def parallel_transport(self, tangent_vec_a, tangent_vec_b, base_point):
r"""Parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector a
along the geodesic defined by exp_(base_point)(tangent_vec_b).
Denoting `tangent_vec_a` by `S`, `base_point` by `A`, let
`B = Exp_A(tangent_vec_b)` and :math: `E = (BA^{- 1})^({ 1 / 2})`.
Then the
parallel transport to `B`is:
..math::
S' = ESE^T
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim + 1]
Tangent vector at base point to be transported.
tangent_vec_b : array-like, shape=[..., dim + 1]
Tangent vector at base point, initial speed of the geodesic along
which the parallel transport is computed.
base_point : array-like, shape=[..., dim + 1]
Point on the manifold of SPD matrices.
Returns
-------
transported_tangent_vec: array-like, shape=[..., dim + 1]
Transported tangent vector at exp_(base_point)(tangent_vec_b).
"""
end_point = self.exp(tangent_vec_b, base_point)
inverse_base_point = GeneralLinear.inverse(base_point)
congruence_mat = GeneralLinear.mul(end_point, inverse_base_point)
congruence_mat = gs.linalg.sqrtm(congruence_mat)
return GeneralLinear.congruent(tangent_vec_a, congruence_mat)
def transp(self, base_point, end_point, tangent):
"""
transports a tangent vector at a base_point
to the tangent space at end_point.
"""
if self.exact_transport:
# https://github.com/geomstats/geomstats/blob/master/geomstats/geometry/spd_matrices.py#L613
inverse_base_point = GeneralLinear.inverse(base_point)
congruence_mat = GeneralLinear.mul(end_point, inverse_base_point)
congruence_mat = gs.linalg.sqrtm(congruence_mat.cpu()).to(tangent)
return GeneralLinear.congruent(tangent, congruence_mat)
# https://github.com/NicolasBoumal/manopt/blob/master/manopt/manifolds/symfixedrank/sympositivedefinitefactory.m#L181
return tangent