1 / 19

Unbiased Matrix Rounding

Unbiased Matrix Rounding. Tobias Friedrich (joint work with B.Doerr, C.Klein, R.Osbild) Max-Planck-Institut für Informatik, Saarbrücken, Germany. Statistics of a rural village. Age of owner. #Animals. Animal. Statistics of a rural village. Age of owner. #Animals. Animal.

hadar
Télécharger la présentation

Unbiased Matrix Rounding

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Unbiased Matrix Rounding Tobias Friedrich (joint work with B.Doerr, C.Klein, R.Osbild) Max-Planck-Institut für Informatik, Saarbrücken, Germany

  2. Statistics of a rural village Age of owner #Animals Animal

  3. Statistics of a rural village Age of owner #Animals Animal

  4. Statistics of a rural village Age of owner #Animals Animal

  5. Statistics of a rural village Rounding to multiples of ten

  6. Statistics of a rural village Rounding to multiples of ten

  7. Statistics of a rural village Rounding to multiples of ten

  8. Totals not preserved! Statistics of a rural village Rounding to multiples of ten

  9. Statistics of a rural village Controlled Rounding

  10. Basic Problem: “Controlled Rounding” • Round a [0,1] matrix to a {0,1} matrix s.t. • rounding errors in row totals are less than one • rounding errors in column totals are less than one • rounding error in grand total is less than one • Classical result: All matrices have controlled roundings • Bacharach ’66, Cox&Ernst ’82: Statistics • Baranyai ’75: Hypergraph coloring

  11. Extension 1:Unbiased Controlled Rounding • “Unbiased” = Randomized: • Pr(yij = 1) = xij, • Pr(yij = 0) = 1 – xij. • Result: Unbiased controlled roundings exist • Cox ’87 • Follows also from GKPS (FOCS ‘02)

  12. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ b b b b ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ P P P P ( ( ( ( ) ) ) ) 8 8 8 8 8 8 8 8 b b 8 8 b b j i j i 1 1 2 2 ¡ ¡ ¡ ¡ < < < < : : : : a a x x x x y y y y i i j j i i j j i i j j i i j j ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ j i j i 1 1 a a = = = = Extension 2:Strongly Controlled Rounding • Small errors in initial intervals of rows/columns: • Observation: Errors less than two in arbitrary intervals. • Allows reliable range queries. • # of pigs owned by 20-59 year olds

  13. Our Result • Unbiased strongly controlled roundings exist • Unbiased strongly controlled roundings exist, i.e., one can round a real matrix to an integer matrix s.t. • rounding errors in row/column/grand totals are less than one • rounding errors in initial row/column intervals are less than one • rounding is unbiased/randomized • It can be generated in time • O((mn)2) • O(mn l), if numbers have binary length at most l • O(mn b2), if numbers are multiples of 1/b

  14. Alternating Cycle Trick • Simplifying assumptions: • Row/column sums integral

  15. 0 1 0 9 0 0 6 4 0 7 0 2 0 9 0 6 ¡ ¡ + + " " " " : : : : : : : 0 0 0 0 3 4 0 1 0 5 0 3 0 2 B C : : : : : : : X = B C 0 9 0 4 0 7 0 2 0 8 @ A : : : : : 0 2 0 8 0 6 0 6 0 4 : : : : : Alternating Cycle Trick • Choose an alternating cycle (of non-zeroes) • Compute possible modifications: εmin= -0.1, εmax= 0.3 • (a) Non-randomized: Modify with any ε [here: ε = εmax](b) Unbiased: Suitable random choice  At least one entry becomes 0 or 1 Time complexity: One iteration O(mn), total O((mn)2).

  16. Fast Alternating Cycle Trick • Additional assumption: • All numbers have finite binary expansion

  17. 0 0 0 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 ¡ ¡ + + + ¡ ¡ + " " " " " " " " : : : : : : : : : : : : : : : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 B B B C C C : : : : : : : : : : : : : : : X X X = = = B B B C C C 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 @ @ @ A A A : : : : : : : : : : : : : : : 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 : : : : : : : : : : : : : : : Fast Alternating Cycle Trick • Choose an alternating cycle (with 1s in last digit) • Allow only modifications ε1= -0.001, ε2= 0.001 • (a) Non-randomized: Modify with either value(b) Unbiased: Pick each value with 50% chance Bit-length in whole cycle reduces Time complexity: Amortized O(mn) to reduce by 1 bit, Total O(mn l) with l denoting bit length

  18. 0 0 0 1 1 1 = = = = = = = = = = = = 1 2 3 4 1 1 2 5 5 5 2 3 3 5 5 5 4 4 4 5 5 5 2 2 2 5 5 5 0 0 0 f g 0 1 + ¡ ¡ + " " " " 5 5 5 5 ; ; ; ; ; = = = = = = = = 3 3 2 5 5 5 4 4 1 5 5 0 0 0 3 3 3 5 5 5 0 0 0 B B B C C C X X X = = = B B B C C C = = = = = = = = = = = = = = = 2 2 2 5 5 5 2 2 2 5 5 5 2 2 2 5 5 5 2 2 2 5 5 5 2 2 2 5 5 5 @ @ @ A A A = = = = = = = = = = = = = = = 4 4 4 5 5 5 1 1 1 5 5 5 4 4 4 5 5 5 3 3 3 5 5 5 3 3 3 5 5 5 Multiples of 1/b (here b=5) • Choose an alternating cycle (of non-zeros) • Allow only modifications ε1= -1/b, ε2= +1/b • (a) Non-randomized: Derandomization(b) Unbiased: Pick each value with 50% chance Entries perform random walk in Time complexity: Amortized O(b2) to round one entry, Total O(mn b2)

  19. Summary • Unbiased strongly controlled roundings: • “randomized roundings” • rounding errors in initial intervals of rows/column: < 1 • Result: Can be generated in time • O((mn)2) • O(mn l), if numbers have binary length at most l • O(mn b2), if numbers are multiples of 1/b Have a good weekend!

More Related