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projective_space.jl
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include("utils.jl")
@testset "ProjectiveSpace" begin
@testset "Real" begin
M = ProjectiveSpace(2)
@testset "Basics" begin
@test repr(M) == "ProjectiveSpace(2, ℝ)"
@test representation_size(M) == (3,)
@test manifold_dimension(M) == 2
@test !is_point(M, [1.0, 0.0, 0.0, 0.0])
@test !is_vector(M, [1.0, 0.0, 0.0, 0.0], [0.0, 1.0, 0.0])
@test_throws DomainError is_point(M, [2.0, 0.0, 0.0], true)
@test !is_point(M, [2.0, 0.0, 0.0])
@test !is_vector(M, [1.0, 0.0, 0.0], [1.0, 0.0, 0.0])
@test_throws DomainError is_vector(M, [1.0, 0.0, 0.0], [1.0, 0.0, 0.0], true)
@test injectivity_radius(M) == π / 2
@test injectivity_radius(M, ExponentialRetraction()) == π / 2
@test injectivity_radius(M, [1.0, 0.0, 0.0]) == π / 2
@test injectivity_radius(M, [1.0, 0.0, 0.0], ExponentialRetraction()) == π / 2
end
types = [Vector{Float64}]
TEST_STATIC_SIZED && push!(types, MVector{3,Float64})
TEST_FLOAT32 && push!(types, Vector{Float32})
basis_types = (DefaultOrthonormalBasis(), ProjectedOrthonormalBasis(:svd))
@testset "Type $T" for T in types
x = [1.0, 0.0, 0.0]
v = [0.0, 1.0, 0.0]
y = exp(M, x, v)
w = [0.0, 1.0, -1.0]
z = exp(M, x, w)
pts = convert.(T, [x, y, z])
test_manifold(
M,
pts,
test_injectivity_radius=false,
test_project_point=true,
test_project_tangent=true,
test_musical_isomorphisms=true,
test_default_vector_transport=true,
vector_transport_methods=[
ParallelTransport(),
ProjectionTransport(),
SchildsLadderTransport(),
PoleLadderTransport(),
],
point_distributions=[Manifolds.uniform_distribution(M, pts[1])],
tvector_distributions=[
Manifolds.normal_tvector_distribution(M, pts[1], 1.0),
],
test_forward_diff=false,
test_reverse_diff=false,
basis_types_vecs=(
DiagonalizingOrthonormalBasis([0.0, 1.0, 2.0]),
basis_types...,
),
basis_types_to_from=basis_types,
test_vee_hat=false,
retraction_methods=[
ProjectionRetraction(),
PolarRetraction(),
QRRetraction(),
],
inverse_retraction_methods=[
ProjectionInverseRetraction(),
PolarInverseRetraction(),
QRInverseRetraction(),
],
is_tangent_atol_multiplier=1,
)
end
@testset "retract/inverse_retract" begin
x = [1.0, 0.0, 0.0]
v = [0.0, 1.0, 0.0]
y = retract(M, x, v, ProjectionRetraction())
v2 = inverse_retract(M, x, y, ProjectionInverseRetraction())
@test v ≈ v2
end
@testset "equivalence" begin
x = [1.0, 0.0, 0.0]
v = [0.0, 1.0, 0.0]
@test isapprox(M, x, -x)
@test isapprox(M, x, exp(M, x, π * v))
@test log(M, x, -x) ≈ zero(v)
@test isapprox(M, -x, vector_transport_to(M, x, v, -x), -v)
end
@testset "Distribution MVector tests" begin
upd_mvector = Manifolds.uniform_distribution(M, @MVector [1.0, 0.0, 0.0])
@test isa(rand(upd_mvector), MVector)
gtpd_mvector =
Manifolds.normal_tvector_distribution(M, (@MVector [1.0, 0.0, 0.0]), 1.0)
@test isa(rand(gtpd_mvector), MVector)
end
end
@testset "Complex" begin
M = ProjectiveSpace(2, ℂ)
@testset "Basics" begin
@test repr(M) == "ProjectiveSpace(2, ℂ)"
@test representation_size(M) == (3,)
@test manifold_dimension(M) == 4
@test Manifolds.allocation_promotion_function(M, exp!, (1,)) == complex
@test !is_point(M, [1.0 + 0im, 0.0, 0.0, 0.0])
@test !is_vector(M, [1.0 + 0im, 0.0, 0.0, 0.0], [0.0 + 0im, 1.0, 0.0])
@test_throws DomainError is_point(M, [1.0, im, 0.0], true)
@test !is_point(M, [1.0, im, 0.0])
@test !is_vector(M, [1.0 + 0im, 0.0, 0.0], [1.0 + 0im, 0.0, 0.0])
@test !is_vector(M, [1.0 + 0im, 0.0, 0.0], [-0.5im, 0.0, 0.0])
@test_throws DomainError is_vector(
M,
[1.0 + 0im, 0.0, 0.0],
[1.0 + 0im, 0.0, 0.0],
true,
)
@test_throws DomainError is_vector(
M,
[1.0 + 0im, 0.0, 0.0],
[-0.5im, 0.0, 0.0],
true,
)
@test injectivity_radius(M) == π / 2
@test injectivity_radius(M, ExponentialRetraction()) == π / 2
@test injectivity_radius(M, [1.0 + 0im, 0.0, 0.0]) == π / 2
@test injectivity_radius(M, [1.0 + 0im, 0.0, 0.0], ExponentialRetraction()) ==
π / 2
end
types = [Vector{ComplexF64}]
@testset "Type $T" for T in types
x = [0.5 + 0.5im, 0.5 + 0.5im, 0]
v = [0.0, 0.0, 1.0 - im]
y = im * exp(M, x, v)
w = [0.5, -0.5, 0.5im]
z = (sqrt(0.5) - sqrt(0.5) * im) * exp(M, x, w)
pts = convert.(T, [x, y, z])
test_manifold(
M,
pts,
test_injectivity_radius=false,
test_project_point=true,
test_project_tangent=true,
test_musical_isomorphisms=true,
test_default_vector_transport=true,
vector_transport_methods=[
ParallelTransport(),
ProjectionTransport(),
SchildsLadderTransport(),
PoleLadderTransport(),
],
test_forward_diff=false,
test_reverse_diff=false,
basis_types_to_from=(DefaultOrthonormalBasis(),),
test_vee_hat=false,
retraction_methods=[
ProjectionRetraction(),
PolarRetraction(),
QRRetraction(),
],
inverse_retraction_methods=[
ProjectionInverseRetraction(),
PolarInverseRetraction(),
QRInverseRetraction(),
],
is_tangent_atol_multiplier=1,
exp_log_atol_multiplier=10.0^3,
retraction_atol_multiplier=10.0,
)
end
@testset "retract/inverse_retract" begin
x = [0.5 + 0.5im, 0.5 + 0.5im, 0]
v = [0.0, 0.0, 1.0 - im]
y = retract(M, x, v, ProjectionRetraction())
v2 = inverse_retract(M, x, y, ProjectionInverseRetraction())
@test v ≈ v2
end
@testset "equivalence" begin
x = [1.0 + 0im, 0.0, 0.0]
v = [0.0, im, 0.0]
s = sqrt(0.5) - sqrt(0.5) * im
@test isapprox(M, x, s * x)
@test isapprox(M, x, exp(M, x, π * v))
@test log(M, x, s * x) ≈ zero(v)
@test isapprox(M, s * x, vector_transport_to(M, x, v, s * x), s * v)
end
end
@testset "Right Quaternion" begin
M = ProjectiveSpace(2, ℍ)
@testset "Basics" begin
@test repr(M) == "ProjectiveSpace(2, ℍ)"
@test representation_size(M) == (3,)
@test manifold_dimension(M) == 8
@test !is_point(M, Quaternion[1.0 + 0im, 0.0, 0.0, 0.0])
@test !is_vector(
M,
Quaternion[1.0 + 0im, 0.0, 0.0, 0.0],
Quaternion[0.0 + 0im, 1.0, 0.0],
)
@test_throws DomainError is_point(M, Quaternion[1.0, im, 0.0], true)
@test !is_point(M, Quaternion[1.0, im, 0.0])
@test !is_vector(
M,
Quaternion[1.0 + 0im, 0.0, 0.0],
Quaternion[1.0 + 0im, 0.0, 0.0],
)
@test !is_vector(
M,
Quaternion[1.0 + 0im, 0.0, 0.0],
Quaternion[-0.5im, 0.0, 0.0],
)
@test_throws DomainError is_vector(
M,
Quaternion[1.0 + 0im, 0.0, 0.0],
Quaternion[1.0 + 0im, 0.0, 0.0],
true,
)
@test_throws DomainError is_vector(
M,
Quaternion[1.0 + 0im, 0.0, 0.0],
Quaternion[-0.5im, 0.0, 0.0],
true,
)
@test injectivity_radius(M) == π / 2
@test injectivity_radius(M, ExponentialRetraction()) == π / 2
@test injectivity_radius(M, Quaternion[1.0 + 0im, 0.0, 0.0]) == π / 2
@test injectivity_radius(
M,
Quaternion[1.0 + 0im, 0.0, 0.0],
ExponentialRetraction(),
) == π / 2
end
types = [Vector{Quaternion{Float64}}]
@testset "Type $T" for T in types
x = [Quaternion(0.5, 0, 0, 0.5), Quaternion(0, 0, 0.5, 0.5), 0]
v = [Quaternion(0), Quaternion(0), Quaternion(0.0, -0.5, -0.5, 0.0)]
y = Quaternion(0, 0, 0, 1) * exp(M, x, v)
w = [
Quaternion(0.25, -0.25, 0.25, 0.25),
Quaternion(0.25, 0.25, -0.25, -0.25),
1,
]
z = Quaternion(0.5, -0.5, 0.5, -0.5) * exp(M, x, w)
pts = convert.(T, [x, y, z])
test_manifold(
M,
pts,
test_injectivity_radius=false,
test_project_point=true,
test_project_tangent=true,
test_musical_isomorphisms=true,
test_default_vector_transport=true,
vector_transport_methods=[
ParallelTransport(),
ProjectionTransport(),
SchildsLadderTransport(),
PoleLadderTransport(),
],
test_forward_diff=false,
test_reverse_diff=false,
basis_types_to_from=(DefaultOrthonormalBasis(),),
test_vee_hat=false,
retraction_methods=[
ProjectionRetraction(),
PolarRetraction(),
QRRetraction(),
],
inverse_retraction_methods=[
ProjectionInverseRetraction(),
PolarInverseRetraction(),
QRInverseRetraction(),
],
is_tangent_atol_multiplier=10,
exp_log_atol_multiplier=10.0^3,
retraction_atol_multiplier=10.0,
)
end
@testset "retract/inverse_retract" begin
x = [Quaternion(0.5, 0, 0, 0.5), Quaternion(0, 0, 0.5, 0.5), 0]
v = [Quaternion(0), Quaternion(0), Quaternion(0.0, -0.5, -0.5, 0.0)]
y = retract(M, x, v, ProjectionRetraction())
v2 = inverse_retract(M, x, y, ProjectionInverseRetraction())
@test v ≈ v2
end
@testset "equivalence" begin
x = Quaternion[1.0 + 0im, 0.0, 0.0]
v = Quaternion[0.0, im, 0.0]
s = Quaternion(0.5, -0.5, 0.5, -0.5)
@test isapprox(M, x, x * s)
@test isapprox(M, x, exp(M, x, π * v))
@test log(M, x, x * s) ≈ zero(v)
@test isapprox(M, x * s, vector_transport_to(M, x, v, x * s), v * s)
end
end
@testset "ArrayProjectiveSpace" begin
M = ArrayProjectiveSpace(2, 2; field=ℝ)
@test manifold_dimension(M) == 3
@test repr(M) == "ArrayProjectiveSpace(2, 2; field = ℝ)"
@test typeof(get_embedding(M)) === Euclidean{Tuple{2,2},ℝ}
@test representation_size(M) == (2, 2)
p = ones(2, 2)
q = project(M, p)
@test is_point(M, q)
Y = [1.0 0.0; 0.0 1.1]
X = project(M, q, Y)
@test is_vector(M, q, X)
M = ArrayProjectiveSpace(2, 2; field=ℂ)
@test manifold_dimension(M) == 6
@test repr(M) == "ArrayProjectiveSpace(2, 2; field = ℂ)"
@test typeof(get_embedding(M)) === Euclidean{Tuple{2,2},ℂ}
@test representation_size(M) == (2, 2)
end
end