# Power manifold

A power manifold is based on a AbstractManifold $\mathcal M$ to build a $\mathcal M^{n_1×n_2 ×⋯×n_m}$. 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 $n_1×n_2$, for example certain types of images.

There are three available representations for points and vectors on a power manifold:

Below are some examples of usage of these representations.

## 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.

### ArrayPowerRepresentation

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 Matrix{Float64}:
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_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.HybridMatrix{3, StaticArraysCore.Dynamic(), Float64, 2, Matrix{Float64}} 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.

### NestedPowerRepresentation

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 Vector{Vector{Float64}}:
[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_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 Vector{Float64}:
0.0
0.0
1.0

### NestedReplacingPowerRepresentation

The final representation is the NestedReplacingPowerRepresentation. It is similar to the NestedPowerRepresentation but it does not perform in-place operations on the points on the underlying manifold. The example below uses this representation to store points on a power manifold of the SpecialEuclidean group in-line in an Vector for improved efficiency. When having a mixture of both, i.e. an array structure that is nested (like ´NestedPowerRepresentation) in the sense that the elements of the main vector are immutable, then changing the elements can not be done in an in-place way and hence NestedReplacingPowerRepresentation has to be used.

using Manifolds, StaticArrays
R2 = Rotations(2)

G = SpecialEuclidean(2)
N = 5
GN = PowerManifold(G, NestedReplacingPowerRepresentation(), N)

q = [1.0 0.0; 0.0 1.0]
p1 = [ArrayPartition(SVector{2,Float64}([i - 0.1, -i]), SMatrix{2,2,Float64}(exp(R2, q, hat(R2, q, i)))) for i in 1:N]
p2 = [ArrayPartition(SVector{2,Float64}([i - 0.1, -i]), SMatrix{2,2,Float64}(exp(R2, q, hat(R2, q, -i)))) for i in 1:N]

X = similar(p1);

log!(GN, X, p1, p2)
5-element Vector{ArrayPartition{Float64, Tuple{StaticArraysCore.SVector{2, Float64}, StaticArraysCore.SMatrix{2, 2, Float64, 4}}}}:
([0.0, 0.0], [0.0 2.0; -2.0 0.0])
([0.0, 0.0], [0.0 -2.2831853071795862; 2.2831853071795862 0.0])
([0.0, 0.0], [0.0 -0.28318530717958645; 0.28318530717958645 0.0])
([0.0, 0.0], [0.0 1.7168146928204135; -1.7168146928204135 0.0])
([0.0, 0.0], [0.0 -2.566370614359173; 2.566370614359173 0.0])

## Types and Functions

Manifolds.PowerFVectorDistributionType
PowerFVectorDistribution([type::VectorSpaceFiber], [x], distr)

Generates a random vector at a point from vector space (a fiber of a tangent bundle) of type type using the power distribution of distr.

Vector space type and point can be automatically inferred from distribution distr.

source
ManifoldDiff.riemannian_HessianMethod
Y = riemannian_Hessian(M::AbstractPowerManifold, p, G, H, X)
riemannian_Hessian!(M::AbstractPowerManifold, Y, p, G, H, X)

Compute the Riemannian Hessian $\operatorname{Hess} f(p)[X]$ given the Euclidean gradient $∇ f(\tilde p)$ in G and the Euclidean Hessian $∇^2 f(\tilde p)[\tilde X]$ in H, where $\tilde p, \tilde X$ are the representations of $p,X$ in the embedding,.

On an abstract power manifold, this decouples and can be computed elementwise.

source
Manifolds.flatMethod
flat(M::AbstractPowerManifold, p, X)

use the musical isomorphism to transform the tangent vector X from the tangent space at p on an AbstractPowerManifold M to a cotangent vector. This can be done elementwise for each entry of X (and p).

source
Manifolds.sharpMethod
sharp(M::AbstractPowerManifold, p, ξ::RieszRepresenterCotangentVector)

Use the musical isomorphism to transform the cotangent vector ξ from the tangent space at p on an AbstractPowerManifold M to a tangent vector. This can be done elementwise for every entry of ξ (and p).

source