def test_parallel_transport(self):
sphere = Hypersphere(dimension=2)
connection = LeviCivitaConnection(sphere.metric)
n_samples = 10
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = sphere.projection_to_tangent_space(
gs.random.rand(n_samples, 3), base_point)
expected = sphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
result = connection.pole_ladder_parallel_transport(
tan_vec_a, tan_vec_b, base_point)
self.assertAllClose(result, expected, rtol=1e-7, atol=1e-5)
class GeomstatsSphere(Manifold):
"""A simple adapter class which proxies calls by pymanopt's solvers to
`Manifold` subclasses to the underlying geomstats `Hypersphere` class.
"""
def __init__(self, ambient_dimension):
self._sphere = Hypersphere(ambient_dimension - 1)
def norm(self, base_vector, tangent_vector):
return self._sphere.metric.norm(tangent_vector, base_point=base_vector)
def inner(self, base_vector, tangent_vector_a, tangent_vector_b):
return self._sphere.metric.inner_product(
tangent_vector_a, tangent_vector_b, base_point=base_vector)
@squeeze_output
def proj(self, base_vector, tangent_vector):
return self._sphere.projection_to_tangent_space(
tangent_vector, base_point=base_vector)
@squeeze_output
def retr(self, base_vector, tangent_vector):
"""The retraction operator, which maps a tangent vector in the tangent
space at a specific point back to the manifold by approximating moving
along a geodesic. Since geomstats's `Hypersphere` class doesn't provide
a retraction we use the exponential map instead (see also
https://hal.archives-ouvertes.fr/hal-00651608/document).
"""
return self._sphere.metric.exp(tangent_vector, base_point=base_vector)
@squeeze_output
def rand(self):
return self._sphere.random_uniform()
class TestHypersphereMethods(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dimension=self.dimension)
self.metric = self.space.metric
self.n_samples = 10
@geomstats.tests.np_and_pytorch_only
def test_random_uniform_and_belongs(self):
"""
Test that the random uniform method samples
on the hypersphere space.
"""
n_samples = self.n_samples
point = self.space.random_uniform(n_samples)
result = self.space.belongs(point)
expected = gs.array([[True]] * n_samples)
self.assertAllClose(expected, result)
@geomstats.tests.np_and_pytorch_only
def test_random_uniform(self):
point = self.space.random_uniform()
self.assertAllClose(gs.shape(point), (1, self.dimension + 1))
def test_projection_and_belongs(self):
point = gs.array([1., 2., 3., 4., 5.])
proj = self.space.projection(point)
result = self.space.belongs(proj)
expected = gs.array([[True]])
self.assertAllClose(expected, result)
def test_intrinsic_and_extrinsic_coords(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([.1, 0., 0., .1])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
point_ext = (1. / (gs.sqrt(6.)) * gs.array([1., 0., 0., 1., 2.]))
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([[.1, 0., 0., .1], [.1, .1, .1, .4],
[.1, .3, 0., .1], [-0.1, .1, -.4, .1],
[0., 0., .1, .1], [.1, .1, .1, .1]])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_log_and_exp_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# General case
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = gs.array([0., 5., 6., 2., -1.])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
self.assertAllClose(result, expected, atol=1e-6)
@geomstats.tests.np_and_pytorch_only
def test_log_and_exp_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# Edge case: two very close points, base_point_2 and point_2,
# form an angle < epsilon
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = (base_point + 1e-12 * gs.array([-1., -2., 1., 1., .1]))
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_exp_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_vec = self.space.random_uniform()
one_base_point = self.space.random_uniform()
n_vecs = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=one_base_point)
result = self.metric.exp(n_tangent_vecs, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, dim))
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=n_base_points)
result = self.metric.exp(one_tangent_vec, n_base_points)
self.assertAllClose(gs.shape(result), (n_samples, dim))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=n_base_points)
result = self.metric.exp(n_tangent_vecs, n_base_points)
self.assertAllClose(gs.shape(result), (n_samples, dim))
@geomstats.tests.np_and_pytorch_only
def test_log_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_base_point = self.space.random_uniform()
one_point = self.space.random_uniform()
n_points = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
result = self.metric.log(one_point, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
result = self.metric.log(n_points, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, dim))
result = self.metric.log(one_point, n_base_points)
self.assertAllClose(gs.shape(result), (n_samples, dim))
result = self.metric.log(n_points, n_base_points)
self.assertAllClose(gs.shape(result), (n_samples, dim))
@geomstats.tests.np_and_pytorch_only
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# TODO(nina): Fix that this test fails, also in numpy
# Riemannian Exp then Riemannian Log
# General case
# NB: Riemannian log gives a regularized tangent vector,
# so we take the norm modulo 2 * pi.
base_point = gs.array([0., -3., 0., 3., 4.])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9., 5., 0., 0., -1.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
# exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
# result = self.metric.log(point=exp, base_point=base_point)
expected = vector
norm_expected = gs.linalg.norm(expected)
regularized_norm_expected = gs.mod(norm_expected, 2 * gs.pi)
expected = expected / norm_expected * regularized_norm_expected
expected = helper.to_vector(expected)
@geomstats.tests.np_and_pytorch_only
def test_exp_and_log_and_projection_to_tangent_space_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# Edge case: tangent vector has norm < epsilon
base_point = gs.array([10., -2., -.5, 34., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = 1e-10 * gs.array([.06, -51., 6., 5., 3.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
expected = helper.to_vector(expected)
self.assertAllClose(result, expected, atol=1e-8)
def test_squared_norm_and_squared_dist(self):
"""
Test that the squared distance between two points is
the squared norm of their logarithm.
"""
point_a = (1. / gs.sqrt(129.) * gs.array([10., -2., -5., 0., 0.]))
point_b = (1. / gs.sqrt(435.) * gs.array([1., -20., -5., 0., 3.]))
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.squared_norm(vector=log)
expected = self.metric.squared_dist(point_a, point_b)
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_squared_dist_vectorization(self):
n_samples = self.n_samples
one_point_a = self.space.random_uniform()
one_point_b = self.space.random_uniform()
n_points_a = self.space.random_uniform(n_samples=n_samples)
n_points_b = self.space.random_uniform(n_samples=n_samples)
result = self.metric.squared_dist(one_point_a, one_point_b)
self.assertAllClose(gs.shape(result), (1, 1))
result = self.metric.squared_dist(n_points_a, one_point_b)
self.assertAllClose(gs.shape(result), (n_samples, 1))
result = self.metric.squared_dist(one_point_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, 1))
result = self.metric.squared_dist(n_points_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, 1))
def test_norm_and_dist(self):
"""
Test that the distance between two points is
the norm of their logarithm.
"""
point_a = (1. / gs.sqrt(129.) * gs.array([10., -2., -5., 0., 0.]))
point_b = (1. / gs.sqrt(435.) * gs.array([1., -20., -5., 0., 3.]))
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.norm(vector=log)
expected = self.metric.dist(point_a, point_b)
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
def test_dist_point_and_itself(self):
# Distance between a point and itself is 0
point_a = (1. / gs.sqrt(129.) * gs.array([10., -2., -5., 0., 0.]))
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0.
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
def test_dist_orthogonal_points(self):
# Distance between two orthogonal points is pi / 2.
point_a = gs.array([10., -2., -.5, 0., 0.])
point_a = point_a / gs.linalg.norm(point_a)
point_b = gs.array([2., 10, 0., 0., 0.])
point_b = point_b / gs.linalg.norm(point_b)
result = gs.dot(point_a, point_b)
result = helper.to_scalar(result)
expected = 0
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
result = self.metric.dist(point_a, point_b)
expected = gs.pi / 2
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_exp_and_dist_and_projection_to_tangent_space(self):
base_point = gs.array([16., -2., -2.5, 84., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9., 0., -1., -2., 1.])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_exp_and_dist_and_projection_to_tangent_space_vec(self):
base_point = gs.array([[16., -2., -2.5, 84., 3.],
[16., -2., -2.5, 84., 3.]])
base_single_point = gs.array([16., -2., -2.5, 84., 3.])
scalar_norm = gs.linalg.norm(base_single_point)
base_point = base_point / scalar_norm
vector = gs.array([[9., 0., -1., -2., 1.], [9., 0., -1., -2., 1]])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec, axis=-1) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_geodesic_and_belongs(self):
n_geodesic_points = 100
initial_point = self.space.random_uniform()
vector = gs.array([2., 0., -1., -2., 1.])
initial_tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=initial_point)
geodesic = self.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 = self.space.belongs(points)
expected = gs.array(n_geodesic_points * [[True]])
self.assertAllClose(expected, result)
def test_inner_product(self):
tangent_vec_a = gs.array([1., 0., 0., 0., 0.])
tangent_vec_b = gs.array([0., 1., 0., 0., 0.])
base_point = gs.array([0., 0., 0., 0., 1.])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = gs.array([[0.]])
self.assertAllClose(expected, result)
@geomstats.tests.np_and_pytorch_only
def test_variance(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.zeros((2, point.shape[0]))
points[0, :] = point
points[1, :] = point
result = self.metric.variance(points)
expected = helper.to_scalar(0.)
self.assertAllClose(expected, result)
@geomstats.tests.np_and_pytorch_only
def test_mean(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.zeros((2, point.shape[0]))
points[0, :] = point
points[1, :] = point
result = self.metric.mean(points)
expected = helper.to_vector(point)
self.assertAllClose(expected, result)
@geomstats.tests.np_only
def test_adaptive_gradientdescent_mean(self):
n_tests = 100
result = gs.zeros(n_tests)
expected = gs.zeros(n_tests)
for i in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.space.random_uniform(n_samples=2)
mean = self.metric.adaptive_gradientdescent_mean(points)
logs = self.metric.log(point=points, base_point=mean)
result[i] = gs.linalg.norm(logs[1, :] + logs[0, :])
self.assertAllClose(expected, result, rtol=1e-10, atol=1e-10)
@geomstats.tests.np_and_pytorch_only
def test_mean_and_belongs(self):
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.zeros((2, point_a.shape[0]))
points[0, :] = point_a
points[1, :] = point_b
mean = self.metric.mean(points)
result = self.space.belongs(mean)
expected = gs.array([[True]])
self.assertAllClose(result, expected)
def test_diameter(self):
dim = 2
sphere = Hypersphere(dim)
point_a = gs.array([[0., 0., 1.]])
point_b = gs.array([[1., 0., 0.]])
point_c = gs.array([[0., 0., -1.]])
result = sphere.metric.diameter(gs.vstack((point_a, point_b, point_c)))
expected = gs.pi
self.assertAllClose(expected, result)
@geomstats.tests.np_and_pytorch_only
def test_closest_neighbor_index(self):
"""
Check that the closest neighbor is one of neighbors.
"""
n_samples = 10
points = self.space.random_uniform(n_samples=n_samples)
point = points[0, :]
neighbors = points[1:, :]
index = self.metric.closest_neighbor_index(point, neighbors)
closest_neighbor = points[index, :]
test = gs.sum(gs.all(points == closest_neighbor, axis=1))
result = test > 0
self.assertTrue(result)
@geomstats.tests.np_and_pytorch_only
def test_sample_von_mises_fisher(self):
"""
Check that the maximum likelihood estimates of the mean and
concentration parameter are close to the real values. A first
estimation of the concentration parameter is obtained by a
closed-form expression and improved through the Newton method.
"""
dim = 2
n_points = 1000000
sphere = Hypersphere(dim)
# check mean value for concentrated distribution
kappa = 10000000
points = sphere.random_von_mises_fisher(kappa, n_points)
sum_points = gs.sum(points, axis=0)
mean = gs.array([0., 0., 1.])
mean_estimate = sum_points / gs.linalg.norm(sum_points)
expected = mean
result = mean_estimate
self.assertTrue(gs.allclose(result, expected,
atol=MEAN_ESTIMATION_TOL))
# check concentration parameter for dispersed distribution
kappa = 1
points = sphere.random_von_mises_fisher(kappa, n_points)
sum_points = gs.sum(points, axis=0)
mean_norm = gs.linalg.norm(sum_points) / n_points
kappa_estimate = (mean_norm * (dim + 1. - mean_norm**2) /
(1. - mean_norm**2))
kappa_estimate = gs.cast(kappa_estimate, gs.float64)
p = dim + 1
n_steps = 100
for i in range(n_steps):
bessel_func_1 = scipy.special.iv(p / 2., kappa_estimate)
bessel_func_2 = scipy.special.iv(p / 2. - 1., kappa_estimate)
ratio = bessel_func_1 / bessel_func_2
denominator = 1. - ratio**2 - (p - 1.) * ratio / kappa_estimate
mean_norm = gs.cast(mean_norm, gs.float64)
kappa_estimate = kappa_estimate - (ratio - mean_norm) / denominator
expected = kappa
result = kappa_estimate
self.assertTrue(
gs.allclose(result, expected, atol=KAPPA_ESTIMATION_TOL))
@geomstats.tests.np_and_pytorch_only
def test_spherical_to_extrinsic(self):
"""
Check vectorization of conversion from spherical
to extrinsic coordinates on the 2-sphere.
"""
dim = 2
sphere = Hypersphere(dim)
points_spherical = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
result = sphere.spherical_to_extrinsic(points_spherical)
expected = gs.array([[1., 0., 0.],
[gs.sqrt(2) / 4,
gs.sqrt(2) / 4,
gs.sqrt(3) / 2]])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_tangent_spherical_to_extrinsic(self):
"""
Check vectorization of conversion from spherical
to extrinsic coordinates for tangent vectors to the
2-sphere.
"""
dim = 2
sphere = Hypersphere(dim)
base_points_spherical = gs.array([[gs.pi / 2, 0], [gs.pi / 2, 0]])
tangent_vecs_spherical = gs.array([[0.25, 0.5], [0.3, 0.2]])
result = sphere.tangent_spherical_to_extrinsic(tangent_vecs_spherical,
base_points_spherical)
expected = gs.array([[0, 0.5, -0.25], [0, 0.2, -0.3]])
self.assertAllClose(result, expected)
def test_christoffels_vectorization(self):
"""
Check vectorization of Christoffel symbols in
spherical coordinates on the 2-sphere.
"""
dim = 2
sphere = Hypersphere(dim)
points_spherical = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
christoffel = sphere.metric.christoffels(points_spherical)
result = christoffel.shape
expected = gs.array([2, dim, dim, dim])
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)