def _addSparseRSVD(U, s, V, X, k=10, kX=None, kRand=None, q=None):
"""
Perform a randomised SVD of the matrix X + U diag(s) V.T. We use th
"""
if kX==None:
kX=k
if kRand==None:
kRand=k
if q==None:
q=1
m, n = X.shape
Us = U*s
kX = numpy.min([m, n, kX])
UX, sX, VX = SparseUtils.svdPropack(X, kX)
omega = numpy.c_[V, VX, numpy.random.randn(n, kRand)]
def rMultA(x):
return Us.dot(V.T.dot(x)) + X.dot(x)
def rMultAT(x):
return V.dot(Us.T.dot(x)) + X.T.dot(x)
Y = rMultA(omega)
for i in range(q):
Y = rMultAT(Y)
Y = rMultA(Y)
Q, R = numpy.linalg.qr(Y)
B = rMultAT(Q).T
U, s, VT = numpy.linalg.svd(B, full_matrices=False)
U, s, V = Util.indSvd(U, s, VT, numpy.flipud(numpy.argsort(s))[:k])
U = Q.dot(U)
return U, s, V
def profilePropackSvd(self):
dataDir = PathDefaults.getDataDir() + "erasm/contacts/"
trainFilename = dataDir + "contacts_train"
trainX = scipy.io.mmread(trainFilename)
trainX = scipy.sparse.csc_matrix(trainX, dtype=numpy.int8)
k = 500
U, s, V = SparseUtils.svdPropack(trainX, k, kmax=k * 5)
print(s)
#Memory consumption is dependent on kmax
print("All done")
def profilePropackSvd(self):
dataDir = PathDefaults.getDataDir() + "erasm/contacts/"
trainFilename = dataDir + "contacts_train"
trainX = scipy.io.mmread(trainFilename)
trainX = scipy.sparse.csc_matrix(trainX, dtype=numpy.int8)
k = 500
U, s, V = SparseUtils.svdPropack(trainX, k, kmax=k * 5)
print(s)
# Memory consumption is dependent on kmax
print("All done")
def testSvdPropack(self):
shape = (500, 100)
r = 5
k = 1000
X, U, s, V = SparseUtils.generateSparseLowRank(shape, r, k, verbose=True)
k2 = 10
U, s, V = SparseUtils.svdPropack(X, k2)
U2, s2, V2 = numpy.linalg.svd(X.todense())
V2 = V2.T
nptst.assert_array_almost_equal(s, s2[0:k2])
nptst.assert_array_almost_equal(numpy.abs(U), numpy.abs(U2[:, 0:k2]), 3)
nptst.assert_array_almost_equal(numpy.abs(V), numpy.abs(V2[:, 0:k2]), 3)
def testSvdPropack(self):
shape = (500, 100)
r = 5
k = 1000
X, U, s, V = SparseUtils.generateSparseLowRank(shape,
r,
k,
verbose=True)
k2 = 10
U, s, V = SparseUtils.svdPropack(X, k2)
U2, s2, V2 = numpy.linalg.svd(X.todense())
V2 = V2.T
nptst.assert_array_almost_equal(s, s2[0:k2])
nptst.assert_array_almost_equal(numpy.abs(U), numpy.abs(U2[:, 0:k2]),
3)
nptst.assert_array_almost_equal(numpy.abs(V), numpy.abs(V2[:, 0:k2]),
3)
def _addSparseProjected(U, s, V, X, k=10):
kk = len(s)
m, n = X.shape
# decompose X as X1 X2.T
inds = scipy.unique(X.nonzero()[1])
if len(inds) > 0:
X1 = numpy.array(X[:,inds].todense())
X2 = numpy.zeros((n, len(inds)))
X2[(inds, numpy.arange(len(inds)))] = 1
nptst.assert_array_almost_equal(X.todense(), X1.dot(X2.T))
# svd decomposition of projections of X1 and X2
UTX1 = U.T.dot(X1)
Q1, R1 = numpy.linalg.qr(X1-U.dot(UTX1))
k1 = Q1.shape[1]
VTX2 = V.T.dot(X2)
Q2, R2 = numpy.linalg.qr(X2-V.dot(VTX2))
k2 = Q2.shape[1]
# construct W
W = scipy.zeros((kk+k1, kk+k2))
W[(numpy.arange(k), numpy.arange(k))] = s
W[:kk,:kk] += UTX1.dot(VTX2.T)
W[:kk,kk:] = UTX1.dot(R2.T)
W[kk:,:kk] = R1.dot(VTX2.T)
W[kk:,kk:] = R1.dot(R2.T)
# svd of W
W = scipy.sparse.csc_matrix(W)
UW, sW, VW = SparseUtils.svdPropack(W, k)
VWT = VW.T
# reconstruct the correct decomposition
Ures = numpy.c_[U, Q1].dot(UW)
sres = sW
Vres = numpy.c_[V, Q2].dot(VWT.T)
return Ures, sres, Vres
def _addSparseProjected(U, s, V, X, k=10):
kk = len(s)
m, n = X.shape
# decompose X as X1 X2.T
inds = scipy.unique(X.nonzero()[1])
if len(inds) > 0:
X1 = numpy.array(X[:, inds].todense())
X2 = numpy.zeros((n, len(inds)))
X2[(inds, numpy.arange(len(inds)))] = 1
nptst.assert_array_almost_equal(X.todense(), X1.dot(X2.T))
# svd decomposition of projections of X1 and X2
UTX1 = U.T.dot(X1)
Q1, R1 = numpy.linalg.qr(X1 - U.dot(UTX1))
k1 = Q1.shape[1]
VTX2 = V.T.dot(X2)
Q2, R2 = numpy.linalg.qr(X2 - V.dot(VTX2))
k2 = Q2.shape[1]
# construct W
W = scipy.zeros((kk + k1, kk + k2))
W[(numpy.arange(k), numpy.arange(k))] = s
W[:kk, :kk] += UTX1.dot(VTX2.T)
W[:kk, kk:] = UTX1.dot(R2.T)
W[kk:, :kk] = R1.dot(VTX2.T)
W[kk:, kk:] = R1.dot(R2.T)
# svd of W
W = scipy.sparse.csc_matrix(W)
UW, sW, VW = SparseUtils.svdPropack(W, k)
VWT = VW.T
# reconstruct the correct decomposition
Ures = numpy.c_[U, Q1].dot(UW)
sres = sW
Vres = numpy.c_[V, Q2].dot(VWT.T)
return Ures, sres, Vres
def _addSparseRSVD(U, s, V, X, k=10, kX=None, kRand=None, q=None):
"""
Perform a randomised SVD of the matrix X + U diag(s) V.T. We use th
"""
if kX == None:
kX = k
if kRand == None:
kRand = k
if q == None:
q = 1
m, n = X.shape
Us = U * s
kX = numpy.min([m, n, kX])
UX, sX, VX = SparseUtils.svdPropack(X, kX)
omega = numpy.c_[V, VX, numpy.random.randn(n, kRand)]
def rMultA(x):
return Us.dot(V.T.dot(x)) + X.dot(x)
def rMultAT(x):
return V.dot(Us.T.dot(x)) + X.T.dot(x)
Y = rMultA(omega)
for i in range(q):
Y = rMultAT(Y)
Y = rMultA(Y)
Q, R = numpy.linalg.qr(Y)
B = rMultAT(Q).T
U, s, VT = numpy.linalg.svd(B, full_matrices=False)
U, s, V = Util.indSvd(U, s, VT, numpy.flipud(numpy.argsort(s))[:k])
U = Q.dot(U)
return U, s, V
def next(self):
X = self.XIterator.next()
logging.debug("Learning on matrix with shape: " +
str(X.shape) + " and " + str(X.nnz) +
" non-zeros")
if self.iterativeSoftImpute.weighted:
#Compute row and col probabilities
up, vp = SparseUtils.nonzeroRowColsProbs(X)
nzuInds = up == 0
nzvInds = vp == 0
u = numpy.sqrt(1 / (up + numpy.array(nzuInds, numpy.int)))
v = numpy.sqrt(1 / (vp + numpy.array(nzvInds, numpy.int)))
u[nzuInds] = 0
v[nzvInds] = 0
if self.rhos != None:
self.iterativeSoftImpute.setRho(self.rhos.next())
if not scipy.sparse.isspmatrix_csc(X):
raise ValueError("X must be a csc_matrix not " +
str(type(X)))
#Figure out what lambda should be
#PROPACK has problems with convergence
Y = scipy.sparse.csc_matrix(X, dtype=numpy.float)
U, s, V = ExpSU.SparseUtils.svdArpack(Y, 1, kmax=20)
del Y
#U, s, V = SparseUtils.svdPropack(X, 1, kmax=20)
maxS = s[0]
logging.debug("Largest singular value : " + str(maxS))
(n, m) = X.shape
if self.j == 0:
self.oldU = numpy.zeros((n, 1))
self.oldS = numpy.zeros(1)
self.oldV = numpy.zeros((m, 1))
else:
oldN = self.oldU.shape[0]
oldM = self.oldV.shape[0]
if self.iterativeSoftImpute.updateAlg == "initial":
if n > oldN:
self.oldU = Util.extendArray(
self.oldU, (n, self.oldU.shape[1]))
elif n < oldN:
self.oldU = self.oldU[0:n, :]
if m > oldM:
self.oldV = Util.extendArray(
self.oldV, (m, self.oldV.shape[1]))
elif m < oldN:
self.oldV = self.oldV[0:m, :]
elif self.iterativeSoftImpute.updateAlg == "zero":
self.oldU = numpy.zeros((n, 1))
self.oldS = numpy.zeros(1)
self.oldV = numpy.zeros((m, 1))
else:
raise ValueError("Unknown SVD update algorithm: " +
self.updateAlg)
rowInds, colInds = X.nonzero()
gamma = self.iterativeSoftImpute.eps + 1
i = 0
self.iterativeSoftImpute.measures = numpy.zeros(
(self.iterativeSoftImpute.maxIterations, 4))
while gamma > self.iterativeSoftImpute.eps:
if i == self.iterativeSoftImpute.maxIterations:
logging.debug("Maximum number of iterations reached")
break
ZOmega = SparseUtilsCython.partialReconstructPQ(
(rowInds, colInds), self.oldU * self.oldS, self.oldV)
Y = X - ZOmega
#Y = Y.tocsc()
#del ZOmega
Y = csarray(Y, storagetype="row")
gc.collect()
#os.system('taskset -p 0xffffffff %d' % os.getpid())
if self.iterativeSoftImpute.svdAlg == "propack":
L = LinOperatorUtils.sparseLowRankOp(Y,
self.oldU,
self.oldS,
self.oldV,
parallel=False)
newU, newS, newV = SparseUtils.svdPropack(
L,
k=self.iterativeSoftImpute.k,
kmax=self.iterativeSoftImpute.kmax)
elif self.iterativeSoftImpute.svdAlg == "arpack":
L = LinOperatorUtils.sparseLowRankOp(Y,
self.oldU,
self.oldS,
self.oldV,
parallel=False)
newU, newS, newV = SparseUtils.svdArpack(
L,
k=self.iterativeSoftImpute.k,
kmax=self.iterativeSoftImpute.kmax)
elif self.iterativeSoftImpute.svdAlg == "svdUpdate":
newU, newS, newV = SVDUpdate.addSparseProjected(
self.oldU, self.oldS, self.oldV, Y,
self.iterativeSoftImpute.k)
elif self.iterativeSoftImpute.svdAlg == "rsvd":
L = LinOperatorUtils.sparseLowRankOp(Y,
self.oldU,
self.oldS,
self.oldV,
parallel=True)
newU, newS, newV = RandomisedSVD.svd(
L,
self.iterativeSoftImpute.k,
p=self.iterativeSoftImpute.p,
q=self.iterativeSoftImpute.q)
elif self.iterativeSoftImpute.svdAlg == "rsvdUpdate":
L = LinOperatorUtils.sparseLowRankOp(Y,
self.oldU,
self.oldS,
self.oldV,
parallel=True)
if self.j == 0:
newU, newS, newV = RandomisedSVD.svd(
L,
self.iterativeSoftImpute.k,
p=self.iterativeSoftImpute.p,
q=self.iterativeSoftImpute.q)
else:
newU, newS, newV = RandomisedSVD.svd(
L,
self.iterativeSoftImpute.k,
p=self.iterativeSoftImpute.p,
q=self.iterativeSoftImpute.qu,
omega=self.oldV)
elif self.iterativeSoftImpute.svdAlg == "rsvdUpdate2":
if self.j == 0:
L = LinOperatorUtils.sparseLowRankOp(Y,
self.oldU,
self.oldS,
self.oldV,
parallel=True)
newU, newS, newV = RandomisedSVD.svd(
L,
self.iterativeSoftImpute.k,
p=self.iterativeSoftImpute.p,
q=self.iterativeSoftImpute.q)
else:
#Need linear operator which is U s V
L = LinOperatorUtils.lowRankOp(
self.oldU, self.oldS, self.oldV)
Y = GeneralLinearOperator.asLinearOperator(
Y, parallel=True)
newU, newS, newV = RandomisedSVD.updateSvd(
L,
self.oldU,
self.oldS,
self.oldV,
Y,
self.iterativeSoftImpute.k,
p=self.iterativeSoftImpute.p)
else:
raise ValueError("Unknown SVD algorithm: " +
self.iterativeSoftImpute.svdAlg)
if self.iterativeSoftImpute.weighted and i == 0:
delta = numpy.diag((u * newU.T).dot(newU))
pi = numpy.diag((v * newV.T).dot(newV))
lmbda = (maxS / numpy.max(
delta * pi)) * self.iterativeSoftImpute.rho
lmbdav = lmbda * delta * pi
elif not self.iterativeSoftImpute.weighted:
lmbda = maxS * self.iterativeSoftImpute.rho
if i == 0:
logging.debug("lambda: " + str(lmbda))
lmbdav = lmbda
newS = newS - lmbdav
#Soft threshold
newS = numpy.clip(newS, 0, numpy.max(newS))
normOldZ = (self.oldS**2).sum()
normNewZmOldZ = (self.oldS**2).sum() + (
newS**2).sum() - 2 * numpy.trace(
(self.oldV.T.dot(newV * newS)).dot(
newU.T.dot(self.oldU * self.oldS)))
#We can get newZ == oldZ in which case we break
if normNewZmOldZ < self.tol:
gamma = 0
elif abs(normOldZ) < self.tol:
gamma = self.iterativeSoftImpute.eps + 1
else:
gamma = normNewZmOldZ / normOldZ
if self.iterativeSoftImpute.verbose:
theta1 = (
self.iterativeSoftImpute.k -
numpy.linalg.norm(self.oldU.T.dot(newU), 'fro')**
2) / self.iterativeSoftImpute.k
theta2 = (
self.iterativeSoftImpute.k -
numpy.linalg.norm(self.oldV.T.dot(newV), 'fro')**
2) / self.iterativeSoftImpute.k
thetaS = numpy.linalg.norm(
newS - self.oldS)**2 / numpy.linalg.norm(newS)**2
self.iterativeSoftImpute.measures[i, :] = numpy.array(
[gamma, theta1, theta2, thetaS])
self.oldU = newU.copy()
self.oldS = newS.copy()
self.oldV = newV.copy()
logging.debug("Iteration " + str(i) + " gamma=" +
str(gamma))
i += 1
if self.iterativeSoftImpute.postProcess:
#Add the mean vectors
previousS = newS
newU = numpy.c_[newU, numpy.array(X.mean(1)).ravel()]
newV = numpy.c_[newV, numpy.array(X.mean(0)).ravel()]
newS = self.iterativeSoftImpute.unshrink(X, newU, newV)
#Note that this increases the rank of U and V by 1
#print("Difference in s after postprocessing: " + str(numpy.linalg.norm(previousS - newS[0:-1])))
logging.debug("Difference in s after postprocessing: " +
str(numpy.linalg.norm(previousS -
newS[0:-1])))
logging.debug("Number of iterations for rho=" +
str(self.iterativeSoftImpute.rho) + ": " +
str(i))
self.j += 1
return (newU, newS, newV)
def next(self):
X = self.XIterator.next()
logging.debug("Learning on matrix with shape: " + str(X.shape) + " and " + str(X.nnz) + " non-zeros")
if self.iterativeSoftImpute.weighted:
#Compute row and col probabilities
up, vp = SparseUtils.nonzeroRowColsProbs(X)
nzuInds = up==0
nzvInds = vp==0
u = numpy.sqrt(1/(up + numpy.array(nzuInds, numpy.int)))
v = numpy.sqrt(1/(vp + numpy.array(nzvInds, numpy.int)))
u[nzuInds] = 0
v[nzvInds] = 0
if self.rhos != None:
self.iterativeSoftImpute.setRho(self.rhos.next())
if not scipy.sparse.isspmatrix_csc(X):
raise ValueError("X must be a csc_matrix not " + str(type(X)))
#Figure out what lambda should be
#PROPACK has problems with convergence
Y = scipy.sparse.csc_matrix(X, dtype=numpy.float)
U, s, V = ExpSU.SparseUtils.svdArpack(Y, 1, kmax=20)
del Y
#U, s, V = SparseUtils.svdPropack(X, 1, kmax=20)
maxS = s[0]
logging.debug("Largest singular value : " + str(maxS))
(n, m) = X.shape
if self.j == 0:
self.oldU = numpy.zeros((n, 1))
self.oldS = numpy.zeros(1)
self.oldV = numpy.zeros((m, 1))
else:
oldN = self.oldU.shape[0]
oldM = self.oldV.shape[0]
if self.iterativeSoftImpute.updateAlg == "initial":
if n > oldN:
self.oldU = Util.extendArray(self.oldU, (n, self.oldU.shape[1]))
elif n < oldN:
self.oldU = self.oldU[0:n, :]
if m > oldM:
self.oldV = Util.extendArray(self.oldV, (m, self.oldV.shape[1]))
elif m < oldN:
self.oldV = self.oldV[0:m, :]
elif self.iterativeSoftImpute.updateAlg == "zero":
self.oldU = numpy.zeros((n, 1))
self.oldS = numpy.zeros(1)
self.oldV = numpy.zeros((m, 1))
else:
raise ValueError("Unknown SVD update algorithm: " + self.updateAlg)
rowInds, colInds = X.nonzero()
gamma = self.iterativeSoftImpute.eps + 1
i = 0
self.iterativeSoftImpute.measures = numpy.zeros((self.iterativeSoftImpute.maxIterations, 4))
while gamma > self.iterativeSoftImpute.eps:
if i == self.iterativeSoftImpute.maxIterations:
logging.debug("Maximum number of iterations reached")
break
ZOmega = SparseUtilsCython.partialReconstructPQ((rowInds, colInds), self.oldU*self.oldS, self.oldV)
Y = X - ZOmega
#Y = Y.tocsc()
#del ZOmega
Y = csarray(Y, storagetype="row")
gc.collect()
#os.system('taskset -p 0xffffffff %d' % os.getpid())
if self.iterativeSoftImpute.svdAlg=="propack":
L = LinOperatorUtils.sparseLowRankOp(Y, self.oldU, self.oldS, self.oldV, parallel=False)
newU, newS, newV = SparseUtils.svdPropack(L, k=self.iterativeSoftImpute.k, kmax=self.iterativeSoftImpute.kmax)
elif self.iterativeSoftImpute.svdAlg=="arpack":
L = LinOperatorUtils.sparseLowRankOp(Y, self.oldU, self.oldS, self.oldV, parallel=False)
newU, newS, newV = SparseUtils.svdArpack(L, k=self.iterativeSoftImpute.k, kmax=self.iterativeSoftImpute.kmax)
elif self.iterativeSoftImpute.svdAlg=="svdUpdate":
newU, newS, newV = SVDUpdate.addSparseProjected(self.oldU, self.oldS, self.oldV, Y, self.iterativeSoftImpute.k)
elif self.iterativeSoftImpute.svdAlg=="rsvd":
L = LinOperatorUtils.sparseLowRankOp(Y, self.oldU, self.oldS, self.oldV, parallel=True)
newU, newS, newV = RandomisedSVD.svd(L, self.iterativeSoftImpute.k, p=self.iterativeSoftImpute.p, q=self.iterativeSoftImpute.q)
elif self.iterativeSoftImpute.svdAlg=="rsvdUpdate":
L = LinOperatorUtils.sparseLowRankOp(Y, self.oldU, self.oldS, self.oldV, parallel=True)
if self.j == 0:
newU, newS, newV = RandomisedSVD.svd(L, self.iterativeSoftImpute.k, p=self.iterativeSoftImpute.p, q=self.iterativeSoftImpute.q)
else:
newU, newS, newV = RandomisedSVD.svd(L, self.iterativeSoftImpute.k, p=self.iterativeSoftImpute.p, q=self.iterativeSoftImpute.qu, omega=self.oldV)
elif self.iterativeSoftImpute.svdAlg=="rsvdUpdate2":
if self.j == 0:
L = LinOperatorUtils.sparseLowRankOp(Y, self.oldU, self.oldS, self.oldV, parallel=True)
newU, newS, newV = RandomisedSVD.svd(L, self.iterativeSoftImpute.k, p=self.iterativeSoftImpute.p, q=self.iterativeSoftImpute.q)
else:
#Need linear operator which is U s V
L = LinOperatorUtils.lowRankOp(self.oldU, self.oldS, self.oldV)
Y = GeneralLinearOperator.asLinearOperator(Y, parallel=True)
newU, newS, newV = RandomisedSVD.updateSvd(L, self.oldU, self.oldS, self.oldV, Y, self.iterativeSoftImpute.k, p=self.iterativeSoftImpute.p)
else:
raise ValueError("Unknown SVD algorithm: " + self.iterativeSoftImpute.svdAlg)
if self.iterativeSoftImpute.weighted and i==0:
delta = numpy.diag((u*newU.T).dot(newU))
pi = numpy.diag((v*newV.T).dot(newV))
lmbda = (maxS/numpy.max(delta*pi))*self.iterativeSoftImpute.rho
lmbdav = lmbda*delta*pi
elif not self.iterativeSoftImpute.weighted:
lmbda = maxS*self.iterativeSoftImpute.rho
if i==0:
logging.debug("lambda: " + str(lmbda))
lmbdav = lmbda
newS = newS - lmbdav
#Soft threshold
newS = numpy.clip(newS, 0, numpy.max(newS))
normOldZ = (self.oldS**2).sum()
normNewZmOldZ = (self.oldS**2).sum() + (newS**2).sum() - 2*numpy.trace((self.oldV.T.dot(newV*newS)).dot(newU.T.dot(self.oldU*self.oldS)))
#We can get newZ == oldZ in which case we break
if normNewZmOldZ < self.tol:
gamma = 0
elif abs(normOldZ) < self.tol:
gamma = self.iterativeSoftImpute.eps + 1
else:
gamma = normNewZmOldZ/normOldZ
if self.iterativeSoftImpute.verbose:
theta1 = (self.iterativeSoftImpute.k - numpy.linalg.norm(self.oldU.T.dot(newU), 'fro')**2)/self.iterativeSoftImpute.k
theta2 = (self.iterativeSoftImpute.k - numpy.linalg.norm(self.oldV.T.dot(newV), 'fro')**2)/self.iterativeSoftImpute.k
thetaS = numpy.linalg.norm(newS - self.oldS)**2/numpy.linalg.norm(newS)**2
self.iterativeSoftImpute.measures[i, :] = numpy.array([gamma, theta1, theta2, thetaS])
self.oldU = newU.copy()
self.oldS = newS.copy()
self.oldV = newV.copy()
logging.debug("Iteration " + str(i) + " gamma="+str(gamma))
i += 1
if self.iterativeSoftImpute.postProcess:
#Add the mean vectors
previousS = newS
newU = numpy.c_[newU, numpy.array(X.mean(1)).ravel()]
newV = numpy.c_[newV, numpy.array(X.mean(0)).ravel()]
newS = self.iterativeSoftImpute.unshrink(X, newU, newV)
#Note that this increases the rank of U and V by 1
#print("Difference in s after postprocessing: " + str(numpy.linalg.norm(previousS - newS[0:-1])))
logging.debug("Difference in s after postprocessing: " + str(numpy.linalg.norm(previousS - newS[0:-1])))
logging.debug("Number of iterations for rho="+str(self.iterativeSoftImpute.rho) + ": " + str(i))
self.j += 1
return (newU, newS, newV)
ps = [0]
qs = [1, 2]
errors = []
times = []
for i, X in enumerate(trainIterator):
print(i)
if i == 10:
break
tempTimes = []
tempErrors = []
startTime = time.time()
U, s, V = SparseUtils.svdPropack(X, k)
tempTimes.append(time.time()-startTime)
tempErrors.append(numpy.linalg.norm(numpy.array(X.todense()) - (U*s).dot(V.T))/numpy.linalg.norm(X.todense()))
for p in ps:
for q in qs:
startTime = time.time()
U2, s2, V2 = RandomisedSVD.svd(X, k, p, q)
tempTimes.append(time.time()-startTime)
tempErrors.append(numpy.linalg.norm(numpy.array(X.todense()) - (U2*s2).dot(V2.T))/numpy.linalg.norm(X.todense()) )
startTime = time.time()
if i == 0:
U3, s3, V3 = RandomisedSVD.svd(X, k, p, q)
else:
U3, s3, V3 = RandomisedSVD.svd(X, k, p, q, omega=lastV)
