def test_exp_np(self):
vec = Matrices.bracket(pi_2 * r_y, gs.array([p_xy, p_yz]))
result = self.metric.exp(vec, gs.array([p_xy, p_yz]))
expected = gs.array([p_yz, p_xy])
self.assertAllClose(result, expected)
vec = Matrices.bracket(pi_2 * gs.array([r_y, r_z]),
gs.array([p_xy, p_yz]))
result = self.metric.exp(vec, gs.array([p_xy, p_yz]))
expected = gs.array([p_yz, p_xz])
self.assertAllClose(result, expected)
def curvature_at_identity(self, tangent_vec_a, tangent_vec_b,
tangent_vec_c):
r"""Compute the curvature at identity.
For three tangent vectors at identity :math: `x,y,z`,
the curvature is defined by
:math: `R(x, y)z = \nabla_{[x,y]}z
- \nabla_x\nabla_y z + \nabla_y\nabla_x z`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
curvature : array-like, shape=[..., n, n]
Tangent vector at identity.
"""
bracket = Matrices.bracket(tangent_vec_a, tangent_vec_b)
bracket_term = self.connection_at_identity(bracket, tangent_vec_c)
left_term = self.connection_at_identity(
tangent_vec_a,
self.connection_at_identity(tangent_vec_b, tangent_vec_c))
right_term = self.connection_at_identity(
tangent_vec_b,
self.connection_at_identity(tangent_vec_a, tangent_vec_c))
return bracket_term - left_term + right_term
def connection_at_identity(self, tangent_vec_a, tangent_vec_b):
r"""Compute the Levi-Civita connection at identity.
For two tangent vectors at identity :math: `x,y`, one can associate
left (respectively right) invariant vector fields :math: `\tilde{x},
\tilde{y}`. Then the vector :math: `(\nabla_\tilde{x}(\tilde{x}))_{
Id}` is computed using the lie bracket and the dual adjoint map. This
is a bilinear map that characterizes the connection [Gallier]_.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
nabla : array-like, shape=[..., n, n]
Tangent vector at identity.
References
----------
.. [Gallier] Gallier, Jean, and Jocelyn Quaintance. Differential
Geometry and Lie Groups: A Computational Perspective.
Geonger International Publishing, 2020.
https://doi.org/10.1007/978-3-030-46040-2.
"""
sign = 1. if self.left_or_right == 'left' else -1.
return sign / 2 * (Matrices.bracket(tangent_vec_a, tangent_vec_b) -
self.dual_adjoint(tangent_vec_a, tangent_vec_b) -
self.dual_adjoint(tangent_vec_b, tangent_vec_a))
def parallel_transport(self, tangent_vec_a, tangent_vec_b, base_point):
r"""Compute the parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector a
along the geodesic defined by :math: `t \mapsto exp_(base_point)(t*
tangent_vec_b)`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at base point to be transported.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at base point, along which the parallel transport
is computed.
base_point : array-like, shape=[..., n, n]
Point on the Grassmann manifold.
Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at `exp_(base_point)(tangent_vec_b)`.
References
----------
.. [BZA20] Bendokat, Thomas, Ralf Zimmermann, and P.-A. Absil.
“A Grassmann Manifold Handbook: Basic Geometry and
Computational Aspects.”
ArXiv:2011.13699 [Cs, Math], November 27, 2020.
http://arxiv.org/abs/2011.13699.
"""
expm = gs.linalg.expm
mul = Matrices.mul
rot = Matrices.bracket(base_point, -tangent_vec_b)
return mul(expm(rot), tangent_vec_a, expm(-rot))
def to_tangent(self, vector, base_point):
"""Project a vector to a tangent space of the manifold.
Compute the bracket (commutator) of the base_point with
the skew-symmetric part of vector.
Parameters
----------
vector : array-like, shape=[..., n, n]
Vector.
base_point : array-like, shape=[..., n, n]
Point on the manifold.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
"""
sym = Matrices.to_symmetric(vector)
return Matrices.bracket(base_point, Matrices.bracket(base_point, sym))
def exp_test_data(self):
smoke_data = [
dict(
n=3,
k=2,
tangent_vec=Matrices.bracket(pi_2 * r_y, gs.array([p_xy,
p_yz])),
base_point=gs.array([p_xy, p_yz]),
expected=gs.array([p_yz, p_xy]),
),
dict(
n=3,
k=2,
tangent_vec=Matrices.bracket(pi_2 * gs.array([r_y, r_z]),
gs.array([p_xy, p_yz])),
base_point=gs.array([p_xy, p_yz]),
expected=gs.array([p_yz, p_xz]),
),
]
return self.generate_tests(smoke_data)
def parallel_transport(self,
tangent_vec,
base_point,
tangent_vec_b=None,
end_point=None):
r"""Compute the 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)`.
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 Grassmann manifold. Point to transport from.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at base point, along which the parallel transport
is computed.
Optional, default: None
end_point : array-like, shape=[..., n, n]
Point on the Grassmann manifold to transport to. Unused if `tangent_vec_b`
is given.
Optional, default: None
Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at `exp_(base_point)(tangent_vec_b)`.
References
----------
.. [BZA20] Bendokat, Thomas, Ralf Zimmermann, and P.-A. Absil.
“A Grassmann Manifold Handbook: Basic Geometry and
Computational Aspects.”
ArXiv:2011.13699 [Cs, Math], November 27, 2020.
https://arxiv.org/abs/2011.13699.
"""
if tangent_vec_b is None:
if end_point is not None:
tangent_vec_b = self.log(end_point, base_point)
else:
raise ValueError(
"Either an end_point or a tangent_vec_b must be given to define the"
" geodesic along which to transport.")
expm = gs.linalg.expm
mul = Matrices.mul
rot = -Matrices.bracket(base_point, tangent_vec_b)
return mul(expm(rot), tangent_vec, expm(-rot))
def exp(self, tangent_vec, base_point, **kwargs):
"""Exponentiate the invariant vector field v from base point p.
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
`tangent_vec` is the bracket of a skew-symmetric matrix with the
base_point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
exp : array-like, shape=[..., n, n]
Riemannian exponential.
"""
expm = gs.linalg.expm
mul = Matrices.mul
rot = Matrices.bracket(base_point, -tangent_vec)
return mul(expm(rot), base_point, expm(-rot))
def structure_constant(self, tangent_vec_a, tangent_vec_b, tangent_vec_c):
r"""Compute the structure constant of the metric.
For three tangent vectors :math: `x, y, z` at identity,
compute :math: `<[x,y], z>`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
structure_constant : array-like, shape=[...,]
"""
bracket = Matrices.bracket(tangent_vec_a, tangent_vec_b)
return self.inner_product_at_identity(bracket, tangent_vec_c)
def log(self, point, base_point, **kwargs):
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.compose(sym2, sym1)
return Matrices.bracket(GLn.log(rot) / 2, base_point)
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)
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.]]])
result = gs.all(self.space.is_symmetric(points))
expected = True
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_bracket(self, mat_a, mat_b, expected):
self.assertAllClose(Matrices.bracket(gs.array(mat_a), gs.array(mat_b)),
gs.array(expected))
def test_log(self):
expected = Matrices.bracket(pi_4 * r_y, p_xy)
result = self.metric.log(self.metric.exp(expected, p_xy), p_xy)
self.assertTrue(self.space.is_tangent(result, p_xy))
self.assertAllClose(result, expected)
