Overlapping Matrix Pattern Visualization: a Hypergraph Approach

1
Overlapping Matrix Pattern Visualization a
Hypergraph Approach

• Ruoming Jin
• Kent State University
• Joint with Yang Xiang, David Fuhry, and Feodor F.
  Dragan (KSU)

2
The Problem

• Given a set of discovered submatrices, how can we
  reorder the rows and columns of the data matrix
  to best display these submatrices and their
  relationship?

3
Motivation Overlapping Bicluster Visualization

• Gene expression profiles (row genes, columns
  conditions, matrix entry expression level)
• Biclustering homogeneous submatrices (genes ?
  conditions)
• Biclustering visualization problem GMM06, KG07

4
Motivation Transactional Data Visualization

• Shopping-basket data (rows transaction, columns
  item, binary matrix)
• Transactional data summarization using a set of
  dense submatrices CK07, WK06, XJFD08

Summarization Cost88521
5
Roadmap

• Problem Definition
• Visualization cost
• Hardness of the visualization problem
• Hypergraph ordering problem
• Minimum linear arrangement (MLA)
• Algorithm
• Leveraging MLA and local convergence
• Experimental Results

6
Submatrix Visualization Cost

• Given a display of the matrix (a fixed row-order
  and column-order), how can we measure the
  goodness of visualization of a submatrix?

t1,t2,t7,t8Xi1,i2,i8,i9
t1,t2,t7,t8Xi1,i2,i8,i9
i1
i2
i8
i3
i4
i5
i6
i7
i9
t1
t8
t2
t7
t3
t6
t4
t5
Why the second one is intuitively better than the
second one?
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Submatrix Visualization Cost
t1,t2,t7,t8Xi1,i2,i8,i9
t1,t2,t7,t8Xi1,i2,i8,i9
i1
i2
i8
i3
i4
i5
i6
i7
i9
t1
t8
t2
t7
t3
t6
t4
t5

• Area 8x8, 6x6,
  4x4, 4x4
• Perimeter 88, 66, 44,
  44
• Given a row order and a column order, the
  visualization cost of a submatrix is the sum of
• difference between its first and last row w.r.t.
  the row order
• difference between its first and last column
  w.r.t. the column order

8
Matrix Visualization Cost

• Given a row order and a column order, and a set
  of submatrices, the matrix visualization cost is
  the sum of these submatrices visualization cost.
  
• Matrix Optimal Visualization Problem
• Find the optimal row order and column order such
  that the matrix visualization cost is minimal.

9
Roadmap

• Problem Definition
• Visualization cost
• Hardness of the visualization problem
• Hypergraph ordering problem
• Minimal linear arrangement (MLA)
• Algorithm
• Leveraging MLA and Local convergence
• Experimental Results

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Hypergraph Ordering

• Hypergraph HG(V,X),
• V is the set of vertices
• Xx1,x2,, is the set of hyperedges, where each
  hyperedge is the set of vertices
• Hyperedge cost and Hypergraph cost
• Hypergraph Ordering Problem

Hyperedge 0,2,3,4 cost 4
0
1
2
3
4
5
6
Hypergraph cost16
Hyperedge 1,3,5 cost 4
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The Link between Matrix Visualization and
Hypergraph Ordering

• Relationship between matrix visualization cost
  and hypergraph cost
• Finding minimum visualization (or hypergraph)
  cost is NP-hard

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Hypergraph Ordering Problem is the Generalization
of MLA

• Graph cost w.r.t. a vertex order
• MLA (Minimal Linear Arrangement) Find an optimal
  vertex ordering to minimize graph cost

0
1
2
3
4
5
6
Graph cost2221143216
0
1
2
3
4
5
6
Graph cost2423421118
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Roadmap

• Problem Definition
• Visualization cost
• Hardness of the visualization problem
• Hypergraph ordering problem
• Minimal linear arrangement
• Algorithm
• Leveraging MLA and Local convergence
• Experimental Results

14
Basic Idea for Hypergraph Ordering

• Many existing work on solving MLA problem
  (heuristic or bounded-approximation)
• Instead of working from scratch for the
  hypergraph ordering problem, can we somehow
  leverage the MLA algorithms?
• The answer is YES!

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Basic Procedure

• Given the hypergraph HG(V,X), and starts with
  a random vertex order ?
• Step 1 Transforming the hypergraph HG into a
  graph G(V,E) based on the vertex order ?
• cost(HG, ?)cost(G, ?)
• Step 2 Run MLA algorithm for graph G to produce
  a new optimal vertex order ?
• cost(G, ?) ?cost(G, ?)
• Step 3 If the new order improve the hypergraph
  cost, cost(HG, ?) gt cost(HG, ?), then use ? as
  the new order (? ?), and repeat Step 1 and 2.
• cost(G, ?) ? cost(HG, ?)

Cost(HG, ? )cost(G, ? ) ?cost(G, ?) ?cost(HG,
?)
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(Step1) Transformation Hyperedge-gtPath
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Hyperedge costpath cost!
17
Step 1-gtStep 2
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Step 1 (Hypergraph-gtGraph) cost(G,
?)2221143216cost(HG, ?)
0
1
2
3
4
5
6
Step 2 (MLA) cost(G, ?)1221212313ltcost(
G, ?)
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Step 1-gtStep 2-gtStep 3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Step 1 (Hypergraph-gtGraph) cost(G, ?)cost(HG,
?)16
Step 2 (MinLA) cost(G, ?)13ltcost(G, ?)
0
1
2
3
4
5
6
0
2
3
5
6
1
4
With the new ordering, hyperedge cost?path cost!
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Step 1-gtStep 2-gtStep 3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Step 1 (Hypergraph-gtGraph) cost(G, ?)cost(HG,
?)16
Step 2 (MinLA) cost(G, ?)13ltcost(G, ?)
0
1
2
3
4
5
6
Step 3 cost(HG, ?)10ltcost(G, ?)13
Cost(HG, ? )cost(G, ? )gtcost(G, ?)gtcost(HG, ?)
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Run Iteratively and Local Convergence
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Other conversions of hyperedge

• Converting hyperedge to cycle
• Converting hyperedge to mulicycles

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Roadmap

• Problem Definition
• Visualization cost
• Hardness of the visualization problem
• Hypergraph ordering
• Algorithm
• Minimum linear arrangement (MLA)
• Leveraging MLA and local convergence
• Experimental Results

23
Visualization effects
24
Visualization effects (continued)
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Visualization effects (continued)
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Cost and running time
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Conclusion

• We found an interesting link from matrix
  visualization problem to a well-know graph
  theoretical problem the minimal linear
  arrangement (MLA) problem.
• Theoretically, we introduce a generalization of
  the MLA problem for the hypergraphs, and develop
  a novel local convergence algorithm
• Our method can be incorporated into an
  interactive visualization environment to allow
  users to focus on different parts of the data and
  patterns.

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Thanks!!