def test_multilogm_complex_positive_definite(self):
shape = (self.k, self.m, self.m)
A = np.random.normal(size=shape) + 1j * np.random.normal(size=shape)
A = A @ multihconj(A)
# Compare fast path for positive definite matrices vs. general slow
# one.
np_testing.assert_allclose(
multilogm(A, positive_definite=True),
multilogm(A, positive_definite=False),
)
def test_dist(self):
# n = self.n
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 exponential metric increasing property
# (see equation (6.8) in [Bha2007]).
logx, logy = multilogm(x), multilogm(y)
assert manifold.dist(x, y) >= np.linalg.norm(logx - logy)
# 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 test_multilogm_singlemat(self):
a = np.diag(np.random.uniform(size=self.m))
q, _ = np.linalg.qr(np.random.normal(size=(self.m, self.m)))
# A is a positive definite matrix
A = q @ a @ q.T
np_testing.assert_allclose(multilogm(A, positive_definite=True),
logm(A))
def dist(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 np.linalg.norm(logm)
def test_multilogm(self):
A = np.zeros((self.k, self.m, self.m))
L = np.zeros((self.k, self.m, self.m))
for i in range(self.k):
a = np.diag(np.random.uniform(size=self.m))
q, _ = np.linalg.qr(np.random.normal(size=(self.m, self.m)))
A[i] = q @ a @ q.T
L[i] = logm(A[i])
np_testing.assert_allclose(multilogm(A, positive_definite=True), L)
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 log(self, point_a, point_b):
return multiskew(multilogm(multitransp(point_a) @ point_b))