Signal Analysis Research (SAR) Group - RNet - Ryerson University

Fig 1 Two-dimensional mapping (Left) clustering using SOFM and @ ight) clustering using SOTM. It is evident that redundant nodes in the
lattice topology of SOFIA can produce unnecessary boundmes by having some of the centres trapped in low-density regions of the input pattern
is declared to be the winner. Every such presentation of
input patterns slightly modifies the winning node's position
in the network: aposition that eventually evolves toward the
centre of mass of the current class. This gradual adaptation
of the node's position is controlled by an exponential
decaying function called the kearnzng rate. The learning rate
resets to its initial value each time a new centre is generated.
Therefore, sufficient time is given to the network to adapt
itself to the presence of new samples, thus, the tree grows
larger and the similarity measurement tends to be more
accurate. The generation of new centres (branches of the
tree) is controlled by a hierarchy function, called the
th~shokd~nction, which decreases over time. If an input
node is encountered whose similarity exceeds this threshold
function (i.e. is significantly different from all nodes in the
current SOTM map) a new node is generated. The new node
is attached as a leaf node of its closest representation in the
current SOTM mapping, thus over time, a tree structure
evolves [q.
Similar to SOFM, SOTM aims at preserving the
topological relationships between patterns in the original
input space. However, unlike SOFM, SOTM classifies a
large group of patterns by building and evolving a tree
structure that tends to form neighborhood relationships by
reflecting a degree of similarity between the new and
already classified patterns.
Although, image indexing with the SOFM was perceived
to be a robust and effective solution that tolerates even very
high input vector dimensionalities [5], the lattice topology of
SOFM network makes it essentially undesirable for
clustering purposes due to concentration of a fraction of
nodes in the map resulting of the best-matching unit
computation [6]. SOFM has nodes that can easily get
trapped in regions of low density and, therefore, can simply
lose its ability to represent the underlying topology of the
input pattern. For instance, let us assume there are two high
density regions in the input space, representing two distinct
clusters. Let us also assume that there are maximum two
nodes in the structure of SOFM to correctly allocate both
regions. If those two nodes were separated by a third node
and each converged to the two adjacent regions of high
density, then the third node could easily get trapped in
between the regions. As a result, it can change the true
boundaries of high density classes by pulling some of the
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samples from the two real clusters and allocating them to the
middle node as is illustrated in Fig. 1. In this figure, the
second node of the SOFM network has dragged some of the
data points from the first cluster and has generated a new but
redundant class. The tree structure of SOTM, however, is
succesdul in determining the high-density regions.
Problem with the SOTM algorithm is two-fold: it
unsuitably decides on the relevant number of classes; and
often loses track of the true query position. Decision on
which clusters are relevant in the SOTM is postponed until
after the algorithm has converged. This is because there is
no innate controlling process available for the algorithm to
influence cluster generation around the query centre (the
SOTM clusters entirely independently). Losing a sense of
query location within the input space can have an undesired
effect on the true structure of the relevant class and can
force the SOTM algorithm to spawn new clusters and form
unnecessary boundaries within the query class as is
illustrated in Fig. 2. In this figure, the SOTM forms a
boundary near the query contaminating relevant samples,
where as some supervision is maintained in the DSOTM
case, preventing unnecessary boundaries fiom forming.
Therefore, retaining some degree of supervision on the
cluster generation around the query class seems to be vital.
Due to the limitations of SOTM, we propose the
Directed SOTM (DSOTM) algorithm in this work. In the
DSOTM algorithm, decision on association of input pattern
to query image is gradually made as each sample is
presented to the system. It also contains a controlling
mechanism that keeps track of the query centre by forcing
the centre of relevant class to remain in the vicinity of the
query position. Therefore, it can dynamically control
generation of new centres and can determine the relevance
of input samples, with respect to the query, as the tree
structure grows. On the other hand, it limits the synaptic
vector adjustments according to its reinforced learning rules
and constrains cluster generation by preventing the
spawning of redundant centres around the query position;
since this part of the map is already occupied by relevant
class centre.
The DSOTM algorithm is summarized as follows:
InihkEzrslion: Choose a mot node, {u3t fTom the
available set of input vectors, { xk)f , in a random manner.
J is the total number of centroids (initially set to 1) and K is
the total number of inputs (i.e. images);
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