Efficient Random-Walk Methods forApproximating Polytope Volume
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This document discusses efficient randomized methods for approximating the volume of polytopes, addressing the challenges and limitations of traditional algorithms. It presents the use of random walks, including techniques like hit-and-run and multiphase Monte Carlo methods, to generate random points within polytopes and estimate their volume. Experimental results demonstrate the effectiveness of these randomized approaches, achieving high accuracy in high-dimensional settings while highlighting the comparative limitations of existing exact algorithms.
![The volume computation problem
Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm
Output: Volume of P
#-P hard for vertex and for halfspace repres. [DyerFrieze’88]](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-2-320.jpg)
![The volume computation problem
Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm
Output: Volume of P
#-P hard for vertex and for halfspace repres. [DyerFrieze’88]
open if both vertex & halfspace representation is available](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-3-320.jpg)
![The volume computation problem
Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm
Output: Volume of P
#-P hard for vertex and for halfspace repres. [DyerFrieze’88]
open if both vertex & halfspace representation is available
no deterministic poly-time algorithm can compute the volume
with less than exponential relative error [Elekes’86]](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-4-320.jpg)
![The volume computation problem
Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm
Output: Volume of P
#-P hard for vertex and for halfspace repres. [DyerFrieze’88]
open if both vertex & halfspace representation is available
no deterministic poly-time algorithm can compute the volume
with less than exponential relative error [Elekes’86]
randomized poly-time algorithm approximates the volume of a
convex body with high probability and arbitrarily small relative
error [DyerFriezeKannan’91] O∗(d23) → O∗(d4) [LovVemp’04]](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-5-320.jpg)
![Implementations
Exact: VINCI [Bueler et al’00], Latte [deLoera et al], Qhull
[Barber et al], LRS [Avis], Normaliz [Bruns et al]
triangulation, sign decomposition methods
cannot compute in high dimensions (e.g. > 20)
Randomized:
[Lov`aszDe`ak’12] cannot compute in > 10 dimensions
Matlab code by Cousins & Vempala based on [LovVemp’04]
Ours: based on [DyerFriezeKannan’91],. . . ,
[KannanLov`aszSimon.’97]](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-6-320.jpg)
![Random Directions Hit-and-Run (RDHR)
P
x Input: point x ∈ P and polytope P ⊂ Rd
Output: a new point in P
1. line through x, uniform on B(x, 1)
2. set x to be a uniform disrtibuted
point on P ∩
Iterate this for W steps and return x.
x is unif. random distrib. in P after W = O∗(d3) steps, where
O∗(·) hides log factors [LovaszVempala’06]
to generate many random points iterate this procedure](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-11-320.jpg)
![Complexity [KannanLS’97]
Assuming B(c, 1) ⊆ P ⊆ B(c, ρ), the volume algorithm returns an
estimation of vol(P), which lies between (1 − )vol(P) and
(1 + )vol(P) with probability ≥ 3/4, making
O∗
(d5
)
oracle calls, where ρ is the radius of a bounding ball for P.
Isotropic sandwitching: ρ = O∗(
√
d) and ball walk.
Runtime
generates d log(ρ) balls
generate N = 400 −2d log d random points in each ball ∩ P
each point is computed after O∗(d3) random walk steps](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-15-320.jpg)
![Experimental results
approximate the volume of a series of polytopes (cubes,
random, cross, birkhoff) up to dimension 100 in less than
2 hours with mean approximation error at most 1%
randomized vs. exact software (VINCI, etc.): there is a
threshold dimension (d < 15) for which exact software halts
computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs.
prob. 3/4 [KLS’97]) up to d = 100](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-23-320.jpg)
![Experimental results
approximate the volume of a series of polytopes (cubes,
random, cross, birkhoff) up to dimension 100 in less than
2 hours with mean approximation error at most 1%
randomized vs. exact software (VINCI, etc.): there is a
threshold dimension (d < 15) for which exact software halts
computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs.
prob. 3/4 [KLS’97]) up to d = 100
CDHR faster and more accurate than RDHR](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-24-320.jpg)
![Experimental results
approximate the volume of a series of polytopes (cubes,
random, cross, birkhoff) up to dimension 100 in less than
2 hours with mean approximation error at most 1%
randomized vs. exact software (VINCI, etc.): there is a
threshold dimension (d < 15) for which exact software halts
computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs.
prob. 3/4 [KLS’97]) up to d = 100
CDHR faster and more accurate than RDHR
Compute the volume of Birkhoff polytopes B11, . . . , B15 in few
hrs whereas exact methods have only computed that of B10 by
specialized software in ∼ 1 year of parallel computation](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-25-320.jpg)
![Volumes of Birkhoff polytopes
n d estimation
asymptotic estimation
asymptotic
exact
exact
asymptotic[CanfieldMcKay09]
3 4 1.12E+000 1.41E+000 0.7932847 1.13E+000 0.7973923
4 9 6.79E-002 7.61E-002 0.8919349 6.21E-002 0.8159296
5 16 1.41E-004 1.69E-004 0.8344350 1.41E-004 0.8341903
6 25 7.41E-009 8.62E-009 0.8598669 7.35E-009 0.8527922
7 36 5.67E-015 6.51E-015 0.8713891 5.64E-015 0.8665047
8 49 4.39E-023 5.03E-023 0.8729497 4.42E-023 0.8778632
9 64 2.62E-033 2.93E-033 0.8960767 2.60E-033 0.8874117
10 81 8.14E-046 9.81E-046 0.8305162 8.78E-046 0.8955491
11 100 1.40E-060 1.49E-060 0.9342584 ??? ???
12 121 7.85E-078 8.38E-078 0.9370513 ??? ???
13 144 1.33E-097 1.43E-097 0.9331517 ??? ???
14 169 5.96E-120 6.24E-120 0.9550089 ??? ???
15 196 5.70E-145 5.94E-145 0.9593786 ??? ???](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-26-320.jpg)
![Ongoing work
1. random walks for polytopes described by optimization oracles
e.g. resultant polytopes [Emiris,F,Konaxis,Penaranda SoCG’12]
2. use approximate oracles (utilizing approximate NN)
3. volume of more general convex bodies
e.g. spectahedra](https://image.slidesharecdn.com/vissarionvolumesocg14slides-200124130419/85/Efficient-Random-Walk-Methods-forApproximating-Polytope-Volume-27-320.jpg)
Efficient Random-Walk Methods forApproximating Polytope Volume
- 1. Efficient Random-Walk Methods for Approximating Polytope Volume Vissarion Fisikopoulos Joint work with I.Z. Emiris Dept. of Informatics & Telecommunications, University of Athens Visiting Scholar at NIMS, South Korea SoCG 10 Jun. 2014
- 2. The volume computation problem Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm Output: Volume of P #-P hard for vertex and for halfspace repres. [DyerFrieze’88]
- 3. The volume computation problem Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm Output: Volume of P #-P hard for vertex and for halfspace repres. [DyerFrieze’88] open if both vertex & halfspace representation is available
- 4. The volume computation problem Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm Output: Volume of P #-P hard for vertex and for halfspace repres. [DyerFrieze’88] open if both vertex & halfspace representation is available no deterministic poly-time algorithm can compute the volume with less than exponential relative error [Elekes’86]
- 5. The volume computation problem Input: Polytope P := {x ∈ Rd | Ax ≤ b} A ∈ Rm×d, b ∈ Rm Output: Volume of P #-P hard for vertex and for halfspace repres. [DyerFrieze’88] open if both vertex & halfspace representation is available no deterministic poly-time algorithm can compute the volume with less than exponential relative error [Elekes’86] randomized poly-time algorithm approximates the volume of a convex body with high probability and arbitrarily small relative error [DyerFriezeKannan’91] O∗(d23) → O∗(d4) [LovVemp’04]
- 6. Implementations Exact: VINCI [Bueler et al’00], Latte [deLoera et al], Qhull [Barber et al], LRS [Avis], Normaliz [Bruns et al] triangulation, sign decomposition methods cannot compute in high dimensions (e.g. > 20) Randomized: [Lov`aszDe`ak’12] cannot compute in > 10 dimensions Matlab code by Cousins & Vempala based on [LovVemp’04] Ours: based on [DyerFriezeKannan’91],. . . , [KannanLov`aszSimon.’97]
- 7. How do we compute a random point in a polytope P? easy for simple shapes like simplex or cube
- 8. How do we compute a random point in a polytope P? easy for simple shapes like simplex or cube BUT for arbitrary polytopes we need random walks e.g. grid walk, ball walk, hit-and-run
- 9. Random Directions Hit-and-Run (RDHR) P x B Input: point x ∈ P and polytope P ⊂ Rd Output: a new point in P 1. line through x, uniform on B(x, 1) 2. set x to be a uniform disrtibuted point on P ∩ Iterate this for W steps and return x.
- 10. Random Directions Hit-and-Run (RDHR) P x Input: point x ∈ P and polytope P ⊂ Rd Output: a new point in P 1. line through x, uniform on B(x, 1) 2. set x to be a uniform disrtibuted point on P ∩ Iterate this for W steps and return x.
- 11. Random Directions Hit-and-Run (RDHR) P x Input: point x ∈ P and polytope P ⊂ Rd Output: a new point in P 1. line through x, uniform on B(x, 1) 2. set x to be a uniform disrtibuted point on P ∩ Iterate this for W steps and return x. x is unif. random distrib. in P after W = O∗(d3) steps, where O∗(·) hides log factors [LovaszVempala’06] to generate many random points iterate this procedure
- 12. Multiphase Monte Carlo (Sequence of balls) B = B(c, r) B = B(c, ρ) P B(c, 2i/d), i = α, α + 1, . . . , β, α = d log r , β = d log ρ Pi := P ∩ B(c, 2i/d), i = α, α + 1, . . . , β, Pα = B(c, 2α/d) ⊆ B(c, r)
- 13. Multiphase Monte Carlo (Generating random points) P0 = B(c, r) B = B(c, ρ) P P1 B(c, 2i/d), i = α, α + 1, . . . , β, α = d log r , β = d log ρ Pi := P ∩ B(c, 2i/d), i = α, α + 1, . . . , β, Pα = B(c, 2α/d) ⊆ B(c, r) 1. Generate rand. points in Pi 2. Count how many rand. points in Pi fall in Pi−1 vol(P) = vol(Pα) β i=α+1 vol(Pi) vol(Pi−1)
- 14. Multiphase Monte Carlo (Generating random points) P0 = B(c, r) B = B(c, ρ) P P1 B(c, 2i/d), i = α, α + 1, . . . , β, α = d log r , β = d log ρ Pi := P ∩ B(c, 2i/d), i = α, α + 1, . . . , β, Pα = B(c, 2α/d) ⊆ B(c, r) 1. Generate rand. points in Pi 2. Count how many rand. points in Pi fall in Pi−1 vol(P) = vol(Pα) β i=α+1 vol(Pi) vol(Pi−1)
- 15. Complexity [KannanLS’97] Assuming B(c, 1) ⊆ P ⊆ B(c, ρ), the volume algorithm returns an estimation of vol(P), which lies between (1 − )vol(P) and (1 + )vol(P) with probability ≥ 3/4, making O∗ (d5 ) oracle calls, where ρ is the radius of a bounding ball for P. Isotropic sandwitching: ρ = O∗( √ d) and ball walk. Runtime generates d log(ρ) balls generate N = 400 −2d log d random points in each ball ∩ P each point is computed after O∗(d3) random walk steps
- 16. Modifications towards practicality W = 10 + d/10 random walk steps (vs. O∗(d3) which hides constant 1011) achieve < 1% error in up to 100 dim. sample partial generations of ≤ N points in each ball ∩ P (starting from the largest ball) coordinate (vs. random) directions hit-and-run (CDHR) implement boundary oracles with O(m) runtime in CDHR
- 17. Iterative rounding 1. compute set S of random points in P P
- 18. Iterative rounding 1. compute set S of random points in P 2. compute (approximate) minimum volume ellipsoid E covers S P E
- 19. Iterative rounding 1. compute set S of random points in P 2. compute (approximate) minimum volume ellipsoid E covers S 3. compute L that maps E to the unit ball B and apply L to P Iterate until the ratio of the maximum over the minumum ellipsoid axis reaches some user-defined threshold. P P E B
- 20. Iterative rounding 1. compute set S of random points in P 2. compute (approximate) minimum volume ellipsoid E covers S 3. compute L that maps E to the unit ball B and apply L to P Iterate until the ratio of the maximum over the minumum ellipsoid axis reaches some user-defined threshold. P P E B Efficiently handle skinny polytopes in practice.
- 21. Experimental results approximate the volume of a series of polytopes (cubes, random, cross, birkhoff) up to dimension 100 in less than 2 hours with mean approximation error at most 1%
- 22. Experimental results approximate the volume of a series of polytopes (cubes, random, cross, birkhoff) up to dimension 100 in less than 2 hours with mean approximation error at most 1% randomized vs. exact software (VINCI, etc.): there is a threshold dimension (d < 15) for which exact software halts
- 23. Experimental results approximate the volume of a series of polytopes (cubes, random, cross, birkhoff) up to dimension 100 in less than 2 hours with mean approximation error at most 1% randomized vs. exact software (VINCI, etc.): there is a threshold dimension (d < 15) for which exact software halts computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs. prob. 3/4 [KLS’97]) up to d = 100
- 24. Experimental results approximate the volume of a series of polytopes (cubes, random, cross, birkhoff) up to dimension 100 in less than 2 hours with mean approximation error at most 1% randomized vs. exact software (VINCI, etc.): there is a threshold dimension (d < 15) for which exact software halts computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs. prob. 3/4 [KLS’97]) up to d = 100 CDHR faster and more accurate than RDHR
- 25. Experimental results approximate the volume of a series of polytopes (cubes, random, cross, birkhoff) up to dimension 100 in less than 2 hours with mean approximation error at most 1% randomized vs. exact software (VINCI, etc.): there is a threshold dimension (d < 15) for which exact software halts computed value always in [(1 − )vol(P), (1 + )vol(P)] (vs. prob. 3/4 [KLS’97]) up to d = 100 CDHR faster and more accurate than RDHR Compute the volume of Birkhoff polytopes B11, . . . , B15 in few hrs whereas exact methods have only computed that of B10 by specialized software in ∼ 1 year of parallel computation
- 26. Volumes of Birkhoff polytopes n d estimation asymptotic estimation asymptotic exact exact asymptotic[CanfieldMcKay09] 3 4 1.12E+000 1.41E+000 0.7932847 1.13E+000 0.7973923 4 9 6.79E-002 7.61E-002 0.8919349 6.21E-002 0.8159296 5 16 1.41E-004 1.69E-004 0.8344350 1.41E-004 0.8341903 6 25 7.41E-009 8.62E-009 0.8598669 7.35E-009 0.8527922 7 36 5.67E-015 6.51E-015 0.8713891 5.64E-015 0.8665047 8 49 4.39E-023 5.03E-023 0.8729497 4.42E-023 0.8778632 9 64 2.62E-033 2.93E-033 0.8960767 2.60E-033 0.8874117 10 81 8.14E-046 9.81E-046 0.8305162 8.78E-046 0.8955491 11 100 1.40E-060 1.49E-060 0.9342584 ??? ??? 12 121 7.85E-078 8.38E-078 0.9370513 ??? ??? 13 144 1.33E-097 1.43E-097 0.9331517 ??? ??? 14 169 5.96E-120 6.24E-120 0.9550089 ??? ??? 15 196 5.70E-145 5.94E-145 0.9593786 ??? ???
- 27. Ongoing work 1. random walks for polytopes described by optimization oracles e.g. resultant polytopes [Emiris,F,Konaxis,Penaranda SoCG’12] 2. use approximate oracles (utilizing approximate NN) 3. volume of more general convex bodies e.g. spectahedra
- 28. Conclusion Practical volume estimation in high dimensions (e.g. 100) Software framework for testing theoretical ideas (e.g. new geometric random walks) Code http://sourceforge.net/projects/randgeom
- 29. Conclusion Practical volume estimation in high dimensions (e.g. 100) Software framework for testing theoretical ideas (e.g. new geometric random walks) Code http://sourceforge.net/projects/randgeom THANK YOU