def retraction(tangent_vec, base_point):
"""Compute the retraction of a tangent vector.
This computation is based on the QR-decomposition.
e.g. :math:`P_x(V) = qf(X + V)`.
Parameters
----------
tangent_vec : array-like, shape=[..., n, p]
Tangent vector at a base point.
base_point : array-like, shape=[..., n, p]
Point in the Stiefel manifold.
Returns
-------
exp : array-like, shape=[..., n, p]
Point in the Stiefel manifold equal to the retraction
of tangent_vec at the base point.
"""
n_tangent_vecs, _, _ = tangent_vec.shape
n_base_points, _, _ = base_point.shape
if not (n_tangent_vecs == n_base_points or n_tangent_vecs == 1
or n_base_points == 1):
raise NotImplementedError
matrix_q, matrix_r = gs.linalg.qr(base_point + tangent_vec)
diagonal = gs.diagonal(matrix_r, axis1=1, axis2=2)
sign = gs.sign(gs.sign(diagonal) + 0.5)
diag = algebra_utils.from_vector_to_diagonal_matrix(sign)
result = gs.einsum('nij,njk->nik', matrix_q, diag)
return result
def retraction(tangent_vec, base_point):
"""Compute the retraction of a tangent vector.
This computation is based on the QR-decomposition.
e.g. :math:`P_x(V) = qf(X + V)`.
Parameters
----------
tangent_vec : array-like, shape=[..., n, p]
Tangent vector at a base point.
base_point : array-like, shape=[..., n, p]
Point in the Stiefel manifold.
Returns
-------
exp : array-like, shape=[..., n, p]
Point in the Stiefel manifold equal to the retraction
of tangent_vec at the base point.
"""
matrix_q, matrix_r = gs.linalg.qr(base_point + tangent_vec)
diagonal = gs.diagonal(matrix_r, axis1=-2, axis2=-1)
sign = gs.sign(gs.sign(diagonal) + 0.5)
diag = algebra_utils.from_vector_to_diagonal_matrix(sign)
result = Matrices.mul(matrix_q, diag)
return result
def retraction(self, tangent_vec, base_point):
"""
Retraction map, based on QR-decomposion:
P_x(V) = qf(X + V)
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
n_tangent_vecs, _, _ = tangent_vec.shape
base_point = gs.to_ndarray(base_point, to_ndim=3)
n_base_points, n, p = base_point.shape
assert (n_tangent_vecs == n_base_points
or n_tangent_vecs == 1
or n_base_points == 1)
if n_base_points == 1:
base_point = gs.tile(base_point, (n_tangent_vecs, 1, 1))
if n_tangent_vecs == 1:
tangent_vec = gs.tile(tangent_vec, (n_base_points, 1, 1))
matrix_q, matrix_r = gs.linalg.qr(base_point+tangent_vec)
diagonal = gs.diagonal(matrix_r, axis1=1, axis2=2)
sign = gs.sign(gs.sign(diagonal) + 0.5)
diag = gs.diag(sign)
result = gs.einsum('nij,njk->nik', matrix_q, diag)
return result
def align_matrices(cls, point, base_point):
"""Align matrices.
Find the optimal rotation R in SO(m) such that the base point and
R.point are well positioned.
Parameters
----------
point : array-like, shape=[..., m, n]
Point on the manifold.
base_point : array-like, shape=[..., m, n]
Point on the manifold.
Returns
-------
aligned : array-like, shape=[..., m, n]
R.point.
"""
mat = gs.matmul(cls.transpose(point), base_point)
left, singular_values, right = gs.linalg.svd(mat)
det = gs.linalg.det(mat)
conditioning = (
singular_values[..., -2] + gs.sign(det) * singular_values[..., -1]
) / singular_values[..., 0]
if gs.any(conditioning < gs.atol):
logging.warning(
f"Singularity close, ill-conditioned matrix "
f"encountered: "
f"{conditioning[conditioning < 1e-10]}"
)
if gs.any(gs.isclose(conditioning, 0.0)):
logging.warning("Alignment matrix is not unique.")
flipped = flip_determinant(cls.transpose(right), det)
return Matrices.mul(point, left, cls.transpose(flipped))
def align(self, point, base_point, **kwargs):
"""Align point to base_point.
Find the optimal rotation R in SO(m) such that the base point and
R.point are well positioned.
Parameters
----------
point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the manifold.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the manifold.
Returns
-------
aligned : array-like, shape=[..., k_landmarks, m_ambient]
R.point.
"""
mat = gs.matmul(Matrices.transpose(point), base_point)
left, singular_values, right = gs.linalg.svd(mat)
det = gs.linalg.det(mat)
conditioning = ((singular_values[..., -2] +
gs.sign(det) * singular_values[..., -1]) /
singular_values[..., 0])
if gs.any(conditioning < 5e-4):
logging.warning(f'Singularity close, ill-conditioned matrix '
f'encountered: {conditioning}')
if gs.any(gs.isclose(conditioning, 0.)):
logging.warning("Alignment matrix is not unique.")
flipped = flip_determinant(Matrices.transpose(right), det)
return Matrices.mul(point, left, Matrices.transpose(flipped))
def retraction(self, tangent_vec, base_point):
"""Compute the retraction of a tangent vector.
This computation is based on the QR-decomposition.
e.g. :math:`P_x(V) = qf(X + V)`.
Parameters
----------
tangent_vec : array-like, shape=[n_samples, n, p]
Tangent vector at a base point.
base_point : array-like, shape=[n_samples, n, p]
Point in the Stiefel manifold.
Returns
-------
exp : array-like, shape=[n_samples, n, p]
Point in the Stiefel manifold equal to the retraction
of tangent_vec at the base point.
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
n_tangent_vecs, _, _ = tangent_vec.shape
base_point = gs.to_ndarray(base_point, to_ndim=3)
n_base_points, n, p = base_point.shape
assert (n_tangent_vecs == n_base_points or n_tangent_vecs == 1
or n_base_points == 1)
if n_base_points == 1:
base_point = gs.tile(base_point, (n_tangent_vecs, 1, 1))
if n_tangent_vecs == 1:
tangent_vec = gs.tile(tangent_vec, (n_base_points, 1, 1))
matrix_q, matrix_r = gs.linalg.qr(base_point + tangent_vec)
diagonal = gs.diagonal(matrix_r, axis1=1, axis2=2)
sign = gs.sign(gs.sign(diagonal) + 0.5)
diag = gs.diag(sign)
result = gs.einsum('nij,njk->nik', matrix_q, diag)
return result
def align(self, point, base_point, **kwargs):
"""Align point to base_point.
Find the optimal rotation R in SO(m) such that the base point and
R.point are well positioned.
Parameters
----------
point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the manifold.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the manifold.
Returns
-------
aligned : array-like, shape=[..., k_landmarks, m_ambient]
R.point.
"""
mat = gs.matmul(Matrices.transpose(point), base_point)
left, singular_values, right = gs.linalg.svd(mat)
det = gs.linalg.det(mat)
conditioning = (
(singular_values[..., -2]
+ gs.sign(det) * singular_values[..., -1]) /
singular_values[..., 0])
if gs.any(conditioning < 5e-4):
logging.warning(f'Singularity close, ill-conditioned matrix '
f'encountered: {conditioning}')
if gs.any(gs.isclose(conditioning, 0.)):
logging.warning("Alignment matrix is not unique.")
if gs.any(det < 0):
ones = gs.ones(self.m_ambient)
reflection_vec = gs.concatenate(
[ones[:-1], gs.array([-1.])], axis=0)
mask = gs.cast(det < 0, gs.float32)
sign = (mask[..., None] * reflection_vec
+ (1. - mask)[..., None] * ones)
j_matrix = from_vector_to_diagonal_matrix(sign)
rotation = Matrices.mul(
Matrices.transpose(right), j_matrix, Matrices.transpose(left))
else:
rotation = gs.matmul(
Matrices.transpose(right), Matrices.transpose(left))
return gs.matmul(point, Matrices.transpose(rotation))
return solver.solve(problem)
if __name__ == '__main__':
if os.environ.get('GEOMSTATS_BACKEND') != 'numpy':
raise SystemExit(
'This example currently only supports the numpy backend')
ambient_dim = 128
mat = gs.random.normal(size=(ambient_dim, ambient_dim))
mat = 0.5 * (mat + mat.T)
eigenvalues, eigenvectors = gs.linalg.eig(mat)
dominant_eigenvector = eigenvectors[:, gs.argmax(eigenvalues)]
dominant_eigenvector_estimate = estimate_dominant_eigenvector(mat)
if (gs.sign(dominant_eigenvector[0]) !=
gs.sign(dominant_eigenvector_estimate[0])):
dominant_eigenvector_estimate = -dominant_eigenvector_estimate
logging.info(
'l2-norm of dominant eigenvector: %s',
gs.linalg.norm(dominant_eigenvector))
logging.info(
'l2-norm of dominant eigenvector estimate: %s',
gs.linalg.norm(dominant_eigenvector_estimate))
error_norm = gs.linalg.norm(
dominant_eigenvector - dominant_eigenvector_estimate)
logging.info('l2-norm of difference vector: %s', error_norm)
logging.info('solution found: %s', gs.isclose(error_norm, 0.0, atol=1e-3))
def rotation_vector_from_matrix(self, rot_mat):
r"""Convert rotation matrix (in 3D) to rotation vector (axis-angle).
Get the angle through the trace of the rotation matrix:
The eigenvalues are:
:math:`\{1, \cos(angle) + i \sin(angle), \cos(angle) - i \sin(angle)\}`
so that:
:math:`trace = 1 + 2 \cos(angle), \{-1 \leq trace \leq 3\}`
Get the rotation vector through the formula:
:math:`S_r = \frac{angle}{(2 * \sin(angle) ) (R - R^T)}`
For the edge case where the angle is close to pi,
the formulation is derived by using the following equality (see the
Axis-angle representation on Wikipedia):
:math:`outer(r, r) = \frac{1}{2} (R + I_3)`
In nD, the rotation vector stores the :math:`n(n-1)/2` values
of the skew-symmetric matrix representing the rotation.
Parameters
----------
rot_mat : array-like, shape=[..., n, n]
Returns
-------
regularized_rot_vec : array-like, shape=[..., 3]
"""
n_rot_mats, _, _ = rot_mat.shape
trace = gs.trace(rot_mat, axis1=1, axis2=2)
trace = gs.to_ndarray(trace, to_ndim=2, axis=1)
trace_num = gs.clip(trace, -1, 3)
angle = gs.arccos(0.5 * (trace_num - 1))
rot_mat_transpose = gs.transpose(rot_mat, axes=(0, 2, 1))
rot_vec_not_pi = self.vector_from_skew_matrix(rot_mat -
rot_mat_transpose)
mask_0 = gs.cast(gs.isclose(angle, 0.), gs.float32)
mask_pi = gs.cast(gs.isclose(angle, gs.pi, atol=1e-2), gs.float32)
mask_else = (1 - mask_0) * (1 - mask_pi)
numerator = 0.5 * mask_0 + angle * mask_else
denominator = (1 - angle**2 /
6) * mask_0 + 2 * gs.sin(angle) * mask_else + mask_pi
rot_vec_not_pi = rot_vec_not_pi * numerator / denominator
vector_outer = 0.5 * (gs.eye(3) + rot_mat)
gs.set_diag(
vector_outer,
gs.maximum(0., gs.diagonal(vector_outer, axis1=1, axis2=2)))
squared_diag_comp = gs.diagonal(vector_outer, axis1=1, axis2=2)
diag_comp = gs.sqrt(squared_diag_comp)
norm_line = gs.linalg.norm(vector_outer, axis=2)
max_line_index = gs.argmax(norm_line, axis=1)
selected_line = gs.get_slice(vector_outer,
(range(n_rot_mats), max_line_index))
signs = gs.sign(selected_line)
rot_vec_pi = angle * signs * diag_comp
rot_vec = rot_vec_not_pi + mask_pi * rot_vec_pi
return self.regularize(rot_vec)
