Statistical analysis on high-dimensional spheres and shape spaces

By Ian L. Dryden. 2005

84 citations recorded by the source corpus.

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available—a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on infinite-dimensional spheres and shape spaces are introduced, and special cases which have important practical consequences are considered. We focus on the high-dimensional real and complex Bingham, uniform, von Mises–Fisher, Fisher–Bingham and the real and complex Watson distributions. Asymptotic distributions in the cases where dimension and sample size are large are discussed. Approximations for practical maximum likelihood based inference are considered, and in particular we discuss an application to brain shape modeling.

Subfields: Applied Mathematics, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e1be4a8a2c7b97294d5e9de9ecc112dc1fede4e3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Xavier Pennec
2. Dominique Picard
3. Gerard Kerkyacharian
4. Axel Munk
5. Thomas Hotzand
6. Georgia Augusta
7. Domenico Marinucci
8. Stephan F. Huckemann

In Artificial Intelligence

1. Dominique Picard
2. Xavier Pennec
3. Domenico Marinucci
4. Gerard Kerkyacharian
5. Axel Munk
6. Stephan F. Huckemann

In Geometry and Topology

1. Stephan F. Huckemann
2. Xavier Pennec
3. Axel Munk
4. Thomas Hotzand
5. Georgia Augusta
6. Domenico Marinucci
7. Benjamin Eltzner
8. Sarang C. Joshi

Citations in the corpus

• INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, background): “Higher dimensional components would then be sub-manifolds spanned by geodesics, totally geodesic at a point, as proposed by Fletcher and Joshi (2007) and computed in approximation: eigenspaces of the respective covariance matrix in the tangent space at the IM are mapped to geodesics and submanifolds”
• Asymptotics for spherical needlets (2006, background): “Examples of spherical data appear also in other areas of the astrophysical sciences [see Angers and Kim (2005)] or outside astrophysics, that is, brain shape modeling and image analysis [see, e.g., Mardia and Patrangenaru (2005), Dryden (2005) and Dette, Melas and Pepelyshev (2005)].”
• Barycentric subspace analysis on manifolds (2016, background): “Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 1643–1665. Eltzner, B., Jung, S. and Huckemann, S. (2015). Dimension Reduction on Polyspheres with Application to Skeletal Representations. In Geometric Science of Information (F. Nielsen and F.”