def log(self, point, base_point):
r"""Compute the Riemannian logarithm of point w.r.t. base_point.
Given :math:`P, P'` in Gr(n, k) the logarithm from :math:`P`
to :math:`P` is given by the infinitesimal rotation [Batzies2015]_:
.. math::
\omega = \frac 1 2 \log \big((2 P' - 1)(2 P - 1)\big)
Parameters
----------
point : array-like, shape=[n_samples, n, n]
Point in the Grassmannian.
base_point : array-like, shape=[n_samples, n, n]
Point in the Grassmannian.
Returns
-------
tangent_vec : array-like, shape=[n_samples, n, n]
Tangent vector at `base_point`.
References
----------
.. [Batzies2015] Batzies, Hüper, Machado, Leite.
"Geometric Mean and Geodesic Regression on Grassmannians"
Linear Algebra and its Applications, 466, 83-101, 2015.
"""
GLn = GeneralLinear(self.n)
id_n = GLn.identity()
sym2 = 2 * point - id_n
sym1 = 2 * base_point - id_n
rot = GLn.mul(sym2, sym1)
return GLn.log(rot) / 2
def log(self, point, base_point=None, **kwargs):
"""Compute logarithm map associated to the canonical metric.
Log map at `base_point` of `point`. The geodesics of this
metric correspond to a direct product metric between rotation and
translation: the translation part is a straight line, while the
rotation part has constant angular velocity (which corresponds to one-
parameter subgroups of the rotation group).
Parameters
----------
point : array-like, shape=[..., n + 1, n + 1]
Point on the manifold.
base_point : array-like, shape=[..., n + 1, n + 1]
Point on the manifold.
Returns
-------
tangent_vec : array-like, shape=[..., n + 1, n + 1]
Tangent vector at the base point.
References
----------
[Zefran98] Zefran, M., V. Kumar, and C.B. Croke.
“On the Generation of Smooth Three-Dimensional Rigid Body
Motions.” IEEE Transactions on Robotics and Automation 14,
no. 4 (August 1998): 576–89.
https://doi.org/10.1109/70.704225.
"""
max_shape = point.shape if point.ndim == 3 else base_point.shape
rotation_bp = base_point[..., :self.n, :self.n]
rotation_p = point[..., :self.n, :self.n]
rotation_log = GeneralLinear.log(rotation_p, rotation_bp)
translation_log = (point[..., :self.n, self.n] -
base_point[..., :self.n, self.n])
log = homogeneous_representation(rotation_log, translation_log,
max_shape, 0.0)
return log
def log(self, point, base_point, **kwargs):
r"""Compute the Riemannian logarithm of point w.r.t. base_point.
Given :math:`P, P'` in Gr(n, k) the logarithm from :math:`P`
to :math:`P` is induced by the infinitesimal rotation [Batzies2015]_:
.. math::
Y = \frac 1 2 \log \big((2 P' - 1)(2 P - 1)\big)
The tangent vector :math:`X` at :math:`P`
is then recovered by :math:`X = [Y, P]`.
Parameters
----------
point : array-like, shape=[..., n, n]
Point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Riemannian logarithm, a tangent vector at `base_point`.
References
----------
.. [Batzies2015] Batzies, Hüper, Machado, Leite.
"Geometric Mean and Geodesic Regression on Grassmannians"
Linear Algebra and its Applications, 466, 83-101, 2015.
"""
GLn = GeneralLinear(self.n)
id_n = GLn.identity
id_n, point, base_point = gs.convert_to_wider_dtype(
[id_n, point, base_point])
sym2 = 2 * point - id_n
sym1 = 2 * base_point - id_n
rot = GLn.compose(sym2, sym1)
return Matrices.bracket(GLn.log(rot) / 2, base_point)
class TestGeneralLinear(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.n_samples = 2
self.group = GeneralLinear(n=self.n)
warnings.simplefilter('ignore', category=ImportWarning)
def test_belongs_shape(self):
mat = gs.eye(3)
result = self.group.belongs(mat)
self.assertAllClose(gs.shape(result), ())
mat = gs.ones((3, 3))
result = self.group.belongs(mat)
self.assertAllClose(gs.shape(result), ())
def test_belongs(self):
mat = gs.eye(3)
result = self.group.belongs(mat)
expected = True
self.assertAllClose(result, expected)
mat = gs.ones((3, 3))
result = self.group.belongs(mat)
expected = False
self.assertAllClose(result, expected)
def test_belongs_vectorization_shape(self):
mats = gs.array([gs.eye(3), gs.ones((3, 3))])
result = self.group.belongs(mats)
self.assertAllClose(gs.shape(result), (2, ))
def test_belongs_vectorization(self):
mats = gs.array([gs.eye(3), gs.ones((3, 3))])
result = self.group.belongs(mats)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
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 = 4
point = self.group.random_uniform(n_samples)
result = self.group.belongs(point)
expected = gs.array([True] * n_samples)
self.assertAllClose(result, expected)
def test_replace_values(self):
points = gs.ones((3, 3, 3))
new_points = gs.zeros((2, 3, 3))
indcs = [True, False, True]
update = self.group._replace_values(points, new_points, indcs)
self.assertAllClose(
update,
gs.stack([gs.zeros((3, 3)),
gs.ones((3, 3)),
gs.zeros((3, 3))]))
def test_compose(self):
mat1 = gs.array([[1., 0.], [0., 2.]])
mat2 = gs.array([[2., 0.], [0., 1.]])
result = self.group.compose(mat1, mat2)
expected = 2. * GeneralLinear(2).identity
self.assertAllClose(result, expected)
def test_inv(self):
mat_a = gs.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 10.]])
imat_a = 1. / 3. * gs.array([[-2., -4., 3.], [-2., 11., -6.],
[3., -6., 3.]])
expected = imat_a
result = self.group.inverse(mat_a)
self.assertAllClose(result, expected)
def test_inv_vectorized(self):
mat_a = gs.array([[0., 1., 0.], [1., 0., 0.], [0., 0., 1.]])
mat_b = -gs.eye(3, 3)
result = self.group.inverse(gs.array([mat_a, mat_b]))
expected = gs.array([mat_a, mat_b])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_log_and_exp(self):
point = 5 * gs.eye(self.n)
group_log = self.group.log(point)
result = self.group.exp(group_log)
expected = point
self.assertAllClose(result, expected)
def test_exp_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.], [0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.], [0., 148.413159, 0.],
[0., 0., 403.42879349]]])
result = self.group.exp(point)
self.assertAllClose(result, expected, rtol=1e-3)
@geomstats.tests.np_and_tf_only
def test_log_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.], [0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.], [0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = self.group.log(point)
self.assertAllClose(result, expected, atol=1e-4)
@geomstats.tests.np_and_tf_only
def test_orbit(self):
point = gs.array([[gs.exp(4.), 0.], [0., gs.exp(2.)]])
sqrt = gs.array([[gs.exp(2.), 0.], [0., gs.exp(1.)]])
idty = GeneralLinear(2).identity
path = GeneralLinear(2).orbit(point)
time = gs.linspace(0., 1., 3)
result = path(time)
expected = gs.array([idty, sqrt, point])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_expm_and_logm_vectorization_symmetric(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
result = self.group.exp(self.group.log(point))
expected = point
self.assertAllClose(result, expected)
class TestGeneralLinear(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.n_samples = 2
self.group = GeneralLinear(n=self.n)
self.group_pos = GeneralLinear(self.n, positive_det=True)
warnings.simplefilter('ignore', category=ImportWarning)
def test_belongs_shape(self):
mat = gs.eye(3)
result = self.group.belongs(mat)
self.assertAllClose(gs.shape(result), ())
mat = gs.ones((3, 3))
result = self.group.belongs(mat)
self.assertAllClose(gs.shape(result), ())
def test_belongs(self):
mat = gs.eye(3)
result = self.group.belongs(mat)
expected = True
self.assertAllClose(result, expected)
mat = gs.ones((3, 3))
result = self.group.belongs(mat)
expected = False
self.assertAllClose(result, expected)
mat = gs.ones(3)
result = self.group.belongs(mat)
expected = False
self.assertAllClose(result, expected)
def test_belongs_vectorization_shape(self):
mats = gs.array([gs.eye(3), gs.ones((3, 3))])
result = self.group.belongs(mats)
self.assertAllClose(gs.shape(result), (2, ))
def test_belongs_vectorization(self):
mats = gs.array([gs.eye(3), gs.ones((3, 3))])
result = self.group.belongs(mats)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_random_and_belongs(self):
for group in [self.group, self.group_pos]:
point = group.random_point()
result = group.belongs(point)
self.assertTrue(result)
def test_random_and_belongs_vectorization(self):
n_samples = 4
expected = gs.array([True] * n_samples)
for group in [self.group, self.group_pos]:
point = group.random_point(n_samples)
result = group.belongs(point)
self.assertAllClose(result, expected)
def test_compose(self):
mat1 = gs.array([[1., 0.], [0., 2.]])
mat2 = gs.array([[2., 0.], [0., 1.]])
result = self.group.compose(mat1, mat2)
expected = 2. * GeneralLinear(2).identity
self.assertAllClose(result, expected)
def test_inv(self):
mat_a = gs.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 10.]])
imat_a = 1. / 3. * gs.array([[-2., -4., 3.], [-2., 11., -6.],
[3., -6., 3.]])
expected = imat_a
result = self.group.inverse(mat_a)
self.assertAllClose(result, expected)
def test_inv_vectorized(self):
mat_a = gs.array([[0., 1., 0.], [1., 0., 0.], [0., 0., 1.]])
mat_b = -gs.eye(3, 3)
result = self.group.inverse(gs.array([mat_a, mat_b]))
expected = gs.array([mat_a, mat_b])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_log_and_exp(self):
point = 5 * gs.eye(self.n)
group_log = self.group.log(point)
result = self.group.exp(group_log)
expected = point
self.assertAllClose(result, expected)
def test_exp_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.], [0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.], [0., 148.413159, 0.],
[0., 0., 403.42879349]]])
expected = gs.cast(expected, gs.float64)
point = gs.cast(point, gs.float64)
result = self.group.exp(point)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_log_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.], [0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.], [0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = self.group.log(point)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_orbit(self):
point = gs.array([[gs.exp(4.), 0.], [0., gs.exp(2.)]])
sqrt = gs.array([[gs.exp(2.), 0.], [0., gs.exp(1.)]])
identity = GeneralLinear(2).identity
path = GeneralLinear(2).orbit(point)
time = gs.linspace(0., 1., 3)
result = path(time)
expected = gs.array([identity, sqrt, point])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_orbit_vectorization(self):
point = gs.array([[gs.exp(4.), 0.], [0., gs.exp(2.)]])
sqrt = gs.array([[gs.exp(2.), 0.], [0., gs.exp(1.)]])
identity = GeneralLinear(2).identity
path = GeneralLinear(2).orbit(gs.stack([point] * 2), identity)
time = gs.linspace(0., 1., 3)
result = path(time)
expected = gs.array([identity, sqrt, point])
expected = gs.stack([expected] * 2)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_expm_and_logm_vectorization_symmetric(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
result = self.group.exp(self.group.log(point))
expected = point
self.assertAllClose(result, expected)
def test_projection_and_belongs(self):
shape = (self.n_samples, self.n, self.n)
result = helper.test_projection_and_belongs(self.group, shape)
for res in result:
self.assertTrue(res)
def test_projection_and_belongs_pos(self):
shape = (self.n_samples, self.n, self.n)
result = helper.test_projection_and_belongs(self.group_pos, shape)
for res in result:
self.assertTrue(res)
class TestGeneralLinearMethods(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.n_samples = 2
self.group = GeneralLinear(n=self.n)
# We generate invertible matrices using so3_group
self.so3_group = SpecialOrthogonal(n=self.n)
warnings.simplefilter('ignore', category=ImportWarning)
@geomstats.tests.np_only
def test_belongs(self):
"""
A rotation matrix belongs to the matrix Lie group
of invertible matrices.
"""
rot_vec = gs.array([0.2, -0.1, 0.1])
rot_mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.belongs(rot_mat)
expected = gs.array([[True]])
self.assertAllClose(result, expected)
def test_compose(self):
# 1. Composition by identity, on the right
# Expect the original transformation
rot_vec = gs.array([0.2, -0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.compose(mat, self.group.identity)
expected = mat
expected = helper.to_matrix(mat)
self.assertAllClose(result, expected)
# 2. Composition by identity, on the left
# Expect the original transformation
rot_vec = gs.array([0.2, 0.1, -0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
result = self.group.compose(self.group.identity, mat)
expected = mat
self.assertAllClose(result, expected)
def test_inverse(self):
mat = gs.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 10.]])
result = self.group.inverse(mat)
expected = 1. / 3. * gs.array([[-2., -4., 3.], [-2., 11., -6.],
[3., -6., 3.]])
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
def test_compose_and_inverse(self):
# 1. Compose transformation by its inverse on the right
# Expect the group identity
rot_vec = gs.array([0.2, 0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
inv_mat = self.group.inverse(mat)
result = self.group.compose(mat, inv_mat)
expected = self.group.identity
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
# 2. Compose transformation by its inverse on the left
# Expect the group identity
rot_vec = gs.array([0.7, 0.1, 0.1])
mat = self.so3_group.matrix_from_rotation_vector(rot_vec)
inv_mat = self.group.inverse(mat)
result = self.group.compose(inv_mat, mat)
expected = self.group.identity
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_group_log_and_exp(self):
point = 5 * gs.eye(self.n)
group_log = self.group.log(point)
result = self.group.exp(group_log)
expected = point
expected = helper.to_matrix(expected)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_group_exp_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[7.38905609, 0., 0.], [0., 20.0855369, 0.],
[0., 0., 54.5981500]],
[[2.718281828, 0., 0.], [0., 148.413159, 0.],
[0., 0., 403.42879349]]])
result = self.group.exp(point)
self.assertAllClose(result, expected, rtol=1e-3)
@geomstats.tests.np_and_tf_only
def test_group_log_vectorization(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
expected = gs.array([[[0.693147180, 0., 0.], [0., 1.09861228866, 0.],
[0., 0., 1.38629436]],
[[0., 0., 0.], [0., 1.609437912, 0.],
[0., 0., 1.79175946]]])
result = self.group.log(point)
self.assertAllClose(result, expected, atol=1e-4)
@geomstats.tests.np_and_tf_only
def test_expm_and_logm_vectorization_symmetric(self):
point = gs.array([[[2., 0., 0.], [0., 3., 0.], [0., 0., 4.]],
[[1., 0., 0.], [0., 5., 0.], [0., 0., 6.]]])
result = self.group.exp(self.group.log(point))
expected = point
self.assertAllClose(result, expected)
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\'.')