A CONTEXT VECTOR-BASED SELF ORGANIZING MAP 
FOR INFORMATION VISUALIZATION 
David A. Rushall, Marc R. Ilgen 
HNC Software, Inc., 5930 Cornerstone Court West, San Diego, CA 92121 USA 
email: dar@hnc.com, mri@hne.eom 
ABSTRACT 
HNC Software, Inc. has developed a system 
called DOCUVERSE for visualizing the information 
content of large textual corpora. The system is built 
around two separate neural network methodologies: 
context vectors and self organizing maps. Context 
vectors (CVs) are high dimensional information 
representations that encode the semantic content of 
the textual entities they represent. Self organizing 
maps (SOMs) are capable of transforming an input, 
high dimensional signal space into a much lower 
(usually two or three) dimensional output space useful 
for visualization. Related information themes 
contained in the corpus, depicted graphically, are 
presented in spatial proximity to one another. Neither 
process requires human intervention, nor an external 
knowledge base. Together, these neural network 
techniques can be utilized to automatically identi~ the 
relevant information themes present in a corpus, and 
present those themes to the user in a intuitive visual 
form. 
1. INTRODUCTION 
In recent years there has been an explosion in 
the amount of information available on-line. Much of 
this explosion has been fueled by the spectacular 
growth of the Internet and especially the World Wide 
Web. Along with this spectacular growth has come 
new challenges for effectively locating on-line 
information, especially when browsing rather than 
performing a directed search for a specific piece of 
information. Key word and linguistically based 
directed search engines offer some ability to present 
relevant information to the user, and have resulted in 
useful Interact products such as Yahoo \[1\], Lycos \[2\], 
and Alta Vista\[3\]. However, these engines suffer fi'om 
the fact that they require the user to specify a query of 
limited length and they offer no visual interface for 
browsing. To solve the problem of browsing the 
information space in order to fred information of 
interest, new techniques for data retrieval and 
presentation must be developed. 
HNC has developed an underlying 
information representation technology and a concept 
for information visualization that can solve the 
problem of effectively browsing large textual corpora. 
As part of HNC's involvement in the ARPA sponsored 
TIPSTER program, HNC has developed a neural 
network technique that can learn word level 
relationships from free text. This capability is based 
upon an approach called context vectors which 
encodes the meaning and context of words and 
documents in the form of unit vectors in a high 
dimensional vector space. Furthermore, as part of 
HNC's involvement in the US intelligence community- 
sponsored P1000 visualization effort, HNC has 
applied a secondary neural network process, the Self 
Organizing Map (SOM) \[4\], which uses the document 
context vectors to build a visual representation of the 
information content of the corpus. The combination of 
these technologies allows users to effectively browse 
the information space, to locate related documents, 
and to discover relationships between different themes 
in the information space. 
The remainder of this paper is organized as 
follows. Section 2 presents an overview of both 
context vectors and the Self Organizing Map. Section 
3 presents the DOCUVERSE system and presents the 
user interface, automatic region fmding and region 
labeling, information retrieval and document 
highlighting, and temporal analysis of the information 
space. Finally, Section 4 presents some concluding 
remarks and directions for future research. 
2. TECHNICAL BACKGROUND 
The DOCUVERSE system is based on two 
technologies: context vectors and the SOM. Context 
vector technology was developed at HNC and has 
been demonstrated to be highly effective for such tasks 
as text retrieval (using a system called MatchPlus) \[5\], 
159 
text routing \[6\], and image retrieval \[7\]. SOM 
technology was originally developed by T. Kohonen 
and has been used throughout the neural network 
community as a method for representing information 
in a manner suitable for visualization \[8\]. The 
following subsections present an overview of each of 
these technologies. 
2.1 Context Vectors 
The key technical feature of context vector 
technology is the representation of terms, documents, 
and queries by high dimensional vectors consisting of 
real-valued numbers or components. These vectors 
are constrained to be unit vectors in the high 
dimensional vector space. Since both terms (words or 
stems) and documents are represented in the same 
flame of reference, this allows several unique 
operations. All operations in MatchPlus are based on 
geometry of these high dimensional spaces \[9\]. 
Specifically, closeness in the space is equivalent to 
closeness in subject content. A neural network-based 
learning algorithm is designed to adjust word vectors 
such that terms that are used in a similar context will 
have vectors that point in similar directions. The 
determination of similar context is based upon the use 
of word stem co-occurrence statistics. Once trained, 
these word stem context vectors are used as building 
blocks for creating document and query context 
vectors. Specifically, context vectors for documents 
and queries are formed as the normalized weighted 
sum of the word stem context vectors for the word 
stems found in the document or query. This process 
results in the fact that context vectors for documents 
with similar subject content will point in similar 
directions. This vector representation of information 
content can thus be used for document retrieval, 
routing, document clustering (self organizing subject 
index) and other text processing. 
2.2 Kohonen's Self Organizing Map 
The concept of self organizing maps was first 
developed by Tuevo Kohonen in 1981 at the 
University of Helsinki. Kohonen demonstrated that a 
system could be taught to organize the data it was 
given, without the need for supervision or external 
intervention, through the use of competitive learning. 
Typically, the SOM consists of a collection of nodes 
arranged in a regular two dimensional grid. Each node 
corresponds to a cluster centroid vector for the high 
dimensional input vector space. A self-organizing 
training process is used to adjust the node vector 
components in an iterative fashion. Upon completion 
of training, the SOM node vectors have the property 
that node vectors that are close in the high dimensional 
vector space will be close in the two dimensional grid 
space, or "map space." This training process is 
described in more detail below. 
2.2.1 Training the SOM 
Assume the input space is comprised of N 
vectors, each of dimension n. Furthermore, assume a 
two dimensional array of "nodes" that will be trained 
to represent the N input vectors. Each of these nodes 
is also a vector of dimension n. Figure 1 depicts this 
arbitrary array of nodes, in this case, a 5-by-5 
regularly spaced array of 25 nodes. Each node is 
uniquely identified by it's (i,j) position. 
©@@@@ 
@@@@@ 
@@@@@ 
@@@@® 
@@@@@ 
Figure 1. An array of map nodes for self 
organization. 
Before training, the node vectors are assigned 
random values. That is, a pseudo random number 
generator is used to assign each node in the map a 
random unit vector of dimension n. In the case where 
n is very large (-300), these initial conditions 
represent a quasi-orthogonal state, i.e., each unit 
vector is approximately orthogonal to each other unit 
vector. 
The SOM training algorithm is based on a 
simple, iterative comparison process, where the N 
input vectors are compared to each of the node vectors 
in the map. In essence, the node vectors are 
competing to have their values adjusted. The idea is 
to find the node vector that is "nearest" (in vector 
space) to the input vector. The node vector that is 
nearest is deemed the "winner" of this competition, 
160 
and is rewarded by having its vector adjusted. The 
adjustment comes in the form of moving the winning 
node vector in the direction of the input vector. The 
SOM extends this simple competitive learning process 
to include updates for the "neighbor" nodes to the 
winning node. These neighbor nodes are nodes that are 
close (in the map space) to the winning node. For 
example, in Figure 1, the neighborhood of node We.3 is 
depicted, and is defined as those nodes that are within 
one row or one column of node W2,3. Neighborhoods 
can be larger or smaller; this is just one example. The 
updates for these neighbor nodes are smaller than the 
updates for the winning node, and the size of the 
neighbor node update is smaller for neighbors that are 
farther away (in map space) ffi'om the winning node. 
The inclusion of neighbor updates results in the 
organization of the information into a form suitable for 
visualization. The algorithm can be represented in 
pseudo-code as: 
For each input vector V 
Find node vector Wid that is closest to V 
(IW~j-Vl<lW,,~-Vl Vk, l) 
Update Widaccording to W 0 = Wi, i + of V- WO) 
Find nodes that are close to node ij in map space 
For each of these "neighbor" nodes It~,t 
Update W t, t according to Wk, t = W ~t + s of V- W t, t ) 
where (0.0 ~s ~ LO) 
End loop over neighbor nodes 
End loop over input vectors 
The size of the adjustment, a, will determine 
how quickly the map space node vectors will converge 
to an accurate representation of the input space 
vectors. One loop through the input vector set is not 
sufficient to train the node vectors. It is necessary to 
perform this loop hundreds, and possibly thousands, of 
times before training is completed. The value of the 
parameter s for updating neighbor nodes is determined 
by a Gaussian function based on the nearness of the 
neighbor node to the winning node. Therefore, close 
neighbors will be updated, or adjusted, more than 
neighbors that are further away. 
2.2.2 Win Frequency and Conscience 
An ideal characteristic of this training is to 
have the map node vectors win the competition with 
equal probabilities. Unfortunately, this is not the case 
for the standard SOM algorithm. A consequence of 
this type of training is that some map nodes may never 
win the competition. This will result in a less useful 
representation of the input space. To eliminate this 
undesirable effect, DeSieno \[10\] has developed an 
improved competitive learning algorithm that makes 
use of the idea of "conscience". The conscience 
mechanism allows nodes that are observed to rarely 
win the competition to subsequently win more often, 
and it prevents nodes that ffi'equently win the 
competition ffi'om subsequently winning too oRen. 
The conscience mechanism is employed as a 
second competition based on the outcome of the first 
competition described above in the self organizing 
algorithm. Before conducting the second competition, 
the conscience mechanism creates a bias factor for 
each node. The value of the bias is determined by the 
running statistics kept on the first competition. Nodes 
that normally lose the first competition are given 
favorable biases, and those that normally win are given 
unfavorable biases. The winner of this second 
competition is determined upon the basis of biased 
distance to the input vector. This biasing enforces an 
equiprobable winning distribution and results in a 
more useful clustering of the input information space. 
2.2.3 Computation Issues 
Earlier we alluded to the fact that these 
algorithms are computationally intensive. This 
intensity depends upon three factors. One is the 
dimensionality of the vectors. Using higher 
dimensioned vectors (-1000) is possible, but adds to 
the timely computation problem. Likewise, the 
number of input vectors, as well as the number of map 
node vectors, will determine the scale of the problem. 
Fortunately, the nature of these algorithms is well 
suited for parallel processing architectures. Therefore, 
scalability of the algorithm depends on the number of 
processors that can be used to compute a solution. 
HNC has developed a hardware architecture 
that is designed to handle neural networks, and in 
particular, the compute intensive processes they 
model. The hardware is a SIMD numerical array 
processor (SNAP), in essence a floating-point parallel 
array processor. The SNAP is ideally suited to 
determine the computationally intensive solution 
required by the SOM algorithms. 
The SNAP comes in a variety of 
configurations. The fastest SNAP available, the 
SNAP-64, has 64 processors, and delivers an 
unmatched price-to-performance ratio of around $20 
per megaflop. At its peak, the SNAP computes at a 
rate of 2.56 gigaflops. This type of performance 
enables HNC to develop and deliver the compute 
intensive solutions to a wide variety of problems, 
including information visualization. 
161 
3. DOCUVERSE 
The DOCUVERSE system, developed as part 
of the IC P1000 research effort at HNC, is an ongoing 
research and development effort with a goal of 
providing users with a tool to quickly and easily assess 
the information content of large textual corpora. 
DOCUVERSE is based on the context vector 
technology foundation developed at HNC over the 
past few years, and additionally provides a visual 
interface that allows the user to browse the 
information space in a visually appealing fashion. 
Figure 2 presents the process by which a text 
corpus is transformed into some intuitive visual 
paradigm that users can easily relate to and 
understand. The initial neural network process is 
shown along the top part of Figure 2. The process 
flow indicates that some set of textual data, the 
training text, is used to obtain context vectors for the 
vocabulary set contained in the training text. This 
process involves a preliminary step of word 
"stemming" and stop list removal. Stemming is the 
process of representing similar word forms as the base 
form of the words (i.e. words like driver, driving, 
drives, driven, and drove are all stemmed to the word 
drive). Stop list removal refers to the removal of 
words with high frequency occurrence and little 
meaning in the training text (i.e. words like the, of, 
and, etc.). Afler preprocessing, context vectors for the 
remaining word stems are learned and stored into a 
database. 
~~ Preprocecsing • 
Stems • Stop List 
~ • Stems • Stop List 
Lesm Stem \] 5~m \] 
Context 
Vectors 
Context Vectors 
Visual Pmdigm 
OvrnlTln ~ 
~ Map Training 
Figure 2. The process of transforming textual data 
into an intuitive, graphical visual. 
At this point, the system is ready to process 
the user's desired corpus: Note that the corpus to be 
visualized' does not need to be the same corpus that the 
system was trained with. It is help~l, however, if the 
training corpus is statistically representative of the 
corpora that will be visualized. 
As shown in Figure 2, the control flow now 
switches to the middle part of the diagram, where the 
user identifies the corpus to be visualized. This text is 
preprocessed in the same exact manner as the training 
text was preprocessed. Afler this step, one pass 
through each document is all that is required to 
calculate a context vector for each document. The 
stem vectors learned from the training text are used to 
compute the context vectors for the user's text. These 
are stored in a document context vector database. It is 
these context vectors that are given to the self 
organizing map for visualization. 
2.3 The Interface 
The DOCUVERSE interface presents the 
user with an array of nodes not unlike the array of 
nodes depicted in Figure 1. The size of the array is 
configurable by the user, but the default is a 20-by-20 
array of nodes. Recall that each of these 400 nodes 
has a context vector associated with it, and that the 
context vectors have been adjusted to represent the 
prevalent themes in the corpus. Therefore each node 
represents an information theme contained in the 
corpus. It is important to note that it is not necessary 
that each node have a different theme. Nodes can 
have similar themes, and in fact, the same theme if 
there is a relatively large amount of information 
pertaining to that particular theme within the corpus. 
This discussion raises the question as to how 
the nodes reflect the amount of information present for 
the theme they represent. Afler experimenting with 
various paradigms such as color or icons, we've 
concluded that the size, or radius, of the nodes best 
conveys this information. Large nodes imply a 
relatively large number of documents for the given 
information theme. Small nodes imply a relatively 
small number of documents for the given information 
theme. 
The way in which the system measures not 
only the amount of information for a theme, but also 
the similarity of themes, documents, words, or any flee 
text, is through the use of the vector dot product 
operation. Recall that each of the aforementioned 
textual entities is associated with a context vector. 
Similar entities have context vectors that point in 
similar directions. The dot product for similar 
direction vectors will be close to 1.0, while dissimilar 
vectors will have dot products that are near zero. 
Figure 3 depicts what we call the "corpus 
integral". This is the broad view of information 
162 
content for the entire corpus. It is an attempt to 
graphically illustrate, through various sized 
information nodes, the entire set of prevalent themes 
contained in the corpus. Again, nodes that are large 
represent the themes that occur with the highest 
fi'equency and volume in the corpus. Nodes that are 
small, or not even visible (such as those in the upper 
left comer of the map), represent themes that occur 
with a much lower ffi'equency and volume, relatively 
speaking. 
• • • - • • = • ogOO-w • 
• = u • g • • • OmOOOOOg 
.... OOOO0000WO w • • • 
..... uoooooOoOOOOO= 
"e=OOOOOOOOOO=- 
• • - eoOOOeoOO000o, 
• t ~ t = m u • • ~mOQO = 0099 • 
• • o m w • =le e=OOO* • =Og • 
- - O0000- wwOOOO! = • • • 
o OgOO0 u g • w g I • ! g * = - • - 
• oeeeeo. ; : :- •- • • .... 
OOOOOOOe ..•••. 000000OOO-*oO0000ooo 
oOOO000oo-*oOOOoOOOO 
The corpus integral is very useful in that the 
user knows, at a glance, which themes are present in 
the corpus. By "mousing" on a node (i.e. clicking the 
mouse button once on a node), a pop-up menu reveals, 
among other choices, the information theme the node 
represents. 
The corpus that was used to generate the 
integral depicted in Figure 3 is a set of over 17,700 
documents. The documents are news reports taken 
directly off the AP News Wire during a four month 
span in 1990. 
Other system capabilities, discussed in the 
sections below, allow the user to do a variety of 
information assimilation and information gathering 
tasks. For example, an undirected information search, 
commonly referred to as browsing, is made even easier 
when using the automatic region finding and labeling 
mechanism. Searching for specific information is also 
supported so that the user can request, in flee text 
form, any desired information. A tool for visualizing 
information in the time domain is also provided. 
Figure 3. The corpus integral. 
The corpus integral is computed by summing 
the dot products of each document context vector with 
each node context vector. The summed dot products 
for each node are used to determine the size of the 
node. In pseudo-code, 
For each node vector It~j 
Node_sizeij = 0 
For each document vector V 
Node_size U += Wsj • V 
End loop over document vectors 
Node size U/= number of document vectors 
End loop over node vectors 
163 
2.4 Automatic Region Finding and 
Labeling 
As we stated earlier, nodes that are near one 
another on the map have the property that they 
represent similar themes of information. We can 
exploit this fact to have the system automatically 
group the nodes into regions of similar themes. This 
can be thought of as a clustering of the clusters. 
Furthermore, the system will automatically generate an 
appropriate name, or label for the region. Consider 
Figures 4a and 4b. 
Figure 4b. The automatically generated labels. 
Figure 4a. The automatically generated regions 
Figures 4a and 4b show one of the many 
ways that a user can make use of the automatic region 
generation. They depict the option of selecting all 
regions found on the map. Figure 4a shows all the 
regions that were found by the system. The regions 
are outlines drawn around sets of nodes. Figure 4b 
shows a second dialog window that is used to display 
the labels for the regions. Although all the labels are 
not visible in the dialog, the region algorithm found a 
total of 84 distinct regions for the self organized map 
of the 17,700 AP News Wire documents. 
Instead of showing all regions at once, a user 
can select regions of interest from either the map or 
the label dialog. If a user is interested in a particular 
area on the map, the region outlines for the nodes of 
interest can be toggled on or off by the pop-up menu 
provided on each node. 
Alternatively, the user can peruse the list of 
region labels and select them directly from the list. 
This will draw the region outline on the map. 
The algorithm for region finding and labeling 
is a two step process. The first step involves finding 
the regions. Initially, each node is given its own 
region. An iterative algorithm compares a region to 
every other region on the map. The comparison is a 
vector dot product operation. If the regions are similar 
enough (i.e., if the dot product between the two 
regions exceeds some threshold), the two regions are 
merged into one region. This process is repeated until 
no region combining occurs. 
The next step involves finding an appropriate 
label for each region. First, a context vector is 
computed for the region. This is done by taking the 
weighted average of the context vectors belonging to 
the nodes in the region. This centroid region context 
164 
vector is compared to all of the stem word context 
vectors in the vocabulary. The stem word vector that 
results in the highest dot product with the centroid 
region vector becomes the label for the region. 
Because of the stemming process, some of the stem 
words are truncated. Therefore, the user is given the 
ability to edit the labels to put them in correct 
grammatical form. 
2.5 Information Retrieval and 
Document Highlighting 
DOCUVERSE makes use of a rich set of 
information retrieval functionality. This functionality 
was inherited from another system developed at HNC 
called MatchPlus \[5\]. MatchPlus focuses on 
information retrieval from large textual corpora. 
When a user desires a more focused search 
for specific information, it is easily accomplished in a 
variety of ways. If a user has identified a map node 
representing an interesting theme of information, the 
user can select, from the node pop-up menu, an option 
to retrieve documents pertaining to the theme. The 
system uses the node's context vector to perform dot 
products with every document context vector in the 
corpus. The user is presented with a ranked list of the 
most relevant documents. The list is presented in a 
window with the document ID, the value of the dot 
product, and the first line of text in the document. 
Figure 5 shows an example of the ranked list. 
DooumentSooroSu~eot~s 
2395 0.442 Christian Militia Re8ro, ~ 
16507 0.442 Rival Christian Forces ill 
16424 0.430 Battles RaBe 8etueen Ch\[I 
2755 0.429 Sniper Fire Persists In II 
539 0.427 Roun Cells Up Reservist:\[\[ 
11059 0.427 53 Bead. 133 Nounded tnll 
1893 0.427 MedIatln 8 Committee Stri\[I 
16721 0.425 F18htin 8 6etseen Chrlst\[I 
827 0.424 Rlval Christian Forces Ill 
17302 0.424 Flghtin 8 in East 6e£rut \[\[ 
2412 0.424 Rlvsl Forces ExchanBe Sill 
16259 0.423 Christian Nllltla Clamp ill 
13665 0.423 Rival Christian Gunners ill 
4350 0.423 Roun's Forces Penetratei\[I 
10735 0.421 fit Least Two Nllled 8s ,!\[\[ 
1940 0.421 Roun Gives Rival 72 Hou\[ll 
13598 0.420 Christian Forces Duel uill 
7069 0.417 Christians Rak Hraui to u 
1790 0.417 Helicopter Base Falls t,l~ 
17409 0.417 Truce Holds In Mountaln,lD 
3401 0.417 Reneued Flghtln8 Thwart: I 
397 0.416 8oun's Forces Head £ntoiB 
110 0.416 Aoun Forces Selze Strat. 1 
..................................................................................................................... i__ B 
Figure 5. A ranked list of documents. 
Alternatively, rather than using a node for a 
query, the user can type free text into a window and 
submit the free text as a query. The system converts 
the free text into a context vector, and the same 
retrieval process is performed. Yet another possibility 
is for the user to use an entire document as a query. 
Regardless of the method, the retrieval is done via 
context vector comparisons. 
Once a ranked list of documents has been 
retrieved, the user can select any document from the 
list to view. The document selected will appear in a 
separate window, along with a highlighting tool to 
further examine the relevant parts of the document. 
The highlight tool segments the document 
into 5-line paragraphs. The user is presented with 
another window containing a histogram, where each 
interval of the histogram corresponds to each of the 
paragraphs in the document, and the height of the 
interval corresponds to the dot product of the 
paragraph with the query that was issued. This tool 
provides the user with a tool to quickly and easily 
locate the most relevant portions of any document. 
2.6 Temporal Analysis 
With corpora comprised of periodically 
released information, it might be useful to visualize the 
information in the time domain. The DOCUVERSE 
temporal analysis tool makes this possible. 
Documents are segmented into user-defined time 
intervals, typically one hour, one day, or one week, 
depending upon the nature of the data. By performing 
a cumulative dot product operation for each document 
in the interval with the corpus integral, we can obtain a 
visual summary of the information content of the 
documents received during that time interval. The 
resulting time series of information themes can be 
viewed in rapid succession. The user is provided a 
media-player type interface with buttons for "play', 
"stop", and stepping forward and backward. Using the 
step buttons, the user can manually step through each 
time increment, or alternatively, the play button will 
rapidly step through the time increments in succession. 
When using the temporal tool on the 17,700 
AP News Wire documents, weekly cycles are easily 
identified. By stepping through the data in one day 
increments, the weekdays (Monday through Friday) 
are identified by three predominate regions pertaining 
to themes like banks, stocks, world news, and taxation. 
The next two days, the weekend, show maps with two 
predominate regions, pertaining to themes like music, 
TV, movies, and various other entertainment themes. 
165 
4. SUMMARY 
As we continue through the information age, 
tools such as DOCUVERSE will no longer be 
considered luxuries, but rather necessities. HNC has 
developed DOCUVERSE as merely a proof of 
concept system. We feel that this technology is far 
from realizing its full potential. As information 
visualization technology evolves and matures, so too 
will tools like DOCUVERSE. 
HNC is in the process of exploring new ways 
to visualize this powerful information representation 
technology. One area of interest is developing three 
dimensional SOMs. The user will be presented with a 
spherical array of nodes, representing the "world" of 
information. Used in conjunction with flat maps, the 
user would have a hierarchical SOM capable 
visualizing information at various levels of resolution. 
It is clear that in terms of visualization, "one 
size fits all" does not apply. What is intuitively 
obvious to one user is unclear and convoluted to 
another. Realizing this, we have identified numerous 
browsing paradigms to appeal to a broader audience. 
Tools like the Virtual Reality Modeling Language 
(VRML) are well suited for use with this technology 
for visualizing information on the World Wide Web, 
and will aid us as we strive to improve information 
visualization. 

REFERENCES 

\[1\] Yahoo!, http://www.yahoo.com/ 

\[2\] Lycos, http://www.lycos.com/ 

\[3\] Alta Vista, http://altavista.digital.corrg 

\[4\] Kohonen, T., Self-Organizing Maps, Springer- 
Verlag, Berlin, 1995. 

\[5\] Gallant, S.I., W. R. Caid, et al, "Feedback and 
Mixing Experiments with MatchPlus", Proceedings 
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MD. Aug. 1993. 

\[6\] Sasseen, R. V., J. L. Carleton, W. R. Caid, 
"CONVECTIS: A Context Vector-Based On-Line 
Indexing System", in Proceedings IEEE Dual-Use 
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\[7\] Pu, K. Q., C. Z. Ren, "Image/Text Automatic 
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Approach," SPIE Volume 2606, 1995. 

\[8\] Kohonen, et. al., http://websom.hut.fffwebsom/ 
166 

\[9\] Watson, G.S., "Statistics on Spheres", John Wiley 
and Sons, 1983. 

\[10\] DeSieno, D., "Adding a Conscience to 
Competitive Learning", in Proceedings of the 
International Conference on Neural Networks, I, IEEE 
Press, NY, 1988. 
