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

Simulation of Conditioned Semimartingales on Riemannian Manifolds

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

\(x_0\)

\(h\)

\(v_1\)

\(v_2\)

MCMC

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

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

Conditioning branching shape processes

By Stefan Sommer

Conditioning branching shape processes

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