def testRank(self):
X = numpy.random.rand(10, 1)
self.assertEquals(Util.rank(X), 1)
X = numpy.random.rand(10, 12)
self.assertEquals(Util.rank(X), 10)
X = numpy.random.rand(31, 12)
self.assertEquals(Util.rank(X), 12)
K = numpy.dot(X, X.T)
self.assertEquals(Util.rank(X), 12)
def testRank(self):
X = numpy.random.rand(10, 1)
self.assertEquals(Util.rank(X), 1)
X = numpy.random.rand(10, 12)
self.assertEquals(Util.rank(X), 10)
X = numpy.random.rand(31, 12)
self.assertEquals(Util.rank(X), 12)
K = numpy.dot(X, X.T)
self.assertEquals(Util.rank(X), 12)
def testEigenRemove(self):
tol = 10**-6
for i in range(10):
m = numpy.random.randint(5, 10)
n = numpy.random.randint(5, 10)
#How many rows/cols to remove
p = numpy.random.randint(1, 5)
A = numpy.random.randn(m, n)
C = A.conj().T.dot(A)
lastError = 100
omega, Q = numpy.linalg.eigh(C)
self.assertTrue(numpy.linalg.norm(C-(Q*omega).dot(Q.conj().T)) < tol )
#
Cprime = C[0:n-p, 0:n-p]
for k in range(1,9):
pi, V, K, Y1, Y2, omega2 = EigenUpdater.eigenRemove(omega, Q, n-p, k, debug=True)
# V is "orthogonal"
self.assertTrue(numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(V.shape[1])) < tol )
# The approximation converges to the exact decomposition
C_k = (V*pi).dot(V.conj().T)
error = numpy.linalg.norm(Cprime-C_k)
if Util.rank(C)<k:
self.assertTrue(error <= tol)
lastError = error
def addRows(U, s, V, B, k=None):
"""
Find the SVD of a matrix [A ; B] where A = U diag(s) V.T. Uses the QR
decomposition to find an orthogonal basis on B.
:param U: The left singular vectors of A
:param s: The singular values of A
:param V: The right singular vectors of A
:param B: The matrix to append to A
"""
if V.shape[0] != B.shape[1]:
raise ValueError("U must have same number of rows as B cols")
if s.shape[0] != U.shape[1]:
raise ValueError("Number of cols of U must be the same size as s")
if s.shape[0] != V.shape[1]:
raise ValueError("Number of cols of V must be the same size as s")
if k == None:
k = U.shape[1]
m, p = U.shape
r = B.shape[0]
C = B.T - V.dot(V.T).dot(B.T)
Q, R = numpy.linalg.qr(C)
rPrime = Util.rank(C)
Q = Q[:, 0:rPrime]
R = R[0:rPrime, :]
D = numpy.c_[numpy.diag(s), numpy.zeros((p, rPrime))]
E = numpy.c_[B.dot(V), R.T]
D = numpy.r_[D, E]
G1 = numpy.c_[U, numpy.zeros((m, r))]
G2 = numpy.c_[numpy.zeros((r, p)), numpy.eye(r)]
G = numpy.r_[G1, G2]
H = numpy.c_[V, Q]
nptst.assert_array_almost_equal(G.T.dot(G), numpy.eye(G.shape[1]))
nptst.assert_array_almost_equal(H.T.dot(H), numpy.eye(H.shape[1]))
nptst.assert_array_almost_equal(
G.dot(D).dot(H.T), numpy.r_[(U * s).dot(V.T), B])
Uhat, sHat, Vhat = numpy.linalg.svd(D, full_matrices=False)
inds = numpy.flipud(numpy.argsort(sHat))[0:k]
Uhat, sHat, Vhat = Util.indSvd(Uhat, sHat, Vhat, inds)
#The best rank k approximation of [A ; B]
Utilde = G.dot(Uhat)
Stilde = sHat
Vtilde = H.dot(Vhat)
return Utilde, Stilde, Vtilde
def addRows(U, s, V, B, k=None):
"""
Find the SVD of a matrix [A ; B] where A = U diag(s) V.T. Uses the QR
decomposition to find an orthogonal basis on B.
:param U: The left singular vectors of A
:param s: The singular values of A
:param V: The right singular vectors of A
:param B: The matrix to append to A
"""
if V.shape[0] != B.shape[1]:
raise ValueError("U must have same number of rows as B cols")
if s.shape[0] != U.shape[1]:
raise ValueError("Number of cols of U must be the same size as s")
if s.shape[0] != V.shape[1]:
raise ValueError("Number of cols of V must be the same size as s")
if k == None:
k = U.shape[1]
m, p = U.shape
r = B.shape[0]
C = B.T - V.dot(V.T).dot(B.T)
Q, R = numpy.linalg.qr(C)
rPrime = Util.rank(C)
Q = Q[:, 0:rPrime]
R = R[0:rPrime, :]
D = numpy.c_[numpy.diag(s), numpy.zeros((p, rPrime))]
E = numpy.c_[B.dot(V), R.T]
D = numpy.r_[D, E]
G1 = numpy.c_[U, numpy.zeros((m, r))]
G2 = numpy.c_[numpy.zeros((r, p)), numpy.eye(r)]
G = numpy.r_[G1, G2]
H = numpy.c_[V, Q]
nptst.assert_array_almost_equal(G.T.dot(G), numpy.eye(G.shape[1]))
nptst.assert_array_almost_equal(H.T.dot(H), numpy.eye(H.shape[1]))
nptst.assert_array_almost_equal(G.dot(D).dot(H.T), numpy.r_[(U*s).dot(V.T), B])
Uhat, sHat, Vhat = numpy.linalg.svd(D, full_matrices=False)
inds = numpy.flipud(numpy.argsort(sHat))[0:k]
Uhat, sHat, Vhat = Util.indSvd(Uhat, sHat, Vhat, inds)
#The best rank k approximation of [A ; B]
Utilde = G.dot(Uhat)
Stilde = sHat
Vtilde = H.dot(Vhat)
return Utilde, Stilde, Vtilde
def testEigenAdd2(self):
tol = 10**-6
for i in range(10):
m = numpy.random.randint(5, 10)
n = numpy.random.randint(5, 10)
p = numpy.random.randint(5, 10)
A = numpy.random.randn(m, n)
Y1 = numpy.random.randn(n, p)
Y2 = numpy.random.randn(n, p)
AA = A.conj().T.dot(A)
Y1Y2 = Y1.dot(Y2.conj().T)
lastError = 100
omega, Q = numpy.linalg.eigh(AA)
self.assertTrue(
numpy.linalg.norm(AA - (Q * omega).dot(Q.conj().T)) < tol)
C = AA + Y1Y2 + Y1Y2.conj().T
for k in range(1, 9):
pi, V, D, DUD = EigenUpdater.eigenAdd2(omega,
Q,
Y1,
Y2,
k,
debug=True)
# V is "orthogonal"
self.assertTrue(
numpy.linalg.norm(V.conj().T.dot(V) -
numpy.eye(V.shape[1])) < tol)
# The approximation converges to the exact decomposition
C_k = (V * pi).dot(V.conj().T)
error = numpy.linalg.norm(C - C_k)
if Util.rank(C) == k:
self.assertTrue(error <= tol)
lastError = error
# DomegaD corresponds to AA_k
omega_k, Q_k = Util.indEig(
omega, Q,
numpy.flipud(numpy.argsort(omega))[0:k])
DomegakD = (D *
numpy.c_[omega_k[numpy.newaxis, :],
numpy.zeros(
(1, max(D.shape[1] - k, 0)))]).dot(
D.conj().T)
self.assertTrue(
numpy.linalg.norm((Q_k * omega_k).dot(Q_k.conj().T) -
DomegakD) < tol)
# DUD is exactly decomposed
self.assertTrue(
numpy.linalg.norm(Y1Y2 + Y1Y2.conj().T -
D.dot(DUD).dot(D.conj().T)) < tol)
def testEigenRemove(self):
tol = 10**-6
for i in range(10):
m = numpy.random.randint(5, 10)
n = numpy.random.randint(5, 10)
#How many rows/cols to remove
p = numpy.random.randint(1, 5)
A = numpy.random.randn(m, n)
C = A.conj().T.dot(A)
lastError = 100
omega, Q = numpy.linalg.eigh(C)
self.assertTrue(
numpy.linalg.norm(C - (Q * omega).dot(Q.conj().T)) < tol)
#
Cprime = C[0:n - p, 0:n - p]
for k in range(1, 9):
pi, V, K, Y1, Y2, omega2 = EigenUpdater.eigenRemove(omega,
Q,
n - p,
k,
debug=True)
# V is "orthogonal"
self.assertTrue(
numpy.linalg.norm(V.conj().T.dot(V) -
numpy.eye(V.shape[1])) < tol)
# The approximation converges to the exact decomposition
C_k = (V * pi).dot(V.conj().T)
error = numpy.linalg.norm(Cprime - C_k)
if Util.rank(C) < k:
self.assertTrue(error <= tol)
lastError = error
def testEigenAdd(self):
for i in range(3):
numCols = numpy.random.randint(5, 10)
numXRows = numpy.random.randint(5, 10)
numYRows = numpy.random.randint(5, 10)
A = numpy.random.rand(numXRows, numCols)
Y = numpy.random.rand(numYRows, numCols)
AA = A.conj().T.dot(A)
AA = (AA + AA.conj().T) / 2
YY = Y.conj().T.dot(Y)
lastError = 1000
for k in range(1, min((numXRows, numCols))):
#Note using eigh since AA is hermatian
omega, Q = numpy.linalg.eigh(AA)
pi, V = EigenUpdater.eigenAdd(omega, Q, Y, k)
Pi = numpy.diag(pi)
tol = 10**-3
t = min(k, Util.rank(AA + YY))
self.assertTrue(pi.shape[0] == t)
self.assertTrue(
numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(t)) < tol)
inds2 = numpy.flipud(numpy.argsort(numpy.abs(omega)))
Q = Q[:, inds2[0:k]]
omega = omega[inds2[0:k]]
AAk = Q.dot(numpy.diag(omega)).dot(Q.conj().T)
AAkpYY = AAk + YY
AApYYEst = V.dot(Pi.dot(V.conj().T))
error = numpy.linalg.norm(AApYYEst - (AA + YY))
self.assertTrue(lastError - error >= -tol)
lastError = error
def testEigenAdd2(self):
tol = 10**-6
for i in range(10):
m = numpy.random.randint(5, 10)
n = numpy.random.randint(5, 10)
p = numpy.random.randint(5, 10)
A = numpy.random.randn(m, n)
Y1 = numpy.random.randn(n, p)
Y2 = numpy.random.randn(n, p)
AA = A.conj().T.dot(A)
Y1Y2 = Y1.dot(Y2.conj().T)
lastError = 100
omega, Q = numpy.linalg.eigh(AA)
self.assertTrue(numpy.linalg.norm(AA-(Q*omega).dot(Q.conj().T)) < tol )
C = AA + Y1Y2 + Y1Y2.conj().T
for k in range(1,9):
pi, V, D, DUD = EigenUpdater.eigenAdd2(omega, Q, Y1, Y2, k, debug = True)
# V is "orthogonal"
self.assertTrue(numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(V.shape[1])) < tol )
# The approximation converges to the exact decomposition
C_k = (V*pi).dot(V.conj().T)
error = numpy.linalg.norm(C-C_k)
if Util.rank(C)==k:
self.assertTrue(error <= tol)
lastError = error
# DomegaD corresponds to AA_k
omega_k, Q_k = Util.indEig(omega, Q, numpy.flipud(numpy.argsort(omega))[0:k])
DomegakD = (D*numpy.c_[omega_k[numpy.newaxis,:],numpy.zeros((1,max(D.shape[1]-k,0)))]).dot(D.conj().T)
self.assertTrue(numpy.linalg.norm((Q_k*omega_k).dot(Q_k.conj().T)-DomegakD) < tol )
# DUD is exactly decomposed
self.assertTrue(numpy.linalg.norm(Y1Y2 + Y1Y2.conj().T - D.dot(DUD).dot(D.conj().T)) < tol )
def testEigenAdd(self):
for i in range(3):
numCols = numpy.random.randint(5, 10)
numXRows = numpy.random.randint(5, 10)
numYRows = numpy.random.randint(5, 10)
A = numpy.random.rand(numXRows, numCols)
Y = numpy.random.rand(numYRows, numCols)
AA = A.conj().T.dot(A)
AA = (AA + AA.conj().T)/2
YY = Y.conj().T.dot(Y)
lastError = 1000
for k in range(1, min((numXRows, numCols))):
#Note using eigh since AA is hermatian
omega, Q = numpy.linalg.eigh(AA)
pi, V = EigenUpdater.eigenAdd(omega, Q, Y, k)
Pi = numpy.diag(pi)
tol = 10**-3
t = min(k, Util.rank(AA+YY))
self.assertTrue(pi.shape[0] == t)
self.assertTrue(numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(t)) < tol)
inds2 = numpy.flipud(numpy.argsort(numpy.abs(omega)))
Q = Q[:, inds2[0:k]]
omega = omega[inds2[0:k]]
AAk = Q.dot(numpy.diag(omega)).dot(Q.conj().T)
AAkpYY = AAk + YY
AApYYEst = V.dot(Pi.dot(V.conj().T))
error = numpy.linalg.norm(AApYYEst - (AA+YY))
self.assertTrue(lastError - error >= -tol)
lastError = error
import numpy
import scipy.sparse
from apgl.graph import GraphUtils
from sandbox.util.Util import Util
numpy.set_printoptions(suppress=True, precision=3)
n = 10
W1 = scipy.sparse.rand(n, n, 0.5).todense()
W1 = W1.T.dot(W1)
W2 = W1.copy()
W2[1, 2] = 1
W2[2, 1] = 1
print("W1=" + str(W1))
print("W2=" + str(W2))
L1 = GraphUtils.normalisedLaplacianSym(scipy.sparse.csr_matrix(W1))
L2 = GraphUtils.normalisedLaplacianSym(scipy.sparse.csr_matrix(W2))
deltaL = L2 - L1
print("L1=" + str(L1.todense()))
print("L2=" + str(L2.todense()))
print("deltaL=" + str(deltaL.todense()))
print("rank(deltaL)=" + str(Util.rank(deltaL.todense())))
def testEigenConcat(self):
tol = 10**-6
for i in range(3):
m = numpy.random.randint(10, 20)
n = numpy.random.randint(5, 10)
p = numpy.random.randint(5, 10)
# A = numpy.zeros((m, n), numpy.complex)
# B = numpy.zeros((m, p), numpy.complex)
# A.real = numpy.random.randn(m, n)
# A.imag = numpy.random.randn(m, n)
# B.real = numpy.random.randn(m, p)
# B.imag = numpy.random.randn(m, p)
A = numpy.random.randn(m, n)
B = numpy.random.randn(m, p)
#logging.debug("m="+str(m)+" n="+str(n)+" p="+str(p))
AcB = numpy.c_[A, B]
ABBA = AcB.conj().T.dot(AcB)
AA = ABBA[0:n, 0:n]
AB = ABBA[0:n, n:]
BB = ABBA[n:, n:]
lastError = 1000
lastError2 = 1000
for k in range(1, n):
#logging.debug("k="+str(k))
#First compute eigen update estimate
omega, Q = numpy.linalg.eig(AA)
pi, V = EigenUpdater.eigenConcat(omega, Q, AB, BB, k)
ABBAEst = V.dot(numpy.diag(pi)).dot(V.conj().T)
t = min(k, Util.rank(ABBA))
self.assertTrue(pi.shape[0] == t)
self.assertTrue(
numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(t)) < tol)
#Second compute another eigen update estimate
omega, Q = numpy.linalg.eig(AA)
pi2, V2, D2, D2UD2 = EigenUpdater.lazyEigenConcatAsUpdate(
omega, Q, AB, BB, k, debug=True)
ABBAEst2 = V2.dot(numpy.diag(pi2)).dot(V2.conj().T)
U = ABBA.copy()
U[0:n, 0:n] = 0
self.assertTrue(
numpy.linalg.norm(U -
D2.dot(D2UD2).dot(D2.conj().T)) < tol)
t = min(k, Util.rank(ABBA))
self.assertTrue(
numpy.linalg.norm(V2.conj().T.dot(V2) -
numpy.eye(pi2.shape[0])) < tol)
#Compute estimate using eigendecomposition of full matrix
sfull, Vfull = numpy.linalg.eig(ABBA)
indsfull = numpy.flipud(numpy.argsort(numpy.abs(sfull)))
Vfull = Vfull[:, indsfull[0:k]]
sfull = sfull[indsfull[0:k]]
ABBAEstfull = Vfull.dot(numpy.diag(sfull)).dot(Vfull.conj().T)
#The errors should reduce
error = numpy.linalg.norm(ABBAEst - ABBA)
if Util.rank(ABBA) == k:
self.assertTrue(error <= tol)
lastError = error
error = numpy.linalg.norm(ABBAEst2 - ABBA)
self.assertTrue(error <= lastError2 + tol)
lastError2 = error
def clusterFromIterator(self, graphListIterator, verbose=False):
"""
Find a set of clusters for the graphs given by the iterator. If verbose
is true the each iteration is timed and bounded the results are returned
as lists.
The difference between a weight matrix and the previous one should be
positive.
"""
clustersList = []
decompositionTimeList = []
kMeansTimeList = []
boundList = []
sinThetaList = []
i = 0
for subW in graphListIterator:
if __debug__:
Parameter.checkSymmetric(subW)
if self.logStep and i % self.logStep == 0:
logging.debug("Graph index: " + str(i))
logging.debug("Clustering graph of size " + str(subW.shape))
if self.alg!="efficientNystrom":
ABBA = GraphUtils.shiftLaplacian(subW)
# --- Eigen value decomposition ---
startTime = time.time()
if self.alg=="IASC":
if i % self.T != 0:
omega, Q = self.approxUpdateEig(subW, ABBA, omega, Q)
if self.computeBound:
inds = numpy.flipud(numpy.argsort(omega))
Q = Q[:, inds]
omega = omega[inds]
bounds = self.pertBound(omega, Q, omegaKbot, AKbot, self.k2)
#boundList.append([i, bounds[0], bounds[1]])
#Now use accurate values of norm of R and delta
rank = Util.rank(ABBA.todense())
gamma, U = scipy.sparse.linalg.eigsh(ABBA, rank-1, which="LM", ncv = ABBA.shape[0])
#logging.debug("gamma=" + str(gamma))
bounds2 = self.realBound(omega, Q, gamma, AKbot, self.k2)
boundList.append([bounds[0], bounds[1], bounds2[0], bounds2[1]])
else:
logging.debug("Computing exact eigenvectors")
self.storeInformation(subW, ABBA)
if self.computeBound:
#omega, Q = scipy.sparse.linalg.eigsh(ABBA, min(self.k2*2, ABBA.shape[0]-1), which="LM", ncv = min(10*self.k2, ABBA.shape[0]))
rank = Util.rank(ABBA.todense())
omega, Q = scipy.sparse.linalg.eigsh(ABBA, rank-1, which="LM", ncv = ABBA.shape[0])
inds = numpy.flipud(numpy.argsort(omega))
omegaKbot = omega[inds[self.k2:]]
QKbot = Q[:, inds[self.k2:]]
AKbot = (QKbot*omegaKbot).dot(QKbot.T)
omegaSort = numpy.flipud(numpy.sort(omega))
boundList.append([0]*4)
else:
omega, Q = scipy.sparse.linalg.eigsh(ABBA, min(self.k2, ABBA.shape[0]-1), which="LM", ncv = min(10*self.k2, ABBA.shape[0]))
elif self.alg == "nystrom":
omega, Q = Nystrom.eigpsd(ABBA, self.k3)
elif self.alg == "exact":
omega, Q = scipy.sparse.linalg.eigsh(ABBA, min(self.k1, ABBA.shape[0]-1), which="LM", ncv = min(15*self.k1, ABBA.shape[0]))
elif self.alg == "efficientNystrom":
omega, Q = EfficientNystrom.eigWeight(subW, self.k2, self.k1)
elif self.alg == "randomisedSvd":
Q, omega, R = RandomisedSVD.svd(ABBA, self.k4)
else:
raise ValueError("Invalid Algorithm: " + str(self.alg))
if self.computeSinTheta:
omegaExact, QExact = scipy.linalg.eigh(ABBA.todense())
inds = numpy.flipud(numpy.argsort(omegaExact))
QExactKbot = QExact[:, inds[self.k1:]]
inds = numpy.flipud(numpy.argsort(omega))
QApproxK = Q[:,inds[:self.k1]]
sinThetaList.append(scipy.linalg.norm(QExactKbot.T.dot(QApproxK)))
decompositionTimeList.append(time.time()-startTime)
if self.alg=="IASC":
self.storeInformation(subW, ABBA)
# --- Kmeans ---
startTime = time.time()
inds = numpy.flipud(numpy.argsort(omega))
standardiser = Standardiser()
#For some very strange reason we get an overflow when computing the
#norm of the rows of Q even though its elements are bounded by 1.
#We'll ignore it for now
try:
V = standardiser.normaliseArray(Q[:, inds[0:self.k1]].real.T).T
except FloatingPointError as e:
logging.warn("FloatingPointError: " + str(e))
V = VqUtils.whiten(V)
if i == 0:
centroids, distortion = vq.kmeans(V, self.k1, iter=self.nb_iter_kmeans)
else:
centroids = self.findCentroids(V, clusters[:subW.shape[0]])
if centroids.shape[0] < self.k1:
nb_missing_centroids = self.k1 - centroids.shape[0]
random_centroids = V[numpy.random.randint(0, V.shape[0], nb_missing_centroids),:]
centroids = numpy.vstack((centroids, random_centroids))
centroids, distortion = vq.kmeans(V, centroids) #iter can only be 1
clusters, distortion = vq.vq(V, centroids)
kMeansTimeList.append(time.time()-startTime)
clustersList.append(clusters)
#logging.debug("subW.shape: " + str(subW.shape))
#logging.debug("len(clusters): " + str(len(clusters)))
#from sandbox.util.ProfileUtils import ProfileUtils
#logging.debug("Total memory usage: " + str(ProfileUtils.memory()/10**6) + "MB")
if ProfileUtils.memory() > 10**9:
ProfileUtils.memDisplay(locals())
i += 1
if verbose:
eigenQuality = {"boundList" : boundList, "sinThetaList" : sinThetaList}
return clustersList, numpy.array((decompositionTimeList, kMeansTimeList)).T, eigenQuality
else:
return clustersList
import numpy
import scipy.sparse
from apgl.graph import GraphUtils
from sandbox.util.Util import Util
numpy.set_printoptions(suppress=True, precision=3)
n = 10
W1 = scipy.sparse.rand(n, n, 0.5).todense()
W1 = W1.T.dot(W1)
W2 = W1.copy()
W2[1, 2] = 1
W2[2, 1] = 1
print("W1="+str(W1))
print("W2="+str(W2))
L1 = GraphUtils.normalisedLaplacianSym(scipy.sparse.csr_matrix(W1))
L2 = GraphUtils.normalisedLaplacianSym(scipy.sparse.csr_matrix(W2))
deltaL = L2 - L1
print("L1="+str(L1.todense()))
print("L2="+str(L2.todense()))
print("deltaL="+str(deltaL.todense()))
print("rank(deltaL)=" + str(Util.rank(deltaL.todense())))
def testEigenConcat(self):
tol = 10**-6
for i in range(3):
m = numpy.random.randint(10, 20)
n = numpy.random.randint(5, 10)
p = numpy.random.randint(5, 10)
# A = numpy.zeros((m, n), numpy.complex)
# B = numpy.zeros((m, p), numpy.complex)
# A.real = numpy.random.randn(m, n)
# A.imag = numpy.random.randn(m, n)
# B.real = numpy.random.randn(m, p)
# B.imag = numpy.random.randn(m, p)
A = numpy.random.randn(m, n)
B = numpy.random.randn(m, p)
#logging.debug("m="+str(m)+" n="+str(n)+" p="+str(p))
AcB = numpy.c_[A, B]
ABBA = AcB.conj().T.dot(AcB)
AA = ABBA[0:n, 0:n]
AB = ABBA[0:n, n:]
BB = ABBA[n:, n:]
lastError = 1000
lastError2 = 1000
for k in range(1,n):
#logging.debug("k="+str(k))
#First compute eigen update estimate
omega, Q = numpy.linalg.eig(AA)
pi, V = EigenUpdater.eigenConcat(omega, Q, AB, BB, k)
ABBAEst = V.dot(numpy.diag(pi)).dot(V.conj().T)
t = min(k, Util.rank(ABBA))
self.assertTrue(pi.shape[0] == t)
self.assertTrue(numpy.linalg.norm(V.conj().T.dot(V) - numpy.eye(t)) < tol)
#Second compute another eigen update estimate
omega, Q = numpy.linalg.eig(AA)
pi2, V2, D2, D2UD2 = EigenUpdater.lazyEigenConcatAsUpdate(omega, Q, AB, BB, k, debug=True)
ABBAEst2 = V2.dot(numpy.diag(pi2)).dot(V2.conj().T)
U = ABBA.copy()
U[0:n, 0:n] = 0
self.assertTrue(numpy.linalg.norm(U - D2.dot(D2UD2).dot(D2.conj().T)) < tol )
t = min(k, Util.rank(ABBA))
self.assertTrue(numpy.linalg.norm(V2.conj().T.dot(V2) - numpy.eye(pi2.shape[0])) < tol)
#Compute estimate using eigendecomposition of full matrix
sfull, Vfull = numpy.linalg.eig(ABBA)
indsfull = numpy.flipud(numpy.argsort(numpy.abs(sfull)))
Vfull = Vfull[:, indsfull[0:k]]
sfull = sfull[indsfull[0:k]]
ABBAEstfull = Vfull.dot(numpy.diag(sfull)).dot(Vfull.conj().T)
#The errors should reduce
error = numpy.linalg.norm(ABBAEst - ABBA)
if Util.rank(ABBA)==k:
self.assertTrue(error <= tol)
lastError = error
error = numpy.linalg.norm(ABBAEst2 - ABBA)
self.assertTrue(error <= lastError2+tol)
lastError2 = error
Xhat1 = X1 - numpy.outer(numpy.mean(X, 1), numpy.ones(numExamples1)) Xhat2 = X2 - numpy.outer(numpy.mean(X, 1), numpy.ones(numExamples2)) Xhat = numpy.c_[Xhat1, Xhat2] sigma = numpy.dot(Xhat.T, Xhat) sigma1 = numpy.dot(Xhat1.T, Xhat1) sigma2 = numpy.dot(Xhat1.T, Xhat2) sigma3 = numpy.dot(Xhat2.T, Xhat2) d, U = numpy.linalg.eig(sigma1) inds = numpy.flipud(numpy.argsort(d)) indsk = inds[0:k] #rank k approximation of sigma sigma1k = numpy.dot(U[:, indsk], numpy.dot(numpy.diag(d[indsk]), U[:, indsk].T )) ell = Util.rank(sigma1) Ptilde1 = numpy.dot(numpy.diag(numpy.sqrt(d[indsk])), U[:, indsk].T) Ptilde1 = numpy.r_[Ptilde1, numpy.zeros((ell-k, numExamples1))] LambdaTildeSq = numpy.diag(d[inds[0:ell]] ** -0.5) Utilde = U[:, inds[0:ell]] Q1 = numpy.dot(LambdaTildeSq, numpy.dot(Utilde.T, sigma2)) Q2 = numpy.zeros((numExamples2, numExamples1)) #Q3 is zero which is odd Q3 = scipy.linalg.sqrtm(sigma3 - numpy.dot(Q1.T, Q1)) Ptilde2 = numpy.r_[Ptilde1, Q2] Y = numpy.r_[Q1, Q3]
def clusterFromIterator(self, graphListIterator, verbose=False):
"""
Find a set of clusters for the graphs given by the iterator. If verbose
is true the each iteration is timed and bounded the results are returned
as lists.
The difference between a weight matrix and the previous one should be
positive.
"""
clustersList = []
decompositionTimeList = []
kMeansTimeList = []
boundList = []
sinThetaList = []
i = 0
for subW in graphListIterator:
if __debug__:
Parameter.checkSymmetric(subW)
if self.logStep and i % self.logStep == 0:
logging.debug("Graph index: " + str(i))
logging.debug("Clustering graph of size " + str(subW.shape))
if self.alg != "efficientNystrom":
ABBA = GraphUtils.shiftLaplacian(subW)
# --- Eigen value decomposition ---
startTime = time.time()
if self.alg == "IASC":
if i % self.T != 0:
omega, Q = self.approxUpdateEig(subW, ABBA, omega, Q)
if self.computeBound:
inds = numpy.flipud(numpy.argsort(omega))
Q = Q[:, inds]
omega = omega[inds]
bounds = self.pertBound(omega, Q, omegaKbot, AKbot,
self.k2)
#boundList.append([i, bounds[0], bounds[1]])
#Now use accurate values of norm of R and delta
rank = Util.rank(ABBA.todense())
gamma, U = scipy.sparse.linalg.eigsh(ABBA,
rank - 1,
which="LM",
ncv=ABBA.shape[0])
#logging.debug("gamma=" + str(gamma))
bounds2 = self.realBound(omega, Q, gamma, AKbot,
self.k2)
boundList.append(
[bounds[0], bounds[1], bounds2[0], bounds2[1]])
else:
logging.debug("Computing exact eigenvectors")
self.storeInformation(subW, ABBA)
if self.computeBound:
#omega, Q = scipy.sparse.linalg.eigsh(ABBA, min(self.k2*2, ABBA.shape[0]-1), which="LM", ncv = min(10*self.k2, ABBA.shape[0]))
rank = Util.rank(ABBA.todense())
omega, Q = scipy.sparse.linalg.eigsh(ABBA,
rank - 1,
which="LM",
ncv=ABBA.shape[0])
inds = numpy.flipud(numpy.argsort(omega))
omegaKbot = omega[inds[self.k2:]]
QKbot = Q[:, inds[self.k2:]]
AKbot = (QKbot * omegaKbot).dot(QKbot.T)
omegaSort = numpy.flipud(numpy.sort(omega))
boundList.append([0] * 4)
else:
omega, Q = scipy.sparse.linalg.eigsh(
ABBA,
min(self.k2, ABBA.shape[0] - 1),
which="LM",
ncv=min(10 * self.k2, ABBA.shape[0]))
elif self.alg == "nystrom":
omega, Q = Nystrom.eigpsd(ABBA, self.k3)
elif self.alg == "exact":
omega, Q = scipy.sparse.linalg.eigsh(
ABBA,
min(self.k1, ABBA.shape[0] - 1),
which="LM",
ncv=min(15 * self.k1, ABBA.shape[0]))
elif self.alg == "efficientNystrom":
omega, Q = EfficientNystrom.eigWeight(subW, self.k2, self.k1)
elif self.alg == "randomisedSvd":
Q, omega, R = RandomisedSVD.svd(ABBA, self.k4)
else:
raise ValueError("Invalid Algorithm: " + str(self.alg))
if self.computeSinTheta:
omegaExact, QExact = scipy.linalg.eigh(ABBA.todense())
inds = numpy.flipud(numpy.argsort(omegaExact))
QExactKbot = QExact[:, inds[self.k1:]]
inds = numpy.flipud(numpy.argsort(omega))
QApproxK = Q[:, inds[:self.k1]]
sinThetaList.append(
scipy.linalg.norm(QExactKbot.T.dot(QApproxK)))
decompositionTimeList.append(time.time() - startTime)
if self.alg == "IASC":
self.storeInformation(subW, ABBA)
# --- Kmeans ---
startTime = time.time()
inds = numpy.flipud(numpy.argsort(omega))
standardiser = Standardiser()
#For some very strange reason we get an overflow when computing the
#norm of the rows of Q even though its elements are bounded by 1.
#We'll ignore it for now
try:
V = standardiser.normaliseArray(Q[:, inds[0:self.k1]].real.T).T
except FloatingPointError as e:
logging.warn("FloatingPointError: " + str(e))
V = VqUtils.whiten(V)
if i == 0:
centroids, distortion = vq.kmeans(V,
self.k1,
iter=self.nb_iter_kmeans)
else:
centroids = self.findCentroids(V, clusters[:subW.shape[0]])
if centroids.shape[0] < self.k1:
nb_missing_centroids = self.k1 - centroids.shape[0]
random_centroids = V[numpy.random.randint(
0, V.shape[0], nb_missing_centroids), :]
centroids = numpy.vstack((centroids, random_centroids))
centroids, distortion = vq.kmeans(
V, centroids) #iter can only be 1
clusters, distortion = vq.vq(V, centroids)
kMeansTimeList.append(time.time() - startTime)
clustersList.append(clusters)
#logging.debug("subW.shape: " + str(subW.shape))
#logging.debug("len(clusters): " + str(len(clusters)))
#from sandbox.util.ProfileUtils import ProfileUtils
#logging.debug("Total memory usage: " + str(ProfileUtils.memory()/10**6) + "MB")
if ProfileUtils.memory() > 10**9:
ProfileUtils.memDisplay(locals())
i += 1
if verbose:
eigenQuality = {
"boundList": boundList,
"sinThetaList": sinThetaList
}
return clustersList, numpy.array(
(decompositionTimeList, kMeansTimeList)).T, eigenQuality
else:
return clustersList
Xhat2 = X2 - numpy.outer(numpy.mean(X, 1), numpy.ones(numExamples2))
Xhat = numpy.c_[Xhat1, Xhat2]
sigma = numpy.dot(Xhat.T, Xhat)
sigma1 = numpy.dot(Xhat1.T, Xhat1)
sigma2 = numpy.dot(Xhat1.T, Xhat2)
sigma3 = numpy.dot(Xhat2.T, Xhat2)
d, U = numpy.linalg.eig(sigma1)
inds = numpy.flipud(numpy.argsort(d))
indsk = inds[0:k]
#rank k approximation of sigma
sigma1k = numpy.dot(U[:, indsk], numpy.dot(numpy.diag(d[indsk]), U[:,
indsk].T))
ell = Util.rank(sigma1)
Ptilde1 = numpy.dot(numpy.diag(numpy.sqrt(d[indsk])), U[:, indsk].T)
Ptilde1 = numpy.r_[Ptilde1, numpy.zeros((ell - k, numExamples1))]
LambdaTildeSq = numpy.diag(d[inds[0:ell]]**-0.5)
Utilde = U[:, inds[0:ell]]
Q1 = numpy.dot(LambdaTildeSq, numpy.dot(Utilde.T, sigma2))
Q2 = numpy.zeros((numExamples2, numExamples1))
#Q3 is zero which is odd
Q3 = scipy.linalg.sqrtm(sigma3 - numpy.dot(Q1.T, Q1))
Ptilde2 = numpy.r_[Ptilde1, Q2]
Y = numpy.r_[Q1, Q3]
