Geometry, stochastics and geometric statistics

Stefan Sommer, University of Copenhagen

Faculty of Science, University of Copenhagen

GeoTop 2022

shapes - geometric statistics - stochastics

w/ Alexis Arnaudon, Darryl Holm, Sarang Joshi, Frank v.d. Meulen, Moritz Schauer, Benjamin Eltzner, Stephan Huckemann, Line Kuhnel, Mathias H. Jensen, Pernille E.H. Hansen, Mads Nielsen, Rasmus Nielsen, Christy Hipsley

Villum foundation

Novo nordisk foundation

University of Copenhagen

Statistical shape analysis

Deformations and shape

E_{s_0,s_1}(\phi)=R(\phi)+\frac1\lambda S(\phi.s_0,s_1)

action: \(\phi.s=\phi\circ s\)         (shapes)
             \(\phi.s=s\circ\phi^{-1}\)     (images)

\( \phi \)

\( \phi \) warp of domain \(\Omega\) (2D or 3D space)

landmarks: \(s=(x_1,\ldots,x_n)\)

curves: \(s: \mathbb S^1\to\mathbb R^2\)

surfaces: \(s: \mathbb S^2\to\mathbb R^3\)

Riemannian view

R(\phi_t)=\int_0^T\|\partial_t \phi_t\|_{\phi_t}^2dt

\( \phi_t:[0,T]\to\mathrm{Diff}(\Omega) \) path of diffeomorphisms

\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\varphi_t

LDDMM: Grenander, Miller, Trouve, Younes, Christensen, Joshi, et al.

Evolution with noise

\partial_t \phi_t = F(\phi_t)\ \to\ d\phi_t=F(\phi_t)dt\color{blue}{+\sigma(\phi_t) dW_t}
\mathrm{Diff}(\Omega)
\mathrm{Id}_{\mathrm{Diff}(\Omega)}
\phi_t

Markussen,CVIU'07; Budhiraja,Dupuis,Maroulas,Bernoulli'10
Trouve,Vialard,QAM'12;Vialard,SPA'13;Marsland/Shardlow,SIIMS'17
Staneva,Younes'17; Sommer,Arnaudon,Kuhnel,Joshi,MFCA'17
Arnaudon,Holm,Sommer,IPMI'17; FoCM'18; JMIV'19
Arnaudon,v.d. Meulen,Schauer,Sommer'21

Stochastic Euler-Poincaré model

Stochastic perturbation of the reconstruction equation

  • right-invariant noise, momentum map preserved
  • stochastic Euler-Poincare principle
  • landmark equations​:
    \(\quad q_i = \frac{\partial h}{\partial p_i} dt+\sum_{l=1}^J\frac{\partial\phi_l}{\partial p_i}\circ dW_t^l\)
    \(\quad p_i = -\frac{\partial h}{\partial q_i} dt+\sum_{l=1}^J\frac{\partial\phi_l}{\partial q_i}\circ dW_t^l\)
  • image equations:
    \(\quad 0=d\mathbf m+\big( (\mathbf u\cdot\nabla)\mathbf m+\mathbf m\cdot(D\mathbf v)^T+\mathrm{div}(\mathbf u)\mathbf m\big)dt\)
    \(\quad\quad+\sum_{l=1}^J\big(\mathbf\sigma_l\cdot\nabla)\mathbf m+\mathbf m\cdot(D\mathbf\sigma_l)^T+\mathrm{div}(\mathbf\sigma_l)\mathbf m\big)\circ dW^l_t\)
    \(\quad dI=-\nabla I\cdot\mathbf udt+\sum_{l=1}^J\nabla I\cdot\mathbf\sigma_l\circ dW_t^l\)
d\frac{\delta l}{\delta u} + \mathrm{ad}^*_{u_t} \frac{\delta l}{\delta u}dt + \color{blue}{\sum_{l=1}^J \mathrm{ad}^*_{\sigma_l} \frac{\delta l}{\delta u}\circ dW^l_t=0}
dg_t = u_t( g_t) dt + \color{blue}{\sum_{l=1}^J \sigma_l( g_t) \circ dW^l_t}

Arnaudon,Holm,Sommer,IPMI'17; FoCM'18; JMIV'19

d\frac{\delta l}{\delta u} + \mathrm{ad}^*_{u_t} \frac{\delta l}{\delta u}dt=0
dg_t = u_t( g_t) dt

Schauer/v.d. Meulen bridges

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

Stochastic morphometry along phylogenies

v.d. Meulen,Schauer,Arnaudon,Sommer, SIIMS 21

Simulation of Conditioned Semimartingales on Riemannian Manifolds

Jensen, Mallasto, Sommer, GSI 2019 ; Jensen, Sommer, GSI 2021; Jensen, Sommer, arxiv 2021

  • 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 \(y\sim x_T\):
    \(\quad\bar{x}_{\mathrm{ML}}=\mathrm{argmax}_\theta\mathcal{L}(\theta)\)

 

 

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

Geometric statistics

Diffusion mean on \(\mathbb S^2\)

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

Brownian motion starting point

Smeariness and uniqueness

Curvature and covariance

Sommer,Svane,JGM'15; Sommer,Entropy,'16

Most probable paths

Grong, Sommer, arXiv, 2021

Grong, 2021

Geometry and deep learning and atlas models

Sommer,Bronstein,TPAMI'20

Examples: MNIST

60,000 handwritten digits

2D latent representation \(\quad F:Z\to\mathbb R^{784}\)

scalar curvature

Ricci curvature (min eigv)

parallel transport in \(Z\)

Geometry, stochastics, geometric statistics

code: http://bitbucket.com/stefansommer/jaxgeometry

slides: https://slides.com/stefansommer

 

References:

  • 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, Huckemann, Sommer: Diffusion Means in Geometric Spaces, 2021, arXiv:2105.12061.
  • Hansen, Eltzner, Sommer: Diffusion Means and Heat Kernel on Manifolds, 2021, arXiv:2103.00588.
  • Højgaard Jensen, Sommer: Simulation of Conditioned Diffusions on Riemannian Manifolds, 2021, arXiv:2105.13190.
  • Højgaard, Joshi, Sommer: Brownian Bridge Simulation and Metric Estimation on Lie Groups and Homogeneous Spaces,
  • 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.

Geometry, stochastics and geometric statistics

By Stefan Sommer

Geometry, stochastics and geometric statistics

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