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 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 projection(self, point):
r"""Project a matrix to the general linear group.
As GL(n) is dense in the space of matrices, this is not a projection
per se, but a regularization if the matrix is not already invertible:
:math:`X + \epsilon I_n` is returned where :math:`\epsilon=gs.atol`
is returned for an input X.
Parameters
----------
point : array-like, shape=[..., dim_embedding]
Point in embedding manifold.
Returns
-------
projected : array-like, shape=[..., dim_embedding]
Projected point.
"""
belongs = self.belongs(point)
regularization = gs.einsum("...,ij->...ij",
gs.where(~belongs, gs.atol, 0.0),
self.identity)
projected = point + regularization
if self.positive_det:
det = gs.linalg.det(point)
return utils.flip_determinant(projected, det)
return projected