def svd_from_eigh(A, eps=10 ** -8, tol=10 ** -8):
"""
Find the SVD of an ill conditioned matrix A. This uses numpy.linalg.eig
but conditions the matrix so is not as precise as numpy.linalg.svd, but
can be useful if svd does not coverge. Uses the eigenvectors of A^T*A and
return singular vectors corresponding to nonzero singular values.
Note: This is slightly different to linalg.svd which returns zero singular
values.
"""
AA = A.conj().T.dot(A)
lmbda, Q = scipy.linalg.eigh(AA + eps * numpy.eye(A.shape[1]))
lmbda = lmbda - eps
inds = numpy.arange(lmbda.shape[0])[lmbda > tol]
lmbda, Q = Util.indEig(lmbda, Q, inds)
sigma = lmbda ** 0.5
P = A.dot(Q) / sigma
Qh = Q.conj().T
if __debug__:
if not scipy.allclose(A, (P * sigma).dot(Qh), atol=tol):
logging.warn(" SVD obtained from EVD is too poor")
Parameter.checkArray(P, softCheck=True, arrayInfo="P in svd_from_eigh()")
if not Parameter.checkOrthogonal(
P, tol=tol, softCheck=True, arrayInfo="P in svd_from_eigh()", investigate=True
):
print("corresponding sigma: ", sigma)
Parameter.checkArray(sigma, softCheck=True, arrayInfo="sigma in svd_from_eigh()")
Parameter.checkArray(Qh, softCheck=True, arrayInfo="Qh in svd_from_eigh()")
if not Parameter.checkOrthogonal(Qh.conj().T, tol=tol, softCheck=True, arrayInfo="Qh.H in svd_from_eigh()"):
print("corresponding sigma: ", sigma)
return P, sigma, Qh
def eigenAdd(omega, Q, Y, k):
"""
Perform an eigen update of the form A*A + Y*Y in which Y is a low-rank matrix
and A^*A = Q Omega Q*. We use the rank-k approximation of A: Q_k Omega_k Q_k^*
and then approximate [A^*A_k Y^*Y]_k.
"""
#logging.debug("< eigenAdd >")
Parameter.checkInt(k, 0, omega.shape[0])
#if not numpy.isrealobj(omega) or not numpy.isrealobj(Q):
# raise ValueError("Eigenvalues and eigenvectors must be real")
if omega.ndim != 1:
raise ValueError("omega must be 1-d array")
if omega.shape[0] != Q.shape[1]:
raise ValueError("Must have same number of eigenvalues and eigenvectors")
if __debug__:
Parameter.checkOrthogonal(Q, tol=EigenUpdater.tol, softCheck=True, arrayInfo="input Q in eigenAdd()")
#Taking the abs of the eigenvalues is correct
inds = numpy.flipud(numpy.argsort(numpy.abs(omega)))
omega, Q = Util.indEig(omega, Q, inds[numpy.abs(omega)>EigenUpdater.tol])
Omega = numpy.diag(omega)
YY = Y.conj().T.dot(Y)
QQ = Q.dot(Q.conj().T)
Ybar = Y - Y.dot(QQ)
Pbar, sigmaBar, Qbar = numpy.linalg.svd(Ybar, full_matrices=False)
inds = numpy.flipud(numpy.argsort(numpy.abs(sigmaBar)))
inds = inds[numpy.abs(sigmaBar)>EigenUpdater.tol]
Pbar, sigmaBar, Qbar = Util.indSvd(Pbar, sigmaBar, Qbar, inds)
SigmaBar = numpy.diag(sigmaBar)
Qbar = Ybar.T.dot(Pbar)
Qbar = Qbar.dot(numpy.diag(numpy.diag(Qbar.T.dot(Qbar))**-0.5))
r = sigmaBar.shape[0]
YQ = Y.dot(Q)
Zeros = numpy.zeros((r, omega.shape[0]))
D = numpy.c_[Q, Qbar]
YYQQ = YY.dot(QQ)
Z = D.conj().T.dot(YYQQ + YYQQ.conj().T).dot(D)
F = numpy.c_[numpy.r_[Omega - YQ.conj().T.dot(YQ), Zeros], numpy.r_[Zeros.T, SigmaBar.conj().dot(SigmaBar)]]
F = F + Z
pi, H = scipy.linalg.eigh(F)
inds = numpy.flipud(numpy.argsort(numpy.abs(pi)))
H = H[:, inds[0:k]]
pi = pi[inds[0:k]]
V = D.dot(H)
#logging.debug("</ eigenAdd >")
return pi, V
def eigenConcat(omega, Q, AB, BB, k):
"""
Find the eigen update of a matrix [A, B]'[A B] where A'A = V diag(s) V*
and AB = A*B, BB = B*B. Q is the set of eigenvectors of A*A and s is the
vector of eigenvalues.
"""
#logging.debug("< eigenConcat >")
Parameter.checkInt(k, 0, omega.shape[0])
if not numpy.isrealobj(omega) or not numpy.isrealobj(Q):
raise ValueError("Eigenvalues and eigenvectors must be real")
if not numpy.isrealobj(AB) or not numpy.isrealobj(BB):
raise ValueError("AB and BB must be real")
if omega.ndim != 1:
raise ValueError("omega must be 1-d array")
if omega.shape[0] != Q.shape[1]:
raise ValueError("Must have same number of eigenvalues and eigenvectors")
if Q.shape[0] != AB.shape[0]:
raise ValueError("Q must have the same number of rows as AB")
if AB.shape[1] != BB.shape[0] or BB.shape[0]!=BB.shape[1]:
raise ValueError("AB must have the same number of cols/rows as BB")
#Check Q is orthogonal
if __debug__:
Parameter.checkOrthogonal(Q, tol=EigenUpdater.tol, softCheck=True, arrayInfo = "input Q in eigenConcat()")
m = Q.shape[0]
p = BB.shape[0]
inds = numpy.flipud(numpy.argsort(numpy.abs(omega)))
Q = Q[:, inds[0:k]]
omega = omega[inds[0:k]]
Omega = numpy.diag(omega)
QAB = Q.conj().T.dot(AB)
F = numpy.c_[numpy.r_[Omega, QAB.conj().T], numpy.r_[QAB, BB]]
D = numpy.c_[numpy.r_[Q, numpy.zeros((p, Q.shape[1]))], numpy.r_[numpy.zeros((m, p)), numpy.eye(p)]]
pi, H = scipy.linalg.eigh(F)
inds = numpy.flipud(numpy.argsort(numpy.abs(pi)))
inds = inds[numpy.abs(pi)>EigenUpdater.tol]
H = H[:, inds[0:k]]
pi = pi[inds[0:k]]
V = numpy.dot(D, H)
#logging.debug("</ eigenConcat >")
return pi, V
def eigenRemove(omega, Q, n, k, debug=False):
"""
Remove a set of rows and columns from a matrix whose eigen-decomposition
is Q diag(omega) Q^T. Keep the first n rows/cols i.e. the rows/cols starting
from n to the end are removed and k is the number of eigenvectors/values
to return for the new matrix. We could generalise this to delete a given
list of rows/cols.
"""
#logging.debug("< eigenRemove >")
Parameter.checkClass(omega, numpy.ndarray)
Parameter.checkClass(Q, numpy.ndarray)
Parameter.checkInt(k, 0, float('inf'))
Parameter.checkInt(n, 0, Q.shape[0])
if omega.ndim != 1:
raise ValueError("omega must be 1-d array")
if omega.shape[0] != Q.shape[1]:
raise ValueError("Must have same number of eigenvalues and eigenvectors")
if __debug__:
Parameter.checkOrthogonal(Q, tol=EigenUpdater.tol, softCheck=True, arrayInfo="input Q in eigenRemove()")
inds = numpy.flipud(numpy.argsort(numpy.abs(omega)))
inds = inds[omega[inds]>EigenUpdater.tol]
omega, Q = Util.indEig(omega, Q, inds[0:k])
AB = (Q[0:n, :]*omega).dot(Q[n:, :].T)
BB = (Q[n:, :]*omega).dot(Q[n:, :].T)
p = BB.shape[0]
Y1 = numpy.r_[numpy.zeros((n, p)), numpy.eye(p)]
Y2 = -numpy.r_[AB, 0.5*BB]
pi, V = EigenUpdater.eigenAdd2(omega, Q, Y1, Y2, k)
#check last rows are zero
if numpy.linalg.norm(V[n:, :]) >= EigenUpdater.tol:
logging.warn("numpy.linalg.norm(V[n:, :])= %s" % str(numpy.linalg.norm(V[n:, :])))
#logging.debug("</ eigenRemove >")
if not debug:
return pi, V[0:n, :]
else:
C = (Q*omega).dot(Q.T)
K = C + Y1.dot(Y2.T) + Y2.dot(Y1.T)
assert numpy.linalg.norm(BB- C[n:, n:]) <= EigenUpdater.tol
assert numpy.linalg.norm(AB - C[0:n, n:]) <= EigenUpdater.tol, "%s \n %s" % (AB, C[0:n, n:])
return pi, V[0:n, :], K, Y1, Y2, omega
def lazyEigenConcatAsUpdate(omega, Q, AB, BB, k, debug= False):
"""
Find the eigen update of a matrix [A, B]'[A B] where
A'A = Q diag(omega) Q* and AB = A*B, BB = B*B. Q is the set of
eigenvectors of A*A and omega is the vector of eigenvalues.
Simply expand Q, and update the eigen decomposition using EigenAdd2.
Computation could be upgraded a bit because of the particular update
type (Y1Bar = Y1 = [0,I]', Y2Bar = [(I-QQ')A'B, 0]').
"""
#logging.debug("< lazyEigenConcatAsUpdate >")
Parameter.checkClass(omega, numpy.ndarray)
Parameter.checkClass(Q, numpy.ndarray)
Parameter.checkClass(AB, numpy.ndarray)
Parameter.checkClass(BB, numpy.ndarray)
Parameter.checkInt(k, 0, AB.shape[0] + BB.shape[0])
if not numpy.isrealobj(omega) or not numpy.isrealobj(Q):
logging.info("Eigenvalues or eigenvectors are not real")
if not numpy.isrealobj(AB) or not numpy.isrealobj(BB):
logging.info("AB or BB are not real")
if omega.ndim != 1:
raise ValueError("omega must be 1-d array")
if omega.shape[0] != Q.shape[1]:
raise ValueError("Must have same number of eigenvalues and eigenvectors")
if Q.shape[0] != AB.shape[0]:
raise ValueError("Q must have the same number of rows as AB")
if AB.shape[1] != BB.shape[0] or BB.shape[0]!=BB.shape[1]:
raise ValueError("AB must have the same number of cols/rows as BB")
if __debug__:
if not Parameter.checkOrthogonal(Q, tol=EigenUpdater.tol, softCheck=True, investigate=True, arrayInfo="input Q in lazyEigenConcatAsUpdate()"):
print("omega:\n", omega)
m = Q.shape[0]
p = BB.shape[0]
Q = numpy.r_[Q, numpy.zeros((p, Q.shape[1]))]
Y1 = numpy.r_[numpy.zeros((m,p)), numpy.eye(p)]
Y2 = numpy.r_[AB, 0.5*BB]
return EigenUpdater.eigenAdd2(omega, Q, Y1, Y2, k, debug=debug)
def eigenAdd2(omega, Q, Y1, Y2, k, debug= False):
"""
Compute an approximation of the eigendecomposition A^*A + Y1Y2^* +Y2Y1^*
in which Y1, Y2 are low rank matrices, Y1^*Y2=0 and A^*A = Q Omega Q*. We
use the rank-k approximation of A^*A: Q_k Omega_k Q_k^* and then find
[A^*A_k + Y1Y2^* + Y2Y1^*]. If debug=False then pi, V are returned which
respectively correspond to all the eigenvalues/eigenvectors of
[A^*A_k + Y1Y2^* + Y2Y1^*].
"""
#logging.debug("< eigenAdd2 >")
Parameter.checkInt(k, 0, float('inf'))
Parameter.checkClass(omega, numpy.ndarray)
Parameter.checkClass(Q, numpy.ndarray)
Parameter.checkClass(Y1, numpy.ndarray)
Parameter.checkClass(Y2, numpy.ndarray)
if not numpy.isrealobj(omega) or not numpy.isrealobj(Q):
logging.warn("Eigenvalues or eigenvectors are not real")
if not numpy.isrealobj(Y1) or not numpy.isrealobj(Y2):
logging.warn("Y1 or Y2 are not real")
if omega.ndim != 1:
raise ValueError("omega must be 1-d array")
if omega.shape[0] != Q.shape[1]:
raise ValueError("Must have same number of eigenvalues and eigenvectors")
if Q.shape[0] != Y1.shape[0]:
raise ValueError("Q must have the same number of rows as Y1 rows")
if Q.shape[0] != Y2.shape[0]:
raise ValueError("Q must have the same number of rows as Y2 rows")
if Y1.shape[1] != Y2.shape[1]:
raise ValueError("Y1 must have the same number of columns as Y2 columns")
if __debug__:
Parameter.checkArray(omega, softCheck=True, arrayInfo="omega as input in eigenAdd2()")
Parameter.checkArray(Q, softCheck=True, arrayInfo="Q as input in eigenAdd2()")
Parameter.checkOrthogonal(Q, tol=EigenUpdater.tol, softCheck=True, arrayInfo="Q as input in eigenAdd2()")
Parameter.checkArray(Y1, softCheck=True, arrayInfo="Y1 as input in eigenAdd2()")
Parameter.checkArray(Y2, softCheck=True, arrayInfo="Y2 as input in eigenAdd2()")
#Get first k eigenvectors/values of A^*A
omega, Q = Util.indEig(omega, Q, numpy.flipud(numpy.argsort(omega))[0:k])
QY1 = Q.conj().T.dot(Y1)
Y1bar = Y1 - Q.dot(QY1)
P1bar, sigma1Bar, Q1bar = Util.safeSvd(Y1bar)
inds = numpy.arange(sigma1Bar.shape[0])[numpy.abs(sigma1Bar)>EigenUpdater.tol]
P1bar, sigma1Bar, Q1bar = Util.indSvd(P1bar, sigma1Bar, Q1bar, inds)
# checks on SVD decomposition of Y1bar
if __debug__:
Parameter.checkArray(QY1, softCheck=True, arrayInfo="QY1 in eigenAdd2()")
Parameter.checkArray(Y1bar, softCheck=True, arrayInfo="Y1bar in eigenAdd2()")
Parameter.checkArray(P1bar, softCheck=True, arrayInfo="P1bar in eigenAdd2()")
if not Parameter.checkOrthogonal(P1bar, tol=EigenUpdater.tol, softCheck=True, arrayInfo="P1bar in eigenAdd2()", investigate=True):
print ("corresponding sigma: ", sigma1Bar)
Parameter.checkArray(sigma1Bar, softCheck=True, arrayInfo="sigma1Bar in eigenAdd2()")
Parameter.checkArray(Q1bar, softCheck=True, arrayInfo="Q1bar in eigenAdd2()")
if not Parameter.checkOrthogonal(Q1bar, tol=EigenUpdater.tol, softCheck=True, arrayInfo="Q1bar in eigenAdd2()"):
print ("corresponding sigma: ", sigma1Bar)
del Y1bar
P1barY2 = P1bar.conj().T.dot(Y2)
QY2 = Q.conj().T.dot(Y2)
Y2bar = Y2 - Q.dot(QY2) - P1bar.dot(P1barY2)
P2bar, sigma2Bar, Q2bar = Util.safeSvd(Y2bar)
inds = numpy.arange(sigma2Bar.shape[0])[numpy.abs(sigma2Bar)>EigenUpdater.tol]
P2bar, sigma2Bar, Q2bar = Util.indSvd(P2bar, sigma2Bar, Q2bar, inds)
# checks on SVD decomposition of Y1bar
if __debug__:
Parameter.checkArray(P1barY2, softCheck=True, arrayInfo="P1barY2 in eigenAdd2()")
Parameter.checkArray(QY2, softCheck=True, arrayInfo="QY2 in eigenAdd2()")
Parameter.checkArray(Y2bar, softCheck=True, arrayInfo="Y2bar in eigenAdd2()")
Parameter.checkArray(P2bar, softCheck=True, arrayInfo="P2bar in eigenAdd2()")
Parameter.checkOrthogonal(P2bar, tol=EigenUpdater.tol, softCheck=True, arrayInfo="P2bar in eigenAdd2()")
Parameter.checkArray(sigma2Bar, softCheck=True, arrayInfo="sigma2Bar in eigenAdd2()")
Parameter.checkArray(Q2bar, softCheck=True, arrayInfo="Q2bar in eigenAdd2()")
Parameter.checkOrthogonal(Q2bar, tol=EigenUpdater.tol, softCheck=True, arrayInfo="Q2bar in eigenAdd2()")
del Y2bar
r = omega.shape[0]
p = Y1.shape[1]
p1 = sigma1Bar.shape[0]
p2 = sigma2Bar.shape[0]
D = numpy.c_[Q, P1bar, P2bar]
del P1bar
del P2bar
# rem: A*s = A.dot(diag(s)) ; A*s[:,new] = diag(s).dot(A)
DStarY1 = numpy.r_[QY1, sigma1Bar[:,numpy.newaxis] * Q1bar.conj().T, numpy.zeros((p2, p))]
DStarY2 = numpy.r_[QY2, P1barY2, sigma2Bar[:,numpy.newaxis] * Q2bar.conj().T]
DStarY1Y2StarD = DStarY1.dot(DStarY2.conj().T)
del DStarY1
del DStarY2
r = omega.shape[0]
F = numpy.zeros((r+p1+p2, r+p1+p2))
F[range(r),range(r)] = omega
F = F + DStarY1Y2StarD + DStarY1Y2StarD.conj().T
#A check to make sure DFD^T is AA_k + Y1Y2 + Y2Y1
#assert numpy.linalg.norm(D.dot(F).dot(D.T) - Q.dot(numpy.diag(omega).dot(Q.T)) - Y1.dot(Y2.T) - Y2.dot(Y1.T)) < 10**-6
# checks on F
if __debug__:
#Parameter.checkArray(DStarY1, softCheck=True, arrayInfo="DStarY1 in eigenAdd2()")
#Parameter.checkArray(DStarY2, softCheck=True, arrayInfo="DStarY2 in eigenAdd2()")
Parameter.checkArray(DStarY1Y2StarD, softCheck=True, arrayInfo="DStarY1Y2StarD in eigenAdd2()")
Parameter.checkArray(F, softCheck=True, arrayInfo="F in eigenAdd2()")
Parameter.checkSymmetric(F, tol=EigenUpdater.tol, softCheck=True, arrayInfo="F in eigenAdd2()")
pi, H = scipy.linalg.eigh(F)
# remove too small eigenvalues
pi, H = Util.indEig(pi, H, numpy.arange(pi.shape[0])[numpy.abs(pi)>EigenUpdater.tol])
# keep greatest eigenvalues
#pi, H = Util.indEig(pi, H, numpy.flipud(numpy.argsort(pi))[:min(k,pi.shape[0])])
V = D.dot(H)
if __debug__:
if not Parameter.checkOrthogonal(D, tol=EigenUpdater.tol, softCheck=True, investigate=True, arrayInfo="D in eigenAdd2()"):
print("pi:\n", pi)
if not Parameter.checkOrthogonal(H, tol=EigenUpdater.tol, softCheck=True, investigate=True, arrayInfo="H in eigenAdd2()"):
print("pi:\n", pi)
if ProfileUtils.memory() > 10**9:
ProfileUtils.memDisplay(locals())
#logging.debug("</ eigenAdd2 >")
if debug:
return pi, V, D, DStarY1Y2StarD + DStarY1Y2StarD.conj().T
else:
return pi, V
