class TestHypersphereMethods(unittest.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dimension=self.dimension)
self.metric = self.space.metric
self.n_samples = 10
def test_belongs(self):
point = self.space.random_uniform()
belongs = self.space.belongs(point)
gs.testing.assert_allclose(belongs.shape, (1, 1))
def test_random_uniform(self):
point = self.space.random_uniform()
gs.testing.assert_allclose(point.shape, (1, self.dimension + 1))
def test_random_uniform_and_belongs(self):
point = self.space.random_uniform()
self.assertTrue(self.space.belongs(point))
def test_projection_and_belongs(self):
point = gs.array([1, 2, 3, 4, 5])
result = self.space.projection(point)
self.assertTrue(self.space.belongs(result))
def test_intrinsic_and_extrinsic_coords(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([.1, 0., 0., .1])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
point_ext = self.space.random_uniform()
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([[.1, 0., 0., .1], [.1, .1, .1, .4],
[.1, .3, 0., .1], [-0.1, .1, -.4, .1],
[0., 0., .1, .1], [.1, .1, .1, .1]])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
n_samples = self.n_samples
point_ext = self.space.random_uniform(n_samples=n_samples)
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_log_and_exp_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# General case
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = gs.array([0., 5., 6., 2., -1])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected, atol=1e-8)
def test_log_and_exp_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# Edge case: two very close points, base_point_2 and point_2,
# form an angle < epsilon
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = base_point + 1e-12 * gs.array([-1., -2., 1., 1., .1])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_exp_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_vec = self.space.random_uniform()
one_base_point = self.space.random_uniform()
n_vecs = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
gs.testing.assert_allclose(result.shape, (1, dim))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=one_base_point)
result = self.metric.exp(n_tangent_vecs, one_base_point)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(n_tangent_vecs[i], one_base_point)
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=n_base_points)
result = self.metric.exp(one_tangent_vec, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(one_tangent_vec[i], n_base_points[i])
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=n_base_points)
result = self.metric.exp(n_tangent_vecs, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(n_tangent_vecs[i], n_base_points[i])
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_log_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_base_point = self.space.random_uniform()
one_point = self.space.random_uniform()
n_points = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
result = self.metric.log(one_point, one_base_point)
gs.testing.assert_allclose(result.shape, (1, dim))
result = self.metric.log(n_points, one_base_point)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
result = self.metric.log(one_point, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
result = self.metric.log(n_points, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# General case
# NB: Riemannian log gives a regularized tangent vector,
# so we take the norm modulo 2 * pi.
base_point = gs.array([0., -3., 0., 3., 4.])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9., 5., 0., 0., -1.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = vector
norm_expected = gs.linalg.norm(expected)
regularized_norm_expected = gs.mod(norm_expected, 2 * gs.pi)
expected = expected / norm_expected * regularized_norm_expected
expected = helper.to_vector(expected)
# TODO(nina): this test fails
# self.assertTrue(
# gs.allclose(result, expected),
# 'result = {}, expected = {}'.format(result, expected))
def test_exp_and_log_and_projection_to_tangent_space_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# Edge case: tangent vector has norm < epsilon
base_point = gs.array([10., -2., -.5, 34., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = 1e-10 * gs.array([.06, -51., 6., 5., 3.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
expected = helper.to_vector(expected)
gs.testing.assert_allclose(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 = self.space.random_uniform()
point_b = self.space.random_uniform()
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.squared_norm(vector=log)
expected = self.metric.squared_dist(point_a, point_b)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_squared_dist_vectorization(self):
n_samples = self.n_samples
one_point_a = self.space.random_uniform()
one_point_b = self.space.random_uniform()
n_points_a = self.space.random_uniform(n_samples=n_samples)
n_points_b = self.space.random_uniform(n_samples=n_samples)
result = self.metric.squared_dist(one_point_a, one_point_b)
gs.testing.assert_allclose(result.shape, (1, 1))
result = self.metric.squared_dist(n_points_a, one_point_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
result = self.metric.squared_dist(one_point_a, n_points_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
result = self.metric.squared_dist(n_points_a, n_points_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
def test_norm_and_dist(self):
"""
Test that the distance between two points is
the norm of their logarithm.
"""
point_a = self.space.random_uniform()
point_b = self.space.random_uniform()
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.norm(vector=log)
expected = self.metric.dist(point_a, point_b)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_dist_point_and_itself(self):
# Distance between a point and itself is 0.
point_a = gs.array([10., -2., -.5, 2., 3.])
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0.
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(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_b = gs.array([2., 10, 0., 0., 0.])
self.assertEqual(gs.dot(point_a, point_b), 0)
result = self.metric.dist(point_a, point_b)
expected = gs.pi / 2
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(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.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.mod(gs.linalg.norm(tangent_vec), 2 * gs.pi)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_geodesic_and_belongs(self):
initial_point = self.space.random_uniform()
vector = gs.array([2., 0., -1., -2., 1.])
initial_tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=initial_point)
geodesic = self.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0, stop=1, num=100)
points = geodesic(t)
self.assertTrue(gs.all(self.space.belongs(points)))
def test_variance(self):
point = self.space.random_uniform()
result = self.metric.variance([point, point])
expected = 0
gs.testing.assert_allclose(result, expected)
def test_mean(self):
point = self.space.random_uniform()
result = self.metric.mean([point, point])
expected = point
gs.testing.assert_allclose(result, expected)
def test_mean_and_belongs(self):
point_a = self.space.random_uniform()
point_b = self.space.random_uniform()
point_c = self.space.random_uniform()
result = self.metric.mean([point_a, point_b, point_c])
self.assertTrue(self.space.belongs(result))
class TestHypersphereOnTensorFlow(tf.test.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dimension=self.dimension)
self.metric = self.space.metric
self.n_samples = 10
self.depth = 3
@classmethod
def setUpClass(cls):
tf.enable_eager_execution()
os.environ['GEOMSTATS_BACKEND'] = 'tensorflow'
importlib.reload(gs)
@classmethod
def tearDownClass(cls):
os.environ['GEOMSTATS_BACKEND'] = 'numpy'
importlib.reload(gs)
def test_random_uniform_and_belongs(self):
pass
"""
Test that the random uniform method samples
on the hypersphere space.
"""
point = self.space.random_uniform()
with self.test_session():
self.assertTrue(gs.eval(self.space.belongs(point)[0, 0]))
def test_exp_and_dist_and_projection_to_tangent_space(self):
with self.test_session():
base_point = gs.array([16., -2., -2.5, 84., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9., 0., -1., -2., 1.])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec,
base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_exp_and_dist_and_projection_to_tangent_space_vec(self):
with self.test_session():
base_point = gs.array([[16., -2., -2.5, 84., 3.],
[16., -2., -2.5, 84., 3.]])
base_single_point = gs.array([16., -2., -2.5, 84., 3.])
scalar_norm = gs.linalg.norm(base_single_point)
base_point = base_point / scalar_norm
vector = gs.array([[9., 0., -1., -2., 1.], [9., 0., -1., -2., 1]])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec,
base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec, axis=-1) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(gs.eval(result), gs.eval(expected))
class TestHypersphereTensorFlow(tf.test.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dimension=self.dimension)
self.metric = self.space.metric
self.n_samples = 3
@classmethod
def setUpClass(cls):
os.environ['GEOMSTATS_BACKEND'] = 'tensorflow'
importlib.reload(gs)
@classmethod
def tearDownClass(cls):
os.environ['GEOMSTATS_BACKEND'] = 'numpy'
importlib.reload(gs)
def test_belongs(self):
point = self.space.random_uniform()
bool_belongs = self.space.belongs(point)
expected = tf.convert_to_tensor([[True]])
with self.test_session():
self.assertAllClose(gs.eval(expected), gs.eval(bool_belongs))
def test_random_uniform(self):
point = self.space.random_uniform()
point_numpy = np.random.uniform(size=(1, self.dimension + 1))
with self.test_session():
self.assertShapeEqual(point_numpy, point)
def test_random_uniform_and_belongs(self):
"""
Test that the random uniform method samples
on the hypersphere space.
"""
point = self.space.random_uniform()
with self.test_session():
self.assertTrue(gs.eval(self.space.belongs(point)[0, 0]))
def test_projection_and_belongs(self):
point = tf.convert_to_tensor([1., 2., 3., 4., 5.])
result = self.space.projection(point)
with self.test_session():
self.assertTrue(gs.eval(self.space.belongs(result)[0, 0]))
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 = tf.convert_to_tensor([.1, 0., 0., .1])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
# TODO(nina): Fix that the test fails if point_ext generated
# with tf.random_uniform
point_ext = (1. / (gs.sqrt(6.)) *
tf.convert_to_tensor([1., 0., 0., 1., 2.]))
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = tf.convert_to_tensor([[.1, 0., 0., .1], [.1, .1, .1, .4],
[.1, .3, 0.,
.1], [-0.1, .1, -.4, .1],
[0., 0., .1, .1], [.1, .1, .1, .1]])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
sqrt_3 = np.sqrt(3.)
point_ext = tf.convert_to_tensor(
[[1. / sqrt_3, 0., 0., 1. / sqrt_3, 1. / sqrt_3],
[1. / sqrt_3, 1. / sqrt_3, 1. / sqrt_3, 0., 0.],
[0., 0., 1. / sqrt_3, 1. / sqrt_3, 1. / sqrt_3]],
dtype=np.float64)
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_log_and_exp_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# General case
base_point = tf.convert_to_tensor([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = tf.convert_to_tensor([0., 5., 6., 2., -1.])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected), atol=1e-8)
def test_log_and_exp_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# Edge case: two very close points, base_point_2 and point_2,
# form an angle < epsilon
base_point = tf.convert_to_tensor([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = (base_point +
1e-12 * tf.convert_to_tensor([-1., -2., 1., 1., .1]))
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_exp_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
with self.test_session():
one_vec = self.space.random_uniform()
one_base_point = self.space.random_uniform()
n_vecs = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
point_numpy = np.random.uniform(size=(1, dim))
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=one_base_point)
result = self.metric.exp(n_tangent_vecs, one_base_point)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=n_base_points)
result = self.metric.exp(one_tangent_vec, n_base_points)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=n_base_points)
result = self.metric.exp(n_tangent_vecs, n_base_points)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
def test_log_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_base_point = self.space.random_uniform()
one_point = self.space.random_uniform()
n_points = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
result = self.metric.log(one_point, one_base_point)
point_numpy = np.random.uniform(size=(1, dim))
with self.test_session():
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
result = self.metric.log(n_points, one_base_point)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
result = self.metric.log(one_point, n_base_points)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
result = self.metric.log(n_points, n_base_points)
point_numpy = np.random.uniform(size=(n_samples, dim))
with self.test_session():
self.assertShapeEqual(point_numpy, result)
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# 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 = tf.convert_to_tensor([0., -3., 0., 3., 4.])
base_point = base_point / gs.linalg.norm(base_point)
vector = tf.convert_to_tensor([9., 5., 0., 0., -1.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = vector
norm_expected = gs.linalg.norm(expected)
regularized_norm_expected = gs.mod(norm_expected, 2 * gs.pi)
expected = expected / norm_expected * regularized_norm_expected
expected = helper.to_vector(expected)
# TODO(nina): Fix that this test fails, in numpy
# with self.test_session():
# self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_exp_and_log_and_projection_to_tangent_space_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# Edge case: tangent vector has norm < epsilon
base_point = tf.convert_to_tensor([10., -2., -.5, 34., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = 1e-10 * tf.convert_to_tensor([.06, -51., 6., 5., 3.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
expected = helper.to_vector(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(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.) *
tf.convert_to_tensor([10., -2., -5., 0., 0.]))
point_b = (1. / gs.sqrt(435.) *
tf.convert_to_tensor([1., -20., -5., 0., 3.]))
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.squared_norm(vector=log)
expected = self.metric.squared_dist(point_a, point_b)
expected = helper.to_scalar(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_squared_dist_vectorization(self):
n_samples = self.n_samples
one_point_a = self.space.random_uniform()
one_point_b = self.space.random_uniform()
n_points_a = self.space.random_uniform(n_samples=n_samples)
n_points_b = self.space.random_uniform(n_samples=n_samples)
result = self.metric.squared_dist(one_point_a, one_point_b)
point_numpy = np.random.uniform(size=(1, 1))
with self.test_session():
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
result = self.metric.squared_dist(n_points_a, one_point_b)
point_numpy = np.random.uniform(size=(n_samples, 1))
with self.test_session():
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
result = self.metric.squared_dist(one_point_a, n_points_b)
point_numpy = np.random.uniform(size=(n_samples, 1))
with self.test_session():
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
result = self.metric.squared_dist(n_points_a, n_points_b)
point_numpy = np.random.uniform(size=(n_samples, 1))
with self.test_session():
# TODO(nina): Fix that this test fails with assertShapeEqual
self.assertAllClose(point_numpy.shape, gs.eval(gs.shape(result)))
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.) *
tf.convert_to_tensor([10., -2., -5., 0., 0.]))
point_b = (1. / gs.sqrt(435.) *
tf.convert_to_tensor([1., -20., -5., 0., 3.]))
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.norm(vector=log)
expected = self.metric.dist(point_a, point_b)
expected = helper.to_scalar(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_dist_point_and_itself(self):
# Distance between a point and itself is 0
point_a = (1. / gs.sqrt(129.) *
tf.convert_to_tensor([10., -2., -5., 0., 0.]))
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0.
expected = helper.to_scalar(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_dist_orthogonal_points(self):
# Distance between two orthogonal points is pi / 2.
point_a = gs.array([10., -2., -.5, 0., 0.])
point_a = point_a / gs.linalg.norm(point_a)
point_b = gs.array([2., 10, 0., 0., 0.])
point_b = point_b / gs.linalg.norm(point_b)
result = gs.dot(point_a, point_b)
result = helper.to_scalar(result)
expected = 0
expected = helper.to_scalar(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
result = self.metric.dist(point_a, point_b)
expected = gs.pi / 2
expected = helper.to_scalar(expected)
with self.test_session():
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_exp_and_dist_and_projection_to_tangent_space(self):
with self.test_session():
base_point = tf.convert_to_tensor([16., -2., -2.5, 84., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = tf.convert_to_tensor([9., 0., -1., -2., 1.])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec,
base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_exp_and_dist_and_projection_to_tangent_space_vec(self):
with self.test_session():
base_point = tf.convert_to_tensor([[16., -2., -2.5, 84., 3.],
[16., -2., -2.5, 84., 3.]])
base_single_point = tf.convert_to_tensor([16., -2., -2.5, 84., 3.])
scalar_norm = gs.linalg.norm(base_single_point)
base_point = base_point / scalar_norm
vector = tf.convert_to_tensor([[9., 0., -1., -2., 1.],
[9., 0., -1., -2., 1]])
tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec,
base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.linalg.norm(tangent_vec, axis=-1) % (2 * gs.pi)
expected = helper.to_scalar(expected)
self.assertAllClose(gs.eval(result), gs.eval(expected))
def test_geodesic_and_belongs(self):
n_geodesic_points = 100
initial_point = self.space.random_uniform()
vector = tf.convert_to_tensor([2., 0., -1., -2., 1.])
initial_tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=initial_point)
geodesic = self.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
bool_belongs = self.space.belongs(points)
expected = tf.convert_to_tensor(n_geodesic_points * [[True]])
with self.test_session():
self.assertAllClose(gs.eval(expected), gs.eval(bool_belongs))
class TestHypersphereMethods(unittest.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(1234)
self.dimension = 4
self.space = Hypersphere(dimension=self.dimension)
self.metric = self.space.metric
self.n_samples = 10
def test_belongs(self):
point = self.space.random_uniform()
belongs = self.space.belongs(point)
gs.testing.assert_allclose(belongs.shape, (1, 1))
def test_random_uniform(self):
point_bound = self.space.random_uniform()
point_nobound = self.space.random_uniform(bound=None)
gs.testing.assert_allclose(point_bound.shape, (1, self.dimension + 1))
gs.testing.assert_allclose(point_nobound.shape,
(1, self.dimension + 1))
def test_random_uniform_and_belongs(self):
point_bound = self.space.random_uniform()
point_nobound = self.space.random_uniform(bound=None)
self.assertTrue(self.space.belongs(point_bound))
self.assertTrue(self.space.belongs(point_nobound))
def test_projection_and_belongs(self):
point = gs.array([1., 2., 3., 4., 5.])
result = self.space.projection(point)
self.assertTrue(self.space.belongs(result))
def test_intrinsic_and_extrinsic_coords(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([.1, 0., 0., .1])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
point_ext = 1. / (gs.sqrt(6.)) * gs.array([1., 0., 0., 1., 2.])
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([[.1, 0., 0., .1], [.1, .1, .1, .4],
[.1, .3, 0., .1], [-0.1, .1, -.4, .1],
[0., 0., .1, .1], [.1, .1, .1, .1]])
point_ext = self.space.intrinsic_to_extrinsic_coords(point_int)
result = self.space.extrinsic_to_intrinsic_coords(point_ext)
expected = point_int
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
n_samples = self.n_samples
point_ext = self.space.random_uniform(n_samples=n_samples)
point_int = self.space.extrinsic_to_intrinsic_coords(point_ext)
result = self.space.intrinsic_to_extrinsic_coords(point_int)
expected = point_ext
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_log_and_exp_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# General case
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = gs.array([0., 5., 6., 2., -1])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected, atol=1e-8)
def test_log_and_exp_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Log then Riemannian Exp
# Edge case: two very close points, base_point_2 and point_2,
# form an angle < epsilon
base_point = gs.array([1., 2., 3., 4., 6.])
base_point = base_point / gs.linalg.norm(base_point)
point = base_point + 1e-12 * gs.array([-1., -2., 1., 1., .1])
point = point / gs.linalg.norm(point)
log = self.metric.log(point=point, base_point=base_point)
result = self.metric.exp(tangent_vec=log, base_point=base_point)
expected = point
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_exp_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_vec = self.space.random_uniform()
one_base_point = self.space.random_uniform()
n_vecs = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
gs.testing.assert_allclose(result.shape, (1, dim))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=one_base_point)
result = self.metric.exp(n_tangent_vecs, one_base_point)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(n_tangent_vecs[i], one_base_point)
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=n_base_points)
result = self.metric.exp(one_tangent_vec, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(one_tangent_vec[i], n_base_points[i])
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=n_base_points)
result = self.metric.exp(n_tangent_vecs, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
expected = gs.zeros((n_samples, dim))
for i in range(n_samples):
expected[i] = self.metric.exp(n_tangent_vecs[i], n_base_points[i])
expected = helper.to_vector(expected)
gs.testing.assert_allclose(result, expected)
def test_log_vectorization(self):
n_samples = self.n_samples
dim = self.dimension + 1
one_base_point = self.space.random_uniform()
one_point = self.space.random_uniform()
n_points = self.space.random_uniform(n_samples=n_samples)
n_base_points = self.space.random_uniform(n_samples=n_samples)
result = self.metric.log(one_point, one_base_point)
gs.testing.assert_allclose(result.shape, (1, dim))
result = self.metric.log(n_points, one_base_point)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
result = self.metric.log(one_point, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
result = self.metric.log(n_points, n_base_points)
gs.testing.assert_allclose(result.shape, (n_samples, dim))
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# General case
# NB: Riemannian log gives a regularized tangent vector,
# so we take the norm modulo 2 * pi.
base_point = gs.array([0., -3., 0., 3., 4.])
base_point = base_point / gs.linalg.norm(base_point)
vector = gs.array([9., 5., 0., 0., -1.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = vector
norm_expected = gs.linalg.norm(expected)
regularized_norm_expected = gs.mod(norm_expected, 2 * gs.pi)
expected = expected / norm_expected * regularized_norm_expected
expected = helper.to_vector(expected)
# TODO(nina): this test fails
# self.assertTrue(
# gs.allclose(result, expected),
# 'result = {}, expected = {}'.format(result, expected))
def test_exp_and_log_and_projection_to_tangent_space_edge_case(self):
"""
Test that the riemannian exponential
and the riemannian logarithm are inverse.
Expect their composition to give the identity function.
NB: points on the n-dimensional sphere are
(n+1)-D vectors of norm 1.
"""
# Riemannian Exp then Riemannian Log
# Edge case: tangent vector has norm < epsilon
base_point = gs.array([10., -2., -.5, 34., 3.])
base_point = base_point / gs.linalg.norm(base_point)
vector = 1e-10 * gs.array([.06, -51., 6., 5., 3.])
vector = self.space.projection_to_tangent_space(vector=vector,
base_point=base_point)
exp = self.metric.exp(tangent_vec=vector, base_point=base_point)
result = self.metric.log(point=exp, base_point=base_point)
expected = self.space.projection_to_tangent_space(
vector=vector, base_point=base_point)
expected = helper.to_vector(expected)
gs.testing.assert_allclose(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 = self.space.random_uniform()
point_b = self.space.random_uniform()
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.squared_norm(vector=log)
expected = self.metric.squared_dist(point_a, point_b)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_squared_dist_vectorization(self):
n_samples = self.n_samples
one_point_a = self.space.random_uniform()
one_point_b = self.space.random_uniform()
n_points_a = self.space.random_uniform(n_samples=n_samples)
n_points_b = self.space.random_uniform(n_samples=n_samples)
result = self.metric.squared_dist(one_point_a, one_point_b)
gs.testing.assert_allclose(result.shape, (1, 1))
result = self.metric.squared_dist(n_points_a, one_point_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
result = self.metric.squared_dist(one_point_a, n_points_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
result = self.metric.squared_dist(n_points_a, n_points_b)
gs.testing.assert_allclose(result.shape, (n_samples, 1))
def test_norm_and_dist(self):
"""
Test that the distance between two points is
the norm of their logarithm.
"""
point_a = self.space.random_uniform()
point_b = self.space.random_uniform()
log = self.metric.log(point=point_a, base_point=point_b)
result = self.metric.norm(vector=log)
expected = self.metric.dist(point_a, point_b)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_dist_point_and_itself(self):
# Distance between a point and itself is 0.
point_a = gs.array([10., -2., -.5, 2., 3.])
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0.
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(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_b = gs.array([2., 10, 0., 0., 0.])
self.assertEqual(gs.dot(point_a, point_b), 0)
result = self.metric.dist(point_a, point_b)
expected = gs.pi / 2
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(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.projection_to_tangent_space(
vector=vector, base_point=base_point)
exp = self.metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = self.metric.dist(base_point, exp)
expected = gs.mod(gs.linalg.norm(tangent_vec), 2 * gs.pi)
expected = helper.to_scalar(expected)
gs.testing.assert_allclose(result, expected)
def test_geodesic_and_belongs(self):
initial_point = self.space.random_uniform()
vector = gs.array([2., 0., -1., -2., 1.])
initial_tangent_vec = self.space.projection_to_tangent_space(
vector=vector, base_point=initial_point)
geodesic = self.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0, stop=1, num=100)
points = geodesic(t)
self.assertTrue(gs.all(self.space.belongs(points)))
def test_variance(self):
point = self.space.random_uniform()
result = self.metric.variance([point, point])
expected = 0
gs.testing.assert_allclose(result, expected)
def test_mean(self):
point = self.space.random_uniform()
result = self.metric.mean([point, point])
expected = point
gs.testing.assert_allclose(result, expected)
def test_mean_and_belongs(self):
point_a = self.space.random_uniform()
point_b = self.space.random_uniform()
point_c = self.space.random_uniform()
result = self.metric.mean([point_a, point_b, point_c])
self.assertTrue(self.space.belongs(result))
def test_diameter(self):
dim = 2
sphere = Hypersphere(dim)
point_a = [0., 0., 1.]
point_b = [1., 0., 0.]
point_c = [0., 0., -1.]
result = sphere.metric.diameter(gs.vstack((point_a, point_b, point_c)))
expected = gs.pi
gs.testing.assert_allclose(result, expected)
gs.testing.assert_allclose(result.size, 1)
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.where((points == closest_neighbor).all(axis=1))
result = test[0].size > 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 = 1000000
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))
p = dim + 1
n_steps = 100
for i in range(n_steps):
bessel_func_1 = scipy.special.iv(p / 2., kappa_estimate)
bessel_func_2 = scipy.special.iv(p / 2. - 1., kappa_estimate)
ratio = bessel_func_1 / bessel_func_2
denominator = 1. - ratio**2 - (p - 1.) * ratio / kappa_estimate
kappa_estimate = kappa_estimate - (ratio - mean_norm) / denominator
expected = kappa
result = kappa_estimate
self.assertTrue(
gs.allclose(result, expected, atol=KAPPA_ESTIMATION_TOL))
def test_optimal_quantization(self):
"""
Check that optimal quantization yields the same result as
the karcher flow algorithm when we look for one center.
"""
dim = 2
n_points = 1000
n_centers = 1
sphere = Hypersphere(dim)
points = sphere.random_von_mises_fisher(kappa=10, n_samples=n_points)
mean = sphere.metric.mean(points)
centers, weights, clusters, n_iterations = sphere.metric.\
optimal_quantization(points=points, n_centers=n_centers)
error = sphere.metric.dist(mean, centers)
diameter = sphere.metric.diameter(points)
result = error / diameter
expected = 0.0
self.assertTrue(
gs.allclose(result, expected, atol=OPTIMAL_QUANTIZATION_TOL))