Global Tangent Spaces

AbstractLieAlgHorMatrix <: AbstractMatrix

AbstractLieAlgHorMatrix is a supertype for various horizontal components of Lie algebras. We usually call this $\mathfrak{g}^\mathrm{hor}$.

See StiefelLieAlgHorMatrix and GrassmannLieAlgHorMatrix for concrete examples.

StiefelLieAlgHorMatrix(A::SkewSymMatrix, B::AbstractMatrix, N::Integer, n::Integer)

Build an instance of StiefelLieAlgHorMatrix based on a skew-symmetric matrix A and an arbitrary matrix B.

An element of StiefelLieAlgMatrix takes the form:

\[\begin{pmatrix} A & B^T \\ B & \mathbb{O} \end{pmatrix},\]

where $A$ is skew-symmetric (this is SkewSymMatrix in GeometricMachineLearning).

Also see GrassmannLieAlgHorMatrix.

Extended help

StiefelLieAlgHorMatrix is the horizontal component of the Lie algebra of skew-symmetric matrices (with respect to the canonical metric).

The projection here is: $\pi:S \to SE$ where

\[E = \begin{bmatrix} \mathbb{I}_{n} \\ \mathbb{O}_{(N-n)\times{}n} \end{bmatrix}.\]

The matrix $E$ is implemented under StiefelProjection in GeometricMachineLearning.

StiefelLieAlgHorMatrix(D::AbstractMatrix, n::Integer)

Take a big matrix as input and build an instance of StiefelLieAlgHorMatrix.

The integer $N$ in $St(n, N)$ is the number of rows of D.

Extended help

If the constructor is called with a big $N\times{}N$ matrix, then the projection is performed the following way:

\[\begin{pmatrix} A & B_1 \\ B_2 & D \end{pmatrix} \mapsto \begin{pmatrix} \mathrm{skew}(A) & -B_2^T \\ B_2 & \mathbb{O} \end{pmatrix}.\]

The operation $\mathrm{skew}:\mathbb{R}^{n\times{}n}\to\mathcal{S}_\mathrm{skew}(n)$ is the skew-symmetrization operation. This is equivalent to calling of SkewSymMatrix with an $n\times{}n$ matrix.

This can also be seen as the operation:

\[D \mapsto \Omega(E, DE) = \mathrm{skew}\left(2 \left(\mathbb{I} - \frac{1}{2} E E^T \right) DE E^T\right).\]

Also see GeometricMachineLearning.Ω.

GrassmannLieAlgHorMatrix(B::AbstractMatrix, N::Integer, n::Integer)

Build an instance of GrassmannLieAlgHorMatrix based on an arbitrary matrix B of size $(N-n)\times{}n$.

GrassmannLieAlgHorMatrix is the horizontal component of the Lie algebra of skew-symmetric matrices (with respect to the canonical metric).

Extended help

The projection here is: $\pi:S \to SE/\sim$ where

\[E = \begin{bmatrix} \mathbb{I}_{n} \\ \mathbb{O}_{(N-n)\times{}n} \end{bmatrix},\]

and the equivalence relation is

\[V_1 \sim V_2 \iff \exists A\in\mathcal{S}_\mathrm{skew}(n) \text{ such that } V_2 = V_1 + \begin{bmatrix} A \\ \mathbb{O} \end{bmatrix}\]

An element of GrassmannLieAlgMatrix takes the form:

\[\begin{pmatrix} \bar{\mathbb{O}} & B^T \\ B & \mathbb{O} \end{pmatrix},\]

where $\bar{\mathbb{O}}\in\mathbb{R}^{n\times{}n}$ and $\mathbb{O}\in\mathbb{R}^{(N - n)\times(N-n)}.$

GrassmannLieAlgHorMatrix(D::AbstractMatrix, n::Integer)

Take a big matrix as input and build an instance of GrassmannLieAlgHorMatrix.

The integer $N$ in $Gr(n, N)$ here is the number of rows of D.

Extended help

If the constructor is called with a big $N\times{}N$ matrix, then the projection is performed the following way:

\[\begin{pmatrix} A & B_1 \\ B_2 & D \end{pmatrix} \mapsto \begin{pmatrix} \bar{\mathbb{O}} & -B_2^T \\ B_2 & \mathbb{O} \end{pmatrix}.\]

This can also be seen as the operation:

\[D \mapsto \Omega(E, DE - EE^TDE),\]

where $\Omega$ is the horizontal lift GeometricMachineLearning.Ω.

GlobalSection(Y)

Construct a global section for Y.

A global section $\lambda$ is a mapping from a homogeneous space $\mathcal{M}$ to the corresponding Lie group $G$ such that

\[\lambda(Y)E = Y,\]

Also see apply_section and global_rep.

Implementation

For an implementation of GlobalSection for a custom array (especially manifolds), the function global_section has to be generalized.

Matrix(λY::GlobalSection)

Put λY into matrix form.

This is not recommended if speed is important!

Use apply_section and global_rep instead!

apply_section(λY::GlobalSection{T, AT}, Y₂::AT) where {T, AT <: StiefelManifold{T}}

Apply λY to Y₂.

Mathematically this is the group action of the element $\lambda{}Y\in{}G$ on the element $Y_2$ of the homogeneous space $\mathcal{M}$.

Internally it calls apply_section!.

apply_section!(Y::AT, λY::GlobalSection{T, AT}, Y₂::AT) where {T, AT<:StiefelManifold{T}}

Apply λY to Y₂ and store the result in Y.

This is the inplace version of apply_section.

λY * Y

Apply the element λY onto Y.

Here λY is an element of a Lie group and Y is an element of a homogeneous space.

global_section(Y::StiefelManifold)

Compute a matrix of size $N\times(N-n)$ whose columns are orthogonal to the columns in Y.

This matrix is also called $Y_\perp$ [20, 22, 23].

Examples

using GeometricMachineLearning
using GeometricMachineLearning: global_section
import Random

Random.seed!(123)

Y = StiefelManifold([1. 0.; 0. 1.; 0. 0.; 0. 0.])

round.(Matrix(global_section(Y)); digits = 3)

# output

4×2 Matrix{Float64}:
 0.0    -0.0
 0.0     0.0
 0.936  -0.353
 0.353   0.936

Further note that we convert the QRCompactWYQ object to a Matrix before we display it.

Implementation

The implementation is done with a QR decomposition (LinearAlgebra.qr!). Internally we do:

A = randn(N, N - n) # or the gpu equivalent
A = A - Y.A * (Y.A' * A)
qr!(A).Q

global_section(Y::GrassmannManifold)

Compute a matrix of size $N\times(N-n)$ whose columns are orthogonal to the columns in Y.

The method global_section for the Grassmann manifold is equivalent to that for the StiefelManifold (we represent the Grassmann manifold as an embedding in the Stiefel manifold).

See the documentation for global_section(Y::StiefelManifold{T}) where T.

global_section(Y::StiefelManifold)

Compute a matrix of size $N\times(N-n)$ whose columns are orthogonal to the columns in Y.

This matrix is also called $Y_\perp$ [20, 22, 23].

Examples

using GeometricMachineLearning
using GeometricMachineLearning: global_section
import Random

Random.seed!(123)

Y = StiefelManifold([1. 0.; 0. 1.; 0. 0.; 0. 0.])

round.(Matrix(global_section(Y)); digits = 3)

# output

4×2 Matrix{Float64}:
 0.0    -0.0
 0.0     0.0
 0.936  -0.353
 0.353   0.936

Further note that we convert the QRCompactWYQ object to a Matrix before we display it.

Implementation

The implementation is done with a QR decomposition (LinearAlgebra.qr!). Internally we do:

A = randn(N, N - n) # or the gpu equivalent
A = A - Y.A * (Y.A' * A)
qr!(A).Q

global_section(Y::GrassmannManifold)

Compute a matrix of size $N\times(N-n)$ whose columns are orthogonal to the columns in Y.

The method global_section for the Grassmann manifold is equivalent to that for the StiefelManifold (we represent the Grassmann manifold as an embedding in the Stiefel manifold).

See the documentation for global_section(Y::StiefelManifold{T}) where T.

global_rep(λY::GlobalSection{T, AT}, Δ::AbstractMatrix{T}) where {T, AT<:StiefelManifold{T}}

Express Δ (an the tangent space of Y) as an instance of StiefelLieAlgHorMatrix.

This maps an element from $T_Y\mathcal{M}$ to an element of $\mathfrak{g}^\mathrm{hor}$.

These two spaces are isomorphic where the isomorphism where the isomorphism is established through $\lambda(Y)\in{}G$ via:

\[T_Y\mathcal{M} \to \mathfrak{g}^{\mathrm{hor}}, \Delta \mapsto \lambda(Y)^{-1}\Omega(Y, \Delta)\lambda(Y).\]

Also see GeometricMachineLearning.Ω.

Examples

using GeometricMachineLearning
using GeometricMachineLearning: _round
import Random 

Random.seed!(123)

Y = rand(StiefelManifold, 6, 3)
Δ = rgrad(Y, randn(6, 3))
λY = GlobalSection(Y)

_round(global_rep(λY, Δ); digits = 3)

# output

6×6 StiefelLieAlgHorMatrix{Float64, SkewSymMatrix{Float64, Vector{Float64}}, Matrix{Float64}}:
  0.0     0.679   1.925   0.981  -2.058   0.4
 -0.679   0.0     0.298  -0.424   0.733  -0.919
 -1.925  -0.298   0.0    -1.815   1.409   1.085
 -0.981   0.424   1.815   0.0     0.0     0.0
  2.058  -0.733  -1.409   0.0     0.0     0.0
 -0.4     0.919  -1.085   0.0     0.0     0.0

Implementation

The function global_rep does in fact not perform the entire map $\lambda(Y)^{-1}\Omega(Y, \Delta)\lambda(Y)$ but only

\[\Delta \mapsto \mathrm{skew}(Y^T\Delta),\]

to get the small skew-symmetric matrix $A\in\mathcal{S}_\mathrm{skew}(n)$ and

\[\Delta \mapsto (\lambda(Y)_{[1:N, n:N]}^T \Delta)_{[1:(N-n), 1:n]},\]

to get the arbitrary matrix $B\in\mathbb{R}^{(N-n)\times{}n}$.

global_rep(λY::GlobalSection{T, AT}, Δ::AbstractMatrix{T}) where {T, AT<:GrassmannManifold{T}}

Express Δ (an element of the tangent space of Y) as an instance of GrassmannLieAlgHorMatrix.

The method global_rep for GrassmannManifold is similar to that for StiefelManifold.

Examples

using GeometricMachineLearning
using GeometricMachineLearning: _round
import Random 

Random.seed!(123)

Y = rand(GrassmannManifold, 6, 3)
Δ = rgrad(Y, randn(6, 3))
λY = GlobalSection(Y)

_round(global_rep(λY, Δ); digits = 3)

# output

6×6 GrassmannLieAlgHorMatrix{Float64, Matrix{Float64}}:
  0.0     0.0     0.0     0.981  -2.058   0.4
  0.0     0.0     0.0    -0.424   0.733  -0.919
  0.0     0.0     0.0    -1.815   1.409   1.085
 -0.981   0.424   1.815   0.0     0.0     0.0
  2.058  -0.733  -1.409   0.0     0.0     0.0
 -0.4     0.919  -1.085   0.0     0.0     0.0