def retr(self, X, U, mode="default"):
if mode == "exp":
return self.exp(X, U)
elif mode == "qr":
YR, YI = frac(X + U)
Q, R = tensor.nlinalg.qr(YR + 1j * YI)
Y = tensor.stack([Q.real, Q.imag])
return Y
elif mode == "svd":
YR, YI = frac(X + U)
U, S, V = tensor.nlinalg.svd(YR + 1j * YI, full_matrices=False)
Y = U.dot(tensor.eye(S.size)).dot(V)
Y = tensor.stack([Y.real, Y.imag])
return Y
elif mode == "cayley":
A = complex_dot(hconj(U), X) + complex_dot(hconj(X), U)
return complex_dot(
complex_matrix_inverse(identity(self._n) + 0.5 * A),
(identity(self._n) - 0.5 * A))
elif mode == "default":
return self.retr(X, U, mode=self.retr_mode)
else:
raise ValueError(
'mode must equal to "svd", "qr", "exp" or "default", but "{}" is given'
.format(mode))
def exp(self, X, U):
# The exponential (in the sense of Lie group theory) of a tangent
# vector U at X.
first = self.concat([X, U], axis=1)
XhU = complex_dot(hconj(X), U)
second = complex_expm(self.concat([self.concat([XhU, -complex_dot(hconj(U), U)], 1),
self.concat([identity(self._n), XhU], 1)], 0))
third = self.concat([complex_expm(-XhU), zeros((self._n, self._n))], 0)
exponential = complex_dot(complex_dot(first, second), third)
return exponential
def exp(self, X, U):
# The exponential (in the sense of Lie group theory) of a tangent
# vector U at X.
first = self.concat([X, U], axis=1)
XhU = complex_dot(hconj(X), U)
second = complex_expm(
self.concat([
self.concat([XhU, -complex_dot(hconj(U), U)], 1),
self.concat([identity(self._n), XhU], 1)
], 0))
third = self.concat([complex_expm(-XhU), zeros((self._n, self._n))], 0)
exponential = complex_dot(complex_dot(first, second), third)
return exponential
def ehess2rhess(self, X, egrad, ehess, H):
XHG = complex_dot(hconj(X), egrad)
#XHG = X.conj().dot(egrad)
herXHG = self.herm(XHG)
HherXHG = complex_dot(H, herXHG)
rhess = self.proj(X, ehess - HherXHG)
return rhess
def ehess2rhess(self, X, egrad, ehess, H):
XHG = complex_dot(hconj(X), egrad)
#XHG = X.conj().dot(egrad)
herXHG = self.herm(XHG)
HherXHG = complex_dot(H, herXHG)
rhess = self.proj(X, ehess - HherXHG)
return rhess
def ehess2rhess(self, X, egrad, ehess, H):
# TODO implement this for future
XHG = complex_dot(hconj(X), egrad)
#XHG = X.conj().dot(egrad)
herXHG = self.herm(XHG)
HherXHG = complex_dot(H, herXHG)
rhess = self.proj(X, ehess - HherXHG)
return rhess
def retr(self, X, U, mode="default"):
if mode == "exp":
return self.exp(X, U)
elif mode == "qr":
YR, YI = frac(X + U)
Q, R = tensor.nlinalg.qr(YR + 1j * YI)
Y = tensor.stack([Q.real, Q.imag])
return Y
elif mode == "svd":
YR, YI = frac(X + U)
U, S, V = tensor.nlinalg.svd(YR + 1j * YI, full_matrices=False)
Y = U.dot(tensor.eye(S.size)).dot(V)
Y = tensor.stack([Y.real, Y.imag])
return Y
elif mode == "cayley":
A = complex_dot(hconj(U), X) + complex_dot(hconj(X), U)
return complex_dot(complex_matrix_inverse(identity(self._n) + 0.5 * A),
(identity(self._n) - 0.5 * A))
elif mode == "default":
return self.retr(X, U, mode=self.retr_mode)
else:
raise ValueError('mode must equal to "svd", "qr", "exp" or "default", but "{}" is given'.format(mode))
def herm(self, X):
Xman = skew_hermitian_expm(X)
XH = hconj(Xman)
return 0.5 * (Xman + XH)
def proj(self, X, U):
XHU = complex_dot(hconj(X), U)
herXHU = self.herm(XHU)
Up = U - complex_dot(X, herXHU)
return Up
def herm(self, X):
XH = hconj(X)
return 0.5 * (X + XH)
def hconj(self, X):
return tuple(hconj(x) for (manifold, x) in zip(self._manifolds, X))
def hconj(self, X):
return tuple(hconj(x) for (manifold, x) in zip(self._manifolds, X))
def proj(self, X, U):
XHU = complex_dot(hconj(X), U)
herXHU = self.herm(XHU)
Up = U - complex_dot(X, herXHU)
return Up
def herm(self, X):
XH = hconj(X)
return 0.5 * (X + XH)