def __init__(self, basef,
translate=True,
rotate=False,
conditioning=None,
asymmetry=None,
oscillate=False,
penalize=None,
):
FunctionEnvironment.__init__(self, basef.xdim, basef.xopt)
self.desiredValue = basef.desiredValue
self.toBeMinimized = basef.toBeMinimized
if translate:
self.xopt = (rand(self.xdim) - 0.5) * 9.8
self._diags = eye(self.xdim)
self._R = eye(self.xdim)
self._Q = eye(self.xdim)
if conditioning is not None:
self._diags = generateDiags(conditioning, self.xdim)
if rotate:
self._R = orth(rand(basef.xdim, basef.xdim))
if conditioning:
self._Q = orth(rand(basef.xdim, basef.xdim))
tmp = lambda x: dot(self._Q, dot(self._diags, dot(self._R, x-self.xopt)))
if asymmetry is not None:
tmp2 = tmp
tmp = lambda x: asymmetrify(tmp2(x), asymmetry)
if oscillate:
tmp3 = tmp
tmp = lambda x: oscillatify(tmp3(x))
self.f = lambda x: basef.f(tmp(x))
def subspace(a, b, deg=True):
"""
Angle between two subspaces specified by the columns of a and b
Ported from MATLAB 'subspace' function
Parameters
----------
a : matrix
b : matrix
deg : bool
return degree or radian
Returns
-------
double
angle
"""
warnings.warn(
"Deprecated. Use scipy.linalg.subspace_angles instead.", FutureWarning
)
oa = linalg.orth(a)
ob = linalg.orth(b)
if oa.shape[1] < ob.shape[1]:
oa, ob = ob.copy(), oa.copy()
ob -= oa @ (oa.T @ ob)
rad = np.arcsin(min(1, linalg.norm(ob, ord=2)))
return np.degrees(rad) if deg else rad
def principal_angles(A, B):
'''Compute the principal angles between subspaces A and B.
The algorithm for computing the principal angles is described in :
A. V. Knyazev and M. E. Argentati,
Principal Angles between Subspaces in an A-Based Scalar Product:
Algorithms and Perturbation Estimates. SIAM Journal on Scientific Computing,
23 (2002), no. 6, 2009-2041.
http://epubs.siam.org/sam-bin/dbq/article/37733
'''
# eps = np.finfo(np.float64).eps**.981
# for i in range(A.shape[1]):
# normi = la.norm(A[:,i],np.inf)
# if normi > eps: A[:,i] = A[:,i]/normi
# for i in range(B.shape[1]):
# normi = la.norm(B[:,i],np.inf)
# if normi > eps: B[:,i] = B[:,i]/normi
QA = sl.orth(A)
QB = sl.orth(B)
_, s, Zs = svd(QA.T.dot(QB), full_matrices=False)
s = np.minimum(s, ones_like(s))
theta = np.maximum(np.arccos(s), np.zeros_like(s))
V = QB.dot(Zs)
idxSmall = s > np.sqrt(2.)/2.
if np.any(idxSmall):
RB = V[:,idxSmall]
_, x, _ = svd(RB-QA.dot(QA.T.dot(RB)),full_matrices=False)
thetaSmall = np.flipud(np.maximum(arcsin(np.minimum(x, ones_like(x))), zeros_like(x)))
theta[idxSmall] = thetaSmall
return theta
def __init__(self, *args, **kwargs):
MultiModalFunction.__init__(self, *args, **kwargs)
self._mu0 = 2.5
self._s = 1 - 1 / (2 * sqrt(self.xdim + 20) - 8.2)
self._mu1 = -sqrt((self._mu0 ** 2 - 1) / self._s)
self._signs = sign(randn(self.xdim))
self._R1 = orth(rand(self.xdim, self.xdim))
self._R2 = orth(rand(self.xdim, self.xdim))
self._diags = generateDiags(100, self.xdim)
def fubini_study(A, B):
'''
fubini_study(A, B) Compute the Fubini-Study distance
Compute the Fubini-Study distance based on principal angles between A and B
as d=\acos{ \prod_i \theta_i}
'''
if A.shape != B.shape:
raise ValueError('Atoms have different dim (', A.shape, ' and ', B.shape,'). Error raised in fubini_study(A, B)')
if np.allclose(A, B): return 0.
return arccos(det(sl.orth(A).T.dot(sl.orth(B))))
def _create_SDP(self):
""" Creates the SDP knockoff of X"""
# Check for rank deficiency (will add later).
# SVD and come up with perpendicular matrix
U, d, V = nplin.svd(self.X,full_matrices=True)
d[d<0] = 0
U_perp = U[:,self.p:(2*self.p)]
if self.randomize:
U_perp = np.dot(U_perp,splin.orth(npran.randn(self.p,self.p)))
# Compute the Gram matrix and its (pseudo)inverse.
G = np.dot(V.T * d**2 ,V)
G_inv = np.dot(V.T * d**-2,V)
# Optimize the parameter s of Equation 1.3 using SDP.
self.s = solve_sdp(G)
self.s[s <= self.zerotol] = 0
# Construct the knockoff according to Equation 1.4:
C_U,C_d,C_V = nplin.svd(2*np.diag(s) - (self.s * G_inv.T).T * self.s)
C_d[C_d < 0] = 0
X_ko = self.X - np.dot(self.X,G_inv*s) + np.dot(U_perp*np.sqrt(C_d),C_V)
self.X_lrg = np.concatenate((self.X,X_ko), axis=1)
def bench_lobpcg_mikota():
print()
print(' lobpcg benchmark using mikota pairs')
print('==============================================================')
print(' shape | blocksize | operation | time ')
print(' | (seconds)')
print('--------------------------------------------------------------')
fmt = ' %15s | %3d | %6s | %6.2f '
m = 10
for n in 128, 256, 512, 1024, 2048:
shape = (n, n)
A, B = _mikota_pair(n)
desired_evs = np.square(np.arange(1, m+1))
tt = time.clock()
X = rand(n, m)
X = orth(X)
LorU, lower = cho_factor(A, lower=0, overwrite_a=0)
M = LinearOperator(shape,
matvec=partial(_precond, LorU, lower),
matmat=partial(_precond, LorU, lower))
eigs, vecs = lobpcg(A, X, B, M, tol=1e-4, maxiter=40)
eigs = sorted(eigs)
elapsed = time.clock() - tt
assert_allclose(eigs, desired_evs)
print(fmt % (shape, m, 'lobpcg', elapsed))
tt = time.clock()
w = eigh(A, B, eigvals_only=True, eigvals=(0, m-1))
elapsed = time.clock() - tt
assert_allclose(w, desired_evs)
print(fmt % (shape, m, 'eigh', elapsed))
def chordal(A, B):
'''
chordal(A, B) Compute the chordal distance
Compute the chordal distance between A and B
as d=\sqrt{K - ||\bar{A}^T\bar{B}||_F^2}
where K is the rank of A and B, || . ||_F is the Frobenius norm,
\bar{A} is the orthogonal basis associated with A and the same goes for B.
'''
if A.shape != B.shape:
raise ValueError('Atoms have not the same dimension (', A.shape, ' and ', B.shape,'). Error raised in chordal(A, B)')
if np.allclose(A, B): return 0.
else:
d2 = A.shape[1] - norm(sl.orth(A).T.dot(sl.orth(B)), 'fro')**2
if d2 < 0.: return sqrt(abs(d2))
else: return sqrt(d2)
def addblock_svd_update( Uarg, Sarg, Varg, Aarg, force_orth = False):
U = Varg
V = Uarg
S = np.eye(len(Sarg),len(Sarg))*Sarg
A = Aarg.T
current_rank = U.shape[1]
m = np.dot(U.T,A)
p = A - np.dot(U,m)
P = lin.orth(p)
Ra = np.dot(P.T,p)
z = np.zeros(m.shape)
K = np.vstack(( np.hstack((S,m)), np.hstack((z.T,Ra)) ))
tUp,tSp,tVp = lin.svd(K);
tUp = tUp[:,:current_rank]
tSp = np.diag(tSp[:current_rank])
tVp = tVp[:,:current_rank]
Sp = tSp
Up = np.dot(np.hstack((U,P)),tUp)
Vp = np.dot(V,tVp[:current_rank,:])
Vp = np.vstack((Vp, tVp[current_rank:tVp.shape[0], :]))
if force_orth:
UQ,UR = lin.qr(Up,mode='economic')
VQ,VR = lin.qr(Vp,mode='economic')
tUp,tSp,tVp = lin.svd( np.dot(np.dot(UR,Sp),VR.T));
tSp = np.diag(tSp)
Up = np.dot(UQ,tUp)
Vp = np.dot(VQ,tVp)
Sp = tSp;
Up1 = Vp;
Vp1 = Up;
return Up1,Sp,Vp1
def fastica_defl(X, nIC=None, guess=None,
nonlinfn = pow3nonlin,
termtol = 5e-7, maxiters = 2e3):
nPC, siglen = X.shape
nIC = nIC or nPC-1
guess = guess or randn(nPC,nIC)
if _orth_loaded:
guess = orth(guess)
B = zeros(guess.shape, np.float64)
errvec = []
icc = 0
while icc < nIC:
w = randn(nPC,1) - 0.5
w -= dot(dot(B, transp(B)), w)
w /= norm(w)
wprev = zeros(w.shape)
for i in xrange(long(maxiters) +1):
w -= dot(dot(B, transp(B)), w)
w /= norm(w)
#wprev = w.copy()
if (norm(w-wprev) < termtol) or (norm(w + wprev) < termtol):
B[:,icc] = transp(w)
icc += 1
break
wprev = w.copy()
return B.real, errvec
def random_walk(G, initial_prob, subspace_dim=3, walk_steps=3):
assert type(initial_prob) == np.ndarray, "Initial probability distribution is not a numpy array"
# Transform the adjacent matrix to a laplacian matrix P
P = adj_to_laplacian(G)
prob_matrix = np.zeros((G.shape[0], subspace_dim))
prob_matrix[:, 0] = initial_prob
for i in range(1, subspace_dim):
prob_matrix[:, i] = np.dot(prob_matrix[:, i - 1], P)
orth_prob_matrix = splin.orth(prob_matrix)
for i in range(walk_steps):
temp = np.dot(orth_prob_matrix.T, P)
orth_prob_matrix = splin.orth(temp.T)
return orth_prob_matrix
def calc_subspace_proj_error(U, U_hat, ortho=False):
"""Calculate the normalized projection error between two orthogonal subspaces.
Keyword arguments:
U: ground truth subspace
U_hat: estimated subspace
"""
if not ortho:
U = splinalg.orth(U)
U_hat = splinalg.orth(U_hat)
I = np.identity(U.shape[0])
top = np.linalg.norm((I - U_hat @ U_hat.T) @ U, ord="fro")
bottom = np.linalg.norm(U, ord="fro")
error = float(top) / float(bottom)
return error
def condex(n, k=4, theta=100):
"""
CONDEX `Counterexamples' to matrix condition number estimators.
CONDEX(N, K, THETA) is a `counterexample' matrix to a condition
estimator. It has order N and scalar parameter THETA (default 100).
If N is not equal to the `natural' size of the matrix then
the matrix is padded out with an identity matrix to order N.
The matrix, its natural size, and the estimator to which it applies
are specified by K (default K = 4) as follows:
K = 1: 4-by-4, LINPACK (RCOND)
K = 2: 3-by-3, LINPACK (RCOND)
K = 3: arbitrary, LINPACK (RCOND) (independent of THETA)
K = 4: N >= 4, SONEST (Higham 1988)
(Note that in practice the K = 4 matrix is not usually a
counterexample because of the rounding errors in forming it.)
References:
A.K. Cline and R.K. Rew, A set of counter-examples to three
condition number estimators, SIAM J. Sci. Stat. Comput.,
4 (1983), pp. 602-611.
N.J. Higham, FORTRAN codes for estimating the one-norm of a real or
complex matrix, with applications to condition estimation
(Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396.
"""
if k == 1: # Cline and Rew (1983), Example B.
a = np.array([[1, -1, -2 * theta, 0], [0, 1, theta, -theta], [0, 1, 1 + theta, -(theta + 1)], [0, 0, 0, theta]])
elif k == 2: # Cline and Rew (1983), Example C.
a = np.array([[1, 1 - 2 / theta ** 2, -2], [0, 1 / theta, -1 / theta], [0, 0, 1]])
elif k == 3: # Cline and Rew (1983), Example D.
a = rogues.triw(n, -1).T
a[-1, -1] = -1
elif k == 4: # Higham (1988), p. 390.
x = np.ones((n, 3)) # First col is e
x[1:n, 1] = np.zeros(n - 1) # Second col is e(1)
# Third col is special vector b in SONEST
x[:, 2] = ((-1) ** np.arange(n)) * (1 + np.arange(n) / (n - 1))
# Q*Q' is now the orthogonal projector onto span(e(1),e,b)).
q = sl.orth(x)
p = np.eye(n) - np.asmatrix(q) * np.asmatrix(q.T)
a = np.eye(n) + theta * p
# Pad out with identity as necessary.
m, m = a.shape
if m < n:
for i in range(n - 1, m, -1):
a[i, i] = 1
return a
def shadow_volume(bases):
_high_dimension = len(bases.T)
_low_dimension = len(bases)
orth_bases = orth(bases.T)
sum = 0.0
for indices in itertools.combinations(range(_high_dimension), _low_dimension):
sub_matrix = orth_bases[indices, :]
sum += abs(det(sub_matrix))
return sum
def __init__(self, *args, **kwargs):
MultiModalFunction.__init__(self, *args, **kwargs)
self._opts = (rand((self.numPeaks, self.xdim)) - 0.5) * 9.8
self._opts[0] = (rand(self.xdim) - 0.5) * 8
alphas = [power(self.maxCond, 2 * i / float(self.numPeaks - 2)) for i in range(self.numPeaks - 1)]
shuffle(alphas)
self._covs = [generateDiags(alpha, self.xdim, shuffled=True) / power(alpha, 0.25) for alpha in [self.optCond] + alphas]
self._R = orth(rand(self.xdim, self.xdim))
self._ws = [10] + [1.1 + 8 * i / float(self.numPeaks - 2) for i in range(self.numPeaks - 1)]
def test_1():
n_roots = 1000
dim = 1000
offset = 0
A = DavTestMat(dim, offset)
guess = LIN.orth(np.random.rand(dim, n_roots))
evals, evecs = solvers.davidson(A, guess)
evals_ref = np.linalg.eigvalsh(A.A)
assert np.allclose(evals, evals_ref)
def test_orth():
m = 4
k = 3
G_temp = np.random.normal(0, 1, [m, k])
G = orth(G_temp)
print G_temp.shape
print G.shape
def main():
"""Main function"""
# Init signal
sig = np.zeros(shape=(SIG_LEN, ))
pos_not_zeros = np.random.randint(0, SIG_LEN, size=NUM_NOT_ZEROS)
sig[pos_not_zeros] = np.sign(np.random.randn(NUM_NOT_ZEROS) - 0.5)
sensing_matrix = np.random.randn(MEASURED_LEN, SIG_LEN)
sensing_matrix = linalg.orth(sensing_matrix.T) # pylint: disable=no-member
sensing_matrix = sensing_matrix.T
measured_sig = np.matmul(sensing_matrix, sig)
sensing_matrix_pinv = linalg.pinv(sensing_matrix)
x_min_norm = np.matmul(sensing_matrix_pinv, measured_sig)
tau = 0.1 * np.max(np.matmul(sensing_matrix.T, measured_sig))
initial_solution = x_min_norm + 0.01 * (np.random.randn(SIG_LEN))
x_gpsr = gpsr.gpsr_bb(initial_solution,
sensing_matrix,
measured_sig,
tau=tau,
alpha0=5,
alpha_lims=(1e-30, 1e+30),
tolerance=0.0001)
x_debiased = gpsr.debaising(x_gpsr,
sensing_matrix,
measured_sig,
tol=0.01,
fix_lev=0.1,
iter_max=12)
print(tau)
_, axes = plt.subplots(2, 2)
axes[0, 0].plot(sig)
axes[0, 0].set_title('True signal')
axes[0, 1].plot(x_debiased, '-r')
axes[0, 1].set_title('Reconstructed after debiasing')
axes[1, 0].plot(x_gpsr, '-b')
axes[1, 0].set_title('Reconstructed after GPSR')
axes[1, 1].plot(sig - x_gpsr, '.b')
axes[1, 1].plot(sig - x_debiased, '.r')
axes[1, 1].set_title('Difference from true signal')
print('MSE:',
np.dot(sig - x_debiased, sig - x_debiased) / SIG_LEN,
np.dot(sig - x_gpsr, sig - x_gpsr) / SIG_LEN)
plt.show()
def generate(designers,
size,
inputs=None,
outputs=None,
is_coupled=True,
random=np.random):
if inputs is None:
# try to assign equally among designers
inputs = [
designers[int(i // (size / len(designers)))]
for i in range(size)
]
num_inputs = [
np.sum(np.array(inputs) == designer).item()
for designer in designers
]
if outputs is None:
# try to assign equally among designers
outputs = [
designers[int(i // (size / len(designers)))]
for i in range(size)
]
num_outputs = [
np.sum(np.array(outputs) == designer).item()
for designer in designers
]
coupling = np.zeros((size, size))
if is_coupled:
# coupling matrix is orthonormal basis of random matrix
coupling = orth(random.rand(size, size))
else:
# coupling matrix has random 1/-1 along diagonal
coupling = np.diag(2 * random.randint(0, 2, size) - 1)
# find a target with no solution values "close" to initial condition
solution = np.zeros(size)
while np.any(np.abs(solution) <= 0.20):
target = orth(2 * random.rand(size, 1) - 1)
# solve using dot product of coupling transpose and target
solution = np.matmul(coupling.T, target)
return Task(designers, num_inputs, num_outputs, coupling.tolist(),
target[:, 0].tolist(), inputs, outputs)
def fit(self, data):
"""
Fit independent components using an iterative fixed-point algorithm
Parameters
----------
data: RDD of (tuple, array) pairs, or RowMatrix
Data to estimate independent components from
Returns
----------
self : returns an instance of self.
"""
if type(data) is not RowMatrix:
data = RowMatrix(data)
# reduce dimensionality
svd = SVD(k=self.k, method=self.svdmethod).calc(data)
# whiten data
whtmat = real(dot(inv(diag(svd.s/sqrt(data.nrows))), svd.v))
unwhtmat = real(dot(transpose(svd.v), diag(svd.s/sqrt(data.nrows))))
wht = data.times(whtmat.T)
# do multiple independent component extraction
if self.seed != 0:
random.seed(self.seed)
b = orth(random.randn(self.k, self.c))
b_old = zeros((self.k, self.c))
iter = 0
minabscos = 0
errvec = zeros(self.maxiter)
while (iter < self.maxiter) & ((1 - minabscos) > self.tol):
iter += 1
# update rule for pow3 non-linearity (TODO: add others)
b = wht.rows().map(lambda x: outer(x, dot(x, b) ** 3)).sum() / wht.nrows - 3 * b
# make orthogonal
b = dot(b, real(sqrtm(inv(dot(transpose(b), b)))))
# evaluate error
minabscos = min(abs(diag(dot(transpose(b), b_old))))
# store results
b_old = b
errvec[iter-1] = (1 - minabscos)
# get un-mixing matrix
w = dot(transpose(b), whtmat)
# get components
sigs = data.times(w.T).rdd
self.w = w
self.sigs = sigs
return self
def test_lobpcg(read_guess=True, tol=1e-5):
X = orth(randn(N, m))
if read_guess:
X = x0
try:
λ, v = lobpcg(A, X, largest=False, M=P, maxiter=100, tol=tol)
assert np.max(np.abs(λref - λ)) < tol
except np.linalg.LinAlgError as e:
print("ERROR: ", str(e))
def get_RndSymPosMatrix(size=FEATURE_SIZE, divide=DIV):
D = np.diag(np.random.random_sample([
size,
])) / divide
V = np.random.rand(size, size)
U = orth(V)
D = mat(D)
U = mat(U)
A = U.I * D * U
return A
def __init__(self, d, q, m=1):
self.D = d
self.Q = q
self.M = m
self.mean = [
100 * orth(np.random.randn(self.D, self.Q)) for i in range(self.M)
]
self.cov = [np.eye(self.Q) for i in range(self.M)]
self.wtw = [np.eye(self.Q) for i in range(self.M)]
self.calc_wtw()
def init(X, init, ncomp):
N, K = X[0].shape[0], len(X)
if init == 'random':
A = orth(np.random.rand(N, ncomp))
elif init == 'nvecs':
S = np.zeros(N, N)
for k in range(K):
S = S + X[k] + X[k].T
_, A = eigsh(S, ncomp)
return A
def init(X, init, ncomp):
N, K = X[0].shape[0], len(X)
if init == 'random':
A = orth(rand(N, ncomp))
elif init == 'nvecs':
S = zeros(N, N)
for k in range(K):
S = S + X[k] + X[k].T
_, A = eigsh(S, ncomp)
return A
def minimize_max_squared_norm(known_bases):
# random_bases = random.rand(low_dimension, high_dimension)
res_bh = basinhopping(objective_function,
known_bases,
T=100,
niter=10,
disp=True)
optimal_bases = res_bh.x.reshape((low_dimension, high_dimension))
orth_optimal_bases = orth(optimal_bases.T).T
return (res_bh.fun, orth_optimal_bases)
def random_walk(G, initial_prob, subspace_dim=3, walk_steps=3):
assert type(
initial_prob
) == np.ndarray, "Initial probability distribution is not a numpy array"
# Transform the adjacent matrix to a laplacian matrix P
P = adj_to_laplacian(G)
prob_matrix = np.zeros((G.shape[0], subspace_dim))
prob_matrix[:, 0] = initial_prob
for i in range(1, subspace_dim):
prob_matrix[:, i] = np.dot(prob_matrix[:, i - 1], P)
orth_prob_matrix = splin.orth(prob_matrix)
for i in range(walk_steps):
temp = np.dot(orth_prob_matrix.T, P)
orth_prob_matrix = splin.orth(temp.T)
return orth_prob_matrix
def Cpp_lrrA(data):
print "Cpp_LRR", "data:", data.shape
Q = linalg.orth(np.transpose(data))
A = np.dot(data, Q)
lam = 0.1
Z, E = lrrA(np.matrix(data), np.matrix(A), lam)
Z = np.dot(Q, Z)
Z = np.abs(Z) + np.abs(np.transpose(Z))
print Z
return [np.array(Z)]
def transform(self, X: NDArray):
# ints as the sensitive index are ok, just have to handle them a little differently
if isinstance(self.sens_idxs, int):
orth_vecs = orth(X[:, self.sens_idxs].reshape(-1, 1))
else:
orth_vecs = orth(X[:, self.sens_idxs])
P = np.zeros((orth_vecs.shape[0], orth_vecs.shape[0]))
for i in range(orth_vecs.shape[1]):
P += orth_vecs[:, i].reshape(-1, 1) @ orth_vecs[:, i].reshape(
1, -1)
R = (np.eye(P.shape[0]) - P) @ X
if self.lmbda:
for j in range(R.shape[1]):
R[:, j] = R[:, j] + self.lmbda * (X[:, j] - R[:, j])
return R
def get_A():
D = np.diag(np.random.random_sample([
size,
])) / divide
V = np.random.rand(size, size)
U = orth(V)
D = mat(D)
U = mat(U)
A = U.I * D * U
return A
def orth(df, cols=None, index=None):
cols = df.columns if cols is None else cols
index = df.index if index is None else index
a2 = df.value
orthed_a = np.array(linalg.orth(a2, rcond=0), dtype=float)
if orthed_a.shape == a2.shape:
return pd.DataFrame(orthed_a, index=index, columns=cols)
else:
raise ValueError('exists highly related factors, please check data!')
def get_gram_T_Matrix(gram, type):
"""
Function used to obtain the T matrix of a gramian matrix.
Calculates the range space and null space of the given matrix and
organizes it in [orth , null]
Requires:
Numpy Array (Gramian matrix)
Returns:
Numpy Array: T Matrix
"""
if type == 'ctrb':
orth_ = spla.orth(gram)
null_ = spla.null_space(gram.transpose())
elif type == 'obsv':
orth_ = spla.orth(gram.transpose())
null_ = spla.null_space(gram)
T = np.hstack([orth_, null_])
return T
def __init__(self, *args, **kwargs):
MultiModalFunction.__init__(self, *args, **kwargs)
print((self.numPeaks, self.xdim))
self._opts = [(rand(self.xdim) - 0.5) * 8]
self._opts.extend([(rand(self.xdim) - 0.5) * 9.8 for _ in range(self.numPeaks-1)])
alphas = [power(self.maxCond, 2 * i / float(self.numPeaks - 2)) for i in range(self.numPeaks - 1)]
shuffle(alphas)
self._covs = [generateDiags(alpha, self.xdim, shuffled=True) / power(alpha, 0.25) for alpha in [self.optCond] + alphas]
self._R = orth(rand(self.xdim, self.xdim))
self._ws = [10] + [1.1 + 8 * i / float(self.numPeaks - 2) for i in range(self.numPeaks - 1)]
def rvs(self, n=1):
"""
It ignores the dimension ``n``.
"""
D = self.D
d = self.d
k = self.k
W_sub = orth(randn(D - k, d - k))
return np.vstack([np.hstack([W_sub, np.zeros((D - k, k))]),
np.hstack([np.zeros((k, d - k)), np.eye(k)])]).flatten()
def ica(data, k, c, svdmethod="direct", maxiter=100, tol=0.000001, seed=0):
"""Perform independent components analysis
:param: data: RDD of data points
:param k: number of principal components to use
:param c: number of independent components to find
:param maxiter: maximum number of iterations (default = 100)
:param: tol: tolerance for change in estimate (default = 0.000001)
:return w: the mixing matrix
:return: sigs: the independent components
TODO: also return unmixing matrix
"""
# get count
n = data.count()
# reduce dimensionality
scores, latent, comps = svd(data, k, meansubtract=0, method=svdmethod)
# whiten data
whtmat = real(dot(inv(diag(latent / sqrt(n))), comps))
unwhtmat = real(dot(transpose(comps), diag(latent / sqrt(n))))
wht = data.mapValues(lambda x: dot(whtmat, x))
# do multiple independent component extraction
if seed != 0:
random.seed(seed)
b = orth(random.randn(k, c))
b_old = zeros((k, c))
iter = 0
minabscos = 0
errvec = zeros(maxiter)
while (iter < maxiter) & ((1 - minabscos) > tol):
iter += 1
# update rule for pow3 non-linearity (TODO: add others)
b = wht.map(lambda (_, v): v).map(
lambda x: outer(x,
dot(x, b)**3)).sum() / n - 3 * b
# make orthogonal
b = dot(b, real(sqrtm(inv(dot(transpose(b), b)))))
# evaluate error
minabscos = min(abs(diag(dot(transpose(b), b_old))))
# store results
b_old = b
errvec[iter - 1] = (1 - minabscos)
# get un-mixing matrix
w = dot(transpose(b), whtmat)
# get components
sigs = data.mapValues(lambda x: dot(w, x))
return w, sigs
def sample_k(self, k=5):
if not hasattr(self,'A'):
self.compute_kernel(kernel_type='cos-sim')
eigen_vals, eigen_vec = eig(self.A)
eigen_vals =np.real(eigen_vals)
eigen_vec =np.real(eigen_vec)
eigen_vec = eigen_vec.T
N =self.A.shape[0]
Z= list(range(N))
if k==-1:
probs = eigen_vals/(eigen_vals+1)
jidx = np.array(np.random.rand(N)<=probs) # set j in paper
else:
jidx = sample_k_eigenvecs(eigen_vals, k)
V = eigen_vec[jidx] # Set of vectors V in paper
num_v = len(V)
Y = []
while num_v>0:
Pr = np.sum(V**2, 0)/np.sum(V**2)
y_i=np.argmax(np.array(np.random.rand() <= np.cumsum(Pr), np.int32))
# pdb.set_trace()
Y.append(y_i)
# Z.remove(Z[y_i])
V =V.T
try:
ri = np.argmax(np.abs(V[y_i]) >0)
except:
print("Error: Check: Matrix PSD/Sym")
exit()
V_r = V[:,ri]
# nidx = list(range(ri)) + list(range(ri+1, len(V)))
# V = V[nidx]
if num_v>0:
try:
V = la.orth(V- np.outer(V_r, (V[y_i,:]/V_r[y_i]) ))
except:
print("Error in Orthogonalization: Check: Matrix PSD/Sym")
pdb.set_trace()
V= V.T
num_v-=1
Y.sort()
out = np.array(Y)
return out
def profile_boundary_conditions(Ya,Yb,s,p): n=s['n'] speed=Ya[3] UL = [1.0/p['beta'], 0, 0]; UR = [0, 0, 0] AM = Flinear( np.array(UL) ,speed,p) AP = Flinear( np.array(UR) ,speed,p) P,Q = EvansBin.projection1(AM,-1,0.0) # Decay manifold at - infty LM = linalg.orth(P.T) P,Q = EvansBin.projection1(AP,0,1e-8) # Growth manifold at + infty LP = linalg.orth(P.T) BCa = list(Ya[0:n]-Ya[n:2*n]) #matching conditions = 4 BCa.append( Ya[0] - 0.5/p['beta']) # phase condition = 1 BCb = list(LM.T.dot(Yb[n:2*n-1] - UL)) # at -infty; = 2 BCb.append( (LP.T.dot(Yb[0:n-1]-UR))[0]) # at +infty; = 1 return np.real(np.array(BCa,dtype = complex)), np.real(np.array(BCb,dtype = complex))
def test_verbosity():
"""Check that nonzero verbosity level code runs.
"""
A, B = ElasticRod(100)
n = A.shape[0]
m = 20
np.random.seed(0)
V = rand(n, m)
X = orth(V)
_, _ = lobpcg(A, X, B=B, tol=1e-5, maxiter=30, largest=False,
verbosityLevel=9)
def fit(self, X, Y, n_features, n_passes):
#Randomized range finder for A^T.B
Qx = np.random.rand(X.shape[1],n_features)
Qy = np.random.rand(Y.shape[1],n_features)
for i in range(n_passes):
Ya = self.RandomMatrix(X, Y, Qy)
Yb = self.RandomMatrix(Y, X, Qx)
Qx = sci_lia.orth(Ya)
Qy = sci_lia.orth(Yb)
finalMatrix,Lx,Ly = self.FinalMatrixCalculation(Qx, Qy, X, Y)
U,D,V = np_lia.svd(finalMatrix)
Xx = ((X.shape[0])**(0.5)) * (np.dot(Qx, np.dot(np_lia.inv(Lx), U)))
Xy = ((Y.shape[0])**(0.5)) * (np.dot(Qy, np.dot(np_lia.inv(Ly), V)))
return Xx, Xy, D
def fit(self, X, y=None):
n_size, n_features = X.shape
S = np.cov(X.T)
eigvals, eigvectors = np.linalg.eig(S)
eigvectors = linalg.orth(eigvectors)
indices = np.argsort(eigvals)[-self.M:][::-1]
self.n_compents = eigvectors[indices]
self.n_compents = self.n_compents.reshape((n_features, -1))
self.eigvals = eigvals[indices]
total = np.sum(eigvals)
self.compent_ratos = self.eigvals / total
def __init__(self, sizes, len_training_data):
self.lmax = 0
self.num_layers = len(sizes)
self.num_training_data = len_training_data
self.sizes = sizes
self.biase = np.random.uniform(-0.2, 0.2, (1, self.sizes[1]))
self.biase = orth(self.biase)
self.biases = np.zeros((len_training_data, self.sizes[1]))
self.W_in = np.random.uniform(-1.0, 1.0, (self.sizes[1], self.sizes[0]))
if self.sizes[1] > self.sizes[0]:
self.W_in = orth(self.W_in)
# print (self.W_in.shape)
else:
self.W_in = orth(self.W_in.T).T
# print (self.W_in.shape)
# self.W_in = orth(self.W_in)
self.W_out = np.zeros((self.sizes[2], self.sizes[1]))
self.H = np.zeros((self.num_training_data, self.sizes[1]))
for i in range(len_training_data):
self.biases[i] = self.biase
def randomize_drone(dist_max,vel_max,ang_max,sim_agent):
start_location = np.random.random(size=[3])*dist_max
start_velocity = np.random.random(size=[3])*vel_max
start_angular = np.random.random(size=[3])*ang_max
#graham schmidt to get orientation, assume full rank... hopefully that works
start_orientation = orth(np.random.random(size=[3,3]))
sim_agent.setOrientation(start_orientation[0],start_orientation[1])
sim_agent.setAngular(start_angular)
sim_agent.setPosition(start_location)
sim_agent.setVelocity(start_velocity)
def run(n, d):
y = np.ones((n, 1))
y[n / 2:] = -1
X = np.random.random((n, d))
idx_row, idx_col = np.where(y == 1)
X[idx_row, 0] = 0.1 + X[idx_row, 0]
idx_row, idx_col = np.where(y == -1)
X[idx_row, 0] = -0.1 - X[idx_row, 0]
U = la.orth(np.random.random((d, d)))
X = np.dot(X, U)
return (X, y)
def fit(self,
*,
timeout: float = None,
iterations: int = None) -> base.CallResult[None]:
assert self._X is not None, "No training data provided."
assert self._X.ndim == 2, "Data is not in the right shape."
assert self._dim_subspaces <= self._X.shape[
1], "Dim_subspaces should be less than ambient dimension."
_X = self._X.T
n_features, n_samples = _X.shape
# randomly initialize subspaces
U_init = np.zeros((self._k, n_features, self._dim_subspaces))
for kk in range(self._k):
U_init[kk] = orth(
self._random_state.randn(n_features, self._dim_subspaces))
# compute residuals
full_residuals = np.zeros((n_samples, self._k))
for kk in range(self._k):
tmp1 = np.dot(U_init[kk].T, _X)
tmp2 = np.dot(U_init[kk], tmp1)
full_residuals[:, kk] = np.linalg.norm(_X - tmp2, ord=2, axis=0)
# label by nearest subspace
estimated_labels = np.argmin(full_residuals, axis=1)
# alternate between subspace estimation and assignment
prev_labels = -1 * np.ones(estimated_labels.shape)
it = 0
while np.sum(estimated_labels != prev_labels) and (iterations is None
or it < iterations):
# first update residuals after labels obtained
U = np.empty((self._k, n_features, self._dim_subspaces))
for kk in range(self._k):
Z = _X[:, estimated_labels == kk]
D, V = np.linalg.eig(np.dot(Z, Z.T))
D_idx = np.argsort(-D) # descending order
U[kk] = V.real[:, D_idx[list(range(self._dim_subspaces))]]
tmp1 = np.dot(U[kk, :].T, _X)
tmp2 = np.dot(U[kk, :], tmp1)
full_residuals[:, kk] = np.linalg.norm(_X - tmp2,
ord=2,
axis=0)
# update prev_labels
prev_labels = estimated_labels
# label by nearest subspace
estimated_labels = np.argmin(full_residuals, axis=1)
it = it + 1
self._U = U
return base.CallResult(None)
def time_mikota(self, n, solver):
m = 10
if solver == 'lobpcg':
X = rand(n, m)
X = orth(X)
LorU, lower = cho_factor(self.A, lower=0, overwrite_a=0)
M = LinearOperator(self.shape,
matvec=partial(_precond, LorU, lower),
matmat=partial(_precond, LorU, lower))
eigs, vecs = lobpcg(self.A, X, self.B, M, tol=1e-4, maxiter=40)
else:
w = eigh(self.A, self.B, eigvals_only=True, eigvals=(0, m-1))
def svd3(data, k, meanSubtract=1):
n = data.count()
d = len(data.first())
if meanSubtract == 1:
data = data.map(lambda x: x - mean(x))
def outerProd(x):
return outer(x, x)
def outerSum(iterator):
yield sum(outer(x, x) for x in iterator)
def outerSum2(iterator, other1, other2):
yield sum(outer(x, dot(dot(x, other1), other2)) for x in iterator)
C = random.rand(k, d)
iterNum = 0
iterMax = 10
error = 100
tol = 0.000001
while (iterNum < iterMax) & (error > tol):
Cold = C
Cinv = dot(transpose(C), inv(dot(C, transpose(C))))
preMult1 = data.context.broadcast(Cinv)
# X = data.times(preMult1.value)
# XX' = X.cov()
XX = data.map(lambda x: outerProd(dot(x, preMult1.value))).reduce(lambda x, y: x + y)
XXinv = inv(XX)
preMult2 = data.context.broadcast(dot(Cinv, XXinv))
# data1 = data.times(dot(Cinv, inv(XX'))
# C = data.times(data1)
C = data.map(lambda x: outer(x, dot(x, preMult2.value))).reduce(lambda x, y: x + y)
C = transpose(C)
error = sum(sum((C-Cold) ** 2))
iterNum += 1
C = transpose(orth(transpose(C)))
# cov = data.times(transpose(C)).cov()
cov = data.map(lambda x: dot(x, transpose(C))).mapPartitions(outerSum).reduce(
lambda x, y: x + y) / n
w, v = eig(cov)
w = real(w)
v = real(v)
inds = argsort(w)[::-1]
latent = w[inds[0:k]]
comps = dot(transpose(v[:, inds[0:k]]), C)
scores = data.map(lambda x: inner(x, comps))
return comps, latent, scores
def __init__(self, basef, rotMat = None):
""" by default the rotation matrix is random. """
FunctionEnvironment.__init__(self, basef.xdim, basef.xopt)
if rotMat == None:
# make a random orthogonal rotation matrix
self.M = orth(rand(basef.xdim, basef.xdim))
else:
self.M = rotMat
if isinstance(basef, FunctionEnvironment):
self.desiredValue = basef.desiredValue
self.xopt = dot(inv(self.M), self.xopt)
self.f = lambda x: basef.f(dot(x,self.M))
def compare_solutions(A, B, m):
"""Check eig vs. lobpcg consistency.
"""
n = A.shape[0]
np.random.seed(0)
V = rand(n, m)
X = orth(V)
eigvals, _ = lobpcg(A, X, B=B, tol=1e-5, maxiter=30, largest=False)
eigvals.sort()
w, _ = eig(A, b=B)
w.sort()
assert_almost_equal(w[:int(m / 2)], eigvals[:int(m / 2)], decimal=2)
def ica(data, k, c, svdmethod="direct", maxiter=100, tol=0.000001, seed=0):
"""perform independent components analysis
arguments:
data - RDD of data points
k - number of principal components to use
c - number of independent components to find
maxiter - maximum number of iterations (default = 100)
tol - tolerance for change in estimate (default = 0.000001)
returns:
w - the mixing matrix
sigs - the independent components
"""
# get count
n = data.count()
# reduce dimensionality
scores, latent, comps = svd(data, k, meansubtract=0, method=svdmethod)
# whiten data
whtmat = real(dot(inv(diag(latent/sqrt(n))), comps))
unwhtmat = real(dot(transpose(comps), diag(latent/sqrt(n))))
wht = data.map(lambda x: dot(whtmat, x))
# do multiple independent component extraction
if seed != 0:
random.seed(seed)
b = orth(random.randn(k, c))
b_old = zeros((k, c))
iter = 0
minabscos = 0
errvec = zeros(maxiter)
while (iter < maxiter) & ((1 - minabscos) > tol):
iter += 1
# update rule for pow3 non-linearity (TODO: add others)
b = wht.map(lambda x: outer(x, dot(x, b) ** 3)).sum() / n - 3 * b
# make orthogonal
b = dot(b, real(sqrtm(inv(dot(transpose(b), b)))))
# evaluate error
minabscos = min(abs(diag(dot(transpose(b), b_old))))
# store results
b_old = b
errvec[iter-1] = (1 - minabscos)
# get un-mixing matrix
w = dot(transpose(b), whtmat)
# get components
sigs = data.map(lambda x: dot(w, x))
return w, sigs
def eigf_bcs(Ya,Yb,s,p): n=s['n'] # = 4 = Dimension of first order profile equation eig_n = 2*n # = 8 = Dimension of first order eigenfunction equation ph = s['ph'] AM = A(s['L'], ((Ya[-2] + 1.0j*Ya[-1]) + (Yb[-2] + 1.0j*Yb[-1]))/2.0,s,p) AP = A(s['R'], ((Ya[-2] + 1.0j*Ya[-1]) + (Yb[-2] + 1.0j*Yb[-1]))/2.0,s,p) # P1,Q1 = projection2(AM,-1,1e-6); LM = linalg.orth(P1.T) P2,Q2 = projection2(AP,+1,1e-6); LP = linalg.orth(P2.T) # BCS = np.zeros(18) BCS[0:8] = (Ya[0:eig_n]-Ya[eig_n:2*eig_n]) # 8 matching conditions BCS[8] = Ya[ph[0,0]]- ph[0,1] # 2 phase conditions BCS[9] = Ya[ph[1,0]]- ph[1,1] # # 8 projective conditions BCS[10:12] = np.real(LP.T.dot( Yb[0:n] + 1.0j*Yb[n:2*n] )) BCS[12:14] = np.imag(LP.T.dot( Yb[0:n] + 1.0j*Yb[n:2*n] )) BCS[14:16] = np.real(LM.T.dot( Yb[eig_n:eig_n+n] + 1.0j*Yb[eig_n+n:eig_n+2*n] )) BCS[16:18] = np.imag(LM.T.dot( Yb[eig_n:eig_n+n] + 1.0j*Yb[eig_n+n:eig_n+2*n] )) return BCS
def test_verbosity():
"""Check that nonzero verbosity level code runs.
"""
A, B = ElasticRod(100)
n = A.shape[0]
m = 20
np.random.seed(0)
V = rand(n,m)
X = linalg.orth(V)
eigs,vecs = lobpcg(A, X, B=B, tol=1e-5, maxiter=30, largest=False,
verbosityLevel=11)
def random_walk(G,initial_prob,subspace_dim=3,walk_steps=3):
"""
Start a random walk with probability distribution p_initial.
Transition matrix needs to be calculated according to adjacent matrix G.
"""
assert type(initial_prob) == np.ndarray, "Initial probability distribution is \
not a numpy array"
# Transform the adjacent matrix to a laplacian matrix P
P = adj_to_Laplacian(G)
Prob_Matrix = np.zeros((G.shape[0], subspace_dim))
Prob_Matrix[:,0] = initial_prob
for i in range(1,subspace_dim):
Prob_Matrix[:,i] = np.dot(Prob_Matrix[:,i-1], P)
Orth_Prob_Matrix = splin.orth(Prob_Matrix)
for i in range(walk_steps):
temp = np.dot(Orth_Prob_Matrix.T, P)
Orth_Prob_Matrix = splin.orth(temp.T)
return Orth_Prob_Matrix
def compare_solutions(A,B,m):
n = A.shape[0]
np.random.seed(0)
V = rand(n,m)
X = linalg.orth(V)
eigs,vecs = lobpcg(A, X, B=B, tol=1e-5, maxiter=30)
eigs.sort()
w,v = eig(A,b=B)
w.sort()
assert_almost_equal(w[:int(m/2)],eigs[:int(m/2)],decimal=2)
def __init__(self, basef, rotMat=None):
""" by default the rotation matrix is random. """
FunctionEnvironment.__init__(self, basef.xdim, basef.xopt)
if rotMat == None:
# make a random orthogonal rotation matrix
self._M = orth(rand(basef.xdim, basef.xdim))
else:
self._M = rotMat
self.desiredValue = basef.desiredValue
self.toBeMinimized = basef.toBeMinimized
self.xopt = dot(inv(self._M), self.xopt)
def rf(x):
if isinstance(x, ParameterContainer):
x = x.params
return basef.f(dot(x, self._M))
self.f = rf
def runtest(self):
k = len(self.rdd.first()[1])
c = 3
n = 1000
B = orth(random.randn(k, c))
Bold = zeros((k, c))
iterNum = 0
errVec = zeros(20)
while (iterNum < 5):
iterNum += 1
B = self.rdd.map(lambda (_, v): v).map(lambda x: outer(x, dot(x, B) ** 3)).reduce(lambda x, y: x + y) / n - 3 * B
B = dot(B, real(sqrtm(inv(dot(transpose(B), B)))))
minAbsCos = min(abs(diag(dot(transpose(B), Bold))))
Bold = B
errVec[iterNum-1] = (1 - minAbsCos)
sigs = self.rdd.mapValues(lambda x: dot(B, x))
def create_random_Sigmas(n, N, L, M, dist=gamma(2, scale=2)):
""" Creates N random nxn covariance matrices s.t. the Lipschitz
constants are uniformly bounded by Lbnd. Here dist is a 'frozen'
scipy.stats probability distribution supported on R+"""
# compute lower bound for eigenvalues
lambdamin = ((2*np.pi)**n*np.e*L**2/M**2)**(-1.0/(n+1))
Sigmas = []
for i in range(N):
# create random orthonormal matrix
V = orth(np.random.uniform(size=(n,n)))
# create n random eigenvalues from the distribution and shift them by lambdamin
lambdas = lambdamin + dist.rvs(n)
Sigma = np.zeros((n,n))
for lbda, v in zip(lambdas, V):
Sigma = Sigma + lbda*np.outer(v,v)
Sigmas.append(Sigma)
return np.array(Sigmas)
def solve_lrr(X, A, lamb, reg=0, alm_type=0, display=False):
Q = orth(A.T)
B = A.dot(Q)
if reg == 0:
if alm_type == 0:
Z, E = exact_alm_lrr_l21v2(X, B, lamb, display=display)
else:
Z, E = inexact_alm_lrr_l21(X, B, lamb, display=display)
else:
if alm_type == 0:
Z, E = exact_alm_lrr_l1v2(X, B, lamb, display=display)
else:
Z, E = inexact_alm_lrr_l1(X, B, lamb, display)
Z = Q.dot(Z)
return (Z, E)
def create_random_Q(domain, mu, L, M, pd=True, H=0, dist=uniform()):
""" Creates random symmetric nxn matrix s.t. the Lipschitz constant of the resulting
quadratic function is bounded by L. Here M is the uniform bound on the maximal loss.
If pd is True, then the matrix is pos. definite and H is a lower bound on the
eigenvalues of Q. Finally, dist is a 'frozen' scipy.stats probability
distribution supported on [0,1]. """
n = domain.n
# compute upper bound for eigenvalues
Dmu = domain.compute_Dmu(mu)
lambdamax = np.min((L/Dmu, 2*M/Dmu**2))
# create random orthonormal matrix
V = orth(np.random.uniform(size=(n,n)))
# create n random eigenvalues from the distribution dist, scale them by lambdamax
if pd:
lambdas = H + (lambdamax-H)*dist.rvs(n)
else:
# randomly assign pos. and negative values to the evals
lambdas = np.random.choice((-1,1), size=n)*lambdamax*dist.rvs(n)
return np.dot(V, np.dot(np.diag(lambdas), V.T)), lambdamax
def haar_matrix(N):
if(not np.log2(N).is_integer()):
print("Invalid Haar Matrix Size")
raise Exception
else:
H = np.ones((1,N))
for k in range(1,N):
p = max(0, np.floor(np.log2(k)))
q = k - (2.0**p) + 1
H_pos_k_upper = np.array((q - 1.0)/(2.0**p) <= np.linspace(0,1,N,False))
H_pos_k_lower = np.array( np.linspace(0,1,N,endpoint=False) < (q - 0.5)/(2.0**p))
H_pos_k = np.logical_and(H_pos_k_upper, H_pos_k_lower)
H_neg_k_upper = np.array((q - 0.5)/(2.0**p) <= np.linspace(0,1,N,endpoint=False))
H_neg_k_lower = np.array(np.linspace(0,1,N,endpoint=False) < (q)/(2.0**p))
H_neg_k = np.logical_and(H_neg_k_upper, H_neg_k_lower)
H_k = np.array(H_pos_k, dtype=np.float) - np.array(H_neg_k, dtype=np.float)
H_k = H_k
H = np.vstack((H,H_k))
return linalg.orth(H)
