Manifolds

Define an interface for working on manifolds, as well as a set of predefined manifolds. The target use is optimization of differential equation integration that stay on a specific manifold at each step of the algorithm.

Example

using Manifolds
M = Sphere()
x = randn(3)
@assert norm(retract(M, x)) ≈ 1

Interface

The following functions are supported:

• retract!(M::Manifold, x): retract x, i.e. find a point close to x that belongs to the specified manifold
• project_tangent!(M::Manifold, g, x): project g onto the tangent space at x as well as out-of-place versions retract(M,x) and project_tangent(M, g, x).

Supported manifolds

The following manifolds are currently supported:

• Flat: Euclidean space, default. The operations are no-ops.
• Sphere: spherical constraint ||x|| = 1.
• Stiefel: Stiefel manifold of N by n matrices with orthogonal columns, i.e. X'*X = I. The default algorithm for retractions on the manifold is to use a SVD, but a faster (although less stable) algorithm is available by calling Stiefel(:CholQR).

The following meta-manifolds construct manifolds out of pre-existing ones:

• PowerManifold: identical copies of a specified manifold
• ProductManifold: product of two (potentially different) manifolds

Implementing new manifolds is as simple as adding methods project_tangent!(M::YourManifold,g,x) and retract!(M::YourManifold,x). If you implement another manifold, please contribute a PR!

References

The Geometry of Algorithms with Orthogonality Constraints, Alan Edelman, Tomás A. Arias, Steven T. Smith, SIAM. J. Matrix Anal. & Appl., 20(2), 303–353

Optimization Algorithms on Matrix Manifolds, P.-A. Absil, R. Mahony, R. Sepulchre, Princeton University Press, 2008