### A Natural Map from Random Walks to Equilateral Polygons in Any Dimension

Clayton Shonkwiler

http://shonkwiler.org

11.07.17

/cmo17

This talk!

### Statistical physics viewpoint

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

Modern polymer physics is based on the analogy between a polymer chain and a random walk.

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

### Sampling random walks in $$\mathbb{R}^d$$ is easy

Generate $$n$$ independent uniform random points on $$S^{d-1}$$ and treat them as an ordered list of edge vectors.

### ...but sampling random polygons was hard

Alvarado, Calvo, Millett, J. Stat. Phys. 143 (2011), 102–138

### Key idea: commuting symmetries

Rotations around $$n-3$$ chords $$d_i$$ by $$n-3$$ angles $$\theta_i$$ commute.

### A polytope

The $$(n-3)$$-dimensional  moment polytope $$\mathcal{P}_n \subset \mathbb{R}^{n-3}$$ is defined by the triangle inequalities

0 \leq d_i \leq 2
$0 \leq d_i \leq 2$
1 \leq d_i + d_{i-1}
$1 \leq d_i + d_{i-1}$
|d_i - d_{i-1}| \leq 1
$|d_i - d_{i-1}| \leq 1$
0 \leq d_{n-3} \leq 2
$0 \leq d_{n-3} \leq 2$

### From action-angle coordinates to polygons

There exists an almost-everywhere defined map $$\alpha: \mathcal{P}_n \times (S^1)^{n-3} \to \text{Pol}_3(n)/SO(3)$$.

This is only sensible as a map to polygons modulo translation and rotation.

### Sampling polygons in $$\mathbb{R}^3$$

Theorem (with Cantarella, 2016)

The map $$\alpha$$ pushes forward the standard probability measure on $$\mathcal{P}_n \times (S^1)^{n-3}$$ to the correct probability measure on $$\text{Pol}_3(n)/SO(3)$$.

Corollary

Independently sampling $$\mathcal{P}_n$$ and $$(S^1)^{n-3}$$ is a perfect sampling algorithm for equilateral $$n$$-gons in $$\mathbb{R}^3$$.

Theorem (with Cantarella, Duplantier, Uehara, 2016)

A direct sampler with expected runtime $$\Theta(n^{5/2})$$.

Kyle Chapman has an even faster sampler he will talk about on Thursday.

Unfortunately, this is all very special to 3 dimensions...

### Strategy

1. Find a natural map $$g:\text{Arm}_d(n) \to \text{Pol}_d(n)$$.
2. Sample points in $$\text{Arm}_d(n)$$ and apply $$g$$.
3. Hope the pushforward measure is almost uniform.

### Efficient closure

Idea: $$g$$ should map each arm to the closed polygon which is closest to it.

Problem: This is not well-defined on all of $$\text{Arm}_d(n)$$.

What is the closest closed polygon?

Relatedly: $$\text{Arm}_d(n) \simeq (S^{d-1})^n$$ and $$\text{Pol}_d(n)$$ have different topologies, so there's no retraction defined on all of $$\text{Arm}_d(n)$$.

### The geometric median

Definition

A geometric median (or Fermat-Weber point) of a collection $$X=\{x_1,\ldots , x_n\}$$ of points in $$\mathbb{R}^d$$ is any point closest to the $$x_i$$:

$$\text{gm}(X)=\text{argmin}_y \sum \|x_i-y\|$$

Definition

A point cloud has a nice geometric median if:

• $$\text{gm}(X)$$ is unique ($$\Leftarrow X$$ is not linear)
• $$\text{gm}(X)$$ is not one of the $$x_i$$

### Geometric median closure

Definition

If the edge cloud $$X$$ of an equilateral arm has a nice geometric median, the geometric median closure $$\text{gmc}(X)$$ recenters the edge cloud at the geometric median.

$$\text{gmc}(X)_i = \frac{x_i-\text{gm}(X)}{\|x_i - \text{gm}(X)\|}$$

### Loop closure

1QMG – Acetohydroxyacid isomeroreductase

### The geometric median closure is closed

Proposition (with Cantarella)

If it exists, the geometric median closure of an arm is a closed polygon.

Proof

$$\text{gm}(X)$$ minimizes the total distance function

$$d_X(y) = \sum_i \|x_i - y\|$$,

which is convex everywhere and smooth away from the $$x_i$$, and

$$\nabla d_X(y) = \sum_i \frac{x_i-y}{\|x_i-y\|}$$.

### The geometric median closure is optimal

Definition

An arm or polygon $$X$$ is given by $$n$$ edge vectors $$x_i \in \mathbb{R}^d$$, or a single point in $$\mathbb{R}^{dn}$$. The distance between $$X$$ and $$Y$$ is the Euclidean distance between these points in $$\mathbb{R}^{dn}$$.

Theorem (with Cantarella)

If $$X$$ is an equilateral arm in $$\mathbb{R}^d$$ with a geometric median closure, then $$\text{gmc}(X)$$ is the closest equilateral polygon to $$X$$.

Proof

Depends on the neat fact that if $$\|x_i\|=\|y_i\|$$, then

$$\langle X, Y -X \rangle \leq 0$$.

### gmc can fail, but not often

Theorem* (with Cantarella)

The fraction of $$\text{Arm}_d(n)$$ without a geometric median closure $$\to 0$$ exponentially fast in $$n$$.

### Proof strategy

$$\text{Ad}(p) = \sqrt{1+\|p\|^2}\ \Gamma\!\left(\frac{d}{2}\right) {}_2F_1\!\!\left(-\frac{1}{4},\frac{1}{4},\frac{d}{2},\frac{4\|p\|^2}{(1+\|p\|^2)^2}\right)$$

For a random point cloud $$X = (x_1, \ldots , x_n)$$ on $$S^{d-1}$$, want to show $$\text{gm}(X)$$ is close to the origin (and therefore $$\text{gmc}(X)$$ exists) with high probability.

Claim: This follows if we can show that $$d_X$$ is $$L^\infty$$ close to $$d \text{Ad}: B^d \to \mathbb{R}$$, where $$\text{Ad}(p)$$ is the average distance from $$S^{d-1}$$ to $$p \in B^d$$.

Lemma (Hjort & Pollard, 1993)

Let $$f$$ be convex and $$g$$ any function with unique argmin at 0. Let $$B_\delta$$ be the ball of radius $$\delta$$, $$M = \sup_{s \in B_\delta}|f(s) -g(s)|$$, $$m = \inf_{s \in \partial B_\delta}|g(s) - g(0)|$$. If $$M < m/2$$, then argmin $$f \in B_\delta$$.

### Concentration of measure

Theorem (Bernstein inequality for Hilbert spaces)

Let $$(\Omega,\mathcal{A},P$$ be a probability space, $$H$$ a separable Hilbert space, $$B > 0, \sigma > 0$$. If $$\xi_1,\ldots , \xi_n:\Omega \to H$$ are independent r.v.'s satisfying $$\mathbb{E}(\xi_i)=0, \|\xi_i\|_\infty \leq B, \mathbb{E}(\|\xi_i\|_H^2)\leq \sigma^2$$, then

$$P\left(\left\|\frac{1}{n}\sum \xi_i\right\|_H \geq \sqrt{\frac{2\sigma^2 \tau}{n}} + \sqrt{\frac{\sigma^2}{n}} + \frac{2B\tau}{3n}\right) \leq e^{-\tau}$$

In our case, let $$\xi_i(p) = \|x_i-p\| - \text{Ad}(p)$$ and $$H = L^2(B^d)$$. Then, e.g.,

$$P\left(\left\|\frac{1}{n}d_X-\text{ad}\right\|_2 > \frac{30}{n^{1/3}}\right) \leq e^{-n^{1/3}}$$

In other words, $$d_X$$ and $$d \text{Ad}$$ are close in $$L^2$$ with high probability

### A reverse bound

Proposition (with Cantarella)

If $$f$$ is differentiable on $$B^d$$ with $$\|\nabla f\|_\infty \leq K$$, then

\|f\|_\infty \leq \frac{2}{\pi^{d/2}\left(\frac{2}{\Gamma(d/2+1)}-\frac{2^dK d \Gamma(d/2)}{(d+1)\sqrt{\pi}\Gamma(d+1/2)}\right)}\|f\|_1.
$\|f\|_\infty \leq \frac{2}{\pi^{d/2}\left(\frac{2}{\Gamma(d/2+1)}-\frac{2^dK d \Gamma(d/2)}{(d+1)\sqrt{\pi}\Gamma(d+1/2)}\right)}\|f\|_1.$

Since $$\|f\|_1 \leq \text{Vol}(B^d) \|f\|_2$$, we can combine this with the concentration result to see that $$d_X$$ is $$L^\infty$$ close to the average distance function with high probability, and hence the Hjort & Pollard lemma applies.

### Is this a good sampler?

If $$X \in (S^{d-1})^n$$ is chosen uniformly, then we've seen that $$\text{gmc}(X)$$ exists with very high probability.

This produces some distribution on closed polygons.

Question

What is this distribution? Is it uniform?

We can test when $$d=3$$…

### A more subtle test

Histogram of chord lengths from ~ 1 million pentagons created by sampling arms and applying gmc, compared to exact pdf of random pentagons

### Exact chord pdfs

Let $$\phi_n(\ell)$$ be the density of the end-to-end distance in an $$n$$-step random flight. From Lord Rayleigh,

\phi_n(\ell) = \frac{2\ell}{\pi}\int_0^\infty x \sin \ell x \text{sinc}^n x \text{d}x.
$\phi_n(\ell) = \frac{2\ell}{\pi}\int_0^\infty x \sin \ell x \text{sinc}^n x \text{d}x.$

This is piecewise-polynomial of degree $$n-3$$.

Proposition

The pdf of the length of the chord connecting $$v_1$$ to $$v_{k+1}$$ in an $$n$$-gon is a constant multiple of

\ell^2 \phi_k(\ell) \phi_{n-k}(\ell)
$\ell^2 \phi_k(\ell) \phi_{n-k}(\ell)$

### A more subtle test

Histogram of chord lengths from ~ 1 million 10-gons created by sampling arms and applying gmc, compared to exact pdf of random 10-gons

### In the limit...

Conjecture

The probability measure generated by geometric median sampling converges (exponentially fast?) to the uniform distribution on equilateral $$n$$-gons as $$n \to \infty$$.

Conjecture

The integral of any function which varies slowly enough with respect to any permutation-invariant probability measure on equilateral $$n$$-gons converges to the integral of the function with respect to the uniform distribution on equilateral $$n$$-gons as $$n \to \infty$$.

### Application: Polygon evolutions

Algorithm

To follow a flow in polygon space:

1. Compute the infinitesimal variation $$V$$ in $$\mathbb{R}^{dn}$$ at $$X^{(n)}$$ tangent to $$\text{Pol}_d(n)$$ (or at least $$\text{Arm}_d(n)$$).
2. Follow the exponential map in $$\text{Arm}_d(n)$$ in direction $$V$$ (to preserve edgelengths exactly).
3. Use geometric median closure to get $$X^{(n+1)} \in \text{Pol}_d(n)$$.

How far can you step?

### A safe step distance

Definition

If $$P$$ is a closed $$n$$-gon in $$\mathbb{R}^d$$ with edge cloud $$X$$ and $$\lambda_1$$ is the first eigenvalue of $$X^TX$$, define

d_{\text{safe}} := \frac{1}{4\sqrt{d}}(d-\lambda_1)
$d_{\text{safe}} := \frac{1}{4\sqrt{d}}(d-\lambda_1)$

Theorem (with Cantarella)

If $$P$$ is a closed equilateral $$n$$-gon, every arm within $$d_{\text{safe}}$$ of $$P$$ has a geometric median closure. This distance bound is positive if $$P$$ is not contained in a line.

### The algorithm, redux

Algorithm

To follow a flow in polygon space:

1. Compute the infinitesimal variation $$V$$ in $$\mathbb{R}^{dn}$$ at $$X^{(n)}$$ tangent to $$\text{Pol}_d(n)$$ (or at least $$\text{Arm}_d(n)$$).
2. Find $$\lambda_1$$ and use it to compute $$d_{\text{safe}}(X^{(n)})$$.
3. Follow the exponential map in $$\text{Arm}_d(n)$$ in direction $$V$$ (to preserve edgelengths exactly), but don't go further than $$d_{\text{safe}}$$.
4. Use geometric median closure to get $$X^{(n+1)} \in \text{Pol}_d(n)$$.

### Recap

In any dimension

• The geometric median provides an optimal loop closure.
• It works on nearly all arms. The failure set is small, but somewhat hard to describe.
• The pushforward measure is almost uniform.
• These tools provide clean computational methods for polygon reconfigurations.

# Thank you!

### References

J. Cantarella & C. Shonkwiler

Annals of Applied Probability   26  (2016), no. 1, 549–596

J. Cantarella, B. Duplantier, C. Shonkwiler, & E. Uehara

Journal of Physics A 49 (2016), no. 27, 275202

A natural map from random walks to closed polygons in any dimension

J. Cantarella & C. Shonkwiler

In preparation

Funding: Simons Foundation

#### From random walks to closed polygons

By Clayton Shonkwiler

# From random walks to closed polygons

A natural map from random walks to equilateral polygons in any dimension

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