Geometry and Statistics: Manifolds and Stratified Spaces

Feragen, Aasa; Nielsen, Mads; Jensen, Eva B. Vedel; Plessis, A. A. Du; Lauze, François

doi:10.1007/s10851-014-0504-5

Cited by 2 publications

(2 citation statements)

References 28 publications

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“…For the landmarkbased representation, the shape space is called Kendall's shape space (Kendall, 1984). Kendall's shape space is in general a stratified space (Goresky and MacPherson, 1988;Feragen et al, 2014), but for the special case of planar shapes, the shape space is the complex projective space, which is a complex Grassmann manifold. The examples mentioned above are often high-dimensional: a YANG AND VEMURI DTI scan usually contains half a million DTs; the shape of the Corpus Callosum (which is used in our experiments) is represented by a several hundreds of boundary points in R 2 .…”

Section: Introductionmentioning

confidence: 99%

Nested Grassmannians for Dimensionality Reduction with Applications

Yang

Vemuri

2022

Melba

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In the recent past, nested structure of Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested spheres. In this paper, we propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifolds. Common examples of homogeneous Riemannian manifolds include the spheres, the Stiefel manifolds, and the Grassmann manifolds. In particular, we focus on applying the proposed framework to the Grassmann manifolds, giving rise to the nested Grassmannians (NG). An important application in which Grassmann manifolds are encountered is planar shape analysis. Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grassmann manifold. Some salient features of our framework are: (i) it explicitly exploits the geometry of the homogeneous Riemannain manifolds and (ii) the nested lower-dimensional submanifolds need not be geodesic. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively. The proposed algorithms are compared with PGA via simulation studies and real data experiments and are shown to achieve a higher ratio of expressed variance compared to PGA.<br>The code is available at <a href='https://github.com/cvgmi/NestedGrassmann'>https://github.com/cvgmi/NestedGrassmann</a>

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Section: Introductionmentioning

confidence: 99%

Nested Grassmannians for Dimensionality Reduction with Applications

Yang

Vemuri

2022

Melba

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“…Most statistical analysis is carried out on data lying in a Euclidean vector space, or more generally, a smooth manifold. Manifold-stratified spaces, however, are playing an increasingly important role in applications and have recently attracted substantial research interest [1,2,3]. In a manifold-stratified space, each stratum is a manifold, and strata are glued together along their boundaries.…”

Section: Introductionmentioning

confidence: 99%

Random walks and Brownian motion on cubical complexes

Nye

2020

Stochastic Processes and their Applications

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Geometry and Statistics: Manifolds and Stratified Spaces

Cited by 2 publications

References 28 publications

Nested Grassmannians for Dimensionality Reduction with Applications

Nested Grassmannians for Dimensionality Reduction with Applications

Random walks and Brownian motion on cubical complexes

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