Manifold bridges and diffusions means

Stefan Sommer, University of Copenhagen

Faculty of Science, University of Copenhagen

Nordstat, 2023

Simulation of Conditioned Semimartingales on Riemannian Manifolds and Lie groups

shapes - geometric statistics - stochastics - diffusion means

w/ Sarang Joshi, Benjamin Eltzner, Stephan Huckemann, Mathias H. Jensen, Pernille E.H. Hansen

Villum foundation

Novo nordisk foundation

University of Copenhagen

Center for Computational Evolutionary Morphometrics

Stochastic morphometry along phylogenies

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

Geometric data

Plane directions:      \(\mathbb{S}^1\)

Geographical data:  \(\mathbb{S}^2\)

3D directions:           \(\mathrm{SO}(3), \mathbb{S}^2\)

Angles:                       \(\mathbb{T}^N\)

Shapes

  • 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,Bernoulli'23

Means in geometric statistics

Least-squares \(\leftrightarrow\) probabilistic

Deterministic:

  • \(\phi_t\) geodesic evolution
  • square distances:
    \(\quad d(s_0,s1)^2\)
  • Riemannian least-
    squares

Stochastic:

  • \(\phi_t\) stochastic process
  • (log) transition density
    \(\quad -\log p_T(s_1; s_0)\)
  • ML/MAP
  • bridge:
    \(\quad \phi_t|\phi_T.s_0=s_1\)
  • bridge + noise in observation:
    \(\quad \phi_t|\phi_T.s_0+\epsilon=s_1\)

Bridges with auxiliary processes

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

Bridges without auxiliary processes

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. \(y_t\)
  • \(\mathbb E_{x_t|x_T=v}[f(x_t)]\propto \mathbb E_{y_t}[f(y_t)\varphi(y_t)]\)

\(v\)

\(x_0\)

\(x_t\)

Frame bundles

  • \(\pi:FM\to M\) is the bundle of linear frames, i.e.
    \(u=(u_1,\ldots,u_d)\in FM\) is an ordered basis for \(T_xM\), \(x=\pi(u)\)
  • \(FM\) is a \(\mathrm{GL}(n)\) principal bundle: \(u:\mathbb R^d\to T_x M\) linear, invertible
  • \(OM\) the subbundle of orthonormal frames (orientations)
  • horizontal lift \(h_u:T_xM\to H_uFM\) and fields: \(H_i:FM\to TFM\)

          \(h_u=(\pi_*|_{H_uFM})^{-1}\)
          \(H_i(u)=h_u(ue_i)\)

Stochastic development

Stochastic development:

    \(dU_t=\sum_{i=1}^d H_i(U_t)\circ_{\mathcal S} dW_t^i\)

\(W_t\) Euclidean Brownian motion

\(X_t=\pi(U_t)\) Riemannian Brownian motion

\(U_t\) is stochastically parallel transported

Fix \(T>0\): \(U_T\) probability distribution in \(FM\)

Rolling without slipping

Simulation of Conditioned Semimartingales on Riemannian Manifolds

Driving semi-martingale:

 

developed process:

 

Fermi bridge:

 

 

Simulation of Conditioned Semimartingales on Riemannian Manifolds

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

Heat kernel approximations

Bridges on Lie groups and homogenous spaces

  • \(A\) quadratic form on \(so(3)\)
  • \(x_t\in SO(3)\) Brownian motion
  • \(\theta=(x_0,A)\)
  • \((\bar{x},\bar{A})=\mathrm{argmax}_\theta\mathcal{L}(\theta)\)

\(\pi\)

Thompson'16, Sommer,Joshi,Højgaard,'22

Left-invariant frame:

      \(V_i(g)=(dL_g)_ev_i,\quad v_i\in\mathfrak g\)

Brownian motion:

      \(dg_t=-\frac12V_0(g_t)dt+V_i(g_t)\circ dW_t^i\)

Fermi bridge:

    \(dg_t=-\frac12V_0(g_t)dt+V_i(g_t)\circ \left(dW_t^i-\frac{\mathrm{log}_{g_t}(v)^i}{T-t}dt\right)\)

Fermi bridge to fiber:

    \(dg_t=-\frac12V_0(g_t)dt+V_i(g_t)\circ \left(dW_t^i-\frac{\left(\nabla d(g_t,\pi^{-1}(v))^2\right)^i}{2(T-t)}dt\right)\)

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

Estimation

Direct optimzation of \(\mathcal{L}(\theta; y_1,\ldots,y_N)\):

  • heat kernel expansion
  • bridge sampling

 

Direct sampling:

  • reverse time and condition on diagonal

Diagonally conditioned distribution

Sommer, Bronstein'19; Jensen,Sommer'22

Fast mean estimation

Fermi bridge:

Coordinate bridge:

One (or few) forward samples - compared to nested optimization

Added variance on top of CLT - gain in computational speed

Sampled landmark means

Diagonally conditioned process:

Frechet mean (green), diffusion mean (blue)

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:

  • Philipp Harms, Peter W. Michor, Xavier Pennec, Stefan Sommer: Geometry of sample spaces, Diff. Geom. and its Appl., 2023, arXiv:2010.08039
  • Hansen, Eltzner, Huckemann, Sommer: Diffusion Means in Geometric Spaces, Bernoulli, in press, 2023, 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, 
  • 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.