def _log_translation_transform(self, rot_vec):
exp_transform = self._exp_translation_transform(rot_vec)
inv_determinant = .5 / utils.taylor_exp_even_func(
rot_vec**2, utils.cosc_close_0, order=4)
transform = gs.einsum('...l, ...jk -> ...jk', inv_determinant,
GeneralLinear.transpose(exp_transform))
return transform
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 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\'.')
def tangent_submersion(tangent_vec, base_point):
product = GeneralLinear.mul(base_point,
GeneralLinear.transpose(tangent_vec))
return 2 * GeneralLinear.to_symmetric(product)
def submersion(point):
return GeneralLinear.mul(point, GeneralLinear.transpose(point))
def _to_lie_algebra(self, tangent_vec):
"""Project vector rotation part onto skew-symmetric matrices."""
tangent_vec = tangent_vec * gs.where(self.translation_mask != 0.,
gs.array(1.), gs.array(0.))
tangent_vec = (tangent_vec - GeneralLinear.transpose(tangent_vec)) / 2.
return tangent_vec * self.translation_mask