def randvec(self, x):
k = self._k
n = self._n
if k == 1:
u = multiherm(rnd.randn(n, n)+1j*rnd.randn(n, n))
else:
u = multiherm(rnd.randn(k, n, n)+1j*rnd.randn(k, n, n))
return u / self.norm(x, u)
def test_proj(self):
man = self.man
x = man.rand()
a = rnd.randn(self.k, self.n, self.n)
+1j * rnd.randn(self.k, self.n, self.n)
np.testing.assert_allclose(man.proj(x, a), multiherm(a))
np.testing.assert_allclose(man.proj(x, a), man.proj(x, man.proj(x, a)))
def test_rand(self):
# Just test that rand returns a point on the manifold and two
# different matrices generated by rand aren't too close together
k = self.k
n = self.n
man = self.man
x = man.rand()
y = man.rand()
assert np.shape(x) == (k, n, n)
assert x.dtype == np.complex
# Check symmetry
np_testing.assert_allclose(x, multiherm(x))
# Check positivity of eigenvalues
w = la.eigvalsh(x)
assert (w > [[0]]).all()
# Check unit determinant
d = np.real(la.det(x))
np_testing.assert_allclose(d, 1)
# Check randomness
assert la.norm(x - y) > 1e-3
def log(self, x, y):
k = self._k
d, q = la.eigh(x)
if k == 1:
x_sqrt = [email protected](np.sqrt(d))@q.conj().T
x_isqrt = [email protected](1/np.sqrt(d))@q.conj().T
else:
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(np.sqrt(d[i, :]))[np.newaxis, :, :]
x_sqrt = multiprod(multiprod(q, temp), multihconj(q))
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(1/np.sqrt(d[i, :]))[np.newaxis, :, :]
x_isqrt = multiprod(multiprod(q, temp), multihconj(q))
d, q = la.eigh(multiprod(multiprod(x_isqrt, y), x_isqrt))
if k == 1:
log = [email protected](np.log(d))@q.conj().T
else:
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(np.log(d[i, :]))[np.newaxis, :, :]
d = temp
log = multiprod(multiprod(q, d), multihconj(q))
xi = multiprod(multiprod(x_sqrt, log), x_sqrt)
xi = multiherm(xi)
return xi
def test_egrad2rgrad(self):
man = self.man
x = man.rand()
u = rnd.randn(self.k, self.n, self.n)
+1j * rnd.randn(self.k, self.n, self.n)
np.testing.assert_allclose(man.egrad2rgrad(x, u),
multiprod(multiprod(x, multiherm(u)), x))
def exp(self, x, u):
k = self._k
d, q = la.eigh(x)
if k == 1:
x_sqrt = [email protected](np.sqrt(d))@q.conj().T
x_isqrt = [email protected](1/np.sqrt(d))@q.conj().T
else:
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(np.sqrt(d[i, :]))[np.newaxis, :, :]
x_sqrt = multiprod(multiprod(q, temp), multihconj(q))
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(1/np.sqrt(d[i, :]))[np.newaxis, :, :]
x_isqrt = multiprod(multiprod(q, temp), multihconj(q))
d, q = la.eigh(multiprod(multiprod(x_isqrt, u), x_isqrt))
if k == 1:
e = [email protected](np.exp(d))@q.conj().T
else:
temp = np.zeros(q.shape, dtype=np.complex)
for i in range(q.shape[0]):
temp[i, :, :] = np.diag(np.exp(d[i, :]))[np.newaxis, :, :]
d = temp
e = multiprod(multiprod(q, d), multihconj(q))
e = multiprod(multiprod(x_sqrt, e), x_sqrt)
e = multiherm(e)
return e
def test_randvec(self):
# Just test that randvec returns an element of the tangent space
# with norm 1 and that two randvecs are different.
man = self.man
x = man.rand()
u = man.randvec(x)
v = man.randvec(x)
np_testing.assert_allclose(multiherm(u), u)
np_testing.assert_almost_equal(1, man.norm(x, u))
assert la.norm(u - v) > 1e-3
def test_proj(self):
man = self.man
x = man.rand()
a = rnd.randn(self.n, self.n)
+1j * rnd.randn(self.n, self.n)
p = man.proj(x, a)
assert np.shape(p) == (self.n, self.n)
np.testing.assert_allclose(p, multiherm(p))
t = np.real(np.trace(la.solve(x, p)))
np_testing.assert_almost_equal(t, 0)
np.testing.assert_allclose(p, man.proj(x, p))
def proj(self, x, u):
n = self._n
k = self._k
# Project matrix on tangent space of HPD.
u = multiherm(u)
# Project on tangent space of SHPD at x.
t = np.trace(la.solve(x, u), axis1=-2, axis2=-1)
if k == 1:
u = u - (1/n) * np.real(t) * x
else:
u = u - (1/n) * np.real(t.reshape(-1, 1, 1)) * x
return u
def test_proj(self):
man = self.man
x = man.rand()
a = rnd.randn(self.k, self.n, self.n)
+1j * rnd.randn(self.k, self.n, self.n)
p = man.proj(x, a)
np.testing.assert_allclose(p, multiherm(p))
t = np.ones(man._k, dtype=np.complex)
temp = la.solve(x, p)
for i in range(man._k):
t[i] = np.real(np.trace(temp[i, :, :]))
np_testing.assert_allclose(t, 0, atol=1e-7)
np.testing.assert_allclose(p, man.proj(x, p))
def test_exp(self):
# exp(x, u) = x + u.
man = self.man
x = man.rand()
u = man.randvec(x)
e = man.exp(x, u)
# Check symmetry
np_testing.assert_allclose(e, multiherm(e))
# Check positivity of eigenvalues
w = la.eigvalsh(e)
assert (w > [0]).all()
u = u * 1e-6
np_testing.assert_allclose(man.exp(x, u), x + u)
def test_rand(self):
# Just test that rand returns a point on the manifold and two
# different matrices generated by rand aren't too close together
n = self.n
man = self.man
x = man.rand()
assert np.shape(x) == (n, n)
assert x.dtype == np.complex
# Check symmetry
np_testing.assert_allclose(x, multiherm(x))
# Check positivity of eigenvalues
w = la.eigvalsh(x)
assert (w > [0]).all()
def test_exp(self):
# Test against manopt implementation, test that for small vectors
# exp(x, u) = x + u.
man = self.man
x = man.rand()
u = man.randvec(x)
e = man.exp(x, u)
# Check symmetry
np_testing.assert_allclose(e, multiherm(e))
# Check positivity of eigenvalues
w = la.eigvalsh(e)
assert (w > [[0]]).all()
u = u * 1e-6
np_testing.assert_allclose(man.exp(x, u), x + u)
def test_retr(self):
# Check that result is on manifold and for small vectors
# retr(x, u) = x + u.
man = self.man
x = man.rand()
u = man.randvec(x)
y = man.retr(x, u)
assert np.shape(y) == (self.k, self.n, self.n)
# Check symmetry
np_testing.assert_allclose(y, multiherm(y))
# Check positivity of eigenvalues
w = la.eigvalsh(y)
assert (w > [[0]]).all()
u = u * 1e-6
np_testing.assert_allclose(man.retr(x, u), x + u)
def test_randvec(self):
# Just test that randvec returns an element of the tangent space
# with norm 1 and that two randvecs are different.
man = self.man
x = man.rand()
u = man.randvec(x)
v = man.randvec(x)
np_testing.assert_allclose(multiherm(u), u)
t = np.empty(man._k, dtype=np.complex)
temp = la.solve(x, u)
for i in range(man._k):
t[i] = np.real(np.trace(temp[i, :, :]))
np_testing.assert_allclose(t, 0, atol=1e-7)
np_testing.assert_almost_equal(1, man.norm(x, u))
assert la.norm(u - v) > 1e-3
def test_randvec(self):
# Just test that randvec returns an element of the tangent space
# with norm 1 and that two randvecs are different.
man = self.man
n = self.n
man = self.man
x = man.rand()
u = man.randvec(x)
v = man.randvec(x)
assert np.shape(x) == (n, n)
assert x.dtype == np.complex
np_testing.assert_allclose(multiherm(u), u)
t = np.real(np.trace(la.solve(x, u)))
np_testing.assert_almost_equal(t, 0)
np_testing.assert_almost_equal(1, man.norm(x, u))
assert la.norm(u - v) > 1e-3
def ehess2rhess(self, x, egrad, ehess, u):
egrad = multiherm(egrad)
hess = multiprod(multiprod(x, multiherm(ehess)), x)
hess += multiherm(multiprod(multiprod(u, egrad), x))
return hess
def egrad2rgrad(self, x, u):
return multiprod(multiprod(x, multiherm(u)), x)
def proj(self, X, G):
return multiherm(G)
