(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 $t$:

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

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

$v$

$x_0$

$x_t$

Simulation of Conditioned Semimartingales on Riemannian Manifolds

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

Simulation of Conditioned Semimartingales on Riemannian Manifolds

From edges to graphs

Bridge:

Leaf conditioning:

$x_0$

$v$

$x_0$

$h$

$v_1$

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

Backwards filtering, forward guiding

$x_0$

$h$

$v_1$

$v_2$

MCMC

Diffusion mean on $\mathbb S^2$

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, doi: 10.1007/978-3-319-19992-4_15.

Generalization of Euclidean statistical notions and techniques.

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

Means in geometric statistics

Uniqueness and asymptotics

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