def time_graphical_lasso(
emp_cov,
alpha=0.01,
rho=1,
beta=1,
max_iter=100,
n_samples=None,
verbose=False,
psi="laplacian",
tol=1e-4,
rtol=1e-4,
return_history=False,
return_n_iter=True,
mode="admm",
compute_objective=True,
stop_at=None,
stop_when=1e-4,
update_rho_options=None,
init="empirical",
):
"""Time-varying graphical lasso solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(S_i, K_i) + alpha*||K_i||_{od,1}
+ beta sum_{i=2}^T Psi(K_i - K_{i-1})
where S_i = (1/n_i) X_i^T \times X_i is the empirical covariance of data
matrix X (training observations by features).
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Empirical covariance of data.
alpha, beta : float, optional
Regularisation parameter.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
n_samples : ndarray
Number of samples available for each time point.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
verbose : bool, default False
Print info at each iteration.
update_rho_options : dict, optional
Arguments for the rho update.
See regain.update_rules.update_rho function for more information.
compute_objective : bool, default True
Choose to compute the objective value.
init : {'empirical', 'zero', ndarray}
Choose how to initialize the precision matrix, with the inverse
empirical covariance, zero matrix or precomputed.
Returns
-------
K : numpy.array, 3-dimensional (T x d x d)
Solution to the problem for each time t=1...T .
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
psi, prox_psi, psi_node_penalty = check_norm_prox(psi)
Z_0 = init_precision(emp_cov, mode=init)
Z_1 = Z_0.copy()[:-1] # np.zeros_like(emp_cov)[:-1]
Z_2 = Z_0.copy()[1:] # np.zeros_like(emp_cov)[1:]
U_0 = np.zeros_like(Z_0)
U_1 = np.zeros_like(Z_1)
U_2 = np.zeros_like(Z_2)
Z_0_old = np.zeros_like(Z_0)
Z_1_old = np.zeros_like(Z_1)
Z_2_old = np.zeros_like(Z_2)
# divisor for consensus variables, accounting for two less matrices
divisor = np.full(emp_cov.shape[0], 3, dtype=float)
divisor[0] -= 1
divisor[-1] -= 1
if n_samples is None:
n_samples = np.ones(emp_cov.shape[0])
checks = [convergence(obj=objective(n_samples, emp_cov, Z_0, Z_0, Z_1, Z_2, alpha, beta, psi))]
for iteration_ in range(max_iter):
# update K
A = Z_0 - U_0
A[:-1] += Z_1 - U_1
A[1:] += Z_2 - U_2
A /= divisor[:, None, None]
# soft_thresholding_ = partial(soft_thresholding, lamda=alpha / rho)
# K = np.array(map(soft_thresholding_, A))
A += A.transpose(0, 2, 1)
A /= 2.0
A *= -rho * divisor[:, None, None] / n_samples[:, None, None]
A += emp_cov
K = np.array([prox_logdet(a, lamda=ni / (rho * div)) for a, div, ni in zip(A, divisor, n_samples)])
# update Z_0
A = K + U_0
A += A.transpose(0, 2, 1)
A /= 2.0
Z_0 = soft_thresholding(A, lamda=alpha / rho)
# other Zs
A_1 = K[:-1] + U_1
A_2 = K[1:] + U_2
if not psi_node_penalty:
prox_e = prox_psi(A_2 - A_1, lamda=2.0 * beta / rho)
Z_1 = 0.5 * (A_1 + A_2 - prox_e)
Z_2 = 0.5 * (A_1 + A_2 + prox_e)
else:
Z_1, Z_2 = prox_psi(
np.concatenate((A_1, A_2), axis=1),
lamda=0.5 * beta / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter,
)
# update residuals
U_0 += K - Z_0
U_1 += K[:-1] - Z_1
U_2 += K[1:] - Z_2
# diagnostics, reporting, termination checks
rnorm = np.sqrt(squared_norm(K - Z_0) + squared_norm(K[:-1] - Z_1) + squared_norm(K[1:] - Z_2))
snorm = rho * np.sqrt(squared_norm(Z_0 - Z_0_old) + squared_norm(Z_1 - Z_1_old) + squared_norm(Z_2 - Z_2_old))
obj = objective(n_samples, emp_cov, Z_0, K, Z_1, Z_2, alpha, beta, psi) if compute_objective else np.nan
# if np.isinf(obj):
# Z_0 = Z_0_old
# break
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(K.size + 2 * Z_1.size) * tol
+ rtol
* max(
np.sqrt(squared_norm(Z_0) + squared_norm(Z_1) + squared_norm(Z_2)),
np.sqrt(squared_norm(K) + squared_norm(K[:-1]) + squared_norm(K[1:])),
),
e_dual=np.sqrt(K.size + 2 * Z_1.size) * tol
+ rtol * rho * np.sqrt(squared_norm(U_0) + squared_norm(U_1) + squared_norm(U_2)),
# precision=Z_0.copy()
)
Z_0_old = Z_0.copy()
Z_1_old = Z_1.copy()
Z_2_old = Z_2.copy()
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f," "eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if stop_at is not None:
if abs(check.obj - stop_at) / abs(stop_at) < stop_when:
break
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho, rnorm, snorm, iteration=iteration_, **(update_rho_options or {}))
# scaled dual variables should be also rescaled
U_0 *= rho / rho_new
U_1 *= rho / rho_new
U_2 *= rho / rho_new
rho = rho_new
# assert is_pos_def(Z_0)
else:
warnings.warn("Objective did not converge.")
covariance_ = np.array([linalg.pinvh(x) for x in Z_0])
return_list = [Z_0, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_ + 1)
return return_list
def kernel_latent_time_graphical_lasso(
emp_cov,
alpha=0.01,
tau=1.0,
rho=1.0,
kernel_psi=None,
kernel_phi=None,
max_iter=100,
verbose=False,
psi="laplacian",
phi="laplacian",
mode="admm",
tol=1e-4,
rtol=1e-4,
assume_centered=False,
n_samples=None,
return_history=False,
return_n_iter=True,
update_rho_options=None,
compute_objective=True,
init="empirical",
):
r"""Time-varying latent variable graphical lasso solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(K_i-L_i) + alpha ||K_i||_{od,1}
+ tau ||L_i||_*
+ sum_{s>t}^T k_psi(s,t) Psi(K_s - K_t)
+ sum_{s>t}^T k_phi(s,t)(L_s - L_t)
where S is the empirical covariance of the data
matrix D (training observations by features).
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Empirical covariance of data.
alpha, tau, beta, eta : float, optional
Regularisation parameters.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
Returns
-------
K, L : numpy.array, 3-dimensional (T x d x d)
Solution to the problem for each time t=1...T .
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
psi, prox_psi, psi_node_penalty = check_norm_prox(psi)
phi, prox_phi, phi_node_penalty = check_norm_prox(phi)
n_times, _, n_features = emp_cov.shape
if kernel_psi is None:
kernel_psi = np.eye(n_times)
if kernel_phi is None:
kernel_phi = np.eye(n_times)
Z_0 = init_precision(emp_cov, mode=init)
W_0 = np.zeros_like(Z_0)
X_0 = np.zeros_like(Z_0)
R_old = np.zeros_like(Z_0)
Z_M, Z_M_old = {}, {}
Y_M = {}
W_M, W_M_old = {}, {}
U_M = {}
for m in range(1, n_times):
Z_L = Z_0.copy()[:-m]
Z_R = Z_0.copy()[m:]
Z_M[m] = (Z_L, Z_R)
W_L = np.zeros_like(Z_L)
W_R = np.zeros_like(Z_R)
W_M[m] = (W_L, W_R)
Y_L = np.zeros_like(Z_L)
Y_R = np.zeros_like(Z_R)
Y_M[m] = (Y_L, Y_R)
U_L = np.zeros_like(W_L)
U_R = np.zeros_like(W_R)
U_M[m] = (U_L, U_R)
Z_L_old = np.zeros_like(Z_L)
Z_R_old = np.zeros_like(Z_R)
Z_M_old[m] = (Z_L_old, Z_R_old)
W_L_old = np.zeros_like(W_L)
W_R_old = np.zeros_like(W_R)
W_M_old[m] = (W_L_old, W_R_old)
if n_samples is None:
n_samples = np.ones(n_times)
checks = []
for iteration_ in range(max_iter):
# update R
A = Z_0 - W_0 - X_0
A += A.transpose(0, 2, 1)
A /= 2.0
A *= -rho / n_samples[:, None, None]
A += emp_cov
# A = emp_cov / rho - A
R = np.array(
[prox_logdet(a, lamda=ni / rho) for a, ni in zip(A, n_samples)])
# update Z_0
A = R + W_0 + X_0
for m in range(1, n_times):
A[:-m] += Z_M[m][0] - Y_M[m][0]
A[m:] += Z_M[m][1] - Y_M[m][1]
A /= n_times
Z_0 = soft_thresholding(A, lamda=alpha / (rho * n_times))
# update W_0
A = Z_0 - R - X_0
for m in range(1, n_times):
A[:-m] += W_M[m][0] - U_M[m][0]
A[m:] += W_M[m][1] - U_M[m][1]
A /= n_times
A += A.transpose(0, 2, 1)
A /= 2.0
W_0 = np.array(
[prox_trace_indicator(a, lamda=tau / (rho * n_times)) for a in A])
# update residuals
X_0 += R - Z_0 + W_0
for m in range(1, n_times):
# other Zs
Y_L, Y_R = Y_M[m]
A_L = Z_0[:-m] + Y_L
A_R = Z_0[m:] + Y_R
if not psi_node_penalty:
prox_e = prox_psi(A_R - A_L,
lamda=2.0 *
np.diag(kernel_psi, m)[:, None, None] / rho)
Z_L = 0.5 * (A_L + A_R - prox_e)
Z_R = 0.5 * (A_L + A_R + prox_e)
else:
Z_L, Z_R = prox_psi(
np.concatenate((A_L, A_R), axis=1),
lamda=0.5 * np.diag(kernel_psi, m)[:, None, None] / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter,
)
Z_M[m] = (Z_L, Z_R)
# update other residuals
Y_L += Z_0[:-m] - Z_L
Y_R += Z_0[m:] - Z_R
# other Ws
U_L, U_R = U_M[m]
A_L = W_0[:-m] + U_L
A_R = W_0[m:] + U_R
if not phi_node_penalty:
prox_e = prox_phi(A_R - A_L,
lamda=2.0 *
np.diag(kernel_phi, m)[:, None, None] / rho)
W_L = 0.5 * (A_L + A_R - prox_e)
W_R = 0.5 * (A_L + A_R + prox_e)
else:
W_L, W_R = prox_phi(
np.concatenate((A_L, A_R), axis=1),
lamda=0.5 * np.diag(kernel_phi, m)[:, None, None] / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter,
)
W_M[m] = (W_L, W_R)
# update other residuals
U_L += W_0[:-m] - W_L
U_R += W_0[m:] - W_R
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(R - Z_0 + W_0) + sum(
squared_norm(Z_0[:-m] - Z_M[m][0]) +
squared_norm(Z_0[m:] - Z_M[m][1]) +
squared_norm(W_0[:-m] - W_M[m][0]) +
squared_norm(W_0[m:] - W_M[m][1]) for m in range(1, n_times)))
snorm = rho * np.sqrt(
squared_norm(R - R_old) + sum(
squared_norm(Z_M[m][0] - Z_M_old[m][0]) +
squared_norm(Z_M[m][1] - Z_M_old[m][1]) +
squared_norm(W_M[m][0] - W_M_old[m][0]) +
squared_norm(W_M[m][1] - W_M_old[m][1])
for m in range(1, n_times)))
obj = (objective(emp_cov, n_samples, R, Z_0, Z_M, W_0, W_M, alpha, tau,
kernel_psi, kernel_phi, psi, phi)
if compute_objective else np.nan)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=n_features * np.sqrt(n_times * (2 * n_times - 1)) * tol +
rtol * max(
np.sqrt(
squared_norm(R) + sum(
squared_norm(Z_M[m][0]) + squared_norm(Z_M[m][1]) +
squared_norm(W_M[m][0]) + squared_norm(W_M[m][1])
for m in range(1, n_times))),
np.sqrt(
squared_norm(Z_0 - W_0) + sum(
squared_norm(Z_0[:-m]) + squared_norm(Z_0[m:]) +
squared_norm(W_0[:-m]) + squared_norm(W_0[m:])
for m in range(1, n_times))),
),
e_dual=n_features * np.sqrt(n_times * (2 * n_times - 1)) * tol +
rtol * rho * np.sqrt(
squared_norm(X_0) + sum(
squared_norm(Y_M[m][0]) + squared_norm(Y_M[m][1]) +
squared_norm(U_M[m][0]) + squared_norm(U_M[m][1])
for m in range(1, n_times))),
)
R_old = R.copy()
for m in range(1, n_times):
Z_M_old[m] = (Z_M[m][0].copy(), Z_M[m][1].copy())
W_M_old[m] = (W_M[m][0].copy(), W_M[m][1].copy())
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
X_0 *= rho / rho_new
for m in range(1, n_times):
Y_L, Y_R = Y_M[m]
Y_L *= rho / rho_new
Y_R *= rho / rho_new
U_L, U_R = U_M[m]
U_L *= rho / rho_new
U_R *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
covariance_ = np.array([linalg.pinvh(x) for x in Z_0])
return_list = [Z_0, W_0, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
def latent_time_graph_lasso(
emp_cov, alpha=1, tau=1, rho=1, beta=1., eta=1., max_iter=1000,
verbose=False, psi='laplacian', phi='laplacian', mode=None,
tol=1e-4, rtol=1e-2, assume_centered=False,
return_history=False, return_n_iter=True):
r"""Time-varying latent variable graphical lasso solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(K_i-L_i) + alpha ||K_i||_{od,1}
+ tau ||L_i||_*
+ beta sum_{i=2}^T Psi(K_i - K_{i-1})
+ eta sum_{i=2}^T Phi(L_i - L_{i-1})
where S is the empirical covariance of the data
matrix D (training observations by features).
Parameters
----------
data_list : list of 2-dimensional matrices.
Input matrices.
alpha, tau : float, optional
Regularisation parameters.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
Returns
-------
K, L : numpy.array, 3-dimensional (T x d x d)
Solution to the problem for each time t=1...T .
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
psi, prox_psi = check_norm_prox(psi)
phi, prox_phi = check_norm_prox(phi)
# S = np.array(map(empirical_covariance, data_list))
# n_samples = np.array([s for s in [1.]])
K = np.zeros_like(emp_cov)
Z_0 = np.zeros_like(K)
Z_1 = np.zeros_like(K)[:-1]
Z_2 = np.zeros_like(K)[1:]
W_0 = np.zeros_like(K)
W_1 = np.zeros_like(K)[:-1]
W_2 = np.zeros_like(K)[1:]
X_0 = np.zeros_like(K)
X_1 = np.zeros_like(K)[:-1]
X_2 = np.zeros_like(K)[1:]
Z_consensus = np.zeros_like(K)
# Z_consensus_old = np.zeros_like(K)
W_consensus = np.zeros_like(K)
# W_consensus_old = np.zeros_like(K)
R_old = np.zeros_like(K)
# divisor for consensus variables, accounting for two less matrices
divisor = np.full(K.shape[0], 3, dtype=float)
divisor[0] -= 1
divisor[-1] -= 1
checks = []
for iteration_ in range(max_iter):
# update R
A = Z_0 - W_0 - X_0
A[:-1] += Z_1 - W_1 - X_1
A[1:] += Z_2 - W_2 - X_2
A /= divisor[:, None, None]
# A += np.array(map(np.transpose, A))
# A /= 2.
# A *= - rho / n_samples[:, None, None]
A *= - rho
A += emp_cov
R = np.array([prox_logdet(a, lamda=1. / rho) for a in A])
# update Z_0
# Zold = Z
# X_hat = alpha * X + (1 - alpha) * Zold
soft_thresholding = partial(soft_thresholding_sign, lamda=alpha / rho)
Z_0 = np.array(map(soft_thresholding, R + W_0 + X_0))
# update Z_1, Z_2
# prox_l = partial(prox_laplacian, beta=2. * beta / rho)
# prox_e = np.array(map(prox_l, K[1:] - K[:-1] + U_2 - U_1))
if beta != 0:
A_1 = R[:-1] + W_1 + X_1
# A_1 = Z_0[:-1].copy()
A_2 = R[1:] + W_2 + X_2
# A_2 = Z_0[1:].copy()
prox_e = prox_psi(A_2 - A_1, lamda=2. * beta / rho)
Z_1 = .5 * (A_1 + A_2 - prox_e)
Z_2 = .5 * (A_1 + A_2 + prox_e)
else:
Z_1 = Z_0[:-1].copy()
Z_2 = Z_0[1:].copy()
# update W_0
A = Z_0 - R - X_0
W_0 = np.array(map(partial(prox_trace_indicator, lamda=tau / rho), A))
# update W_1, W_2
if eta != 0:
A_1 = Z_1 - R[:-1] - X_1
# A_1 = W_0[:-1].copy()
A_2 = Z_2 - R[1:] - X_2
# A_2 = W_0[1:].copy()
prox_e = prox_phi(A_2 - A_1, lamda=2. * eta / rho)
W_1 = .5 * (A_1 + A_2 - prox_e)
W_2 = .5 * (A_1 + A_2 + prox_e)
else:
W_1 = W_0[:-1].copy()
W_2 = W_0[1:].copy()
# update residuals
X_0 += R - Z_0 + W_0
X_1 += R[:-1] - Z_1 + W_1
X_2 += R[1:] - Z_2 + W_2
# diagnostics, reporting, termination checks
X_consensus = X_0.copy()
X_consensus[:-1] += X_1
X_consensus[1:] += X_2
X_consensus /= divisor[:, None, None]
Z_consensus = Z_0.copy()
Z_consensus[:-1] += Z_1
Z_consensus[1:] += Z_2
Z_consensus /= divisor[:, None, None]
W_consensus = W_0.copy()
W_consensus[:-1] += W_1
W_consensus[1:] += W_2
W_consensus /= divisor[:, None, None]
check = convergence(
obj=objective(emp_cov, R, Z_0, Z_1, Z_2, W_0, W_1, W_2,
alpha, tau, beta, eta, psi, phi),
rnorm=np.linalg.norm(R - Z_consensus + W_consensus),
snorm=np.linalg.norm(rho * (R - R_old)),
e_pri=np.sqrt(np.prod(K.shape)) * tol + rtol * max(
np.linalg.norm(R),
np.sqrt(squared_norm(Z_consensus) - squared_norm(W_consensus))),
e_dual=np.sqrt(np.prod(K.shape)) * tol + rtol * np.linalg.norm(
rho * X_consensus)
)
R_old = R.copy()
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check)
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
# if iteration_ % 10 == 0:
# rho = rho * 0.8
else:
warnings.warn("Objective did not converge.")
# return_list = [Z_consensus, W_consensus, emp_cov]
return_list = [Z_consensus, W_0, W_1, W_2, emp_cov]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
def graphical_lasso(
emp_cov,
alpha=0.01,
rho=1,
over_relax=1,
max_iter=100,
verbose=False,
tol=1e-4,
rtol=1e-4,
return_history=False,
return_n_iter=True,
update_rho_options=None,
compute_objective=True,
init="empirical",
):
r"""Graphical lasso solver via ADMM.
Solves the following problem:
minimize trace(S*K) - log det K + alpha ||K||_{od,1}
where S = (1/n) X^T \times X is the empirical covariance of the data
matrix X (training observations by features).
Parameters
----------
emp_cov : array-like
Empirical covariance matrix.
alpha : float, optional
Regularisation parameter.
rho : float, optional
Augmented Lagrangian parameter.
over_relax : float, optional
Over-relaxation parameter (typically between 1.0 and 1.8).
max_iter : int, optional
Maximum number of iterations.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
verbose : bool, default False
Print info at each iteration.
update_rho_options : dict, optional
Arguments for the rho update.
See regain.update_rules.update_rho function for more information.
compute_objective : bool, default True
Choose to compute the objective value.
init : {'empirical', 'zeros', ndarray}, default 'empirical'
How to initialise the inverse covariance matrix. Default is take
the empirical covariance and inverting it.
Returns
-------
precision_ : numpy.array, 2-dimensional
Solution to the problem.
covariance_ : np.array, 2 dimensional
Empirical covariance matrix.
n_iter_ : int
If return_n_iter, returns the number of iterations before convergence.
history_ : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
Z = init_precision(emp_cov, mode=init)
U = np.zeros_like(emp_cov)
Z_old = np.zeros_like(Z)
checks = []
for iteration_ in range(max_iter):
# x-update
A = Z - U
A += A.T
A /= 2.0
K = prox_logdet(emp_cov - rho * A, lamda=1.0 / rho)
# z-update with relaxation
K_hat = over_relax * K - (1 - over_relax) * Z
Z = soft_thresholding_od(K_hat + U, lamda=alpha / rho)
# update residuals
U += K_hat - Z
# diagnostics, reporting, termination checks
obj = objective(emp_cov, K, Z, alpha) if compute_objective else np.nan
rnorm = np.linalg.norm(K - Z, "fro")
snorm = rho * np.linalg.norm(Z - Z_old, "fro")
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(K.size) * tol +
rtol * max(np.linalg.norm(K, "fro"), np.linalg.norm(Z, "fro")),
e_dual=np.sqrt(K.size) * tol + rtol * rho * np.linalg.norm(U),
)
Z_old = Z.copy()
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
U *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [Z, emp_cov]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
def kernel_time_graphical_lasso(
emp_cov,
alpha=0.01,
rho=1,
kernel=None,
max_iter=100,
n_samples=None,
verbose=False,
psi="laplacian",
tol=1e-4,
rtol=1e-4,
return_history=False,
return_n_iter=True,
mode="admm",
update_rho_options=None,
compute_objective=True,
stop_at=None,
stop_when=1e-4,
init="empirical",
):
"""Time-varying graphical lasso solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(K_i-L_i) + alpha ||K_i||_{od,1}
+ sum_{s>t}^T k_psi(s,t) Psi(K_s - K_t)
where S is the empirical covariance of the data
matrix D (training observations by features).
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Empirical covariance of data.
alpha, beta : float, optional
Regularisation parameter.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
init : {'empirical', 'zeros', ndarray}, default 'empirical'
How to initialise the inverse covariance matrix. Default is take
the empirical covariance and inverting it.
Returns
-------
X : numpy.array, 2-dimensional
Solution to the problem.
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
psi, prox_psi, psi_node_penalty = check_norm_prox(psi)
n_times, _, n_features = emp_cov.shape
if kernel is None:
kernel = np.eye(n_times)
Z_0 = init_precision(emp_cov, mode=init)
U_0 = np.zeros_like(Z_0)
Z_0_old = np.zeros_like(Z_0)
Z_M, Z_M_old = {}, {}
U_M = {}
for m in range(1, n_times):
# all possible markovians jumps
Z_L = Z_0.copy()[:-m]
Z_R = Z_0.copy()[m:]
Z_M[m] = (Z_L, Z_R)
U_L = np.zeros_like(Z_L)
U_R = np.zeros_like(Z_R)
U_M[m] = (U_L, U_R)
Z_L_old = np.zeros_like(Z_L)
Z_R_old = np.zeros_like(Z_R)
Z_M_old[m] = (Z_L_old, Z_R_old)
if n_samples is None:
n_samples = np.ones(n_times)
checks = [
convergence(obj=objective(n_samples, emp_cov, Z_0, Z_0, Z_M, alpha,
kernel, psi))
]
for iteration_ in range(max_iter):
# update K
A = Z_0 - U_0
for m in range(1, n_times):
A[:-m] += Z_M[m][0] - U_M[m][0]
A[m:] += Z_M[m][1] - U_M[m][1]
A /= n_times
# soft_thresholding_ = partial(soft_thresholding, lamda=alpha / rho)
# K = np.array(map(soft_thresholding_, A))
A += A.transpose(0, 2, 1)
A /= 2.0
A *= -rho * n_times / n_samples[:, None, None]
A += emp_cov
K = np.array([
prox_logdet(a, lamda=ni / (rho * n_times))
for a, ni in zip(A, n_samples)
])
# update Z_0
A = K + U_0
A += A.transpose(0, 2, 1)
A /= 2.0
Z_0 = soft_thresholding(A, lamda=alpha / rho)
# update residuals
U_0 += K - Z_0
# other Zs
for m in range(1, n_times):
U_L, U_R = U_M[m]
A_L = K[:-m] + U_L
A_R = K[m:] + U_R
if not psi_node_penalty:
prox_e = prox_psi(A_R - A_L,
lamda=2.0 *
np.diag(kernel, m)[:, None, None] / rho)
Z_L = 0.5 * (A_L + A_R - prox_e)
Z_R = 0.5 * (A_L + A_R + prox_e)
else:
Z_L, Z_R = prox_psi(
np.concatenate((A_L, A_R), axis=1),
lamda=0.5 * np.diag(kernel, m)[:, None, None] / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter,
)
Z_M[m] = (Z_L, Z_R)
# update other residuals
U_L += K[:-m] - Z_L
U_R += K[m:] - Z_R
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(K - Z_0) + sum(
squared_norm(K[:-m] - Z_M[m][0]) +
squared_norm(K[m:] - Z_M[m][1]) for m in range(1, n_times)))
snorm = rho * np.sqrt(
squared_norm(Z_0 - Z_0_old) + sum(
squared_norm(Z_M[m][0] - Z_M_old[m][0]) +
squared_norm(Z_M[m][1] - Z_M_old[m][1])
for m in range(1, n_times)))
obj = objective(n_samples, emp_cov, Z_0, K, Z_M, alpha, kernel,
psi) if compute_objective else np.nan
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=n_features * n_times * tol + rtol * max(
np.sqrt(
squared_norm(Z_0) + sum(
squared_norm(Z_M[m][0]) + squared_norm(Z_M[m][1])
for m in range(1, n_times))),
np.sqrt(
squared_norm(K) + sum(
squared_norm(K[:-m]) + squared_norm(K[m:])
for m in range(1, n_times))),
),
e_dual=n_features * n_times * tol + rtol * rho * np.sqrt(
squared_norm(U_0) + sum(
squared_norm(U_M[m][0]) + squared_norm(U_M[m][1])
for m in range(1, n_times))),
)
Z_0_old = Z_0.copy()
for m in range(1, n_times):
Z_M_old[m] = (Z_M[m][0].copy(), Z_M[m][1].copy())
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if stop_at is not None:
if abs(check.obj - stop_at) / abs(stop_at) < stop_when:
break
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
U_0 *= rho / rho_new
for m in range(1, n_times):
U_L, U_R = U_M[m]
U_L *= rho / rho_new
U_R *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
covariance_ = np.array([linalg.pinvh(x) for x in Z_0])
return_list = [Z_0, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_ + 1)
return return_list
def latent_graphical_lasso(
emp_cov,
alpha=1.0,
tau=1.0,
rho=1.0,
max_iter=100,
verbose=False,
tol=1e-4,
rtol=1e-2,
return_history=False,
return_n_iter=True,
update_rho_options=None,
compute_objective=True,
init="empirical",
):
r"""Latent variable graphical lasso solver via ADMM.
Solves the following problem:
min - log_likelihood(S, K-L) + alpha ||K||_{od,1} + tau ||L_i||_*
where S = (1/n) X^T \times X is the empirical covariance of the data
matrix X (training observations by features).
Parameters
----------
emp_cov : array-like
Empirical covariance matrix.
alpha, tau : float, optional
Regularisation parameters.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
verbose : bool, default False
Print info at each iteration.
update_rho_options : dict, optional
Arguments for the rho update.
See regain.update_rules.update_rho function for more information.
compute_objective : bool, default True
Choose to compute the objective value.
init : {'empirical', 'zeros', ndarray}, default 'empirical'
How to initialise the inverse covariance matrix. Default is take
the empirical covariance and inverting it.
Returns
-------
K, L : np.array, 2-dimensional, size (d x d)
Solution to the problem.
S : np.array, 2 dimensional
Empirical covariance matrix.
n_iter : int
If return_n_iter, returns the number of iterations before convergence.
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
K = init_precision(emp_cov, mode=init)
L = np.zeros_like(emp_cov)
U = np.zeros_like(emp_cov)
R_old = np.zeros_like(emp_cov)
checks = []
for iteration_ in range(max_iter):
# update R
A = K - L - U
A += A.T
A /= 2.0
R = prox_logdet(emp_cov - rho * A, lamda=1.0 / rho)
A = L + R + U
K = soft_thresholding(A, lamda=alpha / rho)
A = K - R - U
A += A.T
A /= 2.0
L = prox_trace_indicator(A, lamda=tau / rho)
# update residuals
U += R - K + L
# diagnostics, reporting, termination checks
obj = objective(emp_cov, R, K, L, alpha,
tau) if compute_objective else np.nan
rnorm = np.linalg.norm(R - K + L)
snorm = rho * np.linalg.norm(R - R_old)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(R.size) * tol +
rtol * max(np.linalg.norm(R), np.linalg.norm(K - L)),
e_dual=np.sqrt(R.size) * tol + rtol * rho * np.linalg.norm(U),
)
R_old = R.copy()
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
if check.obj == np.inf:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
U *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
covariance_ = linalg.pinvh(K)
return_list = [K, L, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
def latent_time_graphical_lasso(emp_cov,
alpha=0.01,
tau=1.,
rho=1.,
beta=1.,
eta=1.,
max_iter=100,
n_samples=None,
verbose=False,
psi='laplacian',
phi='laplacian',
mode='admm',
tol=1e-4,
rtol=1e-4,
return_history=False,
return_n_iter=True,
update_rho_options=None,
compute_objective=True,
init='empirical'):
r"""Latent variable time-varying graphical lasso solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(S_i, K_i-L_i) + alpha ||K_i||_{od,1}
+ tau ||L_i||_*
+ beta sum_{i=2}^T Psi(K_i - K_{i-1})
+ eta sum_{i=2}^T Phi(L_i - L_{i-1})
where S_i = (1/n_i) X_i^T \times X_i is the empirical covariance of data
matrix X (training observations by features).
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Empirical covariance of data.
alpha, tau, beta, eta : float, optional
Regularisation parameters.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
n_samples : ndarray
Number of samples available for each time point.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
verbose : bool, default False
Print info at each iteration.
update_rho_options : dict, optional
Arguments for the rho update.
See regain.update_rules.update_rho function for more information.
compute_objective : bool, default True
Choose to compute the objective value.
init : {'empirical', 'zeros', ndarray}, default 'empirical'
How to initialise the inverse covariance matrix. Default is take
the empirical covariance and inverting it.
Returns
-------
K, L : numpy.array, 3-dimensional (T x d x d)
Solution to the problem for each time t=1...T .
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
psi, prox_psi, psi_node_penalty = check_norm_prox(psi)
phi, prox_phi, phi_node_penalty = check_norm_prox(phi)
Z_0 = init_precision(emp_cov, mode=init)
Z_1 = Z_0.copy()[:-1]
Z_2 = Z_0.copy()[1:]
W_0 = np.zeros_like(Z_0)
W_1 = np.zeros_like(Z_1)
W_2 = np.zeros_like(Z_2)
X_0 = np.zeros_like(Z_0)
X_1 = np.zeros_like(Z_1)
X_2 = np.zeros_like(Z_2)
U_1 = np.zeros_like(W_1)
U_2 = np.zeros_like(W_2)
R_old = np.zeros_like(Z_0)
Z_1_old = np.zeros_like(Z_1)
Z_2_old = np.zeros_like(Z_2)
W_1_old = np.zeros_like(W_1)
W_2_old = np.zeros_like(W_2)
# divisor for consensus variables, accounting for two less matrices
divisor = np.full(emp_cov.shape[0], 3, dtype=float)
divisor[0] -= 1
divisor[-1] -= 1
if n_samples is None:
n_samples = np.ones(emp_cov.shape[0])
checks = []
for iteration_ in range(max_iter):
# update R
A = Z_0 - W_0 - X_0
A += A.transpose(0, 2, 1)
A /= 2.
A *= -rho / n_samples[:, None, None]
A += emp_cov
# A = emp_cov / rho - A
R = np.array(
[prox_logdet(a, lamda=ni / rho) for a, ni in zip(A, n_samples)])
# update Z_0
A = R + W_0 + X_0
A[:-1] += Z_1 - X_1
A[1:] += Z_2 - X_2
A /= divisor[:, None, None]
# soft_thresholding_ = partial(soft_thresholding, lamda=alpha / rho)
# Z_0 = np.array(map(soft_thresholding_, A))
Z_0 = soft_thresholding(A,
lamda=alpha / (rho * divisor[:, None, None]))
# update Z_1, Z_2
A_1 = Z_0[:-1] + X_1
A_2 = Z_0[1:] + X_2
if not psi_node_penalty:
prox_e = prox_psi(A_2 - A_1, lamda=2. * beta / rho)
Z_1 = .5 * (A_1 + A_2 - prox_e)
Z_2 = .5 * (A_1 + A_2 + prox_e)
else:
Z_1, Z_2 = prox_psi(np.concatenate((A_1, A_2), axis=1),
lamda=.5 * beta / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter)
# update W_0
A = Z_0 - R - X_0
A[:-1] += W_1 - U_1
A[1:] += W_2 - U_2
A /= divisor[:, None, None]
A += A.transpose(0, 2, 1)
A /= 2.
W_0 = np.array([
prox_trace_indicator(a, lamda=tau / (rho * div))
for a, div in zip(A, divisor)
])
# update W_1, W_2
A_1 = W_0[:-1] + U_1
A_2 = W_0[1:] + U_2
if not phi_node_penalty:
prox_e = prox_phi(A_2 - A_1, lamda=2. * eta / rho)
W_1 = .5 * (A_1 + A_2 - prox_e)
W_2 = .5 * (A_1 + A_2 + prox_e)
else:
W_1, W_2 = prox_phi(np.concatenate((A_1, A_2), axis=1),
lamda=.5 * eta / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter)
# update residuals
X_0 += R - Z_0 + W_0
X_1 += Z_0[:-1] - Z_1
X_2 += Z_0[1:] - Z_2
U_1 += W_0[:-1] - W_1
U_2 += W_0[1:] - W_2
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(R - Z_0 + W_0) + squared_norm(Z_0[:-1] - Z_1) +
squared_norm(Z_0[1:] - Z_2) + squared_norm(W_0[:-1] - W_1) +
squared_norm(W_0[1:] - W_2))
snorm = rho * np.sqrt(
squared_norm(R - R_old) + squared_norm(Z_1 - Z_1_old) +
squared_norm(Z_2 - Z_2_old) + squared_norm(W_1 - W_1_old) +
squared_norm(W_2 - W_2_old))
obj = objective(emp_cov, n_samples, R, Z_0, Z_1, Z_2, W_0, W_1, W_2,
alpha, tau, beta, eta, psi, phi) \
if compute_objective else np.nan
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(R.size + 4 * Z_1.size) * tol + rtol * max(
np.sqrt(
squared_norm(R) + squared_norm(Z_1) + squared_norm(Z_2) +
squared_norm(W_1) + squared_norm(W_2)),
np.sqrt(
squared_norm(Z_0 - W_0) + squared_norm(Z_0[:-1]) +
squared_norm(Z_0[1:]) + squared_norm(W_0[:-1]) +
squared_norm(W_0[1:]))),
e_dual=np.sqrt(R.size + 4 * Z_1.size) * tol + rtol * rho *
(np.sqrt(
squared_norm(X_0) + squared_norm(X_1) + squared_norm(X_2) +
squared_norm(U_1) + squared_norm(U_2))))
R_old = R.copy()
Z_1_old = Z_1.copy()
Z_2_old = Z_2.copy()
W_1_old = W_1.copy()
W_2_old = W_2.copy()
if verbose:
print("obj: %.4f, rnorm: %.4f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
X_0 *= rho / rho_new
X_1 *= rho / rho_new
X_2 *= rho / rho_new
U_1 *= rho / rho_new
U_2 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
covariance_ = np.array([linalg.pinvh(x) for x in Z_0])
return_list = [Z_0, W_0, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list