Barycentric Subspaces and Affine Spans in Manifolds

28/10/2015
Auteurs : Xavier Pennec
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14316

Résumé

This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a “Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter). This definition restores the full symmetry between all parameters of the subspaces, contrarily to the geodesic subspaces which intrinsically privilege one point. We show that this definition defines locally a submanifold of dimension k and that it generalizes in some sense geodesic subspaces. Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean. As this definition relies on points and do not explicitly refer to tangent vectors, it can be extended to non Riemannian geodesic spaces. For instance, principal subspaces may naturally span over several strata in stratified spaces, which is not the case with more classical generalizations of PCA.

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This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a “Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter). This definition restores the full symmetry between all parameters of the subspaces, contrarily to the geodesic subspaces which intrinsically privilege one point. We show that this definition defines locally a submanifold of dimension k and that it generalizes in some sense geodesic subspaces. Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean. As this definition relies on points and do not explicitly refer to tangent vectors, it can be extended to non Riemannian geodesic spaces. For instance, principal subspaces may naturally span over several strata in stratified spaces, which is not the case with more classical generalizations of PCA.

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Barycentric Subspaces and Affine Spans in Manifolds GSI 30-10-2015 Xavier Pennec Asclepios team, INRIA Sophia-Antipolis – Mediterranée, France and Côte d’Azur University (UCA) Statistical Analysis of Geometric Features Computational Anatomy deals with noisy Geometric Measures  Tensors, covariance matrices  Curves, tracts  Surfaces, shapes  Images  Deformations Data live on non-Euclidean manifolds X. Pennec - GSI 2015 2  Manifold dimension reduction  When embedding structure is already manifold (e.g. Riemannian): Not manifold learning (LLE, Isomap,…) but submanifold learning Low dimensional subspace approximation? X. Pennec - GSI 2015 3 Manifold of cerebral ventricles Etyngier, Keriven, Segonne 2007. Manifold of brain images S. Gerber et al, Medical Image analysis, 2009. X. Pennec - GSI 2015 4 Barycentric Subspaces and Affine Spans in Manifolds PCA in manifolds: tPCA / PGA / GPCA / HCA Affine span and barycentric subspaces Conclusion 5 Bases of Algorithms in Riemannian Manifolds Reformulate algorithms with Expx and Logx Vector -> Bi-point (no more equivalence classes) Exponential map (Normal coordinate system):  Expx = geodesic shooting parameterized by the initial tangent  Logx = development of the manifold in the tangent space along geodesics  Geodesics = straight lines with Euclidean distance  Local  global domain: star-shaped, limited by the cut-locus  Covers all the manifold if geodesically complete 6 Statistical tools: Moments Frechet / Karcher mean minimize the variance 