class TestSphereSubspaceComplementIntersectionManifold(unittest.TestCase):
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) as w:
self.man = SphereSubspaceComplementIntersection(self.n, self.U)
def test_dim(self):
self.assertEqual(self.man.dim, 0)
def test_rand(self):
x = self.man.rand()
p = np.array([-1, 1]) / np.sqrt(2)
self.assertTrue(np.allclose(x, p) or np.allclose(x, -p))
def test_proj(self):
h = rnd.randn(self.n)
x = self.man.rand()
p = self.man.proj(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
man = SphereSubspaceComplementIntersection(3, 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(man.dim, 1)
# Check if a random element from the manifold has vanishing
# z-component.
x = man.rand()
np_testing.assert_almost_equal(x[-1], 0)
def test_dim_rand(self):
n = 100
U = rnd.randn(n, n // 3)
# By the rank-nullity theorem the orthogonal complement of span(U) has
# dimension n - rank(U).
dim = n - la.matrix_rank(U) - 1
man = SphereSubspaceComplementIntersection(n, U)
self.assertEqual(man.dim, dim)
# Test if a random element really lies in the left null space of U.
x = man.rand()
np_testing.assert_almost_equal(la.norm(x), 1)
np_testing.assert_array_almost_equal(U.T.dot(x), np.zeros(U.shape[1]))
def test_dim_1(self):
U = np.zeros((3, 1))
U[-1, -1] = 1
man = SphereSubspaceComplementIntersection(3, 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(man.dim, 1)
# Check if a random element from the manifold has vanishing
# z-component.
x = man.rand()
np_testing.assert_almost_equal(x[-1], 0)
def test_dim_rand(self):
n = 100
U = rnd.randn(n, n // 3)
# By the rank-nullity theorem the orthogonal complement of span(U) has
# dimension n - rank(U).
dim = n - la.matrix_rank(U) - 1
man = SphereSubspaceComplementIntersection(n, U)
self.assertEqual(man.dim, dim)
# Test if a random element really lies in the left null space of U.
x = man.rand()
np_testing.assert_almost_equal(la.norm(x), 1)
np_testing.assert_array_almost_equal(U.T.dot(x), np.zeros(U.shape[1]))