This document presents an algorithm called MMBS (Mining Maximally Banded Submatrices) for uncovering multiple, possibly overlapping banded submatrices in binary data. It establishes a correspondence between banded structures and bi-clustering, and introduces an approach using formal concept analysis and concept lattice paths. The MMBS algorithm finds banded submatrices in binary data in three steps. Experimental results on synthetic and real-world data demonstrate the advantages of MMBS over previous approaches.

1. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Mining Maximally Banded Matrices in Binary
Data
Faris Alqadah
Raj Bhatnagar
Anil Jegga
University of Cincinnati
Cincinnati Children’s Hospital

2. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

3. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

4. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Banded Matrices in Data
Banded structures in
binary matrices have
A B C D E
natural interpretations
1 1 1 1 0 0
2 0 1 1 0 0 Bioinformatics (overlapping
3 0 0 1 0 0 roles of genes)
4 0 0 1 1 0 Paleontology (patterns of
5 0 0 0 1 1 species in space)
Social Networks
(community structures)

5. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Motivating Example
k-means multi-way EM bi-cluster subspace
doc1 1 0 1 0 1
doc2 0 1 0 1 0
doc3 0 0 0 0 1
doc4 0 0 0 1 1
doc5 0 0 1 0 1

6. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Motivating Example
k-means EM subspace bi-cluster multi-way
doc1 1 1 1 0 0
doc5 0 1 1 0 0
doc3 0 0 1 0 0
doc4 0 0 1 1 0
doc2 0 0 0 1 1

7. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bi-Clustering Problem
Banded sub-matrices are a form of bi-clusters
Bi-Clustering in binary data focuses on maximally
rectangles full of (or almost full) of 1s

8. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Related Work
Nestedness and segmented nestedness [6]
MBS algorithm [2]
Fix column permutations
Solve the consecutive ones problem
Only find a single band

9. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Contributions
1 Establish correspondence between banded structures and
bi-clustering in binary data
2 Introduce the novel MMBS algorithm to uncover multiple,
possibly overlapping banded sub-matrices
3 Empirical evaluation verifying advantage of MMBS over
previous approaches

10. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Contributions
1 Establish correspondence between banded structures and
bi-clustering in binary data
2 Introduce the novel MMBS algorithm to uncover multiple,
possibly overlapping banded sub-matrices
3 Empirical evaluation verifying advantage of MMBS over
previous approaches

11. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Contributions
1 Establish correspondence between banded structures and
bi-clustering in binary data
2 Introduce the novel MMBS algorithm to uncover multiple,
possibly overlapping banded sub-matrices
3 Empirical evaluation verifying advantage of MMBS over
previous approaches

12. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

13. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Basic Notation
Matrix K with row labels G and column labels M
Think of K as K = (G, M, I)
π permutation of G and τ permutation of M
Kπ
τ
g πi and mτj

14. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Basic Notation
Matrix K with row labels G and column labels M
Think of K as K = (G, M, I)
π permutation of G and τ permutation of M
Kπ
τ
g πi and mτj

15. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Fully Banded Matrix
Definition
A binary matrix K= (G, M, I) is fully banded if there exists a
permutation π of G and permutation τ of M such that (1) for
every row i in Kπ the entries with 1s occur in consecutive
τ
column indices {mi , mi + 1, . . . , mi⋆ } and (2) the values of
starting indices for 1s in successive rows (i and i + 1) satisfy
the conditions mi ≤ mi+1 and mi⋆ ≤ mi+1 . ⋆

16. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Relaxation of Fully Banded
Real data has noise
Subspaces may encompass banded structure
e(Kπ ): number of 1s or 0s that must be flipped to achieve
τ
banded structure
Maximal banded sub-matrix: no more rows or columns can
be added while still preserving bandedness

17. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Relaxation of Fully Banded
Real data has noise
Subspaces may encompass banded structure
e(Kπ ): number of 1s or 0s that must be flipped to achieve
τ
banded structure
Maximal banded sub-matrix: no more rows or columns can
be added while still preserving bandedness

18. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Problem Statement
Given binary matrix K and noise threshold ǫ find all
ˆ
sub-matrices K of K that are ǫ-banded and maximal.

19. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

20. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bi-clustering
Bi-clusters in binary data defined as Formal Concepts
For A ⊆ G, then A′ = {m ∈ M|gIm for all g ∈ A}.
B ⊆ M, we have B ′ = {g ∈ G|gImfor allm ∈ B}
Formal Concept: C = (A, B) such that A′ = B and B ′ = A

21. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bi-clustering
Bi-clusters in binary data defined as Formal Concepts
For A ⊆ G, then A′ = {m ∈ M|gIm for all g ∈ A}.
B ⊆ M, we have B ′ = {g ∈ G|gImfor allm ∈ B}
Formal Concept: C = (A, B) such that A′ = B and B ′ = A

22. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Formal Concepts
m1 m2 m3 m4
g1 0 1 0 1
g2 0 0 1 1
g3 0 0 0 1
g4 1 0 0 0
g5 1 1 1 0
g7 1 1 0 0
g6 0 0 1 0
Maximal rectangles of 1s
Maximal bicliques
Bi-clusters may be ordered by the subset superset
relationship and form a complete lattice
B(G, M, I) denotes the concept or bi-cluster lattice

23. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Formal Concepts
m1 m2 m3 m4
g1 0 1 0 1
g2 0 0 1 1
g3 0 0 0 1
g4 1 0 0 0
g5 1 1 1 0
g7 1 1 0 0
g6 0 0 1 0
Maximal rectangles of 1s
Maximal bicliques
Bi-clusters may be ordered by the subset superset
relationship and form a complete lattice
B(G, M, I) denotes the concept or bi-cluster lattice

24. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Splintering Bands
Trivially a bi-cluster is fully banded

25. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Splintering Bands
Trivially a bi-cluster is fully banded
A B C D E
1 1 1 1 0 0
2 0 1 1 0 0
3 0 0 1 0 0
4 0 0 1 1 0
5 0 0 0 1 1

26. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Splintering Bands
A B C D E
1 1 1 1 0 0
2 0 1 1 0 0
3 0 0 1 0 0
4 0 0 1 1 0
5 0 0 0 1 1
Intuitively, any fully banded matrix can be splintered exactly into
maximal rectangles of 1s or bi-clusters

27. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Ordering Splintered Bands
Let Kπ be fully banded
τ
Γ(g) is a mapping from row g to the bi-clusters g appears
in
The union of all Γ(g) can always be ordered
n-tuple of bi-clusters {C1 , . . . , Cn } having total ordering
{<π1 ,τ1 , . . . , <πn ,τn }
Define lexicographical order <π,τ on C1 × C2 × · · · × Cn .
Considering {C1 , . . . , Cn } in order completely specifies the
permutations π and τ

28. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Ordering Splintered Bands
Let Kπ be fully banded
τ
Γ(g) is a mapping from row g to the bi-clusters g appears
in
The union of all Γ(g) can always be ordered
n-tuple of bi-clusters {C1 , . . . , Cn } having total ordering
{<π1 ,τ1 , . . . , <πn ,τn }
Define lexicographical order <π,τ on C1 × C2 × · · · × Cn .
Considering {C1 , . . . , Cn } in order completely specifies the
permutations π and τ

29. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Ordering Splintered Bands
Let Kπ be fully banded
τ
Γ(g) is a mapping from row g to the bi-clusters g appears
in
The union of all Γ(g) can always be ordered
n-tuple of bi-clusters {C1 , . . . , Cn } having total ordering
{<π1 ,τ1 , . . . , <πn ,τn }
Define lexicographical order <π,τ on C1 × C2 × · · · × Cn .
Considering {C1 , . . . , Cn } in order completely specifies the
permutations π and τ

30. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Ordering Splintered Bands
Let Kπ be fully banded
τ
Γ(g) is a mapping from row g to the bi-clusters g appears
in
The union of all Γ(g) can always be ordered
n-tuple of bi-clusters {C1 , . . . , Cn } having total ordering
{<π1 ,τ1 , . . . , <πn ,τn }
Define lexicographical order <π,τ on C1 × C2 × · · · × Cn .
Considering {C1 , . . . , Cn } in order completely specifies the
permutations π and τ

31. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bands as Sequences of Concepts
Proposition
Given a context K, if permutations π and τ exist such that Kπ is
τ
fully banded then there exists a sequence of bi-clusters
C1 = (A1 , B1 ), . . . , Cn = (An , Bn ) s.t.
π = A1 , A2 A1 , . . . , An An−1
τ = B1 B2 , . . . , Bn−1 Bn , Bn

32. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
An Example
A B C D E
1 1 1 1 0 0
2 0 1 1 0 0
3 0 0 1 0 0
4 0 0 1 1 0
5 0 0 0 1 1
g Γ(g)
1 (1, ABC), (12, BC), (1234, C)
2 (12, BC), (1234, C)
3 (1234, C)
4 (4, CD), (45, D)
5 (5, DE ), (45, D)
F(Kπ )
τ
(1, ABC) < (12, BC) < (1234, C) < (4, CD) < (45, D) < (5, DE )
π = 1, 12 1, . . . , 5 45
= {1, 2, 3, 4, 5}
τ = ABC BC, . . . , D DE , DE
= {A, B, C, D, E }

33. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

34. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Paths in the lattice
Represent B(G, M, I) as G = (V , E )
Edge set define as: C1 , C2 ∈ E ↔ C1 ≺ C2 ∨ C2 ≺ C1
Concept lattice order enforces: Ai+1 ⊆ Ai and Bi ⊆ Bi+1 if
Ci ≺ Ci+1
Dual: Ai ⊆ Ai+1 and Bi+1 ⊆ Bi if Ci ≻ Ci+1

35. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Paths in the lattice
Represent B(G, M, I) as G = (V , E )
Edge set define as: C1 , C2 ∈ E ↔ C1 ≺ C2 ∨ C2 ≺ C1
Concept lattice order enforces: Ai+1 ⊆ Ai and Bi ⊆ Bi+1 if
Ci ≺ Ci+1
Dual: Ai ⊆ Ai+1 and Bi+1 ⊆ Bi if Ci ≻ Ci+1

36. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Construct Partial Bands Via Paths
s
1,2,3,4,5
C
s
1,2,3,4 Ds
B,C 4,5
s
1,2 C,D D,E
s s
4 5
A,B,C
s
1
A,B,C,D,E
s

37. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bound on the error
Key Fact
Each individual edge in a path P is guaranteed to produce a
banded structure

38. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Bound on the error
Proposition

 0
 if n ≤ 1
 ′
 e(P n−1 ) +

 |a ∩ B| if Cn+1 ≻ Cn
e(Pn ) ≤ ˆ
a∈A
|b ′ ∩ A|

n−1 ) + if Cn+1 ≺ Cn
 e(P



 ˆ
b∈B

39. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

40. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Overview
Weigh edges of concept lattice with upper bound of error
Bad news: weights change depending on path
Good news: Error is monotonic along a path, so pruning
with backtracking works!
Three steps:
1 Compute G
2 Search paths of G
3 Determine top bands

41. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Overview
Weigh edges of concept lattice with upper bound of error
Bad news: weights change depending on path
Good news: Error is monotonic along a path, so pruning
with backtracking works!
Three steps:
1 Compute G
2 Search paths of G
3 Determine top bands

42. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Compute G
Many existing algorithms [1, 5, 3, 4, 7]
Incremental vs. non-incremental
Assume availability of G

43. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Search Paths
Potentially exponential number of paths
Any bi-cluster is a valid starting point...but initiate with
upper neighbors of null-element
At each edge add concept to path utilizing previous
procedure
Utilize backtracking, mark previously visited edges

44. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Search Paths
Potentially exponential number of paths
Any bi-cluster is a valid starting point...but initiate with
upper neighbors of null-element
At each edge add concept to path utilizing previous
procedure
Utilize backtracking, mark previously visited edges

45. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Top Bands
Allow user to specify : minRows, minCols, maxOvlp
Quality measure: q(P) = |r (P)| ∗ |c(P)| − w ∗ e(P)
If two bands exceed maxOvlp select the higher quality one

46. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Analysis and Improvements
Running time: O(|U| × |E | × max{X , Y }|)
|U| : size of initial concepts
X , Y : largest symmetric difference between neighboring
concepts
Speed up by reducing size of |U|
Perform simple clustering of U based on maxOvlp
parameter
Good experimental results with this speed up.

47. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Analysis and Improvements
Running time: O(|U| × |E | × max{X , Y }|)
|U| : size of initial concepts
X , Y : largest symmetric difference between neighboring
concepts
Speed up by reducing size of |U|
Perform simple clustering of U based on maxOvlp
parameter
Good experimental results with this speed up.

48. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

49. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Setup
Single band and segmented bands planted in synthetic
data
All experiments:
w =1
maxOvlp = 0.1
minRows = 5
minCols = 5
ǫ = 99

50. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Results
50 20
100 40
150 60
200 80
250 100
300 120
350 140
400 160
450 180
500 200
50 100 150 200 250 300 350 400 450 500 20 40 60 80 100 120 140 160 180 200
Planted Bands
50
100
150
200
250
300
50 100 150 200 250 300

51. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Results
Dataset name Dataset Size p Num. Planted bands Algorithm Quality top ranked Num. bands mined
MMBS 3590 6
MMBS_Fast 3406 4
SynBand100_001 100 × 100 0.01 1
MBS_BD 2507 1
MBS_SD 438 1
MMBS 2278 9
MMBS_Fast 1503 8
SynBand100_005 100 × 100 0.05 1
MBS_BD 1050 1
MBS_SD 1201 1
MMBS 8918 7
MMBS_Fast 8261 6
SynBand500_001 500 × 500 0.01 1
MBS_BD 2822 1
MBS_SD 2145 1
MMBS 3367 2
MMBS_Fast 3367 2
SynMultiBand100_001 100 × 100 0.01 2
MBS 4101 1
MBS_SD 4045 1
MMBS 4054 2
MMBS_Fast 3933 2
SynMultiBand100_001 100 × 100 0.05 2
MBS_BD 3910 1
MBS_SD 3736 1
MMBS 28242 8
MMBS_Fast 21346 5
SynMultiBand500_001 500 × 500 0.01 2
MBS_BD 17498 1
MBS_SD 430 1
MMBS 3311 17
MMBS_Fast 3220 14
SynRandom100_005 100 × 100 0.05 unknown
MBS_BD 2801 1
MBS_SD 1949 1
MMBS 18635 73
MMBS_Fast 16163 64
SynRandom500_001 500 × 500 0.01 unknown
MBS_BD 16771 1
MBS_SD 5229 1

52. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Outline
1 Introduction
Motivation
2 Problem Definition
Preliminaries
3 Bandedness and Bi-Clustering
Formal Concept Analysis
Concept Lattice Paths
4 MMBS Algorithm
Three Steps
5 Experimental Results
Synthetic Data
Real-World Data
6 Conclusion

53. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Dataset Size Sparsity Algorithm Quality top ranked Num. bands mined
MMBS 6665 56
MMBS_Fast 6665 43
Genes_Phenotypes 1910 × 3965 0.008
MBS_BD 5204 1
MBS_SD 3578 1
MMBS 6423 18
MMBS_Fast 6423 13
Genes_Drugs 1608 × 49 0.042
MBS_BD 5346 1
MBS_SD 3047 1
MMBS 72906 42
MMBS_Fast 61410 31
NewsGroups_Mideast_Religion 2000 × 890 0.003
MBS_BD 59781 1
MBS_SD 58713 1
MMBS 93368 5
MMBS_Fast 93368 5
NewsGroups_AllPC 5000 × 2805 0.0001
MBS_BD 89106 1
MBS_SD 74125 1

54. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
early eyelid 1
opening
eyelids open at birth 2
abnormal timing of
postnatal eyelid opening 3
abnormal eyelid
4
morphology
abnormal eye 5
morphology
abnormal homeostasis 6
abnormal ear physiology 7
abnormal hearing
8
physiology
abnormal brainstem audiotry 9
evokedpotential
deafness 10
50 100 150 200 250 300 350 400
Genes_Phenotypes

55. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
1
2
3
4
5
6
7
100 200 300 400 500 600 700 800 900
Genes_Drugs

56. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
100
200
300
400
500
600
700
800
10 20 30 40 50 60 70 80
MideastReligion_SubjectLines

57. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
100
200
300
400
500
600
700
800
900
1000
10 20 30 40 50 60 70 80
AllPC_SubjectLines

58. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion

59. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Performance
4 5
10 10
3 MMBS_fast MMBS_fast
10
MMBS MMBS
4
CPU Time (seconds)
CPU Time (seconds)
MBS 10 MBS
2
10
3
10
1
10
0 2
10 10
0 20 40 60 80 100 0 20 40 60 80 100
epsilon epsilon
5 2
10 10
MMBS_fast
MMBS
MBS
4
10
1
CPU Time (seconds)
CPU Time (seconds) 10
3
10
MMBS_fast
MMBS
MBS 0
10
2
10
1 −1
10 10
0 20 40 60 80 100 0 20 40 60 80 100
epsilon epsilon

60. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Conclusion
Explored connection between bi-clustering and banded
structures in matrices
Banded sub-matrices correspond to paths in the bi-cluster
lattice
MMBS algorithm is based on this correspondence and
ability to bound error
Future work: More efficient search methodologies,
stronger bounds on error
Future work: Quantitative measures of bandedness,
different types of bands desirable in different applications

61. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
Conclusion
Explored connection between bi-clustering and banded
structures in matrices
Banded sub-matrices correspond to paths in the bi-cluster
lattice
MMBS algorithm is based on this correspondence and
ability to bound error
Future work: More efficient search methodologies,
stronger bounds on error
Future work: Quantitative measures of bandedness,
different types of bands desirable in different applications

62. Introduction Problem Definition Bandedness and Bi-Clustering MMBS Algorithm Experimental Results Conclusion
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