def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.manifold = FixedRankEmbedded(m, n, k)
u, s, vt = self.manifold.random_point()
matrix = (u * s) @ vt
@pymanopt.function.autograd(self.manifold)
def cost(u, s, vt):
return np.linalg.norm((u * s) @ vt - matrix) ** 2
self.cost = cost
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
m, n, rank = 5, 4, 2
matrix = rnd.randn(m, n)
cost, egrad = create_cost_egrad(backend, matrix, rank)
manifold = FixedRankEmbedded(m, n, rank)
problem = pymanopt.Problem(manifold, cost=cost, egrad=egrad)
if quiet:
problem.verbosity = 0
solver = ConjugateGradient()
left_singular_vectors, singular_values, right_singular_vectors = \
solver.solve(problem)
low_rank_approximation = (left_singular_vectors @ np.diag(singular_values)
@ right_singular_vectors)
if not quiet:
u, s, vt = la.svd(matrix, full_matrices=False)
indices = np.argsort(s)[-rank:]
low_rank_solution = (
u[:, indices] @ np.diag(s[indices]) @ vt[indices, :])
print("Analytic low-rank solution:")
print()
print(low_rank_solution)
print()
print("Rank-{} approximation:".format(rank))
print()
print(low_rank_approximation)
print()
print("Frobenius norm error:",
la.norm(low_rank_approximation - low_rank_solution))
print()
def fixedrank(A, YT, r):
""" Solves the AX=YT problem on the manifold of r-rank matrices with
"""
# Instantiate a manifold
manifold = FixedRankEmbedded(N, r, r)
# Define the cost function (here using autograd.numpy)
def cost(X):
U = X[0]
cst = 0
for n in range(N):
cst = cst + huber(U[n, :])
Mat = np.matmul(np.matmul(X[0], np.diag(X[1])), X[2])
fidelity = LA.norm(np.subtract(np.matmul(A, Mat), YT))
return cst + lambd * fidelity**2
problem = Problem(manifold=manifold, cost=cost)
solver = ConjugateGradient(maxiter=maxiter)
# Let Pymanopt do the rest
Xopt = solver.solve(problem)
#Solve
Sol = np.dot(np.dot(Xopt[0], np.diag(Xopt[1])), Xopt[2])
return Sol
def setUp(self):
self.m = m = 20
self.n = n = 10
self.rank = rank = 3
A = np.random.normal(size=(m, n))
self.manifold = Product([FixedRankEmbedded(m, n, rank), Euclidean(n)])
@pymanopt.function.autograd(self.manifold)
def cost(u, s, vt, x):
return np.linalg.norm(((u * s) @ vt - A) @ x) ** 2
self.cost = cost
self.gradient = self.cost.get_gradient_operator()
self.hessian = self.cost.get_hessian_operator()
self.problem = pymanopt.Problem(self.manifold, self.cost)
def setUp(self):
self.m = m = 20
self.n = n = 10
self.rank = rank = 3
A = np.random.randn(m, n)
@pymanopt.function.Autograd
def cost(u, s, vt, x):
return np.linalg.norm(((u * s) @ vt - A) @ x)**2
self.cost = cost
self.gradient = self.cost.compute_gradient()
self.hvp = self.cost.compute_hessian_vector_product()
self.manifold = Product([FixedRankEmbedded(m, n, rank), Euclidean(n)])
self.problem = pymanopt.Problem(self.manifold, self.cost)
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
m, n, rank = 5, 4, 2
matrix = np.random.normal(size=(m, n))
manifold = FixedRankEmbedded(m, n, rank)
cost, euclidean_gradient = create_cost_and_derivates(
manifold, matrix, backend
)
problem = pymanopt.Problem(
manifold, cost, euclidean_gradient=euclidean_gradient
)
optimizer = ConjugateGradient(
verbosity=2 * int(not quiet), beta_rule="PolakRibiere"
)
(
left_singular_vectors,
singular_values,
right_singular_vectors,
) = optimizer.run(problem).point
low_rank_approximation = (
left_singular_vectors
@ np.diag(singular_values)
@ right_singular_vectors
)
if not quiet:
u, s, vt = np.linalg.svd(matrix, full_matrices=False)
indices = np.argsort(s)[-rank:]
low_rank_solution = (
u[:, indices] @ np.diag(s[indices]) @ vt[indices, :]
)
print("Analytic low-rank solution:")
print()
print(low_rank_solution)
print()
print(f"Rank-{rank} approximation:")
print()
print(low_rank_approximation)
print()
print(
"Frobenius norm error:",
np.linalg.norm(low_rank_approximation - low_rank_solution),
)
print()
import autograd.numpy as np
from pymanopt import Problem
from pymanopt.solvers import ConjugateGradient
# Let A be a (5 x 4) matrix to be approximated
A = np.random.randn(5, 4)
k = 2
# (a) Instantiation of a manifold
# points on the manifold are parameterized via their singular value
# decomposition (u, s, vt) where
# u is a 5 x 2 matrix with orthonormal columns,
# s is a vector of length 2,
# vt is a 2 x 4 matrix with orthonormal rows,
# so that u*diag(s)*vt is a 5 x 4 matrix of rank 2.
manifold = FixedRankEmbedded(A.shape[0], A.shape[1], k)
# (b) Definition of a cost function (here using autograd.numpy)
# Note that the cost must be defined in terms of u, s and vt, where
# X = u * diag(s) * vt.
def cost(usv):
delta = .5
u = usv[0]
s = usv[1]
vt = usv[2]
X = np.dot(np.dot(u, np.diag(s)), vt)
return np.sum(np.sqrt((X - A)**2 + delta**2) - delta)
# define the Pymanopt problem
class TestFixedRankEmbeddedManifold(unittest.TestCase):
def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.man = FixedRankEmbedded(m, n, k)
def test_dim(self):
assert self.man.dim == (self.m + self.n - self.k) * self.k
def test_typicaldist(self):
assert self.man.dim == self.man.typicaldist
def test_dist(self):
e = self.man
a = e.rand()
x = e.randvec(a)
y = e.randvec(a)
with self.assertRaises(NotImplementedError):
e.dist(x, y)
def test_inner(self):
e = self.man
x = e.rand()
a = e.randvec(x)
b = e.randvec(x)
# First embed in the ambient space
A = x[0].dot(a[1].dot(x[2])) + a[0].dot(x[2]) + x[0].dot(a[2].T)
B = x[0].dot(b[1].dot(x[2])) + b[0].dot(x[2]) + x[0].dot(b[2].T)
trueinner = np.sum(A * B)
np_testing.assert_almost_equal(trueinner, e.inner(x, a, b))
def test_proj_range(self):
m = self.man
x = m.rand()
v = np.random.randn(self.m, self.n)
g = m.proj(x, v)
# Check that g is a true tangent vector
np_testing.assert_allclose(np.dot(g[0].T, x[0]),
np.zeros((self.k, self.k)),
atol=1e-6)
np_testing.assert_allclose(np.dot(g[2].T, x[2].T),
np.zeros((self.k, self.k)),
atol=1e-6)
def test_proj(self):
# Verify that proj gives the closest point within the tangent space
# by displacing the result slightly and checking that this increases
# the distance.
m = self.man
x = self.man.rand()
v = np.random.randn(self.m, self.n)
g = m.proj(x, v)
# Displace g a little
g_disp = g + 0.01 * m.randvec(x)
# Return to the ambient representation
g = m.tangent2ambient(x, g)
g_disp = m.tangent2ambient(x, g_disp)
g = g[0].dot(g[1]).dot(g[2].T)
g_disp = g_disp[0].dot(g_disp[1]).dot(g_disp[2].T)
assert np.linalg.norm(g - v) < np.linalg.norm(g_disp - v)
def test_proj_tangents(self):
# Verify that proj leaves tangent vectors unchanged
e = self.man
x = e.rand()
u = e.randvec(x)
A = e.proj(x, e.tangent2ambient(x, u))
B = u
# diff = [A[k]-B[k] for k in xrange(len(A))]
np_testing.assert_allclose(A[0], B[0])
np_testing.assert_allclose(A[1], B[1])
np_testing.assert_allclose(A[2], B[2])
def test_norm(self):
e = self.man
x = e.rand()
u = e.randvec(x)
np_testing.assert_almost_equal(np.sqrt(e.inner(x, u, u)), e.norm(x, u))
def test_rand(self):
e = self.man
x = e.rand()
y = e.rand()
assert np.shape(x[0]) == (self.m, self.k)
assert np.shape(x[1]) == (self.k, )
assert np.shape(x[2]) == (self.k, self.n)
np_testing.assert_allclose(x[0].T.dot(x[0]), np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(x[2].dot(x[2].T), np.eye(self.k), atol=1e-6)
assert la.norm(x[0] - y[0]) > 1e-6
assert la.norm(x[1] - y[1]) > 1e-6
assert la.norm(x[2] - y[2]) > 1e-6
def test_transp(self):
s = self.man
x = s.rand()
y = s.rand()
u = s.randvec(x)
A = s.transp(x, y, u)
B = s.proj(y, s.tangent2ambient(x, u))
diff = [A[k] - B[k] for k in range(len(A))]
np_testing.assert_almost_equal(s.norm(y, diff), 0)
def test_apply_ambient(self):
m = self.man
z = np.random.randn(self.m, self.n)
# Set u, s, v so that z = u.dot(s).dot(v.T)
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.randn(self.n, self.n)
np_testing.assert_allclose(z.dot(w), m._apply_ambient(z, w))
np_testing.assert_allclose(z.dot(w), m._apply_ambient((u, s, v), w))
def test_apply_ambient_transpose(self):
m = self.man
z = np.random.randn(self.n, self.m)
# Set u, s, v so that z = u.dot(s).dot(v.T)
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.randn(self.n, self.n)
np_testing.assert_allclose(z.T.dot(w),
m._apply_ambient_transpose(z, w))
np_testing.assert_allclose(z.T.dot(w),
m._apply_ambient_transpose((u, s, v), w))
def test_tangent2ambient(self):
m = self.man
x = m.rand()
z = m.randvec(x)
z_ambient = (x[0].dot(z[1]).dot(x[2]) + z[0].dot(x[2]) +
x[0].dot(z[2].T))
u, s, v = m.tangent2ambient(x, z)
np_testing.assert_allclose(z_ambient, u.dot(s).dot(v.T))
def test_ehess2rhess(self):
pass
def test_retr(self):
# Test that the result is on the manifold and that for small
# tangent vectors it has little effect.
x = self.man.rand()
u = self.man.randvec(x)
y = self.man.retr(x, u)
np_testing.assert_allclose(y[0].T.dot(y[0]), np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(y[2].dot(y[2].T), np.eye(self.k), atol=1e-6)
u = u * 1e-6
y = self.man.retr(x, u)
y = y[0].dot(np.diag(y[1])).dot(y[2])
u = self.man.tangent2ambient(x, u)
u = u[0].dot(u[1]).dot(u[2].T)
x = x[0].dot(np.diag(x[1])).dot(x[2])
np_testing.assert_allclose(y, x + u, atol=1e-5)
def test_egrad2rgrad(self):
# Verify that egrad2rgrad and proj are equivalent.
m = self.man
x = m.rand()
u, s, vt = x
i = np.eye(self.k)
f = 1 / (s[..., np.newaxis, :]**2 - s[..., :, np.newaxis]**2 + i)
du = np.random.randn(self.m, self.k)
ds = np.random.randn(self.k)
dvt = np.random.randn(self.k, self.n)
Up = (np.dot(np.dot(np.eye(self.m) - np.dot(u, u.T), du),
np.linalg.inv(np.diag(s))))
M = (np.dot(f * (np.dot(u.T, du) - np.dot(du.T, u)), np.diag(s)) +
np.dot(np.diag(s), f *
(np.dot(vt, dvt.T) - np.dot(dvt, vt.T))) + np.diag(ds))
Vp = (np.dot(np.dot(np.eye(self.n) - np.dot(vt.T, vt), dvt.T),
np.linalg.inv(np.diag(s))))
up, m, vp = m.egrad2rgrad(x, (du, ds, dvt))
U, S, V = self.man.tangent2ambient(x, [Up, M, Vp])
u, s, v = self.man.tangent2ambient(x, [up, m, vp])
T = U.dot(S).dot(V.T)
t = u.dot(s).dot(v.T)
np_testing.assert_allclose(T, t)
def test_randvec(self):
e = self.man
x = e.rand()
u = e.randvec(x)
# Check that u is a tangent vector
assert np.shape(u[0]) == (self.m, self.k)
assert np.shape(u[1]) == (self.k, self.k)
assert np.shape(u[2]) == (self.n, self.k)
np_testing.assert_allclose(np.dot(u[0].T, x[0]),
np.zeros((self.k, self.k)),
atol=1e-6)
np_testing.assert_allclose(np.dot(u[2].T, x[2].T),
np.zeros((self.k, self.k)),
atol=1e-6)
v = e.randvec(x)
np_testing.assert_almost_equal(e.norm(x, u), 1)
assert e.norm(x, u - v) > 1e-6
#if __name__ == '__main__':
# test = TestFixedRankEmbeddedManifold()
# test.setup()
# test.test_egrad2rgrad()
def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.man = FixedRankEmbedded(m, n, k)
m = 4 n = 2 k = 2 X = np.random.rand(m, n) U, S, Vt = np.linalg.svd(X, full_matrices=False) U = U[:, :k] S = S[:k] Vt = Vt[:k, :] grad_f = grad(f) grad_g = grad(g) man = FixedRankEmbedded(m, n, k) dU, dS, dVt = grad_g((U, S, Vt)) Up, M, Vp = man.egrad2rgrad((U, S, Vt), (dU, dS, dVt)) U_, S_, V_ = man.tangent2ambient((U, S, Vt), (Up, M, Vp)) tangent_grad = U_.dot(S_).dot(V_.T) print print print 'X:' print X print print 'U,S,Vt:' print U, S, Vt
#e_correct_prediction = tf.equal(tf.argmax(e_y, 1), tf.argmax(y_, 1))
#e_accuracy = tf.reduce_mean(tf.cast(e_correct_prediction, "float"))
#e_accuracy_summary =tf.summary.scalar("e_accuracy", e_accuracy)
summaries = tf.summary.merge_all()
manifold_b = Euclidean(1,10)
if use_parameterization:
manifold_A = Euclidean(784, k)
manifold_B = Euclidean(k, 10)
manifold_W = Product([manifold_A,manifold_B])
arg = [A, B, b]
else:
manifold_W = FixedRankEmbedded(784, 10, k)
#manifold_W = Product([Stiefel(784,k), Euclidean(k), Stiefel(k,10)])
arg = [A, M, B, b]
manifold = Product([manifold_W, manifold_b])
e_manifold_W = Euclidean(784, 10)
e_arg = [eW, b]
e_manifold = Product([e_manifold_W, manifold_b])
problem = Problem(manifold=manifold, cost=loss, accuracy=accuracy, summary=summaries, arg=arg, data=[x,y_], verbosity=1)
e_problem = Problem(manifold=e_manifold, cost=e_loss, accuracy=accuracy, summary=summaries, arg=e_arg, data=[x, y_],
verbosity=1)
solver = SGD(maxiter=10000000,logverbosity=10)
if not os.path.isdir('result'):
os.makedirs('result')
path = os.path.join('result', experiment_name + '.csv')
m, n, rank = 10, 8, 4
p = 1 / 2
for i in range(n_exp):
matrix = rnd.randn(m, n)
P_omega = np.zeros_like(matrix)
for j in range(m):
for k in range(n):
P_omega[j][k] = random.choice([0, 1])
cost = create_cost(matrix, P_omega)
manifold = FixedRankEmbedded(m, n, rank)
problem = pymanopt.Problem(manifold, cost=cost, egrad=None)
res_list = []
for beta_type in BetaTypes:
solver = ConjugateGradient(beta_type=beta_type, maxiter=10000)
res = solver.solve(problem)
res_list.append(res[1])
res_list.append(res[2])
with open(path, 'a') as f:
writer = csv.writer(f)
writer.writerow(res_list)
class TestFixedRankEmbeddedManifold(ManifoldTestCase):
def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.manifold = FixedRankEmbedded(m, n, k)
u, s, vt = self.manifold.random_point()
matrix = (u * s) @ vt
@pymanopt.function.autograd(self.manifold)
def cost(u, s, vt):
return np.linalg.norm((u * s) @ vt - matrix) ** 2
self.cost = cost
def test_dim(self):
assert self.manifold.dim == (self.m + self.n - self.k) * self.k
def test_typical_dist(self):
assert self.manifold.dim == self.manifold.typical_dist
def test_dist(self):
e = self.manifold
a = e.random_point()
x = e.random_tangent_vector(a)
y = e.random_tangent_vector(a)
with self.assertRaises(NotImplementedError):
e.dist(x, y)
def test_inner_product(self):
e = self.manifold
x = e.random_point()
a = e.random_tangent_vector(x)
b = e.random_tangent_vector(x)
# First embed in the ambient space
A = x[0] @ a[1] @ x[2] + a[0] @ x[2] + x[0] @ a[2].T
B = x[0] @ b[1] @ x[2] + b[0] @ x[2] + x[0] @ b[2].T
trueinner = np.sum(A * B)
np_testing.assert_almost_equal(trueinner, e.inner_product(x, a, b))
def test_proj_range(self):
m = self.manifold
x = m.random_point()
v = np.random.normal(size=(self.m, self.n))
g = m.projection(x, v)
# Check that g is a true tangent vector
np_testing.assert_allclose(
g[0].T @ x[0], np.zeros((self.k, self.k)), atol=1e-6
)
np_testing.assert_allclose(
g[2].T @ x[2].T, np.zeros((self.k, self.k)), atol=1e-6
)
def test_projection(self):
# Verify that proj gives the closest point within the tangent space
# by displacing the result slightly and checking that this increases
# the distance.
m = self.manifold
x = self.manifold.random_point()
v = np.random.normal(size=(self.m, self.n))
g = m.projection(x, v)
# Displace g a little
g_disp = g + 0.01 * m.random_tangent_vector(x)
# Return to the ambient representation
g = m.embedding(x, g)
g_disp = m.embedding(x, g_disp)
g = g[0] @ g[1] @ g[2].T
g_disp = g_disp[0] @ g_disp[1] @ g_disp[2].T
assert np.linalg.norm(g - v) < np.linalg.norm(g_disp - v)
def test_proj_tangents(self):
# Verify that proj leaves tangent vectors unchanged
e = self.manifold
x = e.random_point()
u = e.random_tangent_vector(x)
A = e.projection(x, e.embedding(x, u))
B = u
# diff = [A[k]-B[k] for k in range(len(A))]
np_testing.assert_allclose(A[0], B[0])
np_testing.assert_allclose(A[1], B[1])
np_testing.assert_allclose(A[2], B[2])
def test_norm(self):
e = self.manifold
x = e.random_point()
u = e.random_tangent_vector(x)
np_testing.assert_almost_equal(
np.sqrt(e.inner_product(x, u, u)), e.norm(x, u)
)
def test_random_point(self):
e = self.manifold
x = e.random_point()
y = e.random_point()
assert np.shape(x[0]) == (self.m, self.k)
assert np.shape(x[1]) == (self.k,)
assert np.shape(x[2]) == (self.k, self.n)
np_testing.assert_allclose(x[0].T @ x[0], np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(x[2] @ x[2].T, np.eye(self.k), atol=1e-6)
assert np.linalg.norm(x[0] - y[0]) > 1e-6
assert np.linalg.norm(x[1] - y[1]) > 1e-6
assert np.linalg.norm(x[2] - y[2]) > 1e-6
def test_transport(self):
s = self.manifold
x = s.random_point()
y = s.random_point()
u = s.random_tangent_vector(x)
A = s.transport(x, y, u)
B = s.projection(y, s.embedding(x, u))
diff = [A[k] - B[k] for k in range(len(A))]
np_testing.assert_almost_equal(s.norm(y, diff), 0)
def test_apply_ambient(self):
m = self.manifold
z = np.random.normal(size=(self.m, self.n))
# Set u, s, v so that z = u @ s @ v.T
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.normal(size=(self.n, self.n))
np_testing.assert_allclose(z @ w, m._apply_ambient(z, w))
np_testing.assert_allclose(z @ w, m._apply_ambient((u, s, v), w))
def test_apply_ambient_transpose(self):
m = self.manifold
z = np.random.normal(size=(self.n, self.m))
# Set u, s, v so that z = u @ s @ v.T
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.normal(size=(self.n, self.n))
np_testing.assert_allclose(z.T @ w, m._apply_ambient_transpose(z, w))
np_testing.assert_allclose(
z.T @ w, m._apply_ambient_transpose((u, s, v), w)
)
def test_embedding(self):
m = self.manifold
x = m.random_point()
z = m.random_tangent_vector(x)
z_ambient = x[0] @ z[1] @ x[2] + z[0] @ x[2] + x[0] @ z[2].T
u, s, v = m.embedding(x, z)
np_testing.assert_allclose(z_ambient, u @ s @ v.T)
def test_euclidean_to_riemannian_hessian(self):
pass
def test_retraction(self):
# Test that the result is on the manifold and that for small
# tangent vectors it has little effect.
x = self.manifold.random_point()
u = self.manifold.random_tangent_vector(x)
y = self.manifold.retraction(x, u)
np_testing.assert_allclose(y[0].T @ y[0], np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(y[2] @ y[2].T, np.eye(self.k), atol=1e-6)
u = u * 1e-6
y = self.manifold.retraction(x, u)
y = y[0] @ np.diag(y[1]) @ y[2]
u = self.manifold.embedding(x, u)
u = u[0] @ u[1] @ u[2].T
x = x[0] @ np.diag(x[1]) @ x[2]
np_testing.assert_allclose(y, x + u, atol=1e-5)
def test_euclidean_to_riemannian_gradient(self):
# Verify that euclidean_to_riemannian_gradient and proj are equivalent.
m = self.manifold
x = m.random_point()
u, s, vt = x
i = np.eye(self.k)
f = 1 / (s[..., np.newaxis, :] ** 2 - s[..., :, np.newaxis] ** 2 + i)
du = np.random.normal(size=(self.m, self.k))
ds = np.random.normal(size=self.k)
dvt = np.random.normal(size=(self.k, self.n))
Up = (np.eye(self.m) - u @ u.T) @ du @ np.linalg.inv(np.diag(s))
M = (
f * (u.T @ du - du.T @ u) @ np.diag(s)
+ np.diag(s) @ f * (vt @ dvt.T - dvt @ vt.T)
+ np.diag(ds)
)
Vp = (np.eye(self.n) - vt.T @ vt) @ dvt.T @ np.linalg.inv(np.diag(s))
up, m, vp = m.euclidean_to_riemannian_gradient(x, (du, ds, dvt))
np_testing.assert_allclose(Up, up)
np_testing.assert_allclose(M, m)
np_testing.assert_allclose(Vp, vp)
def test_first_order_function_approximation(self):
self.run_gradient_approximation_test()
def test_random_tangent_vector(self):
e = self.manifold
x = e.random_point()
u = e.random_tangent_vector(x)
# Check that u is a tangent vector
assert np.shape(u[0]) == (self.m, self.k)
assert np.shape(u[1]) == (self.k, self.k)
assert np.shape(u[2]) == (self.n, self.k)
np_testing.assert_allclose(
u[0].T @ x[0], np.zeros((self.k, self.k)), atol=1e-6
)
np_testing.assert_allclose(
u[2].T @ x[2].T, np.zeros((self.k, self.k)), atol=1e-6
)
v = e.random_tangent_vector(x)
np_testing.assert_almost_equal(e.norm(x, u), 1)
assert e.norm(x, u - v) > 1e-6
class TestFixedRankEmbeddedManifold(unittest.TestCase):
def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.man = FixedRankEmbedded(m, n, k)
def test_dim(self):
assert self.man.dim == (self.m + self.n - self.k) * self.k
def test_typicaldist(self):
assert self.man.dim == self.man.typicaldist
def test_dist(self):
e = self.man
a = e.rand()
x = e.randvec(a)
y = e.randvec(a)
with self.assertRaises(NotImplementedError):
e.dist(x, y)
def test_inner(self):
e = self.man
x = e.rand()
a = e.randvec(x)
b = e.randvec(x)
# First embed in the ambient space
A = x[0].dot(a[1].dot(x[2])) + a[0].dot(x[2]) + x[0].dot(a[2].T)
B = x[0].dot(b[1].dot(x[2])) + b[0].dot(x[2]) + x[0].dot(b[2].T)
trueinner = np.sum(A*B)
np_testing.assert_almost_equal(trueinner, e.inner(x, a, b))
def test_proj_range(self):
m = self.man
x = m.rand()
v = np.random.randn(self.m, self.n)
g = m.proj(x, v)
# Check that g is a true tangent vector
np_testing.assert_allclose(np.dot(g[0].T, x[0]),
np.zeros((self.k, self.k)),
atol=1e-6)
np_testing.assert_allclose(np.dot(g[2].T, x[2].T),
np.zeros((self.k, self.k)),
atol=1e-6)
def test_proj(self):
# Verify that proj gives the closest point within the tangent space
# by displacing the result slightly and checking that this increases
# the distance.
m = self.man
x = self.man.rand()
v = np.random.randn(self.m, self.n)
g = m.proj(x, v)
# Displace g a little
g_disp = g + 0.01 * m.randvec(x)
# Return to the ambient representation
g = m.tangent2ambient(x, g)
g_disp = m.tangent2ambient(x, g_disp)
g = g[0].dot(g[1]).dot(g[2].T)
g_disp = g_disp[0].dot(g_disp[1]).dot(g_disp[2].T)
assert np.linalg.norm(g - v) < np.linalg.norm(g_disp - v)
def test_proj_tangents(self):
# Verify that proj leaves tangent vectors unchanged
e = self.man
x = e.rand()
u = e.randvec(x)
A = e.proj(x, e.tangent2ambient(x, u))
B = u
# diff = [A[k]-B[k] for k in xrange(len(A))]
np_testing.assert_allclose(A[0], B[0])
np_testing.assert_allclose(A[1], B[1])
np_testing.assert_allclose(A[2], B[2])
def test_norm(self):
e = self.man
x = e.rand()
u = e.randvec(x)
np_testing.assert_almost_equal(np.sqrt(e.inner(x, u, u)), e.norm(x, u))
def test_rand(self):
e = self.man
x = e.rand()
y = e.rand()
assert np.shape(x[0]) == (self.m, self.k)
assert np.shape(x[1]) == (self.k,)
assert np.shape(x[2]) == (self.k, self.n)
np_testing.assert_allclose(x[0].T.dot(x[0]), np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(x[2].dot(x[2].T), np.eye(self.k), atol=1e-6)
assert la.norm(x[0] - y[0]) > 1e-6
assert la.norm(x[1] - y[1]) > 1e-6
assert la.norm(x[2] - y[2]) > 1e-6
def test_transp(self):
s = self.man
x = s.rand()
y = s.rand()
u = s.randvec(x)
A = s.transp(x, y, u)
B = s.proj(y, s.tangent2ambient(x, u))
diff = [A[k]-B[k] for k in range(len(A))]
np_testing.assert_almost_equal(s.norm(y, diff), 0)
def test_apply_ambient(self):
m = self.man
z = np.random.randn(self.m, self.n)
# Set u, s, v so that z = u.dot(s).dot(v.T)
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.randn(self.n, self.n)
np_testing.assert_allclose(z.dot(w), m._apply_ambient(z, w))
np_testing.assert_allclose(z.dot(w), m._apply_ambient((u, s, v), w))
def test_apply_ambient_transpose(self):
m = self.man
z = np.random.randn(self.n, self.m)
# Set u, s, v so that z = u.dot(s).dot(v.T)
u, s, v = np.linalg.svd(z, full_matrices=False)
s = np.diag(s)
v = v.T
w = np.random.randn(self.n, self.n)
np_testing.assert_allclose(z.T.dot(w),
m._apply_ambient_transpose(z, w))
np_testing.assert_allclose(z.T.dot(w),
m._apply_ambient_transpose((u, s, v), w))
def test_tangent2ambient(self):
m = self.man
x = m.rand()
z = m.randvec(x)
z_ambient = (x[0].dot(z[1]).dot(x[2]) + z[0].dot(x[2]) +
x[0].dot(z[2].T))
u, s, v = m.tangent2ambient(x, z)
np_testing.assert_allclose(z_ambient, u.dot(s).dot(v.T))
def test_ehess2rhess(self):
pass
def test_retr(self):
# Test that the result is on the manifold and that for small
# tangent vectors it has little effect.
x = self.man.rand()
u = self.man.randvec(x)
y = self.man.retr(x, u)
np_testing.assert_allclose(y[0].T.dot(y[0]), np.eye(self.k), atol=1e-6)
np_testing.assert_allclose(y[2].dot(y[2].T), np.eye(self.k), atol=1e-6)
u = u * 1e-6
y = self.man.retr(x, u)
y = y[0].dot(np.diag(y[1])).dot(y[2])
u = self.man.tangent2ambient(x, u)
u = u[0].dot(u[1]).dot(u[2].T)
x = x[0].dot(np.diag(x[1])).dot(x[2])
np_testing.assert_allclose(y, x + u, atol=1e-5)
def test_egrad2rgrad(self):
# Verify that egrad2rgrad and proj are equivalent.
m = self.man
x = m.rand()
u, s, vt = x
i = np.eye(self.k)
f = 1 / (s[..., np.newaxis, :]**2 - s[..., :, np.newaxis]**2 + i)
du = np.random.randn(self.m, self.k)
ds = np.random.randn(self.k)
dvt = np.random.randn(self.k, self.n)
Up = (np.dot(np.dot(np.eye(self.m) - np.dot(u, u.T), du),
np.linalg.inv(np.diag(s))))
M = (np.dot(f * (np.dot(u.T, du) - np.dot(du.T, u)), np.diag(s)) +
np.dot(np.diag(s), f * (np.dot(vt, dvt.T) - np.dot(dvt, vt.T))) +
np.diag(ds))
Vp = (np.dot(np.dot(np.eye(self.n) - np.dot(vt.T, vt), dvt.T),
np.linalg.inv(np.diag(s))))
up, m, vp = m.egrad2rgrad(x, (du, ds, dvt))
np_testing.assert_allclose(Up, up)
np_testing.assert_allclose(M, m)
np_testing.assert_allclose(Vp, vp)
def test_randvec(self):
e = self.man
x = e.rand()
u = e.randvec(x)
# Check that u is a tangent vector
assert np.shape(u[0]) == (self.m, self.k)
assert np.shape(u[1]) == (self.k, self.k)
assert np.shape(u[2]) == (self.n, self.k)
np_testing.assert_allclose(np.dot(u[0].T, x[0]),
np.zeros((self.k, self.k)),
atol=1e-6)
np_testing.assert_allclose(np.dot(u[2].T, x[2].T),
np.zeros((self.k, self.k)),
atol=1e-6)
v = e.randvec(x)
np_testing.assert_almost_equal(e.norm(x, u), 1)
assert e.norm(x, u - v) > 1e-6
def setUp(self):
self.m = m = 10
self.n = n = 5
self.k = k = 3
self.man = FixedRankEmbedded(m, n, k)