Language Independent Methods of Clustering Similar Contexts (with applications)

1
Language Independent Methods of Clustering
Similar Contexts (with applications)

• Ted Pedersen
• University of Minnesota, Duluth
• tpederse_at_d.umn.edu
• http//www.d.umn.edu/tpederse/SCTutorial.html

2
Language Independent Methods

• Do not utilize syntactic information
• No parsers, part of speech taggers, etc. required
• Do not utilize dictionaries or other manually
  created lexical resources
• Based on lexical features selected from corpora
• Assumption word segmentation can be done by
  looking for white spaces between strings
• No manually annotated data of any kind, methods
  are completely unsupervised in the strictest sense

3
Clustering Similar Contexts

• A context is a short unit of text
• often a phrase to a paragraph in length, although
  it can be longer
• Input N contexts
• Output K clusters
• Where each member of a cluster is a context that
  is more similar to each other than to the
  contexts found in other clusters

4
Applications

• Headed contexts (contain target word)
• Name Discrimination
• Word Sense Discrimination
• Headless contexts
• Email Organization
• Document Clustering
• Paraphrase identification
• Clustering Sets of Related Words

5
Tutorial Outline

• Identifying lexical features
• Measures of association tests of significance
• Context representations
• First second order
• Dimensionality reduction
• Singular Value Decomposition
• Clustering
• Partitional techniques
• Cluster stopping
• Cluster labeling
• Evaluation

6
SenseClusters

• A package for clustering contexts
• http//senseclusters.sourceforge.net
• SenseClusters Live! (Knoppix CD)
• Integrates with various other tools
• Ngram Statistics Package
• CLUTO
• SVDPACKC

7
Many thanks

• Amruta Purandare (M.S., 2004)
• Founding developer of SenseClusters (2002-2004)
• Now PhD student in Intelligent Systems at the
  University of Pittsburgh http//www.cs.pitt.edu/a
  mruta/
• Anagha Kulkarni (M.S., 2006, expected)
• Enhancing SenseClusters since Fall 2004!
• Will start as PhD student at CMU/LTI in Fall 2006
  http//www.d.umn.edu/kulka020/
• NSF for supporting Amruta, Anagha and Ted via
  CAREER award 0092784

8
Background and Motivations
9
Headed and Headless Contexts

• A headed context includes a target word
• Our goal is to cluster the target words based on
  their surrounding contexts
• Target word is center of context and our
  attention
• A headless context has no target word
• Our goal is to cluster the contexts based on
  their similarity to each other
• The focus is on the context as a whole

10
Headed Contexts (input)

• I can hear the ocean in that shell.
• My operating system shell is bash.
• The shells on the shore are lovely.
• The shell command line is flexible.
• The oyster shell is very hard and black.

11
Headed Contexts (output)

• Cluster 1
• My operating system shell is bash.
• The shell command line is flexible.
• Cluster 2
• The shells on the shore are lovely.
• The oyster shell is very hard and black.
• I can hear the ocean in that shell.

12
Headless Contexts (input)

• The new version of Linux is more stable and has
  better support for cameras.
• My Chevy Malibu has had some front end troubles.
• Osborne made one of the first personal computers.
• The brakes went out, and the car flew into the
  house.
• With the price of gasoline, I think Ill be
  taking the bus more often!

13
Headless Contexts (output)

• Cluster 1
• The new version of Linux is more stable and
  better support for cameras.
• Osborne made one of the first personal computers.
• Cluster 2
• My Chevy Malibu has had some front end troubles.
• The brakes went out, and the car flew into the
  house.
• With the price of gasoline, I think Ill be
  taking the bus more often!

14
Web Search as Application

• Web search results are headed contexts
• Search term is target word (found in snippets)
• Web search results are often disorganized two
  people sharing same name, two organizations
  sharing same abbreviation, etc. often have their
  pages mixed up
• If you click on search results or follow links in
  pages found, you will encounter headless contexts
  too

15
Email Foldering as Application

• Email (public or private) is made up of headless
  contexts
• Short, usually focused
• Cluster similar email messages together
• Automatic email foldering
• Take all messages from sent-mail file or inbox
  and organize into categories

16
Clustering News as Application

• News articles are headless contexts
• Entire article or first paragraph
• Short, usually focused
• Cluster similar articles together

17
What is it to be similar?

• You shall know a word by the company it keeps
• Firth, 1957 (Studies in Linguistic Analysis)
• Meanings of words are (largely) determined by
  their distributional patterns (Distributional
  Hypothesis)
• Harris, 1968 (Mathematical Structures of
  Language)
• Words that occur in similar contexts will have
  similar meanings (Strong Contextual Hypothesis)
• Miller and Charles, 1991 (Language and Cognitive
  Processes)
• Various extensions
• Similar contexts will have similar meanings, etc.
• Names that occur in similar contexts will refer
  to the same underlying person, etc.

18
General Methodology

• Represent contexts to be clustered using first or
  second order feature vectors
• Lexical features
• Reduce dimensionality to make vectors more
  tractable and/or understandable
• Singular value decomposition
• Cluster the context vectors
• Find the number of clusters
• Label the clusters
• Evaluate and/or use the contexts!

19
Identifying Lexical Features

• Measures of Association and
• Tests of Significance

20
What are features?

• Features represent the (hopefully) salient
  characteristics of the contexts to be clustered
• Eventually we will represent each context as a
  vector, where the dimensions of the vector are
  associated with features
• Vectors/contexts that include many of the same
  features will be similar to each other

21
Where do features come from?

• In unsupervised clustering, it is common for the
  feature selection data to be the same data that
  is to be clustered
• This is not cheating, since data to be clustered
  does not have any labeled classes that can be
  used to assist feature selection
• It may also be necessary, since we may need to
  cluster all available data, and not hold out some
  for a separate feature identification step
• Email or news articles

22
Feature Selection

• Test data the contexts to be clustered
• Assume that the feature selection data is the
  same as the test data, unless otherwise indicated
  
• Training data a separate corpus of held out
  feature selection data (that will not be
  clustered)
• may need to use if you have a small number of
  contexts to cluster (e.g., web search results)
• This sense of training due to Schütze (1998)

23
Lexical Features

• Unigram a single word that occurs more than a
  given number of times
• Bigram an ordered pair of words that occur
  together more often than expected by chance
• Consecutive or may have intervening words
• Co-occurrence an unordered bigram
• Target Co-occurrence a co-occurrence where one
  of the words is the target word

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Bigrams

• fine wine (window size of 2)
• baseball bat
• house of representatives (window size of 3)
• president of the republic (window size of 4)
• apple orchard
• Selected using a small window size (2-4 words),
  trying to capture a regular (localized) pattern
  between two words (collocation?)

25
Co-occurrences

• tropics water
• boat fish
• law president
• train travel
• Usually selected using a larger window (7-10
  words) of context, hoping to capture pairs of
  related words rather than collocations

26
Bigrams and Co-occurrences

• Pairs of words tend to be much less ambiguous
  than unigrams
• bank versus river bank and bank card
• dot versus dot com and dot product
• Three grams and beyond occur much less frequently
  (Ngrams very Zipfian)
• Unigrams are noisy, but bountiful

27
occur together more often than expected by
chance

• Observed frequencies for two words occurring
  together and alone are stored in a 2x2 matrix
• Throw out bigrams that include one or two stop
  words
• Expected values are calculated, based on the
  model of independence and observed values
• How often would you expect these words to occur
  together, if they only occurred together by
  chance?
• If two words occur significantly more often
  than the expected value, then the words do not
  occur together by chance.

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2x2 Contingency Table
Intelligence !Intelligence
Artificial 100.0 000.12 300.0 398.8 400
!Artificial 200.0 298.8 99,400.0 99,301.2 99,600
300 99,700 100,000
29
Measures of Association
30
Interpreting the Scores

• G2 and X2 are asymptotically approximated by
  the chi-squared distribution
• This meansif you fix the marginal totals of a
  table, randomly generate internal cell values in
  the table, calculate the G2 or X2 scores for
  each resulting table, and plot the distribution
  of the scores, you should get

31
Interpreting the Scores

• Values above a certain level of significance can
  be considered grounds for rejecting the null
  hypothesis
• H0 the words in the bigram are independent
• 3.841 is associated with 95 confidence that the
  null hypothesis should be rejected

32
Measures of Association

• There are numerous measures of association that
  can be used to identify bigram and co-occurrence
  features
• Many of these are supported in the Ngram
  Statistics Package (NSP)
• http//www.d.umn.edu/tpederse/nsp.html

33
Summary

• Identify lexical features based on frequency
  counts or measures of association either in the
  data to be clustered or in a separate set of
  feature selection data
• Language independent
• Unigrams usually only selected by frequency
• Remember, no labeled data from which to learn, so
  somewhat less effective as features than in
  supervised case
• Bigrams and co-occurrences can also be selected
  by frequency, or better yet measures of
  association
• Bigrams and co-occurrences need not be
  consecutive
• Stop words should be eliminated
• Frequency thresholds are helpful (e.g.,
  unigram/bigram that occurs once may be too rare
  to be useful)

34
Context Representations

• First and Second Order Methods

35
Once features selected

• We have a set of unigrams, bigrams,
  co-occurrences or target co-occurrences
• We believe/hope that these are descriptive of the
  contexts
• We also have frequency and measure of association
  score that have been used in their selection
• Convert contexts to be clustered into a vector
  representation based on these features

36
First Order Representation

• Each context is represented by a vector with M
  dimensions, each of which indicates whether or
  not a particular feature occurred in that context
• Value may be binary, a frequency count, or an
  association score
• Context by Feature representation

37
Contexts

• Cxt1 There was an island curse of black magic
  cast by that voodoo child.
• Cxt2 Harold, a known voodoo child, was gifted in
  the arts of black magic.
• Cxt3 Despite their military might, it was a
  serious error to attack.
• Cxt4 Military might is no defense against a
  voodoo child or an island curse.

38
Unigram Feature Set

• island 1000
• black 700
• curse 500
• magic 400
• child 200
• (assume these are frequency counts obtained from
  some corpus)

39
First Order Vectors of Unigrams
island black curse magic child
Cxt1 1 1 1 1 1
Cxt2 0 1 0 1 1
Cxt3 0 0 0 0 0
Cxt4 1 0 1 0 1
40
Bigram Feature Set

• island curse 189.2
• black magic 123.5
• voodoo child 120.0
• military might 100.3
• serious error 89.2
• island child 73.2
• voodoo might 69.4
• military error 54.9
• black child 43.2
• serious curse 21.2
• (assume these are log-likelihood scores based on
  frequency counts from some corpus)

41
First Order Vectors of Bigrams
black magic island curse military might serious error voodoo child
Cxt1 1 1 0 0 1
Cxt2 1 0 0 0 1
Cxt3 0 0 1 1 0
Cxt4 0 1 1 0 1
42
First Order Vectors

• Can have binary values or weights associated with
  frequency, etc.
• Forms a context by feature matrix
• May optionally be smoothed/reduced with Singular
  Value Decomposition
• More on that later
• The contexts are ready for clustering
• More on that later

43
Second Order Features

• First order features encode the occurrence of a
  feature in a context
• Feature occurrence represented by binary value
• Second order features encode something extra
  about a feature that occurs in a context
• Feature occurrence represented by word
  co-occurrences
• Feature occurrence represented by context
  occurrences

44
Second Order Representation

• First, build word by word matrix from features
• Based on bigrams or co-occurrences
• First word is row, second word is column, cell is
  score
• (optionally) reduce dimensionality w/SVD
• Each row forms a vector of first order
  co-occurrences
• Second, replace each word in a context with its
  row/vector as found in the word by word matrix
• Average all the word vectors in the context to
  create the second order representation
• Due to Schütze (1998), related to LSI/LSA

45
Word by Word Matrix
magic curse might error child
black 123.5 0 0 0 43.2
island 0 189.2 0 0 73.2
military 0 0 100.3 54.9 0
serious 0 21.2 0 89.2 0
voodoo 0 0 69.4 0 120.0
46
Word by Word Matrix

• can also be used to identify sets of related
  words
• In the case of bigrams, rows represent the first
  word in a bigram and columns represent the second
  word
• Matrix is asymmetric
• In the case of co-occurrences, rows and columns
  are equivalent
• Matrix is symmetric
• The vector (row) for each word represent a set of
  first order features for that word
• Each word in a context to be clustered for which
  a vector exists (in the word by word matrix) is
  replaced by that vector in that context

47
There was an island curse of black magic cast by
that voodoo child.
magic curse might error child
black 123.5 0 0 0 43.2
island 0 189.2 0 0 73.2
voodoo 0 0 69.4 0 120.0
48
Second Order Co-Occurrences

• Word vectors for black and island show
  similarity as both occur with child
• black and island are second order
  co-occurrence with each other, since both occur
  with child but not with each other (i.e.,
  black island is not observed)

49
Second Order Representation

• There was an curse, child curse of magic,
  child magic cast by that might, child child
• curse, child magic, child might, child

50
There was an island curse of black magic cast by
that voodoo child.
magic curse might error child
Cxt1 41.2 63.1 24.4 0 78.8
51
Second Order Representation

• Results in a Context by Feature (Word)
  Representation
• Cell values do not indicate if feature occurred
  in context. Rather, they show the strength of
  association of that feature with other words that
  occur with a word in the context.

52
Summary

• First order representations are intuitive, but
• Can suffer from sparsity
• Contexts represented based on the features that
  occur in those contexts
• Second order representations are harder to
  visualize, but
• Allow a word to be represented by the words it
  co-occurs with (i.e., the company it keeps)
• Allows a context to be represented by the words
  that occur with the words in the context
• Helps combat sparsity

53
Related Work

• Pedersen and Bruce 1997 (EMNLP) presented first
  order method of discrimination
• http//acl.ldc.upenn.edu/W/W97/W97-0322.pdf
• Schütze 1998 (Computational Linguistics)
  introduced second order method
• http//acl.ldc.upenn.edu/J/J98/J98-1004.pdf
• Purandare and Pedersen 2004 (CoNLL) compared
  first and second order methods
• http//acl.ldc.upenn.edu/hlt-naacl2004/conll0
  4/pdf/purandare.pdf
• First order better if you have lots of data
• Second order better with smaller amounts of data

54
Dimensionality Reduction

• Singular Value Decomposition

55
Effect of SVD

• SVD reduces a matrix to a given number of
  dimensions This may convert a word level space
  into a semantic or conceptual space
• If dog and collie and wolf are
  dimensions/columns in a word co-occurrence
  matrix, after SVD they may be a single dimension
  that represents canines

56
Effect of SVD

• The dimensions of the matrix after SVD are
  principal components that represent the meaning
  of concepts
• Similar columns are grouped together
• SVD is a way of smoothing a very sparse matrix,
  so that there are very few zero valued cells
  after SVD

57
How can SVD be used?

• SVD on first order contexts will reduce a context
  by feature representation down to a smaller
  number of features
• Latent Semantic Analysis typically performs SVD
  on a feature by context representation, where the
  contexts are reduced
• SVD used in creating second order context
  representations
• Reduce word by word matrix

58
Word by Word Matrix
apple blood cells ibm data box tissue graphics memory organ plasma
pc 2 0 0 1 3 1 0 0 0 0 0
body 0 3 0 0 0 0 2 0 0 2 1
disk 1 0 0 2 0 3 0 1 2 0 0
petri 0 2 1 0 0 0 2 0 1 0 1
lab 0 0 3 0 2 0 2 0 2 1 3
sales 0 0 0 2 3 0 0 1 2 0 0
linux 2 0 0 1 3 2 0 1 1 0 0
debt 0 0 0 2 3 4 0 2 0 0 0
59
Singular Value DecompositionAUDV
60
Word by Word Matrix After SVD
apple blood cells ibm data tissue graphics memory organ plasma
pc .73 .00 .11 1.3 2.0 .01 .86 .77 .00 .09
body .00 1.2 1.3 .00 .33 1.6 .00 .85 .84 1.5
disk .76 .00 .01 1.3 2.1 .00 .91 .72 .00 .00
germ .00 1.1 1.2 .00 .49 1.5 .00 .86 .77 1.4
lab .21 1.7 2.0 .35 1.7 2.5 .18 1.7 1.2 2.3
sales .73 .15 .39 1.3 2.2 .35 .85 .98 .17 .41
linux .96 .00 .16 1.7 2.7 .03 1.1 1.0 .00 .13
debt 1.2 .00 .00 2.1 3.2 .00 1.5 1.1 .00 .00
61
Second Order Representation

• I got a new disk today!
• What do you think of linux?

apple blood cells ibm data tissue graphics memory organ plasma
disk .76 .00 .01 1.3 2.1 .00 .91 .72 .00 .00
linux .96 .00 .16 1.7 2.7 .03 1.1 1.0 .00 .13

• These two contexts share no words in common, yet
  they are similar! disk and linux both occur with
  Apple, IBM, data, graphics, and memory
  
• The two contexts are similar because they share
  many second order co-occurrences

62
Relationship to LSA

• Latent Semantic Analysis uses feature by context
  first order representation
• Indicates all the contexts in which a feature
  occurs
• Use SVD to reduce dimensions (contexts)
• Cluster features based on similarity of contexts
  in which they occur
• Represent sentences using an average of feature
  vectors

63
Feature by Context Representation
Cxt1 Cxt2 Cxt3 Cxt4
black magic 1 1 0 1
island curse 1 0 0 1
military might 0 0 1 0
serious error 0 0 1 0
voodoo child 1 1 0 1
64
References

• Deerwester, S. and Dumais, S.T. and Furnas, G.W.
  and Landauer, T.K. and Harshman, R., Indexing by
  Latent Semantic Analysis, Journal of the American
  Society for Information Science, vol. 41, 1990
• Landauer, T. and Dumais, S., A Solution to
  Plato's Problem The Latent Semantic Analysis
  Theory of Acquisition, Induction and
  Representation of Knowledge, Psychological
  Review, vol. 104, 1997
• Schütze, H, Automatic Word Sense Discrimination,
  Computational Linguistics, vol. 24, 1998
• Berry, M.W. and Drmac, Z. and Jessup,
  E.R.,Matrices, Vector Spaces, and Information
  Retrieval, SIAM Review, vol 41, 1999

65
Clustering

• Partitional Methods
• Cluster Stopping
• Cluster Labeling

66
Many many methods

• Cluto supports a wide range of different
  clustering methods
• Agglomerative
• Average, single, complete link
• Partitional
• K-means (Direct)
• Hybrid
• Repeated bisections
• SenseClusters integrates with Cluto
• http//www-users.cs.umn.edu/karypis/cluto/

67
General Methodology

• Represent contexts to be clustered in first or
  second order vectors
• Cluster the context vectors directly
• vcluster
• or convert to similarity matrix and then
  cluster
• scluster

68
Partitional Methods

• Randomly create centroids equal to the number of
  clusters you wish to find
• Assign each context to nearest centroid
• After all contexts assigned, re-compute centroids
• best location decided by criterion function
• Repeat until stable clusters found
• Centroids dont shift from iteration to iteration

69
Partitional Methods

• Advantages fast
• Disadvantages
• Results can be dependent on the initial placement
  of centroids
• Must specify number of clusters ahead of time
• maybe not

70
Partitional Criterion Functions

• Intra-Cluster (Internal) similarity/distance
• How close together are members of a cluster?
• Closer together is better
• Inter-Cluster (External) similarity/distance
• How far apart are the different clusters?
• Further apart is better

71
Intra Cluster Similarity

• Ball of String (I1)
• How far is each member from each other member
• Flower (I2)
• How far is each member of cluster from centroid

72
Contexts to be Clustered
73
Ball of String (I1 Internal Criterion Function)
74
Flower(I2 Internal Criterion Function)
75
Inter Cluster Similarity

• The Fan (E1)
• How far is each centroid from the centroid of the
  entire collection of contexts
• Maximize that distance

76
The Fan(E1 External Criterion Function)
77
Hybrid Criterion Functions

• Balance internal and external similarity
• H1 I1/E1
• H2 I2/E1
• Want internal similarity to increase, while
  external similarity decreases
• Want internal distances to decrease, while
  external distances increase

78
Cluster Stopping
79
Cluster Stopping

• Many Clustering Algorithms require that the user
  specify the number of clusters prior to
  clustering
• But, the user often doesnt know the number of
  clusters, and in fact finding that out might be
  the goal of clustering

80
Criterion Functions Can Help

• Run partitional algorithm for k1 to deltaK
• DeltaK is a user estimated or automatically
  determined upper bound for the number of clusters
• Find the value of k at which the criterion
  function does not significantly increase at k1
• Clustering can stop at this value, since no
  further improvement in solution is apparent with
  additional clusters (increases in k)

81
H2 versus kT. Blair V. Putin S. Hussein
82
PK2

• Based on Hartigan, 1975
• When ratio approaches 1, clustering is at a
  plateau
• Select value of k which is closest to but outside
  of standard deviation interval

83
PK2 predicts 3 sensesT. Blair V. Putin S.
Hussein
84
PK3

• Related to Salvador and Chan, 2004
• Inspired by Dice Coefficient
• Values close to 1 mean clustering is improving
• Select value of k which is closest to but outside
  of standard deviation interval

85
PK3 predicts 3 sensesT. Blair V. Putin S.
Hussein
86
References

• Hartigan, J. Clustering Algorithms, Wiley, 1975
• basis for SenseClusters stopping method PK2
• Mojena, R., Hierarchical Grouping Methods and
  Stopping Rules An Evaluation, The Computer
  Journal, vol 20, 1977
• basis for SenseClusters stopping method PK1
• Milligan, G. and Cooper, M., An Examination of
  Procedures for Determining the Number of Clusters
  in a Data Set, Psychometrika, vol. 50, 1985
• Very extensive comparison of cluster stopping
  methods
• Tibshirani, R. and Walther, G. and Hastie, T.,
  Estimating the Number of Clusters in a Dataset
  via the Gap Statistic,Journal of the Royal
  Statistics Society (Series B), 2001
• Pedersen, T. and Kulkarni, A. Selecting the
  "Right" Number of Senses Based on Clustering
  Criterion Functions, Proceedings of the Posters
  and Demo Program of the Eleventh Conference of
  the European Chapter of the Association for
  Computational Linguistics, 2006
• Describes SenseClusters stopping methods

87
Cluster Labeling
88
Cluster Labeling

• Once a cluster is discovered, how can you
  generate a description of the contexts of that
  cluster automatically?
• In the case of contexts, you might be able to
  identify significant lexical features from the
  contents of the clusters, and use those as a
  preliminary label

89
Results of Clustering

• Each cluster consists of some number of contexts
• Each context is a short unit of text
• Apply measures of association to the contents of
  each cluster to determine N most significant
  bigrams
• Use those bigrams as a label for the cluster

90
Label Types

• The N most significant bigrams for each cluster
  will act as a descriptive label
• The M most significant bigrams that are unique to
  each cluster will act as a discriminating label

91
Evaluation Techniques

• Comparison to gold standard data

92
Evaluation

• If Sense tagged text is available, can be used
  for evaluation
• But dont use sense tags for clustering or
  feature selection!
• Assume that sense tags represent true clusters,
  and compare these to discovered clusters
• Find mapping of clusters to senses that attains
  maximum accuracy

93
Evaluation

• Pseudo words are especially useful, since it is
  hard to find data that is discriminated
• Pick two words or names from a corpus, and
  conflate them into one name. Then see how well
  you can discriminate.
• http//www.d.umn.edu/tpederse/tools.html
• Baseline Algorithm group all instances into one
  cluster, this will reach accuracy equal to
  majority classifier

94
Evaluation

• Pseudo words are especially useful, since it is
  hard to find data that is discriminated
• Pick two or more words or names from a corpus,
  and conflate them into one name. Then see how
  well you can discriminate.
• http//www.d.umn.edu/kulka020/kanaghaName.html

95
Baseline Algorithm

• Baseline Algorithm group all instances into one
  cluster, this will reach accuracy equal to
  majority classifier
• What if the clustering said everything should be
  in the same cluster?

96
Baseline Performance
S1 S2 S3 Totals
C1 0 0 0 0
C2 0 0 0 0
C3 80 35 55 170
Totals 80 35 55 170
S3 S2 S1 Totals
C1 0 0 0 0
C2 0 0 0 0
C3 55 35 80 170
Totals 55 35 80 170

• (0055)/170 .32 if C3 is S1
  (0080)/170 .47 if C3 is S3

97
Evaluation

• Suppose that C1 is labeled S1, C2 as S2, and C3
  as S3
• Accuracy (10 0 10) / 170 12
• Diagonal shows how many members of the cluster
  actually belong to the sense given on the column
• Can the columns be rearranged to improve the
  overall accuracy?
• Optimally assign clusters to senses
• 
  
  

S1 S2 S3 Totals
C1 10 30 5 45
C2 20 0 40 60
C3 50 5 10 65
Totals 80 35 55 170
98
Evaluation

• The assignment of C1 to S2, C2 to S3, and C3 to
  S1 results in 120/170 71
• Find the ordering of the columns in the matrix
  that maximizes the sum of the diagonal.
• This is an instance of the Assignment Problem
  from Operations Research, or finding the Maximal
  Matching of a Bipartite Graph from Graph Theory.

S2 S3 S1 Totals
C1 30 5 10 45
C2 0 40 20 60
C3 5 10 50 65
Totals 35 55 80 170
99
Alternatives?

• Unsupervised methods may not discover clusters
  equivalent to the classes learned in supervised
  learning
• Evaluation based on assuming that sense tags
  represent the true cluster are likely a bit
  harsh. Alternatives?
• Humans could look at the members of each cluster
  and determine the nature of the relationship or
  meaning that they all share
• Use the contents of the cluster to generate a
  descriptive label that could be inspected by a
  human

100
Thank you!

• Questions or comments on tutorial or
  SenseClusters are welcome at any time
  tpederse_at_d.umn.edu
• SenseClusters is freely available via LIVE CD,
  the Web, and in source code form
• http//senseclusters.sourceforge.net
• SenseClusters papers available at
• http//www.d.umn.edu/tpederse/senseclusters-pubs.
  html