class TestLieAlgebraMethods(geomstats.tests.TestCase):
def setUp(self):
self.n = 4
self.dim = int(self.n * (self.n - 1) / 2)
self.algebra = SkewSymmetricMatrices(n=self.n)
def test_dimension(self):
result = self.algebra.dim
expected = self.dim
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_matrix_representation_and_belongs(self):
n_samples = 2
point = gs.random.rand(n_samples * self.dim)
point = gs.reshape(point, (n_samples, self.dim))
mat = self.algebra.matrix_representation(point)
result = self.algebra.belongs(mat).all()
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_basis_and_matrix_representation(self):
n_samples = 2
expected = gs.random.rand(n_samples * self.dim)
expected = gs.reshape(expected, (n_samples, self.dim))
mat = self.algebra.matrix_representation(expected)
result = self.algebra.basis_representation(mat)
self.assertAllClose(result, expected)
def test_inner_product_left(self):
lie_algebra = SkewSymmetricMatrices(3)
tangent_vec_a = lie_algebra.matrix_representation(gs.array([1., 0,
2.]))
tangent_vec_a = self.matrix_so3.compose(self.point_1_matrix,
tangent_vec_a)
tangent_vec_b = lie_algebra.matrix_representation(
gs.array([1., 0, 0.5]))
tangent_vec_b = self.matrix_so3.compose(self.point_1_matrix,
tangent_vec_b)
result = self.matrix_left_metric.inner_product(tangent_vec_a,
tangent_vec_b,
self.point_1_matrix)
expected = 4.
self.assertAllClose(result, expected)
tangent_vec_a = lie_algebra.matrix_representation(
gs.array([[1., 0, 2.], [0, 3., 5.]]))
tangent_vec_a = self.matrix_so3.compose(self.point_1_matrix,
tangent_vec_a)
result = self.matrix_left_metric.inner_product(tangent_vec_a,
tangent_vec_b,
self.point_1_matrix)
expected = gs.array([4., 5.])
self.assertAllClose(result, expected)
class TestLieAlgebra(geomstats.tests.TestCase):
def setUp(self):
self.n = 4
self.dim = int(self.n * (self.n - 1) / 2)
self.algebra = SkewSymmetricMatrices(n=self.n)
def test_dimension(self):
result = self.algebra.dim
expected = self.dim
self.assertAllClose(result, expected)
def test_matrix_representation_and_belongs(self):
n_samples = 2
point = gs.random.rand(n_samples * self.dim)
point = gs.reshape(point, (n_samples, self.dim))
mat = self.algebra.matrix_representation(point)
result = gs.all(self.algebra.belongs(mat))
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_basis_and_matrix_representation(self):
n_samples = 2
expected = gs.random.rand(n_samples * self.dim)
expected = gs.reshape(expected, (n_samples, self.dim))
mat = self.algebra.matrix_representation(expected)
result = self.algebra.basis_representation(mat)
self.assertAllClose(result, expected)
def test_orthonormal_basis(self):
group = SpecialOrthogonal(3)
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
basis = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.inner_product_at_identity(basis[0], basis[1])
self.assertAllClose(result, 0.)
result = metric.inner_product_at_identity(basis[1], basis[1])
self.assertAllClose(result, 1.)
metric_mat = from_vector_to_diagonal_matrix(gs.array([1., 2., 3.]))
metric = InvariantMetric(group=group,
algebra=lie_algebra,
metric_mat_at_identity=metric_mat)
basis = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.inner_product_at_identity(basis[0], basis[1])
self.assertAllClose(result, 0.)
result = metric.inner_product_at_identity(basis[1], basis[1])
self.assertAllClose(result, 1.)
def test_inner_product_at_identity(self):
lie_algebra = SkewSymmetricMatrices(3)
tangent_vec_a = lie_algebra.matrix_representation([1., 0, 2.])[0]
tangent_vec_b = lie_algebra.matrix_representation([1., 0, 0.5])[0]
result = self.matrix_left_metric.inner_product_at_identity(
tangent_vec_a, tangent_vec_b)
expected = 4.
self.assertAllClose(result, expected)
tangent_vec_a = lie_algebra.matrix_representation(
[[1., 0, 2.], [0, 3., 5.]])
result = self.matrix_left_metric.inner_product_at_identity(
tangent_vec_a, tangent_vec_b)
expected = gs.array([4., 5.])
self.assertAllClose(result, expected)
class TestLieAlgebra(geomstats.tests.TestCase):
def setup_method(self):
self.n = 4
self.dim = int(self.n * (self.n - 1) / 2)
self.algebra = SkewSymmetricMatrices(n=self.n)
def test_dimension(self):
result = self.algebra.dim
expected = self.dim
self.assertAllClose(result, expected)
def test_matrix_representation_and_belongs(self):
n_samples = 2
point = gs.random.rand(n_samples * self.dim)
point = gs.reshape(point, (n_samples, self.dim))
mat = self.algebra.matrix_representation(point)
result = gs.all(self.algebra.belongs(mat))
self.assertTrue(result)
def test_basis_and_matrix_representation(self):
n_samples = 2
expected = gs.random.rand(n_samples * self.dim)
expected = gs.reshape(expected, (n_samples, self.dim))
mat = self.algebra.matrix_representation(expected)
result = self.algebra.basis_representation(mat)
self.assertAllClose(result, expected)
def test_orthonormal_basis(self):
group = SpecialOrthogonal(3)
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group)
basis = metric.normal_basis(lie_algebra.basis)
result = metric.inner_product_at_identity(basis[0], basis[1])
self.assertAllClose(result, 0.0)
result = metric.inner_product_at_identity(basis[1], basis[1])
self.assertAllClose(result, 1.0)
metric_mat = from_vector_to_diagonal_matrix(gs.array([1.0, 2.0, 3.0]))
metric = InvariantMetric(group=group, metric_mat_at_identity=metric_mat)
basis = metric.normal_basis(lie_algebra.basis)
result = metric.inner_product_at_identity(basis[0], basis[1])
self.assertAllClose(result, 0.0)
result = metric.inner_product_at_identity(basis[1], basis[1])
self.assertAllClose(result, 1.0)
def test_orthonormal_basis_se3(self):
group = SpecialEuclidean(3)
lie_algebra = group.lie_algebra
metric = InvariantMetric(group=group)
basis = metric.normal_basis(lie_algebra.basis)
for i, x in enumerate(basis):
for y in basis[i:]:
result = metric.inner_product_at_identity(x, y)
expected = 0.0 if gs.any(x != y) else 1.0
self.assertAllClose(result, expected)
metric_mat = from_vector_to_diagonal_matrix(
gs.cast(gs.arange(1, group.dim + 1), gs.float32)
)
metric = InvariantMetric(group=group, metric_mat_at_identity=metric_mat)
basis = metric.normal_basis(lie_algebra.basis)
for i, x in enumerate(basis):
for y in basis[i:]:
result = metric.inner_product_at_identity(x, y)
expected = 0.0 if gs.any(x != y) else 1.0
self.assertAllClose(result, expected)
