def reshape_metric_matrix(self):
"""Reshape diagonal metric matrix to a symmetric matrix of size n.
Reshape a diagonal metric matrix of size `dim x dim` into a symmetric
matrix of size `n x n` where :math: `dim= n (n -1) / 2` is the
dimension of the space of skew symmetric matrices. The
non-diagonal coefficients in the output matrix correspond to the
basis matrices of this space. The diagonal is filled with ones.
This useful to compute a matrix inner product.
Parameters
----------
metric_matrix : array-like, shape=[dim, dim]
Diagonal metric matrix.
Returns
-------
symmetric_matrix : array-like, shape=[n, n]
Symmetric matrix.
"""
if Matrices.is_diagonal(self.metric_mat_at_identity):
metric_coeffs = gs.diagonal(self.metric_mat_at_identity)
metric_mat = gs.abs(
self.lie_algebra.matrix_representation(metric_coeffs))
return metric_mat
raise ValueError('This is only possible for a diagonal matrix')
def inner_product_at_identity(self, tangent_vec_a, tangent_vec_b):
"""Compute inner product at tangent space at identity.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
First tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Second tangent vector at identity.
Returns
-------
inner_prod : array-like, shape=[...]
Inner-product of the two tangent vectors.
"""
tan_b = tangent_vec_b
metric_mat = self.metric_mat_at_identity
if (Matrices.is_diagonal(metric_mat) and self.lie_algebra is not None):
tan_b = tangent_vec_b * self.reshaped_metric_matrix
inner_prod = Matrices.frobenius_product(tangent_vec_a, tan_b)
return inner_prod
def inner_product_at_identity(self, tangent_vec_a, tangent_vec_b):
"""Compute inner product at tangent space at identity.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
First tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Second tangent vector at identity.
Returns
-------
inner_prod : array-like, shape=[...]
Inner-product of the two tangent vectors.
"""
aux_prod = tangent_vec_a * tangent_vec_b
metric_mat = self.metric_mat_at_identity
if (Matrices.is_diagonal(metric_mat) and self.lie_algebra is not None):
aux_prod *= self.reshaped_metric_matrix
inner_prod = gs.sum(aux_prod, axis=(-2, -1))
return inner_prod
def inner_product_at_identity(self, tangent_vec_a, tangent_vec_b):
"""Compute inner product at tangent space at identity.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim]
First tangent vector at identity.
tangent_vec_b : array-like, shape=[..., dim]
Second tangent vector at identity.
Returns
-------
inner_prod : array-like, shape=[..., dim]
Inner-product of the two tangent vectors.
"""
geomstats.errors.check_parameter_accepted_values(
self.group.default_point_type, 'default_point_type',
['vector', 'matrix'])
if self.group.default_point_type == 'vector':
inner_product_mat_at_identity = self.metric_mat_at_identity
inner_prod = gs.einsum('...i,...ij->...j', tangent_vec_a,
inner_product_mat_at_identity)
inner_prod = gs.einsum('...j,...j->...', inner_prod, tangent_vec_b)
else:
is_vectorized = \
(gs.ndim(tangent_vec_a) == 3) or (gs.ndim(tangent_vec_b) == 3)
axes = (2, 1) if is_vectorized else (0, 1)
aux_prod = tangent_vec_a * tangent_vec_b
metric_mat = self.metric_mat_at_identity
if (Matrices.is_diagonal(metric_mat)
and self.lie_algebra is not None):
aux_prod *= self.lie_algebra.reshape_metric_matrix(metric_mat)
inner_prod = gs.sum(aux_prod, axis=axes)
return inner_prod
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)
