Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights

1. 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

2. Introducing the problem • Consider all the 2x2 binary matrices:

3. Introducing the problem • Consider all the 2x2 binary matrices: 7 How many binary matrices exist such that number of ones in each row or column is at most ?

4. Applications Memristors

5. Applications Memristors

6. Applications Memristors Drives too much current

7. Applications Memristors

8. Applications Memristors Drives too much current

9. Applications Memristors • Do not want too many memristors in any row or column with low resistance state. 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? Drives too much current

10. Number of bounded row and column matrices First attempt • 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?

11. Are there more bounded row and column matrices Count number of bounded row and column matrices for small . For even : For odd :

12. 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 .

13. 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:

14. Lower bound Canfield , Greenhill and McKay (CGM08) = Set of all binary matrices with row sum equal to column sum equal to . • Theorem[CGM08]:

15. Lower bound Set of bounded row and column matrices: Enumerate bounded row and column matrices that satisfy assumptions of CGM theorem. Number of ones in each row or column is around the mean.

16. Lower bound total number of ones in matrix

17. Lower bound Enlarge the set .

18. Lower bound

19. Lower bound Approximate summation by integration where

20. Lower bound where denotes the real n-dimensional cube ,

21. Lower bound Looks like a multidimensional Gaussian distribution!

22. Lower bound Use saddle point Simulate Gaussians:

23. 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.

24. Upper bound Majorization Lemma Lemma: For any with and and , respectively, majorizing and ,

25. Upper bound Majorization Lemma Lemma: For any with and and , respectively, majorizing and ,

26. Upper bound Majorization Lemma 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 .

27. Upper bound After choosing the appropriate anchor point for majorization and simplification we can show: The Integral is equivalent to Same Gaussian as lower bound. We can compute the expectation using the same techniques as lower bound:

28. Main result Theorem: Let denote the standard normal cumulative distribution function, and then, Proof: Set Lower bound: Upper bound:

29. Future work • Tighter enumeration of bounded row and column matrices. • Efficient mapping of data to bounded row and column matrices that achieves optimal redundancy.