Asymptotic behavior of Diffusion means

Pernille E.H. Hansen

University of Copenhagen

Joint work with Stefan Sommer, Benjamin Eltzner and Stephan Huckemann

Content of talk

Fréchet mean
Diffusion means and estimator
Strong consistency
Smeary central limit theorem
Example of smearieness on hyper spheres

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

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

Conditions for uniqueness, stong consistency and CLT

Diffusion means

Heat kernel on manifolds

Limitations

Let $M$ be a smooth manifold.

A map $p:M\times M\times (0, \infty)\to(0,\infty)$ is a heat kernel on $M$ if

$$1. \enspace p \in \mathcal{C}^\infty(M\times M\times (0,\infty))$$

$$2. \enspace (\partial_t-\Delta_x)p(x,y,t) = 0 $$

$$3. \enspace \lim_{t\to 0} p(x,y,t)= \delta_x(y) $$

1) Existence

Stochastic completeness
$M$ compact

2) Closed form

Euclidean spaces
The hyper spheres
Hyperbolic spaces

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)

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

Diffusion means

For $t>0$, the diffusion $t$-means $E_t(X)$ of a random variable $X: \Omega \to M$ are the minimizers of the log-likelihood function,

$$ L_t(y) = \mathbb{E}[-\ln(p(y,X,t)]$$

Thus,

$$ E_t(X) = \arg\min_{y\in M} \mathbb{E}[-\ln(p(y,X,t)]$$

$$ \lim_{t\to 0}-2t \ln(p(x,y,t)) = \text{dist}(x,y)^2$$

Generalized Fréchet mean*

For a continuous function $ \rho: M\times M \to [0,\infty)$

the Fréchet $\rho$-means of $ X:\Omega\to M$ is the set

E^{\rho}(X) = \arg\min_{y\in M}\mathbb{E}[\rho(y,X)^2]

with estimator

E_n^{\rho}(\omega) = \arg\min_{y\in M} \frac{1}{N} \sum_{i=1}^N \rho(y,X_i(\omega))^2

Diffusion $t$-means: Choosing $\rho_t(x,y) = \sqrt{- \ln(p(x,y,t)\beta_t^{-1})} $ with $ \beta_t > p(x,y,t) $ for all $x,y\in M $. This exist for all $t>0 $ when $M$ has bounded sectional curvature*!

*[Cheng et al., 1981]

*[Huckemann, 2011]

Estimation

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

E_{t,n}(\omega) = \arg\min_{y\in M} L_{t,n}(y)

with sample likelihood means

Fix $t>0$. For $X_1,...,X_n \overset{\text{idd}}{\sim} X$ the sample likelihood function is

We say that $E_{t,n}(\omega) $ is the diffusion estimator.

Diffusion means in $\mathbb{R}^m$?

The heat kernel on $\mathbb{R}^n$ for each $t>0$ is

p(x,y,t) =\frac{1}{(4\pi t)^{m/2}} e^{\frac{-\text{dist}_{\mathbb{R}^m}(x,y)^2}{4t}}

E_t(X)=\arg\min_{y\in \mathbb{R}^m} \int_{\mathbb{R}^m} \frac{2}{m}\ln(4\pi t) - \left(\frac{\text{dist}_{\mathbb{R}^m}(x,y)^2}{4t}\right)d\mathbb{P}_X(x)

= \arg\min_{y\in \mathbb{R}^m} \int_{\R^m} \text{dist}_{\mathbb{R}^m}^2 d\mathbb{P}_X(x) = \mathbb{E}[X]

For $ X:\Omega \to \mathbb{R}^m $, we have

The hyper spheres

p(x,y,t) = \sum_{l=0}^\infty e^{-l(l+m-1)\sqrt{2t}}\frac{2l+m-1}{m-1} \frac{1}{A_{\mathcal{S}}^{m}} C_l^{(m-1)/2}(\langle x,y\rangle_ {\R^m} )

The heat kernel on the sphere $\mathcal{S}^{m}$ for $m\geq 2$ is

Consider $X: \Omega \to \mathcal{S}^m $:

For each $m\geq 2$, $t>0$ and $\alpha\in [0,1/2]$, what are the diffusion $t$-means?

\mu

-\mu

P(X = -\mu) = \alpha

P(X = \mu) = 1-\alpha

Example:

\mu

-\mu

P(X = \mu) = 1-\alpha

P(X = -\mu) = \alpha

For each $m\geq 2$ and $t>\Lambda_{t,m}$ where

there exist $\alpha_m(t)$ such that:

Riemann Center of mass

$\alpha = 0$: Unique Fréchet mean
$ \alpha>0$: Infinite

Diffusion $t$-means

$ \alpha \leq \alpha_m(t) $: Unique diffusion $t$-mean
$ \alpha > \alpha_m(t)$ : Infinitely many

Furthermore, $\alpha_{m}(t) \to 1/2$ as $t\to \infty$

\Lambda_{t,m} = \frac12 \left(\log\left( \frac{8(m+3)}{(m+1)} \right) \frac12 \right)^2 \leq 0.838

Strong consistency

(ZC) of Ziezold if for almost all $\omega \in \Omega$

\cap_{n=1}^\infty \overline{\cup_{k=n}^\infty M_{t,k}(\omega)} \subset M_t

(BPC) Bhattacharya and Patrangenaru if $ \forall \epsilon>0 $ and almost all $\omega\in \Omega,\exist n\in \mathbb{N}:$

\cup_{k=n}^\infty M_{t,k}(\omega) \subset B(M_t,\epsilon)

\mu

\mu_n

\mu_{n-1}

Fix $t>0$. We say that $M_{t,n}(\omega)$ is a strongly consistent estimator of $M_t$ in the sense of

Strong consistency

\mu

\mu_n

\mu_{n-1}

$M$ stoc. complete Riemannian manifold of bounded curvature

Fix $t>0$, $X: \Omega \to M $

$M$ compact Riemannian manifold:

$ E_{t,n} $ satisfies (ZC) and (BPC)

1. $ E_{t,n} $ satisfies (ZC) if either

$X$ has compact support
$\mathbb{E}[-\ln p (x,X,t)]<\infty$ for all $x\in M$ & a continuity property in the second
argument uniform over the first argument

2. $ E_{t,n} $ satisfies (BPC) if $E_t \neq \emptyset$ and

$ E_{t,n} $ satisfies (ZC)
Heine-Borel property of $ \overline{\cup_{n=1}^\infty E_{t,n}}$
A coercivity condition

*[Huckemann, 2011]

Central Limit Theorem

and Smeariness

Central limit theorem & smeariness

(Y_n)

smeary

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

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)

smeary

\phi

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

Let be a chart with

\mu \in U

\Leftrightarrow

X_n

\Leftrightarrow

smeary

*[Eltzner & Huckemann, 2018]

Central limit theorem

Fix

t>0, \enspace M

Riemannian manifold,

X: \Omega \to M.

(Uniqueness):
(LLN): Convergence
(Taylor Expansion): There exist

E_t = \{\mu_t\}

\mu_{t,n}\overset{\mathbb{P}}{\to}\mu_t

L_t(\exp_{\mu_t}(x)) = L_t(\exp_{\mu_t}(0)) + \sum_{i=1}^m T_i |(Rx)_i|^r + o(||x||^r)

\geq 2, R\in\text{SO}(m), T_1,...,T_m\neq 0 \text{ st.}

Assume:

(\mu_{t,n})

smeary

with

k = \quad-2

\mu

-\mu

P(X = \mu) = 1-\alpha

P(X = -\mu) = \alpha

For each $m\geq 2$ and $t>\Lambda_t$ where

$$ \Lambda_t = \sqrt{\log(8(m+2)/m)/2}/2 \leq 0.38$$

there exist $\alpha_m(t)$ such that:

Smearieness of Diffusion $t$-estimator

$ \alpha \leq \alpha_m(t) $: Unique diffusion $t$-mean
$ \alpha > \alpha_m(t)$ : Infinitely many

$ \alpha < \alpha_m(t) $: 0-smeariness
$ \alpha = \alpha_m(t)$ : 2-smearieness

at times of magnetic pole reversal

Magnetic north pole positions

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.

Law of large numbers

Fix

t>0, \enspace M

\mu

\mu_n

\mu_{n-1}

satisfies (ZC)
satisfies (BPC) if

compact Riemannian manifold,

X: \Omega \to M

E_t \neq \emptyset

(E_{t,n})

Central limit theorem

Fix

t>0, \enspace M

Riemannian manifold,

X: \Omega \to M

Then for any measurable selection is holds that

\sqrt{n}(V_1,...,V_n)^T \overset{\mathcal{D}}{\to} \mathcal{N}(0,\Sigma)

Assume (Uniqueness), (LLN) and (Taylor Expansion)

(\mu_{t,n})

where

x_n = \log_{\mu_t}\mu_{t,n}

V_i = (Rx_n)_i|(Rx_n)_i|^{r-2}

T=diag(T_1,...,T_m)

\Sigma = \frac{1}{r^2}T^{-1}\text{Cov}[\text{grad}_x(-\ln p(x,X,t))|_{x=0}]T^{-1}

Asymptotic behavior of Diffusion means

Content of talk

Fréchet mean

Diffusion means

Heat kernel on manifolds

Brownian motion on manifolds

Diffusion means

Generalized Fréchet mean*

Estimation

Diffusion means in \(\mathbb{R}^m\)?

The hyper spheres

Example:

Riemann Center of mass

Diffusion \(t\)-means

Strong consistency

Strong consistency

Central Limit Theorem

and Smeariness

Central limit theorem & smeariness

smeary

smeary

Central limit theorem

Smearieness of Diffusion \(t\)-estimator

Thank you for your attention!

Law of large numbers

Central limit theorem