Asymptotic behavior of Diffusion Means

Pernille E.H. Hansen, Stefan Sommer

University of Copenhagen

Content of talk

  1. Mean values on shape spaces
  2. Diffusion means
  3. Law of Large Numbers
  4. Central Limit Theorem
  5. Example

Statistics on shape spaces

Medical Images

The sphere

\mathcal{S}^2

Manifolds

Mean value?

Y_1,...,Y_N\overset{\text{iid}}{\sim} Y \in
\mu = \mathbb{E}[Y]

Expected value:

Sample estimator:

\mu_N=\frac1N\sum_{i=1}^NY_i

Uniqueness, LLN, CLT

X_1,...,X_N\overset{\text{iid}}{\sim} X \in
\mathbb{R}^n
M
\frac{1}{N} \sum_{i=1}^N X_i, \enspace\mathbb{E}[X]

Fréchet mean

X:\Omega \to M
M = \arg\min_{y\in\mathcal{M}} \mathbb{E}[\text{dist}(y,X)^2]

The Riemann center of mass of a random variable                      is

If                    then      is the Fréchet mean of

M = \{\mu \}
\mu
X.

For                           , the sample Fréchet function

X_1,...,X_N\overset{\text{iid}}{\sim} X
F_N(y) = \frac{1}{N}\sum_{i=1}^{N} \text{dist}(y,X_i)^2
M_N = \arg\min_{y\in M}F_N(y)

and sample Fréchet means

Estimation

Uniqueness, LLN, CLT

Limitation:

  •                    
\text{dist}:M\times M \to \R

is not smooth on all of

M

Diffusion means

Heat kernel on manifolds

p:M \times M \times \mathbb{R}_+ \to \mathbb{R}_+

is called a heat kernel on       if


  •  

  •  
  •                                                    
(\partial_t-\Delta_x)p(x,y,t) = 0
\lim_{t\to 0} p(x,y,t)= \delta_x(y)
  • Stochastic completeness
  • M compact

 

t = 0.1
t = 0.5
t = 1

smooth manifold

M

A map 

p \in \mathcal{C}^\infty(M\times M\times \mathbb{R}_+)
M

Limitations

  • Euclidean spaces
  • The hyper spheres
  • Hyperbolic spaces

2) Closed form

1) Existence 

Brownian motion on manifolds

A Brownian motion on       is a Markov process          with transition density function 

p(x,y,t) \approx
\text{"probability" of hitting }y
\text{ in time }t\text{ when starting in }x
p(x,y,t)
(B_t)
M
x
B_t = y
X_1,...,X_n \overset{\text{iid}}{\sim}X\in M

Given

\mu = \arg\max \frac{1}{n} \sum_{i=1}^n \ln(p(x,X_i,t))

most likely origin points of 

(B_t)
\mu = -\arg\min \mathbb{E}_X[\ln p(x,X,t)]

mean point of

X

Diffusion means

X:\Omega \to \mathcal{M}
E_t = \arg\min_{y\in\mathcal{M}} \mathbb{E}[-\text{ln }p(X,y,t)]

For           , the diffusion t-means       of a random variable

are the minimizers of the likelihood function

t>0
E_t
L_t(y) = \mathbb{E}[-\text{ln }p(X,y,t)]
\lim_{t\to 0} -t\ln p (x,y,t) = \frac{\text{dist}(x,y)^2}{2}

Estimation

L_{t,n}(x) = -\frac{1}{N}\sum_{i=1}^N\ln p(x,X_i,t)
E_{t,n} = \arg\min_{x\in M} L_{t,n}(x)

For                            the sample likelihood function is

X_1,...,X_n \overset{\text{iid}}{\sim} X

with sample likelihood means

Can we say something about:

  • (LLN): Convergence of sample means
  • (CLT): Distribution of sample means

Fix

t>0.

Law of large numbers

Fix

t>0.
(E_{t,n})
\forall \epsilon>0 \exist n\in \mathbb{N}:
\cup_{k=n}^\infty E_{t,k} \subset B(E_t,\epsilon) \quad a.s.
\mu
\mu_n
\mu_{n-1}

We say that            is a strongly consistent estimator of       if

E_t

Diffusion estimator is a SCE on

compact Riemannian manifolds!

Central Limit Theorem

and Smeariness

 

\sqrt{n}(\mu_n-\mu) \overset{\mathcal{D}}{\to} \mathcal{N}(\mu,\Sigma)

CLT:

smeary:

k-
n^{\frac{1}{2(k+1)}}(\mu_n-\mu) \overset{\mathcal{D}}{\to} \mathscr{L}

Note: CLT        0-smeary

\Rightarrow

Central limit theorem & smeariness

(Y_n)
M

smeary

k-

on Riemannian manifold

Define

X_n = \phi(Y_n)-\phi(\mu)\in \R^m

Does it there exist            st                   is                     ?

(\mu_{t,n})
k\geq 0
(Y_n)
k-

smeary

\phi
\phi:U \to \mathbb{R}^m

Let                       be a chart with

\mu \in U
X_n
\Leftrightarrow

smeary

k-

*Stephan Huckemann & Benjamin Eltzner (2018)

Central limit theorem

Fix

t>0, M

stoc., compact  Riemannian        manifold,

X: \Omega \to M.
m-
  • Uniqueness of
  • Existence of Taylor Expansion of log-likelihood function of order 
E_t = \{\mu_t\}

Assume:

(\mu_{t,n})

is

k-

smeary

with

k = \quad-2
r

and

\mathscr{L} \sim \mathcal{N}
r\geq 2

Example

What are the diffusion means and is the estimator smeary?

\alpha\in [0,1]
t>0

Does the answer depend on           and                  ?

\mu
-\mu
\cdot \enspace P(X = \mu) = 1-\alpha
\cdot \enspace P(X = -\mu) = \alpha
X: \Omega \to \mathcal{S}^m
\mu
-\mu
P(X = \mu) = 1-\alpha
P(X = -\mu) = \alpha

For each             and 

 

there exist             such that:

\alpha_m(t)

Unique diffusion mean

and 0-smeary (CLT)

\alpha < \alpha_m(t):

Unique diffusion mean

and 2-smeary

Infinitely many means

\alpha > \alpha_m(t):
\alpha = \alpha_m(t):
m\geq 2
t> \log(8(m+2)/m)/2 \leq 0.562

at times of magnetic pole reversal

Magnetic north pole positions








 

Summary

We have presented

  • An alternative mean value to the Fréchet mean
    • Minimizing smooth function
    • Closed form for heat kernel
  • (LLN) strong consistency of the likelihood estimator
  • (CLT) smeariness of the likelihood estimator with Gaussian limit
  • An example of 0-smeariness
    and 2-smeariness
  • Next step: Estimating t

Thank you for your attention!

[1] Eltzner, Benjamin; Huckemann, Stephan F. A smeary central limit theorem for manifolds with application to high-dimensional spheres. Ann. Statist. 47 (2019), no. 6, 3360--3381. doi:10.1214/18-AOS1781.

 

[2] Huckemann, Stephan F. "INTRINSIC INFERENCE ON THE MEAN GEODESIC OF PLANAR SHAPES AND TREE DISCRIMINATION BY LEAF GROWTH." The Annals of Statistics 39, no. 2 (2011): 1098-124.

Made with Slides.com