def testUpdateSvd(self):
"""
Let's see if the update to the SVD works.
"""
numRuns = 10
for i in range(numRuns):
m, n = numpy.random.randint(10, 100), numpy.random.randint(10, 100)
k = 3
X = numpy.random.rand(m, n)
U, s, V = RandomisedSVD.svd(X, k)
E = numpy.random.randn(m, n) * 0.2
U2, s2, V2 = RandomisedSVD.svd(X + E, k)
U3, s3, V3 = RandomisedSVD.updateSvd(X, U, s, V, E, k)
XE = X + E
error1 = numpy.linalg.norm(XE - (U*s).dot(V.T))
error2 = numpy.linalg.norm(XE - (U2*s2).dot(V2.T))
error3 = numpy.linalg.norm(XE - (U3*s3).dot(V3.T))
self.assertTrue(error1 >= error3)
#print(error1, error2, error3)
#Test use of linear opertors
X = GeneralLinearOperator.asLinearOperator(X)
E = GeneralLinearOperator.asLinearOperator(E)
U3, s3, V3 = RandomisedSVD.updateSvd(X, U, s, V, E, k)
error4 = numpy.linalg.norm(XE - (U2*s2).dot(V2.T))
self.assertEquals(error4, error2)
def updateSvd(A, U, s, V, E, k, p=10):
"""
Given a matrix A whose approximate SVD is U s V.T, compute the SVD
of the new matrix A + E, using previous info. A and E are sparse
matrices. The rank of the approximation is p, and k is an oversampling
parameter.
"""
Parameter.checkInt(k, 1, float("inf"))
Parameter.checkInt(p, 0, float("inf"))
if isinstance(A, GeneralLinearOperator):
L = A
else:
L = GeneralLinearOperator.asLinearOperator(A)
if isinstance(E, GeneralLinearOperator):
M = E
else:
M = GeneralLinearOperator.asLinearOperator(E)
N = GeneralLinearOperator.asLinearOperatorSum(L, M)
n = A.shape[1]
omega = numpy.random.randn(n, p)
Y = U*s + M.matmat(V)
Y = numpy.c_[Y, N.matmat(omega)]
Q, R = numpy.linalg.qr(Y)
del omega
del Y
del R
gc.collect()
B = N.rmatmat(Q).T
U, s, V = numpy.linalg.svd(B, full_matrices=False)
del B
V = V.T
U = Q.dot(U)
U = U[:, 0:k]
s = s[0:k]
V = V[:, 0:k]
return U, s, V
def rsvd(A, k, p=10, q=2, omega=None):
"""
Compute the randomised SVD using the algorithm on page 9 of Halko et al.,
Finding Structure with randomness: stochastic algorithms for constructing
approximate matrix decompositions, 2009.
Finds the partial SVD of a sparse or dense matrix A, resolving the largest k
singular vectors/values, using exponent q and k+p projections. Returns the
left and right singular vectors, and the singular values. The resulting matrix
can be approximated using A ~ U s V.T. To improve the approximation quality
for a fixed k, increase p or q.
:param A: A sparse or dense matrix or GeneralLinearOperator
:param k: The number of singular values and random projections
:param p: The oversampling parameter
:param q: The exponent for the projections.
:param omega: An initial matrix to perform random projections onto with at least k columns
:return U: The left singular vectors
:return s: The singular values
:return V: The right singular vectors
"""
Parameter.checkInt(k, 1, float("inf"))
Parameter.checkInt(p, 0, float("inf"))
Parameter.checkInt(q, 0, float("inf"))
if isinstance(A, GeneralLinearOperator):
L = A
else:
L = GeneralLinearOperator.asLinearOperator(A)
n = L.shape[1]
if omega is None:
omega = numpy.random.randn(n, k + p)
else:
omega = numpy.c_[omega, numpy.random.randn(n, p + k - omega.shape[1])]
Y = L.matmat(omega)
Q, R = numpy.linalg.qr(Y)
del omega
for i in range(q):
Y = L.rmatmat(Q)
Q, R = numpy.linalg.qr(Y)
gc.collect()
Y = L.matmat(Q)
Q, R = numpy.linalg.qr(Y)
gc.collect()
del Y
del R
gc.collect()
B = L.rmatmat(Q).T
U, s, V = numpy.linalg.svd(B, full_matrices=False)
del B
V = V.T
U = Q.dot(U)
U = U[:, 0:k]
s = s[0:k]
V = V[:, 0:k]
return U, s, V
def rsvd(A, k, p=10, q=2, omega=None):
"""
Compute the randomised SVD using the algorithm on page 9 of Halko et al.,
Finding Structure with randomness: stochastic algorithms for constructing
approximate matrix decompositions, 2009.
Finds the partial SVD of a sparse or dense matrix A, resolving the largest k
singular vectors/values, using exponent q and k+p projections. Returns the
left and right singular vectors, and the singular values. The resulting matrix
can be approximated using A ~ U s V.T. To improve the approximation quality
for a fixed k, increase p or q.
:param A: A sparse or dense matrix or GeneralLinearOperator
:param k: The number of singular values and random projections
:param p: The oversampling parameter
:param q: The exponent for the projections.
:param omega: An initial matrix to perform random projections onto with at least k columns
:return U: The left singular vectors
:return s: The singular values
:return V: The right singular vectors
"""
Parameter.checkInt(k, 1, float("inf"))
Parameter.checkInt(p, 0, float("inf"))
Parameter.checkInt(q, 0, float("inf"))
if isinstance(A, GeneralLinearOperator):
L = A
else:
L = GeneralLinearOperator.asLinearOperator(A)
n = L.shape[1]
if omega is None:
omega = numpy.random.randn(n, k + p)
else:
omega = numpy.c_[omega, numpy.random.randn(n, p + k - omega.shape[1])]
Y = L.matmat(omega)
Q, R = numpy.linalg.qr(Y)
del omega
for i in range(q):
Y = L.rmatmat(Q)
Q, R = numpy.linalg.qr(Y)
gc.collect()
Y = L.matmat(Q)
Q, R = numpy.linalg.qr(Y)
gc.collect()
del Y
del R
gc.collect()
B = L.rmatmat(Q).T
U, s, V = numpy.linalg.svd(B, full_matrices=False)
del B
V = V.T
U = Q.dot(U)
U = U[:, 0:k]
s = s[0:k]
V = V[:, 0:k]
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)