def projection_to_tangent_space(self, vector, base_point):
"""Project a vector in the tangent space.
Project a vector in Minkowski space
on the tangent space of the hyperbolic space at a base point.
Parameters
----------
vector : array-like, shape=[n_samples, n_disks, dim + 1]
base_point : array-like, shape=[n_samples, n_disks, dim + 1]
Returns
-------
tangent_vec : array-like, shape=[n_samples, n_disks, dim + 1]
"""
n_disks = base_point.shape[1]
hyperbolic_space = Hyperboloid(2, self.coords_type)
tangent_vec = gs.stack([
hyperbolic_space.projection_to_tangent_space(
vector=vector[:, i_disk, :],
base_point=base_point[:, i_disk, :])
for i_disk in range(n_disks)
],
axis=1)
return tangent_vec
def test_exp_and_belongs(self):
H2 = Hyperboloid(dim=2)
METRIC = H2.metric
base_point = gs.array([1., 0., 0.])
self.assertTrue(H2.belongs(base_point))
tangent_vec = H2.projection_to_tangent_space(
vector=gs.array([1., 2., 1.]),
base_point=base_point)
exp = METRIC.exp(tangent_vec=tangent_vec,
base_point=base_point)
self.assertTrue(H2.belongs(exp))
class TestHyperbolicMethods(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.dimension = 3
self.space = Hyperboloid(dim=self.dimension)
self.metric = self.space.metric
self.ball_manifold = PoincareBall(dim=2)
self.n_samples = 10
def test_random_uniform_and_belongs(self):
point = self.space.random_uniform()
result = self.space.belongs(point)
expected = True
self.assertAllClose(result, expected)
def test_random_uniform(self):
result = self.space.random_uniform()
self.assertAllClose(gs.shape(result), (self.dimension + 1,))
def test_projection_to_tangent_space(self):
base_point = gs.array([1., 0., 0., 0.])
self.assertTrue(self.space.belongs(base_point))
tangent_vec = self.space.projection_to_tangent_space(
vector=gs.array([1., 2., 1., 3.]),
base_point=base_point)
result = self.metric.inner_product(tangent_vec, base_point)
expected = 0.
self.assertAllClose(result, expected)
result = self.space.projection_to_tangent_space(
vector=gs.array([1., 2., 1., 3.]),
base_point=base_point)
expected = tangent_vec
self.assertAllClose(result, expected)
def test_intrinsic_and_extrinsic_coords(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.ones(self.dimension)
point_ext = self.space.from_coordinates(point_int, 'intrinsic')
result = self.space.to_coordinates(point_ext, 'intrinsic')
expected = point_int
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
point_ext = gs.array([2.0, 1.0, 1.0, 1.0])
point_int = self.space.to_coordinates(point_ext, 'intrinsic')
result = self.space.from_coordinates(point_int, 'intrinsic')
expected = point_ext
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
def test_intrinsic_and_extrinsic_coords_vectorization(self):
"""
Test that the composition of
intrinsic_to_extrinsic_coords and
extrinsic_to_intrinsic_coords
gives the identity.
"""
point_int = gs.array([[.1, 0., 0., .1, 0., 0.],
[.1, .1, .1, .4, .1, 0.],
[.1, .3, 0., .1, 0., 0.],
[-0.1, .1, -.4, .1, -.01, 0.],
[0., 0., .1, .1, -0.08, -0.1],
[.1, .1, .1, .1, 0., -0.5]])
point_ext = self.space.from_coordinates(point_int, 'intrinsic')
result = self.space.to_coordinates(point_ext, 'intrinsic')
expected = point_int
expected = helper.to_vector(expected)
self.assertAllClose(result, expected)
point_ext = gs.array([[2., 1., 1., 1.],
[4., 1., 3., math.sqrt(5.)],
[3., 2., 0., 2.]])
point_int = self.space.to_coordinates(point_ext, 'intrinsic')
result = self.space.from_coordinates(point_int, 'intrinsic')
expected = point_ext
expected = helper.to_vector(expected)
self.assertAllClose(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.
"""
# Riemannian Log then Riemannian Exp
# General case
base_point = gs.array([4.0, 1., 3.0, math.sqrt(5.)])
point = gs.array([2.0, 1.0, 1.0, 1.0])
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_general_case_general_dim(self):
"""
Test that the Riemannian exponential
and the Riemannian logarithm are inverse.
Expect their composition to give the identity function.
"""
# Riemannian Log then Riemannian Exp
dim = 5
n_samples = self.n_samples
h5 = Hyperboloid(dim=dim)
h5_metric = h5.metric
base_point = h5.random_uniform()
point = h5.random_uniform()
one_log = h5_metric.log(point=point, base_point=base_point)
result = h5_metric.exp(tangent_vec=one_log, base_point=base_point)
expected = point
self.assertAllClose(result, expected)
# Test vectorization of log
base_point = gs.stack([base_point] * n_samples, axis=0)
point = gs.stack([point] * n_samples, axis=0)
expected = gs.stack([one_log] * n_samples, axis=0)
log = h5_metric.log(point=point, base_point=base_point)
result = log
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
result = h5_metric.exp(tangent_vec=log, base_point=base_point)
expected = point
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
# Test vectorization of exp
tangent_vec = gs.stack([one_log] * n_samples, axis=0)
exp = h5_metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = exp
expected = point
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
def test_exp_and_belongs(self):
H2 = Hyperboloid(dim=2)
METRIC = H2.metric
base_point = gs.array([1., 0., 0.])
self.assertTrue(H2.belongs(base_point))
tangent_vec = H2.projection_to_tangent_space(
vector=gs.array([1., 2., 1.]),
base_point=base_point)
exp = METRIC.exp(tangent_vec=tangent_vec,
base_point=base_point)
self.assertTrue(H2.belongs(exp))
@geomstats.tests.np_and_pytorch_only
def test_exp_vectorization(self):
n_samples = 3
dim = self.dimension + 1
one_vec = gs.array([2.0, 1.0, 1.0, 1.0])
one_base_point = gs.array([4.0, 3., 1.0, math.sqrt(5)])
n_vecs = gs.array([[2., 1., 1., 1.],
[4., 1., 3., math.sqrt(5.)],
[3., 2., 0., 2.]])
n_base_points = gs.array([
[2.0, 0.0, 1.0, math.sqrt(2)],
[5.0, math.sqrt(8), math.sqrt(8), math.sqrt(8)],
[1.0, 0.0, 0.0, 0.0]])
one_tangent_vec = self.space.projection_to_tangent_space(
one_vec, base_point=one_base_point)
result = self.metric.exp(one_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (dim,))
n_tangent_vecs = self.space.projection_to_tangent_space(
n_vecs, base_point=one_base_point)
result = self.metric.exp(n_tangent_vecs, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, dim))
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(gs.array(expected))
self.assertAllClose(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)
self.assertAllClose(gs.shape(result), (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(gs.array(expected))
self.assertAllClose(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)
self.assertAllClose(gs.shape(result), (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(gs.array(expected))
self.assertAllClose(result, expected)
def test_log_vectorization(self):
n_samples = 3
dim = self.dimension + 1
one_point = gs.array([2.0, 1.0, 1.0, 1.0])
one_base_point = gs.array([4.0, 3., 1.0, math.sqrt(5)])
n_points = gs.array([[2.0, 1.0, 1.0, 1.0],
[4.0, 1., 3.0, math.sqrt(5)],
[3.0, 2.0, 0.0, 2.0]])
n_base_points = gs.array([
[2.0, 0.0, 1.0, math.sqrt(2)],
[5.0, math.sqrt(8), math.sqrt(8), math.sqrt(8)],
[1.0, 0.0, 0.0, 0.0]])
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_inner_product(self):
"""
Test that the inner product between two tangent vectors
is the Minkowski inner product.
"""
minkowski_space = Minkowski(self.dimension + 1)
base_point = gs.array(
[1.16563816, 0.36381045, -0.47000603, 0.07381469])
tangent_vec_a = self.space.projection_to_tangent_space(
vector=gs.array([10., 200., 1., 1.]),
base_point=base_point)
tangent_vec_b = self.space.projection_to_tangent_space(
vector=gs.array([11., 20., -21., 0.]),
base_point=base_point)
result = self.metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
expected = minkowski_space.metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
self.assertAllClose(result, expected)
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 = gs.array([2.0, 1.0, 1.0, 1.0])
point_b = gs.array([4.0, 1., 3.0, math.sqrt(5)])
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_norm_and_dist(self):
"""
Test that the distance between two points is
the norm of their logarithm.
"""
point_a = gs.array([2.0, 1.0, 1.0, 1.0])
point_b = gs.array([4.0, 1., 3.0, math.sqrt(5)])
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)
self.assertAllClose(result, expected)
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.
"""
# Riemannian Log then Riemannian Exp
# Edge case: two very close points, base_point_2 and point_2,
# form an angle < epsilon
base_point_intrinsic = gs.array([1., 2., 3.])
base_point =\
self.space.from_coordinates(base_point_intrinsic, 'intrinsic')
point_intrinsic = (base_point_intrinsic +
1e-12 * gs.array([-1., -2., 1.]))
point =\
self.space.from_coordinates(point_intrinsic, 'intrinsic')
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)
@geomstats.tests.np_and_tf_only
def test_exp_and_log_and_projection_to_tangent_space_general_case(self):
"""
Test that the Riemannian exponential
and the Riemannian logarithm are inverse.
Expect their composition to give the identity function.
"""
# Riemannian Exp then Riemannian Log
# General case
base_point = gs.array([4.0, 1., 3.0, math.sqrt(5)])
vector = gs.array([2.0, 1.0, 1.0, 1.0])
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
self.assertAllClose(result, expected)
def test_dist(self):
# Distance between a point and itself is 0.
point_a = gs.array([4.0, 1., 3.0, math.sqrt(5)])
point_b = point_a
result = self.metric.dist(point_a, point_b)
expected = 0
self.assertAllClose(result, expected)
def test_exp_and_dist_and_projection_to_tangent_space(self):
base_point = gs.array([4.0, 1., 3.0, math.sqrt(5)])
vector = gs.array([0.001, 0., -.00001, -.00003])
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)
sq_norm = self.metric.embedding_metric.squared_norm(
tangent_vec)
expected = sq_norm
self.assertAllClose(result, expected, atol=1e-2)
def test_geodesic_and_belongs(self):
# TODO(nina): Fix this tests, as it fails when geodesic goes "too far"
initial_point = gs.array([4.0, 1., 3.0, math.sqrt(5)])
n_geodesic_points = 100
vector = gs.array([1., 0., 0., 0.])
initial_tangent_vec = self.space.projection_to_tangent_space(
vector=vector,
base_point=initial_point)
geodesic = self.metric.geodesic(
initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
t = gs.linspace(start=0., stop=1., num=n_geodesic_points)
points = geodesic(t)
result = self.space.belongs(points)
expected = n_geodesic_points * [True]
self.assertAllClose(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.
"""
# Riemannian Exp then Riemannian Log
# Edge case: tangent vector has norm < epsilon
base_point = gs.array([2., 1., 1., 1.])
vector = 1e-10 * gs.array([.06, -51., 6., 5.])
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)
self.assertAllClose(result, expected, atol=1e-8)
@geomstats.tests.np_only
def test_scaled_inner_product(self):
base_point_intrinsic = gs.array([1, 1, 1])
base_point = self.space.from_coordinates(
base_point_intrinsic, "intrinsic")
tangent_vec_a = gs.array([1, 2, 3, 4])
tangent_vec_b = gs.array([5, 6, 7, 8])
tangent_vec_a = self.space.projection_to_tangent_space(
tangent_vec_a,
base_point)
tangent_vec_b = self.space.projection_to_tangent_space(
tangent_vec_b,
base_point)
scale = 2
default_space = Hyperboloid(dim=self.dimension)
scaled_space = Hyperboloid(dim=self.dimension, scale=2)
inner_product_default_metric = \
default_space.metric.inner_product(
tangent_vec_a,
tangent_vec_b,
base_point)
inner_product_scaled_metric = \
scaled_space.metric.inner_product(
tangent_vec_a,
tangent_vec_b,
base_point)
result = inner_product_scaled_metric
expected = scale ** 2 * inner_product_default_metric
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_scaled_squared_norm(self):
base_point_intrinsic = gs.array([1, 1, 1])
base_point = self.space.from_coordinates(base_point_intrinsic,
'intrinsic')
tangent_vec = gs.array([1, 2, 3, 4])
tangent_vec = self.space.projection_to_tangent_space(
tangent_vec, base_point)
scale = 2
default_space = Hyperboloid(dim=self.dimension)
scaled_space = Hyperboloid(dim=self.dimension, scale=2)
squared_norm_default_metric = default_space.metric.squared_norm(
tangent_vec, base_point)
squared_norm_scaled_metric = scaled_space.metric.squared_norm(
tangent_vec, base_point)
result = squared_norm_scaled_metric
expected = scale ** 2 * squared_norm_default_metric
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_scaled_distance(self):
point_a_intrinsic = gs.array([1, 2, 3])
point_b_intrinsic = gs.array([4, 5, 6])
point_a = self.space.from_coordinates(point_a_intrinsic, 'intrinsic')
point_b = self.space.from_coordinates(point_b_intrinsic, 'intrinsic')
scale = 2
scaled_space = Hyperboloid(dim=self.dimension, scale=2)
distance_default_metric = self.space.metric.dist(point_a, point_b)
distance_scaled_metric = scaled_space.metric.dist(point_a, point_b)
result = distance_scaled_metric
expected = scale * distance_default_metric
self.assertAllClose(result, expected)
