class TestConnection(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 4
self.euc_metric = EuclideanMetric(dim=self.dim)
self.connection = Connection(dim=2)
self.hypersphere = Hypersphere(dim=2)
def test_metric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.metric_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
def test_cometric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_inverse_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_metric_derivative(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_christoffels(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
def test_parallel_transport(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
base_point = gs.cast(base_point, gs.float64)
base_point, tan_vec_a, tan_vec_b = gs.convert_to_wider_dtype(
[base_point, tan_vec_a, tan_vec_b])
for step, alpha in zip(['pole', 'schild'], [1, 2]):
min_n = 1 if step == 'pole' else 50
tol = 1e-5 if step == 'pole' else 1e-2
for n_rungs in [min_n, 11]:
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
tan_vec_b,
base_point,
scheme=step,
n_rungs=n_rungs,
alpha=alpha)
result = ladder['transported_tangent_vec']
result_point = ladder['end_point']
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_parallel_transport_trajectory(self):
n_samples = 2
for step in ['pole', 'schild']:
n_steps = 1 if step == 'pole' else 50
tol = 1e-6 if step == 'pole' else 1e-2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
tan_vec_b,
base_point,
return_geodesics=True,
scheme=step,
n_rungs=n_steps)
result = ladder['transported_tangent_vec']
result_point = ladder['end_point']
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_ladder_alpha(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
self.assertRaises(
ValueError, lambda: self.hypersphere.metric.
ladder_parallel_transport(tan_vec_a,
tan_vec_b,
base_point,
return_geodesics=False,
scheme='pole',
n_rungs=1,
alpha=0.5))
def test_exp_connection_metric(self):
point = gs.array([gs.pi / 2, 0])
vector = gs.array([0.25, 0.5])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
def test_exp_connection_metric_vectorization(self):
point = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
vector = gs.array([[0.25, 0.5], [0.30, 0.2]])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
def test_log_connection_metric(self):
base_point = gs.array([gs.pi / 3, gs.pi / 4])
point = gs.array([1.0, gs.pi / 2])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk',
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)
def test_log_connection_metric_vectorization(self):
base_point = gs.array([[gs.pi / 3, gs.pi / 4], [gs.pi / 2, gs.pi / 4]])
point = gs.array([[1.0, gs.pi / 2], [gs.pi / 6, gs.pi / 3]])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk',
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)
def test_geodesic_and_coincides_exp_hypersphere(self):
n_geodesic_points = 10
initial_point = self.hypersphere.random_uniform(2)
vector = gs.array([[2., 0., -1.]] * 2)
initial_tangent_vec = self.hypersphere.to_tangent(
vector=vector, base_point=initial_point)
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = points[-1]
expected = self.hypersphere.metric.exp(vector, initial_point)
self.assertAllClose(expected, result)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = self.hypersphere.metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_and_coincides_exp_son(self):
n_geodesic_points = 10
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = points[-1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(result, expected)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_invalid_initial_conditions(self):
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
end_point = space.random_uniform(2)
self.assertRaises(
RuntimeError, lambda: space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec,
end_point=end_point))
class TestRiemannianMetric(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(0)
self.dim = 2
self.euc = Euclidean(dim=self.dim)
self.sphere = Hypersphere(dim=self.dim)
self.euc_metric = EuclideanMetric(dim=self.dim)
self.sphere_metric = HypersphereMetric(dim=self.dim)
def _euc_metric_matrix(base_point):
"""Return matrix of Euclidean inner-product."""
dim = base_point.shape[-1]
return gs.eye(dim)
def _sphere_metric_matrix(base_point):
"""Return sphere's metric in spherical coordinates."""
theta = base_point[..., 0]
mat = gs.array([[1.0, 0.0], [0.0, gs.sin(theta) ** 2]])
return mat
new_euc_metric = RiemannianMetric(dim=self.dim)
new_euc_metric.metric_matrix = _euc_metric_matrix
new_sphere_metric = RiemannianMetric(dim=self.dim)
new_sphere_metric.metric_matrix = _sphere_metric_matrix
self.new_euc_metric = new_euc_metric
self.new_sphere_metric = new_sphere_metric
def test_cometric_matrix(self):
base_point = self.euc.random_point()
result = self.euc_metric.metric_inverse_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_metric_derivative_euc_metric(self):
base_point = self.euc.random_point()
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim,) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_metric_derivative_new_euc_metric(self):
base_point = self.euc.random_point()
result = self.new_euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim,) * 3)
self.assertAllClose(result, expected)
def test_inner_product_new_euc_metric(self):
base_point = self.euc.random_point()
tan_a = self.euc.random_point()
tan_b = self.euc.random_point()
expected = gs.dot(tan_a, tan_b)
result = self.new_euc_metric.inner_product(tan_a, tan_b, base_point=base_point)
self.assertAllClose(result, expected)
def test_inner_product_new_sphere_metric(self):
base_point = gs.array([gs.pi / 3.0, gs.pi / 5.0])
tan_a = gs.array([0.3, 0.4])
tan_b = gs.array([0.1, -0.5])
expected = -0.12
result = self.new_sphere_metric.inner_product(
tan_a, tan_b, base_point=base_point
)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_christoffels_eucl_metric(self):
base_point = self.euc.random_point()
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim,) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_christoffels_new_eucl_metric(self):
base_point = self.euc.random_point()
result = self.new_euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim,) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_christoffels_sphere_metrics(self):
base_point = gs.array([gs.pi / 10.0, gs.pi / 9.0])
expected = self.sphere_metric.christoffels(base_point)
result = self.new_sphere_metric.christoffels(base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_exp_new_eucl_metric(self):
base_point = self.euc.random_point()
tan = self.euc.random_point()
expected = base_point + tan
result = self.new_euc_metric.exp(tan, base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_log_new_eucl_metric(self):
base_point = self.euc.random_point()
point = self.euc.random_point()
expected = point - base_point
result = self.new_euc_metric.log(point, base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_exp_new_sphere_metric(self):
base_point = gs.array([gs.pi / 10.0, gs.pi / 9.0])
tan = gs.array([gs.pi / 2.0, 0.0])
expected = gs.array([gs.pi / 10.0 + gs.pi / 2.0, gs.pi / 9.0])
result = self.new_sphere_metric.exp(tan, base_point)
self.assertAllClose(result, expected)
class TestConnectionMethods(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter('ignore', category=UserWarning)
self.dimension = 4
self.euc_metric = EuclideanMetric(dimension=self.dimension)
self.connection = Connection(dimension=2)
self.hypersphere = Hypersphere(dimension=2)
def test_metric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_matrix(base_point)
expected = gs.array([gs.eye(self.dimension)])
with self.session():
self.assertAllClose(result, expected)
def test_cometric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_inverse_matrix(base_point)
expected = gs.array([gs.eye(self.dimension)])
with self.session():
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_metric_derivative(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((1, ) + (self.dimension, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_christoffels(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((1, ) + (self.dimension, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_parallel_transport(self):
n_samples = 10
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
result = self.hypersphere.metric.pole_ladder_parallel_transport(
tan_vec_a, tan_vec_b, base_point)
self.assertAllClose(result, expected, rtol=1e-7, atol=1e-5)
@geomstats.tests.np_only
def test_exp(self):
point = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
vector = gs.array([[0.25, 0.5], [0.30, 0.2]])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
@geomstats.tests.np_only
def test_log(self):
base_point = gs.array([[gs.pi / 3, gs.pi / 4], [gs.pi / 2, gs.pi / 4]])
point = gs.array([[1.0, gs.pi / 2], [gs.pi / 6, gs.pi / 3]])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk')
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)
class TestConnectionMethods(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 4
self.euc_metric = EuclideanMetric(dim=self.dim)
self.connection = Connection(dim=2)
self.hypersphere = Hypersphere(dim=2)
def test_metric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
def test_cometric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_inverse_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_metric_derivative(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_christoffels(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_parallel_transport(self):
n_samples = 2
for step in ['pole', 'schild']:
n_steps = 1 if step == 'pole' else 100
tol = 1e-6 if step == 'pole' else 1e-1
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a, tan_vec_b, base_point, step=step, n_steps=n_steps)
result = ladder['transported_tangent_vec']
self.assertAllClose(result, expected, rtol=tol, atol=tol)
@geomstats.tests.np_and_pytorch_only
def test_parallel_transport_trajectory(self):
n_samples = 2
for step in ['pole', 'schild']:
n_steps = 1 if step == 'pole' else 100
rtol = 1e-6 if step == 'pole' else 1e-1
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
tan_vec_b,
base_point,
return_geodesics=True,
step=step,
n_steps=n_steps)
result = ladder['transported_tangent_vec']
self.assertAllClose(result, expected, rtol=rtol)
@geomstats.tests.np_only
def test_exp(self):
point = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
vector = gs.array([[0.25, 0.5], [0.30, 0.2]])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
@geomstats.tests.np_only
def test_log(self):
base_point = gs.array([[gs.pi / 3, gs.pi / 4], [gs.pi / 2, gs.pi / 4]])
point = gs.array([[1.0, gs.pi / 2], [gs.pi / 6, gs.pi / 3]])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk')
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)