def test_to_grassmanniann_vectorized(self):
inf_rots = gs.array([gs.pi * r_z / n for n in [2, 3, 4]])
rots = GeneralLinear.exp(inf_rots)
points = Matrices.mul(rots, point1)
result = Stiefel.to_grassmannian(points)
expected = gs.array([p_xy, p_xy, p_xy])
self.assertAllClose(result, expected)
def random_uniform(self, n_samples=1):
"""Define a log-uniform random sample of SPD matrices."""
n = self.n
size = (n_samples, n, n) if n_samples != 1 else (n, n)
mat = 2 * gs.random.rand(*size) - 1
spd_mat = GeneralLinear.exp(mat + Matrices.transpose(mat))
return spd_mat
def to_grassmannian_test_data(self):
point1 = gs.array([[1.0, -1.0], [1.0, 1.0], [0.0, 0.0]]) / gs.sqrt(2.0)
batch_points = Matrices.mul(
GeneralLinear.exp(gs.array([gs.pi * r_z / n for n in [2, 3, 4]])),
point1,
)
smoke_data = [
dict(point=point1, expected=p_xy),
dict(point=batch_points, expected=gs.array([p_xy, p_xy, p_xy])),
]
return self.generate_tests(smoke_data)
def exp(self,
tangent_vec,
base_point=None,
n_steps=10,
step="rk4",
**kwargs):
"""Exponential map associated to the cannonical metric.
Exponential map at `base_point` of `tangent_vec`. 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
----------
tangent_vec : array-like, shape=[..., n + 1, n + 1]
Tangent vector at the base point.
base_point : array-like, shape=[..., n + 1, n + 1]
Point on the manifold.
Returns
-------
exp : array-like, shape=[..., n + 1, n + 1]
Point on the manifold.
See Also
--------
examples.plot_geodesics_se2
"""
group = self.group
if base_point is None:
base_point = group.identity
inf_rotation = tangent_vec[..., :self.n, :self.n]
rotation = base_point[..., :self.n, :self.n]
rotation_exp = GeneralLinear.exp(inf_rotation, rotation)
translation_exp = (tangent_vec[..., :self.n, self.n] +
base_point[..., :self.n, self.n])
exp = homogeneous_representation(rotation_exp, translation_exp,
tangent_vec.shape, 1.0)
return exp
def random_uniform(self, n_samples=1):
"""Sample in SPD(n) from the log-uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
Returns
-------
samples : array-like, shape=[..., n, n]
Points sampled in SPD(n).
"""
n = self.n
size = (n_samples, n, n) if n_samples != 1 else (n, n)
mat = 2 * gs.random.rand(*size) - 1
spd_mat = GeneralLinear.exp(mat + Matrices.transpose(mat))
return spd_mat
def random_point(self, n_samples=1, bound=1.0):
r"""Sample in PSD(n,k) from the log-uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound : float
Bound of the interval in which to sample in the tangent space.
Optional, default: 1.
Returns
-------
samples : array-like, shape=[..., n, n]
Points sampled in PSD(n,k).
"""
n = self.n
size = (n_samples, n, n) if n_samples != 1 else (n, n)
mat = bound * (2 * gs.random.rand(*size) - 1)
spd_mat = GeneralLinear.exp(Matrices.to_symmetric(mat))
return self.projection(spd_mat)
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)
def random_uniform(self, n_samples=1):
"""Define a log-uniform random sample of SPD matrices."""
mat = 2 * gs.random.rand(n_samples, self.n, self.n) - 1
spd_mat = GeneralLinear.exp(mat + Matrices.transpose(mat))
return spd_mat
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 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\'.')