# Conditioning branching shape processes

Stefan Sommer, University of Copenhagen

Faculty of Science, University of Copenhagen

# (simple) shape processes

Stochastic shape process:

$dX_t=K(X_t)\circ dW_t$

Kernel matrix:

$K(X_t)^i_j=k(x_i,x_j)$

$$X_t$$ landmarks at time $$t$$:

$X_t=\begin{pmatrix}x_{1,t}\\y_{1,t}\\\vdots\\x_{n,t}\\y_{n,t}\end{pmatrix}$

$$X_0$$

$$t=\frac12$$

$$t=3$$

# Bridges: conditioned processes

Conditioning on hitting target $$v$$ at time $$T>0$$:

$X_t|X_T=v$

Ito stochastic process:

$dx_t=b(t,x_t)dt\qquad\qquad\qquad\\+\sigma(t,x_t)dW_t$

Bridge process:

$dx^*_t=b(t,x^*_t)dt+a(t,x^*_t)\nabla_x\log \rho_t(x^*_t)dt\\+\sigma(t,x^*_t)dW_t$

Score $$\nabla_x\log \rho_t$$ intractable....

$\rho_t(x)=p_{T-t}(v;x)$

$a(t,x)=\sigma(t,x)\sigma(t,x)^T$

black: $$X_0$$, red: $$v$$

# Approximate simulation

Auxilary process:

$d\tilde{x}_t=\tilde{b}(t,\tilde{x}_t)dt+\tilde{\sigma}(t,\tilde{x}_t)dW_t$

Approximate bridge:

$d\tilde{x}_t=\tilde{x}(t,\tilde{x}_t)dt\\+\tilde{a}(t,\tilde{x}_t)\nabla_x\log \tilde{\rho}_t(\tilde{x})dt+\tilde{\sigma}(t,\tilde{x}_t)dW_t$

E.g. linear process so that score $$\nabla_x\log \tilde{\rho}_t$$ is known in closed from

(almost) explicitly computable likelihood ratio:

$\frac{d\mathbb P^*}{d\tilde\mathbb P}=\frac{\tilde{\rho}_0(v)}{\rho_0(v)}\Psi(\tilde{x}_t)$

van der Meulen, Schauer et al.

Ito stochastic process:

$dx_t=b(t,x_t)dt+\sigma(t,x_t)dW_t$

Bridge process:

$dx^*_t=b(t,x^*_t)dt\\+a(t,x^*_t)\nabla_x\log \rho_t(x^*_t)dt+\sigma(t,x^*_t)dW_t$

Score $$\nabla_x\log \rho_t$$ intractable....

# v.d. Meulen/Schauer bridges

v.d. Meulen,Schauer,Arnaudon,Sommer,arxiv'21

# Explicit guiding term

dx_t = b(t,x_t)dt +\sigma(t,x_t)dW_t

Delyon/Hu 2006:

$$\sigma$$ invertible:

• guided bridge proposal$$dy_t = b(t,y_t)dt - \frac{y_t-v}{T-t}dt + \sigma(t,y_t)dW_t$$
• $$y_T=v$$ a.s.
• $$x_t|x_T=v$$ absolute continuous wrt. $$y_t$$
• $$\mathbb E_{x_t|x_T=v}[f(x_t)]\propto \mathbb E_{y_t}[f(y_t)\varphi(y_t)]$$

$$v$$

$$x_0$$

$$x_t$$

# Simulation of Conditioned Semimartingales on Riemannian Manifolds

Jensen, Mallasto, Sommer 2019 ; Jensen, Sommer 2021, 2022

# From edges to graphs

Bridge:

Leaf conditioning:

$$x_0$$

$$v$$

$$x_0$$

$$h$$

$$v_1$$

$$v_2$$

van der Meulen, Schauer + Soustrup, Nielsen, van der Meulen, Sommer

$$v_2$$

# Backwards filtering, forward guiding

$$x_0$$

$$h$$

$$v_1$$

$$v_2$$

# Diffusion mean on $$\mathbb S^2$$

• $$x_t\in M$$ Brownian motion
• $$\theta=x_0$$, $$y\sim x_T$$
• $$\bar{x}_{\mathrm{diffusion}}=\mathrm{argmax}_\theta\mathcal{L}(\theta)$$
dx_t= -\frac12g(x_t)^{kl}\Gamma(x_t)_{kl}dt + \sqrt{g(x_t)^*}dW_t

Brownian motion starting point

# Geometry, stochastics, geometric statistics

code: http://bitbucket.com/stefansommer/jaxgeometry                          Centre for Computational Evolutionary Morphometry: http://www.ccem.dk

slides: https://slides.com/stefansommer                                                     Stochastic Morphometry: https://www.ccem.dk/stochastic-morphometry/

References:

• Eltzner, Hansen, Huckemann, Sommer: Diffusion Means in Geometric Spaces, Bernoulli, 2022, arXiv:2105.12061.
• Grong, Sommer: Most probable paths for anisotropic Brownian motions on manifolds, FoCM 2022, arXiv:2110.15634
• Højgaard, Joshi, Sommer: Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation, Algorithms, 2022, https://www.mdpi.com/1999-4893/15/8/290
• Jensen, Sommer: Mean Estimation on the Diagonal of Product Manifolds, Algorithms, 2022, https://www.mdpi.com/1999-4893/15/3/92
• Arnaudon, v.d. Meulen, Schauer, Sommer: Diffusion bridges for stochastic Hamiltonian systems and shape evolutions, SIIM, 2022, arXiv:2002.00885
• Hansen, Eltzner, Sommer: Diffusion Means and Heat Kernel on Manifolds, 2021, GSI 2021, arXiv:2103.00588.
• Højgaard Jensen, Sommer: Simulation of Conditioned Diffusions on Riemannian Manifolds, 2021, arXiv:2105.13190.
• Sommer, Bronstein: Horizontal Flows and Manifold Stochastics in Geometric Deep Learning, TPAMI, 2020, doi: 10.1109/TPAMI.2020.2994507
• Arnaudon, Holm, Sommer: A Geometric Framework for Stochastic Shape Analysis, Foundations of Computational Mathematics, 2019, arXiv:1703.09971.
• Højgaard Jensen, Mallasto, Sommer: Simulation of Conditioned Diffusions on the Flat Torus, GSI 2019., arXiv:1906.09813.
• Sommer, Svane: Modelling Anisotropic Covariance using Stochastic Development and Sub-Riemannian Frame Bundle Geometry, JoGM, 2017, arXiv:1512.08544.
• Sommer: Anisotropically Weighted and Nonholonomically Constrained Evolutions, Entropy, 2017, arXiv:1609.00395 .
• Sommer, Svane: Modelling Anisotropic Covariance using Stochastic Development and Sub-Riemannian Frame Bundle Geometry, JoGM, 2017, arXiv:1512.08544.
• Sommer: Anisotropically Weighted and Nonholonomically Constrained Evolutions, Entropy, 2017, arXiv:1609.00395 .
• Arnaudon, Holm, Sommer: A Stochastic Large Deformation Model for Computational Anatomy, IPMI 2017, arXiv:1612.05323.
• Sommer: Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths, IPMI 2015,
• parametric families of probability distributions $$\mu_\theta$$
• likelihood from density:
$$\quad\mathcal{L}(\theta; y_1,\ldots,y_N)=\prod_{i=1}^Np_\theta(y_i)$$
• ML/MAP estimates:
$$\quad\bar{\theta}=\mathrm{argmax}_\theta\mathcal{L}(\theta; y_1,\ldots,y_N)$$
• Diffusion mean:
$$\quad x_t\in M$$ Brownian motion
$$\quad\theta=x_0$$
• assume $$y\sim x_T$$:
$$\quad\bar{x}_{\mathrm{diffusion}}=\mathrm{argmax}_\theta\mathcal{L}(\theta)$$

Generalization of Euclidean statistical notions and techniques.

• i.i.d. samples $$y_1,\ldots,y_N\in M$$
• Fréchet mean:
$$\bar{x}=\mathrm{argmin}_{x\in M}\sum_{i=1}^Nd(x,y_i)^2$$

Nye, White, JMIV'14;
Sommer,IPMI'15; Sommer,Svane,JGM'15;
Hansen,Eltzner,Huckemann,Sommer,GSI'21,'21

# Estimating diffusion mean and diffusion variance

smeary at optimal $$t$$?

Brownian motion samples

two-pole distribution

# Finite sample size smeariness

variance modulation:

# Can we avoid directional smeariness?

non-trivial covariance: fit anisotropic normal distributions

By Stefan Sommer

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