def test_inverse_metric_matrix(self, dim, base_point):
pullback_metric = PullbackMetric(dim=dim,
embedding_dim=dim + 1,
immersion=immersion)
result = pullback_metric.cometric_matrix(base_point)
expected = _expected_inverse_metric_matrix(base_point)
self.assertAllClose(result, expected)
class TestPullbackMetric(geomstats.tests.TestCase):
def setup_method(self):
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(0)
self.dim = 2
self.sphere = Hypersphere(dim=self.dim)
self.sphere_metric = self.sphere.metric
def _sphere_immersion(spherical_coords):
theta = spherical_coords[..., 0]
phi = spherical_coords[..., 1]
return gs.array([
gs.cos(phi) * gs.sin(theta),
gs.sin(phi) * gs.sin(theta),
gs.cos(theta),
])
self.immersion = _sphere_immersion
self.pullback_metric = PullbackMetric(dim=self.dim,
embedding_dim=self.dim + 1,
immersion=self.immersion)
def test_immersion(self):
expected = gs.array([0.0, 0.0, 1.0])
result = self.immersion(gs.array([0.0, 0.0]))
self.assertAllClose(result, expected)
expected = gs.array([0.0, 0.0, -1.0])
result = self.immersion(gs.array([gs.pi, 0.0]))
self.assertAllClose(result, expected)
expected = gs.array([-1.0, 0.0, 0.0])
result = self.immersion(gs.array([gs.pi / 2.0, gs.pi]))
self.assertAllClose(result, expected)
def test_immersion_and_spherical_to_extrinsic(self):
point = gs.array([0.0, 0.0])
expected = self.immersion(point)
result = self.sphere.spherical_to_extrinsic(point)
self.assertAllClose(result, expected)
point = gs.array([0.2, 0.1])
expected = self.immersion(point)
result = self.sphere.spherical_to_extrinsic(point)
self.assertAllClose(result, expected)
def test_jacobian_immersion(self):
def _expected_jacobian_immersion(point):
theta = point[..., 0]
phi = point[..., 1]
jacobian = gs.array([
[gs.cos(phi) * gs.cos(theta), -gs.sin(phi) * gs.sin(theta)],
[gs.sin(phi) * gs.cos(theta),
gs.cos(phi) * gs.sin(theta)],
[-gs.sin(theta), 0.0],
])
return jacobian
pole = gs.array([0.0, 0.0])
result = self.pullback_metric.jacobian_immersion(pole)
expected = _expected_jacobian_immersion(pole)
self.assertAllClose(result, expected)
base_point = gs.array([0.22, 0.1])
result = self.pullback_metric.jacobian_immersion(base_point)
expected = _expected_jacobian_immersion(base_point)
self.assertAllClose(result, expected)
base_point = gs.array([0.1, 0.88])
result = self.pullback_metric.jacobian_immersion(base_point)
expected = _expected_jacobian_immersion(base_point)
self.assertAllClose(result, expected)
def test_tangent_immersion(self):
point = gs.array([gs.pi / 2.0, gs.pi / 2.0])
tangent_vec = gs.array([1.0, 0.0])
result = self.pullback_metric.tangent_immersion(tangent_vec, point)
expected = gs.array([0.0, 0.0, -1.0])
self.assertAllClose(result, expected)
tangent_vec = gs.array([0.0, 1.0])
result = self.pullback_metric.tangent_immersion(tangent_vec, point)
expected = gs.array([-1.0, 0.0, 0.0])
self.assertAllClose(result, expected)
point = gs.array([gs.pi / 2.0, 0.0])
tangent_vec = gs.array([1.0, 0.0])
result = self.pullback_metric.tangent_immersion(tangent_vec, point)
expected = gs.array([0.0, 0.0, -1.0])
self.assertAllClose(result, expected)
tangent_vec = gs.array([0.0, 1.0])
result = self.pullback_metric.tangent_immersion(tangent_vec, point)
expected = gs.array([0.0, 1.0, 0.0])
self.assertAllClose(result, expected)
def test_metric_matrix(self):
def _expected_metric_matrix(point):
theta = point[..., 0]
mat = gs.array([[1.0, 0.0], [0.0, gs.sin(theta)**2]])
return mat
base_point = gs.array([0.0, 0.0])
result = self.pullback_metric.metric_matrix(base_point)
expected = _expected_metric_matrix(base_point)
self.assertAllClose(result, expected)
base_point = gs.array([1.0, 1.0])
result = self.pullback_metric.metric_matrix(base_point)
expected = _expected_metric_matrix(base_point)
self.assertAllClose(result, expected)
base_point = gs.array([0.3, 0.8])
result = self.pullback_metric.metric_matrix(base_point)
expected = _expected_metric_matrix(base_point)
self.assertAllClose(result, expected)
def test_inverse_metric_matrix(self):
def _expected_inverse_metric_matrix(point):
theta = point[..., 0]
mat = gs.array([[1.0, 0.0], [0.0, gs.sin(theta)**(-2)]])
return mat
base_point = gs.array([0.6, -1.0])
result = self.pullback_metric.cometric_matrix(base_point)
expected = _expected_inverse_metric_matrix(base_point)
self.assertAllClose(result, expected)
base_point = gs.array([0.8, -0.8])
result = self.pullback_metric.cometric_matrix(base_point)
expected = _expected_inverse_metric_matrix(base_point)
self.assertAllClose(result, expected)
def test_inner_product_and_sphere_inner_product(self):
"""Test consistency between sphere's inner-products.
The inner-product of the class Hypersphere is
defined in terms of extrinsic coordinates.
The inner-product of pullback_metric is defined in terms
of the spherical coordinates.
"""
tangent_vec_a = gs.array([0.0, 1.0])
tangent_vec_b = gs.array([0.0, 1.0])
base_point = gs.array([gs.pi / 2.0, 0.0])
immersed_base_point = self.immersion(base_point)
jac_immersion = self.pullback_metric.jacobian_immersion(base_point)
immersed_tangent_vec_a = gs.matmul(jac_immersion, tangent_vec_a)
immersed_tangent_vec_b = gs.matmul(jac_immersion, tangent_vec_b)
result = self.pullback_metric.inner_product(tangent_vec_a,
tangent_vec_b,
base_point=base_point)
expected = self.sphere_metric.inner_product(
immersed_tangent_vec_a,
immersed_tangent_vec_b,
base_point=immersed_base_point,
)
self.assertAllClose(result, expected)
tangent_vec_a = gs.array([0.4, 1.0])
tangent_vec_b = gs.array([0.2, 0.6])
base_point = gs.array([gs.pi / 2.0, 0.1])
immersed_base_point = self.immersion(base_point)
jac_immersion = self.pullback_metric.jacobian_immersion(base_point)
immersed_tangent_vec_a = gs.matmul(jac_immersion, tangent_vec_a)
immersed_tangent_vec_b = gs.matmul(jac_immersion, tangent_vec_b)
result = self.pullback_metric.inner_product(tangent_vec_a,
tangent_vec_b,
base_point=base_point)
expected = self.sphere_metric.inner_product(
immersed_tangent_vec_a,
immersed_tangent_vec_b,
base_point=immersed_base_point,
)
self.assertAllClose(result, expected)
# def test_christoffels_and_sphere_christoffels(self):
# """Test consistency between sphere's christoffels.
# The christoffels of the class Hypersphere are
# defined in terms of spherical coordinates.
# The christoffels of pullback_metric are also defined
# in terms of the spherical coordinates.
# """
# base_point = gs.array([0.1, 0.2])
# result = self.pullback_metric.christoffels(base_point=base_point)
# expected = self.sphere_metric.christoffels(point=base_point)
# self.assertAllClose(result, expected)
# base_point = gs.array([0.7, 0.233])
# result = self.pullback_metric.christoffels(base_point=base_point)
# expected = self.sphere_metric.christoffels(point=base_point)
# self.assertAllClose(result, expected)
def test_exp_and_sphere_exp(self):
"""Test consistency between sphere's Riemannian exp.
The exp map of the class Hypersphere is
defined in terms of extrinsic coordinates.
The exp map of pullback_metric is defined
in terms of the spherical coordinates.
"""
base_point = gs.array([gs.pi / 2.0, 0.0])
tangent_vec_a = gs.array([0.0, 1.0])
immersed_base_point = self.immersion(base_point)
jac_immersion = self.pullback_metric.jacobian_immersion(base_point)
immersed_tangent_vec_a = gs.matmul(jac_immersion, tangent_vec_a)
result = self.pullback_metric.exp(tangent_vec_a, base_point=base_point)
result = self.sphere.spherical_to_extrinsic(result)
expected = self.sphere.metric.exp(immersed_tangent_vec_a,
base_point=immersed_base_point)
self.assertAllClose(result, expected)
base_point = gs.array([gs.pi / 2.0, 0.1])
tangent_vec_a = gs.array([0.4, 1.0])
immersed_base_point = self.immersion(base_point)
jac_immersion = self.pullback_metric.jacobian_immersion(base_point)
immersed_tangent_vec_a = gs.matmul(jac_immersion, tangent_vec_a)
result = self.pullback_metric.exp(tangent_vec_a, base_point=base_point)
result = self.sphere.spherical_to_extrinsic(result)
expected = self.sphere.metric.exp(immersed_tangent_vec_a,
base_point=immersed_base_point)
self.assertAllClose(result, expected, atol=1e-1)
def test_parallel_transport_and_sphere_parallel_transport(self):
"""Test consistency between sphere's parallel transports.
The parallel transport of the class Hypersphere is
defined in terms of extrinsic coordinates.
The parallel transport of pullback_metric is defined
in terms of the spherical coordinates.
"""
tangent_vec_a = gs.array([0.0, 1.0])
tangent_vec_b = gs.array([0.0, 1.0])
base_point = gs.array([gs.pi / 2.0, 0.0])
immersed_base_point = self.immersion(base_point)
jac_immersion = self.pullback_metric.jacobian_immersion(base_point)
immersed_tangent_vec_a = gs.matmul(jac_immersion, tangent_vec_a)
immersed_tangent_vec_b = gs.matmul(jac_immersion, tangent_vec_b)
result_dict = self.pullback_metric.ladder_parallel_transport(
tangent_vec_a, base_point=base_point, direction=tangent_vec_b)
result = result_dict["transported_tangent_vec"]
end_point = result_dict["end_point"]
result = self.pullback_metric.tangent_immersion(v=result, x=end_point)
expected = self.sphere_metric.parallel_transport(
immersed_tangent_vec_a,
base_point=immersed_base_point,
direction=immersed_tangent_vec_b,
)
self.assertAllClose(result, expected, atol=1e-5)