class SquareMatrices(MatrixLieAlgebra):
"""Lie algebra of the general linear group.
This is the space of matrices.
Parameters
----------
n : int
Integer representing the shape of the matrices: n x n.
"""
def __init__(self, n):
super(SquareMatrices, self).__init__(n=n, dim=n**2)
self.mat_space = Matrices(n, n)
def _create_basis(self):
"""Create the canonical basis of the space of matrices."""
return self.mat_space.basis
def basis_representation(self, matrix_representation):
"""Compute the coefficient in the usual matrix basis.
This simply flattens the input.
Parameters
----------
matrix_representation : array-like, shape=[..., n, n]
Matrix.
Returns
-------
basis_representation : array-like, shape=[..., dim]
Representation in the basis.
"""
return self.mat_space.flatten(matrix_representation)
def matrix_representation(self, basis_representation):
"""Compute the matrix representation for the given basis coefficients.
This simply reshapes the input into a square matrix.
Parameters
----------
basis_representation : array-like, shape=[..., dim]
Coefficients in the basis.
Returns
-------
matrix_representation : array-like, shape=[..., n, n]
Matrix.
"""
return self.mat_space.reshape(basis_representation)
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_and_autograd_only
def test_mul(self):
a = gs.eye(3, 3, 1)
b = gs.eye(3, 3, -1)
c = gs.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 0.0]])
d = gs.array([[0.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]])
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_and_autograd_only
def test_bracket(self):
x = gs.array([[0.0, 0.0, 0.0], [0.0, 0.0, -1.0], [0.0, 1.0, 0.0]])
y = gs.array([[0.0, 0.0, 1.0], [0.0, 0.0, 0.0], [-1.0, 0.0, 0.0]])
z = gs.array([[0.0, -1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 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_and_autograd_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([[1.0, 2.0], [2.0, 1.0], [3.0, 1.0]])
result = self.space.is_symmetric(not_squared)
expected = False
self.assertAllClose(result, expected)
sym_mat = gs.array([[1.0, 2.0], [2.0, 1.0]])
result = self.space.is_symmetric(sym_mat)
expected = gs.array(True)
self.assertAllClose(result, expected)
not_a_sym_mat = gs.array([[1.0, 0.6, -3.0], [6.0, -7.0, 0.0],
[0.0, 7.0, 8.0]])
result = self.space.is_symmetric(not_a_sym_mat)
expected = gs.array(False)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_autograd_only
def test_is_skew_symmetric(self):
skew_mat = gs.array([[0, -2.0], [2.0, 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, 0.6, -3.0], [6.0, -7.0, 0.0],
[0.0, 7.0, 8.0]])
result = self.space.is_skew_symmetric(not_a_sym_mat)
expected = gs.array(False)
self.assertAllClose(result, expected)
@geomstats.tests.np_autograd_and_tf_only
def test_is_symmetric_vectorization(self):
points = gs.array([
[[1.0, 2.0], [2.0, 1.0]],
[[3.0, 4.0], [4.0, 5.0]],
[[1.0, 2.0], [3.0, 4.0]],
])
result = self.space.is_symmetric(points)
expected = [True, True, False]
self.assertAllClose(result, expected)
@geomstats.tests.np_autograd_and_torch_only
def test_make_symmetric(self):
sym_mat = gs.array([[1.0, 2.0], [2.0, 1.0]])
result = self.space.to_symmetric(sym_mat)
expected = sym_mat
self.assertAllClose(result, expected)
mat = gs.array([[1.0, 2.0, 3.0], [0.0, 0.0, 0.0], [3.0, 1.0, 1.0]])
result = self.space.to_symmetric(mat)
expected = gs.array([[1.0, 1.0, 3.0], [1.0, 0.0, 0.5], [3.0, 0.5,
1.0]])
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_autograd_and_tf_only
def test_make_symmetric_and_is_symmetric_vectorization(self):
points = gs.array([[[1.0, 2.0], [3.0, 4.0]], [[5.0, 6.0], [4.0, 9.0]]])
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([[1.0, 2.0, 3.0], [0.0, 0.0, 0.0],
[3.0, 1.0, 1.0]])
tangent_vector_1 = gs.array([[1.0, 2.0, 3.0], [0.0, -10.0, 0.0],
[30.0, 1.0, 1.0]])
tangent_vector_2 = gs.array([[1.0, 4.0, 3.0], [5.0, 0.0, 0.0],
[3.0, 1.0, 1.0]])
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([[1.0, 2.0, 3.0], [0.0, 0.0, 0.0],
[3.0, 1.0, 1.0]])
tangent_vector = gs.array([[1.0, 2.0, 3.0], [0.0, -10.0, 0.0],
[30.0, 1.0, 1.0]])
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([[1.0, 2.0, 3.0], [0.0, 0.0, 0.0],
[3.0, 1.0, 1.0]])
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)**0.5
self.assertAllClose(result, expected)
def test_flatten_reshape(self):
matrix_list = self.space_nonsquare.random_point(n_samples=1)
result = self.space_nonsquare.reshape(
self.space_nonsquare.flatten(matrix_list))
self.assertAllClose(result, matrix_list)
matrix_list = self.space_nonsquare.random_point(n_samples=2)
result = self.space_nonsquare.reshape(
self.space_nonsquare.flatten(matrix_list))
self.assertAllClose(result, matrix_list)
def test_diagonal(self):
mat = gs.eye(3)
result = Matrices.diagonal(mat)
expected = gs.ones(3)
self.assertAllClose(result, expected)
mat = gs.stack([mat] * 2)
result = Matrices.diagonal(mat)
expected = gs.ones((2, 3))
self.assertAllClose(result, expected)
def test_flatten_reshape(self, m, n, mat):
cls_mn = Matrices(m, n)
self.assertAllClose(cls_mn.reshape(cls_mn.flatten(gs.array(mat))),
gs.array(mat))
