def test_parallel_transport_trajectory(self, dim, n_samples):
sphere = Hypersphere(dim)
for step in ["pole", "schild"]:
n_steps = 1 if step == "pole" else 50
tol = 1e-6 if step == "pole" else 1e-2
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = sphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
expected = sphere.metric.parallel_transport(
tan_vec_a, base_point, tan_vec_b)
expected_point = sphere.metric.exp(tan_vec_b, base_point)
ladder = sphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=n_steps,
scheme=step,
return_geodesics=True,
)
result = ladder["transported_tangent_vec"]
result_point = ladder["end_point"]
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_geodesic_vectorization(self):
space = Hypersphere(2)
metric = space.metric
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 3)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
end_point = space.random_uniform(2)
time = gs.linspace(0, 1, 10)
geo = metric.geodesic(initial_point, initial_tangent_vec)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
geo = metric.geodesic(initial_point, end_point=end_point)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
geo = metric.geodesic(initial_point, end_point=end_point[0])
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point[0])
geo = metric.geodesic(initial_point[0], initial_tangent_vec)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
def test_parallel_transport_vectorization(self):
sphere = Hypersphere(2)
n_samples = 4
def is_isometry(tan_a, trans_a, endpoint):
is_tangent = gs.isclose(sphere.metric.inner_product(
endpoint, trans_a),
0.,
atol=1e-6)
is_equinormal = gs.isclose(gs.linalg.norm(trans_a, axis=-1),
gs.linalg.norm(tan_a, axis=-1))
return gs.logical_and(is_tangent, is_equinormal)
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
tan_vec_b = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
end_point = sphere.metric.exp(tan_vec_b, base_point)
transported = sphere.metric.parallel_transport(tan_vec_a, tan_vec_b,
base_point)
result = is_isometry(tan_vec_a, transported, end_point)
self.assertTrue(gs.all(result))
base_point = base_point[0]
tan_vec_a = sphere.to_tangent(tan_vec_a, base_point)
tan_vec_b = sphere.to_tangent(tan_vec_b, base_point)
end_point = sphere.metric.exp(tan_vec_b, base_point)
transported = sphere.metric.parallel_transport(tan_vec_a, tan_vec_b,
base_point)
result = is_isometry(tan_vec_a, transported, end_point)
self.assertTrue(gs.all(result))
one_tan_vec_a = tan_vec_a[0]
transported = sphere.metric.parallel_transport(one_tan_vec_a,
tan_vec_b, base_point)
result = is_isometry(one_tan_vec_a, transported, end_point)
self.assertTrue(gs.all(result))
one_tan_vec_b = tan_vec_b[0]
end_point = end_point[0]
transported = sphere.metric.parallel_transport(tan_vec_a,
one_tan_vec_b,
base_point)
result = is_isometry(tan_vec_a, transported, end_point)
self.assertTrue(gs.all(result))
transported = sphere.metric.parallel_transport(one_tan_vec_a,
one_tan_vec_b,
base_point)
result = is_isometry(one_tan_vec_a, transported, end_point)
self.assertTrue(result)
def test_parallel_transport(self, dim, n_samples):
sphere = Hypersphere(dim)
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
tan_vec_b = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
expected = sphere.metric.parallel_transport(tan_vec_a, base_point,
tan_vec_b)
expected_point = sphere.metric.exp(tan_vec_b, base_point)
base_point = gs.cast(base_point, gs.float64)
base_point, tan_vec_a, tan_vec_b = gs.convert_to_wider_dtype(
[base_point, tan_vec_a, tan_vec_b])
for step, alpha in zip(["pole", "schild"], [1, 2]):
min_n = 1 if step == "pole" else 50
tol = 1e-5 if step == "pole" else 1e-2
for n_rungs in [min_n, 11]:
ladder = sphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=n_rungs,
scheme=step,
alpha=alpha,
)
result = ladder["transported_tangent_vec"]
result_point = ladder["end_point"]
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
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)
def proj(self, base_vector, tangent_vector):
return self._sphere.to_tangent(
tangent_vector, base_point=base_vector)
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)
def rand(self):
return self._sphere.random_uniform()
def tangent_extrinsic_to_spherical_raises_test_data(self):
smoke_data = []
dim_list = [2, 3]
for dim in dim_list:
space = Hypersphere(dim)
base_point = space.random_point()
tangent_vec = space.to_tangent(space.random_point(), base_point)
if dim == 2:
expected = does_not_raise()
smoke_data.append(
dict(
dim=2,
tangent_vec=tangent_vec,
base_point=None,
base_point_spherical=None,
expected=pytest.raises(ValueError),
)
)
else:
expected = pytest.raises(NotImplementedError)
smoke_data.append(
dict(
dim=dim,
tangent_vec=tangent_vec,
base_point=base_point,
base_point_spherical=None,
expected=expected,
)
)
return self.generate_tests(smoke_data)
def sectional_curvature_test_data(self):
dim_list = [1]
n_samples_list = random.sample(range(1, 4), 2)
random_data = []
for dim, n_samples in zip(dim_list, n_samples_list):
sphere = Hypersphere(dim)
base_point = sphere.random_uniform()
tangent_vec_a = sphere.to_tangent(
gs.random.rand(n_samples, sphere.dim + 1), base_point)
tangent_vec_b = sphere.to_tangent(
gs.random.rand(n_samples, sphere.dim + 1), base_point)
expected = gs.ones(n_samples) # try shape here
random_data.append(
dict(
dim=dim,
tangent_vec_a=tangent_vec_a,
tangent_vec_b=tangent_vec_b,
base_point=base_point,
expected=expected,
), )
return self.generate_tests(random_data)
def test_ladder_alpha(self, dim, n_samples):
sphere = Hypersphere(dim)
base_point = sphere.random_uniform(n_samples)
tan_vec_a = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
tan_vec_b = sphere.to_tangent(gs.random.rand(n_samples, 3), base_point)
with pytest.raises(ValueError):
sphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=1,
scheme="pole",
alpha=0.5,
return_geodesics=False,
)
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)
class GeomstatsSphere(EuclideanEmbeddedSubmanifold):
"""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):
dim = ambient_dimension - 1
self._sphere = Hypersphere(dim)
super().__init__('{}-dimensional Hypersphere'.format(dim), dim)
def norm(self, base_point, tangent_vector):
return self._sphere.metric.norm(tangent_vector, base_point=base_point)
def inner(self, base_point, tangent_vector_a, tangent_vector_b):
return self._sphere.metric.inner_product(tangent_vector_a,
tangent_vector_b,
base_point=base_point)
def proj(self, base_point, ambient_vector):
return self._sphere.to_tangent(ambient_vector, base_point=base_point)
def retr(self, base_point, 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_point)
def rand(self):
return self._sphere.random_uniform()
def randvec(self, base_point):
random_point = gs.random.normal(size=self.dim + 1)
random_tangent_vector = self.proj(base_point, random_point)
return random_tangent_vector / gs.linalg.norm(random_tangent_vector)
def zerovec(self, base_point):
return gs.zeros_like(self.rand())
class TestConnection(geomstats.tests.TestCase):
def setup_method(self):
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(0)
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([0.0, 1.0, 0.0, 0.0])
result = self.euc_metric.metric_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
def test_parallel_transport(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, base_point, tan_vec_b)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
base_point = gs.cast(base_point, gs.float64)
base_point, tan_vec_a, tan_vec_b = gs.convert_to_wider_dtype(
[base_point, tan_vec_a, tan_vec_b])
for step, alpha in zip(["pole", "schild"], [1, 2]):
min_n = 1 if step == "pole" else 50
tol = 1e-5 if step == "pole" else 1e-2
for n_rungs in [min_n, 11]:
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=n_rungs,
scheme=step,
alpha=alpha,
)
result = ladder["transported_tangent_vec"]
result_point = ladder["end_point"]
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_parallel_transport_trajectory(self):
n_samples = 2
for step in ["pole", "schild"]:
n_steps = 1 if step == "pole" else 50
tol = 1e-6 if step == "pole" else 1e-2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, base_point, tan_vec_b)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=n_steps,
scheme=step,
return_geodesics=True,
)
result = ladder["transported_tangent_vec"]
result_point = ladder["end_point"]
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_ladder_alpha(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
with pytest.raises(ValueError):
self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
base_point,
tan_vec_b,
n_rungs=1,
scheme="pole",
alpha=0.5,
return_geodesics=False,
)
def test_exp_connection_metric(self):
point = gs.array([gs.pi / 2, 0])
vector = gs.array([0.25, 0.5])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step="rk4")
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected)
def test_exp_connection_metric_vectorization(self):
point = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
vector = gs.array([[0.25, 0.5], [0.30, 0.2]])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step="rk4")
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_log_connection_metric(self):
base_point = gs.array([gs.pi / 3, gs.pi / 4])
point = gs.array([1.0, gs.pi / 2])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step="rk4",
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_log_connection_metric_vectorization(self):
base_point = gs.array([[gs.pi / 3, gs.pi / 4], [gs.pi / 2, gs.pi / 4]])
point = gs.array([[1.0, gs.pi / 2], [gs.pi / 6, gs.pi / 3]])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step="rk4",
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, atol=1e-6)
def test_geodesic_and_coincides_exp_hypersphere(self):
n_geodesic_points = 10
initial_point = self.hypersphere.random_uniform(2)
vector = gs.array([[2.0, 0.0, -1.0]] * 2)
initial_tangent_vec = self.hypersphere.to_tangent(
vector=vector, base_point=initial_point)
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0.0, stop=1.0, num=n_geodesic_points)
points = geodesic(t)
result = points[:, -1]
expected = self.hypersphere.metric.exp(vector, initial_point)
self.assertAllClose(expected, result)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = self.hypersphere.metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_and_coincides_exp_son(self):
n_geodesic_points = 10
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0.0, stop=1.0, num=n_geodesic_points)
points = geodesic(t)
result = points[:, -1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(result, expected)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_invalid_initial_conditions(self):
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
end_point = space.random_uniform(2)
with pytest.raises(RuntimeError):
space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec,
end_point=end_point,
)
def test_geodesic_vectorization(self):
space = Hypersphere(2)
metric = space.metric
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 3)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
end_point = space.random_uniform(2)
time = gs.linspace(0, 1, 10)
geo = metric.geodesic(initial_point, initial_tangent_vec)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
geo = metric.geodesic(initial_point, end_point=end_point)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
geo = metric.geodesic(initial_point, end_point=end_point[0])
path = geo(time)
result = path.shape
expected = (2, 10, 3)
self.assertAllClose(result, expected)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point[0])
geo = metric.geodesic(initial_point[0], initial_tangent_vec)
path = geo(time)
result = path.shape
expected = (2, 10, 3)
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.
"""
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)
class TestWrappedGaussianProcess(geomstats.tests.TestCase):
def setup_method(self):
gs.random.seed(1234)
self.n_samples = 20
# Set up for hypersphere
self.dim_sphere = 2
self.shape_sphere = (self.dim_sphere + 1, )
self.sphere = Hypersphere(dim=self.dim_sphere)
self.intercept_sphere_true = gs.array([0.0, -1.0, 0.0])
self.coef_sphere_true = gs.array([1.0, 0.0, 0.5])
# set up the prior
self.prior = lambda x: self.sphere.metric.exp(
x * self.coef_sphere_true,
base_point=self.intercept_sphere_true,
)
self.kernel = ConstantKernel(1.0, (1e-3, 1e3)) * RBF(10.0, (1e-2, 1e2))
# generate data
X = gs.linspace(0.0, 1.5 * gs.pi, self.n_samples)
self.X_sphere = gs.reshape((X - gs.mean(X)), (-1, 1))
# generate the geodesic
y = self.prior(self.X_sphere)
# Then add orthogonal sinusoidal oscillations
o = (1.0 / 20.0) * gs.array([-0.5, 0.0, 1.0])
o = self.sphere.to_tangent(o, base_point=y)
s = self.X_sphere * gs.sin(5.0 * gs.pi * self.X_sphere)
self.y_sphere = self.sphere.metric.exp(s * o, base_point=y)
def test_fit_hypersphere(self):
"""Test the fit method"""
wgpr = WrappedGaussianProcess(self.sphere,
metric=self.sphere.metric,
prior=self.prior,
kernel=self.kernel)
wgpr.fit(self.X_sphere, self.y_sphere)
self.assertAllClose(wgpr.score(self.X_sphere, self.y_sphere), 1)
def test_predict_hypersphere(self):
"""Test the predict method"""
wgpr = WrappedGaussianProcess(self.sphere,
metric=self.sphere.metric,
prior=self.prior,
kernel=self.kernel)
wgpr.fit(self.X_sphere, self.y_sphere)
y, std = wgpr.predict(self.X_sphere, return_tangent_std=True)
self.assertAllClose(std, gs.zeros(std.shape), atol=1e-4)
self.assertAllClose(y, self.y_sphere, atol=1e-4)
def test_samples_y_hypersphere(self):
"""Test the samples_y method"""
wgpr = WrappedGaussianProcess(self.sphere,
metric=self.sphere.metric,
prior=self.prior,
kernel=self.kernel)
wgpr.fit(self.X_sphere, self.y_sphere)
y = wgpr.sample_y(self.X_sphere, n_samples=100)
y_ = gs.reshape(gs.transpose(y, [0, 2, 1]), (-1, y.shape[1]))
self.assertTrue(gs.all(self.sphere.belongs(y_)))
class TestGeodesicRegression(geomstats.tests.TestCase):
_multiprocess_can_split_ = True
def setup_method(self):
gs.random.seed(1234)
self.n_samples = 20
# Set up for euclidean
self.dim_eucl = 3
self.shape_eucl = (self.dim_eucl, )
self.eucl = Euclidean(dim=self.dim_eucl)
X = gs.random.rand(self.n_samples)
self.X_eucl = X - gs.mean(X)
self.intercept_eucl_true = self.eucl.random_point()
self.coef_eucl_true = self.eucl.random_point()
self.y_eucl = (self.intercept_eucl_true +
self.X_eucl[:, None] * self.coef_eucl_true)
self.param_eucl_true = gs.vstack(
[self.intercept_eucl_true, self.coef_eucl_true])
self.param_eucl_guess = gs.vstack([
self.y_eucl[0],
self.y_eucl[0] + gs.random.normal(size=self.shape_eucl)
])
# Set up for hypersphere
self.dim_sphere = 4
self.shape_sphere = (self.dim_sphere + 1, )
self.sphere = Hypersphere(dim=self.dim_sphere)
X = gs.random.rand(self.n_samples)
self.X_sphere = X - gs.mean(X)
self.intercept_sphere_true = self.sphere.random_point()
self.coef_sphere_true = self.sphere.projection(
gs.random.rand(self.dim_sphere + 1))
self.y_sphere = self.sphere.metric.exp(
self.X_sphere[:, None] * self.coef_sphere_true,
base_point=self.intercept_sphere_true,
)
self.param_sphere_true = gs.vstack(
[self.intercept_sphere_true, self.coef_sphere_true])
self.param_sphere_guess = gs.vstack([
self.y_sphere[0],
self.sphere.to_tangent(gs.random.normal(size=self.shape_sphere),
self.y_sphere[0]),
])
# Set up for special euclidean
self.se2 = SpecialEuclidean(n=2)
self.metric_se2 = self.se2.left_canonical_metric
self.metric_se2.default_point_type = "matrix"
self.shape_se2 = (3, 3)
X = gs.random.rand(self.n_samples)
self.X_se2 = X - gs.mean(X)
self.intercept_se2_true = self.se2.random_point()
self.coef_se2_true = self.se2.to_tangent(
5.0 * gs.random.rand(*self.shape_se2), self.intercept_se2_true)
self.y_se2 = self.metric_se2.exp(
self.X_se2[:, None, None] * self.coef_se2_true[None],
self.intercept_se2_true,
)
self.param_se2_true = gs.vstack([
gs.flatten(self.intercept_se2_true),
gs.flatten(self.coef_se2_true),
])
self.param_se2_guess = gs.vstack([
gs.flatten(self.y_se2[0]),
gs.flatten(
self.se2.to_tangent(gs.random.normal(size=self.shape_se2),
self.y_se2[0])),
])
# Set up for discrete curves
n_sampling_points = 8
self.curves_2d = DiscreteCurves(R2)
self.metric_curves_2d = self.curves_2d.srv_metric
self.metric_curves_2d.default_point_type = "matrix"
self.shape_curves_2d = (n_sampling_points, 2)
X = gs.random.rand(self.n_samples)
self.X_curves_2d = X - gs.mean(X)
self.intercept_curves_2d_true = self.curves_2d.random_point(
n_sampling_points=n_sampling_points)
self.coef_curves_2d_true = self.curves_2d.to_tangent(
5.0 * gs.random.rand(*self.shape_curves_2d),
self.intercept_curves_2d_true)
# Added because of GitHub issue #1575
intercept_curves_2d_true_repeated = gs.tile(
gs.expand_dims(self.intercept_curves_2d_true, axis=0),
(self.n_samples, 1, 1),
)
self.y_curves_2d = self.metric_curves_2d.exp(
self.X_curves_2d[:, None, None] * self.coef_curves_2d_true[None],
intercept_curves_2d_true_repeated,
)
self.param_curves_2d_true = gs.vstack([
gs.flatten(self.intercept_curves_2d_true),
gs.flatten(self.coef_curves_2d_true),
])
self.param_curves_2d_guess = gs.vstack([
gs.flatten(self.y_curves_2d[0]),
gs.flatten(
self.curves_2d.to_tangent(
gs.random.normal(size=self.shape_curves_2d),
self.y_curves_2d[0])),
])
def test_loss_euclidean(self):
"""Test that the loss is 0 at the true parameters."""
gr = GeodesicRegression(
self.eucl,
metric=self.eucl.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
loss = gr._loss(
self.X_eucl,
self.y_eucl,
self.param_eucl_true,
self.shape_eucl,
)
self.assertAllClose(loss.shape, ())
self.assertTrue(gs.isclose(loss, 0.0))
def test_loss_hypersphere(self):
"""Test that the loss is 0 at the true parameters."""
gr = GeodesicRegression(
self.sphere,
metric=self.sphere.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
loss = gr._loss(
self.X_sphere,
self.y_sphere,
self.param_sphere_true,
self.shape_sphere,
)
self.assertAllClose(loss.shape, ())
self.assertTrue(gs.isclose(loss, 0.0))
@geomstats.tests.autograd_and_tf_only
def test_loss_se2(self):
"""Test that the loss is 0 at the true parameters."""
gr = GeodesicRegression(
self.se2,
metric=self.metric_se2,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
loss = gr._loss(self.X_se2, self.y_se2, self.param_se2_true,
self.shape_se2)
self.assertAllClose(loss.shape, ())
self.assertTrue(gs.isclose(loss, 0.0))
@geomstats.tests.autograd_only
def test_loss_curves_2d(self):
"""Test that the loss is 0 at the true parameters."""
gr = GeodesicRegression(
self.curves_2d,
metric=self.metric_curves_2d,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
loss = gr._loss(
self.X_curves_2d,
self.y_curves_2d,
self.param_curves_2d_true,
self.shape_curves_2d,
)
self.assertAllClose(loss.shape, ())
self.assertTrue(gs.isclose(loss, 0.0))
@geomstats.tests.autograd_tf_and_torch_only
def test_value_and_grad_loss_euclidean(self):
gr = GeodesicRegression(
self.eucl,
metric=self.eucl.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
regularization=0,
)
def loss_of_param(param):
return gr._loss(self.X_eucl, self.y_eucl, param, self.shape_eucl)
# Without numpy conversion
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param)
loss_value, loss_grad = objective_with_grad(self.param_eucl_guess)
expected_grad_shape = (2, self.dim_eucl)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(gs.isnan(loss_value))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
# With numpy conversion
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
loss_value, loss_grad = objective_with_grad(self.param_eucl_guess)
# Convert back to arrays/tensors
loss_value = gs.array(loss_value)
loss_grad = gs.array(loss_grad)
expected_grad_shape = (2, self.dim_eucl)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(gs.isnan(loss_value))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
@geomstats.tests.autograd_tf_and_torch_only
def test_value_and_grad_loss_hypersphere(self):
gr = GeodesicRegression(
self.sphere,
metric=self.sphere.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
regularization=0,
)
def loss_of_param(param):
return gr._loss(self.X_sphere, self.y_sphere, param,
self.shape_sphere)
# Without numpy conversion
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param)
loss_value, loss_grad = objective_with_grad(self.param_sphere_guess)
expected_grad_shape = (2, self.dim_sphere + 1)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(gs.isnan(loss_value))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
# With numpy conversion
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
loss_value, loss_grad = objective_with_grad(self.param_sphere_guess)
# Convert back to arrays/tensors
loss_value = gs.array(loss_value)
loss_grad = gs.array(loss_grad)
expected_grad_shape = (2, self.dim_sphere + 1)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(gs.isnan(loss_value))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
@geomstats.tests.autograd_and_tf_only
def test_value_and_grad_loss_se2(self):
gr = GeodesicRegression(
self.se2,
metric=self.metric_se2,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
def loss_of_param(param):
return gr._loss(self.X_se2, self.y_se2, param, self.shape_se2)
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param)
loss_value, loss_grad = objective_with_grad(self.param_se2_true)
expected_grad_shape = (
2,
self.shape_se2[0] * self.shape_se2[1],
)
self.assertTrue(gs.isclose(loss_value, 0.0))
loss_value, loss_grad = objective_with_grad(self.param_se2_guess)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
loss_value, loss_grad = objective_with_grad(self.param_se2_guess)
expected_grad_shape = (
2,
self.shape_se2[0] * self.shape_se2[1],
)
self.assertAllClose(loss_value.shape, ())
self.assertAllClose(loss_grad.shape, expected_grad_shape)
self.assertFalse(gs.isclose(loss_value, 0.0))
self.assertFalse(gs.isnan(loss_value))
self.assertFalse(
gs.all(gs.isclose(loss_grad, gs.zeros(expected_grad_shape))))
self.assertTrue(gs.all(~gs.isnan(loss_grad)))
@geomstats.tests.autograd_tf_and_torch_only
def test_loss_minimization_extrinsic_euclidean(self):
"""Minimize loss from noiseless data."""
gr = GeodesicRegression(self.eucl, regularization=0)
def loss_of_param(param):
return gr._loss(self.X_eucl, self.y_eucl, param, self.shape_eucl)
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
initial_guess = gs.flatten(self.param_eucl_guess)
res = minimize(
objective_with_grad,
initial_guess,
method="CG",
jac=True,
tol=10 * gs.atol,
options={
"disp": True,
"maxiter": 50
},
)
self.assertAllClose(gs.array(res.x).shape, (self.dim_eucl * 2, ))
self.assertAllClose(res.fun, 0.0, atol=1000 * gs.atol)
# Cast required because minimization happens in scipy in float64
param_hat = gs.cast(gs.array(res.x), self.param_eucl_true.dtype)
intercept_hat, coef_hat = gs.split(param_hat, 2)
coef_hat = self.eucl.to_tangent(coef_hat, intercept_hat)
self.assertAllClose(intercept_hat, self.intercept_eucl_true)
tangent_vec_of_transport = self.eucl.metric.log(
self.intercept_eucl_true, base_point=intercept_hat)
transported_coef_hat = self.eucl.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat,
self.coef_eucl_true,
atol=10 * gs.atol)
@geomstats.tests.autograd_tf_and_torch_only
def test_loss_minimization_extrinsic_hypersphere(self):
"""Minimize loss from noiseless data."""
gr = GeodesicRegression(self.sphere, regularization=0)
def loss_of_param(param):
return gr._loss(self.X_sphere, self.y_sphere, param,
self.shape_sphere)
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
initial_guess = gs.flatten(self.param_sphere_guess)
res = minimize(
objective_with_grad,
initial_guess,
method="CG",
jac=True,
tol=10 * gs.atol,
options={
"disp": True,
"maxiter": 50
},
)
self.assertAllClose(
gs.array(res.x).shape, ((self.dim_sphere + 1) * 2, ))
self.assertAllClose(res.fun, 0.0, atol=5e-3)
# Cast required because minimization happens in scipy in float64
param_hat = gs.cast(gs.array(res.x), self.param_sphere_true.dtype)
intercept_hat, coef_hat = gs.split(param_hat, 2)
intercept_hat = self.sphere.projection(intercept_hat)
coef_hat = self.sphere.to_tangent(coef_hat, intercept_hat)
self.assertAllClose(intercept_hat,
self.intercept_sphere_true,
atol=5e-2)
tangent_vec_of_transport = self.sphere.metric.log(
self.intercept_sphere_true, base_point=intercept_hat)
transported_coef_hat = self.sphere.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat,
self.coef_sphere_true,
atol=0.6)
@geomstats.tests.autograd_and_tf_only
def test_loss_minimization_extrinsic_se2(self):
gr = GeodesicRegression(
self.se2,
metric=self.metric_se2,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
def loss_of_param(param):
return gr._loss(self.X_se2, self.y_se2, param, self.shape_se2)
objective_with_grad = gs.autodiff.value_and_grad(loss_of_param,
to_numpy=True)
res = minimize(
objective_with_grad,
gs.flatten(self.param_se2_guess),
method="CG",
jac=True,
options={
"disp": True,
"maxiter": 50
},
)
self.assertAllClose(gs.array(res.x).shape, (18, ))
self.assertAllClose(res.fun, 0.0, atol=1e-6)
# Cast required because minimization happens in scipy in float64
param_hat = gs.cast(gs.array(res.x), self.param_se2_true.dtype)
intercept_hat, coef_hat = gs.split(param_hat, 2)
intercept_hat = gs.reshape(intercept_hat, self.shape_se2)
coef_hat = gs.reshape(coef_hat, self.shape_se2)
intercept_hat = self.se2.projection(intercept_hat)
coef_hat = self.se2.to_tangent(coef_hat, intercept_hat)
self.assertAllClose(intercept_hat, self.intercept_se2_true, atol=1e-4)
tangent_vec_of_transport = self.se2.metric.log(
self.intercept_se2_true, base_point=intercept_hat)
transported_coef_hat = self.se2.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat, self.coef_se2_true, atol=0.6)
@geomstats.tests.autograd_tf_and_torch_only
def test_fit_extrinsic_euclidean(self):
gr = GeodesicRegression(
self.eucl,
metric=self.eucl.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
initialization="random",
regularization=0.9,
)
gr.fit(self.X_eucl, self.y_eucl, compute_training_score=True)
training_score = gr.training_score_
intercept_hat, coef_hat = gr.intercept_, gr.coef_
self.assertAllClose(intercept_hat.shape, self.shape_eucl)
self.assertAllClose(coef_hat.shape, self.shape_eucl)
self.assertAllClose(training_score, 1.0, atol=500 * gs.atol)
self.assertAllClose(intercept_hat, self.intercept_eucl_true)
tangent_vec_of_transport = self.eucl.metric.log(
self.intercept_eucl_true, base_point=intercept_hat)
transported_coef_hat = self.eucl.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat, self.coef_eucl_true)
@geomstats.tests.autograd_tf_and_torch_only
def test_fit_extrinsic_hypersphere(self):
gr = GeodesicRegression(
self.sphere,
metric=self.sphere.metric,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
initialization="random",
regularization=0.9,
)
gr.fit(self.X_sphere, self.y_sphere, compute_training_score=True)
training_score = gr.training_score_
intercept_hat, coef_hat = gr.intercept_, gr.coef_
self.assertAllClose(intercept_hat.shape, self.shape_sphere)
self.assertAllClose(coef_hat.shape, self.shape_sphere)
self.assertAllClose(training_score, 1.0, atol=500 * gs.atol)
self.assertAllClose(intercept_hat,
self.intercept_sphere_true,
atol=5e-3)
tangent_vec_of_transport = self.sphere.metric.log(
self.intercept_sphere_true, base_point=intercept_hat)
transported_coef_hat = self.sphere.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat,
self.coef_sphere_true,
atol=0.6)
@geomstats.tests.autograd_and_tf_only
def test_fit_extrinsic_se2(self):
gr = GeodesicRegression(
self.se2,
metric=self.metric_se2,
center_X=False,
method="extrinsic",
max_iter=50,
init_step_size=0.1,
verbose=True,
initialization="warm_start",
)
gr.fit(self.X_se2, self.y_se2, compute_training_score=True)
intercept_hat, coef_hat = gr.intercept_, gr.coef_
training_score = gr.training_score_
self.assertAllClose(intercept_hat.shape, self.shape_se2)
self.assertAllClose(coef_hat.shape, self.shape_se2)
self.assertTrue(gs.isclose(training_score, 1.0))
self.assertAllClose(intercept_hat, self.intercept_se2_true, atol=1e-4)
tangent_vec_of_transport = self.se2.metric.log(
self.intercept_se2_true, base_point=intercept_hat)
transported_coef_hat = self.se2.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat, self.coef_se2_true, atol=0.6)
@geomstats.tests.autograd_tf_and_torch_only
def test_fit_riemannian_euclidean(self):
gr = GeodesicRegression(
self.eucl,
metric=self.eucl.metric,
center_X=False,
method="riemannian",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
gr.fit(self.X_eucl, self.y_eucl, compute_training_score=True)
intercept_hat, coef_hat = gr.intercept_, gr.coef_
training_score = gr.training_score_
self.assertAllClose(intercept_hat.shape, self.shape_eucl)
self.assertAllClose(coef_hat.shape, self.shape_eucl)
self.assertAllClose(training_score, 1.0, atol=0.1)
self.assertAllClose(intercept_hat, self.intercept_eucl_true)
tangent_vec_of_transport = self.eucl.metric.log(
self.intercept_eucl_true, base_point=intercept_hat)
transported_coef_hat = self.eucl.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat,
self.coef_eucl_true,
atol=1e-2)
@geomstats.tests.autograd_tf_and_torch_only
def test_fit_riemannian_hypersphere(self):
gr = GeodesicRegression(
self.sphere,
metric=self.sphere.metric,
center_X=False,
method="riemannian",
max_iter=50,
init_step_size=0.1,
verbose=True,
)
gr.fit(self.X_sphere, self.y_sphere, compute_training_score=True)
intercept_hat, coef_hat = gr.intercept_, gr.coef_
training_score = gr.training_score_
self.assertAllClose(intercept_hat.shape, self.shape_sphere)
self.assertAllClose(coef_hat.shape, self.shape_sphere)
self.assertAllClose(training_score, 1.0, atol=0.1)
self.assertAllClose(intercept_hat,
self.intercept_sphere_true,
atol=1e-2)
tangent_vec_of_transport = self.sphere.metric.log(
self.intercept_sphere_true, base_point=intercept_hat)
transported_coef_hat = self.sphere.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat,
self.coef_sphere_true,
atol=0.6)
@geomstats.tests.autograd_and_tf_only
def test_fit_riemannian_se2(self):
init = (self.y_se2[0], gs.zeros_like(self.y_se2[0]))
gr = GeodesicRegression(
self.se2,
metric=self.metric_se2,
center_X=False,
method="riemannian",
max_iter=50,
init_step_size=0.1,
verbose=True,
initialization=init,
)
gr.fit(self.X_se2, self.y_se2, compute_training_score=True)
intercept_hat, coef_hat = gr.intercept_, gr.coef_
training_score = gr.training_score_
self.assertAllClose(intercept_hat.shape, self.shape_se2)
self.assertAllClose(coef_hat.shape, self.shape_se2)
self.assertAllClose(training_score, 1.0, atol=1e-4)
self.assertAllClose(intercept_hat, self.intercept_se2_true, atol=1e-4)
tangent_vec_of_transport = self.se2.metric.log(
self.intercept_se2_true, base_point=intercept_hat)
transported_coef_hat = self.se2.metric.parallel_transport(
tangent_vec=coef_hat,
base_point=intercept_hat,
direction=tangent_vec_of_transport,
)
self.assertAllClose(transported_coef_hat, self.coef_se2_true, atol=0.6)
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):
point = gs.array([1., 2., 3., 4., 5.])
proj = self.space.projection(point)
result = self.space.belongs(proj)
expected = 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
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, atol=1e-6)
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-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
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_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.
"""
# 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.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
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-10 * 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)
expected = self.space.to_tangent(vector=vector, base_point=base_point)
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)
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_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 = 100
initial_point = self.space.random_uniform()
vector = gs.array([2., 0., -1., -2., 1.])
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 = 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 = 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)
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(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 _ 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_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)
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 TestConnection(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 4
self.euc_metric = EuclideanMetric(dim=self.dim)
self.connection = Connection(dim=2)
self.hypersphere = Hypersphere(dim=2)
def test_metric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.metric_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
def test_cometric_matrix(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_inverse_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_metric_derivative(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_christoffels(self):
base_point = gs.array([0., 1., 0., 0.])
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
def test_parallel_transport(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
base_point = gs.cast(base_point, gs.float64)
base_point, tan_vec_a, tan_vec_b = gs.convert_to_wider_dtype(
[base_point, tan_vec_a, tan_vec_b])
for step, alpha in zip(['pole', 'schild'], [1, 2]):
min_n = 1 if step == 'pole' else 50
tol = 1e-5 if step == 'pole' else 1e-2
for n_rungs in [min_n, 11]:
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
tan_vec_b,
base_point,
scheme=step,
n_rungs=n_rungs,
alpha=alpha)
result = ladder['transported_tangent_vec']
result_point = ladder['end_point']
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_parallel_transport_trajectory(self):
n_samples = 2
for step in ['pole', 'schild']:
n_steps = 1 if step == 'pole' else 50
tol = 1e-6 if step == 'pole' else 1e-2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
tan_vec_b = self.hypersphere.to_tangent(
gs.random.rand(n_samples, 3), base_point)
expected = self.hypersphere.metric.parallel_transport(
tan_vec_a, tan_vec_b, base_point)
expected_point = self.hypersphere.metric.exp(tan_vec_b, base_point)
ladder = self.hypersphere.metric.ladder_parallel_transport(
tan_vec_a,
tan_vec_b,
base_point,
return_geodesics=True,
scheme=step,
n_rungs=n_steps)
result = ladder['transported_tangent_vec']
result_point = ladder['end_point']
self.assertAllClose(result, expected, rtol=tol, atol=tol)
self.assertAllClose(result_point, expected_point)
def test_ladder_alpha(self):
n_samples = 2
base_point = self.hypersphere.random_uniform(n_samples)
tan_vec_a = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
tan_vec_b = self.hypersphere.to_tangent(gs.random.rand(n_samples, 3),
base_point)
self.assertRaises(
ValueError, lambda: self.hypersphere.metric.
ladder_parallel_transport(tan_vec_a,
tan_vec_b,
base_point,
return_geodesics=False,
scheme='pole',
n_rungs=1,
alpha=0.5))
def test_exp_connection_metric(self):
point = gs.array([gs.pi / 2, 0])
vector = gs.array([0.25, 0.5])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
def test_exp_connection_metric_vectorization(self):
point = gs.array([[gs.pi / 2, 0], [gs.pi / 6, gs.pi / 4]])
vector = gs.array([[0.25, 0.5], [0.30, 0.2]])
point_ext = self.hypersphere.spherical_to_extrinsic(point)
vector_ext = self.hypersphere.tangent_spherical_to_extrinsic(
vector, point)
self.connection.christoffels = self.hypersphere.metric.christoffels
expected = self.hypersphere.metric.exp(vector_ext, point_ext)
result_spherical = self.connection.exp(vector,
point,
n_steps=50,
step='rk4')
result = self.hypersphere.spherical_to_extrinsic(result_spherical)
self.assertAllClose(result, expected, rtol=1e-6)
def test_log_connection_metric(self):
base_point = gs.array([gs.pi / 3, gs.pi / 4])
point = gs.array([1.0, gs.pi / 2])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk',
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)
def test_log_connection_metric_vectorization(self):
base_point = gs.array([[gs.pi / 3, gs.pi / 4], [gs.pi / 2, gs.pi / 4]])
point = gs.array([[1.0, gs.pi / 2], [gs.pi / 6, gs.pi / 3]])
self.connection.christoffels = self.hypersphere.metric.christoffels
vector = self.connection.log(point=point,
base_point=base_point,
n_steps=75,
step='rk',
tol=1e-10)
result = self.hypersphere.tangent_spherical_to_extrinsic(
vector, base_point)
p_ext = self.hypersphere.spherical_to_extrinsic(base_point)
q_ext = self.hypersphere.spherical_to_extrinsic(point)
expected = self.hypersphere.metric.log(base_point=p_ext, point=q_ext)
self.assertAllClose(result, expected, rtol=1e-5, atol=1e-5)
def test_geodesic_and_coincides_exp_hypersphere(self):
n_geodesic_points = 10
initial_point = self.hypersphere.random_uniform(2)
vector = gs.array([[2., 0., -1.]] * 2)
initial_tangent_vec = self.hypersphere.to_tangent(
vector=vector, base_point=initial_point)
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = points[-1]
expected = self.hypersphere.metric.exp(vector, initial_point)
self.assertAllClose(expected, result)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = self.hypersphere.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = self.hypersphere.metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_and_coincides_exp_son(self):
n_geodesic_points = 10
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = points[-1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(result, expected)
initial_point = initial_point[0]
initial_tangent_vec = initial_tangent_vec[0]
geodesic = space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
points = geodesic(t)
result = points[-1]
expected = space.bi_invariant_metric.exp(initial_tangent_vec,
initial_point)
self.assertAllClose(expected, result)
def test_geodesic_invalid_initial_conditions(self):
space = SpecialOrthogonal(n=4)
initial_point = space.random_uniform(2)
vector = gs.random.rand(2, 4, 4)
initial_tangent_vec = space.to_tangent(vector=vector,
base_point=initial_point)
end_point = space.random_uniform(2)
self.assertRaises(
RuntimeError, lambda: space.bi_invariant_metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec,
end_point=end_point))
class TestRiemannianMetric(geomstats.tests.TestCase):
def setup_method(self):
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(0)
self.dim = 2
self.euc = Euclidean(dim=self.dim)
self.sphere = Hypersphere(dim=self.dim)
self.euc_metric = EuclideanMetric(dim=self.dim)
self.sphere_metric = HypersphereMetric(dim=self.dim)
def _euc_metric_matrix(base_point):
"""Return matrix of Euclidean inner-product."""
dim = base_point.shape[-1]
return gs.eye(dim)
def _sphere_metric_matrix(base_point):
"""Return sphere's metric in spherical coordinates."""
theta = base_point[..., 0]
mat = gs.array([[1.0, 0.0], [0.0, gs.sin(theta)**2]])
return mat
new_euc_metric = RiemannianMetric(dim=self.dim)
new_euc_metric.metric_matrix = _euc_metric_matrix
new_sphere_metric = RiemannianMetric(dim=self.dim)
new_sphere_metric.metric_matrix = _sphere_metric_matrix
self.new_euc_metric = new_euc_metric
self.new_sphere_metric = new_sphere_metric
def test_cometric_matrix(self):
base_point = self.euc.random_point()
result = self.euc_metric.cometric_matrix(base_point)
expected = gs.eye(self.dim)
self.assertAllClose(result, expected)
def test_inner_coproduct(self):
base_point = self.euc.random_point()
cotangent_vec_a = gs.array([1.0, 2.0])
cotangent_vec_b = gs.array([1.0, 2.0])
result = self.euc_metric.inner_coproduct(cotangent_vec_a,
cotangent_vec_b, base_point)
expected = 5.0
self.assertAllClose(result, expected)
base_point = gs.array([0.0, 0.0, 1.0])
cotangent_vec_a = self.sphere.to_tangent(gs.array([1.0, 2.0, 0.0]),
base_point)
cotangent_vec_b = self.sphere.to_tangent(gs.array([1.0, 3.0, 0.0]),
base_point)
result = self.sphere_metric.inner_coproduct(cotangent_vec_a,
cotangent_vec_b,
base_point)
expected = 7.0
self.assertAllClose(result, expected)
def test_hamiltonian_euc_metric(self):
position = gs.array([1.0, 2.0])
momentum = gs.array([1.0, 2.0])
state = position, momentum
result = self.euc_metric.hamiltonian(state)
expected = 2.5
self.assertAllClose(result, expected)
def test_hamiltonian_sphere_metric(self):
position = gs.array([0.0, 0.0, 1.0])
momentum = gs.array([1.0, 2.0, 1.0])
state = position, momentum
result = self.sphere_metric.hamiltonian(state)
expected = 3.0
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_metric_derivative_euc_metric(self):
base_point = self.euc.random_point()
result = self.euc_metric.inner_product_derivative_matrix(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_metric_derivative_new_euc_metric(self):
base_point = self.euc.random_point()
result = self.new_euc_metric.inner_product_derivative_matrix(
base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
def test_inner_product_new_euc_metric(self):
base_point = self.euc.random_point()
tan_a = self.euc.random_point()
tan_b = self.euc.random_point()
expected = gs.dot(tan_a, tan_b)
result = self.new_euc_metric.inner_product(tan_a,
tan_b,
base_point=base_point)
self.assertAllClose(result, expected)
def test_inner_product_new_sphere_metric(self):
base_point = gs.array([gs.pi / 3.0, gs.pi / 5.0])
tan_a = gs.array([0.3, 0.4])
tan_b = gs.array([0.1, -0.5])
expected = -0.12
result = self.new_sphere_metric.inner_product(tan_a,
tan_b,
base_point=base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_christoffels_eucl_metric(self):
base_point = self.euc.random_point()
result = self.euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_christoffels_new_eucl_metric(self):
base_point = self.euc.random_point()
result = self.new_euc_metric.christoffels(base_point)
expected = gs.zeros((self.dim, ) * 3)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_christoffels_sphere_metrics(self):
base_point = gs.array([gs.pi / 10.0, gs.pi / 9.0])
expected = self.sphere_metric.christoffels(base_point)
result = self.new_sphere_metric.christoffels(base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_exp_new_eucl_metric(self):
base_point = self.euc.random_point()
tan = self.euc.random_point()
expected = base_point + tan
result = self.new_euc_metric.exp(tan, base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_and_torch_only
def test_log_new_eucl_metric(self):
base_point = self.euc.random_point()
point = self.euc.random_point()
expected = point - base_point
result = self.new_euc_metric.log(point, base_point)
self.assertAllClose(result, expected)
@geomstats.tests.autograd_tf_and_torch_only
def test_exp_new_sphere_metric(self):
base_point = gs.array([gs.pi / 10.0, gs.pi / 9.0])
tan = gs.array([gs.pi / 2.0, 0.0])
expected = gs.array([gs.pi / 10.0 + gs.pi / 2.0, gs.pi / 9.0])
result = self.new_sphere_metric.exp(tan, base_point)
self.assertAllClose(result, expected)
