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 Cost=8+8+5=21

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 t1t1 t2t2 t3t3 t6t6 t4t4 t5t5 t7t7 t8t8 i1i1 i2i2 i3i3 i4i4 i5i5 i6i6 i7i7 i8i8 i9i9 t1t1 t2t2 t3t3 t6t6 t4t4 t5t5 t7t7 t8t8 i1i1 i2i2 i3i3 i4i4 i5i5 i6i6 i7i7 i8i8 i9i9 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,t8}X{i1,i2,i8,i9} Why the second one is intuitively better than the second one?

7. Submatrix Visualization Cost t1t1 t2t2 t3t3 t6t6 t4t4 t5t5 t7t7 t8t8 i1i1 i2i2 i3i3 i4i4 i5i5 i6i6 i7i7 i8i8 i9i9 t1t1 t2t2 t3t3 t6t6 t4t4 t5t5 t7t7 t8t8 i1i1 i2i2 i3i3 i4i4 i5i5 i6i6 i7i7 i8i8 i9i9 Area: 8x8, 6x6, 4x4, 4x4 Perimeter: 8+8, 6+6, 4+4, 4+4 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 {t1,t2,t7,t8}X{i1,i2,i8,i9}

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

10. Hypergraph Ordering Hypergraph HG=(V,X), –V is the set of vertices –X={x1,x2,…,} is the set of hyperedges, where each hyperedge is the set of vertices Hyperedge cost and Hypergraph cost Hypergraph Ordering Problem 0123456 Hyperedge {0,2,3,4} cost = 4 Hyperedge {1,3,5} cost = 4 Hypergraph cost=16

11. 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 t1t1 t2t2 t3t3 t6t6 t4t4 t5t5 t7t7 t8t8 i1i1 i2i2 i3i3 i4i4 i5i5 i6i6 i7i7 i8i8 i9i9 i1i1 i2i2 i3i3 i7i7 i8i8 i9i9 t1t1 t2t2 t3t3 t6t6 t7t7 t8t8 i4i4 i5i5 i6i6 t5t5 t4t4 HG 1 HG 2

12. 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 0123456 0123456 Graph cost=2+2+2*1+1+4+3+2=16 Graph cost=2+4+2*3+4+2+1+1=18

13. 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!

15. 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,  ) > 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,  ’)

16. (Step1) Transformation: Hyperedge->Path 0123456 0123456 0123456 Hyperedge cost=path cost!

17. Step 1->Step 2 0123456 0 1 23 4 56 Step 1 (Hypergraph->Graph): cost(G,  )=2+2+2*1+1+4+3+2=16=cost(HG,  ) Step 2 (MLA): cost(G,  ’)=1+2+2*1+2+1+2+3=13<cost(G,  ) 0123456

18. Step 1->Step 2->Step 3 0123456 0 1 23 4 56 0 1 23 4 56 0 1 23 4 56 Step 1 (Hypergraph->Graph): cost(G,  )=cost(HG,  )=16 Step 2 (MinLA): cost(G,  ’)=13<cost(G,  ) With the new ordering, hyperedge cost  path cost!

19. Step 1->Step 2->Step 3 0123456 0 1 23 4 56 0 1 23 4 56 Step 1 (Hypergraph->Graph): cost(G,  )=cost(HG,  )=16 Step 2 (MinLA): cost(G,  ’)=13<cost(G,  ) Step 3: cost(HG,  ’)=10<cost(G,  ’)=13 0123456 Cost(HG,  )=cost(G,  )>cost(G,  ’)>cost(HG,  ’)

20. Run Iteratively and Local Convergence

21. Other conversions of hyperedge Converting hyperedge to cycle Converting hyperedge to mulicycles

22. 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)

26. Cost and running time

27. 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.

28. Thanks!!