Database Reference
In-Depth Information
Algorithm 5
soft-moVMF
Require:
Set
d
−
1
X
of data points on
S
Ensure:
A soft clustering of
over a mixture of
k
vMF distributions
Initialize all
α
h
,μ
h
,κ
h
,h
=1
,
X
···
,k
repeat
{
The E (Expectation) step of EM
}
for
i
=1to
n
do
for
h
=1to
k
do
f
h
(
x
i
|
c
d
(
κ
h
)
e
κ
h
μ
h
x
i
for
h
=1to
k
do
p
(
h
θ
h
)
←
θ
h
)
l
=1
α
l
f
l
(
x
i
|
α
h
f
h
(
x
i
|
|
x
i
,
Θ)
←
θ
l
)
{
The M (Maximization) step of EM
}
for
h
=1to
k
do
α
h
←
n
i
=1
p
(
h
1
|
x
i
,
Θ)
μ
h
←
i
=1
x
i
p
(
h
|
x
i
,
Θ)
r
←
μ
h
/
(
nα
h
)
μ
h
←
μ
h
/
μ
h
rd−r
3
1
−r
2
until
convergence
κ
h
←
{
α
h
,μ
h
,κ
h
}
h
=1
of the
k
vMFs
clustering of the data and the parameters Θ =
X
that model the input dataset
.
Finally, we show that by enforcing certain restrictive assumptions on the
special case of both the
soft-moVMF
and
hard-moVMF
algorithms. In a mixture
of vMF model, assume that the priors of all the components are equal, i.e.,
α
h
=1
/k,
h
, and that all the components have (equal) infinite concentration
parameters, i.e.,
κ
h
=
κ
∀
h
. Under these assumptions the E-step in the
soft-moVMF
algorithm reduces to assigning a point to its
nearest
cluster, where
nearness is computed as a cosine similarity between the point and the cluster
representative, i.e., a point
x
i
will be assigned to cluster
h
∗
=argmax
h
x
i
μ
h
,
since
→∞
,
∀
T
i
μ
h
∗
h
=1
e
κ
x
e
κ
x
p
(
h
∗
|
x
i
,
Θ) = lim
κ→∞
=1
,
T
i
μ
h
=
h
∗
.
To show that
spkmeans
can also be seen as a special case of the
hard-moVMF
,
in addition to assuming the priors of the components to be equal, we further
assume that the concentration parameters of all the components are equal,
i.e.,
κ
h
=
κ
for all
h
. With these assumptions on the model, the estimation
of the common concentration parameter becomes unessential since the hard
assignment will depend only on the value of the cosine similarity
x
i
μ
h
,and
hard-moVMF
reduces to
spkmeans
.
and
p
(
h
|
x
i
,
Θ)
→
0, as
κ
→∞
for all
h
Search WWH ::
Custom Search