def test_dist(self):
man = self.man
x = man.rand()
y = man.rand()
# Test separability
np_testing.assert_almost_equal(man.dist(x, x), 0.)
# Test symmetry
np_testing.assert_almost_equal(man.dist(x, y), man.dist(y, x))
# Test alternative implementation
# from Eq 6.14 of "Positive definite matrices"
d = np.sqrt((np.log(sp.linalg.eigvalsh(x, y))**2).sum())
np_testing.assert_almost_equal(man.dist(x, y), d)
# check that dist is consistent with log
np_testing.assert_almost_equal(man.dist(x, y),
man.norm(x, man.log(x, y)))
# Test invariance under inversion
np_testing.assert_almost_equal(man.dist(x, y),
man.dist(la.inv(y), la.inv(x)))
# Test congruence-invariance
a = rnd.randn(self.n, self.n) # must be invertible
axa = multiprod(multiprod(a, x), multitransp(a))
aya = multiprod(multiprod(a, y), multitransp(a))
np_testing.assert_almost_equal(man.dist(x, y), man.dist(axa, aya))
def log(self, point_a, point_b):
ytx = multitransp(point_b) @ point_a
At = multitransp(point_b) - ytx @ multitransp(point_a)
Bt = np.linalg.solve(ytx, At)
u, s, vt = np.linalg.svd(multitransp(Bt), full_matrices=False)
arctan_s = np.expand_dims(np.arctan(s), -2)
return (u * arctan_s) @ vt
def log(self, point_a, point_b):
c = np.linalg.cholesky(point_a)
c_inv = np.linalg.inv(c)
logm = multilogm(
c_inv @ point_b @ multitransp(c_inv),
positive_definite=True,
)
return c @ logm @ multitransp(c)
def log(self, X, Y):
ytx = multiprod(multitransp(Y), X)
At = multitransp(Y) - multiprod(ytx, multitransp(X))
Bt = np.linalg.solve(ytx, At)
u, s, vt = svd(multitransp(Bt), full_matrices=False)
arctan_s = np.expand_dims(np.arctan(s), -2)
U = multiprod(u * arctan_s, vt)
return U
def log(self, X, Y):
ytx = multiprod(multitransp(Y), X)
At = multitransp(Y) - multiprod(ytx, multitransp(X))
Bt = np.linalg.solve(ytx, At)
u, s, vt = svd(multitransp(Bt), full_matrices=False)
arctan_s = np.expand_dims(np.arctan(s), -2)
U = multiprod(u * arctan_s, vt)
return U
def test_proj(self):
# Construct a random point X on the manifold.
X = self.man.rand()
# Construct a vector H in the ambient space.
H = rnd.randn(self.k, self.m, self.n)
# Compare the projections.
Hproj = H - multiprod(X, multiprod(multitransp(X), H) + multiprod(multitransp(H), X)) / 2
np_testing.assert_allclose(Hproj, self.man.proj(X, H))
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_proj(self):
# Construct a random point X on the manifold.
X = self.man.rand()
# Construct a vector H in the ambient space.
H = rnd.randn(self.k, self.m, self.n)
# Compare the projections.
Hproj = H - multiprod(X, multiprod(multitransp(X), H) +
multiprod(multitransp(H), X)) / 2
np_testing.assert_allclose(Hproj, self.man.proj(X, H))
def geodesic(point_a, point_b, alpha):
if alpha < 0 or 1 < alpha:
raise ValueError("Exponent must be in [0,1]")
c = np.linalg.cholesky(point_a)
c_inv = np.linalg.inv(c)
log_cbc = multilogm(
c_inv @ point_b @ multitransp(c_inv),
positive_definite=True,
)
powm = multiexpm(alpha * log_cbc, symmetric=False)
return c @ powm @ multitransp(c)
def test_dist(self):
manifold = self.manifold
x = manifold.random_point()
y = manifold.random_point()
z = manifold.random_point()
# Test separability
np_testing.assert_almost_equal(manifold.dist(x, x), 0.0)
# Test symmetry
np_testing.assert_almost_equal(
manifold.dist(x, y), manifold.dist(y, x)
)
# Test triangle inequality
assert manifold.dist(x, y) <= manifold.dist(x, z) + manifold.dist(z, y)
# Test alternative implementation (see equation (6.14) in [Bha2007]).
d = np.sqrt((np.log(eigvalsh(x, y)) ** 2).sum())
np_testing.assert_almost_equal(manifold.dist(x, y), d)
# Test exponential metric increasing property
# (see equation (6.8) in [Bha2007]).
assert manifold.dist(x, y) >= np.linalg.norm(logm(x) - logm(y))
# check that dist is consistent with log
np_testing.assert_almost_equal(
manifold.dist(x, y), manifold.norm(x, manifold.log(x, y))
)
# Test invariance under inversion
np_testing.assert_almost_equal(
manifold.dist(x, y),
manifold.dist(np.linalg.inv(y), np.linalg.inv(x)),
)
# Test congruence-invariance
a = np.random.normal(size=(self.n, self.n)) # must be invertible
axa = a @ x @ multitransp(a)
aya = a @ y @ multitransp(a)
np_testing.assert_almost_equal(
manifold.dist(x, y), manifold.dist(axa, aya)
)
# Test proportionality (see equation (6.12) in [Bha2007]).
alpha = np.random.uniform()
np_testing.assert_almost_equal(
manifold.dist(x, geodesic(x, y, alpha)),
alpha * manifold.dist(x, y),
)
def inner(self, x, u, v):
xinvu = la.solve(x, u)
if u is v:
xinvv = xinvu
else:
xinvv = la.solve(x, v)
return np.tensordot(xinvu, multitransp(xinvv), axes=x.ndim)
def norm(self, x, u):
# This implementation is as fast as np.linalg.solve_triangular and is
# more stable, as the above solver tends to output non positive
# definite results.
c = la.cholesky(x)
c_inv = la.inv(c)
return la.norm(multiprod(multiprod(c_inv, u), multitransp(c_inv)))
def log(self, X, Y):
U = multiprod(multitransp(X), Y)
if self._k == 1:
return multiskew(np.real(logm(U)))
for i in range(self._k):
U[i] = np.real(logm(U[i]))
return multiskew(U)
def euclidean_to_riemannian_hessian(self, point, euclidean_gradient,
euclidean_hessian, tangent_vector):
Xt = multitransp(point)
Xtegrad = Xt @ euclidean_gradient
symXtegrad = multisym(Xtegrad)
Xtehess = Xt @ euclidean_hessian
return multiskew(Xtehess - tangent_vector @ symXtegrad)
def test_rand(self):
# Just make sure that things generated are on the manifold and that
# if you generate two they are not equal.
X = self.man.rand()
np_testing.assert_allclose(multiprod(multitransp(X), X), multieye(self.k, self.n), atol=1e-10)
Y = self.man.rand()
assert np.linalg.norm(X - Y) > 1e-6
def test_multitransp(self):
A = rnd.randn(self.k, self.m, self.n)
C = np.zeros((self.k, self.n, self.m))
for i in range(self.k):
C[i] = A[i].T
np_testing.assert_array_equal(C, multitransp(A))
def test_random_point(self):
point = self.so.random_point()
assert point.shape == (self.n, self.n)
np_testing.assert_almost_equal(point.T @ point - point @ point.T, 0)
np_testing.assert_almost_equal(point.T @ point, np.eye(self.n))
assert np.allclose(np.linalg.det(point), 1)
point = self.so_product.random_point()
assert point.shape == (self.k, self.n, self.n)
np_testing.assert_almost_equal(
multitransp(point) @ point - point @ multitransp(point),
0,
)
np_testing.assert_almost_equal(
multitransp(point) @ point, multieye(self.k, self.n)
)
assert np.allclose(np.linalg.det(point), 1)
def test_rand(self):
# Just make sure that things generated are on the manifold and that
# if you generate two they are not equal.
X = self.man.rand()
np_testing.assert_allclose(multiprod(multitransp(X), X),
multieye(self.k, self.n), atol=1e-10)
Y = self.man.rand()
assert np.linalg.norm(X - Y) > 1e-6
def test_multitransp(self):
A = rnd.randn(self.k, self.m, self.n)
C = np.zeros((self.k, self.n, self.m))
for i in range(self.k):
C[i] = A[i].T
np_testing.assert_array_equal(C, multitransp(A))
def test_multitransp(self):
A = np.random.normal(size=(self.k, self.m, self.n))
C = np.zeros((self.k, self.n, self.m))
for i in range(self.k):
C[i] = A[i].T
np_testing.assert_array_equal(C, multitransp(A))
def dist(self, x, y):
# Adapted from equation 6.13 of "Positive definite matrices". Chol
# decomp gives the same result as matrix sqrt. There may be a more
# efficient way to compute this!
c = np.linalg.cholesky(x)
c_inv = np.linalg.inv(c)
l = multilog(multiprod(multiprod(c_inv, y), multitransp(c_inv)),
pos_def=True)
return la.norm(multiprod(multiprod(c, l), c_inv))
def inner_product(self, point, tangent_vector_a, tangent_vector_b):
p_inv_tv_a = np.linalg.solve(point, tangent_vector_a)
if tangent_vector_a is tangent_vector_b:
p_inv_tv_b = p_inv_tv_a
else:
p_inv_tv_b = np.linalg.solve(point, tangent_vector_b)
return np.tensordot(p_inv_tv_a,
multitransp(p_inv_tv_b),
axes=tangent_vector_a.ndim)
def exp(self, point, tangent_vector):
pt_tv = multitransp(point) @ tangent_vector
if self._k == 1:
identity = np.eye(self._p)
else:
identity = multieye(self._k, self._p)
a = np.block([point, tangent_vector])
b = multiexpm(
np.block([
[
pt_tv,
-multitransp(tangent_vector) @ tangent_vector,
],
[identity, pt_tv],
]))[..., :self._p]
c = multiexpm(-pt_tv)
return a @ (b @ c)
def dist(self, x, y):
# Adapted from equation 6.13 of "Positive definite matrices". The
# Cholesky decomposition gives the same result as matrix sqrt. There
# may be more efficient ways to compute this.
c = la.cholesky(x)
c_inv = la.inv(c)
logm = multilog(multiprod(multiprod(c_inv, y), multitransp(c_inv)),
pos_def=True)
return la.norm(logm)
def dist(self, x, y):
# Adapted from equation 6.13 of "Positive definite matrices". Chol
# decomp gives the same result as matrix sqrt. There may be a more
# efficient way to compute this!
c = np.linalg.cholesky(x)
c_inv = np.linalg.inv(c)
l = multilog(multiprod(multiprod(c_inv, y), multitransp(c_inv)),
pos_def=True)
return la.norm(multiprod(multiprod(c, l), c_inv))
def randskew(n, N=1):
idxs = np.triu_indices(n, 1)
S = np.zeros((N, n, n))
for i in range(N):
S[i][idxs] = rnd.randn(int(n * (n - 1) / 2))
S = S - multitransp(S)
if N == 1:
return S.reshape(n, n)
return S
def test_inner(self):
man = self.man
x = man.rand()
a = man.randvec(x)
b = man.randvec(x)
# b is not symmetric, it is Hermitian
np.testing.assert_almost_equal(
np.tensordot(a, multitransp(b), axes=a.ndim),
man.inner(x, multiprod(x, a), multiprod(x, b)))
assert man.inner(x, a, b).dtype == np.float
def test_randvec(self):
# Make sure things generated are in tangent space and if you generate
# two then they are not equal.
X = self.man.rand()
U = self.man.randvec(X)
np_testing.assert_allclose(multisym(multiprod(multitransp(X), U)),
np.zeros((self.k, self.n, self.n)),
atol=1e-10)
V = self.man.randvec(X)
assert la.norm(U - V) > 1e-6
def random_tangent_vector(self, point):
n, k = self._n, self._k
inds = np.triu_indices(n, 1)
vector = np.zeros((k, n, n))
for i in range(k):
vector[i][inds] = np.random.normal(size=int(n * (n - 1) / 2))
vector = vector - multitransp(vector)
if k == 1:
vector = vector[0]
return vector / np.sqrt(np.tensordot(vector, vector, axes=vector.ndim))
def rand(self):
# The way this is done is arbitrary. I think the space of p.d.
# matrices would have infinite measure w.r.t. the Riemannian metric
# (cf. integral 0-inf [ln(x)] dx = inf) so impossible to have a
# 'uniform' distribution.
# Generate eigenvalues between 1 and 2
d = np.ones((self._k, self._n, 1)) + rnd.rand(self._k, self._n, 1)
# Generate an orthogonal matrix. Annoyingly qr decomp isn't
# vectorized so need to use a for loop. Could be done using
# svd but this is slower for bigger matrices.
u = np.zeros((self._k, self._n, self._n))
for i in range(self._k):
u[i], r = la.qr(rnd.randn(self._n, self._n))
if self._k == 1:
return multiprod(u, d * multitransp(u))[0]
return multiprod(u, d * multitransp(u))
def test_randvec(self):
# Make sure things generated are in tangent space and if you generate
# two then they are not equal.
X = self.man.rand()
U = self.man.randvec(X)
np_testing.assert_allclose(multisym(multiprod(multitransp(X), U)),
np.zeros((self.k, self.n, self.n)),
atol=1e-10)
V = self.man.randvec(X)
assert la.norm(U - V) > 1e-6
def random_point(self):
# Generate eigenvalues between 1 and 2.
d = 1.0 + np.random.uniform(size=(self._k, self._n, 1))
# Generate an orthogonal matrix.
q, _ = multiqr(np.random.normal(size=(self._n, self._n)))
point = q @ (d * multitransp(q))
if self._k == 1:
return point[0]
return point
def test_random_tangent_vector(self):
point = self.so.random_point()
tangent_vector = self.so.random_tangent_vector(point)
np_testing.assert_almost_equal(tangent_vector, -tangent_vector.T)
point = self.so_product.random_point()
tangent_vector = self.so_product.random_tangent_vector(point)
np_testing.assert_almost_equal(
tangent_vector, -multitransp(tangent_vector)
)
def exp(self, point, tangent_vector):
u, s, vt = np.linalg.svd(tangent_vector, full_matrices=False)
cos_s = np.expand_dims(np.cos(s), -2)
sin_s = np.expand_dims(np.sin(s), -2)
Y = point @ (multitransp(vt) * cos_s) @ vt + (u * sin_s) @ vt
# From numerical experiments, it seems necessary to re-orthonormalize.
# This is quite expensive.
q, _ = multiqr(Y)
return q
def rand(self):
# The way this is done is arbitrary. I think the space of p.d.
# matrices would have infinite measure w.r.t. the Riemannian metric
# (c.f. integral 0-inf [ln(x)] dx = inf) so impossible to have a
# 'uniform' distribution.
# Generate eigenvalues between 1 and 2
d = np.ones((self._k, self._n, 1)) + rnd.rand(self._k, self._n, 1)
# Generate an orthogonal matrix. Annoyingly qr decomp isn't
# vectorized so need to use a for loop. Could be done using
# svd but this is slower for bigger matrices.
u = np.zeros((self._k, self._n, self._n))
for i in range(self._k):
u[i], r = la.qr(rnd.randn(self._n, self._n))
if self._k == 1:
return multiprod(u, d * multitransp(u))[0]
else:
return multiprod(u, d * multitransp(u))
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
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)
xretru = self.man.retr(x, u)
np_testing.assert_allclose(multiprod(multitransp(xretru), xretru), multieye(self.k, self.n), atol=1e-10)
u = u * 1e-6
xretru = self.man.retr(x, u)
np_testing.assert_allclose(xretru, x + 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.man
x = s.rand()
u = s.randvec(x)
xexpu = s.exp(x, u)
np_testing.assert_allclose(multiprod(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)
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.man
x = s.rand()
u = s.randvec(x)
xexpu = s.exp(x, u)
np_testing.assert_allclose(multiprod(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)
def dist(self, X, Y):
if self._k == 1:
u, s, v = np.linalg.svd(np.dot(X.T, Y))
s[s > 1] = 1
s = np.arccos(s)
return np.linalg.norm(s)
else:
XtY = multiprod(multitransp(X), Y)
square_d = 0
for i in xrange(self._k):
s = np.linalg.svd(XtY[i], compute_uv=False)
# Ensure that -1 <= s <= 1
s = np.fmin(s, [1])
s = np.fmax(s, [-1])
square_d = square_d + np.linalg.norm(np.arccos(s))**2
return np.sqrt(square_d)
def exp(self, X, U):
u, s, vt = svd(U, full_matrices=False)
cos_s = np.expand_dims(np.cos(s), -2)
sin_s = np.expand_dims(np.sin(s), -2)
Y = (multiprod(multiprod(X, multitransp(vt) * cos_s), vt) +
multiprod(u * sin_s, vt))
# From numerical experiments, it seems necessary to
# re-orthonormalize. This is overall quite expensive.
if self._k == 1:
Y, unused = np.linalg.qr(Y)
return Y
else:
for i in range(self._k):
Y[i], unused = np.linalg.qr(Y[i])
return Y
def log(self, x, y):
c = la.cholesky(x)
c_inv = la.inv(c)
l = multilog(multiprod(multiprod(c_inv, y), multitransp(c_inv)),
pos_def=True)
return multiprod(multiprod(c, l), multitransp(c))
def proj(self, X, U):
UNew = U - multiprod(
X, multiprod(multitransp(X), U) + multiprod(multitransp(U), X)) / 2
return UNew
def test_multitransp_singlemat(self):
A = rnd.randn(self.m, self.n)
np_testing.assert_array_equal(A.T, multitransp(A))
def ehess2rhess(self, X, egrad, ehess, H):
# Convert Euclidean into Riemannian Hessian.
XtG = multiprod(multitransp(X), egrad)
symXtG = multisym(XtG)
HsymXtG = multiprod(H, symXtG)
return self.proj(X, ehess - HsymXtG)
def dist(self, X, Y):
u, s, v = svd(multiprod(multitransp(X), Y))
s[s > 1] = 1
s = np.arccos(s)
return np.linalg.norm(s)
def proj(self, X, U):
return U - multiprod(X, multiprod(multitransp(X), U))
def ehess2rhess(self, X, egrad, ehess, H):
# Convert Euclidean into Riemannian Hessian.
PXehess = self.proj(X, ehess)
XtG = multiprod(multitransp(X), egrad)
HXtG = multiprod(H, XtG)
return PXehess - HXtG