a new algorithm for sampling closed equilateral random walksclayton/research/... · : p tn...

A New Algorithm for Sampling ClosedEquilateral Random Walks

Jason Cantarella, Clayton Shonkwiler, and Erica Uehara

University of Georgia, Colorado State University, and Ochanomizu University

Applied Math SeminarNovember 13, 2014

Random Walks (and Polymer Physics)

Physics QuestionWhat is the average shape of a polymer in solution or in melt?

Protonated P2VPRoiter/MinkoClarkson University

Plasmid DNAAlonso-Sarduy, Dietler LabEPF Lausanne

Random Walks

Let Arm(n) be the moduli space of random walks in R3

consisting of n unit-length steps up to translation.

Then Arm(n) ∼= S2(1)× . . .× S2(1)︸︷︷︸n

.

Let Pol(n) ⊂ Arm(n) be the submanifold of closed randomwalks (or random polygons); i.e., those walks which satisfy

n∑

i=1

~ei = ~0.

A Random Walk with 3,500 Steps

A Closed Random Walk with 3,500 Steps

Main Problem

Main QuestionHow can we sample closed equilateral random walks accordingto the codimension-3 Hausdorff measure on Pol(n) ⊂ Arm(n)?

Motivating QuestionHow do we understand Pol(n)? What is this space?

Point of TalkNew direct sampling algorithms backed by symplecticgeometry. Sample ensembles are guaranteed to be perfect.O(n3) complexity, but reasonably fast in practice.

(Incomplete?) History of Sampling Algorithms

• Markov Chain Algorithms• crankshaft (Vologoskii 1979, Klenin 1988)• polygonal fold (Millett 1994)

• Direct Sampling Algorithms• triangle method (Moore 2004)• generalized hedgehog method (Varela 2009)• sinc integral method (Moore 2005, Diao 2011)

(Incomplete?) History of Sampling Algorithms

• Markov Chain Algorithms• crankshaft (Vologoskii et al. 1979, Klenin et al. 1988)

• convergence to correct distribution unproved

• polygonal fold (Millett 1994)• convergence to correct distribution unproved

• Direct Sampling Algorithms• triangle method (Moore et al. 2004)

• samples a subset of closed polygons

• generalized hedgehog method (Varela et al. 2009)• unproved whether this is correct distribution

• sinc integral method (Moore et al. 2005, Diao et al. 2011)• requires sampling complicated 1-d polynomial densities

Action-Angle Coordinates

DefinitionA triangulation T of the n-gon has n− 3 nonintersecting chords.The lengths of these chords obey triangle inequalities, so theylie in a convex polytope in Rn−3 called the moment polytope P.

DefinitionIf T n−3 = (S1)n−3, there are action-angle coordinates:

α : P × T n−3 → Pol(n)/SO(3)14

d1d2

✓1

✓2

FIG. 2: This figure shows how to construct an equilateral pentagon in cPol(5;~1) using the action-angle map.First, we pick a point in the moment polytope shown in Figure 3 at center. We have now specified diagonalsd1 and d2 of the pentagon, so we may build the three triangles in the triangulation from their side lengths,as in the picture at left. We then choose dihedral angles ✓1 and ✓2 independently and uniformly, and jointhe triangles along the diagonals d1 and d2, as in the middle picture. The right hand picture shows the finalspace polygon, which is the boundary of this triangulated surface.

Arm3(n;~r) admits a Hamiltonian action by the Lie group SO(3) given by rotating the polygonalarm in space (this is the diagonal SO(3) action on the product of spheres) whose moment mapµ gives the vector joining the ends of the polygon. The closed polygons Pol3(n;~r) are the fiberµ�1(~0) of this map. While the group action does not generally preserve fibers of this moment map,it does preserve µ�1(~0) = Pol3(n;~r) and in this situation, we can perform what is known as asymplectic reduction (or Marsden–Weinstein–Meyer reduction [49, 50]) to produce a symplecticstructure on the quotient of the fiber µ�1(~0) by the group action. This yields a symplectic structureon the (2n � 6)-dimensional moduli space cPol3(n;~r). The symplectic measure induced by thissymplectic structure is equal to the standard measure given by pushing forward the subspace mea-sure on Pol3(n;~r) to cPol3(n;~r) because the “parent” symplectic manifold Arm3(n;~r) is a Kahlermanifold [33].

The polygon space cPol3(n;~r) is singular if

"I(~r) :=X

i2I

ri �X

j /2I

rj

is zero for some I ⇢ {1, . . . , n}. Geometrically, this means it is possible to construct a linearpolygon with edgelengths given by ~r. Since linear polygons are fixed by rotations around theaxis on which they lie, the action of SO(3) is not free in this case and the symplectic reductiondevelops singularities. Nonetheless, the reduction cPol3(n;~r) is a complex analytic space withisolated singularities; in particular, the complement of the singularities is a symplectic (in factKahler) manifold to which Theorem 13 applies.

Both the volume and the cohomology ring of cPol3(n;~r) are well-understood from this sym-plectic perspective [11, 32, 36, 38, 39, 46, 66]. For example:

Main Theorem

Theorem (with Cantarella, arXiv:1310.5924)α pushes the standard probability measure on P × T n−3

forward to the correct probability measure on Pol(n)/SO(3).

Ingredients of the Proof.Kapovich–Millson toric symplectic structure on polygon space +Duistermaat–Heckman theorem + Hitchin’s theorem oncompatibility of Riemannian and symplectic volume onsymplectic reductions of Kähler manifolds +Howard–Manon–Millson analysis of polygon space.

CorollarySampling P and T n−3 independently is a perfect samplingalgorithm for Pol(n).

Understanding the Triangulation Polytope

If the chord lengths are di then the triangle inequalities are

0 ≤ d1 ≤ 21 ≤ di + di+1|di − di+1| ≤ 1

0 ≤ dn−3 ≤ 2

and the moment polytope is

d1

d2

00

1

2

1 2



0 ≤ d1 ≤ 21 ≤ di + di+1|di − di+1| ≤ 1

0 ≤ dn−3 ≤ 2


d1

d2 d1 + d2 ≥ 1

d1 ≤ 2

d2 ≤ 2

d1 ≤ d2 + 1

d2 ≤ d1 + 1

00

1

2

1 2



0 ≤ d1 ≤ 21 ≤ di + di+1|di − di+1| ≤ 1

0 ≤ dn−3 ≤ 2


d1

d2

d3

(2,3,2)

(0,0,0)(2,1,0)

Hit-and-Run

TheoremFor any convex polytope P, the hit-and-run Markov chain isuniformly ergodic with respect to Lebesgue measure on P.

A Markov Chain for Closed Random Walks

Definition (TSMCMC(β))If xk = (~pk , ~θk ) ∈ P × T n−3, define xk+1 by:• With probability β, update ~pk by a hit-and-run step on P.• With probability 1− β, replace ~θk with a new uniformly

sampled point in T n−3.At each step, construct the corresponding closed random walkα(xk ) using action-angle coordinates.

Proposition (with Cantarella)TSMCMC(β) is uniformly ergodic with respect to the standardprobability measure on Pol(n)/SO(3).

A change of coordinates

If we introduce a fake chordlength d0 = 1, and make the lineartransformation

ai = di − di+1, for 0 ≤ i ≤ n − 4, an−3 = dn−3 − d0

then our inequalities

0 ≤ d1 ≤ 21 ≤ di + di+1|di − di+1| ≤ 1

0 ≤ dn−3 ≤ 2

become

−1 ≤ ai ≤ 1,∑

ai = 0︸︷︷︸

|di−di+1|≤1

, 2i−1∑

j=0

aj + ai ≤ 1

︸︷︷︸di+di+1≥1

A change of coordinates

If we introduce a fake chordlength d0 = 1, and make the lineartransformation

ai = di − di+1, for 0 ≤ i ≤ n − 4, an−3 = dn−3 − d0

then our inequalities

0 ≤ d1 ≤ 21 ≤ di + di+1|di − di+1| ≤ 1

0 ≤ dn−3 ≤ 2

become

−1 ≤ ai ≤ 1,∑

ai = 0︸︷︷︸

easy conditions

, 2i−1∑

j=0

aj + ai ≤ 1

︸︷︷︸hard conditions

Basic Idea

DefinitionThe n-dimensional cross-polytope C is the slice of thehypercube [−1,1]n+1 by the plane x1 + · · ·+ xn+1 = 0.

→

0 1 2

0

1

2

0 1 2

0

1

2

IdeaSample points in the cross polytope, which all obey the “easyconditions”, and reject any samples which fail to obey the “hardconditions”.

Relative volumes

Theorem (Marichal-Mossinghoff)The volume of the projection of the (n − 3)-dimensionalcross-polytope C is

∑b n−22 c

j=0 (−1)j(n − 2j − 2)n−3(n−2j

)

(n − 3)!

Theorem (Khoi, Takakura, Mandini)The volume of the (n − 3)-dimensional moment polytope forn-edge equilateral polygons P is

−∑b n

2cj=0 (−1)j(n − 2j)n−3(n

j

)

2(n − 3)!

Runtime of algorithm depends on acceptance ratio

Acceptance ratio = Vol(P)Vol(C) is conjectured ∼ 6

n+2 . It is certainlybounded below by 1

n .

100 200 300 400 500

0.02

0.04

0.06

0.08

0.10

0.12

0.14

graph of 6n+2 (in red)

number of edges in polygon

Vol(P)Vol(C)

Sampling the Cross Polytope

DefinitionThe hypersimplex ∆k ,n is the slice of the cube [0,1]n with∑n

i=1 xi = k or the slab of [0,1]n−1 with k − 1 ≤∑n−1i=1 xi ≤ k .

Theorem (Stanley)There is a unimodular triangulation1 of ∆k ,n in [0,1]n−1 indexedby permutations of (1, . . . ,n − 1) with k − 1 descents.

Standard triangulation

ψ−1

−→

Stanley triangulation1a decomposition into disjoint simplices of equal volume

Parity Issues

Recall that we’re interested in the cross polytope determined by

a0 + a1 + . . .+ an−3 = 0

in the cube [−1,1]n−2. After translation and scaling, thiscorresponds to the slice

x0 + x1 + . . .+ xn−3 =n − 2

2

in the cube [0,1]n−2.

For even n this is just the hypersimplex ∆(n−2)/2,n−2 andStanley’s triangulation applies.

For odd n this is not a hypersimplex.

The Algorithm (for n even)

Moment Polytope Sampling Algorithm (with Cantarellaand Uehara)

1 Generate a permutation of n − 3 numbers with n/2− 2descents. O(n2) time .

2 Generate a point in the corresponding simplex of theStanley triangulation for ∆(n−2)/2,n−2.

3 Project up to the cross polytope in [0,1]n−2, over to thecross polytope in [−1,1]n−2 and down to a set of diagonals.

4 Test the proposed set of diagonals against the “hard”conditions. acceptance ratio > 1/n

5 Generate dihedral angles from T n−3.6 Build sample polygon in action-angle coordinates.

Example MPSA 160-gon

Example Sampling Calculation

We ran 10,000 samples of 48-gons using MPSA and 10,000samples of 48-gons using TSMCMC and computed totalcurvature:

Algorithm Result 95% confidence intervalMPSA 76.5568 (76.4763, 76.6572)TSMCMC(β) 78.0419 (76.4848, 79.545)TSMCMC(β, δ) 76.6142945311 (76.5163, 76.7122)Exact Value 76.6014

TSMCMC(β) (no permutations) used lagged autocovariancesup to lag 2018 in computing the error estimate, whileTSMCMC(β, δ) (90% permutations) used up to lag 4. MPSAdoesn’t need to compute any lagged autocovariances and justuses the sample variance to find the confidence interval.

A Combinatorial Mystery

Recall the volumes of the cross polytope and moment polytope:

Vol(C) =

∑b n−22 c

j=0 (−1)j(n − 2j − 2)n−3(n−2j

)

(n − 3)!

Vol(P) = −∑b n

2cj=0 (−1)j(n − 2j)n−3(n

j

)

2(n − 3)!

where⟨m

k

⟩is the Eulerian number which gives the number of

permutations of {1, . . . ,m} with exactly k descents. TheEulerian numbers satisfy the recurrence relation

⟨mk

⟩= (k + 1)

⟨m − 1

k

⟩+ (m − k)

⟨m − 1k − 1

⟩



Vol(C) =

∑b n−22 c

j=0 (−1)j(n − 2j − 2)n−3(n−2j

)

(n − 3)!

=2n−3

(n − 3)!

⟨n − 3

n/2− 2

⟩,

where⟨m

k

⟩is the Eulerian number which gives the number of

permutations of {1, . . . ,m} with exactly k descents. TheEulerian numbers satisfy the recurrence relation

⟨mk

⟩= (k + 1)

⟨m − 1

k

⟩+ (m − k)

⟨m − 1k − 1

⟩

Eulerian Numbers⟨m

k

⟩counts the number of permutations of {1, . . . ,m} with

exactly k descents and satisfies the recurrence relation

⟨mk

⟩= (k + 1)

⟨m − 1

k

⟩+ (m − k)

⟨m − 1k − 1

⟩

⟨mk

⟩k

11 11 4 11 11 11 1

m 1 26 66 26 11 57 302 302 57 11 120 1191 2416 1191 120 11 247 4293 15619 15619 4293 247 11 502 14608 88234 156190 88234 14608 502 1



Vol(C) =

∑b n−22 c

j=0 (−1)j(n − 2j − 2)n−3(n−2j

)

(n − 3)!

Vol(P) = −∑b n

2cj=0 (−1)j(n − 2j)n−3(n

j

)

2(n − 3)!

where Cm is the mth Catalan number

Cm =1

m + 1

(2mm

)=

Γ(2m + 1)

Γ(m + 2)Γ(m + 1).



Vol(P) = −∑b n

2cj=0 (−1)j(n − 2j)n−3(n

j

)

2(n − 3)!

'conj.

3√

38

Cn/2

where Cm is the mth Catalan number

Cm =1

m + 1

(2mm

)=

Γ(2m + 1)

Γ(m + 2)Γ(m + 1).

Catalan Numbers and Moment Polytope Volumes

The ratio of Vol(P) to Cn/2:

0 20 40 60 80 100

0.2

0.4

0.6

0.8

1.

3√

38 (in red)


Vol(P)Cn/2

A Recurrence Relation for Vol(P)

The normalized volume (n − 3)! Vol(P) of the moment polytopeis the k = 1 case of a two-parameter family V (n, k) given by therecurrence relation

V (n, k) = (n− k −1)V (n−1, k −1) + (n + k −1)V (n−1, k + 1)

subject to the boundary conditions

V (n,0) = 0, V (3,1) = 1, V (3,2) =12.

cf. OEIS A012249 “Volume of a certain rational polytope. . . ”



V (n, k) = (n− k −1)V (n−1, k −1) + (n + k −1)V (n−1, k + 1)

0 1 12

0000000




V (n, k) = (n− k −1)V (n−1, k −1) + (n + k −1)V (n−1, k + 1)

0 1 12

0 2 10 5 4 10 24 22 8 10 154 160 75 16 10 1280 1445 800 236 32 10 13005 15680 9821 3584 721 64 10 156800 199066 137088 58478 15232 2178 128 1


Catalan Numbers and Eulerian Numbers

Conjecture

Vol(P) ' 3√

38

Cn/2.

Combining the conjecture with Vol(C) = 2n−3

(n−3)!

⟨n−3

n/2−2

⟩,

Stirling’s approximation, and

Theorem (Giladi–Keller ’94)

⟨n − 3

n/2− 2

⟩'√

6π(n − 2)

(n − 3)!

would suffice to prove

Vol(P)

Vol(C)' 6√

n − 2n3/2 .

6√

n−2n3/2 vs. 6

n+2

6√

n−2n3/2 and 6

n+2 are basically indistinguishable as asymptotic

estimates for Vol(P)Vol(C) :

10 20 30 40 50

0.2

0.4

0.6

0.8

1.

graph of 6n+2 (in blue)

graph of 6√

n−2n3/2 (in yellow)


Vol(P)Vol(C)

Future Directions

• Prove conjectured volume Vol(P) ' 3√

38 Cn/2, which

implies the volume ratio estimate.

• Extend technique to odd numbers of edges. (Requires anew method for sampling the cross-polytope.)

• Improve runtime to O(n2) using a reordering method.

• Improve algorithm for generating random k -descentpermutations.

• Reference implementation and reference ensembles ofpolygons.

Thank you!

Thank you for listening!

Supported by:

Simons Foundation NSF DMS–1105699

Background

• Probability Theory of Random Polygons from theQuaternionic ViewpointJason Cantarella, Tetsuo Deguchi, and Clayton ShonkwilerCommunications on Pure and Applied Mathematics 67(2014), no. 10, 1658–1699.

• The Expected Total Curvature of Random PolygonsJason Cantarella, Alexander Y. Grosberg, Robert Kusner,and Clayton ShonkwilerarXiv:1210.6537.To appear in American Journal of Mathematics.

• The Symplectic Geometry of Closed Equilateral RandomWalks in 3-spaceJason Cantarella and Clayton ShonkwilerarXiv:1310.5924.

a new algorithm for sampling closed equilateral random walksclayton/research/... · : p tn...

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