Manifold-definitions function

Manifold definitions

Get definitions for simple manifolds

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)

Returns

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

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 $\{X \in R^{n \times n} : X^T X = I\}$

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.