class TestSphereSubspaceComplementIntersectionManifold(ManifoldTestCase):
def setUp(self):
self.n = 2
# Define the 1-sphere intersected with the 1-dimensional subspace
# orthogonal to the line passing through (1, 1) / sqrt(2). This creates
# a 0-dimensional manifold as it only consits of isolated points in
# R^2.
self.U = np.ones((self.n, 1)) / np.sqrt(2)
with warnings.catch_warnings(record=True):
self.manifold = SphereSubspaceComplementIntersection(self.U)
super().setUp()
def test_dim(self):
self.assertEqual(self.manifold.dim, 0)
def test_random_point(self):
x = self.manifold.random_point()
p = np.array([-1, 1]) / np.sqrt(2)
self.assertTrue(np.allclose(x, p) or np.allclose(x, -p))
def test_projection(self):
h = np.random.normal(size=self.n)
x = self.manifold.random_point()
p = self.manifold.projection(x, h)
# Since the manifold is 0-dimensional, the tangent at each point is
# simply the 0-dimensional space {0}.
np_testing.assert_array_almost_equal(p, np.zeros(self.n))
def test_dim_1(self):
U = np.zeros((3, 1))
U[-1, -1] = 1
manifold = SphereSubspaceComplementIntersection(U)
# U spans the z-axis with its orthogonal complement being the x-y
# plane, therefore the manifold consists of the 1-sphere in the x-y
# plane, and has dimension 1.
self.assertEqual(manifold.dim, 1)
# Check if a random element from the manifold has vanishing
# z-component.
x = manifold.random_point()
np_testing.assert_almost_equal(x[-1], 0)
def test_dim_rand(self):
n = 100
U = np.random.normal(size=(n, n // 3))
# By the rank-nullity theorem the orthogonal complement of span(U) has
# dimension n - rank(U).
dim = n - np.linalg.matrix_rank(U) - 1
manifold = SphereSubspaceComplementIntersection(U)
self.assertEqual(manifold.dim, dim)
# Test if a random element really lies in the left null space of U.
x = manifold.random_point()
np_testing.assert_almost_equal(np.linalg.norm(x), 1)
np_testing.assert_array_almost_equal(U.T @ x, np.zeros(U.shape[1]))
def test_dim_1(self):
U = np.zeros((3, 1))
U[-1, -1] = 1
manifold = SphereSubspaceComplementIntersection(U)
# U spans the z-axis with its orthogonal complement being the x-y
# plane, therefore the manifold consists of the 1-sphere in the x-y
# plane, and has dimension 1.
self.assertEqual(manifold.dim, 1)
# Check if a random element from the manifold has vanishing
# z-component.
x = manifold.random_point()
np_testing.assert_almost_equal(x[-1], 0)
def test_dim_rand(self):
n = 100
U = np.random.normal(size=(n, n // 3))
# By the rank-nullity theorem the orthogonal complement of span(U) has
# dimension n - rank(U).
dim = n - np.linalg.matrix_rank(U) - 1
manifold = SphereSubspaceComplementIntersection(U)
self.assertEqual(manifold.dim, dim)
# Test if a random element really lies in the left null space of U.
x = manifold.random_point()
np_testing.assert_almost_equal(np.linalg.norm(x), 1)
np_testing.assert_array_almost_equal(U.T @ x, np.zeros(U.shape[1]))