def testSvdSoft(self):
A = scipy.sparse.rand(10, 10, 0.2)
A = A.tocsc()
lmbda = 0.1
k = 6
U, s, V = SoftImpute.svdSoft(A, lmbda, k)
ATilde = U.dot(numpy.diag(s)).dot(V.T)
#Now comput the same matrix using numpy
#Pick first k singular vectors/values
A = A.todense()
U2, s2, V2 = numpy.linalg.svd(A)
inds = numpy.flipud(numpy.argsort(s2))[0:k]
U2, s2, V2 = Util.indSvd(U2, s2, V2, inds)
s2 = s2 - lmbda
s2 = numpy.clip(s, 0, numpy.max(s2))
ATilde2 = U2.dot(numpy.diag(s2)).dot(V2.T)
nptst.assert_array_almost_equal(s, s)
nptst.assert_array_almost_equal(ATilde, ATilde2)
def testSvd(self):
n = 100
A = scipy.sparse.rand(n, n, 0.1)
ks = [10, 20, 30, 40]
q = 2
lastError = numpy.linalg.norm(A.todense())
for k in ks:
U, s, V = RandomisedSVD.svd(A, k, q)
nptst.assert_array_almost_equal(U.T.dot(U), numpy.eye(k))
nptst.assert_array_almost_equal(V.T.dot(V), numpy.eye(k))
A2 = (U*s).dot(V.T)
error = numpy.linalg.norm(A - A2)
self.assertTrue(error <= lastError)
lastError = error
#Compare versus exact svd
U, s, V = numpy.linalg.svd(numpy.array(A.todense()))
inds = numpy.flipud(numpy.argsort(s))[0:k*2]
U, s, V = Util.indSvd(U, s, V, inds)
Ak = (U*s).dot(V.T)
error2 = numpy.linalg.norm(A - Ak)
self.assertTrue(error2 <= error)
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 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
