def test_projection_and_belongs(self):
gs.random.seed(3)
group = SpecialOrthogonal(n=4)
mat = gs.random.rand(4, 4)
point = group.projection(mat)
result = group.belongs(point)
self.assertTrue(result)
mat = gs.random.rand(2, 4, 4)
point = group.projection(mat)
result = group.belongs(point, atol=1e-4)
self.assertTrue(gs.all(result))
class _SpecialEuclideanMatrices(MatrixLieGroup, LevelSet):
"""Class for special Euclidean group.
Parameters
----------
n : int
Integer dimension of the underlying Euclidean space. Matrices will
be of size: (n+1) x (n+1).
Attributes
----------
rotations : SpecialOrthogonal
Subgroup of rotations of size n.
translations : Euclidean
Subgroup of translations of size n.
left_canonical_metric : InvariantMetric
The left invariant metric that corresponds to the Frobenius inner
product at the identity.
right_canonical_metric : InvariantMetric
The right invariant metric that corresponds to the Frobenius inner
product at the identity.
metric : MatricesMetric
The Euclidean (Frobenius) inner product.
"""
def __init__(self, n, **kwargs):
super().__init__(n=n + 1,
dim=int((n * (n + 1)) / 2),
embedding_space=GeneralLinear(n + 1,
positive_det=True),
submersion=submersion,
value=gs.eye(n + 1),
tangent_submersion=tangent_submersion,
lie_algebra=SpecialEuclideanMatrixLieAlgebra(n=n),
**kwargs)
self.rotations = SpecialOrthogonal(n=n)
self.translations = Euclidean(dim=n)
self.n = n
self.left_canonical_metric = SpecialEuclideanMatrixCannonicalLeftMetric(
group=self)
if self._metric is None:
self._metric = self.left_canonical_metric
@property
def identity(self):
"""Return the identity matrix."""
return gs.eye(self.n + 1, self.n + 1)
def random_point(self, n_samples=1, bound=1.0):
"""Sample in SE(n) from the uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound: float
Bound of the interval in which to sample each entry of the
translation part.
Optional, default: 1.
Returns
-------
samples : array-like, shape=[..., n + 1, n + 1]
Sample in SE(n).
"""
random_translation = self.translations.random_point(n_samples)
random_rotation = self.rotations.random_uniform(n_samples)
output_shape = ((n_samples, self.n + 1,
self.n + 1) if n_samples != 1 else (self.n + 1, ) * 2)
random_point = homogeneous_representation(random_rotation,
random_translation,
output_shape)
return random_point
@classmethod
def inverse(cls, point):
"""Return the inverse of a point.
Parameters
----------
point : array-like, shape=[..., n + 1, n + 1]
Point to be inverted.
Returns
-------
inverse : array-like, shape=[..., n + 1, n + 1]
Inverse of point.
"""
n = point.shape[-1] - 1
transposed_rot = Matrices.transpose(point[..., :n, :n])
translation = point[..., :n, -1]
translation = gs.einsum("...ij,...j->...i", transposed_rot,
translation)
return homogeneous_representation(transposed_rot, -translation,
point.shape)
def projection(self, mat):
"""Project a matrix on SE(n).
The upper-left n x n block is projected to SO(n) by minimizing the
Frobenius norm. The last columns is kept unchanged and used as the
translation part. The last row is discarded.
Parameters
----------
mat : array-like, shape=[..., n + 1, n + 1]
Matrix.
Returns
-------
projected : array-like, shape=[..., n + 1, n + 1]
Rotation-translation matrix in homogeneous representation.
"""
n = mat.shape[-1] - 1
projected_rot = self.rotations.projection(mat[..., :n, :n])
translation = mat[..., :n, -1]
return homogeneous_representation(projected_rot, translation,
mat.shape)
class TestSpecialOrthogonal2(geomstats.tests.TestCase):
def setup_method(self):
warnings.simplefilter("ignore", category=ImportWarning)
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(1234)
self.group = SpecialOrthogonal(n=2, point_type="vector")
# -- Set attributes
self.n_samples = 4
def test_projection(self):
# Test 2D and nD cases
rot_mat = gs.eye(2)
delta = 1e-12 * gs.ones((2, 2))
rot_mat_plus_delta = rot_mat + delta
result = self.group.projection(rot_mat_plus_delta)
expected = rot_mat
self.assertAllClose(result, expected)
def test_projection_vectorization(self):
n_samples = self.n_samples
mats = gs.ones((n_samples, 2, 2))
result = self.group.projection(mats)
self.assertAllClose(gs.shape(result), (n_samples, 2, 2))
def test_skew_matrix_from_vector(self):
rot_vec = gs.array([0.9])
skew_matrix = self.group.skew_matrix_from_vector(rot_vec)
result = gs.matmul(skew_matrix, skew_matrix)
diag = gs.array([-0.81, -0.81])
expected = algebra_utils.from_vector_to_diagonal_matrix(diag)
self.assertAllClose(result, expected)
def test_skew_matrix_and_vector(self):
rot_vec = gs.array([0.8])
skew_mat = self.group.skew_matrix_from_vector(rot_vec)
result = self.group.vector_from_skew_matrix(skew_mat)
expected = rot_vec
self.assertAllClose(result, expected)
def test_skew_matrix_from_vector_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
result = self.group.skew_matrix_from_vector(rot_vecs)
self.assertAllClose(gs.shape(result), (n_samples, 2, 2))
def test_random_uniform_shape(self):
result = self.group.random_uniform()
self.assertAllClose(gs.shape(result), (self.group.dim, ))
def test_random_and_belongs(self):
point = self.group.random_uniform()
result = self.group.belongs(point)
expected = True
self.assertAllClose(result, expected)
def test_random_and_belongs_vectorization(self):
n_samples = self.n_samples
points = self.group.random_uniform(n_samples=n_samples)
result = self.group.belongs(points)
expected = gs.array([True] * n_samples)
self.assertAllClose(result, expected)
def test_regularize(self):
angle = 2 * gs.pi + 1
result = self.group.regularize(gs.array([angle]))
expected = gs.array([1.0])
self.assertAllClose(result, expected)
def test_regularize_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
result = self.group.regularize(rot_vecs)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_matrix_from_rotation_vector(self):
angle = gs.pi / 3
expected = gs.array([[1.0 / 2, -gs.sqrt(3.0) / 2],
[gs.sqrt(3.0) / 2, 1.0 / 2]])
result = self.group.matrix_from_rotation_vector(gs.array([angle]))
self.assertAllClose(result, expected)
def test_matrix_from_rotation_vector_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
rot_mats = self.group.matrix_from_rotation_vector(rot_vecs)
self.assertAllClose(gs.shape(rot_mats),
(n_samples, self.group.n, self.group.n))
def test_rotation_vector_from_matrix(self):
angle = 0.12
rot_mat = gs.array([[gs.cos(angle), -gs.sin(angle)],
[gs.sin(angle), gs.cos(angle)]])
result = self.group.rotation_vector_from_matrix(rot_mat)
expected = gs.array([0.12])
self.assertAllClose(result, expected)
def test_rotation_vector_and_rotation_matrix(self):
"""
This tests that the composition of
rotation_vector_from_matrix
and
matrix_from_rotation_vector
is the identity.
"""
# TODO(nguigs): bring back a 1d representation of SO2
point = gs.array([0.78])
rot_mat = self.group.matrix_from_rotation_vector(point)
result = self.group.rotation_vector_from_matrix(rot_mat)
expected = point
self.assertAllClose(result, expected)
def test_rotation_vector_and_rotation_matrix_vectorization(self):
rot_vecs = gs.array([[2.0], [1.3], [0.8], [0.03]])
rot_mats = self.group.matrix_from_rotation_vector(rot_vecs)
result = self.group.rotation_vector_from_matrix(rot_mats)
expected = self.group.regularize(rot_vecs)
self.assertAllClose(result, expected)
def test_compose(self):
point_a = gs.array([0.12])
point_b = gs.array([-0.15])
result = self.group.compose(point_a, point_b)
expected = self.group.regularize(gs.array([-0.03]))
self.assertAllClose(result, expected)
def test_compose_and_inverse(self):
angle = 0.986
point = gs.array([angle])
inv_point = self.group.inverse(point)
result = self.group.compose(point, inv_point)
expected = self.group.identity
self.assertAllClose(result, expected)
result = self.group.compose(inv_point, point)
expected = self.group.identity
self.assertAllClose(result, expected)
def test_compose_vectorization(self):
point_type = "vector"
self.group.default_point_type = point_type
n_samples = self.n_samples
n_points_a = self.group.random_uniform(n_samples=n_samples)
n_points_b = self.group.random_uniform(n_samples=n_samples)
one_point = self.group.random_uniform(n_samples=1)
result = self.group.compose(one_point, n_points_a)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
result = self.group.compose(n_points_a, one_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
result = self.group.compose(n_points_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_inverse_vectorization(self):
n_samples = self.n_samples
points = self.group.random_uniform(n_samples=n_samples)
result = self.group.inverse(points)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_exp(self):
"""
The Riemannian exp and log are inverse functions of each other.
This test is the inverse of test_log's.
"""
rot_vec_base_point = gs.array([gs.pi / 5])
rot_vec = gs.array([2 * gs.pi / 5])
expected = gs.array([3 * gs.pi / 5])
result = self.group.exp(base_point=rot_vec_base_point,
tangent_vec=rot_vec)
self.assertAllClose(result, expected)
def test_group_exp_vectorization(self):
n_samples = self.n_samples
one_tangent_vec = self.group.random_uniform(n_samples=1)
one_base_point = self.group.random_uniform(n_samples=1)
n_tangent_vec = self.group.random_uniform(n_samples=n_samples)
n_base_point = self.group.random_uniform(n_samples=n_samples)
# Test with the 1 base point, and n tangent vecs
result = self.group.exp(n_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the several base point, and one tangent vec
result = self.group.exp(one_tangent_vec, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the same number n of base point and n tangent vec
result = self.group.exp(n_tangent_vec, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_log(self):
"""
The Riemannian exp and log are inverse functions of each other.
This test is the inverse of test_exp's.
"""
rot_vec_base_point = gs.array([gs.pi / 5])
rot_vec = gs.array([2 * gs.pi / 5])
expected = gs.array([1 * gs.pi / 5])
result = self.group.log(point=rot_vec, base_point=rot_vec_base_point)
self.assertAllClose(result, expected)
def test_group_log_vectorization(self):
n_samples = self.n_samples
one_point = self.group.random_uniform(n_samples=1)
one_base_point = self.group.random_uniform(n_samples=1)
n_point = self.group.random_uniform(n_samples=n_samples)
n_base_point = self.group.random_uniform(n_samples=n_samples)
# Test with the 1 base point, and several different points
result = self.group.log(n_point, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the several base point, and 1 point
result = self.group.log(one_point, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the same number n of base point and point
result = self.group.log(n_point, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_exp_then_log_from_identity(self):
"""
Test that the group exponential
and the group logarithm are inverse.
Expect their composition to give the identity function.
"""
tangent_vec = gs.array([0.12])
result = helper.group_exp_then_log_from_identity(
group=self.group, tangent_vec=tangent_vec)
expected = self.group.regularize(tangent_vec)
self.assertAllClose(result, expected)
def test_group_log_then_exp_from_identity(self):
"""
Test that the group exponential
and the group logarithm are inverse.
Expect their composition to give the identity function.
"""
point = gs.array([0.12])
result = helper.group_log_then_exp_from_identity(group=self.group,
point=point)
expected = self.group.regularize(point)
self.assertAllClose(result, expected)
def test_group_exp_then_log(self):
"""
This tests that the composition of
log and exp gives identity.
"""
base_point = gs.array([0.12])
tangent_vec = gs.array([0.35])
result = helper.group_exp_then_log(group=self.group,
tangent_vec=tangent_vec,
base_point=base_point)
expected = self.group.regularize_tangent_vec(tangent_vec=tangent_vec,
base_point=base_point)
self.assertAllClose(result, expected)
def test_group_log_then_exp(self):
"""
This tests that the composition of
log and exp gives identity.
"""
base_point = gs.array([0.12])
point = gs.array([0.35])
result = helper.group_log_then_exp(group=self.group,
point=point,
base_point=base_point)
expected = self.group.regularize(point)
self.assertAllClose(result, expected)