Cartesian Contour: A Concise Representation for a Collection of Frequent Sets

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Cartesian Contour A Concise Representation for a
Collection of Frequent Sets

• Ruoming Jin
• Kent State University

Joint work with Yang Xiang and Lin Liu (KSU)
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Frequent Pattern Mining

• Summarizing the underlying datasets, providing
  key insights
• Key building block for data mining toolbox
• Association rule mining
• Classification
• Clustering
• Change Detection
• etc
• Application Domains
• Business, biology, chemistry, WWW,
  computer/networing security, software
  engineering,

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The Problem

• The number of patterns is too large
• Attempt
• Maximal Frequent Itemsets
• Closed Frequent Itemsets
• Non-Derivable Itemsets
• Compressed or Top-k Patterns
• Tradeoff
• Significant Information Loss
• Large Size

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Pattern Summarization

• Using a small number of itemsets to best
  represent the entire collection of frequent
  itemsets
• The Spanning Set Approach Afrati-Gionis-Mannila,
  KDD04
• Exact Description Maximal Frequent Itemsets
• Our problem
• Can we find a concise representation which can
  allow both exact and approximate summarization of
  a collection of frequent itemsets?

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Basic Idea
A,B,G,H, A,B,I,J, A,B,K,L C,D,G,H,
C,D,I,J, C,D,K,L E,F,G,H, E,F,I,J,
E,F,K,L
9 itemsets, 36 items.
Covering
Picturing
A,B,C,D,E,F
Cartesian Product
G,H,I,J,K,L
1 biclique, 6 itemsets, 12 items
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Cartesian Covering
Non-frequent itemsets
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Problem Formulation

• Cartesian product
• e.g.
• Cost of a Cartesian product
• e.g. 1 biclique, 3 itemsets, and 5 items
• Covering
• e.g.

How can we use Cartesian products to concisely
represent a collection of frequent itemsets?
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Exact and Approximate Covering
Exact Representation
Cost 2 biclique, 4 itemsets, 6 items False
positive none
Approximate Representation
Cost 1 biclique, 3 itemsets, 5 items False
positive G,C,G,D,G,C,D
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Covering Maximal Frequent Itemsets
MNOVWX
CDEJKL
CDEVWX
MNOGHI
CDEGHI
PQRJKL
CDESTU
GHI, JKL
ABCGHI
ABCSTU
STU, VWX
ABC, CDE
MNO, PQR
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Problem Reformulation

• Given Maximal Frequent Itemsets

Exact representation
Approximate representation
Frequent Itemsets
C1 C2
C1 C2
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Minimal Biclique Set Cover Problem
Ground Set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
1, 2,3,4,6,7,8,9
5,10,11
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NP-hardness

• By reducing the Minimal Biclique Set Cover into
  our problem, we can easily prove our problem1
  (exact) and problem2 (approximate) are NP-hard.
• Minimal Biclique Set Cover is a Variant of the
  Classical Set Cover Problem

Can we use the standard set-cover greedy
algorithm?
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Naïve greedy algorithm

• Greedy algorithm
• Each time choose a biclique with the lowest price
  .
• is the cost.
• This method has a logarithmic approximation
  bound.
• The problem?
• The number of candidate bicliques are 2XY !!

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Candidate Reduction

• Assume one side of the biclique candidate is
  known, how to choose the other side?

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Greedy Algorithm
Biclique Candidate
Split and sort
Covering 4
Covering 3
Covering 3
Add 1st single Y-vertex Biclique
Add 2nd single Y-vertex Biclique
Add 3th single Y-vertex Biclique
Fixed!
Cheapest sub-biclique!
Cost 1
Cost 5/7
Cost 6/8
gt 5/7
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Approximation Bound of the Greedy Algorithm

• The greedy SubBiclique procedure can find a
  sub-biclique whose price is less than or equal to
  e/(1-e) of the price of the optimal sub-biclique
  (cheapest price)!

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Further Reduction

• Only using the IDEA1, the time complexity is
  still exponential .
• How to reduce this further??
• Are all the combinations equally
  important?
• No, because some are more likely to connect to
  the Y side.
• Our solution Frequent itemset mining!

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Using Frequent Itemset Mining
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Overall Algorithm

• Step 1 Use the Frequent Itemset Mining tool to
  find all the (one-side maximal) biclique
  candidates
• Step 2 Calculate the cheapest sub-biclique for
  each candidate using the greedy procedure
• Step 3 Compare all the sub-bicliques, choose the
  cheapest one
• Step 4 if MFI totally covered, done else go to
  Step 2.

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Approximation Bound

• Our algorithm has e/(1-e) (ln (n)1)
  approximation ratio with respect to the candidate
  set (all the sub-bicliques with one sides coming
  from the frequent itemset mining).

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Speed-up techniques (1)

• Using Closed itemsets for X and Y
• Initially X and Y contain all the FI,
  respectively.
• Using to cover MFI is similar to factorizing
  MFI
• MFIs maximal factor itemsets are closed
  itemsets, whose number is much smaller!

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Speed-up techniques (2)
Dense Graph
Sparse Graph
TRADEOFF
Frequent Itemset
Supporting Transaction
Frequent itemsets is small Valuable biclique
candidates are not be fully used!
Frequent itemsets is big Handling those
candidates are too slow!
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Speed-up techniques (3)

• Iterative procedure
• A large number of closed itemsets
• To cover MFI in one time can produce a huge
  number of biclique candidates
• So to cover MFI in several times
• Support level is reduced gradually!

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Experiments

• Data sets

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Conclusion

• We propose an interesting summarization problem
  which consider the interaction between frequent
  patterns
• We transform this problem into a generalized
  minimal biclique covering problem and design an
  approximate algorithm with bound
• The experimental results demonstrate the
  effective and efficiency of our approach

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• Thank you !!!

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Reference

• Bayardo98 Roberto J. Bayardo Jr. Efficiently
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• Pasquier99 Nicolas Pasquier, Yves Bastide,
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  ICDT99.
• Calder07 Toon Calder and Bart Goethals.
  Non-derivable itemset mining. Data Min. Knowl.
  Discover. 07.
• Han02 Jiawei Han, Jianyong Wang, Ying Lu and
  Petre Tzvetkov. Mining top-k frequent closed
  patterns without minimum support. ICDM02.
• Xin06 Dong Xin, Hong Cheng, Xifeng Yan, and
  Jiawei Han. Extracting redundancy-aware top-k
  patterns. KDD06.
• Xin05 Dong Xin, Jiawei Han, Xifeng Yan, and
  Hong Cheng. Mining compressed frequent-pattern
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• Afrati04 Foto Afrati, Aristides Gionis, and
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• Yan05 Xifeng Yan, Hong Cheng, Jiawei Han, and
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• Wang06 Chao Wang and Srinivasan Parthasarathy.
  Summarizing itemset patterns using probabilistic
  models. KDD06.
• Jin08 Ruoming Jin, Muad Abu-Ata, Yang Xiang,
  and Ning Ruan. Effective and efficient itemset
  pattern summarization regression-based
  approaches. KDD08.
• Xiang08 Yang Xiang, Ruoming Jin, David Fuhy,
  and Feodor F. Dragan. Succinct Summarization of
  transactional databases an overlapped
  hyperrectangle scheme. KDD08.

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Related Work

• K-itemset approximation Afrati04.
• Difference
• their work is a special case of our work
• their work is expensive for exact description
• Our work use set cover and max-k cover methods.
• Restoring the frequency of frequent itemsets
  Yan05, Wang06, Jin08.
• Hyperrectangle covering problem Xiang08.