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 log(self, point, base_point):
r"""Compute the Riemannian logarithm of point w.r.t. base_point.
Given :math:`P, P'` in Gr(n, k) the logarithm from :math:`P`
to :math:`P` is given by the infinitesimal rotation [Batzies2015]_:
.. math::
\omega = \frac 1 2 \log \big((2 P' - 1)(2 P - 1)\big)
Parameters
----------
point : array-like, shape=[n_samples, n, n]
Point in the Grassmannian.
base_point : array-like, shape=[n_samples, n, n]
Point in the Grassmannian.
Returns
-------
tangent_vec : array-like, shape=[n_samples, n, n]
Tangent vector at `base_point`.
References
----------
.. [Batzies2015] Batzies, Hüper, Machado, Leite.
"Geometric Mean and Geodesic Regression on Grassmannians"
Linear Algebra and its Applications, 466, 83-101, 2015.
"""
GLn = GeneralLinear(self.n)
id_n = GLn.identity()
sym2 = 2 * point - id_n
sym1 = 2 * base_point - id_n
rot = GLn.mul(sym2, sym1)
return GLn.log(rot) / 2
def test_horizontal_projection(self):
mat = self.bundle.total_space.random_uniform()
vec = self.bundle.total_space.random_uniform()
horizontal_vec = self.bundle.horizontal_projection(vec, mat)
product = GeneralLinear.mul(horizontal_vec, GeneralLinear.inverse(mat))
is_horizontal = GeneralLinear.is_symmetric(product)
self.assertTrue(is_horizontal)
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
def log(self, point, base_point):
r"""Compute the Riemannian logarithm of point w.r.t. base_point.
Given :math:`P, P'` in Gr(n, k) the logarithm from :math:`P`
to :math:`P` is induced by the infinitesimal rotation [Batzies2015]_:
.. math::
Y = \frac 1 2 \log \big((2 P' - 1)(2 P - 1)\big)
The tangent vector :math:`X` at :math:`P`
is then recovered by :math:`X = [Y, P]`.
Parameters
----------
point : array-like, shape=[..., n, n]
Point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Riemannian logarithm, a tangent vector at `base_point`.
References
----------
.. [Batzies2015] Batzies, Hüper, Machado, Leite.
"Geometric Mean and Geodesic Regression on Grassmannians"
Linear Algebra and its Applications, 466, 83-101, 2015.
"""
GLn = GeneralLinear(self.n)
id_n = GLn.identity
id_n, point, base_point = gs.convert_to_wider_dtype(
[id_n, point, base_point])
sym2 = 2 * point - id_n
sym1 = 2 * base_point - id_n
rot = GLn.mul(sym2, sym1)
return Matrices.bracket(GLn.log(rot) / 2, base_point)
def horizontal_lift(tangent_vec, point, base_point=None):
if base_point is None:
base_point = submersion(point)
sylvester = gs.linalg.solve_sylvester(base_point, base_point,
tangent_vec)
return GeneralLinear.mul(sylvester, point)
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))