1. Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations ECCV 2010 Stefan Sommer 1, François Lauze 1, Søren Hauberg 1, Mads Nielsen 1,2 1 : Department of Computer Science, University of Copenhagen 2 : Nordic Bioscience Imaging, Herlev, Denmark

2. 2 A Glimpse of Manifold Valued Statistics ● Importance of Non-Linearity in Modeling ● Statistics and Computations on Manifolds ● Principal Geodesic Analysis (PGA) ● Computing it Right: Exact PGA ● The Indicators: How Linear is It? ● Example: Modeling Human Poses in End- Effector Space

3. 3 Non-linearity in Modeling ● (Riemannian) manifolds generalizes Euclidean spaces ● Examples: m-rep models, motion capture data, diffeomorphisms in neuroimaging, shapes... ● When non-linearity is present, manifolds are needed: linearity is not compatible Images from: Fletcher et al. '04, Sommer et al. '09/'10, Wikipedia

4. 4 Statistics on Manifolds is Complex Problems: ● Generalizing Euclidean statistics ● no vector space structure ● kNN, PGA, SVM, regression ● lots of unsolved problems ● Computations ● exponential map, geodesics, ● distances ● optimization on geodesics: ● Sommer et al. 2010b ● provides optimization tools

5. 5 Principal Geodesic Analysis (PGA) ● PGA generalizes PCA ● dependent on manifold: ● curvature, metric, etc. ● finds geodesic subspaces ● (PCA finds linear subspaces) ● localized to intrinsic mean ● PCA: ● orthogonal projections in vector space ● PGA: ● projections using manifold distances

6. 6 Computing it Right: Exact PGA ● standard approximation: linearized PGA ● approximation amounts to PCA in tangent space: ● no approximation: exact PGA ● optimization problem in tangent space ● previously unsolved for general manifolds ● Sommer et. al 2010b provides optimization tools – calculate gradients using IVPs for Jacobi fields and 2. order derivatives – perform gradient descent or similar optimization ● computationally heavy ● reveals numerous interesting non-Euclidean effects ● (residual/variance formulation, decreasing variance, orthogonality, ● see Sommer et. al 2010a/b) first principal direction (variance formulation) data points

7. 7 ● difference between exact and linearized PGA ● curvature in relation to spread of data ● can we estimate the magnitude? ● save resources when linearized is sufficient ● improve precision when difference is significant ● understand effect of curvature on statistics ● indicators for fast prediction ● indicator: orthogonal/manifold projection difference ● indicator: exact/linearized difference The Indicators: How Linear is It? geodesic distance tangent space distance manifold projection orthogonal projection

8. 8 Example: Human Pose Manifold ● spatial coordinates of end-effectors ● fixed bone lengths introduces constraints ● implicitly represented manifold ● indicators against true differences: Camera output superimposed with tracking result Pose with end-effectors

9. 9 Example: Human Pose Manifold ● Dataset: movement of right arm ● First principal direction: Difference: exact PGA: blue linearized PGA: red movement along first exact principal direction

10. 10 Concluding Remarks ● solve general optimization problems involving ● geodesics, including exact PGA ● pitfalls, e.g. orthogonality and variance: ● difference between exact and linearized PGA when both curvature and spread is present ● indicators predict the difference ● linear/exact: commonly occuring theme in manifold statistics References: ( available at http://image.diku.dk/sommer/)http://image.diku.dk/sommer/ ● Sommer et al. 2010a: “Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations” Stefan Sommer, François Lauze, Søren Hauberg, Mads Nielsen ECCV 2010 ● Sommer et al. 2010b: “The Differential of the Exponential Map, Jacobi Fields, and Exact Principal Geodesic Analysis” Stefan Sommer, François Lauze, Mads Nielsen (Preprint)