def reconstructLowRank(U, s, V, k): """ Take the SVD of a low rank matrix and partially compute it with at most k values. If k is an array of values [0, U.shape[0]*V.shape[0]] then these indices are used for reconstruction. """ (m, n) = (U.shape[0], V.shape[0]) if type(k) == numpy.ndarray: inds = k inds = numpy.unique(inds) rowInds, colInds = numpy.unravel_index(inds, (m, n)) elif type(k) == tuple: rowInds, colInds = k else: inds = numpy.random.randint(0, n * m, k) inds = numpy.unique(inds) rowInds, colInds = numpy.unravel_index(inds, (m, n)) U = numpy.ascontiguousarray(U) V = numpy.ascontiguousarray(V) X = SparseUtilsCython.partialReconstructPQ((rowInds, colInds), U * s, V) return X
def reconstructLowRankPQ(P, Q, inds): """ Given an array of unique indices inds in [0, U.shape[0]*V.shape[0]-1], partially reconstruct $P*Q^T$. The returned matrix is a scipy csc_matrix. """ (m, n) = (P.shape[0], Q.shape[0]) if type(inds) == tuple: rowInds, colInds = inds rowInds = numpy.array(rowInds, numpy.int) colInds = numpy.array(colInds, numpy.int) else: rowInds, colInds = numpy.unravel_index(inds, (m, n)) X = SparseUtilsCython.partialReconstructPQ((rowInds, colInds), P, Q) return X
def reconstructLowRank(U, s, V, k): """ Take the SVD of a low rank matrix and partially compute it with at most k values. If k is an array of values [0, U.shape[0]*V.shape[0]] then these indices are used for reconstruction. """ (m, n) = (U.shape[0], V.shape[0]) if type(k) == numpy.ndarray: inds = k inds = numpy.unique(inds) rowInds, colInds = numpy.unravel_index(inds, (m, n)) elif type(k) == tuple: rowInds, colInds = k else: inds = numpy.random.randint(0, n*m, k) inds = numpy.unique(inds) rowInds, colInds = numpy.unravel_index(inds, (m, n)) U = numpy.ascontiguousarray(U) V = numpy.ascontiguousarray(V) X = SparseUtilsCython.partialReconstructPQ((rowInds, colInds), U*s, V) return X
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 learnModel(self, X, fullMatrices=True): """ Learn the matrix completion using a sparse matrix X. This is the simple version of the soft impute algorithm in which we store the entire matrices, newZ and oldZ. """ if not scipy.sparse.isspmatrix_csc(X): raise ValueError("Input matrix must be csc_matrix") (n, m) = X.shape oldU = numpy.zeros((n, 1)) oldS = numpy.zeros(1) oldV = numpy.zeros((m, 1)) omega = X.nonzero() tol = 10**-6 rowInds = numpy.array(omega[0], numpy.int) colInds = numpy.array(omega[1], numpy.int) ZList = [] for rho in self.rhos: gamma = self.eps + 1 i = 0 Y = scipy.sparse.csc_matrix(X, dtype=numpy.float) U, s, V = ExpSU.SparseUtils.svdArpack(Y, 1, kmax=20) lmbda = rho*numpy.max(s) while gamma > self.eps: ZOmega = SparseUtilsCython.partialReconstructPQ((rowInds, colInds), oldU*oldS, oldV) Y = X - ZOmega Y = Y.tocsc() newU, newS, newV = ExpSU.SparseUtils.svdSparseLowRank(Y, oldU, oldS, oldV) #Soft threshold newS = newS - lmbda newS = numpy.clip(newS, 0, numpy.max(newS)) normOldZ = (oldS**2).sum() normNewZmOldZ = (oldS**2).sum() + (newS**2).sum() - 2*numpy.trace((oldV.T.dot(newV*newS)).dot(newU.T.dot(oldU*oldS))) #We can get newZ == oldZ in which case we break if normNewZmOldZ < tol: gamma = 0 elif abs(normOldZ) < tol: gamma = self.eps + 1 else: gamma = normNewZmOldZ/normOldZ oldU = newU.copy() oldS = newS.copy() oldV = newV.copy() logging.debug("Iteration " + str(i) + " gamma="+str(gamma)) i += 1 logging.debug("Number of iterations for lambda="+str(rho) + ": " + str(i)) if fullMatrices: newZ = scipy.sparse.lil_matrix((newU*newS).dot(newV.T)) ZList.append(newZ) else: ZList.append((newU,newS,newV)) if self.rhos.shape[0] != 1: return ZList else: return ZList[0]
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)