asymptotic enumeration of binary matrices with bounded row and column weights
DESCRIPTION
Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights. Farzad Parvaresh HP Labs, Palo Alto Joint work with Erik Ordentlich and Ron M. Roth Novermber 2011. Introducing the problem. Consider all the 2x2 binary matrices:. Introducing the problem. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/1.jpg)
Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights
Farzad ParvareshHP Labs, Palo Alto
Joint work with Erik Ordentlich and Ron M. RothNovermber 2011
![Page 2: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/2.jpg)
2
Introducing the problem
Consider all the 2x2 binary matrices:
0 0
0 0
1 0
0 0
0 1
0 0
0 0
0 1
0 0
1 0
1 1
0 0
1 0
0 1
1 0
1 0
1 1
0 1
1 0
1 1
0 1
0 1
0 1
1 0
0 1
1 1
1 1
1 0
0 0
1 1
1 1
1 1
![Page 3: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/3.jpg)
3
Introducing the problem
Consider all the 2x2 binary matrices:
0 0
0 0
1 0
0 0
0 1
0 0
0 0
0 1
0 0
1 0
1 0
0 1
0 1
1 0
7
How many binary matrices exist such that number of ones in each row or column is at most ?
![Page 4: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/4.jpg)
4
Memristors
Applications
![Page 5: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/5.jpg)
5
Memristors
Applications
![Page 6: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/6.jpg)
6
Memristors
Applications
Drives too much current
![Page 7: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/7.jpg)
7
Memristors
Applications
![Page 8: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/8.jpg)
8
Memristors
Applications
Drives too much current
![Page 9: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/9.jpg)
9
Memristors
Applications
Do not want too many memristors in any
row or column with low resistance state.
Drives too much current
Map binary data into matrices such
that number of ones in each row or
column is at most .
Each one in the matrix corresponds to
a low resistance state.
How many bits can be stored in an memory with this restriction?
![Page 10: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/10.jpg)
10
First attempt
Number of bounded row and column matrices
E. Ordentlich, and R.M. Roth, “Low complexity two-dimensional weight-constrained codes”, ISIT, August, 2011.
Efficient one-to-one mapping of bits to binary bounded row and column weight matrices.
Are there more bounded row and column weight matrices?
![Page 11: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/11.jpg)
17
4 0.982601
5 1.409983
6 1.136897
7 1.424295
8 1.195001
9 1.424725
10 1.227649
11 1.424964
12 1.249322
13 1.425054
14 1.265102
15 1.425093
Are there more bounded row and column matricesCount number of bounded row and column matrices for small .
For even :
For odd :
![Page 12: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/12.jpg)
18
Main result
Theorem:Let denote the standard normal cumulative distribution function, and
then,
Proof: In two parts. Show a lower bound and an upper bound for .
![Page 13: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/13.jpg)
19
Previous work
• B.D. McKay, I.M. Wanless, and N.C. Wormald, “Asymptotic enumeration of graphs with a given bound on the maximum degree,” Comb. Probab. Comput., 2002.
• E.C. Posner and R.J. McEliece, “The number of stable points of and infinite-range spin glass memory,” Jet Propulsion Laboratory, Tech. Rep., 1985.
Expected number of solutions to:
![Page 14: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/14.jpg)
20
Canfield , Greenhill and McKay (CGM08)
Lower bound
Theorem[CGM08]:
1 0 0 1 1 1 0
1 1 0 0 0 0 1
0 0 0 1 1 0 0
0 0 1 0 1 0 0
1 1 1 1 0 1 0
0 1 1 0 1 1 1
1 0 0 0 0 0 0
1 1 1 0 0 1 0
= Set of all binary matrices with row sum equal to column sum equal to .
![Page 15: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/15.jpg)
21
Lower bound
Enumerate bounded row and column matrices that satisfy assumptions of CGM theorem.Number of ones in each row or column is around the mean.
Set of bounded row and column matrices:
![Page 16: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/16.jpg)
22
Lower bound total number of ones in matrix
![Page 17: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/17.jpg)
23
Lower bound Enlarge the set .
![Page 18: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/18.jpg)
24
Lower bound
![Page 19: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/19.jpg)
25
Lower bound
where
Approximate summation by integration
![Page 20: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/20.jpg)
26
Lower bound
where
denotes the real n-dimensional cube ,
![Page 21: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/21.jpg)
27
Lower bound Looks like a multidimensional Gaussian distribution!
![Page 22: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/22.jpg)
28
Lower bound
Simulate Gaussians:
Use saddle point
![Page 23: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/23.jpg)
29
Upper bound
Set of bounded row and column matrices:
We have to enumerate rest of the matrices that do not satisfy assumptions of CGM theorem.
![Page 24: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/24.jpg)
30
Majorization Lemma
Upper bound
Lemma:For any with and and , respectively, majorizing and ,
![Page 25: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/25.jpg)
31
Majorization Lemma
Upper bound
Lemma:For any with and and , respectively, majorizing and ,
![Page 26: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/26.jpg)
32
Majorization Lemma
Upper bound
Lemma:For any with and and , respectively, majorizing and ,
For any and find and that are majorized by and , and satisfy the assumptions of CGM theorem. Then use the Lemma to upper bound .
![Page 27: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/27.jpg)
33
Upper bound
After choosing the appropriate anchor point for majorization and simplification we can show:
The Integral is equivalent to
We can compute the expectation using the same techniques as lower bound:
Same Gaussian as lower bound.
![Page 28: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/28.jpg)
34
Main result
Theorem:Let denote the standard normal cumulative distribution function, and
then,
Proof:
Lower bound:
Upper bound:
Set
![Page 29: Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights](https://reader036.vdocuments.mx/reader036/viewer/2022062518/56813fce550346895daaac8d/html5/thumbnails/29.jpg)
35
Future work
• Tighter enumeration of bounded row and column matrices.
• Efficient mapping of data to bounded row and column matrices that achieves optimal redundancy.