Evolution Equations with Anisotropic Distributions and Diffusion PCA

28/10/2015
Auteurs : Stefan Sommer
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14315

Résumé

This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracket-generating) sub-Riemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the sub-Riemannian geometry are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

Evolution Equations with Anisotropic Distributions and Diffusion PCA

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This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracket-generating) sub-Riemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the sub-Riemannian geometry are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

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Faculty of Science Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations GSI 2015, Paris, France Stefan Sommer Department of Computer Science, University of Copenhagen October 29, 2015 Slide 1/21 Intrinsic Statistics in Geometric Spaces Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 2/21 Statistics on Manifolds • Frech´et mean: argminx∈M 1 N ∑N i=1 d(x,yi)2 • PGA (Fletcher et al., ’04); GPCA (Huckeman et al., ’10); HCA (Sommer, ’13); PNS (Jung et al., ’12); BS (Pennec, ’15) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 3/21 PGA GPCA HCA Infinitesimally defined Distributions; MLE • aim: construct a family NM(µ,Σ) of anisotropic Gaussian-like distributions; fit by MLE/MAP • in Rn , Gaussian distributions are transition distributions of diffusion processes dXt = dWt • on (M,g), Brownian motion is transition distribution of stochastic process (Eells-Elworthy-Malliavin construction), or solution to heat diffusion equation ∂ ∂t p(t,x) = 1 2 ∆p(t,x) • infinitesimal dXt vs. global pt (x;y) ∝ e− x−y 2 Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 4/21 MLE of Diffusion Processes • Eells-Elworthy-Malliavin construction gives map Diff : FM → Dens(M) • Diff(FM) = NM ⊂ Dens(M): the set of (normalized) transition densities from FM diffusions • γ = Diff(x,Xα) = pγγ0, the log-likelihood lnL(x,Xα) = lnL(γ) = N ∑ i=1 lnpγ(yi) • Estimated Template: argmax(x,Xα)∈FM lnL(x,Xα) • MLE of data yi under the assumption y ∼ γ ∈ NM • Diffusion PCA (Sommer ’14): argmax lnL(x,Xα +εI) generalizing Probabilistic PCA (Tipping, Bishop, ’99; Zhang, Fletcher ’13) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 5/21 Most Probable Paths to Samples • Euclidean: • density pt (x;y) ∝ e−(x−y)T Σ−1 (x−y) • transition density of diffusion processes with stationary generator • x −y most probable path from y to x • Manifolds: • which distributions correspond to anisotropic Gaussian distributions N(x,Σ)? • what is the most probable path from y to x? Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 6/21 Anisotropic Diffusions and Holonomy • driftless diffusion SDE in Rn , stationary generator: dXt = σdWt , σ ∈ Mn×d • diffusion field σ, infinitesimal generator σσT • curvature: stationary field/generator cannot be defined due to holonomy Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 7/21 Stochastic Development: Eells-Elworthy-Malliavin Construction • Xt : Rn valued Brownian motion (driving process) • Ut : FM valued (sub-elliptic) diffusion • Yt : M valued stochastic process (target process) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 8/21 The Frame Bundle • the manifold and frames (bases) for the tangent spaces TpM • F(M) consists of pairs u = (x,Xα), x ∈ M, Xα frame for Tx M • curves in the horizontal part of F(M) correspond to curves in M and parallel transport of frames Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 9/21 Driving process, FM valued process and Target process • Hi, i = 1...,n horizontal vector fields on F(M): Hi(u) = π−1 ∗ (ui) • SDE in Rn (driving): dXt = IdndBt , X0 = 0 • SDE in FM: dUt = Hi(Ut )◦dXi t , U0 = (x0,Xα) , Xα ∈ GL(Rn ,Tx0M) • Process on M (target): Yt = πFM(Ut ) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 10/21 Ut: Frame Bundle Diffusion Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 11/21 Estimated Templates Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 12/21 MLE template Most Probable Paths • in Rn , straight lines are most probable for stationary diffusion processes • Onsager-Machlup functional, σt curve on M: L(σt ) = − 1 2 σ (t) 2 g + 1 12 R(σ(t)) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 13/21 Most Probable Paths • in Rn , straight lines are most probable for stationary diffusion processes • Onsager-Machlup functional, σt curve on M: L(σt ) = − 1 2 σ (t) 2 g + 1 12 R(σ(t)) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 13/21 Most Probable Paths • in Rn , straight lines are most probable for stationary diffusion processes • Onsager-Machlup functional, σt curve on M: L(σt ) = − 1 2 σ (t) 2 g + 1 12 R(σ(t)) • MPP for target process Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 13/21 Most Probable Paths • in Rn , straight lines are most probable for stationary diffusion processes • Onsager-Machlup functional, σt curve on M: L(σt ) = − 1 2 σ (t) 2 g + 1 12 R(σ(t)) • MPP for driving process Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 13/21 R=0 Definition (MPPs for Driving Process) Let Xt be the driving process for the diffusion Yt and x ∈ M, i.e. Yt = π(φ(Xt )). Then σ is a most probable path for the driving process if it satisfies σ = argminc∈H(Rd ),φ(c)(1)=x 1 0 −L(ct )dt Proposition Let Yα be a frame for Ty M, and let Yt = π(φ(y,Yα)(Xt )), i.e. Yt is the development of Xt starting at (y,Yα). Then MPPs for the driving process Xt maps to geodesics of a lifted sub-Riemannian metric on FM: w, ˜w FM = X−1 α π∗w,X−1 α π∗ ˜w Rn . • isotropic case, MPPs for drv. process maps to geodesics • if −lnL(x,Xα) ≈ c + 1 N ∑N i=1 p(MPP(x,yi )). Then Frech´et mean ≈ MLE, isotropic case Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 14/21 MPPs on S2 increasing anisotropy −→ (a) cov. diag(1,1) (b) cov. diag(2,.5) (c) cov. diag(4,.25) Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 15/21 Sub-Riemannian Geometry on FM • Xα : Rn → Tx M gives inner-product v,w Xα = X−1 α v,X−1 α w Rn • optimal control problem with nonholonomic constraints xt = arg min ct ,c0=x,c1=y 1 0 ˙ct 2 Xα,t dt • let ˜v, ˜w HFM = X−1 α,t π∗(˜v),X−1 α,t π∗(˜w) Rn on H(xt ,Xα,t )FM. This defines a sub-Riemannian metric G on TFM and equivalent problem (xt ,Xα,t ) = arg min (ct ,Cα,t ),c0=x,c1=y 1 0 (˙ct , ˙Cα,t ) 2 HFMdt with horizontality constraint (˙ct , ˙Cα,t ) ∈ H(ct ,Cα,t )FM Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 16/21 MPP Evolution Equations • sub-Riemannian Hamilton-Jacobi equations ˙yk t = Gkj (yt )ξt,j , ˙ξt,k = − 1 2 ∂Gpq ∂yk ξt,pξt,q • in coordinates (xi ) for M, Xi α for Xα, and W encoding the inner product Wkl = δαβXk αXl β: ˙xi = Wij ξj −Wih Γ jβ h ξjβ , ˙Xi α = −Γiα h Whj ξj +Γiα k Wkh Γ jβ h ξjβ ˙ξi = Whl Γ kδ l,i ξhξkδ − 1 2 Γ hγ k,iWkh Γ kδ h +Γ hγ k Wkh Γ kδ h,i ξhγξkδ ˙ξiα = Γ hγ k,iα Wkh Γ kδ h ξhγξkδ − Whl ,iα Γ kδ l +Whl Γ kδ l,iα ξhξkδ − 1 2 Whk ,iα ξhξk +Γ hγ k Wkh ,iα Γ kδ h ξhγξkδ Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 17/21 Landmark LDDMM • Christoffel symbols (Michelli et al. ’08) Γk ij = 1 2 gir gkl grs ,l −gsl grk ,l −grl gks ,l gsj • mix of transported frame and cometric: Fd M bundle of rank d linear maps Rd → Tx M, ξ,˜ξ ∈ T∗Fd M, cometric gFd M +λgR: ξ,˜ξ = δαβ (ξ|π−1 ∗ Xα)(˜ξ|π−1 ∗ Xβ)+λ ξ,˜ξ gR • the whole frame need not be transported Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 18/21 LDDMM Landmark MPPs Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 19/21 + horz. var. isotropic + vert. var. Statistical Manifold: Geometry of Γ • Densities Dens(M) = {γ ∈ Ωn (M) : M γ = 1,γ > 0} • Fisher-Rao metric: GFR γ (α,β) = M α γ β γ γ • Γ finite dim. subset of Dens(M) Diff : FM → Dens(M) • naturally defined on bundle of symmetric positive T0 2 tensors Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 20/21 Summary • infinitesimal definition of anisotropic normal distributions NM(µ,Σ) on M • diffusion map Diff : FM → Dens(M) from Eells-Elworthy-Malliavin construction, stoch. develop. • MLE of template / covariance (in FM) • MPPs for driving processes generalize geodesics being sub-Riemannian geodesics 1 Sommer: Diffusion Processes and PCA on Manifolds, Oberwolfach extended abstract (Asymptotic Statistics on Stratified Spaces), 2014. 2 Sommer: Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths, Information Processing in Medical Imaging (IPMI) 2015. 3 Sommer: Evolution Equations with Anisotropic Distributions and Diffusion PCA, Geometric Science of Information (GSI) 2015. 4 Svane, Sommer: Similarities, SDEs, and Most Probable Paths, SIMBAD15 extended abstract. 5 Sommer, Svane: Holonomy, Curvature, and Anisotropic Diffusions, MOTR15 extended abstract. Stefan Sommer (sommer@diku.dk) (Department of Computer Science, University of Copenhagen) — Anisotropic Distributions on Manifolds, Diffusion PCA, and Evolution Equations Slide 21/21