Manifold definitions {ManifoldOptim}

Manifold definitions

Description

Get definitions for simple manifolds

Usage

get.stiefel.defn(n, p, numofmani = 1L, ParamSet = 1L)

get.grassmann.defn(n, p, numofmani = 1L, ParamSet = 1L)

get.spd.defn(n, numofmani = 1L, ParamSet = 1L)

get.sphere.defn(n, numofmani = 1L, ParamSet = 1L)

get.euclidean.defn(n, m, numofmani = 1L, ParamSet = 1L)

get.lowrank.defn(n, m, p, numofmani = 1L, ParamSet = 1L)

get.orthgroup.defn(n, numofmani = 1L, ParamSet = 1L)

Arguments

n

Dimension for manifold object (see Details)

p

Dimension for manifold object (see Details)

numofmani

Multiplicity of this space. For example, use numofmani = 2 if problem requires 2 points from this manifold

ParamSet

A positive integer indicating a set of properties for the manifold which can be used by the solver. See Huang et al (2016b) for details.

m

Dimension for manifold object (see Details)

Details

The functions define manifolds as follows:

• get.stiefel.defn: Stiefel manifold \{X \in R^{n \times p} : X^T X = I\}

• get.grassmann.defn: Grassmann manifold of p-dimensional subspaces in R^n

• get.spd.defn: Manifold of n \times n symmetric positive definite matrices

• get.sphere.defn: Manifold of n-dimensional vectors on the unit sphere

• get.euclidean.defn: Euclidean R^{n \times m} space

• get.lowrank.defn: Low-rank manifold \{ X \in R^{n \times m} : \textrm{rank}(X) = p \}

• get.orthgroup.defn: Orthonormal group \{X \in R^{n \times n} : X^T X = I\}

Value

List containing input arguments and name field denoting the type of manifold

References

Wen Huang, P.A. Absil, K.A. Gallivan, Paul Hand (2016a). "ROPTLIB: an object-oriented C++ library for optimization on Riemannian manifolds." Technical Report FSU16-14, Florida State University.

Wen Huang, Kyle A. Gallivan, and P.A. Absil (2016b). Riemannian Manifold Optimization Library. URL https://www.math.fsu.edu/~whuang2/pdf/USER_MANUAL_for_2016-04-29.pdf

S. Martin, A. Raim, W. Huang, and K. Adragni (2020). "ManifoldOptim: An R Interface to the ROPTLIB Library for Riemannian Manifold Optimization." Journal of Statistical Software, 93(1):1-32.