def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""Test change of coordinates.
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
space = Hypersphere(dim=2)
point_int = gs.array([
[0.1, 0.1],
[0.1, 0.4],
[0.1, 0.3],
[0.0, 0.0],
[0.1, 0.5],
])
point_ext = space.intrinsic_to_extrinsic_coords(point_int)
result = space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
self.assertAllClose(result, expected)
point_int = space.extrinsic_to_intrinsic_coords(point_ext)
result = space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
self.assertAllClose(result, expected)
def test_intrinsic_and_extrinsic_coords(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
space = Hypersphere(dim=2)
point_int = gs.array([0.1, 0.0])
point_ext = space.intrinsic_to_extrinsic_coords(point_int)
result = space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
self.assertAllClose(result, expected)
point_ext = 1. / (gs.sqrt(2.)) * gs.array([1.0, 0.0, 1.0])
point_int = space.extrinsic_to_intrinsic_coords(point_ext)
result = space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
self.assertAllClose(result, expected)
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 TestHypersphere(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dim=self.dimension)
self.metric = self.space.metric
self.n_samples = 10
def test_random_uniform_and_belongs(self):
"""Test random uniform and belongs.
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)
def test_random_uniform(self):
point = self.space.random_uniform()
self.assertAllClose(gs.shape(point), (self.dimension + 1, ))
def test_replace_values(self):
points = gs.ones((3, 5))
new_points = gs.zeros((2, 5))
indcs = [True, False, True]
update = self.space._replace_values(points, new_points, indcs)
self.assertAllClose(update,
gs.stack([gs.zeros(5),
gs.ones(5),
gs.zeros(5)]))
def test_projection_and_belongs(self):
shape = (self.n_samples, self.dimension + 1)
result = helper.test_projection_and_belongs(self.space, shape)
for res in result:
self.assertTrue(res)
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
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
self.assertAllClose(result, expected)
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""Test change of coordinates.
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
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
self.assertAllClose(result, expected)
def test_log_and_exp_general_case(self):
"""Test Log and Exp.
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
self.assertAllClose(result, expected)
def test_log_and_exp_edge_case(self):
"""Test Log and Exp.
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-4 * 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
self.assertAllClose(result, expected)
def test_exp_vectorization_single_samples(self):
dim = self.dimension + 1
one_vec = self.space.random_uniform()
one_base_point = self.space.random_uniform()
one_tangent_vec = self.space.to_tangent(one_vec,
base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (dim, ))
one_base_point = gs.to_ndarray(one_base_point, to_ndim=2)
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
one_tangent_vec = gs.to_ndarray(one_tangent_vec, to_ndim=2)
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
one_base_point = self.space.random_uniform()
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
def test_exp_vectorization_n_samples(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)
n_tangent_vecs = self.space.to_tangent(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.to_tangent(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.to_tangent(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))
def test_log_vectorization_single_samples(self):
dim = self.dimension + 1
one_base_point = self.space.random_uniform()
one_point = self.space.random_uniform()
result = self.metric.log(one_point, one_base_point)
self.assertAllClose(gs.shape(result), (dim, ))
one_base_point = gs.to_ndarray(one_base_point, to_ndim=2)
result = self.metric.log(one_point, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
one_point = gs.to_ndarray(one_base_point, to_ndim=2)
result = self.metric.log(one_point, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
one_base_point = self.space.random_uniform()
result = self.metric.log(one_point, one_base_point)
self.assertAllClose(gs.shape(result), (1, dim))
def test_log_vectorization_n_samples(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), (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))
def test_exp_log_are_inverse(self):
initial_point = self.space.random_uniform(2)
end_point = self.space.random_uniform(2)
vec = self.space.metric.log(point=end_point, base_point=initial_point)
result = self.space.metric.exp(vec, initial_point)
self.assertAllClose(end_point, result)
def test_log_extreme_case(self):
initial_point = self.space.random_uniform(2)
vec = 1e-4 * gs.random.rand(*initial_point.shape)
vec = self.space.to_tangent(vec, initial_point)
point = self.space.metric.exp(vec, base_point=initial_point)
result = self.space.metric.log(point, initial_point)
self.assertAllClose(vec, result)
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""Test Log and Exp.
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
# 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([3., 2., 0., 0., -1.])
vector = self.space.to_tangent(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
# The Log can be the opposite vector on the tangent space,
# whose Exp gives the base_point
are_close = gs.allclose(result, expected)
norm_2pi = gs.isclose(gs.linalg.norm(result - expected), 2 * gs.pi)
self.assertTrue(are_close or norm_2pi)
def test_exp_and_log_and_projection_to_tangent_space_edge_case(self):
"""Test Log and Exp.
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-4 * gs.array([.06, -51., 6., 5., 3.])
vector = self.space.to_tangent(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)
self.assertAllClose(result, vector)
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)
self.assertAllClose(result, expected)
def test_squared_dist_vectorization_single_sample(self):
one_point_a = self.space.random_uniform()
one_point_b = self.space.random_uniform()
result = self.metric.squared_dist(one_point_a, one_point_b)
self.assertAllClose(gs.shape(result), ())
one_point_a = gs.to_ndarray(one_point_a, to_ndim=2)
result = self.metric.squared_dist(one_point_a, one_point_b)
self.assertAllClose(gs.shape(result), (1, ))
one_point_b = gs.to_ndarray(one_point_b, to_ndim=2)
result = self.metric.squared_dist(one_point_a, one_point_b)
self.assertAllClose(gs.shape(result), (1, ))
one_point_a = self.space.random_uniform()
result = self.metric.squared_dist(one_point_a, one_point_b)
self.assertAllClose(gs.shape(result), (1, ))
def test_squared_dist_vectorization_n_samples(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), ())
result = self.metric.squared_dist(n_points_a, one_point_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
result = self.metric.squared_dist(one_point_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
result = self.metric.squared_dist(n_points_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
one_point_a = gs.to_ndarray(one_point_a, to_ndim=2)
one_point_b = gs.to_ndarray(one_point_b, to_ndim=2)
result = self.metric.squared_dist(n_points_a, one_point_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
result = self.metric.squared_dist(one_point_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
result = self.metric.squared_dist(n_points_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, ))
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)
self.assertAllClose(gs.shape(log), (5, ))
result = self.metric.norm(vector=log)
self.assertAllClose(gs.shape(result), ())
expected = self.metric.dist(point_a, point_b)
self.assertAllClose(gs.shape(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.
self.assertAllClose(result, expected)
def test_dist_pairwise(self):
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.]))
point = gs.array([point_a, point_b])
result = self.metric.dist_pairwise(point)
expected = gs.array([[0., 1.24864502], [1.24864502, 0.]])
self.assertAllClose(result, expected, rtol=1e-3)
def test_dist_pairwise_parallel(self):
n_samples = 15
points = self.space.random_uniform(n_samples)
result = self.metric.dist_pairwise(points, n_jobs=2, prefer='threads')
is_sym = Matrices.is_symmetric(result)
belongs = Matrices(n_samples, n_samples).belongs(result)
self.assertTrue(is_sym)
self.assertTrue(belongs)
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)
expected = 0
self.assertAllClose(result, expected)
result = self.metric.dist(point_a, point_b)
expected = gs.pi / 2
self.assertAllClose(result, expected)
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.to_tangent(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)
self.assertAllClose(result, expected)
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.to_tangent(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)
self.assertAllClose(result, expected)
def test_geodesic_and_belongs(self):
n_geodesic_points = 10
initial_point = self.space.random_uniform(2)
vector = gs.array([[2., 0., -1., -2., 1.]] * 2)
initial_tangent_vec = self.space.to_tangent(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 = gs.stack([self.space.belongs(pt) for pt in points])
self.assertTrue(gs.all(result))
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = self.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = self.space.belongs(points)
expected = gs.array(n_geodesic_points * [True])
self.assertAllClose(expected, result)
def test_geodesic_end_point(self):
n_geodesic_points = 10
initial_point = self.space.random_uniform(4)
geodesic = self.metric.geodesic(initial_point=initial_point[:2],
end_point=initial_point[2:])
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = points[:, -1]
expected = initial_point[2:]
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 = 0.
self.assertAllClose(expected, result)
def test_inner_product_vectorization_single_samples(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 = 0.
self.assertAllClose(expected, result)
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)
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)
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)
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)
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)
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)
def test_sample_von_mises_fisher_arbitrary_mean(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.
"""
for dim in [2, 9]:
n_points = 10000
sphere = Hypersphere(dim)
# check mean value for concentrated distribution for different mean
kappa = 1000.
mean = sphere.random_uniform()
points = sphere.random_von_mises_fisher(mu=mean,
kappa=kappa,
n_samples=n_points)
sum_points = gs.sum(points, axis=0)
result = sum_points / gs.linalg.norm(sum_points)
expected = mean
self.assertAllClose(result, expected, atol=MEAN_ESTIMATION_TOL)
def test_random_von_mises_kappa(self):
# check concentration parameter for dispersed distribution
kappa = 1.
n_points = 100000
for dim in [2, 9]:
sphere = Hypersphere(dim)
points = sphere.random_von_mises_fisher(kappa=kappa,
n_samples=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 _ 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)
result = kappa_estimate
expected = kappa
self.assertAllClose(result, expected, atol=KAPPA_ESTIMATION_TOL)
def test_random_von_mises_general_dim_mean(self):
for dim in [2, 9]:
sphere = Hypersphere(dim)
n_points = 100000
# check mean value for concentrated distribution
kappa = 10
points = sphere.random_von_mises_fisher(kappa=kappa,
n_samples=n_points)
sum_points = gs.sum(points, axis=0)
expected = gs.array([1.] + [0.] * dim)
result = sum_points / gs.linalg.norm(sum_points)
self.assertAllClose(result, expected, atol=KAPPA_ESTIMATION_TOL)
def test_random_von_mises_one_sample_belongs(self):
for dim in [2, 9]:
sphere = Hypersphere(dim)
point = sphere.random_von_mises_fisher()
self.assertAllClose(point.shape, (dim + 1, ))
result = sphere.belongs(point)
self.assertTrue(result)
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])
result = sphere.spherical_to_extrinsic(points_spherical)
expected = gs.array([1., 0., 0.])
self.assertAllClose(result, expected)
def test_spherical_to_extrinsic_vectorization(self):
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)
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)
result = sphere.tangent_spherical_to_extrinsic(
tangent_vecs_spherical[0], base_points_spherical[0])
self.assertAllClose(result, expected[0])
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)
def test_parallel_transport_vectorization(self):
sphere = Hypersphere(2)
metric = sphere.metric
shape = (4, 3)
results = helper.test_parallel_transport(sphere, metric, shape)
for res in results:
self.assertTrue(res)
def test_is_tangent(self):
space = self.space
vec = space.random_uniform()
result = space.is_tangent(vec, vec)
self.assertFalse(result)
base_point = space.random_uniform()
tangent_vec = space.to_tangent(vec, base_point)
result = space.is_tangent(tangent_vec, base_point)
self.assertTrue(result)
base_point = space.random_uniform(2)
vec = space.random_uniform(2)
tangent_vec = space.to_tangent(vec, base_point)
result = space.is_tangent(tangent_vec, base_point)
self.assertAllClose(gs.shape(result), (2, ))
self.assertTrue(gs.all(result))
def test_sectional_curvature(self):
n_samples = 4
sphere = self.space
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.to_tangent(
gs.random.rand(n_samples, sphere.dim + 1), base_point)
tan_vec_b = sphere.to_tangent(
gs.random.rand(n_samples, sphere.dim + 1), base_point)
result = sphere.metric.sectional_curvature(tan_vec_a, tan_vec_b,
base_point)
expected = gs.ones(result.shape)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_riemannian_normal_and_belongs(self):
mean = self.space.random_uniform()
cov = gs.eye(self.space.dim)
sample = self.space.random_riemannian_normal(mean, cov, 10)
result = self.space.belongs(sample)
self.assertTrue(gs.all(result))
@geomstats.tests.np_and_pytorch_only
def test_riemannian_normal_mean(self):
space = self.space
mean = space.random_uniform()
precision = gs.eye(space.dim) * 10
sample = space.random_riemannian_normal(mean, precision, 10000)
estimator = FrechetMean(space.metric, method='adaptive')
estimator.fit(sample)
estimate = estimator.estimate_
self.assertAllClose(estimate, mean, atol=1e-2)
