout.
Since all the approximations depend on r (which implicitly depends on κ
and d ), it is illustrative to also plot the approximation errors as r is allowed
to vary. Figure 6.3 shows how the three approximations perform as r ranges
from 0 . 05 to 0 . 95. Let f ( d, r ), g ( d, r ), and h ( d, r ) represent the approxima-
tions to κ using (6.14), (6.15), and (6.17), respectively. Figure 6.3 displays
|

≥

for the varying r
values. Note that the y -axis is on a log-scale to appreciate the differences
between the three approximations. We see that up to r

A d ( f ( d, r )) − r | , | A d ( g ( d, r )) − r | ,and | A d ( h ( d, r )) − r |

0 . 18 (dashed line
on the plot), the approximation yielded by (6.15) has lower error. Thereafter,
approximation (6.17) becomes better.

≈