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 = Matrices.mul(end_point, inverse_base_point)
congruence_mat = gs.linalg.sqrtm(congruence_mat)
return Matrices.congruent(tangent_vec_a, congruence_mat)
def parallel_transport(self,
tangent_vec,
base_point,
direction=None,
end_point=None):
r"""Parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector
along the geodesic between two points `base_point` and `end_point`
or alternatively defined by :math:`t \mapsto exp_{(base\_point)}(
t*direction)`.
Denoting `tangent_vec_a` by `S`, `base_point` by `A`, and `end_point`
by `B` or `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 : array-like, shape=[..., n, n]
Tangent vector at base point to be transported.
base_point : array-like, shape=[..., n, n]
Point on the manifold of SPD matrices. Point to transport from
direction : array-like, shape=[..., n, n]
Tangent vector at base point, initial speed of the geodesic along
which the parallel transport is computed. Unused if `end_point` is given.
Optional, default: None.
end_point : array-like, shape=[..., n, n]
Point on the manifold of SPD matrices. Point to transport to.
Optional, default: None.
Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at exp_(base_point)(tangent_vec_b).
"""
if end_point is None:
end_point = self.exp(direction, base_point)
# compute B^1/2(B^-1/2 A B^-1/2)B^-1/2 instead of sqrtm(AB^-1)
sqrt_bp, inv_sqrt_bp = SymmetricMatrices.powerm(
base_point, [1.0 / 2, -1.0 / 2])
pdt = SymmetricMatrices.powerm(
Matrices.mul(inv_sqrt_bp, end_point, inv_sqrt_bp), 1.0 / 2)
congruence_mat = Matrices.mul(sqrt_bp, pdt, inv_sqrt_bp)
return Matrices.congruent(tangent_vec, congruence_mat)
def parallel_transport(self,
tangent_vec,
base_point,
direction=None,
end_point=None):
r"""Parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector
along the geodesic between two points `base_point` and `end_point`
or alternatively defined by :math:`t\mapsto exp_(base_point)(
t*direction)`.
Denoting `tangent_vec_a` by `S`, `base_point` by `A`, and `end_point`
by `B` or `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 : array-like, shape=[..., n, n]
Tangent vector at base point to be transported.
base_point : array-like, shape=[..., n, n]
Point on the manifold of SPD matrices. Point to transport from
direction : array-like, shape=[..., n, n]
Tangent vector at base point, initial speed of the geodesic along
which the parallel transport is computed. Unused if `end_point` is given.
Optional, default: None.
end_point : array-like, shape=[..., n, n]
Point on the manifold of SPD matrices. Point to transport to.
Optional, default: None.
Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at exp_(base_point)(tangent_vec_b).
"""
if end_point is None:
end_point = self.exp(direction, base_point)
inverse_base_point = GeneralLinear.inverse(base_point)
congruence_mat = Matrices.mul(end_point, inverse_base_point)
congruence_mat = gs.linalg.sqrtm(congruence_mat)
return Matrices.congruent(tangent_vec, congruence_mat)
class TestMatrices(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.m = 2
self.n = 3
self.space = Matrices(m=self.n, n=self.n)
self.space_nonsquare = Matrices(m=self.m, n=self.n)
self.metric = self.space.metric
self.n_samples = 2
@geomstats.tests.np_only
def test_mul(self):
a = gs.eye(3, 3, 1)
b = gs.eye(3, 3, -1)
c = gs.array([
[1., 0., 0.],
[0., 1., 0.],
[0., 0., 0.]])
d = gs.array([
[0., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
result = self.space.mul([a, b], [b, a])
expected = gs.array([c, d])
self.assertAllClose(result, expected)
result = self.space.mul(a, [a, b])
expected = gs.array([gs.eye(3, 3, 2), c])
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_bracket(self):
x = gs.array([
[0., 0., 0.],
[0., 0., -1.],
[0., 1., 0.]])
y = gs.array([
[0., 0., 1.],
[0., 0., 0.],
[-1., 0., 0.]])
z = gs.array([
[0., -1., 0.],
[1., 0., 0.],
[0., 0., 0.]])
result = self.space.bracket([x, y], [y, z])
expected = gs.array([z, x])
self.assertAllClose(result, expected)
result = self.space.bracket(x, [x, y, z])
expected = gs.array([gs.zeros((3, 3)), z, -y])
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_transpose(self):
tr = self.space.transpose
ar = gs.array
a = gs.eye(3, 3, 1)
b = gs.eye(3, 3, -1)
self.assertAllClose(tr(a), b)
self.assertAllClose(tr(ar([a, b])), ar([b, a]))
def test_is_symmetric(self):
not_squared = gs.array([[1., 2.], [2., 1.], [3., 1.]])
result = self.space.is_symmetric(not_squared)
expected = False
self.assertAllClose(result, expected)
sym_mat = gs.array([[1., 2.], [2., 1.]])
result = self.space.is_symmetric(sym_mat)
expected = gs.array(True)
self.assertAllClose(result, expected)
not_a_sym_mat = gs.array([[1., 0.6, -3.],
[6., -7., 0.],
[0., 7., 8.]])
result = self.space.is_symmetric(not_a_sym_mat)
expected = gs.array(False)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_is_skew_symmetric(self):
skew_mat = gs.array([[0, - 2.],
[2., 0]])
result = self.space.is_skew_symmetric(skew_mat)
expected = gs.array(True)
self.assertAllClose(result, expected)
not_a_sym_mat = gs.array([[1., 0.6, -3.],
[6., -7., 0.],
[0., 7., 8.]])
result = self.space.is_skew_symmetric(not_a_sym_mat)
expected = gs.array(False)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_is_symmetric_vectorization(self):
points = gs.array([
[[1., 2.],
[2., 1.]],
[[3., 4.],
[4., 5.]],
[[1., 2.],
[3., 4.]]])
result = self.space.is_symmetric(points)
expected = [True, True, False]
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_make_symmetric(self):
sym_mat = gs.array([[1., 2.],
[2., 1.]])
result = self.space.to_symmetric(sym_mat)
expected = sym_mat
self.assertAllClose(result, expected)
mat = gs.array([[1., 2., 3.],
[0., 0., 0.],
[3., 1., 1.]])
result = self.space.to_symmetric(mat)
expected = gs.array([[1., 1., 3.],
[1., 0., 0.5],
[3., 0.5, 1.]])
self.assertAllClose(result, expected)
mat = gs.array([[1e100, 1e-100, 1e100],
[1e100, 1e-100, 1e100],
[1e-100, 1e-100, 1e100]])
result = self.space.to_symmetric(mat)
res = 0.5 * (1e100 + 1e-100)
expected = gs.array([[1e100, res, res],
[res, 1e-100, res],
[res, res, 1e100]])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_make_symmetric_and_is_symmetric_vectorization(self):
points = gs.array([
[[1., 2.],
[3., 4.]],
[[5., 6.],
[4., 9.]]])
sym_points = self.space.to_symmetric(points)
result = gs.all(self.space.is_symmetric(sym_points))
expected = True
self.assertAllClose(result, expected)
def test_inner_product(self):
base_point = gs.array([
[1., 2., 3.],
[0., 0., 0.],
[3., 1., 1.]])
tangent_vector_1 = gs.array([
[1., 2., 3.],
[0., -10., 0.],
[30., 1., 1.]])
tangent_vector_2 = gs.array([
[1., 4., 3.],
[5., 0., 0.],
[3., 1., 1.]])
result = self.metric.inner_product(
tangent_vector_1,
tangent_vector_2,
base_point=base_point)
expected = gs.trace(
gs.matmul(
gs.transpose(tangent_vector_1),
tangent_vector_2))
self.assertAllClose(result, expected)
def test_cong(self):
base_point = gs.array([
[1., 2., 3.],
[0., 0., 0.],
[3., 1., 1.]])
tangent_vector = gs.array([
[1., 2., 3.],
[0., -10., 0.],
[30., 1., 1.]])
result = self.space.congruent(tangent_vector, base_point)
expected = gs.matmul(
tangent_vector, gs.transpose(base_point))
expected = gs.matmul(base_point, expected)
self.assertAllClose(result, expected)
def test_belongs(self):
base_point_square = gs.zeros((self.n, self.n))
base_point_nonsquare = gs.zeros((self.m, self.n))
result = self.space.belongs(base_point_square)
expected = True
self.assertAllClose(result, expected)
result = self.space_nonsquare.belongs(base_point_square)
expected = False
self.assertAllClose(result, expected)
result = self.space.belongs(base_point_nonsquare)
expected = False
self.assertAllClose(result, expected)
result = self.space_nonsquare.belongs(base_point_nonsquare)
expected = True
self.assertAllClose(result, expected)
result = self.space.belongs(gs.zeros((2, 2, 3)))
self.assertFalse(gs.all(result))
result = self.space.belongs(gs.zeros((2, 3, 3)))
self.assertTrue(gs.all(result))
def test_is_diagonal(self):
base_point = gs.array([
[1., 2., 3.],
[0., 0., 0.],
[3., 1., 1.]])
result = self.space.is_diagonal(base_point)
expected = False
self.assertAllClose(result, expected)
diagonal = gs.eye(3)
result = self.space.is_diagonal(diagonal)
self.assertTrue(result)
base_point = gs.stack([base_point, diagonal])
result = self.space.is_diagonal(base_point)
expected = gs.array([False, True])
self.assertAllClose(result, expected)
base_point = gs.stack([diagonal] * 2)
result = self.space.is_diagonal(base_point)
self.assertTrue(gs.all(result))
base_point = gs.reshape(gs.arange(6), (2, 3))
result = self.space.is_diagonal(base_point)
self.assertTrue(~result)
def test_norm(self):
for n_samples in [1, 2]:
mat = self.space.random_point(n_samples)
result = self.metric.norm(mat)
expected = self.space.frobenius_product(mat, mat) ** .5
self.assertAllClose(result, expected)
def test_congruent(self, mat_a, mat_b, expected):
self.assertAllClose(
Matrices.congruent(gs.array(mat_a), gs.array(mat_b)),
gs.array(expected))
class TestMatricesMethods(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.n = 3
self.space = Matrices(m=self.n, n=self.n)
self.metric = self.space.metric
self.n_samples = 2
@geomstats.tests.np_only
def test_mul(self):
a = gs.eye(3, 3, 1)
b = gs.eye(3, 3, -1)
c = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 0.]])
d = gs.array([[0., 0., 0.], [0., 1., 0.], [0., 0., 1.]])
result = self.space.mul([a, b], [b, a])
expected = gs.array([c, d])
self.assertAllClose(result, expected)
result = self.space.mul(a, [a, b])
expected = gs.array([gs.eye(3, 3, 2), c])
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_bracket(self):
x = gs.array([[0., 0., 0.], [0., 0., -1.], [0., 1., 0.]])
y = gs.array([[0., 0., 1.], [0., 0., 0.], [-1., 0., 0.]])
z = gs.array([[0., -1., 0.], [1., 0., 0.], [0., 0., 0.]])
result = self.space.bracket([x, y], [y, z])
expected = gs.array([z, x])
self.assertAllClose(result, expected)
result = self.space.bracket(x, [x, y, z])
expected = gs.array([gs.zeros((3, 3)), z, -y])
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_transpose(self):
tr = self.space.transpose
ar = gs.array
a = gs.eye(3, 3, 1)
b = gs.eye(3, 3, -1)
self.assertAllClose(tr(a), b)
self.assertAllClose(tr(ar([a, b])), ar([b, a]))
@geomstats.tests.np_only
def test_is_symmetric(self):
sym_mat = gs.array([[1., 2.], [2., 1.]])
result = self.space.is_symmetric(sym_mat)
expected = gs.array(True)
self.assertAllClose(result, expected)
not_a_sym_mat = gs.array([[1., 0.6, -3.], [6., -7., 0.], [0., 7., 8.]])
result = self.space.is_symmetric(not_a_sym_mat)
expected = gs.array(False)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_is_symmetric_vectorization(self):
points = gs.array([[[1., 2.], [2., 1.]], [[3., 4.], [4., 5.]],
[[1., 2.], [3., 4.]]])
result = self.space.is_symmetric(points)
expected = [True, True, False]
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_make_symmetric(self):
sym_mat = gs.array([[1., 2.], [2., 1.]])
result = self.space.make_symmetric(sym_mat)
expected = sym_mat
self.assertAllClose(result, expected)
mat = gs.array([[1., 2., 3.], [0., 0., 0.], [3., 1., 1.]])
result = self.space.make_symmetric(mat)
expected = gs.array([[1., 1., 3.], [1., 0., 0.5], [3., 0.5, 1.]])
self.assertAllClose(result, expected)
mat = gs.array([[1e100, 1e-100, 1e100], [1e100, 1e-100, 1e100],
[1e-100, 1e-100, 1e100]])
result = self.space.make_symmetric(mat)
res = 0.5 * (1e100 + 1e-100)
expected = gs.array([[1e100, res, res], [res, 1e-100, res],
[res, res, 1e100]])
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_make_symmetric_and_is_symmetric_vectorization(self):
points = gs.array([[[1., 2.], [3., 4.]], [[5., 6.], [4., 9.]]])
sym_points = self.space.make_symmetric(points)
result = gs.all(self.space.is_symmetric(sym_points))
expected = True
self.assertAllClose(result, expected)
def test_inner_product(self):
base_point = gs.array([[1., 2., 3.], [0., 0., 0.], [3., 1., 1.]])
tangent_vector_1 = gs.array([[1., 2., 3.], [0., -10., 0.],
[30., 1., 1.]])
tangent_vector_2 = gs.array([[1., 4., 3.], [5., 0., 0.], [3., 1., 1.]])
result = self.metric.inner_product(tangent_vector_1,
tangent_vector_2,
base_point=base_point)
expected = gs.trace(
gs.matmul(gs.transpose(tangent_vector_1), tangent_vector_2))
expected = helper.to_scalar(expected)
self.assertAllClose(result, expected)
def test_cong(self):
base_point = gs.array([[1., 2., 3.], [0., 0., 0.], [3., 1., 1.]])
tangent_vector = gs.array([[1., 2., 3.], [0., -10., 0.], [30., 1.,
1.]])
result = self.space.congruent(tangent_vector, base_point)
expected = gs.matmul(tangent_vector, gs.transpose(base_point))
expected = gs.matmul(base_point, expected)
self.assertAllClose(result, expected)
