Information Technology Reference
In-Depth Information
y
data space
2
f(x
3
f(x
5
y
y
1
latent space
x
x
x
x
x
x
1
2
3
4
5
6
Fig. 1.
UNN: illustration of embedding of a low-dimensional point to a discrete latent space
topology w.r.t. the DSRE testing all
N
+1
intermediate positions
3. choose the latent position that minimizes
E
(
X
)
, and embed
y
,
4. remove
y
from
Y
,add
y
to
Y
, and repeat from Step 1 until all patterns have been
embedded (
N
=0
).
Figure 1 illustrates the
N
+1
possible embeddings of a pattern into an existing order
of points in latent space (yellow/bright circles). The position of element
x
3
results in
a lower DSRE with
K
=2
than the position of
x
5
, as the mean of the two nearest
neighbors of
x
3
is closer to
y
than the mean of the two nearest neighbors of
x
5
.
3.3
Greedy Strategy
The iterative approach introduced in the last section tests all intermediate positions of
previously embedded latent points. We proposed a second iterative variant (UNN
g
)that
only tests the neighbored intermediate positions in latent space of the nearest embedded
point
y
∗
from
Y
in data space [17]. The second iterative strategy works as follows:
1. Choose a pattern
y
from
Y
,
2. look for the nearest
y
∗
from
Y
that has already been embedded (w.r.t. distance
measure like Euclidean distance),
3. choose the latent position next to
x
∗
that minimizes
E
(
X
)
and embed
y
,
4. remove
y
from
Y
,add
y
to
Y
, and repeat from Step 1 until all patterns have been
embedded.
Figure 2 illustrates the embedding of a 2-dimensional point
y
(yellow) left or right of
the nearest point
y
∗
in data space. The position with the lowest DSRE is chosen. In
comparison to UNN,
N
distance comparisons in data space have to be computed, but
only two positions have to be tested w.r.t. the data space reconstruction error.
