def test_curvature(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
x, y, z = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.curvature_at_identity(x, y, x)
expected = 1. / 8 * y
self.assertAllClose(result, expected)
tan_a = gs.stack([x, x])
tan_b = gs.stack([y] * 2)
result = metric.curvature(tan_a, tan_b, tan_a)
self.assertAllClose(result, gs.array([expected] * 2))
point = group.random_uniform()
translation_map = group.tangent_translation_map(point)
tan_a = translation_map(x)
tan_b = translation_map(y)
result = metric.curvature(tan_a, tan_b, tan_a, point)
expected = translation_map(expected)
self.assertAllClose(result, expected)
result = metric.curvature(y, y, z)
expected = gs.zeros_like(z)
self.assertAllClose(result, expected)
def test_sectional_curvature(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
x, y, z = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.sectional_curvature(x, y)
expected = 1. / 8
self.assertAllClose(result, expected)
point = group.random_uniform()
translation_map = group.tangent_translation_map(point)
tan_a = translation_map(-z)
tan_b = translation_map(y)
result = metric.sectional_curvature(tan_a, tan_b, point)
self.assertAllClose(result, expected)
tan_a = gs.stack([x, y])
tan_b = gs.stack([z] * 2)
result = metric.sectional_curvature(tan_a, tan_b)
self.assertAllClose(result, gs.array([expected] * 2))
result = metric.sectional_curvature(y, y)
expected = 0.
self.assertAllClose(result, expected)
def test_structure_constant(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
basis = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
x, y, z = basis
result = metric.structure_constant(-z, y, -x)
expected = 2.**.5 / 2.
self.assertAllClose(result, expected)
result = -metric.structure_constant(y, -z, -x)
self.assertAllClose(result, expected)
result = metric.structure_constant(y, -x, -z)
self.assertAllClose(result, expected)
result = -metric.structure_constant(-x, y, -z)
self.assertAllClose(result, expected)
result = metric.structure_constant(-x, -z, y)
self.assertAllClose(result, expected)
result = -metric.structure_constant(-z, -x, y)
self.assertAllClose(result, expected)
result = metric.structure_constant(x, x, z)
expected = 0.
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_dual_adjoint(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
basis = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
for x in basis:
for y in basis:
for z in basis:
result = metric.inner_product_at_identity(
metric.dual_adjoint(x, y), z)
expected = metric.structure_constant(x, z, y)
self.assertAllClose(result, expected)
def test_connection(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
x, y, z = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.connection(-z, y)
expected = -1. / 2**.5 / 2. * x
self.assertAllClose(result, expected)
point = group.random_uniform()
translation_map = group.tangent_translation_map(point)
tan_a = translation_map(-z)
tan_b = translation_map(y)
result = metric.connection(tan_a, tan_b, point)
expected = translation_map(expected)
self.assertAllClose(result, expected)
def test_curvature_derivative(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
x, y, z = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = metric.curvature_derivative(x, y, z, x)
expected = gs.zeros_like(x)
self.assertAllClose(result, expected)
point = group.random_uniform()
translation_map = group.tangent_translation_map(point)
tan_a = translation_map(x)
tan_b = translation_map(y)
tan_c = translation_map(z)
result = metric.curvature_derivative(tan_a, tan_b, tan_c, tan_a, point)
expected = gs.zeros_like(x)
self.assertAllClose(result, expected)
def test_curvature_derivative_at_identity(self):
group = self.matrix_so3
lie_algebra = SkewSymmetricMatrices(3)
metric = InvariantMetric(group=group, algebra=lie_algebra)
basis = lie_algebra.orthonormal_basis(metric.metric_mat_at_identity)
result = True
for x in basis:
for i, y in enumerate(basis):
for z in basis[i:]:
for t in basis:
nabla_r = metric.curvature_derivative_at_identity(
x, y, z, t)
if not gs.all(gs.isclose(nabla_r, 0., atol=1e-5)):
print(nabla_r)
result = False
self.assertTrue(result)
