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Transcript of Fragrances Grimal 15-11-10
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Reunion FRAGRANCES
Clement Grimal (sup. Eric Gaussier et Gilles Bisson)
15 novembre 2010
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Outline
1 Improvements of -Sim
2 Experiments
3 Tensorial extension
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Improvements of -Sim Experiments Tensorial extension
The text mining context
Document#1:
A contruction found in villagesand in the suburbs of biggertown, used to house a family.
Document#2:
A building which main purposeis to provide accomodation tohuman beings.
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Improvements of -Sim Experiments Tensorial extension
The text mining context
Document#1:
A contruction found in villagesand in the suburbs of biggertown, used to house a family.
Document#2:
A building which main purposeis to provide accomodation tohuman beings.
With a classical clustering approach:No shared terms between the two documents
Similarity(Document#1, Document#2) = 0
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Improvements of -Sim Experiments Tensorial extension
The text mining context
Document#1:
A contruction found in villagesand in the suburbs of biggertown, used to house a family.
Document#2:
A building which main purposeis to provide accomodation tohuman beings.
With a classical clustering approach:No shared terms between the two documents
Similarity(Document#1, Document#2) = 0
Using co-clustering:Clustering of the terms
Similarity(Document#1, Document#2) > 0
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I f S E i T i l i
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Improvements of -Sim Experiments Tensorial extension
Notations
M: documents/words matrix of r rows and c columns
mi: = [mi1 . . .mic]: row vector describing document i
m:j = [m1j . . .mrj ]: column vector describing word j
SR: square similarity matrix (documents) of size r, with srij [0, 1]
SC: square similarity matrix (words) of size c, with scij [0, 1]
Fs(mij , mkl) [0, 1]: similarity function between two elements of the datamatrix M
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I t f S E i t T i l t i
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Improvements of -Sim Experiments Tensorial extension
Notations
M: documents/words matrix of r rows and c columns
mi: = [mi1 . . .mic]: row vector describing document i
m:j = [m1j . . .mrj ]: column vector describing word j
SR: square similarity matrix (documents) of size r, with srij [0, 1]
SC: square similarity matrix (words) of size c, with scij [0, 1]
Fs(mij , mkl) [0, 1]: similarity function between two elements of the datamatrix M
Two documents are similar if they contain similar words. Two words are similar if they appear in similar documents.
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Improvements of -Sim Experiments Tensorial extension
Notations
M: documents/words matrix of r rows and c columns
mi: = [mi1 . . .mic]: row vector describing document i
m:j = [m1j . . .mrj ]: column vector describing word j
SR: square similarity matrix (documents) of size r, with srij [0, 1]
SC: square similarity matrix (words) of size c, with scij [0, 1]
Fs(mij , mkl) [0, 1]: similarity function between two elements of the datamatrix M
Two documents are similar if they contain similar words. Two words are similar if they appear in similar documents.
Joint construction of the two similarity matrices SR and SC.
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Improvements of -Sim Experiments Tensorial extension
Similarity between two documents
Classical approach: similarity = f(shared words)
Sim(mi:,mj:) = Fs(mi1, mj1) + + Fs(mic, mjc)
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p S m p
Similarity between two documents
Classical approach: similarity = f(shared words)
Sim(mi:,mj:) = Fs(mi1, mj1) + + Fs(mic, mjc)
Using SC (usually, scii = 1) and the k-norm:
Sim(mi:,mj:) =k
c
l=1
(Fs(mil, mjl))k scll
Now comparing every pair of words:
Simk(mi:,mj:) =k
c
l=1
cn=1
(Fs (mil, mjn))k scln
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p p
If Fs(mij , mkl) = mij mkl:
Simk(mi:,mj:) =k
(mi:)k SC mT
j:
k
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If Fs(mij , mkl) = mij mkl:
Simk(mi:,mj:) =k
(mi:)k SC mT
j:
k
Then we need to normalize this similarity:
srij =
k(mi:)
k
SCm
T
j:k
N(mi:,mj:) [0, 1]
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If Fs(mij , mkl) = mij mkl:
Simk(mi:,mj:) =k
(mi:)k SC mT
j:
k
Then we need to normalize this similarity:
srij =
k(mi:)
k
SCm
T
j:k
N(mi:,mj:) [0, 1]
With special values for k, SC and N, we have: Jaccard: SC = I, k = 1, N = mi:1 + mj:1 mi:m
T
j:
Dice: SC = 2I, k = 1, N = mi:1 + mj:1
Generalized Cosine: SC > 0, k = 1, N =mi:SC
mj:SC
Classical -Sim : k = 1, N = |mi:| |mj:|
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Parameter k
The generalized -Sim
i, j 1..r, srij =Simk(mi:,mj:)
Simk(mi:,mi:)
Simk(mj:,mj:)
i, j 1..c, scij =Simk(m:i,m:j)
Simk(m:i,m:i)
Simk(m:j ,m:j)
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Parameter k
The generalized -Sim
i, j 1..r, srij =Simk(mi:,mj:)
Simk(mi:,mi:)
Simk(mj:,mj:)
i, j 1..c, scij =Simk(m:i,m:j)
Simk(m:i,m:i)
Simk(m:j ,m:j)
For k = 1, SR and SC are not positive semi-definite...
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Parameter k
The generalized -Sim
i, j 1..r, srij =Simk(mi:,mj:)
Simk(mi:,mi:)
Simk(mj:,mj:)
i, j 1..c, scij =Simk(m:i,m:j)
Simk(m:i,m:i)
Simk(m:j ,m:j)
For k = 1, SR and SC are not positive semi-definite...We are not defining aninner product so it is not a generalized cosine measure...
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Another step: pruning
In such a corpus...
Many words are not specific enough, and creates a lot of irrelevant similarities.
These similarities can be considered as noise.
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Another step: pruning
In such a corpus...
Many words are not specific enough, and creates a lot of irrelevant similarities.
These similarities can be considered as noise.
How to deal with it?
Hypothetis: these irrelevant similarities are small.
At each iteration, we remove the smallest p% of the similarity matrices.
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Meaning of an iteration
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Meaning of an iteration
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Meaning of an iteration
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Meaning of an iteration
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Methods
Five similarity measures -Sim (with or without k and p) [Hussain et al.(2010)] Cosine LSA (Latent Semantic Analysis) [Deerwester et al.(1990)] SNOS (Similarity in Non-Othogonal Space) [Liu et al.(2004)] CTK (Commute Time Kernel) [Yen et al.(2009)]
+ Ascendant Hierarchical Clustering, with Wards index
Three co-clustering methods ITCC (Information Theoric Co-Clustering) [Dhillon et al.(2003)] BVD (Block Value Decomposition) [Long et al.(2005)] RSN (k-partite graph partioning algorithm) [Long et al.(2006)]
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Methodology and Data
Methodology We randomly select subsets of documents already labeled
We measure the quality of the clusters using the micro-averaged precision
The subsets:Name Newsgroups included #clusters. #docs.
M2 talk.politics.mideast, talk.politics.misc 2 500M5 comp.graphics, rec.motorcycles, rec.sport.baseball, sci.space,
talk.politics.mideast5 500
M10 alt.atheism, comp.sys.mac.hardware, misc.forsale, rec.autos,rec.sport.hockey, sci.crypt, sci.electronics, sci.med, sci.space,talk.politics.gun
10 500
NG1 rec.sports.baseball, rec.sports.hockey 2 400NG2 comp.os.ms-windows.misc, comp.windows.x, rec.motorcycles,
sci.crypt, sci.space5 1000
NG3 comp.os.ms-windows.misc, comp.windows.x, misc.forsale,rec.motorcycles, sci.crypt, sci.space, talk.politics.mideast,talk.religion.misc
8 1600
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Results
M2 M5 M10 NG1 NG2 NG3
Cosine Pr 0.600.00 0.630.07 0.490.06 0.900.11 0.600.10 0.590.04
LSA Pr 0.920.02 0.870.06 0.590.07 0.960.01 0.820.03 0.740.03
ITCC Pr 0.790.06 0.490.10 0.290.02 0.690.09 0.630.06 0.590.05
BVD Pr best: 0.95 best: 0.93 best: 0.67 - - -
RSN NMI - - - 0.640.16 0.750.07 0.700.04
SNOS Pr 0.550.02 0.250.02 0.240.06 0.510.01 0.240.02 0.220.05
CTK Pr 0.940.01 0.940.01 0.710.01 0.960.01 0.900.01 0.870.01
CTK* Pr 0.95 0.95 0.80 0.98 0.91 0.87
-SimPr 0.910.09 0.960.00 0.690.05 0.960.01 0.920.01 0.790.06
NMI 0.760.06 0.790.02 0.720.03
-SimpPr 0.940.01 0.960.00 0.730.03 0.970.01 0.920.01 0.840.05
NMI 0.780.05 0.790.02 0.730.02
-Sim1 Pr 0.95 0.00 0.960.02 0.780.03 0.970.02 0.94 0.01 0.860.05
NMI 0.850.07 0.830.03 0.790.03
-Sim1pPr 0.95 0.00 0.97 0.01 0.780.03 0.98 0.01 0.94 0.01 0.870.05
NMI 0.860.04 0.830.03 0.800.02
-Sim0.8Pr 0.95 0.00 0.97 0.01 0.790.02 0.98 0.01 0.94 0.01 0.90 0.01
NMI 0.870.05 0.840.02 0.810.02
-Sim0.8pPr 0.95 0.00 0.97 0.01 0.80 0.04 0.98 0.00 0.94 0.01 0.90 0.02
NMI 0.880.03 0.850.02 0.810.03
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An other preprocessing
Too many words...
Initially, we have 100.000 words, which is too much, both consideringspatial and time complexity.
We decided to select only 2.000 words.
Previsously, it was done using mutual information and the documentslabels...
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An other preprocessing
Too many words...
Initially, we have 100.000 words, which is too much, both consideringspatial and time complexity.
We decided to select only 2.000 words.
Previsously, it was done using mutual information and the documentslabels... A supervised preprocessing for an unsupervised task?!
Clement Grimal (LIG) Reunion FRAGRANCES 15 novembre 2010 13 / 25
Improvements of -Sim Experiments Tensorial extension
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An other preprocessing
Too many words...
Initially, we have 100.000 words, which is too much, both consideringspatial and time complexity.
We decided to select only 2.000 words.
Previsously, it was done using mutual information and the documentslabels... A supervised preprocessing for an unsupervised task?!
Unsupervised feature selection
We decided to choose the 2.000 words with the PAM (Partitioning AroundMedods) algorithm.
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Influence ofk
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Influence ofp
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Conclusion for -Sim
Generalization of the method
Exploration of different normed spaces (k)
Pruning of the similarity matrices (p)
Very good experimental results
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Preliminaries
DataFolksonomy data with users, ressources and tags:
Mi,j,k = 1 if user i describes ressource j using tag k.
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Preliminaries
DataFolksonomy data with users, ressources and tags:
Mi,j,k = 1 if user i describes ressource j using tag k.
Additonal notations M: data tensor describing the relation between users, ressources and tags.
Mi::: slice of the tensor describing user i M:j:: slice of the tensor describing ressource j M::k: slice of the tensor describing tag k
SU: square similarity matrix for users
SR: square similarity matrix for ressources
ST: square similarity matrix for tags
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Improvements of -Sim Experiments Tensorial extension
T i l i f S
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Tensorial extension of -Sim
-Sim computes similarities between vectors:
Having a data tensor users, ressources, tags, we want to compute thesimilarities between users, described by matrices (slices of the tensor). Rows areressourcres and columns are tags.
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Th i l h
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The simplest approach
Two users are similar if they use the same tags to describe the same ressources.
Sim(ui, ui ) =
nr
j=1
nt
k=1
Fs(mijk ,mijk) srjj stkk
=
nr
j=1
nt
k=1
Fs(mijk ,mijk)
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Th i l t h
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The simplest approach
Two users are similar if they use the same tags to describe the same ressources.
Sim(ui, ui ) =
nr
j=1
nt
k=1
Fs(mijk ,mijk) srjj stkk
=
nr
j=1
nt
k=1
Fs(mijk ,mijk)
Complexity: n2 comparisons for every pair of users n4
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Improvements of -Sim
Experiments Tensorial extension
Th f ll h
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The full approach
Two users are similar if they use similar tags to describe similar ressources.
Sim(ui, u
i
) =nr
j=1
nt
k=1
nr
j=1
nt
k=1
Fs(mijk ,mijk) srjj stkk
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Improvements of -Sim
Experiments Tensorial extension
The full approach
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The full approach
Two users are similar if they use similar tags to describe similar ressources.
Sim(ui, u
i
) =nr
j=1
nt
k=1
nr
j=1
nt
k=1
Fs(mijk ,mijk) srjj stkk
Complexity: n4 comparisons for every pair of users n6
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Improvements of -Sim
Experiments Tensorial extension
The intermediate approach
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The intermediate approach
Two users are similar if they use similar tags to describe the same ressources.
Sim(ui, ui ) =
nr
j=1
nt
k=1
nt
k=1
Fs(mijk ,mijk ) srjj stkk
=
nr
j=1
nt
k=1
nt
k=1
Fs(mijk ,mijk ) stkk
Complexity: n3 comparisons for every pair of users n5
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Improvements of -Sim
Experiments Tensorial extension
The intermediate approach
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The intermediate approach
Two users are similar if they use the same tags to describe similar ressources.
Sim(ui, ui ) =
nr
j=1
nt
k=1
nr
j=1
Fs(mijk ,mijk) srjj stkk
=
nr
j=1
nt
k=1
nr
j=1
Fs(mijk ,mijk) srjj
Complexity: n3 comparisons for every pair of users n5
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Improvements of -Sim
Experiments Tensorial extension
A few ideas
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A few ideas...
By considering that Fs(, ) is just the product, we should be able to reducethe time complexity (as for -Sim actually).
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Improvements of -Sim
Experiments Tensorial extension
A few ideas
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A few ideas...
By considering that Fs(, ) is just the product, we should be able to reducethe time complexity (as for -Sim actually).
It would be interesting to use the full approach on users, and only thesimplest or intermediate approach on ressources and tags.
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Improvements of-Sim
Experiments Tensorial extension
A few ideas
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A few ideas...
By considering that Fs(, ) is just the product, we should be able to reducethe time complexity (as for -Sim actually).
It would be interesting to use the full approach on users, and only thesimplest or intermediate approach on ressources and tags.
Maybe SR (and ST) should be computed beforehand with a specificmeasure depending on the type of the ressources (images, etc.).
Clement Grimal (LIG) Reunion FRAGRANCES 15 novembre 2010 23 / 25
Improvements of -Sim Experiments Tensorial extension
A few ideas
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A few ideas...
By considering that Fs(, ) is just the product, we should be able to reducethe time complexity (as for -Sim actually).
It would be interesting to use the full approach on users, and only thesimplest or intermediate approach on ressources and tags.
Maybe SR (and ST) should be computed beforehand with a specificmeasure depending on the type of the ressources (images, etc.).
Finding a good normalization scheme will be challenging...
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Thank you very much!
Improvements of -Sim Experiments Tensorial extension
References
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References
S. Deerwester, S. T. Dumais, G. W. Furnas, Thomas, and R. Harshman. Indexing by latent semantic
analysis.Journal of the American Society for Information Science, 41:391407, 1990.
I. S. Dhillon, S. Mallela, and D. S. Modha. Information-theoretic co-clustering.
In Proceedings of the Ninth ACM SIGKDD, pages 8998, 2003.
S. F. Hussain, C. Grimal, and G. Bisson. An improved co-similarity measure for document clustering.
In International Conference on Machine Learning and Applications, 2010.
N. Liu, B. Zhang, J. Yan, Q. Yang, S. Yan, Z. Chen, F. Bai, and W. ying Ma. Learning similarity
measures in non-orthogonal space.In Proceedings of the 13th ACM CIKM, pages 334341. ACM Press, 2004.
B. Long, Z. M. Zhang, and P. S. Yu. Co-clustering by block value decomposition.
In Proceedings of the Eleventh ACM SIGKDD, pages 635640, New York, NY, USA, 2005. ACM.
B. Long, Z. M. Zhang, X. Wu, and P. S. Yu. Spectral clustering for multi-type relational data.
In ICML 06: Proceedings of the 23rd international conference on Machine learning, pages 585592,New York, NY, USA, 2006. ACM.
L. Yen, F. Fouss, C. Decaestecker, P. Francq, and M. Saerens. Graph nodes clustering with the
sigmoid commute-time kernel: A comparative study.Data Knowl. Eng., 68(3):338361, 2009.
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