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.
"""
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)
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_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([1.0, 2.0, 3.0, 4.0, 6.0])
base_point = base_point / gs.linalg.norm(base_point)
point = gs.array([0.0, 5.0, 6.0, 2.0, -1.0])
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([1.0, 2.0, 3.0, 4.0, 6.0])
base_point = base_point / gs.linalg.norm(base_point)
point = base_point + 1e-4 * gs.array([-1.0, -2.0, 1.0, 1.0, 0.1])
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([0.0, -3.0, 0.0, 3.0, 4.0])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([3.0, 2.0, 0.0, 0.0, -1.0])
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([10.0, -2.0, -0.5, 34.0, 3.0])
base_point = base_point / gs.linalg.norm(base_point)
vector = 1e-4 * gs.array([0.06, -51.0, 6.0, 5.0, 3.0])
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 = 1.0 / gs.sqrt(129.0) * gs.array([10.0, -2.0, -5.0, 0.0, 0.0])
point_b = 1.0 / gs.sqrt(435.0) * gs.array([1.0, -20.0, -5.0, 0.0, 3.0])
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 = 1.0 / gs.sqrt(129.0) * gs.array([10.0, -2.0, -5.0, 0.0, 0.0])
point_b = 1.0 / gs.sqrt(435.0) * gs.array([1.0, -20.0, -5.0, 0.0, 3.0])
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 = 1.0 / gs.sqrt(129.0) * gs.array([10.0, -2.0, -5.0, 0.0, 0.0])
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0.0
self.assertAllClose(result, expected)
def test_dist_pairwise(self):
point_a = 1.0 / gs.sqrt(129.0) * gs.array([10.0, -2.0, -5.0, 0.0, 0.0])
point_b = 1.0 / gs.sqrt(435.0) * gs.array([1.0, -20.0, -5.0, 0.0, 3.0])
point = gs.array([point_a, point_b])
result = self.metric.dist_pairwise(point)
expected = gs.array([[0.0, 1.24864502], [1.24864502, 0.0]])
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([10.0, -2.0, -0.5, 0.0, 0.0])
point_a = point_a / gs.linalg.norm(point_a)
point_b = gs.array([2.0, 10, 0.0, 0.0, 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([16.0, -2.0, -2.5, 84.0, 3.0])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9.0, 0.0, -1.0, -2.0, 1.0])
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([[16.0, -2.0, -2.5, 84.0, 3.0],
[16.0, -2.0, -2.5, 84.0, 3.0]])
base_single_point = gs.array([16.0, -2.0, -2.5, 84.0, 3.0])
scalar_norm = gs.linalg.norm(base_single_point)
base_point = base_point / scalar_norm
vector = gs.array([[9.0, 0.0, -1.0, -2.0, 1.0],
[9.0, 0.0, -1.0, -2.0, 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, 0.0, -1.0, -2.0, 1.0]] * 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=0.0, stop=1.0, 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=0.0, stop=1.0, 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, 0.0, 0.0, 0.0])
tangent_vec_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
base_point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = 0.0
self.assertAllClose(expected, result)
def test_inner_product_vectorization_single_samples(self):
tangent_vec_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
tangent_vec_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
base_point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = 0.0
self.assertAllClose(expected, result)
tangent_vec_a = gs.array([[1.0, 0.0, 0.0, 0.0, 0.0]])
tangent_vec_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
base_point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = gs.array([0.0])
self.assertAllClose(expected, result)
tangent_vec_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
tangent_vec_b = gs.array([[0.0, 1.0, 0.0, 0.0, 0.0]])
base_point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = gs.array([0.0])
self.assertAllClose(expected, result)
tangent_vec_a = gs.array([[1.0, 0.0, 0.0, 0.0, 0.0]])
tangent_vec_b = gs.array([[0.0, 1.0, 0.0, 0.0, 0.0]])
base_point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = gs.array([0.0])
self.assertAllClose(expected, result)
tangent_vec_a = gs.array([[1.0, 0.0, 0.0, 0.0, 0.0]])
tangent_vec_b = gs.array([[0.0, 1.0, 0.0, 0.0, 0.0]])
base_point = gs.array([[0.0, 0.0, 0.0, 0.0, 1.0]])
result = self.metric.inner_product(tangent_vec_a, tangent_vec_b,
base_point)
expected = gs.array([0.0])
self.assertAllClose(expected, result)
def test_diameter(self):
dim = 2
sphere = Hypersphere(dim)
point_a = gs.array([[0.0, 0.0, 1.0]])
point_b = gs.array([[1.0, 0.0, 0.0]])
point_c = gs.array([[0.0, 0.0, -1.0]])
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 = 1000.0
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 = 1.0
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 + 1.0 - mean_norm**2) /
(1.0 - 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 / 2.0, kappa_estimate)
bessel_func_2 = scipy.special.iv(p / 2.0 - 1.0, kappa_estimate)
ratio = bessel_func_1 / bessel_func_2
denominator = 1.0 - ratio**2 - (p -
1.0) * 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] + [0.0] * 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, 0.0, 0.0])
self.assertAllClose(result, expected)
def test_extrinsic_to_spherical(self):
"""
Check vectorization of conversion from spherical
to extrinsic coordinates on the 2-sphere.
"""
dim = 2
sphere = Hypersphere(dim)
points_extrinsic = gs.array([1.0, 0.0, 0.0])
result = sphere.extrinsic_to_spherical(points_extrinsic)
expected = gs.array([gs.pi / 2, 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, 0.0, 0.0],
[gs.sqrt(2.0) / 4.0,
gs.sqrt(2.0) / 4.0,
gs.sqrt(3.0) / 2.0],
])
self.assertAllClose(result, expected)
def test_extrinsic_to_spherical_vectorization(self):
dim = 2
sphere = Hypersphere(dim)
expected = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
point_extrinsic = gs.array([
[1.0, 0.0, 0.0],
[gs.sqrt(2.0) / 4.0,
gs.sqrt(2.0) / 4.0,
gs.sqrt(3.0) / 2.0],
])
result = sphere.extrinsic_to_spherical(point_extrinsic)
self.assertAllClose(result, expected)
def test_spherical_to_extrinsic_and_inverse(self):
dim = 2
n_samples = 5
sphere = Hypersphere(dim)
points = gs.random.rand(n_samples, 2) * gs.pi * gs.array([1., 2.
])[None, :]
extrinsic = sphere.spherical_to_extrinsic(points)
result = sphere.extrinsic_to_spherical(extrinsic)
self.assertAllClose(result, points)
points_extrinsic = sphere.random_uniform(n_samples)
spherical = sphere.extrinsic_to_spherical(points_extrinsic)
result = sphere.spherical_to_extrinsic(spherical)
self.assertAllClose(result, points_extrinsic)
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_tangent_extrinsic_to_spherical(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]])
expected = gs.array([[0.25, 0.5], [0.3, 0.2]])
tangent_vecs = gs.array([[0, 0.5, -0.25], [0, 0.2, -0.3]])
result = sphere.tangent_extrinsic_to_spherical(
tangent_vecs, base_point_spherical=base_points_spherical)
self.assertAllClose(result, expected)
result = sphere.tangent_extrinsic_to_spherical(tangent_vecs[0],
base_point=gs.array(
[1., 0., 0.]))
self.assertAllClose(result, expected[0])
def test_tangent_spherical_and_extrinsic_inverse(self):
dim = 2
n_samples = 5
sphere = Hypersphere(dim)
points = gs.random.rand(n_samples, 2) * gs.pi * gs.array([1., 2.
])[None, :]
tangent_spherical = gs.random.rand(n_samples, 2)
tangent_extrinsic = sphere.tangent_spherical_to_extrinsic(
tangent_spherical, points)
result = sphere.tangent_extrinsic_to_spherical(
tangent_extrinsic, base_point_spherical=points)
self.assertAllClose(result, tangent_spherical)
points_extrinsic = sphere.random_uniform(n_samples)
vector = gs.random.rand(n_samples, dim + 1)
tangent_extrinsic = sphere.to_tangent(vector, points_extrinsic)
tangent_spherical = sphere.tangent_extrinsic_to_spherical(
tangent_extrinsic, base_point=points_extrinsic)
spherical = sphere.extrinsic_to_spherical(points_extrinsic)
result = sphere.tangent_spherical_to_extrinsic(tangent_spherical,
spherical)
self.assertAllClose(result, tangent_extrinsic)
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_autograd_and_torch_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_autograd_and_torch_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)
def test_raises(self):
space = self.space
point = space.random_uniform()
self.assertRaises(NotImplementedError,
lambda: space.extrinsic_to_spherical(point))
self.assertRaises(
NotImplementedError,
lambda: space.tangent_extrinsic_to_spherical(point, point))
sphere = Hypersphere(2)
self.assertRaises(ValueError,
lambda: sphere.tangent_extrinsic_to_spherical(point))
def test_angle_to_extrinsic(self):
space = Hypersphere(1)
point = gs.pi / 4
result = space.angle_to_extrinsic(point)
expected = gs.array([1., 1.]) / gs.sqrt(2.)
self.assertAllClose(result, expected)
point = gs.array([1. / 3, 0.]) * gs.pi
result = space.angle_to_extrinsic(point)
expected = gs.array([[1. / 2, gs.sqrt(3.) / 2], [1., 0.]])
self.assertAllClose(result, expected)
def test_extrinsic_to_angle(self):
space = Hypersphere(1)
point = gs.array([1., 1.]) / gs.sqrt(2.)
result = space.extrinsic_to_angle(point)
expected = gs.pi / 4
self.assertAllClose(result, expected)
point = gs.array([[1. / 2, gs.sqrt(3.) / 2], [1., 0.]])
result = space.extrinsic_to_angle(point)
expected = gs.array([1. / 3, 0.]) * gs.pi
self.assertAllClose(result, expected)
def test_extrinsic_to_angle_inverse(self):
space = Hypersphere(1)
point = space.random_uniform()
angle = space.extrinsic_to_angle(point)
result = space.angle_to_extrinsic(angle)
self.assertAllClose(result, point)
space = Hypersphere(1, default_coords_type='intrinsic')
angle = space.random_uniform()
extrinsic = space.angle_to_extrinsic(angle)
result = space.extrinsic_to_angle(extrinsic)
self.assertAllClose(result, angle)
