def rejection_sampling_z(N, y, W1, W2):
"""
A rejection sampling method for sampling the inactive variables from a
polytope.
**See Also**
domains.sample_z
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# Build a box around z for uniform sampling
qps = QPSolver()
A = np.vstack((W2, -W2))
b = np.vstack((-1-s, -1+s)).reshape((2*m, 1))
lbox, ubox = np.zeros((1,m-n)), np.zeros((1,m-n))
for i in range(m-n):
clb = np.zeros((m-n,1))
clb[i,0] = 1.0
lbox[0,i] = qps.linear_program_ineq(clb, A, b)[i,0]
cub = np.zeros((m-n,1))
cub[i,0] = -1.0
ubox[0,i] = qps.linear_program_ineq(cub, A, b)[i,0]
bn = BoundedNormalizer(lbox, ubox)
Zbox = bn.unnormalize(np.random.uniform(-1.0,1.0,size=(50*N,m-n)))
ind = np.all(np.dot(A, Zbox.T) >= b, axis=0)
if np.sum(ind) >= N:
Z = Zbox[ind,:]
return Z[:N,:].reshape((N,m-n))
else:
return None
def hit_and_run_z(N, y, W1, W2):
m, n = W1.shape
# get an initial feasible point
qps = QPSolver()
lb = -np.ones((m,1))
ub = np.ones((m,1))
c = np.zeros((m,1))
x0 = qps.linear_program_eq(c, W1.T, y.reshape((n,1)), lb, ub)
z0 = np.dot(W2.T, x0).reshape((m-n, 1))
# define the polytope A >= b
s = np.dot(W1, y).reshape((m, 1))
A = np.vstack((W2, -W2))
b = np.vstack((-1-s, -1+s)).reshape((2*m, 1))
# tolerance
ztol = 1e-6
eps0 = ztol/4.0
Z = np.zeros((N, m-n))
for i in range(N):
# random direction
bad_dir = True
count, maxcount = 0, 50
while bad_dir:
d = np.random.normal(size=(m-n,1))
bad_dir = np.any(np.dot(A, z0 + eps0*d) <= b)
count += 1
if count >= maxcount:
warnings.warn('There are no more directions worth pursuing in hit and run. Got {:d} samples.'.format(i))
Z[i:,:] = np.tile(z0, (1,N-i)).transpose()
return Z
# find constraints that impose lower and upper bounds on eps
f, g = b - np.dot(A,z0), np.dot(A, d)
# find an upper bound on the step
min_ind = np.logical_and(g<=0, f < -np.sqrt(np.finfo(np.float).eps))
eps_max = np.amin(f[min_ind]/g[min_ind])
# find a lower bound on the step
max_ind = np.logical_and(g>0, f < -np.sqrt(np.finfo(np.float).eps))
eps_min = np.amax(f[max_ind]/g[max_ind])
# randomly sample eps
eps1 = np.random.uniform(eps_min, eps_max)
# take a step along d
z1 = z0 + eps1*d
Z[i,:] = z1.reshape((m-n, ))
# update temp vars
z0, eps0 = z1, eps1
return Z
def rejection_sampling_z(N, y, W1, W2):
"""A rejection sampling method for sampling the from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# Build a box around z for uniform sampling
qps = QPSolver()
A = np.vstack((W2, -W2))
b = np.vstack((-1 - s, -1 + s)).reshape((2 * m, 1))
lbox, ubox = np.zeros((1, m - n)), np.zeros((1, m - n))
for i in range(m - n):
clb = np.zeros((m - n, 1))
clb[i, 0] = 1.0
lbox[0, i] = qps.linear_program_ineq(clb, A, b)[i, 0]
cub = np.zeros((m - n, 1))
cub[i, 0] = -1.0
ubox[0, i] = qps.linear_program_ineq(cub, A, b)[i, 0]
bn = BoundedNormalizer(lbox, ubox)
Zbox = bn.unnormalize(np.random.uniform(-1.0, 1.0, size=(50 * N, m - n)))
ind = np.all(np.dot(A, Zbox.T) >= b, axis=0)
if np.sum(ind) >= N:
Z = Zbox[ind, :]
return Z[:N, :].reshape((N, m - n))
else:
return None
def rejection_sampling_z(N, y, W1, W2):
"""A rejection sampling method for sampling the from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# Build a box around z for uniform sampling
qps = QPSolver()
A = np.vstack((W2, -W2))
b = np.vstack((-1-s, -1+s)).reshape((2*m, 1))
lbox, ubox = np.zeros((1,m-n)), np.zeros((1,m-n))
for i in range(m-n):
clb = np.zeros((m-n,1))
clb[i,0] = 1.0
lbox[0,i] = qps.linear_program_ineq(clb, A, b)[i,0]
cub = np.zeros((m-n,1))
cub[i,0] = -1.0
ubox[0,i] = qps.linear_program_ineq(cub, A, b)[i,0]
bn = BoundedNormalizer(lbox, ubox)
Zbox = bn.unnormalize(np.random.uniform(-1.0,1.0,size=(50*N,m-n)))
ind = np.all(np.dot(A, Zbox.T) >= b, axis=0)
if np.sum(ind) >= N:
Z = Zbox[ind,:]
return Z[:N,:].reshape((N,m-n))
else:
return None
def regularize_z(self, Y, N=1):
"""Train the global quadratic for the regularization.
Parameters
----------
Y : ndarray
N-by-n matrix of points in the space of active variables
N : int, optional
merely there satisfy the interface of `regularize_z`. It should not
be anything other than 1
Returns
-------
Z : ndarray
N-by-(m-n)-by-1 matrix that contains a value of the inactive
variables for each value of the inactive variables
Notes
-----
In contrast to the `regularize_z` in BoundedActiveVariableMap and
UnboundedActiveVariableMap, this implementation of `regularize_z` uses
a quadratic program to find a single value of the inactive variables
for each value of the active variables.
"""
if N != 1:
raise Exception('MinVariableMap needs N=1.')
W1, W2 = self.domain.subspaces.W1, self.domain.subspaces.W2
m, n = W1.shape
NY = Y.shape[0]
qps = QPSolver()
Zlist = []
A_ineq = np.vstack((W2, -W2))
for y in Y:
c = self._bz.reshape((m-n, 1)) + np.dot(self._zAy, y).reshape((m-n, 1))
b_ineq = np.vstack((
-1-np.dot(W1, y).reshape((m, 1)),
-1+np.dot(W1, y).reshape((m, 1))
))
z = qps.quadratic_program_ineq(c, self._zAz, A_ineq, b_ineq)
Zlist.append(z)
Z = np.array(Zlist).reshape((NY, m-n, N))
return Z
def regularize_z(self, Y, N=1):
"""Train the global quadratic for the regularization.
Parameters
----------
Y : ndarray
N-by-n matrix of points in the space of active variables
N : int, optional
merely there satisfy the interface of `regularize_z`. It should not
be anything other than 1
Returns
-------
Z : ndarray
N-by-(m-n)-by-1 matrix that contains a value of the inactive
variables for each value of the inactive variables
Notes
-----
In contrast to the `regularize_z` in BoundedActiveVariableMap and
UnboundedActiveVariableMap, this implementation of `regularize_z` uses
a quadratic program to find a single value of the inactive variables
for each value of the active variables.
"""
if N != 1:
raise Exception('MinVariableMap needs N=1.')
W1, W2 = self.domain.subspaces.W1, self.domain.subspaces.W2
m, n = W1.shape
NY = Y.shape[0]
qps = QPSolver()
Zlist = []
A_ineq = np.vstack((W2, -W2))
for y in Y:
c = self._bz.reshape((m-n, 1)) + np.dot(self._zAy, y).reshape((m-n, 1))
b_ineq = np.vstack((
-1-np.dot(W1, y).reshape((m, 1)),
-1+np.dot(W1, y).reshape((m, 1))
))
z = qps.quadratic_program_ineq(c, self._zAz, A_ineq, b_ineq)
Zlist.append(z)
Z = np.array(Zlist).reshape((NY, m-n, N))
return Z
def random_walk_z(N, y, W1, W2):
"""
A random walk method for sampling the inactive variables from a polytope.
**See Also**
domains.sample_z
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# linear program to get starting z0
if np.all(np.zeros((m, 1)) <= 1-s) and np.all(np.zeros((m, 1)) >= -1-s):
z0 = np.zeros((m-n, 1))
else:
qps = QPSolver()
lb = -np.ones((m,1))
ub = np.ones((m,1))
c = np.zeros((m,1))
x0 = qps.linear_program_eq(c, W1.T, y.reshape((n,1)), lb, ub)
z0 = np.dot(W2.T, x0).reshape((m-n, 1))
# get MCMC step size
sig = 0.1*np.minimum(
np.linalg.norm(np.dot(W2, z0) + s - 1),
np.linalg.norm(np.dot(W2, z0) + s + 1))
# burn in
for i in range(10*N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
# sample
Z = np.zeros((m-n, N))
for i in range(N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
Z[:,i] = z0.reshape((z0.shape[0], ))
return Z.reshape((N, m-n))
def regularize_z(self, Y, N=1):
if N != 1:
raise Exception('MinVariableMap needs N=1.')
W1, W2 = self.domain.subspaces.W1, self.domain.subspaces.W2
m, n = W1.shape
NY = Y.shape[0]
qps = QPSolver()
Zlist = []
A_ineq = np.vstack((W2, -W2))
for y in Y:
c = self.bz.reshape((m-n, 1)) + np.dot(self.zAy, y).reshape((m-n, 1))
b_ineq = np.vstack((
-1-np.dot(W1, y).reshape((m, 1)),
-1+np.dot(W1, y).reshape((m, 1))
))
z = qps.quadratic_program_ineq(c, self.zAz, A_ineq, b_ineq)
Zlist.append(z)
return np.array(Zlist).reshape((NY, m-n, N))
def random_walk_z(N, y, W1, W2):
"""
I tried getting a random walk to work for MCMC sampling from the polytope
that z lives in. But it doesn't seem to work very well. Gonna try rejection
sampling.
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# linear program to get starting z0
if np.all(np.zeros((m, 1)) <= 1-s) and np.all(np.zeros((m, 1)) >= -1-s):
z0 = np.zeros((m-n, 1))
else:
qps = QPSolver()
lb = -np.ones((m,1))
ub = np.ones((m,1))
c = np.zeros((m,1))
x0 = qps.linear_program_eq(c, W1.T, y.reshape((n,1)), lb, ub)
z0 = np.dot(W2.T, x0).reshape((m-n, 1))
# get MCMC step size
sig = 0.1*np.minimum(
np.linalg.norm(np.dot(W2, z0) + s - 1),
np.linalg.norm(np.dot(W2, z0) + s + 1))
# burn in
for i in range(10*N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
# sample
Z = np.zeros((m-n, N))
for i in range(N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
Z[:,i] = z0.reshape((z0.shape[0], ))
return Z.reshape((N, m-n))
def hit_and_run_z(N, y, W1, W2):
"""
A hit and run method for sampling the inactive variables from a polytope.
**See Also**
domains.sample_z
"""
m, n = W1.shape
# get an initial feasible point using the Chebyshev center. huge props to
# David Gleich for showing me the Chebyshev center.
s = np.dot(W1, y).reshape((m, 1))
normW2 = np.sqrt( np.sum( np.power(W2, 2), axis=1 ) ).reshape((m,1))
A = np.hstack(( np.vstack((W2, -W2.copy())), np.vstack((normW2, normW2.copy())) ))
b = np.vstack((1-s, 1+s)).reshape((2*m, 1))
c = np.zeros((m-n+1,1))
c[-1] = -1.0
qps = QPSolver()
zc = qps.linear_program_ineq(c, -A, -b)
z0 = zc[:-1].reshape((m-n, 1))
# define the polytope A >= b
s = np.dot(W1, y).reshape((m, 1))
A = np.vstack((W2, -W2))
b = np.vstack((-1-s, -1+s)).reshape((2*m, 1))
# tolerance
ztol = 1e-6
eps0 = ztol/4.0
Z = np.zeros((N, m-n))
for i in range(N):
# random direction
bad_dir = True
count, maxcount = 0, 50
while bad_dir:
d = np.random.normal(size=(m-n,1))
bad_dir = np.any(np.dot(A, z0 + eps0*d) <= b)
count += 1
if count >= maxcount:
logging.getLogger(__name__).warn('There are no more directions worth pursuing in hit and run. Got {:d} samples.'.format(i))
Z[i:,:] = np.tile(z0, (1,N-i)).transpose()
return Z
# find constraints that impose lower and upper bounds on eps
f, g = b - np.dot(A,z0), np.dot(A, d)
# find an upper bound on the step
min_ind = np.logical_and(g<=0, f < -np.sqrt(np.finfo(np.float).eps))
eps_max = np.amin(f[min_ind]/g[min_ind])
# find a lower bound on the step
max_ind = np.logical_and(g>0, f < -np.sqrt(np.finfo(np.float).eps))
eps_min = np.amax(f[max_ind]/g[max_ind])
# randomly sample eps
eps1 = np.random.uniform(eps_min, eps_max)
# take a step along d
z1 = z0 + eps1*d
Z[i,:] = z1.reshape((m-n, ))
# update temp var
z0 = z1.copy()
return Z
def random_walk_z(N, y, W1, W2):
"""A random walk method for sampling from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# linear program to get starting z0
if np.all(np.zeros((m, 1)) <= 1 - s) and np.all(
np.zeros((m, 1)) >= -1 - s):
z0 = np.zeros((m - n, 1))
else:
qps = QPSolver()
lb = -np.ones((m, 1))
ub = np.ones((m, 1))
c = np.zeros((m, 1))
x0 = qps.linear_program_eq(c, W1.T, y.reshape((n, 1)), lb, ub)
z0 = np.dot(W2.T, x0).reshape((m - n, 1))
# get MCMC step size
sig = 0.1 * np.minimum(np.linalg.norm(np.dot(W2, z0) + s - 1),
np.linalg.norm(np.dot(W2, z0) + s + 1))
# burn in
for i in range(10 * N):
zc = z0 + sig * np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1 - s) and np.all(
np.dot(W2, zc) >= -1 - s):
z0 = zc
# sample
Z = np.zeros((m - n, N))
for i in range(N):
zc = z0 + sig * np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1 - s) and np.all(
np.dot(W2, zc) >= -1 - s):
z0 = zc
Z[:, i] = z0.reshape((z0.shape[0], ))
return Z.reshape((N, m - n))
def hit_and_run_z(N, y, W1, W2):
"""A hit and run method for sampling the inactive variables from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
# get an initial feasible point using the Chebyshev center. huge props to
# David Gleich for showing me the Chebyshev center.
s = np.dot(W1, y).reshape((m, 1))
normW2 = np.sqrt(np.sum(np.power(W2, 2), axis=1)).reshape((m, 1))
A = np.hstack((np.vstack(
(W2, -W2.copy())), np.vstack((normW2, normW2.copy()))))
b = np.vstack((1 - s, 1 + s)).reshape((2 * m, 1))
c = np.zeros((m - n + 1, 1))
c[-1] = -1.0
qps = QPSolver()
zc = qps.linear_program_ineq(c, -A, -b)
z0 = zc[:-1].reshape((m - n, 1))
# define the polytope A >= b
s = np.dot(W1, y).reshape((m, 1))
A = np.vstack((W2, -W2))
b = np.vstack((-1 - s, -1 + s)).reshape((2 * m, 1))
# tolerance
ztol = 1e-6
eps0 = ztol / 4.0
Z = np.zeros((N, m - n))
for i in range(N):
# random direction
bad_dir = True
count, maxcount = 0, 50
while bad_dir:
d = np.random.normal(size=(m - n, 1))
bad_dir = np.any(np.dot(A, z0 + eps0 * d) <= b)
count += 1
if count >= maxcount:
Z[i:, :] = np.tile(z0, (1, N - i)).transpose()
return Z
# find constraints that impose lower and upper bounds on eps
f, g = b - np.dot(A, z0), np.dot(A, d)
# find an upper bound on the step
min_ind = np.logical_and(g <= 0, f < -np.sqrt(np.finfo(np.float).eps))
eps_max = np.amin(f[min_ind] / g[min_ind])
# find a lower bound on the step
max_ind = np.logical_and(g > 0, f < -np.sqrt(np.finfo(np.float).eps))
eps_min = np.amax(f[max_ind] / g[max_ind])
# randomly sample eps
eps1 = np.random.uniform(eps_min, eps_max)
# take a step along d
z1 = z0 + eps1 * d
Z[i, :] = z1.reshape((m - n, ))
# update temp var
z0 = z1.copy()
return Z
def random_walk_z(N, y, W1, W2):
"""A random walk method for sampling from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
s = np.dot(W1, y).reshape((m, 1))
# linear program to get starting z0
if np.all(np.zeros((m, 1)) <= 1-s) and np.all(np.zeros((m, 1)) >= -1-s):
z0 = np.zeros((m-n, 1))
else:
qps = QPSolver()
lb = -np.ones((m,1))
ub = np.ones((m,1))
c = np.zeros((m,1))
x0 = qps.linear_program_eq(c, W1.T, y.reshape((n,1)), lb, ub)
z0 = np.dot(W2.T, x0).reshape((m-n, 1))
# get MCMC step size
sig = 0.1*np.minimum(
np.linalg.norm(np.dot(W2, z0) + s - 1),
np.linalg.norm(np.dot(W2, z0) + s + 1))
# burn in
for i in range(10*N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
# sample
Z = np.zeros((m-n, N))
for i in range(N):
zc = z0 + sig*np.random.normal(size=z0.shape)
if np.all(np.dot(W2, zc) <= 1-s) and np.all(np.dot(W2, zc) >= -1-s):
z0 = zc
Z[:,i] = z0.reshape((z0.shape[0], ))
return Z.reshape((N, m-n))
def hit_and_run_z(N, y, W1, W2):
"""A hit and run method for sampling the inactive variables from a polytope.
Parameters
----------
N : int
the number of inactive variable samples
y : ndarray
the value of the active variables
W1 : ndarray
m-by-n matrix that contains the eigenvector bases of the n-dimensional
active subspace
W2 : ndarray
m-by-(m-n) matrix that contains the eigenvector bases of the
(m-n)-dimensional inactive subspace
Returns
-------
Z : ndarray
N-by-(m-n) matrix that contains values of the inactive variable that
correspond to the given `y`
See Also
--------
domains.sample_z
Notes
-----
The interface for this implementation is written specifically for
`domains.sample_z`.
"""
m, n = W1.shape
# get an initial feasible point using the Chebyshev center. huge props to
# David Gleich for showing me the Chebyshev center.
s = np.dot(W1, y).reshape((m, 1))
normW2 = np.sqrt( np.sum( np.power(W2, 2), axis=1 ) ).reshape((m,1))
A = np.hstack(( np.vstack((W2, -W2.copy())), np.vstack((normW2, normW2.copy())) ))
b = np.vstack((1-s, 1+s)).reshape((2*m, 1))
c = np.zeros((m-n+1,1))
c[-1] = -1.0
qps = QPSolver()
zc = qps.linear_program_ineq(c, -A, -b)
z0 = zc[:-1].reshape((m-n, 1))
# define the polytope A >= b
s = np.dot(W1, y).reshape((m, 1))
A = np.vstack((W2, -W2))
b = np.vstack((-1-s, -1+s)).reshape((2*m, 1))
# tolerance
ztol = 1e-6
eps0 = ztol/4.0
Z = np.zeros((N, m-n))
for i in range(N):
# random direction
bad_dir = True
count, maxcount = 0, 50
while bad_dir:
d = np.random.normal(size=(m-n,1))
bad_dir = np.any(np.dot(A, z0 + eps0*d) <= b)
count += 1
if count >= maxcount:
Z[i:,:] = np.tile(z0, (1,N-i)).transpose()
return Z
# find constraints that impose lower and upper bounds on eps
f, g = b - np.dot(A,z0), np.dot(A, d)
# find an upper bound on the step
min_ind = np.logical_and(g<=0, f < -np.sqrt(np.finfo(np.float).eps))
eps_max = np.amin(f[min_ind]/g[min_ind])
# find a lower bound on the step
max_ind = np.logical_and(g>0, f < -np.sqrt(np.finfo(np.float).eps))
eps_min = np.amax(f[max_ind]/g[max_ind])
# randomly sample eps
eps1 = np.random.uniform(eps_min, eps_max)
# take a step along d
z1 = z0 + eps1*d
Z[i,:] = z1.reshape((m-n, ))
# update temp var
z0 = z1.copy()
return Z
