class TestMultiStiefelManifold(ManifoldTestCase):
def setUp(self):
self.m = m = 10
self.n = n = 3
self.k = k = 3
self.manifold = Stiefel(m, n, k=k)
self.manifold_polar = Stiefel(m, n, k=k, retraction="polar")
super().setUp()
def test_dim(self):
assert self.manifold.dim == 0.5 * self.k * self.n * (2 * self.m -
self.n - 1)
def test_typical_dist(self):
np_testing.assert_almost_equal(self.manifold.typical_dist,
np.sqrt(self.n * self.k))
def test_inner_product(self):
X = self.manifold.random_point()
A = self.manifold.random_tangent_vector(X)
B = self.manifold.random_tangent_vector(X)
np_testing.assert_allclose(np.sum(A * B),
self.manifold.inner_product(X, A, B))
def test_projection(self):
# Construct a random point X on the manifold.
X = self.manifold.random_point()
# Construct a vector H in the ambient space.
H = np.random.normal(size=(self.k, self.m, self.n))
# Compare the projections.
Hproj = H - X @ (multitransp(X) @ H + multitransp(H) @ X) / 2
np_testing.assert_allclose(Hproj, self.manifold.projection(X, H))
def test_first_order_function_approximation(self):
self.run_gradient_approximation_test()
def test_second_order_function_approximation(self):
self.run_hessian_approximation_test()
def test_random_point(self):
# Just make sure that things generated are on the manifold and that
# if you generate two they are not equal.
X = self.manifold.random_point()
np_testing.assert_allclose(multitransp(X) @ X,
multieye(self.k, self.n),
atol=1e-10)
Y = self.manifold.random_point()
assert np.linalg.norm(X - Y) > 1e-6
def test_random_tangent_vector(self):
# Make sure things generated are in tangent space and if you generate
# two then they are not equal.
X = self.manifold.random_point()
U = self.manifold.random_tangent_vector(X)
np_testing.assert_allclose(
multisym(multitransp(X) @ U),
np.zeros((self.k, self.n, self.n)),
atol=1e-10,
)
V = self.manifold.random_tangent_vector(X)
assert np.linalg.norm(U - V) > 1e-6
@params("manifold", "manifold_polar")
def test_retraction(self, manifold_attribute):
manifold = getattr(self, manifold_attribute)
# Test that the result is on the manifold and that for small
# tangent vectors it has little effect.
x = manifold.random_point()
u = manifold.random_tangent_vector(x)
xretru = manifold.retraction(x, u)
np_testing.assert_allclose(
multitransp(xretru) @ xretru,
multieye(self.k, self.n),
atol=1e-10,
)
u = u * 1e-6
xretru = manifold.retraction(x, u)
np_testing.assert_allclose(xretru, x + u)
def test_norm(self):
x = self.manifold.random_point()
u = self.manifold.random_tangent_vector(x)
np_testing.assert_almost_equal(self.manifold.norm(x, u),
np.linalg.norm(u))
def test_exp(self):
# Check that exp lies on the manifold and that exp of a small vector u
# is close to x + u.
s = self.manifold
x = s.random_point()
u = s.random_tangent_vector(x)
xexpu = s.exp(x, u)
np_testing.assert_allclose(
multitransp(xexpu) @ xexpu,
multieye(self.k, self.n),
atol=1e-10,
)
u = u * 1e-6
xexpu = s.exp(x, u)
np_testing.assert_allclose(xexpu, x + u)
class TestSingleStiefelManifold(ManifoldTestCase):
def setUp(self):
self.m = m = 20
self.n = n = 2
self.k = k = 1
self.manifold = Stiefel(m, n, k=k)
self.manifold_polar = Stiefel(m, n, k=k, retraction="polar")
self.projection = lambda x, u: u - x @ (x.T @ u + u.T @ x) / 2
super().setUp()
def test_dim(self):
assert self.manifold.dim == 0.5 * self.n * (2 * self.m - self.n - 1)
def test_inner_product(self):
X = np.linalg.qr(np.random.normal(size=(self.m, self.n)))[0]
A, B = np.random.normal(size=(2, self.m, self.n))
np_testing.assert_allclose(np.sum(A * B),
self.manifold.inner_product(X, A, B))
def test_projection(self):
# Construct a random point X on the manifold.
X = np.random.normal(size=(self.m, self.n))
X = np.linalg.qr(X)[0]
# Construct a vector H in the ambient space.
H = np.random.normal(size=(self.m, self.n))
# Compare the projections.
Hproj = H - X @ (X.T @ H + H.T @ X) / 2
np_testing.assert_allclose(Hproj, self.manifold.projection(X, H))
def test_first_order_function_approximation(self):
self.run_gradient_approximation_test()
def test_second_order_function_approximation(self):
self.run_hessian_approximation_test()
def test_random_point(self):
# Just make sure that things generated are on the manifold and that
# if you generate two they are not equal.
X = self.manifold.random_point()
np_testing.assert_allclose(X.T @ X, np.eye(self.n), atol=1e-10)
Y = self.manifold.random_point()
assert np.linalg.norm(X - Y) > 1e-6
def test_random_tangent_vector(self):
# Make sure things generated are in tangent space and if you generate
# two then they are not equal.
X = self.manifold.random_point()
U = self.manifold.random_tangent_vector(X)
np_testing.assert_allclose(multisym(X.T @ U),
np.zeros((self.n, self.n)),
atol=1e-10)
V = self.manifold.random_tangent_vector(X)
assert np.linalg.norm(U - V) > 1e-6
@params("manifold", "manifold_polar")
def test_retraction(self, manifold_attribute):
manifold = getattr(self, manifold_attribute)
# Test that the result is on the manifold and that for small
# tangent vectors it has little effect.
x = manifold.random_point()
u = manifold.random_tangent_vector(x)
xretru = manifold.retraction(x, u)
np_testing.assert_allclose(xretru.T @ xretru,
np.eye(self.n, self.n),
atol=1e-10)
u = u * 1e-6
xretru = manifold.retraction(x, u)
np_testing.assert_allclose(xretru, x + u)
def test_euclidean_to_riemannian_hessian(self):
# Test this function at some randomly generated point.
x = self.manifold.random_point()
u = self.manifold.random_tangent_vector(x)
egrad = np.random.normal(size=(self.m, self.n))
ehess = np.random.normal(size=(self.m, self.n))
np_testing.assert_allclose(
testing.euclidean_to_riemannian_hessian(self.projection)(x, egrad,
ehess, u),
self.manifold.euclidean_to_riemannian_hessian(x, egrad, ehess, u),
)
def test_norm(self):
x = self.manifold.random_point()
u = self.manifold.random_tangent_vector(x)
np_testing.assert_almost_equal(self.manifold.norm(x, u),
np.linalg.norm(u))
def test_exp(self):
# Check that exp lies on the manifold and that exp of a small vector u
# is close to x + u.
s = self.manifold
x = s.random_point()
u = s.random_tangent_vector(x)
xexpu = s.exp(x, u)
np_testing.assert_allclose(xexpu.T @ xexpu,
np.eye(self.n, self.n),
atol=1e-10)
u = u * 1e-6
xexpu = s.exp(x, u)
np_testing.assert_allclose(xexpu, x + u)