def setUp(self):
self.so3 = SpecialOrthogonalGroup(n=3)
self.n_samples = 10
self.X = self.so3.random_uniform(n_samples=self.n_samples)
self.metric = self.so3.bi_invariant_metric
self.n_components = 2
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.n_samples = 2
self.group = GeneralLinearGroup(n=self.n)
# We generate invertible matrices using so3_group
self.so3_group = SpecialOrthogonalGroup(n=self.n)
warnings.simplefilter('ignore', category=ImportWarning)
def setUp(self):
self.n_samples = 10
self.SO3_GROUP = SpecialOrthogonalGroup(n=3)
self.SE3_GROUP = SpecialEuclideanGroup(n=3)
self.S1 = Hypersphere(dimension=1)
self.S2 = Hypersphere(dimension=2)
self.H2 = HyperbolicSpace(dimension=2)
plt.figure()
class TestTangentPCA(geomstats.tests.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
self.so3 = SpecialOrthogonalGroup(n=3)
self.n_samples = 10
self.X = self.so3.random_uniform(n_samples=self.n_samples)
self.metric = self.so3.bi_invariant_metric
self.n_components = 2
@geomstats.tests.np_only
def test_tangent_pca_error(self):
X = self.X
trans = TangentPCA(self.metric, n_components=self.n_components)
trans.fit(X)
X_diff_size = gs.ones((self.n_samples, gs.shape(X)[1] + 1))
self.assertRaises(ValueError, trans.transform, X_diff_size)
@geomstats.tests.np_only
def test_tangent_pca(self):
X = self.X
trans = TangentPCA(self.metric, n_components=gs.shape(X)[1])
trans.fit(X)
self.assertEquals(trans.n_features_, gs.shape(X)[1])
def __init__(self, n, point_type=None, epsilon=0.):
assert isinstance(n, int) and n > 1
self.n = n
self.dimension = int((n * (n - 1)) / 2 + n)
self.epsilon = epsilon
self.default_point_type = point_type
if point_type is None:
self.default_point_type = 'vector' if n == 3 else 'matrix'
super(SpecialEuclideanGroup, self).__init__(dimension=self.dimension)
self.rotations = SpecialOrthogonalGroup(n=n, epsilon=epsilon)
self.translations = EuclideanSpace(dimension=n)
class TestVisualizationMethods(geomstats.tests.TestCase):
def setUp(self):
self.n_samples = 10
self.SO3_GROUP = SpecialOrthogonalGroup(n=3)
self.SE3_GROUP = SpecialEuclideanGroup(n=3)
self.S1 = Hypersphere(dimension=1)
self.S2 = Hypersphere(dimension=2)
self.H2 = HyperbolicSpace(dimension=2)
plt.figure()
@geomstats.tests.np_only
def test_plot_points_so3(self):
points = self.SO3_GROUP.random_uniform(self.n_samples)
visualization.plot(points, space='SO3_GROUP')
@geomstats.tests.np_only
def test_plot_points_se3(self):
points = self.SE3_GROUP.random_uniform(self.n_samples)
visualization.plot(points, space='SE3_GROUP')
@geomstats.tests.np_only
def test_plot_points_s1(self):
points = self.S1.random_uniform(self.n_samples)
visualization.plot(points, space='S1')
@geomstats.tests.np_only
def test_plot_points_s2(self):
points = self.S2.random_uniform(self.n_samples)
visualization.plot(points, space='S2')
@geomstats.tests.np_only
def test_plot_points_h2_poincare_disk(self):
points = self.H2.random_uniform(self.n_samples)
visualization.plot(points, space='H2_poincare_disk')
@geomstats.tests.np_only
def test_plot_points_h2_poincare_half_plane(self):
points = self.H2.random_uniform(self.n_samples)
visualization.plot(points, space='H2_poincare_half_plane')
@geomstats.tests.np_only
def test_plot_points_h2_klein_disk(self):
points = self.H2.random_uniform(self.n_samples)
visualization.plot(points, space='H2_klein_disk')
def setUp(self):
warnings.simplefilter('ignore', category=ImportWarning)
self.so3_group = SpecialOrthogonalGroup(n=3)
self.n_samples = 2
class TestBackendNumpy(unittest.TestCase):
_multiprocess_can_split_ = True
@classmethod
def setUpClass(cls):
cls.initial_backend = os.environ['GEOMSTATS_BACKEND']
os.environ['GEOMSTATS_BACKEND'] = 'numpy'
importlib.reload(gs)
@classmethod
def tearDownClass(cls):
os.environ['GEOMSTATS_BACKEND'] = cls.initial_backend
importlib.reload(gs)
def setUp(self):
warnings.simplefilter('ignore', category=ImportWarning)
self.so3_group = SpecialOrthogonalGroup(n=3)
self.n_samples = 2
def test_logm(self):
point = gs.array([[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]])
result = gs.linalg.logm(point)
expected = gs.array([[0.693147180, 0., 0.], [0., 1.098612288, 0.],
[0., 0., 1.38629436]])
self.assertTrue(gs.allclose(result, expected))
def test_expm_and_logm(self):
point = gs.array([[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]])
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertTrue(gs.allclose(result, expected))
def test_expm_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.], [0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.], [0., 148.413159, 0.],
[0., 0., 403.42879349]]])
result = gs.linalg.expm(point)
self.assertTrue(gs.allclose(result, expected))
def test_logm_vectorization_diagonal(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.], [0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.], [0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = gs.linalg.logm(point)
self.assertTrue(gs.allclose(result, expected))
def test_expm_and_logm_vectorization_random_rotation(self):
point = self.so3_group.random_uniform(self.n_samples)
point = self.so3_group.matrix_from_rotation_vector(point)
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertTrue(gs.allclose(result, expected))
def test_expm_and_logm_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertTrue(gs.allclose(result, expected))
class TestGeneralLinearGroupMethods(geomstats.tests.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.n_samples = 2
self.group = GeneralLinearGroup(n=self.n)
# We generate invertible matrices using so3_group
self.so3_group = SpecialOrthogonalGroup(n=self.n)
warnings.simplefilter('ignore', category=ImportWarning)
@geomstats.tests.np_only
def test_belongs(self):
"""
A rotation matrix belongs to the matrix Lie group
of invertible matrices.
"""
rot_vec = gs.array([0.2, -0.1, 0.1])
rot_mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.belongs(rot_mat)
expected = gs.array([True])
self.assertAllClose(result, expected)
def test_compose(self):
# 1. Composition by identity, on the right
# Expect the original transformation
rot_vec = gs.array([0.2, -0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.compose(mat, self.group.identity)
expected = mat
expected = helper.to_matrix(mat)
self.assertAllClose(result, expected)
# 2. Composition by identity, on the left
# Expect the original transformation
rot_vec = gs.array([0.2, 0.1, -0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.compose(self.group.identity, mat)
expected = mat
self.assertAllClose(result, expected)
def test_inverse(self):
mat = gs.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 10.]])
result = self.group.inverse(mat)
expected = 1. / 3. * gs.array([[-2., -4., 3.], [-2., 11., -6.],
[3., -6., 3.]])
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
def test_compose_and_inverse(self):
# 1. Compose transformation by its inverse on the right
# Expect the group identity
rot_vec = gs.array([0.2, 0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
inv_mat = self.group.inverse(mat)
result = self.group.compose(mat, inv_mat)
expected = self.group.identity
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
# 2. Compose transformation by its inverse on the left
# Expect the group identity
rot_vec = gs.array([0.7, 0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
inv_mat = self.group.inverse(mat)
result = self.group.compose(inv_mat, mat)
expected = self.group.identity
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_group_log_and_exp(self):
point = 5 * gs.eye(self.n)
group_log = self.group.group_log(point)
result = self.group.group_exp(group_log)
expected = point
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_group_exp_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.], [0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.], [0., 148.413159, 0.],
[0., 0., 403.42879349]]])
result = self.group.group_exp(point)
self.assertAllClose(result, expected, rtol=1e-3)
@geomstats.tests.np_and_tf_only
def test_group_log_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.], [0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.], [0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = self.group.group_log(point)
self.assertAllClose(result, expected, atol=1e-4)
@geomstats.tests.np_and_tf_only
def test_expm_and_logm_vectorization_symmetric(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
result = self.group.group_exp(self.group.group_log(point))
expected = point
self.assertAllClose(result, expected)
"""
Compute the mean of a data set of 3D rotations.
Performs tangent PCA at the mean.
"""
import matplotlib.pyplot as plt
import numpy as np
import geomstats.visualization as visualization
from geomstats.learning.pca import TangentPCA
from geomstats.geometry.special_orthogonal_group import SpecialOrthogonalGroup
SO3_GROUP = SpecialOrthogonalGroup(n=3)
METRIC = SO3_GROUP.bi_invariant_metric
N_SAMPLES = 10
N_COMPONENTS = 2
def main():
fig = plt.figure(figsize=(15, 5))
data = SO3_GROUP.random_uniform(n_samples=N_SAMPLES)
mean = METRIC.mean(data)
tpca = TangentPCA(metric=METRIC, n_components=N_COMPONENTS)
tpca = tpca.fit(data, base_point=mean)
tangent_projected_data = tpca.transform(data)
print('Coordinates of the Log of the first 5 data points at the mean, '
'projected on the principal components:')
class SpecialEuclideanGroup(LieGroup):
"""
Class for the special euclidean group SE(n),
i.e. the Lie group of rigid transformations.
"""
def __init__(self, n, point_type=None, epsilon=0.):
assert isinstance(n, int) and n > 1
self.n = n
self.dimension = int((n * (n - 1)) / 2 + n)
self.epsilon = epsilon
self.default_point_type = point_type
if point_type is None:
self.default_point_type = 'vector' if n == 3 else 'matrix'
super(SpecialEuclideanGroup, self).__init__(dimension=self.dimension)
self.rotations = SpecialOrthogonalGroup(n=n, epsilon=epsilon)
self.translations = EuclideanSpace(dimension=n)
def get_identity(self, point_type=None):
"""
Get the identity of the group,
as a vector if point_type == 'vector',
as a matrix if point_type == 'matrix'.
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dimension)
if self.default_point_type == 'matrix':
identity = gs.eye(self.n)
return identity
identity = property(get_identity)
def belongs(self, point, point_type=None):
"""
Evaluate if a point belongs to SE(n).
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
point = gs.to_ndarray(point, to_ndim=2)
n_points, point_dim = point.shape
belongs = point_dim == self.dimension
belongs = gs.to_ndarray(belongs, to_ndim=1)
belongs = gs.to_ndarray(belongs, to_ndim=2, axis=1)
belongs = gs.tile(belongs, (n_points, 1))
elif point_type == 'matrix':
point = gs.to_ndarray(point, to_ndim=3)
raise NotImplementedError()
return belongs
def regularize(self, point, point_type=None):
"""
Regularize a point to the canonical representation
chosen for SE(n).
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
point = gs.to_ndarray(point, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_vec = point[:, :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec,
point_type=point_type)
translation = point[:, dim_rotations:]
regularized_point = gs.concatenate(
[regularized_rot_vec, translation], axis=1)
elif point_type == 'matrix':
point = gs.to_ndarray(point, to_ndim=3)
regularized_point = gs.copy(point)
return regularized_point
def regularize_tangent_vec_at_identity(self,
tangent_vec,
metric=None,
point_type=None):
if point_type is None:
point_type = self.default_point_type
return self.regularize_tangent_vec(tangent_vec,
self.identity,
metric,
point_type=point_type)
def regularize_tangent_vec(self,
tangent_vec,
base_point,
metric=None,
point_type=None):
if point_type is None:
point_type = self.default_point_type
if metric is None:
metric = self.left_canonical_metric
if point_type == 'vector':
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_tangent_vec = tangent_vec[:, :dim_rotations]
rot_base_point = base_point[:, :dim_rotations]
metric_mat = metric.inner_product_mat_at_identity
rot_metric_mat = metric_mat[:, :dim_rotations, :dim_rotations]
rot_metric = InvariantMetric(
group=rotations,
inner_product_mat_at_identity=rot_metric_mat,
left_or_right=metric.left_or_right)
regularized_vec = gs.zeros_like(tangent_vec)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric,
point_type=point_type)
regularized_vec = gs.concatenate(
[rotations_vec, tangent_vec[:, dim_rotations:]], axis=1)
elif point_type == 'matrix':
regularized_vec = tangent_vec
return regularized_vec
def compose(self, point_1, point_2, point_type=None):
"""
Compose two elements of SE(n).
Formula:
point_1 . point_2 = [R1 * R2, (R1 * t2) + t1]
where:
R1, R2 are rotation matrices,
t1, t2 are translation vectors.
"""
if point_type is None:
point_type = self.default_point_type
rotations = self.rotations
dim_rotations = rotations.dimension
point_1 = self.regularize(point_1, point_type=point_type)
point_2 = self.regularize(point_2, point_type=point_type)
if point_type == 'vector':
n_points_1, _ = point_1.shape
n_points_2, _ = point_2.shape
assert (point_1.shape == point_2.shape or n_points_1 == 1
or n_points_2 == 1)
if n_points_1 == 1:
point_1 = gs.stack([point_1[0]] * n_points_2)
if n_points_2 == 1:
point_2 = gs.stack([point_2[0]] * n_points_1)
rot_vec_1 = point_1[:, :dim_rotations]
rot_mat_1 = rotations.matrix_from_rotation_vector(rot_vec_1)
rot_vec_2 = point_2[:, :dim_rotations]
rot_mat_2 = rotations.matrix_from_rotation_vector(rot_vec_2)
translation_1 = point_1[:, dim_rotations:]
translation_2 = point_2[:, dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_1, rot_mat_2)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum('ij,ikj->ik', translation_2,
rot_mat_1) + translation_1
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
composition = self.regularize(composition, point_type=point_type)
return composition
def inverse(self, point, point_type=None):
"""
Compute the group inverse in SE(n).
Formula:
(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))
"""
if point_type is None:
point_type = self.default_point_type
rotations = self.rotations
dim_rotations = rotations.dimension
point = self.regularize(point)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
inverse_point = gs.zeros_like(point)
inverse_rotation = -rot_vec
inv_rot_mat = rotations.matrix_from_rotation_vector(
inverse_rotation)
inverse_translation = gs.einsum(
'ni,nij->nj', -translation,
gs.transpose(inv_rot_mat, axes=(0, 2, 1)))
inverse_point = gs.concatenate(
[inverse_rotation, inverse_translation], axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
inverse_point = self.regularize(inverse_point, point_type=point_type)
return inverse_point
def jacobian_translation(self,
point,
left_or_right='left',
point_type=None):
"""
Compute the jacobian matrix of the differential
of the left/right translations from the identity to point in SE(n).
"""
if point_type is None:
point_type = self.default_point_type
assert left_or_right in ('left', 'right')
dim = self.dimension
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dimension
dim_translations = translations.dimension
point = self.regularize(point, point_type=point_type)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
jacobian = gs.zeros((n_points, ) + (dim, ) * 2)
jacobian_rot = self.rotations.jacobian_translation(
point=rot_vec,
left_or_right=left_or_right,
point_type=point_type)
block_zeros_1 = gs.zeros(
(n_points, dim_rotations, dim_translations))
jacobian_block_line_1 = gs.concatenate(
[jacobian_rot, block_zeros_1], axis=2)
if left_or_right == 'left':
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
jacobian_trans = rot_mat
block_zeros_2 = gs.zeros(
(n_points, dim_translations, dim_rotations))
jacobian_block_line_2 = gs.concatenate(
[block_zeros_2, jacobian_trans], axis=2)
else:
inv_skew_mat = -self.rotations.skew_matrix_from_vector(rot_vec)
eye = gs.to_ndarray(gs.eye(self.n), to_ndim=3)
eye = gs.tile(eye, [n_points, 1, 1])
jacobian_block_line_2 = gs.concatenate([inv_skew_mat, eye],
axis=2)
jacobian = gs.concatenate(
[jacobian_block_line_1, jacobian_block_line_2], axis=1)
assert gs.ndim(jacobian) == 3
elif point_type == 'matrix':
raise NotImplementedError()
return jacobian
def group_exp_from_identity(self, tangent_vec, point_type=None):
"""
Compute the group exponential of the tangent vector at the identity.
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_vec = tangent_vec[:, :dim_rotations]
rot_vec = self.rotations.regularize(rot_vec, point_type=point_type)
translation = tangent_vec[:, dim_rotations:]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_0 = gs.equal(angle, 0.)
mask_close_0 = gs.isclose(angle, 0.) & ~mask_0
mask_else = ~mask_0 & ~mask_close_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * 1. / 2. * gs.ones_like(angle)
coef_2 += mask_0_float * 1. / 6. * gs.ones_like(angle)
coef_1 += mask_close_0_float * (
TAYLOR_COEFFS_1_AT_0[0] + TAYLOR_COEFFS_1_AT_0[2] * angle**2 +
TAYLOR_COEFFS_1_AT_0[4] * angle**4 +
TAYLOR_COEFFS_1_AT_0[6] * angle**6)
coef_2 += mask_close_0_float * (
TAYLOR_COEFFS_2_AT_0[0] + TAYLOR_COEFFS_2_AT_0[2] * angle**2 +
TAYLOR_COEFFS_2_AT_0[4] * angle**4 +
TAYLOR_COEFFS_2_AT_0[6] * angle**6)
coef_1 += mask_else_float * ((1. - gs.cos(angle)) / angle**2)
coef_2 += mask_else_float * ((angle - gs.sin(angle)) / angle**3)
n_tangent_vecs, _ = tangent_vec.shape
group_exp_translation = gs.zeros((n_tangent_vecs, self.n))
for i in range(n_tangent_vecs):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_mat[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_mat[i]))
mask_i_float = gs.get_mask_i_float(i, n_tangent_vecs)
group_exp_translation += mask_i_float * (translation_i +
term_1_i + term_2_i)
group_exp = gs.concatenate([rot_vec, group_exp_translation],
axis=1)
group_exp = self.regularize(group_exp, point_type=point_type)
return group_exp
elif point_type == 'matrix':
raise NotImplementedError()
def group_log_from_identity(self, point, point_type=None):
"""
Compute the group logarithm of the point at the identity.
"""
if point_type is None:
point_type = self.default_point_type
point = self.regularize(point, point_type=point_type)
rotations = self.rotations
dim_rotations = rotations.dimension
if point_type == 'vector':
rot_vec = point[:, :dim_rotations]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
translation = point[:, dim_rotations:]
skew_rot_vec = rotations.skew_matrix_from_vector(rot_vec)
sq_skew_rot_vec = gs.matmul(skew_rot_vec, skew_rot_vec)
mask_close_0 = gs.isclose(angle, 0.)
mask_close_pi = gs.isclose(angle, gs.pi)
mask_else = ~mask_close_0 & ~mask_close_pi
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_close_pi_float = gs.cast(mask_close_pi, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
mask_0 = gs.isclose(angle, 0., atol=1e-6)
mask_0_float = gs.cast(mask_0, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = -0.5 * gs.ones_like(angle)
coef_2 = gs.zeros_like(angle)
coef_2 += mask_close_0_float * (1. / 12. + angle**2 / 720. +
angle**4 / 30240. +
angle**6 / 1209600.)
delta_angle = angle - gs.pi
coef_2 += mask_close_pi_float * (
1. / PI2 + (PI2 - 8.) * delta_angle / (4. * PI3) -
((PI2 - 12.) * delta_angle**2 / (4. * PI4)) +
((-192. + 12. * PI2 + PI4) * delta_angle**3 / (48. * PI5)) -
((-240. + 12. * PI2 + PI4) * delta_angle**4 / (48. * PI6)) +
((-2880. + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**5 /
(480. * PI7)) -
((-3360 + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**6 /
(480. * PI8)))
psi = 0.5 * angle * gs.sin(angle) / (1 - gs.cos(angle))
coef_2 += mask_else_float * (1 - psi) / (angle**2)
n_points, _ = point.shape
group_log_translation = gs.zeros((n_points, self.n))
for i in range(n_points):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_rot_vec[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_rot_vec[i]))
mask_i_float = gs.get_mask_i_float(i, n_points)
group_log_translation += mask_i_float * (translation_i +
term_1_i + term_2_i)
group_log = gs.concatenate([rot_vec, group_log_translation],
axis=1)
assert gs.ndim(group_log) == 2
elif point_type == 'matrix':
raise NotImplementedError()
return group_log
def random_uniform(self, n_samples=1, point_type=None):
"""
Sample in SE(n) with the uniform distribution.
"""
if point_type is None:
point_type = self.default_point_type
random_rot_vec = self.rotations.random_uniform(n_samples,
point_type=point_type)
random_translation = self.translations.random_uniform(n_samples)
if point_type == 'vector':
random_transfo = gs.concatenate(
[random_rot_vec, random_translation], axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
random_transfo = self.regularize(random_transfo, point_type=point_type)
return random_transfo
def exponential_matrix(self, rot_vec):
"""
Compute the exponential of the rotation matrix represented by rot_vec.
"""
rot_vec = self.rotations.regularize(rot_vec)
n_rot_vecs, _ = rot_vec.shape
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_rot_vec = self.rotations.skew_matrix_from_vector(rot_vec)
coef_1 = gs.empty_like(angle)
coef_2 = gs.empty_like(coef_1)
mask_0 = gs.equal(angle, 0)
mask_0 = gs.squeeze(mask_0, axis=1)
mask_close_to_0 = gs.isclose(angle, 0)
mask_close_to_0 = gs.squeeze(mask_close_to_0, axis=1)
mask_else = ~mask_0 & ~mask_close_to_0
coef_1[mask_close_to_0] = (1. / 2. - angle[mask_close_to_0]**2 / 24.)
coef_2[mask_close_to_0] = (1. / 6. - angle[mask_close_to_0]**3 / 120.)
# TODO(nina): Check if the discountinuity at 0 is expected.
coef_1[mask_0] = 0
coef_2[mask_0] = 0
coef_1[mask_else] = (angle[mask_else]**(-2) *
(1. - gs.cos(angle[mask_else])))
coef_2[mask_else] = (angle[mask_else]**(-2) *
(1. -
(gs.sin(angle[mask_else]) / angle[mask_else])))
term_1 = gs.zeros((n_rot_vecs, self.n, self.n))
term_2 = gs.zeros_like(term_1)
for i in range(n_rot_vecs):
term_1[i] = gs.eye(self.n) + skew_rot_vec[i] * coef_1[i]
term_2[i] = gs.matmul(skew_rot_vec[i], skew_rot_vec[i]) * coef_2[i]
exponential_mat = term_1 + term_2
assert exponential_mat.ndim == 3
return exponential_mat
def group_exponential_barycenter(self,
points,
weights=None,
point_type=None):
"""
Compute the group exponential barycenter in SE(n).
"""
if point_type is None:
point_type = self.default_point_type
n_points = points.shape[0]
assert n_points > 0
if weights is None:
weights = gs.ones((n_points, 1))
weights = gs.to_ndarray(weights, to_ndim=2, axis=1)
n_weights, _ = weights.shape
assert n_points == n_weights
dim = self.dimension
rotations = self.rotations
dim_rotations = rotations.dimension
if point_type == 'vector':
rotation_vectors = points[:, :dim_rotations]
translations = points[:, dim_rotations:dim]
assert rotation_vectors.shape == (n_points, dim_rotations)
assert translations.shape == (n_points, self.n)
mean_rotation = rotations.group_exponential_barycenter(
points=rotation_vectors, weights=weights)
mean_rotation_mat = rotations.matrix_from_rotation_vector(
mean_rotation)
matrix = gs.zeros((1, ) + (self.n, ) * 2)
translation_aux = gs.zeros((1, self.n))
inv_rot_mats = rotations.matrix_from_rotation_vector(
-rotation_vectors)
matrix_aux = gs.matmul(mean_rotation_mat, inv_rot_mats)
assert matrix_aux.shape == (n_points, ) + (dim_rotations, ) * 2
vec_aux = rotations.rotation_vector_from_matrix(matrix_aux)
matrix_aux = self.exponential_matrix(vec_aux)
matrix_aux = gs.linalg.inv(matrix_aux)
for i in range(n_points):
matrix += weights[i] * matrix_aux[i]
translation_aux += weights[i] * gs.dot(
gs.matmul(matrix_aux[i], inv_rot_mats[i]), translations[i])
mean_translation = gs.dot(
translation_aux,
gs.transpose(gs.linalg.inv(matrix), axes=(0, 2, 1)))
exp_bar = gs.zeros((1, dim))
exp_bar[0, :dim_rotations] = mean_rotation
exp_bar[0, dim_rotations:dim] = mean_translation
elif point_type == 'matrix':
vector_points = self.rotation_vector_from_matrix(points)
vector_exp_bar = self.group_exponential_barycenter(
vector_points, weights, point_type='vector')
exp_bar = self.matrix_from_rotation_vector(vector_exp_bar)
return exp_bar
"""
Predict on SE3: losses.
"""
import os
import tensorflow as tf
import geomstats.backend as gs
import geomstats.geometry.lie_group as lie_group
from geomstats.geometry.special_euclidean_group import SpecialEuclideanGroup
from geomstats.geometry.special_orthogonal_group import SpecialOrthogonalGroup
SE3 = SpecialEuclideanGroup(n=3)
SO3 = SpecialOrthogonalGroup(n=3)
def loss(y_pred,
y_true,
metric=SE3.left_canonical_metric,
representation='vector'):
"""
Loss function given by a riemannian metric on a Lie group,
by default the left-invariant canonical metric.
"""
if gs.ndim(y_pred) == 1:
y_pred = gs.expand_dims(y_pred, axis=0)
if gs.ndim(y_true) == 1:
y_true = gs.expand_dims(y_true, axis=0)
if representation == 'quaternion':
than the scikit-learn version, while being close to the actual value.
"""
import timeit
import matplotlib.pyplot as plt
import geomstats.backend as gs
from geomstats.geometry.skew_symmetric_matrices import SkewSymmetricMatrices
from geomstats.geometry.special_orthogonal_group import SpecialOrthogonalGroup
N = 3
MAX_ORDER = 10
GROUP = SpecialOrthogonalGroup(n=N)
GROUP.default_point_type = 'matrix'
DIM = int(N * (N - 1) / 2)
ALGEBRA = SkewSymmetricMatrices(n=N)
def main():
norm_rv_1 = gs.normal(size=DIM)
tan_rv_1 = ALGEBRA.matrix_representation(
norm_rv_1 / gs.norm(norm_rv_1, axis=0) / 2
)
exp_1 = gs.linalg.expm(tan_rv_1)
norm_rv_2 = gs.normal(size=DIM)
tan_rv_2 = ALGEBRA.matrix_representation(
class TestBackendNumpy(geomstats.tests.TestCase):
def setUp(self):
warnings.simplefilter('ignore', category=ImportWarning)
self.so3_group = SpecialOrthogonalGroup(n=3)
self.n_samples = 2
def test_logm(self):
point = gs.array([[2., 0., 0.],
[0., 3., 0.],
[0., 0., 4.]])
result = gs.linalg.logm(point)
expected = gs.array([[0.693147180, 0., 0.],
[0., 1.098612288, 0.],
[0., 0., 1.38629436]])
self.assertAllClose(result, expected)
def test_expm_and_logm(self):
point = gs.array([[2., 0., 0.],
[0., 3., 0.],
[0., 0., 4.]])
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertAllClose(result, expected)
def test_expm_vectorization(self):
point = gs.array([[[2., 0., 0.],
[0., 3., 0.],
[0., 0., 4.]],
[[1., 0., 0.],
[0., 5., 0.],
[0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.],
[0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.],
[0., 148.413159, 0.],
[0., 0., 403.42879349]]])
result = gs.linalg.expm(point)
self.assertAllClose(result, expected)
def test_logm_vectorization_diagonal(self):
point = gs.array([[[2., 0., 0.],
[0., 3., 0.],
[0., 0., 4.]],
[[1., 0., 0.],
[0., 5., 0.],
[0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.],
[0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.],
[0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = gs.linalg.logm(point)
self.assertAllClose(result, expected)
def test_expm_and_logm_vectorization_random_rotation(self):
point = self.so3_group.random_uniform(self.n_samples)
point = self.so3_group.matrix_from_rotation_vector(point)
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertAllClose(result, expected)
def test_expm_and_logm_vectorization(self):
point = gs.array([[[2., 0., 0.],
[0., 3., 0.],
[0., 0., 4.]],
[[1., 0., 0.],
[0., 5., 0.],
[0., 0., 6.]]])
result = gs.linalg.expm(gs.linalg.logm(point))
expected = point
self.assertAllClose(result, expected)
def test_powerm_diagonal(self):
power = .5
point = gs.array([[1., 0., 0.],
[0., 4., 0.],
[0., 0., 9.]])
result = gs.linalg.powerm(point, power)
expected = gs.array([[1., 0., 0.],
[0., 2., 0.],
[0., 0., 3.]])
self.assertAllClose(result, expected)
def test_powerm(self):
power = 2.4
point = gs.array([[1., 0., 0.],
[0., 2.5, 1.5],
[0., 1.5, 2.5]])
result = gs.linalg.powerm(point, power)
result = gs.linalg.powerm(result, 1/power)
expected = point
self.assertAllClose(result, expected)
def test_powerm_vectorized(self):
power = 2.4
points = gs.array([[[1., 0., 0.],
[0., 4., 0.],
[0., 0., 9.]],
[[1., 0., 0.],
[0., 2.5, 1.5],
[0., 1.5, 2.5]]])
result = gs.linalg.powerm(points, power)
result = gs.linalg.powerm(result, 1/power)
expected = points
self.assertAllClose(result, expected)
