def cos_similarity(vec1,vec2):
dot_product=np.dot(vec1,vec2)
norm_vec1=linalg.norm(vec1)
norm_vec2=linalg.norm(vec2)
if norm_vec1*norm_vec2!=0:
return(dot_product/(norm_vec1*norm_vec2))
else: return 0
def corrSparsityCliques(x, obj, constraints):
n = len(x) # Independent variable vector size
C = identity(
n, format='lil'
) # Initialising appropriate identity matrix (lil_matrix format)
# Retrieve cross-terms of objective function and update C matrix
objectiveCrossDependencies = getObjectiveCrossDependencies(obj, x)
updateCWithCrossDependencies(objectiveCrossDependencies, C)
# Retrieve co-dependent terms for every constraint and update C matrix
for constraint in constraints:
constraintCrossDependencies = getConstraintCodependencies(
constraint, x)
updateCWithCrossDependencies(constraintCrossDependencies, C)
C = csc_matrix(
C) # Convert into CSC structure, most efficient for next steps
C = C + norm(C, ord=1) * identity(
n, format='csc') # Ensure strict diagonal dominance for C
cliqueStructure = cliquesFromSpMatD(
C) # Large function that will generate clique structure
return cliqueStructure
def l2_normalize(self):
'''
L2-normalize all vectors in the matrix.
'''
l2norm = linalg.norm(self.matrix, axis=1, ord=2)
l2norm[l2norm==0.0] = 1.0 # Convert 0 values to 1
self.matrix = csr_matrix(self.matrix/l2norm.reshape(len(l2norm),1))
def get_pagerank(adj_mat, theta=.85, epsilon=1e-03, max_iter=20):
"""
Returns the vector of pagerank scores
:param adj_mat: (scipy.sparse.csc.csc_matrix) n x n
:param theta: (numeric) damping factor
:param epsilon: (numeric) convergence parameter
:return: vector of pagerank scores n x 1
"""
n = adj_mat.shape[0]
g_mat = get_gmat(adj_mat, theta)
pr_vec = sparse.csc_matrix(np.ones(n)) / n
norm_iter = adj_mat.shape[0]
n = adj_mat.shape[0]
i = 0
norm = []
while (norm_iter > epsilon * n) and (i < max_iter):
pr_iter = pr_vec.dot(g_mat)
norm_iter = linalg.norm(pr_vec - pr_iter)
pr_vec = pr_iter
i += 1
norm += [norm_iter]
print("iter {0}: {1}".format(i, norm_iter))
return pr_vec.T
def normalize_sparse(M, norm="frag", order=1, iterations=3):
"""Applies a normalization type to a sparse matrix.
"""
try:
from scipy.sparse import csr_matrix
except ImportError as e:
print(str(e))
print("I am peforming dense normalization by default.")
return normalize_dense(M.todense())
r = csr_matrix(M)
if norm == "SCN":
for _ in range(1, iterations):
row_sums = np.array(r.sum(axis=1)).flatten()
col_sums = np.array(r.sum(axis=0)).flatten()
row_indices, col_indices = r.nonzero()
r.data /= row_sums[row_indices] * col_sums[col_indices]
elif norm == "global":
try:
from scipy.sparse import linalg
r = linalg.norm(M, ord=order)
except (ImportError, AttributeError) as e:
print(str(e))
print("I can't import linalg tools for sparse matrices.")
print("Please upgrade your scipy version to 0.16.0.")
elif callable(norm):
r = norm(M)
else:
print("Unknown norm. Returning input as fallback")
return r
def normalize_sparse(M, norm="frag", order=1, iterations=3):
"""Applies a normalization type to a sparse matrix."""
try:
from scipy.sparse import csr_matrix
except ImportError as e:
print(str(e))
print("I am peforming dense normalization by default.")
return normalize_dense(M.todense())
r = csr_matrix(M)
if norm == "SCN":
for iteration in range(1, iterations):
row_sums = np.array(r.sum(axis=1)).flatten()
col_sums = np.array(r.sum(axis=0)).flatten()
row_indices, col_indices = r.nonzero()
r.data /= row_sums[row_indices] * col_sums[col_indices]
elif norm == "global":
try:
from scipy.sparse import linalg
r = linalg.norm(M, ord=order)
except (ImportError, AttributeError) as e:
print(str(e))
print("I can't import linalg tools for sparse matrices.")
print("Please upgrade your scipy version to 0.16.0.")
elif callable(norm):
r = norm(M)
else:
print("I don't recognize this norm, I am returning input matrix by default.")
return r
def run_iterative(T, I, neg_percent_tr, links_te, signs_te, c,
convergence_threshold):
R = T.copy().asformat('csc') #intialize with T
R_ = T.copy().asformat('csc') #itialize with T
norm2 = 999999999
it = 0
while norm2 > convergence_threshold:
if it % 2 == 0:
R_ = c * (T.dot(R)) + (1 - c) * I
else:
R = c * (T.dot(R_)) + (1 - c) * I
norm2 = norm(R - R_)
print('Iteration {} and difference {}'.format(it, norm2))
it += 1
#uncomment to see progress while converging
#if it % 2 == 0:
# evaluate(R.copy().asformat('dok'), neg_percent_tr, links_te, signs_te)
#else:
# evaluate(R_.copy().asformat('dok'), neg_percent_tr, links_te, signs_te)
#get final evaluation
#if we quit when it = 2 that means it = 1 was the last to execute
#and so R was the last result
if it % 2 == 0:
evaluate(R.copy().asformat('dok'), neg_percent_tr, links_te, signs_te)
else:
evaluate(R_.copy().asformat('dok'), neg_percent_tr, links_te, signs_te)
def normalize_test(self, K_test, feats_test): #K_test unormalized
m = K_test.shape[0]
#norms_test = np.sum(feats_test*feats_test,axis=1)
norms_test = norm(feats_test,axis=1)
matrix_norms = np.outer(norms_test, self.norms_train) #+ 1e-40 # matrix sqrt(K(xtest,xtest)*K(xtrain,xtrain))
K_test = np.divide(K_test, matrix_norms)
return K_test
def to_svd(beg=200, end=2613, jump=300, with_norm=True):
"""
Wykonuje SVD na rzadkiej macierzy wczytanej z with_idf
Zapisuej dla róznych wartosci k pod nazwa k, gdzie k = 1,2,3...A.shape-1 A-macierz
:param beg: od ktorego k
:param end: do jakiego k
:param jump: o jakie k
:param with_norm: noramlizować przed zapisem
:return: zapisuje macierze rzadkie do plikow
"""
spare = load_sparse_csr('usage_files/with_idf.npz')
x = []
for k in range(beg, end, jump):
print("Starting: " + str(k))
U, sigma, V = svds(spare, k=k)
print("Done svd")
reconsign = csc_matrix(U) * diags(sigma, format='csc') * csc_matrix(V)
print("Done multiplication")
if with_norm:
x = [(1 / norm(reconsign.getcol(i)))
for i in range(reconsign.shape[1])]
reconsign = reconsign.multiply(csc_matrix(x))
x = []
save_sparse('svd/' + str(k), csc_matrix(reconsign))
print("Saved " + str(k))
def lbfgs_step(gfk, k, vecs, props):
m = props.get("lbfgsMemory", 10)
q = gfk
a = {}
if k == 0:
return -gfk / linalg.norm(gfk, numpy.inf)
k = numpy.max(vecs.keys())+1
bl = max(0, k-m)
for i in range(k-1, bl-1, -1):
(sk, yk, rhok) = vecs[i]
a[i] = rhok * numpy.dot(sk, q)
q = q - a[i]*yk
(sk, yk, rhok) = vecs[k-1]
gammak = numpy.dot(sk,yk)/(numpy.dot(yk,yk))
r = gammak * q
for i in range(bl, k):
(sk, yk, rhok) = vecs[i]
beta = rhok * numpy.dot(yk, r)
r = r + sk*(a[i]-beta)
return -r
def orthogonality(A, g):
"""Measure orthogonality between a vector and the null space of a matrix.
Compute a measure of orthogonality between the null space
of the (possibly sparse) matrix ``A`` and a given vector ``g``.
The formula is a simplified (and cheaper) version of formula (3.13)
from [1]_.
``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``.
References
----------
.. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal.
"On the solution of equality constrained quadratic
programming problems arising in optimization."
SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395.
"""
# Compute vector norms
norm_g = np.linalg.norm(g)
# Compute Frobenius norm of the matrix A
if issparse(A):
norm_A = linalg.norm(A, ord='fro')
else:
norm_A = np.linalg.norm(A, ord='fro')
# Check if norms are zero
if norm_g == 0 or norm_A == 0:
return 0
norm_A_g = np.linalg.norm(A.dot(g))
# Orthogonality measure
orth = norm_A_g / (norm_A * norm_g)
return orth
def baseline():
filename = './data/GSM3067191_08hpf.csv' # 8hpf Zebrafish embryos
df = pd.read_csv(filename)
# Each row now corresponds to a cell (example) and each column to a gene (feature)
data = df.values[:, 1:].astype(np.float).T
# Make data sparse
sata = sparse.dok_matrix(data)
k = 50
lsvec, svals, rsvect = la.svd(data)
dnorm = sla.norm(sata)
approx = lsvec.dot(np.diag(svals)).dot(rsvect)
print("SVD reconstruction error:", la.norm(sata - approx) / dnorm)
avgs = np.sum(data[:], axis=0) / data.shape[0]
plt.plot(avgs)
plt.show()
expression_counts = np.sum(sata, axis=0)
best = np.array(np.argsort(-expression_counts))[0]
common = best[:k]
uncommon = best[k:]
common_norm = la.norm(data[:, common])
print("Baseline reconstruction error:", np.sqrt(dnorm**2 - common_norm** 2) / dnorm)
def cosine_similarities(raw_texts,
dictionary_path='dictionary/AmericanDictionary.csv'):
dictionary = read_csv(dictionary_path, sep=';', error_bad_lines=False)
documents_per_code = dictionary.groupby('Icd1')['DiagnosisText'].agg(
lambda x: ' '.join(x))
tfidf_mapper = TfidfVectorizer()
dictionary_vectors = tfidf_mapper.fit_transform(documents_per_code)
raw_texts_vectors = tfidf_mapper.transform(raw_texts)
products = raw_texts_vectors.dot(dictionary_vectors.T)
raw_norms = norm(raw_texts_vectors, axis=1)
dictionary_norms = norm(dictionary_vectors, axis=1)
raw_norms[raw_norms == 0] = 1.0
dictionary_norms[dictionary_norms == 0] = 1.0
d = products / np.expand_dims(raw_norms, axis=1)
d /= np.expand_dims(dictionary_norms, axis=0)
return d
def gs_solve(D,b,lam):
"Solve the smoothing problem with Gauss-Seidel iteration. Variables same as assignment."
(m,n) = np.shape(D)
ident = spsp.identity(n)
A = lam * D.T @ D + ident
# the matrix a is our target, we want to solve lx = b. gauss-seidel splits
# this into two components: lower and strictly-upper triangular
L = spsp.tril(A, format='csc')
U = A - L
assert(spspla.norm( A - (L + U), inf) < 1e-3 )
Linv = spspla.inv(L)
guess = spsp.csc_matrix(np.zeros(pred_shape(A)))
err = 10
while err > tol:
temp = (b - U @ guess)
guess = Linv @ temp
result = A @ guess
err = npla.norm(result - b, 2)
# print(err)
return guess
def propagation(M, adj, alpha=0.7, tol=10e-6): # TODO equation, M, alpha
"""Network propagation iterative process
Iterative algorithm for apply propagation using random walk on a network:
Initialize::
X1 = M
Repeat::
X2 = alpha * X1.A + (1-alpha) * M
X1 = X2
Until::
norm(X2-X1) < tol
Where::
A : degree-normalized adjacency matrix
Parameters
----------
M : sparse matrix
Data matrix to be diffused.
adj : sparse matrix
Adjacency matrice.
alpha : float, default: 0.7
Diffusion/propagation factor with 0 <= alpha <= 1.
For alpha = 0 : no diffusion.
For alpha = 1 :
tol : float, default: 10e-6
Convergence threshold.
Returns
-------
X2 : sparse matrix
Smoothed matrix.
"""
n = adj.shape[0]
# diagonal = 1 -> degree
# TODO to set diagonal = 0 before applying eye
adj = adj + sp.eye(n, dtype=np.float32)
d = sp.dia_matrix((np.array(adj.sum(axis=0))**-1, [0]),
shape=(n, n),
dtype=np.float32)
A = adj.dot(d)
X1 = M.astype(np.float32)
X2 = alpha * X1.dot(A) + (1 - alpha) * M
i = 0
while norm(X2 - X1) > tol:
X1 = X2
X2 = alpha * X1.dot(A) + (1 - alpha) * M
i += 1
print(' Propagation iteration = {} ----- {}'.format(
i,
datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")))
return X2
def katz_hungarian_connection(adj,
nodes,
n_perturbations,
threshold=0.000001,
nsteps=10000,
dataset_name=None):
graph = nx.from_scipy_sparse_matrix(adj, create_using=nx.Graph)
rows = nodes[:n_perturbations]
cols = nodes[n_perturbations:]
precomputed_path = f'data/tmp/{dataset_name}_katz.pkl'
if dataset_name is not None and os.path.exists(precomputed_path):
print("Loading precomputed_katz...")
with open(precomputed_path, 'r') as ff:
sigma = pickle.load(open(precomputed_path, 'rb'))
else:
D = nx.linalg.laplacianmatrix.laplacian_matrix(graph) + adj
D_inv = spalg.inv(D)
D_invA = D_inv * adj
l, v = spalg.eigs(D_invA, k=1, which="LR")
lmax = l[0].real
alpha = (1 / lmax) * 0.9
sigma = sp.csr_matrix(D_invA.shape, dtype=np.float)
print('Calculate sigma matrix')
for i in range(nsteps):
sigma_new = alpha * D_invA * sigma + sp.identity(
adj.shape[0], dtype=np.float, format='csr')
diff = abs(spalg.norm(sigma, 1) - spalg.norm(sigma_new, 1))
sigma = sigma_new
print(diff)
if diff < threshold:
break
print('Number of steps taken: ' + str(i))
sigma = sigma.toarray().astype('float')
if dataset_name is not None:
pickle.dump(sigma, open(precomputed_path, "wb"))
similarity = {u: {v: sigma[u][v] for v in cols} for u in rows}
mtx = np.array(
[np.array(list(similarity[u].values())) for u in similarity])
i_u = {i: u for i, u in enumerate(similarity)}
i_v = {i: v for i, v in enumerate(similarity[list(similarity.keys())[0]])}
u, v = linear_sum_assignment(+mtx)
return [[i_u[i], i_v[j]] for i, j in zip(u, v)]
def propagation(M, adj, alpha=0.7, tol=10e-6): # TODO equation, M, alpha
"""Network propagation iterative process
Iterative algorithm for apply propagation using random walk on a network:
Initialize::
X1 = M
Repeat::
X2 = alpha * X1.A + (1-alpha) * M
X1 = X2
Until::
norm(X2-X1) < tol
Where::
A : degree-normalized adjacency matrix
Parameters
----------
M : sparse matrix
Data matrix to be diffused.
adj : sparse matrix
Adjacency matrice.
alpha : float, default: 0.7
Diffusion/propagation factor with 0 <= alpha <= 1.
For alpha = 0 : no diffusion.
For alpha = 1 :
tol : float, default: 10e-6
Convergence threshold.
Returns
-------
X2 : sparse matrix
Smoothed matrix.
"""
n = adj.shape[0]
# diagonal = 1 -> degree
# TODO to set diagonal = 0 before applying eye
adj = adj+sp.eye(n, dtype=np.float32)
d = sp.dia_matrix((np.array(adj.sum(axis=0))**-1, [0]),
shape=(n, n),
dtype=np.float32)
A = adj.dot(d)
X1 = M.astype(np.float32)
X2 = alpha * X1.dot(A) + (1-alpha) * M
i = 0
while norm(X2-X1) > tol:
X1 = X2
X2 = alpha * X1.dot(A) + (1-alpha) * M
i += 1
print(' Propagation iteration = {} ----- {}'.format(
i, datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")))
return X2
def _compute_cost(self):
pattern = self._R != 0
proj_ABt = pattern.multiply((self._A).dot(self._B.T))
cost = splinalg.norm((proj_ABt - self._R).multiply(
self._ptwise_sqrt_W_sparse), 'fro') ** 2
norm_A = self._Lambda * (np.linalg.norm(self._A, 'fro')) ** 2
norm_B = self._Lambda * (np.linalg.norm(self._B, 'fro')) ** 2
return cost + norm_A + norm_B
def sparse_cosine(x_mat: _SPARSE_SCIPY_TYPES,
y_mat: _SPARSE_SCIPY_TYPES) -> 'np.ndarray':
"""Cosine distance between each row in x_mat and each row in y_mat.
:param x_mat: scipy.sparse like array with ndim=2
:param y_mat: scipy.sparse like array with ndim=2
:return: np.ndarray with ndim=2
"""
from scipy.sparse.linalg import norm
# we need the np.asarray otherwise we get a np.matrix object that iterates differently
return 1 - np.clip(
np.asarray(
x_mat.dot(y_mat.T) /
(np.outer(norm(x_mat, axis=1), norm(y_mat, axis=1)))),
-1,
1,
)
def _test_symmetry_W(problem_filename):
problem = read_fclib_format(problem_filename)[1]
A = csr_matrix(N.SBM_to_sparse(problem.M)[1])
#print("A=", A)
#print("A=", np.transpose(A))
#print("A-A^T",A-np.transpose(A))
print("norm A-A^T", linalg.norm(A - np.transpose(A)))
is_symmetric = False
symmetry_test = linalg.norm(A - np.transpose(A), np.inf)
print("symmetry_test === ", symmetry_test)
is_symmetric = (symmetry_test <= 1e-12)
is_dd, dd_test = dd(A)
print("is_dd", is_dd)
print("is_symmetric", is_symmetric)
return is_symmetric, symmetry_test, is_dd, dd_test
def kernel_function(self, x, y, p=2, sigma=5.0):
if self.kernel == 'linear_kernel':
return np.dot(x, y)
if self.kernel == 'polynomial_kernel':
return (1 + np.dot(x, y))**p
if self.kernel == 'gaussian_kernel':
return np.exp(-linalg.norm(x - y)**2 / (2 * (sigma**2)))
def searchDirection(gk, Hk, epsilon=1e-8):
pk = -lsqr(Hk.tocsr(),
gk.toarray().ravel())[0] # compute the search direction
if -pk @ gk <= epsilon * norm(gk) * np.linalg.norm(pk):
gk = gk.toarray() # needed here to not get fail later
return -gk.ravel(
) # ensure that the directional derivative is negative in this direction
return pk
def __relative_error(A, W, H, trace_AT_A):
'''Calculate relative error over sparse A matrix'''
norm = trace_AT_A - 2 * __trace_of_product(
H.T, W.T @ A) + __trace_of_product(H.T, W.T @ W @ H)
relative_norm = norm / slg.norm(A)
return relative_norm
def testit(problem='cylinderwake', N=None, nu=None, Re=None,
nnwtnstps=9, npcrdstps=5, palpha=1e-5):
vel_nwtn_tol = 1e-14
# prefix for data files
data_prfx = problem
# dir to store data
ddir = 'data/'
# paraview output
ParaviewOutput = True
proutdir = 'results/'
femp, stokesmatsc, rhsd = dnsps.get_sysmats(problem=problem, N=N, Re=Re,
nu=nu, scheme='TH',
mergerhs=True, bccontrol=True)
proutdir = 'results/'
ddir = 'data/'
import scipy.sparse.linalg as spsla
print 'get expmats: ||A|| = {0}'.format(spsla.norm(stokesmatsc['A']))
print 'get expmats: ||Arob|| = {0}'.format(spsla.norm(stokesmatsc['Arob']))
stokesmatsc['A'] = stokesmatsc['A'] + 1./palpha*stokesmatsc['Arob']
b_mat = 0.*1./palpha*stokesmatsc['Brob']
brhs = 1.5*b_mat[:, :1] - 1.5*b_mat[:, 1:]
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
soldict.update(fv=rhsd['fv']+brhs, fp=rhsd['fp'],
N=N, nu=nu,
vel_nwtn_stps=nnwtnstps,
vel_pcrd_stps=npcrdstps,
vel_nwtn_tol=vel_nwtn_tol,
ddir=ddir, get_datastring=None,
clearprvdata=True,
data_prfx=data_prfx,
paraviewoutput=ParaviewOutput,
vfileprfx=proutdir+'vel_',
pfileprfx=proutdir+'p_')
#
# compute the uncontrolled steady state Navier-Stokes solution
#
v_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(**soldict)
def wolfeLineSearch(l,
thetas_k,
p,
fk,
gk,
pk,
c1,
c2,
rho,
ak=1.0,
nmaxls=100):
pkgk = pk @ gk
# Increase the step length until the Armijo rule is (almost) not satisfied
while 0.5 * norm(r(l, thetas_k + rho * ak * pk,
p)) < fk + c1 * rho * ak * pkgk[0]:
ak *= rho
# Use bisection to find the optimal step length
aU = ak # upper step length limit
aL = 0 # lower step length limit
for i in range(nmaxls):
# Find the midpoint of aU and aL
ak = 0.5 * (aU + aL)
if 0.5 * norm(r(l, thetas_k + ak * pk, p)) > fk + c1 * ak * pkgk:
# Armijo condition is not satisfied, decrease the upper limit
aU = ak
continue
gk_ak_pk = Jacobi(l, thetas_k + ak * pk).T @ r(l, thetas_k + ak * pk,
p)
if pk @ gk_ak_pk > -c2 * pkgk:
# Upper Wolfe condition is not satisfied, decrease the upper limit
aU = ak
continue
if pk @ gk_ak_pk < c2 * pkgk:
# Lower Wolfe condition is not satisfied, increase the lower limit
aL = ak
continue
# Otherwise, all conditions are satisfied, stop the search
break
return ak
def diffusion(M, adj, alpha=0.7, tol=10e-6): # TODO equation, M, alpha
"""
Network propagation iterative process
Iterative algorithm for apply propagation using random walk on a network:
Initialize::
X1 = M
Repeat::
X2 = alpha * X1.A + (1-alpha) * M
X1 = X2
Until::
norm(X2-X1) < tol
Where::
A : degree-normalized adjacency matrix
Parameters
----------
M : sparse matrix
Data matrix to be diffused.
adj : sparse matrix
Adjacency matrice.
alpha : float, default: 0.7
Diffusion/propagation factor with 0 <= alpha <= 1.
For alpha = 0 : no diffusion.
For alpha = 1 :
tol : float, default: 10e-6
Convergence threshold.
Returns
-------
X2 : sparse matrix
Smoothed matrix.
Notes
-----
Copied from the stratipy Python library
"""
n = adj.shape[0]
adj = adj + sp.eye(n)
d = sp.dia_matrix((np.array(adj.sum(axis=0))**-1, [0]), shape=(n, n))
A = adj.dot(d)
X1 = M
X2 = alpha * X1.dot(A) + (1 - alpha) * M
i = 0
while norm(X2 - X1) > tol:
X1 = X2
X2 = alpha * X1.dot(A) + (1 - alpha) * M
i += 1
return X2
def dominate_value(mx):
x1 = csr_matrix(random.rand(mx.shape[0], 1))
x2 = x1
print(x1.todense())
x1 = mx * x1
print(norm(x1, ord=float('inf')))
x1 = x1 / norm(x1, ord=float('inf'))
while abs(norm(x1 - x2)) > 0.0000001:
print(x1.todense())
print('\n')
x2 = x1
x1 = mx.dot(x1)
x1 = x1 / norm(x1, ord=float('inf'))
print((x1 / norm(x1)).todense())
def test_sparse_varfd_1d(self):
n_points = 100
degree = 10
quad = hm.Quad.gauss_hermite(n_points)
mat1 = quad.varfd('1', degree, [0, 0], sparse=True).matrix
mat2 = quad.varfd('x', degree, [0], sparse=True).matrix
bk_ou = mat1 - mat2
off_diag = bk_ou - sp.diags(bk_ou.diagonal())
self.assertAlmostEqual(las.norm(off_diag), 0)
def _calculate_neighbor_weight_matrix(self,
train_r: csc_matrix) -> np.ndarray:
l2norm = norm(train_r, ord=2, axis=1)
l2norm[l2norm == 0] = 1 # handel 0 vector
U: csc_matrix = train_r.multiply(1 / l2norm.reshape(-1, 1))
UUT = U.dot(U.transpose()).toarray() # dense
W = np.exp(self._tau * np.power(1 - UUT, self._k))
np.fill_diagonal(W, 0)
return W
def norm_svd(di, q):
"""
Oblicza norma miedzy di a q (q ma norme 1)
:param di:
:param q:
:return:
"""
a = norm(di)
return ((q.dot(di) / a).toarray())[0][0]
def test_basics(self):
x = np.array([[-2, -1], [-2, 1], [2, 1], [2, -1]])
knn = KNNDense(n_neighbors=1)
knn.fit(x)
truth = np.zeros(16).reshape((4, 4))
truth[0, 1] = 1
truth[2, 3] = 1
truth = sparse.csr_matrix(truth + truth.T)
self.assertAlmostEqual(norm(truth - knn.adjacency_), 0)
def euclidean_distance(vector1, vector2):
from scipy.sparse import csr_matrix
import numpy as np
from scipy.sparse.linalg import norm
vector1 = csr_matrix(vector1)
vector2 = csr_matrix(vector2)
result = norm(vector1 - vector2)
#print(result)
return result
def test_basics(self):
x = np.array([[-2, -1], [-2, 1], [2, 1], [2, -1]])
knn = KNeighborsTransformer(n_neighbors=1)
knn.fit(x)
gt = np.zeros(16).reshape((4, 4))
gt[0, 1] = 1
gt[2, 3] = 1
gt = sparse.csr_matrix(gt + gt.T)
self.assertAlmostEqual(norm(gt - knn.adjacency_), 0)
def solve(A, name, use_umfpack=False):
# Compute b such that the exact solution of Ax = b is xe = [1, 1, ...]
n_rows, n_columns = A.shape
p_non_zero = A.nnz / (n_rows * n_columns)
xe = scipy.ones(n_rows)
b = A * xe
# Solve Ax = b
start_time = datetime.now()
x = spsolve(A, b, use_umfpack=use_umfpack)
t = datetime.now() - start_time
# || xe - x || / ||xe||
relative_error = norm(x - xe, 2) / norm(xe, 2)
row = f"{name},{n_rows},{A.nnz},{relative_error}," + \
f"{t.seconds}.{t.microseconds},{platform.system()}"
print(row)
def cosine_simil2(A, B): """ Returns the cosine similarity between the rows of A and B. The output is a Mx1 vector where M is the number of rows of A. A and B are sparse matrices. """ C = A.dot(B.T) AN = spla.norm(A, axis = 1).reshape((A.shape[0],1)) # Reshape into a column vector BN = np.linalg.norm(B) C = C * (1. / (AN * BN)) return C
def cosine_simil(A, B): """ Returns the cosine similarity between the rows of A and B. The output is a Mx1 vector where M is the number of rows of A. A and B are sparse matrices. """ if len(A.shape) == 1: A = A.reshape((1,A.shape[0])) if len(B.shape) == 1: B = B.reshape((1,B.shape[0])) C = A.dot(B.T) # Reshape into a column vector AN = spla.norm(A, axis = 1).reshape((A.shape[0],1)) if issparse(A) else np.linalg.norm(A, axis = 1).reshape((A.shape[0],1)) BN = spla.norm(B) if issparse(B) else np.linalg.norm(B) if issparse(C): C = C.multiply(1. / (AN * BN)) else: if len(C.shape) == 1: C = C.reshape(C.shape[0],1) AN *= BN C /= AN return C
def genRandomDocInTopic( topic ):
K = 20
maxlen = -10.0
id = -1
tc.vec.set_params( norm=None )
for i in range( K ):
tid = random.choice( tc.inv_topic[ topic ] )
d = tc.vec.transform( [ tc.corpus[ tid ] ] )
len = norm( d )
if len > maxlen:
maxlen , id = len , tid
tc.vec.set_params( norm='l2' )
return tc.datas[ id ][ 'body' ]
def test_sketch_preserves_frobenius_norm(self):
# Given the probabilistic nature of the sketches
# we run the test multiple times and check that
# we pass all/almost all the tries.
n_errors = 0
for A in self.test_matrices:
if issparse(A):
true_norm = norm(A)
else:
true_norm = np.linalg.norm(A)
for seed in self.seeds:
sketch = clarkson_woodruff_transform(
A, self.n_sketch_rows, seed=seed,
)
if issparse(sketch):
sketch_norm = norm(sketch)
else:
sketch_norm = np.linalg.norm(sketch)
if np.abs(true_norm - sketch_norm) > 0.1 * true_norm:
n_errors += 1
assert_(n_errors == 0)
def test_norm_axis(self):
a = np.array([[ 1, 2, 3],
[-1, 1, 4]])
c = csr_matrix(a)
#Frobenius norm
assert_equal(norm(c, axis=0), np.sqrt(np.power(np.asmatrix(a), 2).sum(axis=0)))
assert_equal(norm(c, axis=1), np.sqrt(np.power(np.asmatrix(a), 2).sum(axis=1)))
assert_equal(norm(c, np.inf, axis=0), max(abs(np.asmatrix(a)).sum(axis=0)))
assert_equal(norm(c, np.inf, axis=1), max(abs(np.asmatrix(a)).sum(axis=1)))
assert_equal(norm(c, -np.inf, axis=0), min(abs(np.asmatrix(a)).sum(axis=0)))
assert_equal(norm(c, -np.inf, axis=1), min(abs(np.asmatrix(a)).sum(axis=1)))
assert_equal(norm(c, 1, axis=0), abs(np.asmatrix(a)).sum(axis=0))
assert_equal(norm(c, 1, axis=1), abs(np.asmatrix(a)).sum(axis=1))
assert_equal(norm(c, -1, axis=0), min(abs(np.asmatrix(a)).sum(axis=0)) )
assert_equal(norm(c, -1, axis=1), min(abs(np.asmatrix(a)).sum(axis=1)) )
#_multi_svd_norm is not implemented for sparse matrix
assert_raises(NotImplementedError, norm, c, 2, 0)
def ksvd(AA,S,max_iter=10):
"""
Apply KSVD from the starting dictionary AA, given a set of signals S.
"""
M,N = AA.shape
P = S.shape[1]
A = AA.copy()
X = zeros((N,P))
#max_iter = 10
i = 0
while i < max_iter:
# SMV, and slow.
#for p in range(P):
# X[:,p] = mp.omp(A,S[:,p].reshape(M,1),0.1).flatten()
# MMV
# Default: no residual tolerance, max nonzeros 10%
# Assume dictionary A is normalized!
X = sklearn.linear_model.orthogonal_mp(A,S)
print i,linalg.norm(S - dot(A,X)), sum([sparse.zero_norm(X[:,p],0.01) for p in range(P)])
for l in range(N):
nonzeros = sparse.nonzero_idx(X[l,:]) # xs with nonzero in row 'l'
if len(nonzeros) == 0:
continue
E_w = S[:,nonzeros] - dot(A,X[:,nonzeros]) + dot(A[:,l].reshape(M,1),X[l,nonzeros].reshape(1,len(nonzeros)))
try:
u,s,v = scipy.sparse.linalg.svds(E_w, k = 1)
except:
continue
A[:,l] = u.flatten()
X[l,nonzeros] = s*v.flatten()
i += 1
return A, X
def test_norm(self):
a = np.arange(9) - 4
b = a.reshape((3, 3))
b = csr_matrix(b)
#Frobenius norm is the default
assert_equal(norm(b), 7.745966692414834)
assert_equal(norm(b, 'fro'), 7.745966692414834)
assert_equal(norm(b, np.inf), 9)
assert_equal(norm(b, -np.inf), 2)
assert_equal(norm(b, 1), 7)
assert_equal(norm(b, -1), 6)
#_multi_svd_norm is not implemented for sparse matrix
assert_raises(NotImplementedError, norm, b, 2)
assert_raises(NotImplementedError, norm, b, -2)
def sp_expm(A, p=13, sparse=False):
"""
Sparse matrix exponential.
Reference
---------
Expokit, ACM-Transactions on Mathematical Software, 24(1):130-156, 1998
"""
if _isdiag(A.indices, A.indptr, A.shape[0]):
A.data = np.exp(A.data)
return A
N = A.shape[0]
c = np.zeros(p+1,dtype=float)
# Pade coefficients
c[0] = 1
for k in range(p):
c[k+1] = c[k]*((p-k)/((k+1.0)*(2.0*p-k)))
# Scaling
if sparse:
A = A.tocsc()
nrm = spla.norm(A, np.inf)
else:
A = A.toarray()
nrm = la.norm(A, np.inf)
if nrm > 0.5:
nrm = max(0, np.fix(np.log(nrm)/np.log(2))+2)
A = 2.0**(-nrm)*A
# Horner evaluation of the irreducible fraction
if sparse:
I = sp.identity(N, dtype=complex, format='csc')
else:
I = np.identity(N, dtype=complex)
A2 = A.dot(A)
Q = c[-1]*I
P = c[p]*I
odd = 1
for k in range(p-2,-1,-1):
if odd == 1:
Q = Q.dot(A2) +c[k]*I
else:
P = P.dot(A2) +c[k]*I
odd = 1-odd
if odd == 1:
Q = Q.dot(A)
Q = Q-P
if sparse:
E = -(I+2.0*spla.spsolve(Q,P))
else:
E = -(I+2.0*la.solve(Q,P))
else:
P = P.dot(A)
Q = Q-P
if sparse:
E = I+2.0*spla.spsolve(Q,P)
else:
E = I+2.0*la.solve(Q,P)
# Squaring
for k in range(int(nrm)):
E = E.dot(E)
return sp.csr_matrix(E)
def _ASD(self, M, r = None, reltol=1e-5, maxiter=5000):
"""
Alternating Steepest Descent (ASD)
Taken from Low rank matrix completion by alternating steepest descent methods
Jared Tanner and Ke Wei
SIAM J. IMAGING SCIENCES (2014)
We have a matrix M with incomplete entries,
and want to estimate the full matrix
Solves the following relaxation of the problem:
minimize_{X,Y} \frac{1}{2} ||P_{\Omega}(Z^0) - P_\{Omega}(XY)||_F^2
Where \Omega represents the set of m observed entries of the matrix M
and P_{\Omega}() is an operator that represents the observed data.
Inputs:
M := Incomplete matrix, with NaN on the unknown matrix
r := hypothesized rank of the matrix
Usage:
Just call the function _ASD(M)
"""
# Get shape and Omega
m, n = M.shape
if r == None:
r = min(m-1, n-1, 50)
# Set relative error
I, J = [], []
M_list = []
for i, j in M.keys():
I.append(i)
J.append(j)
M_list.append(M[i,j])
M_list = np.array(M_list)
Omega = [I,J]
frob_norm_data = linalg_s.norm(M)
relres = reltol * frob_norm_data
# Initialize
M_omega = M.tocsc()
U, s, V = linalg_s.svds(M_omega, r)
S = np.diag(s)
X = np.dot(U, S)
Y = V
itres = np.zeros((maxiter+1, 1))
XY = np.dot(X, Y)
diff_on_omega = M_list - XY[Omega]
res = linalg.norm(diff_on_omega)
iter_c = 0
itres[iter_c] = res/frob_norm_data
while iter_c < maxiter and res >= relres:
# Gradient for X
diff_on_omega_matrix = np.zeros((m,n))
diff_on_omega_matrix[Omega] = diff_on_omega
grad_X = np.dot(diff_on_omega_matrix, np.transpose(Y))
# Stepsize for X
delta_XY = np.dot(grad_X, Y)
tx = linalg.norm(grad_X,'fro')**2/linalg.norm(delta_XY)**2
# Update X
X = X + tx*grad_X;
diff_on_omega = diff_on_omega-tx*delta_XY[Omega]
# Gradient for Y
diff_on_omega_matrix = np.zeros((m,n))
diff_on_omega_matrix[Omega] = diff_on_omega
Xt = np.transpose(X)
grad_Y = np.dot(Xt, diff_on_omega_matrix)
# Stepsize for Y
delta_XY = np.dot(X, grad_Y)
ty = linalg.norm(grad_Y,'fro')**2/linalg.norm(delta_XY)**2
# Update Y
Y = Y + ty*grad_Y
diff_on_omega = diff_on_omega-ty*delta_XY[Omega]
res = linalg.norm(diff_on_omega)
iter_c = iter_c + 1
itres[iter_c] = res/frob_norm_data
M_out = np.dot(X, Y)
out_info = [iter_c, itres]
return M_out, out_info
def cosine_distance(x, y):
xy = x.dot(y.T) # should be 1x1 mat
dist = xy / norm(x) / norm(y)
return 1 - dist[0, 0] # ~ arccos()
def get_svd_res(mat, components): usvt= svds(mat,k=components,which='LM') ## compute svd if usvt[1][-1]==0: usvt=flip_svd_res(usvt) enrgy=(usvt[1][-1]/norm(mat)) ## energy of first principal component var=enrgy**2 ## variance of first principal component return usvt, enrgy, var
def compute_depth(mask_array, normal_array, threshold=100): index_map = -np.ones(mask_array.shape) x = [] y = [] ind = 0 for (xT, value) in np.ndenumerate(mask_array): if(value > threshold): index_map[xT] = ind x.append(xT[1]) y.append(-xT[0]) ind = ind+1 row = [] col = [] data = [] b = [] i = 0 for (xT, value) in np.ndenumerate(index_map): if value >= 0: normal = normal_array[xT]/128.0 - 1.0 normal = normal/linalg.norm(normal) if not np.isnan(np.sum(normal)): #x iother = index_map[xT[0], xT[1]+1] if abs(normal[2]) > 0.01: if iother >= 0: row.append(i) col.append(value) data.append(-normal[2]) row.append(i) col.append(iother) data.append(normal[2]) b.append(-normal[0]) i = i+1 #y iother = index_map[xT[0]-1, xT[1]] if iother >= 0: row.append(i) col.append(value) data.append(-normal[2]) row.append(i) col.append(iother) data.append(normal[2]) b.append(-normal[1]) i = i+1 mat = scipy.sparse.coo_matrix((data, (row, col)), shape=(i, ind)).tocsc() b = np.array(b) z = scipy.sparse.linalg.lsqr(mat,b, iter_lim=1000, show=True) ind = 0 for (xT, value) in np.ndenumerate(index_map): if value >= 0: index_map[xT] = z[0][ind] ind = ind+1 return index_map
def cosine_distance(x, y):
xy = x.dot(y.T)
dist = xy/(norm(x)*norm(y))
return 1-dist[0,0]
def lbfgs(f, xk, gfk, k, vecs, props):
logger = logging.getLogger("phf.innersolve")
solve_fraction = props.get("solveFraction", 0.2)
stepFactor = props.get("innerSolveStepFactor", 0.5)
average = props.get("innerSolveAverage", False)
n = len(xk)
w = lbfgs_step(gfk, k, vecs, props)
wsum = zeros(n)
wsum_count = 0
gnorm = linalg.norm(gfk)
# Each iteration requires two evaluations, so we half the max iters here
maxiter = int(ceil(solve_fraction*f.parts/2.0))
pkHpk = 0
wHw = 0
for i in range(maxiter):
mv = f.make_mv_rand(xk)
Hw = mv(w)
ri = Hw + gfk
pk = -lbfgs_step(ri, k+i, vecs, props)
mpk = mv(pk)
Hpk = mpk
pkHpk = dot(pk, Hpk)
sst = stepFactor*dot(ri, pk) / pkHpk
wHw = dot(w, Hw)
###### Update quasi-newton approximation
kmax = max_key(vecs)
if pkHpk < 0:
raise Exception("Hessian is not positive semi-definite. " +
"Try using the Gauss-Newton approximation to the hessian." +
"If your problem is convex, your gradient " +
"calculation may just be wrong")
else:
vecs[kmax] = (pk, Hpk, 1.0 / numpy.dot(pk,Hpk))
kmax = kmax + 1
if wHw > 0:
vecs[kmax] = (w, Hw, 1.0 / numpy.dot(w,Hw))
wp = w - sst*pk
cosdirection = dot(wp, gfk) / (linalg.norm(wp) * gnorm)
if cosdirection < 0:
w = wp
if i == 0 or (i % max(1,maxiter / 10) == 0):
logger.debug("w: %s", w[0:min(5, n)])
else:
logger.debug("Skipping w update to ensure w is a descent direction")
wnorm = linalg.norm(w)
if i > maxiter/2:
wsum += w
wsum_count += 1
if average:
wavg = wsum / wsum_count
return wavg
else:
return w
def ncutW(W, num_eigs=10, kmeans_iters=10, offset = 0.5):
"""Run the normalized cut algorithm on the affinity matrix, W.
(as implemented in Ng, Jordan, and Weiss, 2002)
Parameters
----------
W : scipy sparse matrix
Square matrix with high values for edges to be preserved, and low
values for edges to be cut.
num_eigs : int, optional
Number of eigenvectors of the affinity matrix to use for clustering.
kmeans_iters : int, optional
Number of iterations of the k-means algorithm to run when clustering
eigenvectors.
offset : float, optional
Diagonal offset used to stabilise the eigenvector computation.
Returns
-------
labels : array of int
`labels[i]` is an integer value mapping node/row `i` to the cluster
ID `labels[i]`.
eigenvectors : list of array of float
The computed eigenvectors of `W + offset * I`, where `I` is the
identity matrix of same size as `W`.
eigenvalues : array of float
The corresponding eigenvalues.
"""
n, m = W.shape
# Add an offset in case some rows are zero
# We also add the offset below to the diagonal matrix. See (Yu, 2001),
# "Understanding Popout through Repulsion" for more information. This
# helps to stabilize the eigenvector computation.
W = W + sparse.diags(np.full(n, offset))
d = np.ravel(W.sum(axis=1))
Dinv2 = sparse.diags(1 / (np.sqrt(d) + offset*np.ones(n)))
P = Dinv2 @ W @ Dinv2
# Get the eigenvectors and sort by eigenvalue
eigvals, U = eigs(P, num_eigs, which='LR')
eigvals = np.real(eigvals) # it should be real anyway
U = np.real(U)
ind = np.argsort(eigvals)[::-1]
eigvals = eigvals[ind]
U = U[:, ind]
# Normalize
for i in range(n):
U[i, :] /= norm(U[i, :])
# Cluster them into labels, running k-means multiple times
labels_list = []
distortion_list = []
for _iternum in range(kmeans_iters):
# Cluster
centroid, labels = vq.kmeans2(U, num_eigs, minit='points')
# Calculate distortion
distortion = 0
for j in range(num_eigs):
numvals = np.sum(labels == j)
if numvals == 0:
continue
distortion += np.mean([norm(v - centroid[j])**2 for (i, v) in
enumerate(U) if labels[i] == j])
# Save values
labels_list.append(labels)
distortion_list.append(distortion)
# Use lowest distortion
labels = labels_list[np.argmin(distortion_list)]
return labels, U, eigvals
