Sample-Limited $$L_p$$ Barycentric Subspace Analysis on Constant Curvature Spaces

Pennec, Xavier

doi:10.1007/978-3-319-68445-1_3

Cited by 3 publications

(7 citation statements)

References 11 publications

(15 reference statements)

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“…The L p variant of the forward (FBS), backward (PBS) and BSA algorithms were evaluated on synthetically generated data on spheres and hyperbolic spaces in [38]. The projection of a point of a sphere on a subsphere is almost always unique and corresponds to the renormalization of the projection on the Euclidean subspace containing the subsphere.…”

Section: Example Applications Of Barycentric Subspace Analysis 41 Exmentioning

confidence: 99%

“…The main drawback is the combinatorial explosion of the computational complexity with the dimension for the optimal order-k flag of affine spans, which is involving O(N k+1 ) operations, where N is the number of data points. In [38] we perform an exhaustive search, but approximate optima can be sought using a limited number of randomly sampled points [12].…”

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“…While changing the power does not change the subspace definition, it has a drastic impact on its estimation: minimizing the sum of L p distance to the subspace for non-vanishing residuals obviously changes the relative influence of points [38]. It is well known that medians are more robust than least-squares estimators: the intuitive idea is to minimize the power of residuals with 1 ≤ p ≤ 2 to minimize the influence of outliers.…”

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confidence: 99%

“…On manifolds, this would lead to a generalization of tangent-PCA maximizing the explained p-variance. In contrast, we proposed in [38] to minimize the Accumulated Unexplained p-Variance (L p AUV) over all the subspaces of the flag under consideration. Since the subspace definition is not affected by the power p, we can compare the subspaces' parameters (the reference points) for different powers.…”

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confidence: 99%

“…The three reference points of 2-PBS are represented with dark green solid circles, and the ones of 2-BSA with deep pink solid boxes. Reproduced, with permission from [38].…”

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Advances in Geometric Statistics for Manifold Dimension Reduction

Pennec

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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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 0-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. * Author pre-print of chapter to appear in Handbook of Variational Methods for Nonlinear Geometric Data, P.

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Section: Example Applications Of Barycentric Subspace Analysis 41 Exmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

“…The three reference points of 2-PBS are represented with dark green solid circles, and the ones of 2-BSA with deep pink solid boxes. Reproduced, with permission from [38].…”

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confidence: 99%

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Advances in Geometric Statistics for Manifold Dimension Reduction

Pennec

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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On the Robust PCA and Weiszfeld’s Algorithm

Neumayer

Nimmer

Setzer³

et al. 2019

Appl Math Optim

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The principal component analysis (PCA) is a powerful standard tool for reducing the dimensionality of data. Unfortunately, it is sensitive to outliers so that various robust PCA variants were proposed in the literature. This paper addresses the robust PCA by successively determining the directions of lines having minimal Euclidean distances from the data points. The corresponding energy functional is not differentiable at a finite number of directions which we call anchor directions. We derive a Weiszfeld-like algorithm for minimizing the energy functional which has several advantages over existing algorithms. Special attention is paid to the careful handling of the anchor directions, where we take the relation between local minima and one-sided derivatives of Lipschitz continuous functions on submanifolds of R d into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka-Lojasiewicz property of the energy functional, we prove global convergence of the whole sequence of iterates generated by the algorithm to a critical point of the energy functional. Numerical examples demonstrate the very good performance of our algorithm.

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The geometry of Riemannian submersions from compact Lie groups. Application to flag manifolds

Rabenoro¹,

Pennec²

2023

Preprint

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We study the geometry of Riemannian manifolds of finite dimension which are isometric to a quotient of a compact Lie group endowed with the quotient of a bi-invariant metric. Then, we apply the results obtained to enlight the geometry of flag manifolds.By the preceding equivalences, (2.3) yields the decomposition of QQ * Ω into its vertical and horizontal parts in T QQ G, which concludes the proof.

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Sample-Limited $$L_p$$ Barycentric Subspace Analysis on Constant Curvature Spaces

Cited by 3 publications

References 11 publications

Advances in Geometric Statistics for Manifold Dimension Reduction

Advances in Geometric Statistics for Manifold Dimension Reduction

On the Robust PCA and Weiszfeld’s Algorithm

The geometry of Riemannian submersions from compact Lie groups. Application to flag manifolds

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