stochastic shapes and probabilistic geometric statistics

w/ Alexis Arnaudon, Darryl Holm, Sarang Joshi, Tom Fletcher, Frank v.d. Meulen, Moritz Schauer, Benjamin Eltzner, Line Kuhnel, Mathias H. Jensen, Pernille E.H. Hansen

Novo nordisk foundation

Villum foundation

Carlsberg foundation

Statistical shape analysis

Geometric statistics

Generalization of Euclidean statistical notions and techniques to spaces without vector space structure

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

Geometric data: Directional statistics

Plane directions:      $\mathbb{S}^1$

Geographical data:  $\mathbb{S}^2$

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

Angles:                       $\mathbb{T}^N$

Geometric data: Diffusion MRI

Positive, symmetric 3 tensors:   SPD(3)

Geometric data: Functional signals

 continuous curves: $\mathrm{Emb}(\mathbb S^1,\mathbb R^2)/\mathrm{Diff}(\mathbb S^1)$

Mieritz et al.,JCEM'15

Shape matching

$\phi\in\mathrm{Diff}(\Omega)$ diffeomorphism of domain $\Omega$
action: $\phi.s=\phi\circ s$         (shapes)
             $\phi.s=s\circ\phi^{-1}$     (images)

$\phi$

Riemannian view

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

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

Evolution with noise

Trouve,Vialard,QAM'12;Vialard,SPA'13;Marsland/Shardlow,SIIMS'17
Arnaudon,Holm,Sommer,IPMI'17; FoCM'18; JMIV'19

Least-squares $\leftrightarrow$ probabilistic

Deterministic:

• $\phi_t$ geodesic evolution
• exact matching:
  $\quad\mathrm{argmin}_{\phi_1.s_0=s_1} E(\phi_t)$
• inexact matching:
  $\quad\mathrm{argmin}_{\phi_t} E(\phi_t)$
• distance:
  $\quad d(s_0,s1)^2=E(\phi_t)$
• Riemannian least-
  squares

Stochastic:

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

Inferring geometry of landmarks spaces

• Landmarks $\mathbf{q}=(q_1,\ldots,q_n),\,q_i\in\Omega\subseteq\mathbb R^d$
• LDDMM cometric $\left<\xi,\eta\right>_{\mathbf q}=\xi^T K(\mathbf q,\mathbf q)\eta$
• kernel $K(\mathbf q_1,\mathbf q_2)=\alpha e^{-1\frac12 x^T\Sigma^{-1}x}\mathrm{Id}_d$
• Riemannian Brownian motion
  $$dq_t^i=g^{kl}\Gamma(\mathbf q_t)^i_{kl}dt+\sqrt{g^*(\mathbf q_t)}^idW_t$$

Sommer,Arnaudon,Kuhnel,Josh,MFCA'17

Generalization of Euclidean statistical notions and techniques.

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

Geometric statistics

Diffusion mean on $\mathbb S^2$

Brownian motion starting point

Approximating $p_T(v)$: Stochastic matching

bridge sampling

$v$

$x_0$

$x_t$

Højgaard,Sommer,in preparation

Højgaard,Sommer,2021,arXiv:2105.13190

Simulation of Conditioned Semimartingales on Riemannian Manifolds

From means to covariance
$SO(3)$ and $\mathbb S^2$

$\pi$

Sommer,Joshi,Højgaard, in preparation

Beyond the mean: PCA

Non-Euclidean generalizations of PCA:

• Principal Geodesic Analysis (PGA, Fletcher et al., ’04)
• Geodesic PCA (GPCA, Huckeman et al., ’10)
• Horizontal Component Analysis (HCA, Sommer, ’13)
• Principal Nested Spheres ((C)PNS, Jung et al., ’12)
• Barycentric Subspaces (BS, Pennec, ’15)

Covariance and principal components

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: $H_i:FM\to TFM$

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

Transporting along all paths

Stochastic development:

    $dU_t=\sum_{i=1}^d H_i\circ_{\mathcal S} dW_t^i$

$W_t$ Euclidean Brownian motion

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

$X_t$ supports stochastic parallel transport

Fix $T>0$: $U_T$ probability distribution in $FM$

... as opposed to geodesics only

Need measure on path space  $W ([0, T ], M )$

Covariance transport

Most probable paths

• in $\mathbb R^d$, straight lines are most probable for stationary (isotropic) diffusion processes
• Onsager-Machlup functional, $\gamma_t\in M$:$$L_M(\gamma(t),\dot{\gamma}(t))=-\frac{1}{2}\|\dot{\gamma}(t)\|^2_g+\frac{1}{12}S(\gamma(t))$$
• most probable paths maximize $L_M(\gamma)$:$$\int_0^1 L_M(\gamma(t),\dot{\gamma}(t))dt = -E(\gamma)+ \frac{1}{12}\int_0^1S(\gamma(t))dt$$

sub-Riemannian geodesics

Curvature and covariance

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

sub-Riemannian structure

Hörmander condition

Smooth density

Small time asymptotics

MPPs revisited

Grong, Sommer, in preparation, 2021

Grong, 2021

http://ipmi2021.org

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

           http://bitbucket.com/stefansommer/jaxgeometry

slides: https://slides.com/stefansommer

References:

• 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, arXiv:2106.03431, 2021.
• Arnaudon, v.d. Meulen, Schauer, Sommer: Diffusion bridges for stochastic Hamiltonian systems and shape evolutions, 2021, arXiv:2002.00885.
• 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.

Sub-Riemannian geometry and geometric statistics

• smooth, global support
• Composition:
  $k_2\ast_{W_{T/2}} (k_1\ast_{W_{T/2}} f)(u)=$
      $\mathrm{E}[k_2(-W_{T/2})k_1(-(W_T-W_{T/2}))f(U_T^u)]$
• Tensors: $k^n_m$, $f:OM\to\mathbb R^m$
       $y^n=\mathrm{E}[k^n(-W_t)f(U_T^u)]$
• Equivariance: $a\in O(d)$
      $k\ast_{W_T} (a.f)(u)=a.(k\ast_{W_T} f)(u)$
• Non-linearities $\phi_i$:
      $\phi_n(k_n\ast_{W_{T/n}} \phi_{n-1}(\cdots \phi_1(k_1\ast_{W_{T/n}} f))(u)$
• precomputed density $\rho$

Geometry and deep learning:

Intrinsic convolutions

Sommer,Bronstein,TPAMI'20