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))
def test_load_poses_only_rotations(self):
"""Test that the poses belong to SO(3)."""
so3 = SpecialOrthogonal(n=3, point_type="vector")
data, _ = data_utils.load_poses()
result = so3.belongs(data)
self.assertTrue(gs.all(result))
class _SpecialEuclideanVectors(LieGroup):
"""Base Class for the special Euclidean groups in 2d and 3d in vector form.
i.e. the Lie group of rigid transformations. Elements of SE(2), SE(3) can
either be represented as vectors (in 2d or 3d) or as matrices in general.
The matrix representation corresponds to homogeneous coordinates. This
class is specific to the vector representation of rotations. For the matrix
representation use the SpecialEuclidean class and set `n=2` or `n=3`.
Parameter
---------
epsilon : float
Precision to use for calculations involving potential
division by 0 in rotations.
Optional, default: 0.
"""
def __init__(self, n, epsilon=0.0):
dim = n * (n + 1) // 2
LieGroup.__init__(
self,
dim=dim,
shape=(dim, ),
default_point_type="vector",
lie_algebra=Euclidean(dim),
)
self.n = n
self.epsilon = epsilon
self.rotations = SpecialOrthogonal(n=n,
point_type="vector",
epsilon=epsilon)
self.translations = Euclidean(dim=n)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}
The point_type of the returned value.
Optional, default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dim)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
def belongs(self, point, atol=gs.atol):
"""Evaluate if a point belongs to SE(2) or SE(3).
Parameters
----------
point : array-like, shape=[..., dim]
Point to check.
Returns
-------
belongs : array-like, shape=[...,]
Boolean indicating whether point belongs to SE(2) or SE(3).
"""
point_dim = point.shape[-1]
point_ndim = point.ndim
belongs = gs.logical_and(point_dim == self.dim, point_ndim < 3)
belongs = gs.logical_and(
belongs,
self.rotations.belongs(point[..., :self.rotations.dim], atol=atol))
return belongs
def projection(self, point):
"""Project a point to the group.
The point is regularized, so that the norm of the rotation part lie in [0, pi).
Parameters
----------
point: array-like, shape[..., dim]
Point.
Returns
-------
projected: array-like, shape[..., dim]
Regularized point.
"""
return self.regularize(point)
def regularize(self, point):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[..., dim]
Point to regularize.
Returns
-------
point : array-like, shape=[..., dim]
Regularized point.
"""
rotations = self.rotations
dim_rotations = rotations.dim
regularized_point = gs.copy(point)
rot_vec = regularized_point[..., :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec)
translation = regularized_point[..., dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=-1)
@geomstats.vectorization.decorator(["else", "vector", "else"])
def regularize_tangent_vec_at_identity(self, tangent_vec, metric=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., dim]
Tangent vector at base point.
metric : RiemannianMetric
Metric.
Optional, default: None.
Returns
-------
regularized_vec : array-like, shape=[..., dim]
Regularized vector.
"""
return self.regularize_tangent_vec(tangent_vec, self.identity, metric)
@geomstats.vectorization.decorator(["else", "vector"])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec : array-like, shape=[..., dim]
Vector.
Returns
-------
mat : array-like, shape=[..., n+1, n+1]
Matrix.
"""
vec = self.regularize(vec)
output_shape = ((vec.shape[0], self.n + 1,
self.n + 1) if vec.ndim == 2 else (self.n + 1, ) * 2)
rot_vec = vec[..., :self.rotations.dim]
trans_vec = vec[..., self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
return homogeneous_representation(rot_mat, trans_vec, output_shape)
@geomstats.vectorization.decorator(["else", "vector", "vector"])
def compose(self, point_a, point_b):
r"""Compose two elements of SE(2) or SE(3).
Parameters
----------
point_a : array-like, shape=[..., dim]
Point of the group.
point_b : array-like, shape=[..., dim]
Point of the group.
Equation
--------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition : array-like, shape=[..., dim]
Composition of point_a and point_b.
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a)
point_b = self.regularize(point_b)
rot_vec_a = point_a[..., :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[..., :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[..., dim_rotations:]
translation_b = point_b[..., dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = (
gs.einsum("...j,...kj->...k", translation_b, rot_mat_a) +
translation_a)
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition)
@geomstats.vectorization.decorator(["else", "vector"])
def inverse(self, point):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[..., dim]
Point.
Returns
-------
inverse_point : array-like, shape=[..., dim]
Inverted point.
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
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)
return self.regularize(inverse_point)
@geomstats.vectorization.decorator(["else", "vector"])
def exp_from_identity(self, tangent_vec):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
Tangent vector at base point.
Returns
-------
group_exp: array-like, shape=[..., 3]
Group exponential of the tangent vectors computed
at the identity.
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = tangent_vec[..., :dim_rotations]
rot_vec_regul = self.rotations.regularize(rot_vec)
rot_vec_regul = gs.to_ndarray(rot_vec_regul, to_ndim=2, axis=1)
transform = self._exp_translation_transform(rot_vec_regul)
translation = tangent_vec[..., dim_rotations:]
exp_translation = gs.einsum("ijk, ik -> ij", transform, translation)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp)
return group_exp
@geomstats.vectorization.decorator(["else", "vector"])
def log_from_identity(self, point):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[..., 3]
Point.
Returns
-------
group_log: array-like, shape=[..., 3]
Group logarithm in the Lie algebra.
"""
point = self.regularize(point)
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
transform = self._log_translation_transform(rot_vec)
log_translation = gs.einsum("ijk, ik -> ij", transform, translation)
return gs.concatenate([rot_vec, log_translation], axis=1)
def random_point(self, n_samples=1, bound=1.0, **kwargs):
r"""Sample in SE(n) with the uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound : float
Upper bound for the translation part of the sample.
Optional, default: 1.
Returns
-------
random_point : array-like, shape=[..., dim]
Sample.
"""
random_translation = self.translations.random_point(n_samples, bound)
random_rot_vec = self.rotations.random_uniform(n_samples)
return gs.concatenate([random_rot_vec, random_translation], axis=-1)
class TestFrechetMean(geomstats.tests.TestCase):
_multiprocess_can_split_ = True
def setUp(self):
gs.random.seed(123)
self.sphere = Hypersphere(dim=4)
self.hyperbolic = Hyperboloid(dim=3)
self.euclidean = Euclidean(dim=2)
self.minkowski = Minkowski(dim=2)
self.so3 = SpecialOrthogonal(n=3, point_type='vector')
self.so_matrix = SpecialOrthogonal(n=3)
def test_logs_at_mean_default_gradient_descent_sphere(self):
n_tests = 10
estimator = FrechetMean(
metric=self.sphere.metric, method='default', lr=1.)
result = []
for _ in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.sphere.random_uniform(n_samples=2)
estimator.fit(points)
mean = estimator.estimate_
logs = self.sphere.metric.log(point=points, base_point=mean)
result.append(gs.linalg.norm(logs[1, :] + logs[0, :]))
result = gs.stack(result)
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result)
def test_logs_at_mean_adaptive_gradient_descent_sphere(self):
n_tests = 10
estimator = FrechetMean(metric=self.sphere.metric, method='adaptive')
result = []
for _ in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.sphere.random_uniform(n_samples=2)
estimator.fit(points)
mean = estimator.estimate_
logs = self.sphere.metric.log(point=points, base_point=mean)
result.append(gs.linalg.norm(logs[1, :] + logs[0, :]))
result = gs.stack(result)
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result)
def test_estimate_shape_default_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(
metric=self.sphere.metric, method='default', verbose=True)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim,))
def test_estimate_shape_adaptive_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method='adaptive')
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim,))
def test_estimate_and_belongs_default_gradient_descent_sphere(self):
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method='default')
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_estimate_default_gradient_descent_so3(self):
points = self.so3.random_uniform(2)
mean_vec = FrechetMean(
metric=self.so3.bi_invariant_metric, method='default', lr=1.)
mean_vec.fit(points)
logs = self.so3.bi_invariant_metric.log(points, mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected)
def test_estimate_and_belongs_default_gradient_descent_so3(self):
point = self.so3.random_uniform(10)
mean_vec = FrechetMean(
metric=self.so3.bi_invariant_metric, method='default')
mean_vec.fit(point)
result = self.so3.belongs(mean_vec.estimate_)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_estimate_default_gradient_descent_so_matrix(self):
points = self.so_matrix.random_uniform(2)
mean_vec = FrechetMean(
metric=self.so_matrix.bi_invariant_metric, method='default',
lr=1.)
mean_vec.fit(points)
logs = self.so_matrix.bi_invariant_metric.log(
points, mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected, atol=1e-5)
@geomstats.tests.np_and_tf_only
def test_estimate_and_belongs_default_gradient_descent_so_matrix(self):
point = self.so_matrix.random_uniform(10)
mean = FrechetMean(
metric=self.so_matrix.bi_invariant_metric, method='default')
mean.fit(point)
result = self.so_matrix.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_estimate_and_belongs_adaptive_gradient_descent_so_matrix(self):
point = self.so_matrix.random_uniform(10)
mean = FrechetMean(
metric=self.so_matrix.bi_invariant_metric, method='adaptive',
verbose=True, lr=.5)
mean.fit(point)
result = self.so_matrix.belongs(mean.estimate_)
self.assertTrue(result)
@geomstats.tests.np_and_tf_only
def test_estimate_and_coincide_default_so_vec_and_mat(self):
point = self.so_matrix.random_uniform(3)
mean = FrechetMean(
metric=self.so_matrix.bi_invariant_metric, method='default')
mean.fit(point)
expected = mean.estimate_
mean_vec = FrechetMean(
metric=self.so3.bi_invariant_metric, method='default')
point_vec = self.so3.rotation_vector_from_matrix(point)
mean_vec.fit(point_vec)
result_vec = mean_vec.estimate_
result = self.so3.matrix_from_rotation_vector(result_vec)
self.assertAllClose(result, expected)
def test_estimate_and_belongs_adaptive_gradient_descent_sphere(self):
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method='adaptive')
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_variance_sphere(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.array([point, point])
result = variance(
points, base_point=point, metric=self.sphere.metric)
expected = gs.array(0.)
self.assertAllClose(expected, result)
def test_estimate_default_gradient_descent_sphere(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.array([point, point])
mean = FrechetMean(metric=self.sphere.metric, method='default')
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_adaptive_gradient_descent_sphere(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.array([point, point])
mean = FrechetMean(metric=self.sphere.metric, method='adaptive')
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_spd(self):
point = SPDMatrices(3).random_point()
points = gs.array([point, point])
mean = FrechetMean(metric=SPDMetricAffine(3), point_type='matrix')
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_variance_hyperbolic(self):
point = gs.array([2., 1., 1., 1.])
points = gs.array([point, point])
result = variance(
points, base_point=point, metric=self.hyperbolic.metric)
expected = gs.array(0.)
self.assertAllClose(result, expected)
def test_estimate_hyperbolic(self):
point = gs.array([2., 1., 1., 1.])
points = gs.array([point, point])
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
expected = point
result = mean.estimate_
self.assertAllClose(result, expected)
def test_estimate_and_belongs_hyperbolic(self):
point_a = self.hyperbolic.random_point()
point_b = self.hyperbolic.random_point()
point_c = self.hyperbolic.random_point()
points = gs.stack([point_a, point_b, point_c], axis=0)
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
result = self.hyperbolic.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_mean_euclidean_shape(self):
dim = 2
point = gs.array([1., 4.])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim,))
def test_mean_euclidean(self):
point = gs.array([1., 4.])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([
[1., 2.],
[2., 3.],
[3., 4.],
[4., 5.]])
weights = [1., 2., 1., 2.]
mean = FrechetMean(metric=self.euclidean.metric)
mean.fit(points, weights=weights)
result = mean.estimate_
expected = gs.array([16. / 6., 22. / 6.])
self.assertAllClose(result, expected)
def test_variance_euclidean(self):
points = gs.array([
[1., 2.],
[2., 3.],
[3., 4.],
[4., 5.]])
weights = gs.array([1., 2., 1., 2.])
base_point = gs.zeros(2)
result = variance(
points, weights=weights, base_point=base_point,
metric=self.euclidean.metric)
# we expect the average of the points' sq norms.
expected = gs.array((1 * 5. + 2 * 13. + 1 * 25. + 2 * 41.) / 6.)
self.assertAllClose(result, expected)
def test_mean_matrices_shape(self):
m, n = (2, 2)
point = gs.array([
[1., 4.],
[2., 3.]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type='matrix')
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (m, n))
def test_mean_matrices(self):
m, n = (2, 2)
point = gs.array([
[1., 4.],
[2., 3.]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type='matrix')
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
def test_mean_minkowski_shape(self):
dim = 2
point = gs.array([2., -math.sqrt(3)])
points = [point, point, point]
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim,))
def test_mean_minkowski(self):
point = gs.array([2., -math.sqrt(3)])
points = [point, point, point]
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([
[1., 0.],
[2., math.sqrt(3)],
[3., math.sqrt(8)],
[4., math.sqrt(24)]])
weights = gs.array([1., 2., 1., 2.])
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points, weights=weights)
result = mean.estimate_
result = self.minkowski.belongs(result)
expected = gs.array(True)
self.assertAllClose(result, expected)
def test_variance_minkowski(self):
points = gs.array([
[1., 0.],
[2., math.sqrt(3)],
[3., math.sqrt(8)],
[4., math.sqrt(24)]])
weights = gs.array([1., 2., 1., 2.])
base_point = gs.array([-1., 0.])
var = variance(
points, weights=weights, base_point=base_point,
metric=self.minkowski.metric)
result = var != 0
# we expect the average of the points' Minkowski sq norms.
expected = True
self.assertAllClose(result, expected)
def test_one_point(self):
point = gs.array([0., 0., 0., 0., 1.])
mean = FrechetMean(metric=self.sphere.metric, method='default')
mean.fit(X=point)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
mean = FrechetMean(
metric=self.sphere.metric, method='frechet-poincare-ball')
mean.fit(X=point)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
class _SpecialEuclideanMatrices(GeneralLinear, LieGroup):
"""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):
super().__init__(n=n + 1,
dim=int((n * (n + 1)) / 2),
default_point_type='matrix',
lie_algebra=SpecialEuclideanMatrixLieAlgebra(n=n))
self.rotations = SpecialOrthogonal(n=n)
self.translations = Euclidean(dim=n)
self.n = n
self.left_canonical_metric = \
SpecialEuclideanMatrixCannonicalLeftMetric(group=self)
def get_identity(self):
"""Return the identity matrix."""
return gs.eye(self.n + 1, self.n + 1)
identity = property(get_identity)
def belongs(self, point, atol=gs.atol):
"""Check whether point is of the form rotation, translation.
Parameters
----------
point : array-like, shape=[..., n, n].
Point to be checked.
atol : float
Tolerance threshold.
Returns
-------
belongs : array-like, shape=[...,]
Boolean denoting if point belongs to the group.
"""
n = self.n
belongs = Matrices(n + 1, n + 1).belongs(point)
if gs.all(belongs):
rotation = point[..., :n, :n]
belongs = self.rotations.belongs(rotation, atol=atol)
last_line_except_last_term = point[..., n:, :-1]
all_but_last_zeros = ~gs.any(last_line_except_last_term,
axis=(-2, -1))
belongs = gs.logical_and(belongs, all_but_last_zeros)
last_term = point[..., n, n]
belongs = gs.logical_and(belongs,
gs.isclose(last_term, 1., atol=atol))
return belongs
def random_point(self, n_samples=1, bound=1.):
"""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, n]
Point to be inverted.
"""
n = point.shape[-1] - 1
transposed_rot = cls.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)
class _SpecialEuclideanMatrices(GeneralLinear, LieGroup):
"""Class for special orthogonal groups.
Parameters
----------
n : int
Integer representing the shape of the matrices: n x n.
"""
def __init__(self, n):
super(_SpecialEuclideanMatrices,
self).__init__(default_point_type='matrix', n=n + 1)
self.rotations = SpecialOrthogonal(n=n)
self.translations = Euclidean(dim=n)
self.n = n
self.dim = int((n * (n + 1)) / 2)
def get_identity(self):
"""Return the identity matrix."""
return gs.eye(self.n + 1, self.n + 1)
identity = property(get_identity)
def belongs(self, point):
"""Check whether point is of the form rotation, translation.
Parameters
----------
point : array-like, shape=[..., n, n].
Point to be checked.
Returns
-------
belongs : array-like, shape=[...,]
Boolean denoting if point belongs to the group.
"""
point_dim1, point_dim2 = point.shape[-2:]
belongs = (point_dim1 == point_dim2 == self.n + 1)
rotation = point[..., :self.n, :self.n]
rot_belongs = self.rotations.belongs(rotation)
belongs = gs.logical_and(belongs, rot_belongs)
last_line_except_last_term = point[..., self.n:, :-1]
all_but_last_zeros = ~gs.any(last_line_except_last_term, axis=(-2, -1))
belongs = gs.logical_and(belongs, all_but_last_zeros)
last_term = point[..., self.n:, self.n:]
belongs = gs.logical_and(belongs, gs.all(last_term == 1,
axis=(-2, -1)))
if point.ndim == 2:
return gs.squeeze(belongs)
return gs.flatten(belongs)
def _is_in_lie_algebra(self, tangent_vec, atol=TOLERANCE):
"""Project vector rotation part onto skew-symmetric matrices."""
point_dim1, point_dim2 = tangent_vec.shape[-2:]
belongs = (point_dim1 == point_dim2 == self.n + 1)
rotation = tangent_vec[..., :self.n, :self.n]
rot_belongs = self.is_skew_symmetric(rotation, atol=atol)
belongs = gs.logical_and(belongs, rot_belongs)
last_line = tangent_vec[..., -1, :]
all_zeros = ~gs.any(last_line, axis=-1)
belongs = gs.logical_and(belongs, all_zeros)
return belongs
def _to_lie_algebra(self, tangent_vec):
"""Project vector rotation part onto skew-symmetric matrices."""
translation_mask = gs.hstack(
[gs.ones((self.n, ) * 2), 2 * gs.ones((self.n, 1))])
translation_mask = gs.concatenate(
[translation_mask, gs.zeros((1, self.n + 1))], axis=0)
tangent_vec = tangent_vec * gs.where(translation_mask != 0.,
gs.array(1.), gs.array(0.))
tangent_vec = (tangent_vec - GeneralLinear.transpose(tangent_vec)) / 2.
return tangent_vec * translation_mask
def random_uniform(self, n_samples=1, tol=1e-6):
"""Sample in SE(n) from the uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
tol : unused
Returns
-------
samples : array-like, shape=[..., n + 1, n + 1]
Sample in SE(n).
"""
random_translation = self.translations.random_uniform(n_samples)
random_rotation = self.rotations.random_uniform(n_samples)
random_rotation = gs.to_ndarray(random_rotation, to_ndim=3)
random_translation = gs.to_ndarray(random_translation, to_ndim=2)
random_translation = gs.transpose(
gs.to_ndarray(random_translation, to_ndim=3, axis=1), (0, 2, 1))
random_point = gs.concatenate((random_rotation, random_translation),
axis=2)
last_line = gs.zeros((n_samples, 1, self.n + 1))
random_point = gs.concatenate((random_point, last_line), axis=1)
random_point = gs.assignment(random_point, 1, (-1, -1), axis=0)
if gs.shape(random_point)[0] == 1:
random_point = gs.squeeze(random_point, axis=0)
return random_point
class TestFrechetMean(geomstats.tests.TestCase):
_multiprocess_can_split_ = True
def setup_method(self):
gs.random.seed(123)
self.sphere = Hypersphere(dim=4)
self.hyperbolic = Hyperboloid(dim=3)
self.euclidean = Euclidean(dim=2)
self.minkowski = Minkowski(dim=2)
self.so3 = SpecialOrthogonal(n=3, point_type="vector")
self.so_matrix = SpecialOrthogonal(n=3)
self.curves_2d = DiscreteCurves(R2)
self.elastic_metric = ElasticMetric(a=1, b=1, ambient_manifold=R2)
def test_logs_at_mean_curves_2d(self):
n_tests = 10
metric = self.curves_2d.srv_metric
estimator = FrechetMean(metric=metric, init_step_size=1.0)
result = []
for _ in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.curves_2d.random_point(n_samples=2)
estimator.fit(points)
mean = estimator.estimate_
# Expand and tile are added because of GitHub issue 1575
mean = gs.expand_dims(mean, axis=0)
mean = gs.tile(mean, (2, 1, 1))
logs = metric.log(point=points, base_point=mean)
logs_srv = metric.aux_differential_srv_transform(logs, curve=mean)
# Note that the logs are NOT inverse, only the logs_srv are.
result.append(gs.linalg.norm(logs_srv[1, :] + logs_srv[0, :]))
result = gs.stack(result)
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result, atol=1e-5)
def test_logs_at_mean_default_gradient_descent_sphere(self):
n_tests = 10
estimator = FrechetMean(metric=self.sphere.metric,
method="default",
init_step_size=1.0)
result = []
for _ in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.sphere.random_uniform(n_samples=2)
estimator.fit(points)
mean = estimator.estimate_
logs = self.sphere.metric.log(point=points, base_point=mean)
result.append(gs.linalg.norm(logs[1, :] + logs[0, :]))
result = gs.stack(result)
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result)
def test_logs_at_mean_adaptive_gradient_descent_sphere(self):
n_tests = 10
estimator = FrechetMean(metric=self.sphere.metric, method="adaptive")
result = []
for _ in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.sphere.random_uniform(n_samples=2)
estimator.fit(points)
mean = estimator.estimate_
logs = self.sphere.metric.log(point=points, base_point=mean)
result.append(gs.linalg.norm(logs[1, :] + logs[0, :]))
result = gs.stack(result)
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result)
def test_estimate_shape_default_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
point_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric,
method="default",
verbose=True)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_estimate_shape_adaptive_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
point_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method="adaptive")
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_estimate_shape_elastic_metric(self):
points = self.curves_2d.random_point(n_samples=2)
mean = FrechetMean(metric=self.elastic_metric)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (points.shape[1:]))
def test_estimate_shape_metric(self):
points = self.curves_2d.random_point(n_samples=2)
mean = FrechetMean(metric=self.curves_2d.srv_metric)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (points.shape[1:]))
def test_estimate_and_belongs_default_gradient_descent_sphere(self):
point_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
point_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method="default")
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_estimate_and_belongs_curves_2d(self):
points = self.curves_2d.random_point(n_samples=2)
mean = FrechetMean(metric=self.curves_2d.srv_metric)
mean.fit(points)
result = self.curves_2d.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_estimate_default_gradient_descent_so3(self):
points = self.so3.random_uniform(2)
mean_vec = FrechetMean(metric=self.so3.bi_invariant_metric,
method="default",
init_step_size=1.0)
mean_vec.fit(points)
logs = self.so3.bi_invariant_metric.log(points, mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected)
def test_estimate_and_belongs_default_gradient_descent_so3(self):
point = self.so3.random_uniform(10)
mean_vec = FrechetMean(metric=self.so3.bi_invariant_metric,
method="default")
mean_vec.fit(point)
result = self.so3.belongs(mean_vec.estimate_)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_autograd_and_tf_only
def test_estimate_default_gradient_descent_so_matrix(self):
points = self.so_matrix.random_uniform(2)
mean_vec = FrechetMean(
metric=self.so_matrix.bi_invariant_metric,
method="default",
init_step_size=1.0,
)
mean_vec.fit(points)
logs = self.so_matrix.bi_invariant_metric.log(points,
mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected, atol=1e-5)
@geomstats.tests.np_autograd_and_tf_only
def test_estimate_and_belongs_default_gradient_descent_so_matrix(self):
point = self.so_matrix.random_uniform(10)
mean = FrechetMean(metric=self.so_matrix.bi_invariant_metric,
method="default")
mean.fit(point)
result = self.so_matrix.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_autograd_and_tf_only
def test_estimate_and_belongs_adaptive_gradient_descent_so_matrix(self):
point = self.so_matrix.random_uniform(10)
mean = FrechetMean(
metric=self.so_matrix.bi_invariant_metric,
method="adaptive",
init_step_size=0.5,
verbose=True,
)
mean.fit(point)
result = self.so_matrix.belongs(mean.estimate_)
self.assertTrue(result)
@geomstats.tests.np_autograd_and_tf_only
def test_estimate_and_coincide_default_so_vec_and_mat(self):
point = self.so_matrix.random_uniform(3)
mean = FrechetMean(metric=self.so_matrix.bi_invariant_metric,
method="default")
mean.fit(point)
expected = mean.estimate_
mean_vec = FrechetMean(metric=self.so3.bi_invariant_metric,
method="default")
point_vec = self.so3.rotation_vector_from_matrix(point)
mean_vec.fit(point_vec)
result_vec = mean_vec.estimate_
result = self.so3.matrix_from_rotation_vector(result_vec)
self.assertAllClose(result, expected)
def test_estimate_and_belongs_adaptive_gradient_descent_sphere(self):
point_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
point_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method="adaptive")
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_variance_sphere(self):
point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
points = gs.array([point, point])
result = variance(points, base_point=point, metric=self.sphere.metric)
expected = gs.array(0.0)
self.assertAllClose(expected, result)
def test_estimate_default_gradient_descent_sphere(self):
point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
points = gs.array([point, point])
mean = FrechetMean(metric=self.sphere.metric, method="default")
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_elastic_metric(self):
point = self.curves_2d.random_point(n_samples=1)
points = gs.array([point, point])
mean = FrechetMean(metric=self.elastic_metric)
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_curves_2d(self):
point = self.curves_2d.random_point(n_samples=1)
points = gs.array([point, point])
mean = FrechetMean(metric=self.curves_2d.srv_metric)
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_adaptive_gradient_descent_sphere(self):
point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
points = gs.array([point, point])
mean = FrechetMean(metric=self.sphere.metric, method="adaptive")
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_spd(self):
point = SPDMatrices(3).random_point()
points = gs.array([point, point])
mean = FrechetMean(metric=SPDMetricAffine(3), point_type="matrix")
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_spd_two_samples(self):
space = SPDMatrices(3)
metric = SPDMetricAffine(3)
point = space.random_point(2)
mean = FrechetMean(metric)
mean.fit(point)
result = mean.estimate_
expected = metric.exp(metric.log(point[0], point[1]) / 2, point[1])
self.assertAllClose(expected, result)
def test_variance_hyperbolic(self):
point = gs.array([2.0, 1.0, 1.0, 1.0])
points = gs.array([point, point])
result = variance(points,
base_point=point,
metric=self.hyperbolic.metric)
expected = gs.array(0.0)
self.assertAllClose(result, expected)
def test_estimate_hyperbolic(self):
point = gs.array([2.0, 1.0, 1.0, 1.0])
points = gs.array([point, point])
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
expected = point
result = mean.estimate_
self.assertAllClose(result, expected)
def test_estimate_and_belongs_hyperbolic(self):
point_a = self.hyperbolic.random_point()
point_b = self.hyperbolic.random_point()
point_c = self.hyperbolic.random_point()
points = gs.stack([point_a, point_b, point_c], axis=0)
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
result = self.hyperbolic.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_mean_euclidean_shape(self):
dim = 2
point = gs.array([1.0, 4.0])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_mean_euclidean(self):
point = gs.array([1.0, 4.0])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([[1.0, 2.0], [2.0, 3.0], [3.0, 4.0], [4.0, 5.0]])
weights = [1.0, 2.0, 1.0, 2.0]
mean = FrechetMean(metric=self.euclidean.metric)
mean.fit(points, weights=weights)
result = mean.estimate_
expected = gs.array([16.0 / 6.0, 22.0 / 6.0])
self.assertAllClose(result, expected)
def test_variance_euclidean(self):
points = gs.array([[1.0, 2.0], [2.0, 3.0], [3.0, 4.0], [4.0, 5.0]])
weights = gs.array([1.0, 2.0, 1.0, 2.0])
base_point = gs.zeros(2)
result = variance(points,
weights=weights,
base_point=base_point,
metric=self.euclidean.metric)
# we expect the average of the points' sq norms.
expected = gs.array((1 * 5.0 + 2 * 13.0 + 1 * 25.0 + 2 * 41.0) / 6.0)
self.assertAllClose(result, expected)
def test_mean_matrices_shape(self):
m, n = (2, 2)
point = gs.array([[1.0, 4.0], [2.0, 3.0]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type="matrix")
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (m, n))
def test_mean_matrices(self):
m, n = (2, 2)
point = gs.array([[1.0, 4.0], [2.0, 3.0]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type="matrix")
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
def test_mean_minkowski_shape(self):
dim = 2
point = gs.array([2.0, -math.sqrt(3)])
points = [point, point, point]
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_mean_minkowski(self):
point = gs.array([2.0, -math.sqrt(3)])
points = [point, point, point]
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([[1.0, 0.0], [2.0, math.sqrt(3)],
[3.0, math.sqrt(8)], [4.0, math.sqrt(24)]])
weights = gs.array([1.0, 2.0, 1.0, 2.0])
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points, weights=weights)
result = self.minkowski.belongs(mean.estimate_)
self.assertTrue(result)
def test_variance_minkowski(self):
points = gs.array([[1.0, 0.0], [2.0, math.sqrt(3)],
[3.0, math.sqrt(8)], [4.0, math.sqrt(24)]])
weights = gs.array([1.0, 2.0, 1.0, 2.0])
base_point = gs.array([-1.0, 0.0])
var = variance(points,
weights=weights,
base_point=base_point,
metric=self.minkowski.metric)
result = var != 0
# we expect the average of the points' Minkowski sq norms.
expected = True
self.assertAllClose(result, expected)
def test_one_point(self):
point = gs.array([0.0, 0.0, 0.0, 0.0, 1.0])
mean = FrechetMean(metric=self.sphere.metric, method="default")
mean.fit(X=point)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
mean = FrechetMean(metric=self.sphere.metric, method="default")
mean.fit(X=point)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_batched(self):
space = SPDMatrices(3)
metric = SPDMetricAffine(3)
point = space.random_point(4)
mean_batch = FrechetMean(metric, method="batch", verbose=True)
data = gs.stack([point[:2], point[2:]], axis=1)
mean_batch.fit(data)
result = mean_batch.estimate_
mean = FrechetMean(metric)
mean.fit(data[:, 0])
expected_1 = mean.estimate_
mean.fit(data[:, 1])
expected_2 = mean.estimate_
expected = gs.stack([expected_1, expected_2])
self.assertAllClose(expected, result)
@geomstats.tests.np_and_autograd_only
def test_stiefel_two_samples(self):
space = Stiefel(3, 2)
metric = space.metric
point = space.random_point(2)
mean = FrechetMean(metric)
mean.fit(point)
result = mean.estimate_
expected = metric.exp(metric.log(point[0], point[1]) / 2, point[1])
self.assertAllClose(expected, result)
@geomstats.tests.np_and_autograd_only
def test_stiefel_n_samples(self):
space = Stiefel(3, 2)
metric = space.metric
point = space.random_point(2)
mean = FrechetMean(metric,
method="default",
init_step_size=0.5,
verbose=True)
mean.fit(point)
result = space.belongs(mean.estimate_)
self.assertTrue(result)
def test_circle_mean(self):
space = Hypersphere(1)
points = space.random_uniform(10)
mean_circle = FrechetMean(space.metric)
mean_circle.fit(points)
estimate_circle = mean_circle.estimate_
# set a wrong dimension so that the extrinsic coordinates are used
metric = space.metric
metric.dim = 2
mean_extrinsic = FrechetMean(metric)
mean_extrinsic.fit(points)
estimate_extrinsic = mean_extrinsic.estimate_
self.assertAllClose(estimate_circle, estimate_extrinsic)
class _SpecialEuclidean3Vectors(LieGroup):
"""Class for the special euclidean group in 3d, SE(3).
i.e. the Lie group of rigid transformations. Elements of SE(3) can either
be represented as vectors (in 3d) or as matrices in general. The matrix
representation corresponds to homogeneous coordinates.This class is
specific to the vector representation of rotations. For the matrix
representation use the SpecialEuclidean class and set `n=3`.
Parameter
---------
epsilon : float, optional (defaults to 0)
Precision to use for calculations involving potential
division by 0 in rotations.
"""
def __init__(self, epsilon=0.):
super(_SpecialEuclidean3Vectors,
self).__init__(dim=6, default_point_type='vector')
self.n = 3
self.epsilon = epsilon
self.rotations = SpecialOrthogonal(n=3,
point_type='vector',
epsilon=epsilon)
self.translations = Euclidean(dim=3)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}, optional
The point_type of the returned value.
default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dim)
if point_type == 'matrix':
identity = gs.eye(self.n + 1)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
def belongs(self, point):
"""Evaluate if a point belongs to SE(3).
Parameters
----------
point : array-like, shape=[..., 3]
The point of which to check whether it belongs to SE(3).
Returns
-------
belongs : array-like, shape=[..., 1]
Boolean indicating whether point belongs to SE(3).
"""
point_dim = point.shape[-1]
point_ndim = point.ndim
belongs = gs.logical_and(point_dim == self.dim, point_ndim < 3)
belongs = gs.logical_and(belongs,
self.rotations.belongs(point[..., :self.n]))
return belongs
def regularize(self, point):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[..., 3]
The point to regularize.
Returns
-------
point : array-like, shape=[..., 3]
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[..., :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec)
translation = point[..., dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=-1)
@geomstats.vectorization.decorator(['else', 'vector', 'else'])
def regularize_tangent_vec_at_identity(self, tangent_vec, metric=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
metric : RiemannianMetric, optional
Returns
-------
regularized_vec : the regularized tangent vector
"""
return self.regularize_tangent_vec(tangent_vec, self.identity, metric)
def regularize_tangent_vec(self, tangent_vec, base_point, metric=None):
"""Regularize a tangent vector at a base point.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
base_point : array-like, shape=[..., 3]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
Returns
-------
regularized_vec : the regularized tangent vector
"""
if metric is None:
metric = self.left_canonical_metric
rotations = self.rotations
dim_rotations = rotations.dim
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)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric)
return gs.concatenate(
[rotations_vec, tangent_vec[..., dim_rotations:]], axis=-1)
@geomstats.vectorization.decorator(['else', 'vector'])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec: array-like, shape=[..., 3]
Returns
-------
mat: array-like, shape=[..., n+1, n+1]
"""
vec = self.regularize(vec)
n_vecs, _ = vec.shape
rot_vec = vec[:, :self.rotations.dim]
trans_vec = vec[:, self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
trans_vec = gs.reshape(trans_vec, (n_vecs, self.n, 1))
mat = gs.concatenate((rot_mat, trans_vec), axis=2)
last_lines = gs.array(gs.get_mask_i_float(self.n, self.n + 1))
last_lines = gs.to_ndarray(last_lines, to_ndim=2)
last_lines = gs.to_ndarray(last_lines, to_ndim=3)
mat = gs.concatenate((mat, last_lines), axis=1)
return mat
@geomstats.vectorization.decorator(['else', 'vector', 'vector'])
def compose(self, point_a, point_b):
r"""Compose two elements of SE(3).
Parameters
----------
point_a : array-like, shape=[..., 3]
Point of the group.
point_b : array-like, shape=[..., 3]
Point of the group.
Equation
---------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition :
The composition of point_a and point_b.
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a)
point_b = self.regularize(point_b)
rot_vec_a = point_a[..., :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[..., :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[..., dim_rotations:]
translation_b = point_b[..., dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum('...j,...kj->...k', translation_b,
rot_mat_a) + translation_a
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition)
@geomstats.vectorization.decorator(['else', 'vector'])
def inverse(self, point):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[..., 3]
Returns
-------
inverse_point : array-like, shape=[..., 3]
The inverted point.
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
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)
return self.regularize(inverse_point)
@geomstats.vectorization.decorator(['else', 'vector', 'else'])
def jacobian_translation(self, point, left_or_right='left'):
"""Compute the Jacobian matrix resulting from translation.
Compute the matrix of the differential of the left/right translations
from the identity to point in SE(3).
Parameters
----------
point: array-like, shape=[..., 3]
left_or_right: str, {'left', 'right'}, optional
Whether to compute the jacobian of the left or right translation.
Returns
-------
jacobian : array-like, shape=[..., 3]
The jacobian of the left / right translation.
"""
if left_or_right not in ('left', 'right'):
raise ValueError('`left_or_right` must be `left` or `right`.')
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dim
dim_translations = translations.dim
point = self.regularize(point)
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
jacobian_rot = self.rotations.jacobian_translation(
point=rot_vec, left_or_right=left_or_right)
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)
return gs.concatenate([jacobian_block_line_1, jacobian_block_line_2],
axis=1)
@geomstats.vectorization.decorator(['else', 'vector'])
def exp_from_identity(self, tangent_vec):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
Returns
-------
group_exp: array-like, shape=[..., 3]
The group exponential of the tangent vectors calculated
at the identity.
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = tangent_vec[..., :dim_rotations]
rot_vec = self.rotations.regularize(rot_vec)
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
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)
exp_translation += gs.outer(mask_i_float,
translation_i + term_1_i + term_2_i)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp)
return group_exp
@geomstats.vectorization.decorator(['else', 'vector'])
def log_from_identity(self, point):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[..., 3]
Returns
-------
group_log: array-like, shape=[..., 3]
the group logarithm in the Lie algbra
"""
point = self.regularize(point)
rotations = self.rotations
dim_rotations = rotations.dim
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
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)
log_translation += gs.outer(mask_i_float,
translation_i + term_1_i + term_2_i)
return gs.concatenate([rot_vec, log_translation], axis=1)
def random_uniform(self, n_samples=1):
"""Sample in SE(3) with the uniform distribution.
Parameters
----------
n_samples : int, optional
default : 1
Returns
-------
random_point : array-like, shape=[..., 3]
An array of random elements in SE(3) having the given.
"""
random_translation = self.translations.random_uniform(n_samples)
random_rot_vec = self.rotations.random_uniform(n_samples)
return gs.concatenate([random_rot_vec, random_translation], axis=-1)
def _exponential_matrix(self, rot_vec):
"""Compute exponential of rotation matrix represented by rot_vec.
Parameters
----------
rot_vec : array-like, shape=[..., 3]
Returns
-------
exponential_mat : The matrix exponential of rot_vec
"""
# TODO(nguigs): find usecase for this method
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 discontinuity 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
return exponential_mat
class TestSpecialOrthogonal(geomstats.tests.TestCase):
def setUp(self):
self.n = 2
self.group = SpecialOrthogonal(n=self.n)
self.n_samples = 4
def test_dim(self):
for n in [2, 3, 4, 5, 6]:
group = SpecialOrthogonal(n=n)
result = group.dim
expected = n * (n - 1) / 2
self.assertAllClose(result, expected)
def test_belongs(self):
theta = gs.pi / 3
point_1 = gs.array([[gs.cos(theta), - gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_1)
self.assertTrue(result)
point_2 = gs.array([[gs.cos(theta), gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_2)
self.assertFalse(result)
point = gs.array([point_1, point_2])
expected = gs.array([True, False])
result = self.group.belongs(point)
self.assertAllClose(result, expected)
point = point_1[0]
result = self.group.belongs(point)
self.assertFalse(result)
point = gs.zeros((2, 3))
result = self.group.belongs(point)
self.assertFalse(result)
point = gs.zeros((2, 2, 3))
result = self.group.belongs(point)
self.assertFalse(gs.all(result))
def test_random_uniform_and_belongs(self):
point = self.group.random_uniform()
result = self.group.belongs(point)
expected = True
self.assertAllClose(result, expected)
point = self.group.random_uniform(self.n_samples)
result = self.group.belongs(point)
expected = gs.array([True] * self.n_samples)
self.assertAllClose(result, expected)
def test_identity(self):
result = self.group.identity
expected = gs.eye(self.n)
self.assertAllClose(result, expected)
def test_is_in_lie_algebra(self):
theta = gs.pi / 3
vec_1 = gs.array([[0., - theta],
[theta, 0.]])
result = self.group.is_tangent(vec_1)
self.assertTrue(result)
vec_2 = gs.array([[0., - theta],
[theta, 1.]])
result = self.group.is_tangent(vec_2)
self.assertFalse(result)
vec = gs.array([vec_1, vec_2])
result = self.group.is_tangent(vec)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_is_tangent(self):
point = self.group.random_uniform()
theta = 1.
vec_1 = gs.array([[0., - theta],
[theta, 0.]])
vec_1 = self.group.compose(point, vec_1)
result = self.group.is_tangent(vec_1, point)
self.assertTrue(result)
vec_2 = gs.array([[0., - theta],
[theta, 1.]])
vec_2 = self.group.compose(point, vec_2)
result = self.group.is_tangent(vec_2, point)
self.assertFalse(result)
vec = gs.array([vec_1, vec_2])
point = gs.array([point, point])
expected = gs.array([True, False])
result = self.group.is_tangent(vec, point)
self.assertAllClose(result, expected)
def test_to_tangent(self):
theta = 1.
vec_1 = gs.array([[0., - theta],
[theta, 0.]])
result = self.group.to_tangent(vec_1)
expected = vec_1
self.assertAllClose(result, expected)
n_samples = self.n_samples
base_points = self.group.random_uniform(n_samples=n_samples)
tangent_vecs = self.group.compose(base_points, vec_1)
result = self.group.to_tangent(tangent_vecs, base_points)
expected = tangent_vecs
self.assertAllClose(result, expected)
def test_projection_and_belongs(self):
gs.random.seed(4)
shape = (self.n_samples, self.n, self.n)
result = helper.test_projection_and_belongs(
self.group, shape, gs.atol * 100)
for res in result:
self.assertTrue(res)
def test_skew_to_vec_and_back(self):
group = SpecialOrthogonal(n=4)
vec = gs.random.rand(group.dim)
mat = group.skew_matrix_from_vector(vec)
result = group.vector_from_skew_matrix(mat)
self.assertAllClose(result, vec)
def test_parallel_transport(self):
metric = self.group.bi_invariant_metric
shape = (self.n_samples, self.group.n, self.group.n)
results = helper.test_parallel_transport(self.group, metric, shape)
for res in results:
self.assertTrue(res)
def test_metric_left_invariant(self):
group = self.group
point = group.random_point()
tangent_vec = self.group.lie_algebra.basis[0]
expected = group.bi_invariant_metric.norm(
tangent_vec)
translated = group.tangent_translation_map(point)(tangent_vec)
result = group.bi_invariant_metric.norm(translated)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_distance_broadcast(self):
group = self.group
point = group.random_point(5)
result = group.bi_invariant_metric.dist_broadcast(point[:3], point)
expected = []
for a in point[:3]:
expected.append(group.bi_invariant_metric.dist(a, point))
expected = gs.stack(expected)
self.assertAllClose(result, expected)
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)
class TestSpecialOrthogonal(geomstats.tests.TestCase):
def setUp(self):
self.n = 2
self.group = SpecialOrthogonal(n=self.n)
self.n_samples = 4
def test_belongs(self):
theta = gs.pi / 3
point_1 = gs.array([[gs.cos(theta), -gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_1)
expected = True
self.assertAllClose(result, expected)
point_2 = gs.array([[gs.cos(theta), gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_2)
expected = False
self.assertAllClose(result, expected)
point = gs.array([point_1, point_2])
expected = gs.array([True, False])
result = self.group.belongs(point)
self.assertAllClose(result, expected)
def test_random_uniform_and_belongs(self):
point = self.group.random_uniform()
result = self.group.belongs(point)
expected = True
self.assertAllClose(result, expected)
point = self.group.random_uniform(self.n_samples)
result = self.group.belongs(point)
expected = gs.array([True] * self.n_samples)
self.assertAllClose(result, expected)
def test_identity(self):
result = self.group.identity
expected = gs.eye(self.n)
self.assertAllClose(result, expected)
def test_is_in_lie_algebra(self):
theta = gs.pi / 3
vec_1 = gs.array([[0., -theta], [theta, 0.]])
result = self.group.is_tangent(vec_1)
expected = True
self.assertAllClose(result, expected)
vec_2 = gs.array([[0., -theta], [theta, 1.]])
result = self.group.is_tangent(vec_2)
expected = False
self.assertAllClose(result, expected)
vec = gs.array([vec_1, vec_2])
expected = gs.array([True, False])
result = self.group.is_tangent(vec)
self.assertAllClose(result, expected)
def test_is_tangent(self):
point = self.group.random_uniform()
theta = 1.
vec_1 = gs.array([[0., -theta], [theta, 0.]])
vec_1 = self.group.compose(point, vec_1)
result = self.group.is_tangent(vec_1, point)
expected = True
self.assertAllClose(result, expected)
vec_2 = gs.array([[0., -theta], [theta, 1.]])
vec_2 = self.group.compose(point, vec_2)
result = self.group.is_tangent(vec_2, point)
expected = False
self.assertAllClose(result, expected)
vec = gs.array([vec_1, vec_2])
point = gs.array([point, point])
expected = gs.array([True, False])
result = self.group.is_tangent(vec, point)
self.assertAllClose(result, expected)
def test_to_tangent(self):
theta = 1.
vec_1 = gs.array([[0., -theta], [theta, 0.]])
result = self.group.to_tangent(vec_1)
expected = vec_1
self.assertAllClose(result, expected)
n_samples = self.n_samples
base_points = self.group.random_uniform(n_samples=n_samples)
tangent_vecs = self.group.compose(base_points, vec_1)
result = self.group.to_tangent(tangent_vecs, base_points)
expected = tangent_vecs
self.assertAllClose(result, expected)
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))
def test_skew_to_vec_and_back(self):
group = SpecialOrthogonal(n=4)
vec = gs.random.rand(group.dim)
mat = group.skew_matrix_from_vector(vec)
result = group.vector_from_skew_matrix(mat)
self.assertAllClose(result, vec)
class TestExponentialBarycenter(geomstats.tests.TestCase):
def setUp(self):
logger = logging.getLogger()
logger.disabled = True
self.se_mat = SpecialEuclidean(n=3)
self.so_vec = SpecialOrthogonal(n=3, point_type='vector')
self.so = SpecialOrthogonal(n=3)
self.n_samples = 4
@geomstats.tests.np_only
def test_estimate_and_belongs_se(self):
point = self.se_mat.random_point(self.n_samples)
estimator = ExponentialBarycenter(self.se_mat)
estimator.fit(point)
barexp = estimator.estimate_
result = self.se_mat.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
def test_estimate_one_sample_se(self):
point = self.se_mat.random_point()
estimator = ExponentialBarycenter(self.se_mat)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_reach_max_iter_se(self):
point = self.se_mat.random_point(1)
estimator = ExponentialBarycenter(self.se_mat, max_iter=2)
points = gs.array([point, point])
estimator.fit(points)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec, max_iter=2)
points = gs.array([point, point])
estimator.fit(points)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_so_matrix(self):
points = self.so.random_uniform(2)
mean_vec = ExponentialBarycenter(group=self.so)
mean_vec.fit(points)
logs = self.so.log(points, mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_belongs_so(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_one_sample_so(self):
point = self.so.random_uniform(1)
estimator = ExponentialBarycenter(self.so)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_reach_max_iter_so(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, max_iter=2)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec, max_iter=2)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_coincides_with_frechet_so(self):
gs.random.seed(0)
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, max_iter=40, epsilon=1e-10)
estimator.fit(point)
result = estimator.estimate_
frechet_estimator = FrechetMean(self.so.bi_invariant_metric,
max_iter=40,
epsilon=1e-10,
lr=1.,
method='adaptive')
frechet_estimator.fit(point)
expected = frechet_estimator.estimate_
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_weights(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, verbose=True)
weights = gs.arange(self.n_samples)
estimator.fit(point, weights=weights)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point, weights=weights)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
def test_linear_mean(self):
euclidean = Euclidean(3)
point = euclidean.random_point(self.n_samples)
estimator = ExponentialBarycenter(euclidean)
estimator.fit(point)
result = estimator.estimate_
expected = gs.mean(point, axis=0)
self.assertAllClose(result, expected)
class TestSpecialOrthogonal(geomstats.tests.TestCase):
def setUp(self):
self.n = 2
self.group = SpecialOrthogonal(n=self.n)
self.n_samples = 4
def test_belongs(self):
theta = gs.pi / 3
point_1 = gs.array([[gs.cos(theta), -gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_1)
expected = True
self.assertAllClose(result, expected)
point_2 = gs.array([[gs.cos(theta), gs.sin(theta)],
[gs.sin(theta), gs.cos(theta)]])
result = self.group.belongs(point_2)
expected = False
self.assertAllClose(result, expected)
point = gs.array([point_1, point_2])
expected = gs.array([True, False])
result = self.group.belongs(point)
self.assertAllClose(result, expected)
def test_random_uniform_and_belongs(self):
point = self.group.random_uniform()
result = self.group.belongs(point)
expected = True
self.assertAllClose(result, expected)
point = self.group.random_uniform(self.n_samples)
result = self.group.belongs(point)
expected = gs.array([True] * self.n_samples)
self.assertAllClose(result, expected)
def test_identity(self):
result = self.group.identity
expected = gs.eye(self.n)
self.assertAllClose(result, expected)
def test_is_in_lie_algebra(self):
theta = gs.pi / 3
vec_1 = gs.array([[0., -theta], [theta, 0.]])
result = self.group.is_tangent(vec_1)
expected = True
self.assertAllClose(result, expected)
vec_2 = gs.array([[0., -theta], [theta, 1.]])
result = self.group.is_tangent(vec_2)
expected = False
self.assertAllClose(result, expected)
vec = gs.array([vec_1, vec_2])
expected = gs.array([True, False])
result = self.group.is_tangent(vec)
self.assertAllClose(result, expected)
def test_is_tangent(self):
point = self.group.random_uniform()
theta = 1.
vec_1 = gs.array([[0., -theta], [theta, 0.]])
vec_1 = self.group.compose(point, vec_1)
result = self.group.is_tangent(vec_1, point, atol=1e-6)
expected = True
self.assertAllClose(result, expected)
vec_2 = gs.array([[0., -theta], [theta, 1.]])
vec_2 = self.group.compose(point, vec_2)
result = self.group.is_tangent(vec_2, point, atol=1e-6)
expected = False
self.assertAllClose(result, expected)
vec = gs.array([vec_1, vec_2])
point = gs.array([point, point])
expected = gs.array([True, False])
result = self.group.is_tangent(vec, point, atol=1e-6)
self.assertAllClose(result, expected)
def test_to_tangent(self):
theta = 1.
vec_1 = gs.array([[0., -theta], [theta, 0.]])
result = self.group.to_tangent(vec_1)
expected = vec_1
self.assertAllClose(result, expected)
n_samples = self.n_samples
base_points = self.group.random_uniform(n_samples=n_samples)
tangent_vecs = self.group.compose(base_points, vec_1)
result = self.group.to_tangent(tangent_vecs, base_points)
expected = tangent_vecs
self.assertAllClose(result, expected)
class SpecialEuclidean(LieGroup):
"""Class for the special euclidean group SE(n).
i.e. the Lie group of rigid transformations. Elements of SE(n) can either
be represented as vectors (in 3d) or as matrices in general. The matrix
representation corresponds to homogeneous coordinates.
"""
def __init__(self, n, default_point_type=None, epsilon=0.):
"""Initiate an object of class SpecialEuclidean.
Parameter
---------
n : int
the dimension of the euclidean space that SE(n) acts upon
point_type : str, {'vector', 'matrix'}, optional
whether to represent elmenents of SE(n) by vectors or matrices
if None is given, point_type is set to 'vector' for dimension 3
and 'matrix' otherwise
epsilon : float, optional
precision to use for calculations involving potential division by
rotations
default: 0
"""
if not (isinstance(n, int) and n > 1):
raise ValueError('n must be an integer > 1.')
self.n = n
self.dimension = int((n * (n - 1)) / 2 + n)
self.epsilon = epsilon
super(SpecialEuclidean,
self).__init__(dim=self.dimension,
default_point_type=default_point_type)
if default_point_type is None:
self.default_point_type = 'vector' if n == 3 else 'matrix'
self.rotations = SpecialOrthogonal(
n=n, epsilon=epsilon, default_point_type=default_point_type)
self.translations = Euclidean(dim=n)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}, optional
the point_type of the returned value
default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dimension)
if point_type == 'matrix':
identity = gs.eye(self.n + 1)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def belongs(self, point, point_type=None):
"""Evaluate if a point belongs to SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the point of which to check whether it belongs to SE(n)
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
belongs : array-like, shape=[n_samples, 1]
array of booleans indicating whether point belongs to SE(n)
"""
if point_type == 'vector':
n_points, vec_dim = gs.shape(point)
belongs = vec_dim == self.dimension
belongs = gs.tile([belongs], (point.shape[0], ))
belongs = gs.logical_and(belongs,
self.rotations.belongs(point[:, :self.n]))
return gs.flatten(belongs)
if point_type == 'matrix':
n_points, point_dim1, point_dim2 = point.shape
belongs = (point_dim1 == point_dim2 == self.n + 1)
belongs = [belongs] * n_points
rotation = point[:, :self.n, :self.n]
rot_belongs = self.rotations.belongs(rotation,
point_type=point_type)
belongs = gs.logical_and(belongs, rot_belongs)
last_line_except_last_term = point[:, self.n:, :-1]
all_but_last_zeros = ~gs.any(last_line_except_last_term,
axis=(1, 2))
belongs = gs.logical_and(belongs, all_but_last_zeros)
last_term = point[:, self.n:, self.n:]
belongs = gs.logical_and(belongs,
gs.all(last_term == 1, axis=(1, 2)))
return gs.flatten(belongs)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def regularize(self, point, point_type=None):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the point which should be regularized
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
"""
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[:, :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec,
point_type=point_type)
translation = point[:, dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=1)
if point_type == 'matrix':
return gs.to_ndarray(point, to_ndim=3)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'else', 'point_type'])
def regularize_tangent_vec_at_identity(self,
tangent_vec,
metric=None,
point_type=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if point_type == 'vector':
return self.regularize_tangent_vec(tangent_vec,
self.identity,
metric,
point_type=point_type)
if point_type == 'matrix':
translation_mask = gs.hstack(
[gs.ones((self.n, ) * 2), 2 * gs.ones((self.n, 1))])
translation_mask = gs.concatenate(
[translation_mask, gs.zeros((1, self.n + 1))], axis=0)
tangent_vec = tangent_vec * gs.where(translation_mask != 0.,
gs.array(1.), gs.array(0.))
tangent_vec = (tangent_vec -
GeneralLinear.transpose(tangent_vec)) / 2.
return tangent_vec * translation_mask
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(
['else', 'point', 'point', 'else', 'point_type'])
def regularize_tangent_vec(self,
tangent_vec,
base_point,
metric=None,
point_type=None):
"""Regularize a tangent vector at a base point.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
base_point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if metric is None:
metric = self.left_canonical_metric
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
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)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric,
point_type=point_type)
return gs.concatenate(
[rotations_vec, tangent_vec[:, dim_rotations:]], axis=1)
if point_type == 'matrix':
tangent_vec_at_id = self.compose(self.inverse(base_point),
tangent_vec)
regularized = self.regularize_tangent_vec_at_identity(
tangent_vec_at_id, point_type=point_type)
return self.compose(base_point, regularized)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'vector'])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec: array-like, shape=[n_samples, dim]
Returns
-------
mat: array-like, shape=[n_samples, {dim, [n+1, n+1]}]
"""
vec = self.regularize(vec, point_type='vector')
n_vecs, _ = vec.shape
rot_vec = vec[:, :self.rotations.dim]
trans_vec = vec[:, self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
trans_vec = gs.reshape(trans_vec, (n_vecs, self.n, 1))
mat = gs.concatenate((rot_mat, trans_vec), axis=2)
last_lines = gs.array(gs.get_mask_i_float(self.n, self.n + 1))
last_lines = gs.to_ndarray(last_lines, to_ndim=2)
last_lines = gs.to_ndarray(last_lines, to_ndim=3)
mat = gs.concatenate((mat, last_lines), axis=1)
return mat
@geomstats.vectorization.decorator(
['else', 'point', 'point', 'point_type'])
def compose(self, point_a, point_b, point_type=None):
r"""Compose two elements of SE(n).
Parameters
----------
point_1 : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_2 : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Equation
---------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition : the composition of point_1 and point_2
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a, point_type=point_type)
point_b = self.regularize(point_b, point_type=point_type)
if point_type == 'vector':
n_points_a, _ = point_a.shape
n_points_b, _ = point_b.shape
if not (point_a.shape == point_b.shape or n_points_a == 1
or n_points_b == 1):
raise ValueError()
rot_vec_a = point_a[:, :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[:, :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[:, dim_rotations:]
translation_b = point_b[:, dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum(
'...j,...kj->...k', translation_b, rot_mat_a) + translation_a
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition, point_type=point_type)
if point_type == 'matrix':
return GeneralLinear.compose(point_a, point_b)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def inverse(self, point, point_type=None):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
inverse_point : array-like,
shape=[n_samples, {dim, [n + 1, n + 1]}]
the inverted point
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
if point_type == 'vector':
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
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)
return self.regularize(inverse_point, point_type=point_type)
if point_type == 'matrix':
inv_rot = gs.transpose(point[:, :self.n, :self.n], axes=(0, 2, 1))
inv_trans = gs.matmul(inv_rot, -point[:, :self.n, self.n:])
last_line = point[:, self.n:, :]
inverse_point = gs.concatenate((inv_rot, inv_trans), axis=2)
return gs.concatenate((inverse_point, last_line), axis=1)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'else', 'point_type'])
def jacobian_translation(self,
point,
left_or_right='left',
point_type=None):
"""Compute the Jacobian matrix resulting from translation.
Compute the matrix of the differential
of the left/right translations from the identity to point in SE(n).
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
left_or_right: str, {'left', 'right'}, optional
default: 'left'
whether to compute the jacobian of the left or right translation
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
jacobian : array-like, shape=[n_samples, dim]
The jacobian of the left / right translation
"""
if point_type is None:
point_type = self.default_point_type
if left_or_right not in ('left', 'right'):
raise ValueError('`left_or_right` must be `left` or `right`.')
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dim
dim_translations = translations.dim
point = self.regularize(point, point_type=point_type)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
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)
return gs.concatenate(
[jacobian_block_line_1, jacobian_block_line_2], axis=1)
if point_type == 'matrix':
return point
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def exp_from_identity(self, tangent_vec, point_type=None):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_exp: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the group exponential of the tangent vectors calculated
at the identity
"""
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
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
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)
exp_translation += gs.outer(
mask_i_float, translation_i + term_1_i + term_2_i)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp, point_type=point_type)
return group_exp
if point_type == 'matrix':
return GeneralLinear.exp(tangent_vec)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def log_from_identity(self, point, point_type=None):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_log: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the group logarithm in the Lie algbra
"""
point = self.regularize(point, point_type=point_type)
rotations = self.rotations
dim_rotations = rotations.dim
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
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)
log_translation += gs.outer(
mask_i_float, translation_i + term_1_i + term_2_i)
return gs.concatenate([rot_vec, log_translation], axis=1)
if point_type == 'matrix':
return GeneralLinear.log(point)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
def random_uniform(self, n_samples=1, point_type=None):
"""Sample in SE(n) with the uniform distribution.
Parameters
----------
n_samples: int, optional
default: 1
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
random_point: array-like,
shape=[n_samples, {dim, [n + 1, n + 1]}]
An array of random elements in SE(n) having the given point_type.
"""
if point_type is None:
point_type = self.default_point_type
random_translation = self.translations.random_uniform(n_samples)
if point_type == 'vector':
random_rot_vec = self.rotations.random_uniform(
n_samples, point_type=point_type)
return gs.concatenate([random_rot_vec, random_translation],
axis=-1)
if point_type == 'matrix':
random_rotation = self.rotations.random_uniform(
n_samples, point_type=point_type)
random_rotation = gs.to_ndarray(random_rotation, to_ndim=3)
random_translation = gs.to_ndarray(random_translation, to_ndim=2)
random_translation = gs.transpose(
gs.to_ndarray(random_translation, to_ndim=3, axis=1),
(0, 2, 1))
random_point = gs.concatenate(
(random_rotation, random_translation), axis=2)
last_line = gs.zeros((n_samples, 1, self.n + 1))
random_point = gs.concatenate((random_point, last_line), axis=1)
random_point = gs.assignment(random_point, 1, (-1, -1), axis=0)
if gs.shape(random_point)[0] == 1:
random_point = gs.squeeze(random_point, axis=0)
return random_point
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
def _exponential_matrix(self, rot_vec):
"""Compute exponential of rotation matrix represented by rot_vec.
Parameters
----------
rot_vec : array-like, shape=[n_samples, dim]
Returns
-------
exponential_mat : The matrix exponential of rot_vec
"""
# TODO(nguigs): find usecase for this method
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 discontinuity 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
return exponential_mat
class TestExponentialBarycenter(geomstats.tests.TestCase):
def setUp(self):
self.se_mat = SpecialEuclidean(n=3, default_point_type='matrix')
self.so_vec = SpecialOrthogonal(n=3, default_point_type='vector')
self.so = SpecialOrthogonal(n=3, default_point_type='matrix')
self.n_samples = 3
@geomstats.tests.np_only
def test_estimate_and_belongs_se(self):
point = self.se_mat.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.se_mat)
estimator.fit(point)
barexp = estimator.estimate_
result = self.se_mat.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
def test_estimate_one_sample_se(self):
point = self.se_mat.random_uniform(1)
estimator = ExponentialBarycenter(self.se_mat)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_reach_max_iter_se(self):
point = self.se_mat.random_uniform(1)
estimator = ExponentialBarycenter(self.se_mat, max_iter=2)
points = gs.array([point, point])
estimator.fit(points)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec, max_iter=2)
points = gs.array([point, point])
estimator.fit(points)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_belongs_so(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_one_sample_so(self):
point = self.so.random_uniform(1)
estimator = ExponentialBarycenter(self.so)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(1)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point)
result = estimator.estimate_
expected = point
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_and_reach_max_iter_so(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, max_iter=2)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec, max_iter=2)
estimator.fit(point)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_coincides_with_frechet_so(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, max_iter=32, epsilon=1e-12)
estimator.fit(point)
result = estimator.estimate_
print(self.so.default_point_type)
so_vector = SpecialOrthogonal(3, default_point_type='vector')
frechet_estimator = FrechetMean(so_vector.bi_invariant_metric,
max_iter=32,
epsilon=1e-10,
point_type='vector')
vector_point = so_vector.rotation_vector_from_matrix(point)
frechet_estimator.fit(vector_point)
mean = frechet_estimator.estimate_
expected = so_vector.matrix_from_rotation_vector(mean)
result = estimator.estimate_
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_estimate_weights(self):
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, verbose=True)
weights = gs.arange(self.n_samples)
estimator.fit(point, weights=weights)
barexp = estimator.estimate_
result = self.so.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
point = self.so_vec.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so_vec)
estimator.fit(point, weights=weights)
barexp = estimator.estimate_
result = self.so_vec.belongs(barexp)
expected = True
self.assertAllClose(result, expected)
def test_linear_mean(self):
euclidean = Euclidean(3)
point = euclidean.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(euclidean)
estimator.fit(point)
result = estimator.estimate_
expected = gs.mean(point, axis=0)
self.assertAllClose(result, expected)
class _SpecialEuclideanMatrices(GeneralLinear, LieGroup):
"""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).
"""
def __init__(self, n):
super().__init__(
n=n + 1, dim=int((n * (n + 1)) / 2), default_point_type='matrix',
lie_algebra=SpecialEuclideanMatrixLieAlgebra(n=n))
self.rotations = SpecialOrthogonal(n=n)
self.translations = Euclidean(dim=n)
self.n = n
self.left_canonical_metric = \
SpecialEuclideanMatrixCannonicalLeftMetric(group=self)
def get_identity(self):
"""Return the identity matrix."""
return gs.eye(self.n + 1, self.n + 1)
identity = property(get_identity)
def belongs(self, point):
"""Check whether point is of the form rotation, translation.
Parameters
----------
point : array-like, shape=[..., n, n].
Point to be checked.
Returns
-------
belongs : array-like, shape=[...,]
Boolean denoting if point belongs to the group.
"""
point_dim1, point_dim2 = point.shape[-2:]
belongs = (point_dim1 == point_dim2 == self.n + 1)
rotation = point[..., :self.n, :self.n]
rot_belongs = self.rotations.belongs(rotation)
belongs = gs.logical_and(belongs, rot_belongs)
last_line_except_last_term = point[..., self.n:, :-1]
all_but_last_zeros = ~ gs.any(
last_line_except_last_term, axis=(-2, -1))
belongs = gs.logical_and(belongs, all_but_last_zeros)
last_term = point[..., self.n, self.n]
belongs = gs.logical_and(belongs, gs.isclose(last_term, 1.))
if point.ndim == 2:
return gs.squeeze(belongs)
return gs.flatten(belongs)
def random_uniform(self, n_samples=1, tol=1e-6):
"""Sample in SE(n) from the uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
tol : unused
Returns
-------
samples : array-like, shape=[..., n + 1, n + 1]
Sample in SE(n).
"""
random_translation = self.translations.random_uniform(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, n]
Point to be inverted.
"""
n = point.shape[-1] - 1
transposed_rot = cls.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)
