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Advances in Geometric Statistics for Manifold Dimension Reduction

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

Geometric statistics aim at shifting the classical paradigm for inference from points in a Euclidean space to objects living in a non-linear space, in a consistent way with the underlying geometric structure considered. In this chapter, we illustrate some recent advances of geometric statistics for dimension reduction in manifolds. Beyond the mean value (the best zero-dimensional summary statistics of our data), we want to estimate higher dimensional approximation spaces fitting our data. We first define a family of natural parametric geometric subspaces in manifolds that generalize the now classical geodesic subspaces: barycentric subspaces are implicitly defined as the locus of weighted means of k + 1 reference points with positive or negative weights summing up to one. Depending on the definition of the mean, we obtain the Fréchet, Karcher or Exponential Barycentric subspaces (FBS/KBS/EBS). The completion of the EBS, called the affine span of the points in a manifold is the most interesting notion as it defines complete sub-(pseudo)-spheres in constant curvature spaces. Barycentric subspaces can be characterized very similarly to the Euclidean case by the singular value decomposition of a certain matrix or by the diagonalization of the covariance and the Gram matrices. This shows that they are stratified spaces that are locally manifolds of dimension k at regular points. Barycentric subspaces can naturally be nested by defining an ordered series of reference points in the manifold. This allows the construction of inductive forward or backward properly nested sequences of subspaces approximating data points. These flags of barycentric subspaces generalize the sequence of nested linear subspaces (flags) appearing in the classical Principal Component Analysis. We propose a criterion on the space of flags, the accumulated unexplained variance (AUV), whose optimization exactly lead to the PCA decomposition in Euclidean spaces. This procedure is called barycentric subspace analysis (BSA). We illustrate the power of barycentric subspaces in the context of cardiac imaging with the estimation, analysis and reconstruction of cardiac motion from sequences of images.

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Notes

  1. 1.

    Page 259: “It is not certain that such an element exists nor that it is unique.”

  2. 2.

    Note III on normal spaces with negative or null Riemannian curvature, p. 267.

  3. 3.

    Appliquons au point origine O les différents déplacements définis par les transformations de γ. Le groupe γ étant clos, nous obtenons ainsi une variété fermée V (qui peut se réduire à un point). Or, dans un espace de Riemann sans point singulier à distance finie, simplement connexe, a courbure negative ou nulle, on peut trouver, étant donnés des points en nombre fini, un point fixe invariant par tous les déplacements qui échangent entre eux les points donnés: c’est le point pour lequel la somme des carrés des distances au point donné est minima [4, p. 267]. Cette propriété est encore vraie si, au lieu d’un nombre fini de points, on en a une infinité formant une variété fermée: nous arrivons donc à la conclusion que le groupe γ qui laisse évidemment invariante la variété V, laisse invariant un point fixe de l’espace, il fait done partie du groupe des rotations (isométriques) autour de ce point. Mais ce groupe est homologue à g dans le groupe adjoint continu, ce qui démontre le théoreme.

  4. 4.

    p-jets are equivalent classes of functions up to order p. Thus, a p-jet specifies the Taylor expansion of a smooth function up to order p. Non-local jets, or multijets, generalize subspaces of the tangent spaces to higher differential orders with multiple base points.

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Acknowledgements

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant G-Statistics No 786854).

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Authors and Affiliations

  1. Universtié Côte d’Azur, Nice, France

    Xavier Pennec

  2. Inria, Sophia-Antipolis, France

    Xavier Pennec

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Correspondence to Xavier Pennec .

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  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Pennec, X. (2020). Advances in Geometric Statistics for Manifold Dimension Reduction. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_11

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