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Benjamin Doerr
Max-Planck-Institut für Informatik Saarbrücken
Component-by-Component Construction of Low-Discrepancy Point Sets
joint work with Michael Gnewuch (Kiel), Peter Kritzer (Salzburg), and Friedrich Pillichshammer (Linz)
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Benjamin Doerr
Reminder: Star Discrepancy Definition:
– s ∈ NN “dimension” (Austrian notation)
– P = {p0, p1, ..., pN-1} multi-set of N points in [0,1)s
– Discrepancy function: For x ∈ [0,1]s, Δ(x,P) := λ([0,x)) – #{i : pi ∈ [0,x)} / N “(how many points should be in [0,x) – how many
actually are there) normalized by N”
– Star discrepancy: D*(P) := sup{ |Δ(x,P)| : x ∈ [0,1]s}
Measure of how evenly P is distributed in [0,1)s
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Benjamin Doerr
Reminder: Star Discrepancy Application: Numerical Integration
– Given f : [0,1]s → R
– Compute/estimate ∫[0,1]s f(x) dx !
Hope: ∫[0,1]s f(x) dx ≈ (1/N) ∑i f(pi)
Koksma-Hlawka inequality:
| ∫[0,1]s f(x) dx - (1/N) ∑i f(pi) | ≤ V(f)
D*(P)– V(f): Variation in the sense of Hardy and Krause
Low star discrepancy = good integration
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Benjamin Doerr
Reminder: Star Discrepancy How good?
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Benjamin Doerr
Reminder: Star Discrepancy Very good! There are N-point sets P with
D*(P) ≤ Cs (log N)s-1 / N
“More points = drastically reduced integration error” Really?
Note: All constants ‘C’ may be different. They never depend on N. If they depend on s, I call them ‘Cs’.
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Benjamin Doerr
Reminder: Star Discrepancy Very good! – There are N-point sets P with
D*(P) ≤ Cs (log N)s-1 / N
“More points = drastically reduced integration error” Really? No!
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Benjamin Doerr
Problem: Only good for many points!
– Increasing for N ≤ e10 (more points = worse integration?)– ≥ 1 (trivial bound), if N ≤ 1010
– ≥ D*(random point set), if N ≤ 102∙10
Need for “small” low-discrepancy point sets!
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Benjamin Doerr
Motivation, Outline
Previous slides: – O((log N)s-1/N) bounds only good for many points
many: at least exponential in dimension.
– Otherwise: Random points have better guarantees.
Plan for this talk: – Be inspired by random points– ...and use this to construct better point sets
Note: Almost all ugly details omitted in this talk!– For many technicalities, the sharpest bounds and
more results see the full paper (MCMAppl, to appear).
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Benjamin Doerr
Previous Work (1)
Heinrich, Novak, Wasilkowski, Woźniakowski (Acta Arith., 2002): – There are point sets with D*(P) ≤ C (s/N)1/2
randomized construction Talagrand inequality
– Good bounds for N polynomial in s– Existential result only, implicit constants not
known
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Benjamin Doerr
Previous Work (2)
We build on previous results by D., Gnewuch, Srivastav (JoC ‘05, MCQMC ’06): – D*(P) ≤ C (s/N)1/2 (log N)1/2, C small– via randomized rounding:
discrepancy guarantee holds with high probability
– derandomization: deterministic construction of P in run-time (CN)s+2
computes the exact star discrepancy on the way– wait for Michael’s talk (next talk) to see how difficult
computing the star discrepancy can be...
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”)
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N
vol(B)
3.0625
1.96875
1.96875
1.2656..
1.09375
0.875
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N
vol(B) Round these numbers to integers yB ...
3.0625
1.96875
1.96875
1.2656..
1.09375
0.875
3 2
1
1
1
2
0.7031..
0.5625
1
0 1
1
1
1 1
0
0
0
0.31..
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)
Round these numbers to integers yB such that for all aligned corners C,yC := ∑B⊆CyB is close to xC := ∑B⊆CxB.
3.0625
1.96875
1.96875
1.2656..
1.09375
0.875
3 2
1
1
1
2
0.7031..
0.5625
1
0 1
1
1
1 1
0
0
0
0.31..
xC=12.25 yC=12
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Benjamin Doerr
Rounding Approach
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)
Round these numbers to integers yB such that for all aligned corners C,yC := ∑B⊆CyB is close to xC := ∑B⊆CxB. Then put yB points in B arbitrarily.
3.0625
1.96875
1.96875
1.2656..
1.09375
0.875
3 2
1
1
1
2
0.7031..
0.5625
1
0 1
1
1
1 1
0
0
0
0.31..
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Benjamin Doerr
Classical Rounding Theory Let x1, ..., xn be numbers, N:=||x||1 and I1, ..., Im ⊆ {1, ...,n}.
Randomized Rounding: – If xi is an integer, yi := xi
– If not, then yi := ⌈xi⌉ with probability equal to the fractional part of xi and yi := ⌊xi⌋ otherwise
Theorem: With probability 1-ε, we have for all 1 ≤ k ≤ m
(*) | ∑i ∈ Ik yi - ∑i ∈ Ik xi | ≤ (0.5 N log(2m/ε))1/2
Derandomization: A rounding (yi) satisfying (*) with ε=1 can be computed deterministically in time O(mn).
Experiment: Derandomization yields smaller rounding errors.
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Benjamin Doerr
Rounding Approach (continued)
Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)
Round these numbers to integers yB such that for all aligned corners C,| ∑B⊆CyB - ∑B⊆CxB | ≤ (0.5 N log(2 #boxes))1/2 ...
3.0625
1.96875
1.96875
1.2656..
1.09375
0.875
3 2
1
1
1
2
0.7031..
0.5625
1
0 1
1
1
1 1
0
0
0
0.31..
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Benjamin Doerr
New Result: The same can be done in a component-by-component
way: – Compoment-by-compoment: Given an (s-1)-dimensional low-
discrepancy point set, add an sth component to each point.– Adjust the randomized-rounding approach accordingly.
Advantage: – Fewer variables to be rounded in each iteration.– Total run-time (over all s iterations) roughly N(s+3)/2 instead of Ns+2.
Surprise: Discrepancy increases only by a factor of s.– Roughly C s3/2 N-1/2 log(N)1/2 instead of C s1/2 N-1/2 log(N)1/2
That’s OK, because we can now compute N2 points in roughly the same time as needed for N points before.
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Benjamin Doerr
Summary and Conclusion Result: Component-by-Component derandomized
construction is much faster and yields only slightly higher discrepancies compared to “all at once”.
Outlook: Could also be useful if components are of different importance. E.g., do the expensive derandomization only for few components.
Open problem: Come up with something really efficient.... (instead of NCs).
Merci/Thanks!