Power manifold

A power manifold is based on a Manifold $\mathcal M$ to build a $\mathcal M^{n_1 \times n_2 \times \cdots \times n_m}$. In the case where $m=1$ we can represent a manifold-valued vector of data of length $n_1$, for example a time series. The case where $m=2$ is useful for representing manifold-valued matrices of data of size $n_1 \times n_2$, for example certain types of images.

Example

There are two ways to store the data: in a multidimensional array or in a nested array.

Let's look at an example for both. Let $\mathcal M$ be Sphere(2) the 2-sphere and we want to look at vectors of length 4.

For the default, the ArrayPowerRepresentation, we store the data in a multidimensional array,

using Manifolds
M = PowerManifold(Sphere(2), 4)
p = cat([1.0, 0.0, 0.0],
        [1/sqrt(2.0), 1/sqrt(2.0), 0.0],
        [1/sqrt(2.0), 0.0, 1/sqrt(2.0)],
        [0.0, 1.0, 0.0]
    ,dims=2)

3×4 Array{Float64,2}:
 1.0  0.707107  0.707107  0.0
 0.0  0.707107  0.0       1.0
 0.0  0.0       0.707107  0.0

which is a valid point i.e.

is_manifold_point(M, p)

true

This can also be used in combination with HybridArrays.jl and StaticArrays.jl, by setting

using HybridArrays, StaticArrays
q = HybridArray{Tuple{3,StaticArrays.Dynamic()},Float64,2}(p)

3×4 HybridArrays.HybridArray{Tuple{3,StaticArrays.Dynamic()},Float64,2,2,Array{Float64,2}} with indices SOneTo(3)×Base.OneTo(4):
 1.0  0.707107  0.707107  0.0
 0.0  0.707107  0.0       1.0
 0.0  0.0       0.707107  0.0

which is still a valid point on M and PowerManifold works with these, too.

An advantage of this representation is that it is quite efficient, especially when a HybridArray (from the HybridArrays.jl package) is used to represent a point on the power manifold. A disadvantage is not being able to easily identify parts of the multidimensional array that correspond to a single point on the base manifold. Another problem is, that accessing a single point is p[:, 1] which might be unintuitive.

For the NestedPowerRepresentation we can now do

using Manifolds
M = PowerManifold(Sphere(2), NestedPowerRepresentation(), 4)
p = [ [1.0, 0.0, 0.0],
      [1/sqrt(2.0), 1/sqrt(2.0), 0.0],
      [1/sqrt(2.0), 0.0, 1/sqrt(2.0)],
      [0.0, 1.0, 0.0],
    ]

4-element Array{Array{Float64,1},1}:
 [1.0, 0.0, 0.0]
 [0.7071067811865475, 0.7071067811865475, 0.0]
 [0.7071067811865475, 0.0, 0.7071067811865475]
 [0.0, 1.0, 0.0]

which is again a valid point so is_manifold_point(M, p) here also yields true. A disadvantage might be that with nested arrays one loses a little bit of performance. The data however is nicely encapsulated. Accessing the first data item is just p[1].

For accessing points on power manifolds in both representations you can use get_component and set_component! functions. They work work both point representations.

using Manifolds
M = PowerManifold(Sphere(2), NestedPowerRepresentation(), 4)
p = [ [1.0, 0.0, 0.0],
      [1/sqrt(2.0), 1/sqrt(2.0), 0.0],
      [1/sqrt(2.0), 0.0, 1/sqrt(2.0)],
      [0.0, 1.0, 0.0],
    ]
set_component!(M, p, [0.0, 0.0, 1.0], 4)
get_component(M, p, 4)

3-element view(::Array{Float64,1}, :) with eltype Float64:
 0.0
 0.0
 1.0

Types and Functions

Manifolds.AbstractPowerManifold — Type

AbstractPowerManifold{𝔽,M,TPR} <: Manifold{𝔽}

An abstract Manifold to represent manifolds that are build as powers of another Manifold M with representation type TPR, a subtype of AbstractPowerRepresentation.