def infimal_convolution(
S,
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,
):
r"""Latent variable graphical lasso solver.
Solves the following problem via ADMM:
min - log_likelihood(S, K-L) + alpha ||K||_{od,1} + tau ||L_i||_*
where S is the empirical covariance of the data
matrix D (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.
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 = np.zeros_like(S)
L = np.zeros_like(S)
U = np.zeros_like(S)
R_old = np.zeros_like(S)
checks = []
for iteration_ in range(max_iter):
# update R
A = K - L - U
A += A.T
A /= 2.0
R = prox_laplacian(S + rho * A, lamda=rho / 2.0)
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(S, 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 prox_node_penalty(A_12, lamda, rho=1, tol=1e-4, rtol=1e-2, max_iter=500):
"""Lamda = beta / (2. * rho).
A_12 = np.vstack((A_1, A_2))
"""
n_time, _, n_dim = A_12.shape
U_1 = np.full((A_12.shape[0], n_dim, n_dim), 1.0 / n_dim, dtype=float)
U_2 = np.copy(U_1)
Y_1 = np.copy(U_1)
Y_2 = np.copy(U_1)
C = np.hstack((np.eye(n_dim), -np.eye(n_dim), np.eye(n_dim)))
inverse = np.linalg.inv(C.T.dot(C) + 2 * np.eye(3 * n_dim))
V = np.zeros_like(U_1)
W = np.zeros_like(U_1)
V_old = np.zeros_like(U_1)
W_old = np.zeros_like(U_1)
for iteration_ in range(max_iter):
A = (Y_1 - Y_2 - W - U_1 + (W.transpose(0, 2, 1) - U_2).transpose(0, 2, 1)) / 2.0
V = blockwise_soft_thresholding_symmetric(A, lamda=lamda)
A = np.concatenate(((V + U_2).transpose(0, 2, 1), A_12), axis=1)
D = V + U_1
# Z = np.linalg.solve(C.T*C + eta*np.identity(3*n), - C.T*D + eta* A)
Z = np.empty_like(A)
for i, (A_i, D_i) in enumerate(zip(A, D)):
Z[i] = inverse.dot(2 * A_i - C.T.dot(D_i))
W, Y_1, Y_2 = (Z[:, i * n_dim : (i + 1) * n_dim, :] for i in range(3))
# update residuals
delta_U_1 = V + W - (Y_1 - Y_2)
delta_U_2 = V - W.transpose(0, 2, 1)
U_1 += delta_U_1
U_2 += delta_U_2
# diagnostics
rnorm = np.sqrt(squared_norm(delta_U_1) + squared_norm(delta_U_2))
snorm = rho * np.sqrt(squared_norm(W - W_old) + squared_norm(V + W - V_old - W_old))
check = convergence(
obj=np.nan,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(2 * V.size) * tol
+ rtol
* max(np.sqrt(squared_norm(W) + squared_norm(V + W)), np.sqrt(squared_norm(V) + squared_norm(Y_1 - Y_2))),
e_dual=np.sqrt(2 * V.size) * tol + rtol * rho * np.sqrt(squared_norm(U_1) + squared_norm(U_2)),
)
W_old = W.copy()
V_old = V.copy()
# if np.linalg.norm(delta_U_1, 'fro') < tol and \
# np.linalg.norm(delta_U_2, 'fro') < tol:
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual:
break
rho_new = update_rho(rho, rnorm, snorm, iteration=iteration_)
# scaled dual variables should be also rescaled
U_1 *= rho / rho_new
U_2 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Node norm did not converge.")
return Y_1, Y_2
def gradient_equal_time_graphical_lasso(S, K_init, max_iter, loss, C, theta,
rho, mult, weights, m, eps, psi, gamma,
tol, rtol, verbose, return_history,
return_n_iter, mode, compute_objective,
stop_at, stop_when,
update_rho_options):
"""Equality constrained time-varying graphical LASSO solver.
Solves the following problem via ADMM:
min sum_{i=1}^T ||K_i||_{od,1} + beta sum_{i=2}^T Psi(K_i - K_{i-1})
s.t. objective = c_i for i = 1, ..., T
where S_i = (1/n_i) X_i^T 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.
gamma: float, optional
Kernel parameter when psi is chosen to be 'kernel'.
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)
if loss == 'LL':
loss_func = neg_logl
else:
loss_func = dtrace
T = S.shape[0]
I = np.eye(S.shape[1])
Z_0 = K_init
out_obj = []
checks = [convergence(obj=penalty_objective(Z_0, Z_0, Z_0, psi, theta))]
def _Z_0(x1, x2, Z_0, loss_res, nabla_con, nabla_pen):
A = Z_0 - x2 * (1 - theta) * nabla_pen
# A = Z_0 - x1 * nabla_con - x2 * (1 - theta) * nabla_pen
A -= x1 * loss_res[:, None, None] * nabla_con
return soft_thresholding_od(A, lamda=x2 * theta), A
# constrained optimisation via line search
def _f(x, _Z_0, Z_0, loss_res, nabla_con, nabla_pen, loss_func, S, C):
_Z_0, A = _Z_0(x[0], x[1], Z_0, loss_res, nabla_con, nabla_pen)
loss_res = loss_gen(loss_func, S, _Z_0) - C
# loss_res_A = loss_gen(loss_func, S, A) - C
# return squared_norm(loss_res) + squared_norm(loss_res - loss_res_A)
return squared_norm(loss_res) + squared_norm(_Z_0 - A) / (S.shape[1] *
S.shape[2])
loss_res = loss_gen(loss_func, S, Z_0) - C
for iteration_ in range(max_iter):
if loss_func.__name__ == 'neg_logl':
nabla_con = np.array(
[S_t - np.linalg.inv(A_t) for (S_t, A_t) in zip(S, Z_0)])
# nabla = np.array([S_t - np.linalg.inv(Z_0_t) for (S_t, Z_0_t) in zip(S, Z_0_pre)])
elif loss_func.__name__ == 'dtrace':
nabla_con = np.array([(2 * A_t @ S_t - I)
for (S_t, A_t) in zip(S, Z_0)])
# nabla = np.array([(2 * Z_0_t @ S_t - I) for (S_t, Z_0_t) in zip(S, Z_0_pre)])
nabla_pen = grad_laplacian(Z_0)
out = minimize(partial(_f,
_Z_0=_Z_0,
Z_0=Z_0,
loss_res=loss_res,
nabla_con=nabla_con,
nabla_pen=nabla_pen,
loss_func=loss_func,
S=S,
C=C),
x0=np.zeros(2),
method='Nelder-Mead',
tol=1e-4)
Z_0, _ = _Z_0(out.x[0], out.x[1], Z_0, loss_res, nabla_con, nabla_pen)
loss_res = loss_gen(loss_func, S, Z_0) - C
out_obj.append(penalty_objective(Z_0, Z_0[:-1], Z_0[1:], psi, theta))
if not iteration_ % 100:
print(iteration_)
print(np.max(loss_res), np.mean(loss_res))
print(out_obj[-1])
# print(out_obj[-1], np.max(loss_res), np.mean(loss_res))
else:
warnings.warn("Objective did not converge.")
print(iteration_, out_obj[-1])
# print(check.rnorm, check.e_pri)
# print(check.snorm, check.e_dual)
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_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
def lasso_kernel_admm(K,
y,
lamda=0.01,
rho=1.,
max_iter=100,
verbose=0,
rtol=1e-4,
tol=1e-4,
return_n_iter=True,
update_rho_options=None,
sample_weight=None):
"""Elastic Net kernel learning.
Solve the following problem via ADMM:
min sum_{i=1}^p 1/2 ||y_i - alpha_i * sum_{k=1}^{n_k} w_k * K_{ik}||^2
+ lamda ||w||_1 + beta sum_{j=1}^{c_i}||alpha_j||_2^2
"""
n_kernels, n_samples, n_features = K.shape
coef = np.ones(n_kernels)
# alpha = [np.zeros(K[j].shape[2]) for j in range(n_patients)]
# u = [np.zeros(K[j].shape[1]) for j in range(n_patients)]
w_1 = coef.copy()
u_1 = np.zeros(n_kernels)
# x_old = [np.zeros(K[0].shape[1]) for j in range(n_patients)]
w_1_old = w_1.copy()
Y = y[:, None].dot(y[:, None].T)
checks = []
for iteration_ in range(max_iter):
# update w
KK = 2 * np.tensordot(K, K.T, axes=([1, 2], [0, 1]))
yy = 2 * np.tensordot(Y, K, axes=([0, 1], [1, 2]))
yy += rho * (w_1 - u_1)
coef = _solve_cholesky_kernel(KK, yy[..., None], rho).ravel()
w_1 = soft_thresholding(coef + u_1, lamda / rho)
# w_2 = prox_laplacian(coef + u_2, beta / rho)
u_1 += coef - w_1
# diagnostics, reporting, termination checks
rnorm = np.sqrt(squared_norm(coef - w_1))
snorm = rho * np.sqrt(squared_norm(w_1 - w_1_old))
obj = lasso_objective(Y, coef, K, w_1, lamda)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(coef.size) * tol + rtol *
max(np.sqrt(squared_norm(coef)), np.sqrt(squared_norm(w_1))),
e_dual=np.sqrt(coef.size) * tol + rtol * rho *
(np.sqrt(squared_norm(u_1))))
w_1_old = w_1.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 and iteration_ > 1:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
u_1 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [coef]
if return_n_iter:
return_list.append(iteration_)
return return_list
def inequality_time_graphical_lasso(S, K_init, max_iter, loss, C, theta,
c_prox, rho, div, psi, gamma, tol, rtol,
verbose, return_history, return_n_iter,
mode, compute_objective, stop_at,
stop_when, update_rho_options, init):
"""Inequality constrained time-varying graphical LASSO solver.
Solves the following problem via ADMM:
min sum_{i=1}^T ||K_i||_{od,1} + beta sum_{i=2}^T Psi(K_i - K_{i-1})
s.t. objective =< c_i for i = 1, ..., T
where S_i = (1/n_i) X_i^T 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.
gamma: float, optional
Kernel parameter when psi is chosen to be 'kernel'.
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)
psi_name = psi.__name__
if loss == 'LL':
loss_function = neg_logl
else:
loss_function = dtrace
Z_0 = K_init # init_precision(S, mode=init)
Z_1 = Z_0.copy()[:-1]
Z_2 = Z_0.copy()[1:]
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 one less matrix for t = 0 and t = T
divisor = np.full(S.shape[0], 2, dtype=float)
divisor[0] -= 1
divisor[-1] -= 1
out_obj = []
checks = [convergence(obj=penalty_objective(Z_0, Z_1, Z_2, psi, theta))]
for iteration_ in range(max_iter):
A_K_pen = np.zeros_like(Z_0)
A_K_pen[:-1] += Z_1 - U_1
A_K_pen[1:] += Z_2 - U_2
A_K_pen += A_K_pen.transpose(0, 2, 1)
A_K_pen /= 2.
Z_0 = soft_thresholding_od(A_K_pen / divisor[:, None, None],
lamda=theta / (rho * divisor))
# check feasibility and perform line search if necessary
losses_all = loss_gen(loss_function, S, Z_0)
feasibility_check = losses_all > C
infeasible_indices = list(
compress(range(len(feasibility_check)), feasibility_check))
for i in infeasible_indices:
if c_prox == 'cvx':
Z_0[i], loss_i = prox_cvx(loss_function, S[i], Z_0[i],
Z_0_old[i], C[i], div)
elif c_prox == 'grad':
if i > 0:
Z_0[i], loss_i = prox_grad(loss_function, S[i], Z_0[i],
Z_0_old[i], C[i], 0.)
else:
Z_0[i], loss_i = prox_grad(loss_function, S[i], Z_0[i],
Z_0_old[i], C[i], 0.)
# break if losses post-correction blow up
losses_all_new = loss_gen(loss_function, S, Z_0)
if np.inf in losses_all_new:
print(iteration_, 'Inf')
covariance_ = np.array([linalg.pinvh(x) for x in Z_0_old])
return_list = [Z_0_old, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
# other Zs
A_1 = Z_0[:-1] + U_1
A_2 = Z_0[1:] + U_2
if not psi_node_penalty:
prox_e = prox_psi(A_2 - A_1, lamda=2. * (1 - theta) / 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 * (1 - theta) / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter)
# update residuals
U_1 += Z_0[:-1] - Z_1
U_2 += Z_0[1:] - Z_2
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(Z_0[:-1] - Z_1) + squared_norm(Z_0[1:] - Z_2))
snorm = rho * np.sqrt(
squared_norm(Z_1 - Z_1_old) + squared_norm(Z_2 - Z_2_old))
obj = penalty_objective(Z_0, Z_1, Z_2, psi, theta)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(losses_all_new.size + 2 * Z_1.size) * tol + rtol *
(max(np.sqrt(squared_norm(losses_all_new)), np.sqrt(
squared_norm(C))) +
max(np.sqrt(squared_norm(Z_1)), np.sqrt(squared_norm(Z_0[:-1]))) +
max(np.sqrt(squared_norm(Z_2)), np.sqrt(squared_norm(Z_0[1:])))),
e_dual=np.sqrt(2 * Z_1.size) * tol +
rtol * rho * np.sqrt(squared_norm(U_1) + squared_norm(U_2)))
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])
out_obj.append(penalty_objective(Z_0, Z_0[:-1], Z_0[1:], psi, theta))
checks.append(check)
# if len(out_obj) > 100 and c_prox == 'grad':
# if (np.mean(out_obj[-11:-1]) - np.mean(out_obj[-10:])) < stop_when:
# print('obj break')
# break
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_,
# mu=1e2, tau_inc=1.01, tau_dec=1.01)
# # **(update_rho_options or {}))
# # scaled dual variables should be also rescaled
# U_1 *= rho / rho_new
# U_2 *= rho / rho_new
# rho = rho_new
else:
warnings.warn("Objective did not converge.")
print(iteration_, out_obj[-1])
# print(out_obj)
print(check.rnorm, check.e_pri)
print(check.snorm, check.e_dual)
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 enet_kernel_learning_admm2(
K, y, lamda=0.01, beta=0.01, rho=1., max_iter=100, verbose=0, rtol=1e-4,
tol=1e-4, return_n_iter=True, update_rho_options=None):
"""Elastic Net kernel learning.
Solve the following problem via ADMM:
min sum_{i=1}^p 1/2 ||y_i - alpha_i * sum_{k=1}^{n_k} w_k * K_{ik}||^2
+ lamda ||w||_1 + beta sum_{j=1}^{c_i}||alpha_j||_2^2
"""
n_patients = len(K)
n_kernels = len(K[0])
coef = np.ones(n_kernels)
alpha = [np.zeros(K[j].shape[2]) for j in range(n_patients)]
u = [np.zeros(K[j].shape[1]) for j in range(n_patients)]
u_1 = np.zeros(n_kernels)
w_1 = np.zeros(n_kernels)
x_old = [np.zeros(K[0].shape[1]) for j in range(n_patients)]
w_1_old = w_1.copy()
# w_2_old = w_2.copy()
checks = []
for iteration_ in range(max_iter):
# update x
A = [K[j].T.dot(coef) for j in range(n_patients)]
x = [prox_laplacian(y[j] + rho * (A[j].T.dot(alpha[j]) - u[j]), rho / 2.)
for j in range(n_patients)]
# update alpha
# solve (AtA + 2I)^-1 (Aty) with A = wK
KK = [rho * A[j].dot(A[j].T) for j in range(n_patients)]
yy = [rho * A[j].dot(x[j] + u[j]) for j in range(n_patients)]
alpha = [_solve_cholesky_kernel(
KK[j], yy[j][..., None], 2 * beta).ravel() for j in range(n_patients)]
# equivalent to alpha_dot_K
# solve (sum(AtA) + 2*rho I)^-1 (sum(Aty) + rho(w1+w2-u1-u2))
# with A = K * alpha
A = [K[j].dot(alpha[j]) for j in range(n_patients)]
KK = sum(A[j].dot(A[j].T) for j in range(n_patients))
yy = sum(A[j].dot(x[j] + u[j]) for j in range(n_patients))
yy += w_1 - u_1
coef = _solve_cholesky_kernel(KK, yy[..., None], 1).ravel()
w_1 = soft_thresholding(coef + u_1, lamda / rho)
# w_2 = prox_laplacian(coef + u_2, beta / rho)
# update residuals
alpha_coef_K = [
alpha[j].dot(K[j].T.dot(coef)) for j in range(n_patients)]
residuals = [x[j] - alpha_coef_K[j] for j in range(n_patients)]
u = [u[j] + residuals[j] for j in range(n_patients)]
u_1 += coef - w_1
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(coef - w_1) +
sum(squared_norm(residuals[j]) for j in range(n_patients)))
snorm = rho * np.sqrt(
squared_norm(w_1 - w_1_old) +
sum(squared_norm(x[j] - x_old[j]) for j in range(n_patients)))
obj = objective_admm2(x, y, alpha, lamda, beta, w_1)
check = convergence(
obj=obj, rnorm=rnorm, snorm=snorm,
e_pri=np.sqrt(coef.size + sum(
x[j].size for j in range(n_patients))) * tol + rtol * max(
np.sqrt(squared_norm(coef) + sum(squared_norm(
alpha_coef_K[j]) for j in range(n_patients))),
np.sqrt(squared_norm(w_1) + sum(squared_norm(
x[j]) for j in range(n_patients)))),
e_dual=np.sqrt(coef.size + sum(
x[j].size for j in range(n_patients))) * tol + rtol * rho * (
np.sqrt(squared_norm(u_1) + sum(squared_norm(
u[j]) for j in range(n_patients)))))
w_1_old = w_1.copy()
x_old = [x[j].copy() for j in range(n_patients)]
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 and iteration_ > 1:
break
rho_new = update_rho(rho, rnorm, snorm, iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
u = [u[j] * (rho / rho_new) for j in range(n_patients)]
u_1 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [alpha, coef]
if return_n_iter:
return_list.append(iteration_)
return return_list
def enet_kernel_learning_admm(K,
y,
lamda=0.01,
beta=0.01,
rho=1.,
max_iter=100,
verbose=0,
rtol=1e-4,
tol=1e-4,
return_n_iter=True,
update_rho_options=None):
"""Elastic Net kernel learning.
Solve the following problem via ADMM:
min sum_{i=1}^p 1/2 ||alpha_i * w * K_i - y_i||^2 + lamda ||w||_1 +
+ beta||w||_2^2
"""
n_patients = len(K)
n_kernels = len(K[0])
coef = np.ones(n_kernels)
u_1 = np.zeros(n_kernels)
u_2 = np.zeros(n_kernels)
w_1 = np.zeros(n_kernels)
w_2 = np.zeros(n_kernels)
w_1_old = w_1.copy()
w_2_old = w_2.copy()
checks = []
for iteration_ in range(max_iter):
# update alpha
# solve (AtA + 2I)^-1 (Aty) with A = wK
A = [K[j].T.dot(coef) for j in range(n_patients)]
KK = [A[j].dot(A[j].T) for j in range(n_patients)]
yy = [y[j].dot(A[j]) for j in range(n_patients)]
alpha = [
_solve_cholesky_kernel(KK[j], yy[j][..., None], 2).ravel()
for j in range(n_patients)
]
# alpha = [_solve_cholesky_kernel(
# K_dot_coef[j], y[j][..., None], 0).ravel() for j in range(n_patients)]
w_1 = soft_thresholding(coef + u_1, lamda / rho)
w_2 = prox_laplacian(coef + u_2, beta / rho)
# equivalent to alpha_dot_K
# solve (sum(AtA) + 2*rho I)^-1 (sum(Aty) + rho(w1+w2-u1-u2))
# with A = K * alpha
A = [K[j].dot(alpha[j]) for j in range(n_patients)]
KK = sum(A[j].dot(A[j].T) for j in range(n_patients))
yy = sum(y[j].dot(A[j].T) for j in range(n_patients))
yy += rho * (w_1 + w_2 - u_1 - u_2)
coef = _solve_cholesky_kernel(KK, yy[..., None], 2 * rho).ravel()
# update residuals
u_1 += coef - w_1
u_2 += coef - w_2
# diagnostics, reporting, termination checks
rnorm = np.sqrt(squared_norm(coef - w_1) + squared_norm(coef - w_2))
snorm = rho * np.sqrt(
squared_norm(w_1 - w_1_old) + squared_norm(w_2 - w_2_old))
obj = objective_admm(K, y, alpha, lamda, beta, coef, w_1, w_2)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(2 * coef.size) * tol +
rtol * max(np.sqrt(squared_norm(coef) + squared_norm(coef)),
np.sqrt(squared_norm(w_1) + squared_norm(w_2))),
e_dual=np.sqrt(2 * coef.size) * tol + rtol * rho *
(np.sqrt(squared_norm(u_1) + squared_norm(u_2))))
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)
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual and iteration_ > 1:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
u_1 *= rho / rho_new
u_2 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [alpha, coef]
if return_n_iter:
return_list.append(iteration_)
return return_list
def enet_kernel_learning_admm2(K,
y,
lamda=0.01,
beta=0.01,
rho=1.,
max_iter=100,
verbose=0,
rtol=1e-4,
tol=1e-4,
return_n_iter=True,
update_rho_options=None):
"""Elastic Net kernel learning.
Solve the following problem via ADMM:
min sum_{i=1}^p 1/2 ||y_i - alpha_i * sum_{k=1}^{n_k} w_k * K_{ik}||^2
+ lamda ||w||_1 + beta sum_{j=1}^{c_i}||alpha_j||_2^2
"""
n_patients = len(K)
n_kernels = len(K[0])
coef = np.ones(n_kernels)
alpha = [np.zeros(K[j].shape[2]) for j in range(n_patients)]
u = [np.zeros(K[j].shape[1]) for j in range(n_patients)]
u_1 = np.zeros(n_kernels)
w_1 = np.zeros(n_kernels)
x_old = [np.zeros(K[0].shape[1]) for j in range(n_patients)]
w_1_old = w_1.copy()
# w_2_old = w_2.copy()
checks = []
for iteration_ in range(max_iter):
# update x
A = [K[j].T.dot(coef) for j in range(n_patients)]
x = [
prox_laplacian(y[j] + rho * (A[j].T.dot(alpha[j]) - u[j]),
rho / 2.) for j in range(n_patients)
]
# update alpha
# solve (AtA + 2I)^-1 (Aty) with A = wK
KK = [rho * A[j].dot(A[j].T) for j in range(n_patients)]
yy = [rho * A[j].dot(x[j] + u[j]) for j in range(n_patients)]
alpha = [
_solve_cholesky_kernel(KK[j], yy[j][..., None], 2 * beta).ravel()
for j in range(n_patients)
]
# equivalent to alpha_dot_K
# solve (sum(AtA) + 2*rho I)^-1 (sum(Aty) + rho(w1+w2-u1-u2))
# with A = K * alpha
A = [K[j].dot(alpha[j]) for j in range(n_patients)]
KK = sum(A[j].dot(A[j].T) for j in range(n_patients))
yy = sum(A[j].dot(x[j] + u[j]) for j in range(n_patients))
yy += w_1 - u_1
coef = _solve_cholesky_kernel(KK, yy[..., None], 1).ravel()
w_1 = soft_thresholding(coef + u_1, lamda / rho)
# w_2 = prox_laplacian(coef + u_2, beta / rho)
# update residuals
alpha_coef_K = [
alpha[j].dot(K[j].T.dot(coef)) for j in range(n_patients)
]
residuals = [x[j] - alpha_coef_K[j] for j in range(n_patients)]
u = [u[j] + residuals[j] for j in range(n_patients)]
u_1 += coef - w_1
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
squared_norm(coef - w_1) +
sum(squared_norm(residuals[j]) for j in range(n_patients)))
snorm = rho * np.sqrt(
squared_norm(w_1 - w_1_old) +
sum(squared_norm(x[j] - x_old[j]) for j in range(n_patients)))
obj = objective_admm2(x, y, alpha, lamda, beta, w_1)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(coef.size + sum(x[j].size
for j in range(n_patients))) * tol +
rtol * max(
np.sqrt(
squared_norm(coef) + sum(
squared_norm(alpha_coef_K[j])
for j in range(n_patients))),
np.sqrt(
squared_norm(w_1) +
sum(squared_norm(x[j]) for j in range(n_patients)))),
e_dual=np.sqrt(coef.size + sum(x[j].size
for j in range(n_patients))) * tol +
rtol * rho * (np.sqrt(
squared_norm(u_1) +
sum(squared_norm(u[j]) for j in range(n_patients)))))
w_1_old = w_1.copy()
x_old = [x[j].copy() for j in range(n_patients)]
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 and iteration_ > 1:
break
rho_new = update_rho(rho,
rnorm,
snorm,
iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
u = [u[j] * (rho / rho_new) for j in range(n_patients)]
u_1 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [alpha, coef]
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 taylor_time_graphical_lasso(
S, K_init, max_iter, loss, C, theta, rho, mult,
weights, m, eps, psi, gamma, tol, rtol, verbose,
return_history, return_n_iter, mode, compute_objective,
stop_at, stop_when, update_rho_options
):
"""Equality constrained time-varying graphical LASSO solver.
Solves the following problem via ADMM:
min sum_{i=1}^T ||K_i||_{od,1} + beta sum_{i=2}^T Psi(K_i - K_{i-1})
s.t. objective = c_i for i = 1, ..., T
where S_i = (1/n_i) X_i^T 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.
gamma: float, optional
Kernel parameter when psi is chosen to be 'kernel'.
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)
if loss == 'LL':
loss_func = neg_logl
else:
loss_func = dtrace
T = S.shape[0]
S_flat = S.copy().reshape(T, S.shape[1] * S.shape[2])
I_flat = np.diagflat(S.shape[1]).ravel()
K = K_init.copy()
Z_0 = K_init.copy()
Z_1 = Z_0.copy()[:-1]
Z_2 = Z_0.copy()[1:]
u = np.zeros(T)
U_0 = np.zeros_like(Z_0)
U_1 = np.zeros_like(Z_1)
U_2 = np.zeros_like(Z_2)
Z_0_old = Z_0.copy()
Z_1_old = np.zeros_like(Z_1)
Z_2_old = np.zeros_like(Z_2)
# divisor for consensus variables, accounting for one less matrix for t = 0 and t = T
divisor = np.full(T, 3, dtype=float)
divisor[0] -= 1
divisor[-1] -= 1
rho = rho * np.ones(T)
if weights[0] is not None:
if weights[0] == 'rbf':
weights = rbf_weights(T, weights[1], mult)
elif weights[0] == 'exp':
weights = exp_weights(T, weights[1], mult)
elif weights[0] == 'lin':
weights = lin_weights(T, weights[1], mult)
con_obj = {}
for t in range(T):
con_obj[t] = []
con_obj_mean = []
con_obj_max = []
# loss residuals
loss_res = np.zeros(T)
loss_init = loss_gen(loss_func, S, Z_0_old)
loss_res_old = loss_init - C
# loss_diff = C - loss_init
# C_ = C - loss_diff
out_obj = []
checks = [
convergence(
obj=penalty_objective(Z_0, Z_1, Z_2, psi, theta))
]
def _K(x, A_t, g_t, nabla_t, nabla_t_T_A_t, nabla_t_T_nabla_t, rho_t, divisor_t):
_K_t = (A_t + x * g_t * nabla_t -
(x * nabla_t_T_A_t + x ** 2 * g_t * nabla_t_T_nabla_t) * nabla_t /
(divisor_t * rho_t + x * nabla_t_T_nabla_t)
).reshape(S.shape[1], S.shape[2])
_K_t /= (rho_t * divisor_t)
return 0.5 * (_K_t + _K_t.transpose(1, 0))
# def _K(x, A_t, nabla_t):
# _A_t = A_t - x * nabla_t
# return _A_t
# constrained optimisation via line search
def _f(x, _K, A_t, g_t, nabla_t, nabla_t_T_A_t, nabla_t_T_nabla_t, rho_t, divisor_t,
loss_func, S_t, c_t, loss_res_old_t, nabla_t_T_K_old_t):
_K_t = _K(x, A_t, g_t, nabla_t, nabla_t_T_A_t, nabla_t_T_nabla_t, rho_t, divisor_t)
loss_res_t = loss_func(S_t, _K_t) - c_t
return loss_res_t ** 2 + (loss_res_t - loss_res_old_t - nabla_t @ _K_t.ravel() + nabla_t_T_K_old_t) ** 2
# # constrained optimisation via line search
# def _f(x, _K, A_t, nabla_t, loss_func, S_t, c_t, loss_res_old_t):
# _K_t = _K(x, A_t, nabla_t)
# loss_res_t = loss_func(S_t, _K_t) - c_t
# return loss_res_t ** 2 + (loss_res_t - loss_res_old_t - np.sum(nabla_t * (_K_t - A_t))) ** 2
for iteration_ in range(max_iter):
# update K
A = rho[:, None, None] * (Z_0 - U_0)
A[:-1] += rho[:-1, None, None] * (Z_1 - U_1)
A[1:] += rho[1:, None, None] * (Z_2 - U_2)
# A += A.transpose(0, 2, 1)
# A /= 2.
# A /= (rho * divisor)[:, None, None]
# loss_res_pre = loss_gen(loss_func, S, A) - C
if loss_func.__name__ == 'neg_logl':
nabla = np.array([S_t - np.linalg.inv(K_t).ravel() for (S_t, K_t) in zip(S_flat, K)])
# nabla = np.array([S_t - np.linalg.inv(K_t) for (S_t, K_t) in zip(S, A)])
elif loss_func.__name__ == 'dtrace':
nabla = np.array([(2 * K_t.ravel() @ S_t - I) for (S_t, K_t) in zip(S_flat, K)])
# nabla = np.array([(2 * K_t @ S_t - I) for (S_t, K_t) in zip(S, K)])
nabla_T_K_old = np.array([nabla_t @ K_t.ravel() for (nabla_t, K_t) in zip(nabla, K)])
# nabla_T_K_old = np.array([np.sum(nabla_t * K_t) for (nabla_t, K_t) in zip(nabla, K)])
g = nabla_T_K_old - loss_res_old
nabla_T_A = np.array([nabla_t @ A_t.ravel() for (nabla_t, A_t) in zip(nabla, A)])
nabla_T_nabla = np.einsum('ij,ij->i', nabla, nabla)
if iteration_ == 0:
nabla = np.zeros_like(S_flat)
# nabla = np.zeros_like(S)
nabla_T_K_old = np.zeros(T)
g = np.zeros(T)
nabla_T_A = np.zeros(T)
nabla_T_nabla = np.zeros(T)
col = []
for t in range(T):
out = minimize_scalar(
partial(_f, _K=_K, A_t=A[t].ravel(), g_t=g[t], nabla_t=nabla[t],
nabla_t_T_A_t=nabla_T_A[t], nabla_t_T_nabla_t=nabla_T_nabla[t],
rho_t=rho[t], divisor_t=divisor[t], loss_func=loss_func,
S_t=S[t], c_t=C[t], loss_res_old_t=loss_res_old[t],
nabla_t_T_K_old_t=nabla_T_K_old[t])
)
# out = minimize_scalar(
# partial(_f, _K=_K, A_t=A[t], nabla_t=nabla[t], loss_func=loss_func,
# S_t=S[t], c_t=C[t], loss_res_old_t=loss_res_pre[t])
# )
K[t] = _K(out.x, A[t].ravel(), g[t], nabla[t], nabla_T_A[t], nabla_T_nabla[t], rho[t], divisor[t])
# K[t] = _K(out.x, A[t], nabla[t])
loss_res[t] = loss_func(S[t], K[t]) - C[t]
# u[t] += loss_res[t]
if weights[0] is not None:
con_obj[t].append(loss_res[t] ** 2)
if len(con_obj[t]) > m and np.mean(con_obj[t][-m:-int(m/2)]) < np.mean(con_obj[t][-int(m/2):]) and loss_res[t] > eps:
col.append(t)
# update Z_0
_Z_0 = K + U_0
_Z_0 += _Z_0.transpose(0, 2, 1)
_Z_0 /= 2.
Z_0 = soft_thresholding_od(_Z_0, lamda=theta / rho[:, None, None])
# update Z_1, Z_2
A_1 = Z_0[:-1] + U_1
A_2 = Z_0[1:] + U_2
if not psi_node_penalty:
A_add = A_2 + A_1
A_sub = A_2 - A_1
prox_e_1 = prox_psi(A_sub, lamda=2. * (1 - theta) / rho[:-1, None, None])
prox_e_2 = prox_psi(A_sub, lamda=2. * (1 - theta) / rho[1:, None, None])
Z_1 = .5 * (A_add - prox_e_1)
Z_2 = .5 * (A_add + prox_e_2)
# TODO: Fix for rho vector
# else:
# if weights is not None:
# Z_1, Z_2 = prox_psi(
# np.concatenate((A_1, A_2), axis=1), lamda=.5 * (1 - theta) / rho[t],
# rho=rho[t], tol=tol, rtol=rtol, max_iter=max_iter)
# update residuals
con_obj_mean.append(np.mean(loss_res) ** 2)
con_obj_max.append(np.max(loss_res))
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)
)
loss_res_old = loss_res.copy()
snorm = np.sqrt(
squared_norm(rho[:, None, None] * (Z_0 - Z_0_old)) +
squared_norm(rho[:-1, None, None] * (Z_1 - Z_1_old)) +
squared_norm(rho[1:, None, None] * (Z_2 - Z_2_old))
)
e_dual = np.sqrt(Z_0.size + 2 * Z_1.size) * tol + rtol * np.sqrt(
squared_norm(rho[:, None, None] * U_0) +
squared_norm(rho[:-1, None, None] * U_1) +
squared_norm(rho[1:, None, None] * U_2)
)
obj = objective(loss_res, Z_0, Z_1, Z_2, psi, theta)
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(loss_res.size + Z_0.size + 2 * Z_1.size) * tol + rtol *
(
max(np.sqrt(squared_norm(Z_0)), np.sqrt(squared_norm(K))) +
max(np.sqrt(squared_norm(Z_1)), np.sqrt(squared_norm(K[:-1]))) +
max(np.sqrt(squared_norm(Z_2)), np.sqrt(squared_norm(K[1:])))
),
e_dual=e_dual
)
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])
out_obj.append(penalty_objective(Z_0, Z_0[:-1], Z_0[1:], psi, theta))
if not iteration_ % 100:
print(iteration_)
print(np.max(con_obj_max[-1]), np.mean(loss_res))
print(out_obj[-1])
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
if weights[0] is None:
if len(con_obj_mean) > m:
if np.mean(con_obj_mean[-m:-int(m/2)]) < np.mean(con_obj_mean[-int(m/2):]) and np.max(loss_res) > eps:
# or np.mean(con_obj_max[-100:-50]) < np.mean(con_obj_max[-50:])) # np.mean(loss_res) > 0.25:
print("Rho Mult", mult * rho[0], iteration_, np.mean(loss_res), con_obj_max[-1])
# loss_diff /= 5
# C_ = C - loss_diff
# resscale scaled dual variables
rho = mult * rho
# u /= mult
U_0 /= mult
U_1 /= mult
U_2 /= mult
con_obj_mean = []
con_obj_max = []
else:
for t in col:
rho *= weights[t]
# u /= weights[t]
U_0 /= weights[t][:, None, None]
U_1 /= weights[t][:-1, None, None]
U_2 /= weights[t][1:, None, None]
con_obj[t] = []
print('Mult', iteration_, t, rho[t])
else:
warnings.warn("Objective did not converge.")
print(iteration_, out_obj[-1])
# print(out_obj)
print(check.rnorm, check.e_pri)
print(check.snorm, check.e_dual)
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 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 time_graph_lasso(data_list,
lamda=1,
beta=1,
max_iter=1000,
verbose=False,
tol=1e-4,
rtol=1e-2,
return_history=False):
"""Time-varying graphical lasso solver.
Solves the following problem via ADMM:
minimize trace(S*X) - log det X + lambda*||X||_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.
lamda : float, optional
Regularisation parameter.
rho : float, optional
Augmented Lagrangian parameter.
alpha : 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.
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.
"""
S = np.array(map(empirical_covariance, data_list))
n_samples = np.array([s.shape[0] for s in data_list])
K = np.zeros_like(S)
# divisor for consensus variables, accounting for two less matrices
divisor = np.zeros(S.shape[0]) + 3
divisor[0] -= 1
divisor[-1] -= 1
checks = []
alpha = 1
Kold = np.ones_like(S) + 5000
for _ in range(max_iter):
for k in range(S.shape[0]):
K[k].flat[::K.shape[1] + 1] = 1
alpha_old = alpha
# choose a gamma
gamma = .75
# total variation
Y = _J(K, beta, lamda, gamma, 1, S, n_samples)
alpha = choose_alpha(alpha_old, K, S, n_samples, beta, lamda, gamma)
alpha = 1
K = Kold + alpha * (Y - Kold)
check = convergence(
obj=objective(n_samples, S, K, lamda, beta, l1_norm),
rnorm=np.linalg.norm(K - Kold),
snorm=20, # np.linalg.norm(
# rho * (Z_consensus - Z_consensus_old)),
e_pri=30, # np.sqrt(np.prod(K.shape)) * tol + rtol * max(
# np.linalg.norm(K), np.linalg.norm(Z_consensus)),
e_dual=30 # np.sqrt(np.prod(K.shape)) * tol + rtol * np.linalg.norm(
# rho * U_consensus)
)
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 check.rnorm <= tol:
break
Kold = K.copy()
else:
warnings.warn("Objective did not converge.")
if return_history:
return K, S, checks
return K, S
def _fit_time_poisson_model(
X,
alpha=0.01,
rho=1,
kernel=None,
max_iter=100,
verbose=False,
psi="laplacian",
gamma=0.1,
tol=1e-4,
rtol=1e-4,
return_history=False,
return_n_iter=True,
compute_objective=True,
stop_at=None,
stop_when=1e-4,
n_cores=-1,
):
"""Time-varying graphical model solver.
Solves the following problem via ADMM:
min sum_{i=1}^T -n_i log_likelihood(K_i, X_i) + alpha ||K_i||_{od,1}
+ sum_{s>t}^T k(s,t) Psi(K_s - K_t)
where X is a matrix n_i x D, the observations at time i and the
log-likelihood changes according to the distribution.
Parameters
----------
X : ndarray, shape (n_times, n_samples, n_features)
Data matrix. It has to contain two values: 0 or 1, -1 or 1.
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_samples, n_features = X.shape
n_samples = np.array([n_samples] * n_times)
if kernel is None:
kernel = np.eye(n_times)
K = np.zeros((n_times, n_features, n_features))
Z_M = {}
U_M = {}
Z_M_old = {}
for m in range(1, n_times):
# all possible non markovians jumps
Z_L = K.copy()[:-m]
Z_R = K.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)
checks = [convergence(obj=objective(X, K, Z_M, alpha, kernel, psi))]
for iteration_ in range(max_iter):
# update K
A = np.zeros_like(K)
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
A += A.transpose(0, 2, 1)
A /= 2.0
# K_new = np.zeros_like(K)
for t in range(n_times):
thetas_pred = []
for v in range(n_features):
inner_verbose = max(0, verbose - 1)
res = fit_each_variable(X[t, :, :],
v,
alpha,
tol=tol,
verbose=inner_verbose,
A=A[t, :, :],
T=n_times,
rho=rho)
thetas_pred.append(res[0])
K[t, :, :] = build_adjacency_matrix(thetas_pred, "union")
# 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(
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(
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(X, 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(
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(
sum(
squared_norm(U_M[m][0]) + squared_norm(U_M[m][1])
for m in range(1, n_times))),
)
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.")
return_list = [K]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_ + 1)
return return_list
def time_latent_graph_lasso_v2(emp_cov,
alpha=1.,
tau=1.,
rho=1.,
beta=1.,
eta=1.,
max_iter=1000,
verbose=False,
psi='laplacian',
phi='laplacian',
assume_centered=False,
tol=1e-4,
rtol=1e-2,
return_history=False,
return_n_iter=True,
mode=None):
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.
"""
if psi == 'laplacian':
prox_psi = prox_laplacian
psi = squared_norm
elif psi == 'l1':
prox_psi = soft_thresholding_sign
psi = l1_norm
elif psi == 'l2':
prox_psi = blockwise_soft_thresholding
psi = np.linalg.norm
elif psi == 'linf':
prox_psi = prox_linf
psi = partial(np.linalg.norm, ord=np.inf)
else:
raise ValueError("Value of `psi` not understood.")
if phi == 'laplacian':
prox_phi = prox_laplacian
phi = squared_norm
elif phi == 'l1':
prox_phi = soft_thresholding_sign
phi = l1_norm
elif phi == 'l2':
prox_phi = blockwise_soft_thresholding
phi = np.linalg.norm
elif phi == 'linf':
prox_phi = prox_linf
phi = partial(np.linalg.norm, ord=np.inf)
else:
raise ValueError("Value of `phi` not understood.")
# emp_cov = np.array([empirical_covariance(
# x, assume_centered=assume_centered) for x in data_list])
n_samples = np.array([s for s in [1.]])
K = np.zeros_like(emp_cov)
L = np.zeros_like(emp_cov)
X = 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:]
X_0 = np.zeros_like(K)
X_1 = np.zeros_like(K)[:-1]
X_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:]
U_0 = np.zeros_like(emp_cov)
U_1 = np.zeros_like(emp_cov)[:-1]
U_2 = np.zeros_like(emp_cov)[1:]
Y_0 = np.zeros_like(emp_cov)
Y_1 = np.zeros_like(emp_cov)[:-1]
Y_2 = np.zeros_like(emp_cov)[1:]
U_consensus = np.zeros_like(emp_cov)
Y_consensus = np.zeros_like(emp_cov)
Z_consensus = np.zeros_like(emp_cov)
Z_consensus_old = np.zeros_like(emp_cov)
W_consensus = np.zeros_like(emp_cov)
W_consensus_old = np.zeros_like(emp_cov)
R_old = np.zeros_like(emp_cov)
# divisor for consensus variables, accounting for two less matrices
divisor = np.zeros(emp_cov.shape[0]) + 3
divisor[0] -= 1
divisor[-1] -= 1
# eta = np.divide(n_samples, divisor * rho)
checks = []
for iteration_ in range(max_iter):
# update R
A = K - L - X
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] + 1
A += np.array(map(np.transpose, A))
A /= 2.
A *= -rho # / n_samples[:, None, None]
A += emp_cov
R = np.array(map(prox_logdet, A, n_samples / (rho)))
# update K, L
K = L + R + X + Z_0 - U_0
K[:-1] += Z_1 - U_1
K[1:] += Z_2 - U_2
K /= divisor[:, None, None] + 1
L = K - R - X + W_0 - Y_0
L[:-1] += W_1 - Y_1
L[1:] += W_2 - Y_2
L /= divisor[:, None, None] + 1
# update Z_0
soft_thresholding = partial(soft_thresholding_sign, lamda=alpha / rho)
Z_0 = np.array(map(soft_thresholding, (K + U_0 + R + W_0 + X_0) / 2.))
# update Z_1, Z_2
A_1 = (K[:-1] + U_1 + R[:-1] + W_1 + X_1) / 2.
A_2 = (K[1:] + U_2 + R[1:] + W_2 + X_2) / 2.
prox_e = prox_psi(A_2 - A_1, lamda=2. * beta / rho)
Z_1 = .5 * (A_2 + A_1 - prox_e)
Z_2 = .5 * (A_2 + A_1 + prox_e)
# update W_0
A = (L + Y_0 - R + Z_0 - X_0) / 2.
W_0 = np.array(map(partial(prox_trace_indicator, lamda=tau / rho), A))
# update W_1, W_2
A_1 = (L[:-1] + Y_1 - R[:-1] + Z_1 - X_1) / 2.
A_2 = (L[1:] + Y_2 - R[1:] + Z_2 - X_2) / 2.
prox_e = prox_phi(A_2 - A_1, lamda=2. * eta / rho)
W_1 = .5 * (A_2 + A_1 - prox_e)
W_2 = .5 * (A_2 + A_1 + prox_e)
# update residuals
X += R - K + L
X_0 += (R - Z_0 + W_0)
X_1 += (R[:-1] - Z_1 + W_1)
X_2 += (R[1:] - Z_2 + W_2)
U_0 += (K - Z_0)
U_1 += (K[:-1] - Z_1)
U_2 += (K[1:] - Z_2)
Y_0 += (L - W_0)
Y_1 += (L[:-1] - W_1)
Y_2 += (L[1:] - W_2)
# diagnostics, reporting, termination checks
Z_consensus = Z_0.copy()
Z_consensus[:-1] += Z_1
Z_consensus[1:] += Z_2
Z_consensus /= divisor[:, None, None]
U_consensus = U_0.copy()
U_consensus[:-1] += U_1
U_consensus[1:] += U_2
U_consensus /= divisor[:, None, None]
W_consensus = W_0.copy()
W_consensus[:-1] += W_1
W_consensus[1:] += W_2
W_consensus /= divisor[:, None, None]
X_consensus = X_0.copy()
X_consensus[:-1] += X_1
X_consensus[1:] += X_2
X_consensus /= divisor[:, None, None]
Y_consensus = Y_0.copy()
Y_consensus[:-1] += Y_1
Y_consensus[1:] += Y_2
Y_consensus /= divisor[:, None, None]
check = convergence(
obj=objective(n_samples, emp_cov, R, Z_0, Z_1, Z_2, W_0, W_1, W_2,
alpha, tau, beta, eta, psi, phi),
rnorm=np.sqrt(
squared_norm(K - Z_consensus) + squared_norm(L - W_consensus) +
squared_norm(Z_consensus - W_consensus - R) +
squared_norm(K - L - R)),
snorm=np.sqrt(
squared_norm(rho * (Z_consensus - Z_consensus_old)) +
squared_norm(rho * (W_consensus - W_consensus_old)) +
squared_norm(rho * (Z_consensus - W_consensus -
(Z_consensus_old - W_consensus_old))) +
squared_norm(rho * (R - R_old))),
e_pri=np.sqrt(np.prod(K.shape) * 4) * tol + rtol * max(
np.sqrt(
squared_norm(K) + squared_norm(L) + squared_norm(K - L) +
squared_norm(Z_consensus - W_consensus)),
np.sqrt(
squared_norm(Z_consensus) + squared_norm(W_consensus) +
squared_norm(R) + squared_norm(R))),
e_dual=np.sqrt(np.prod(K.shape) * 4) * tol + rtol * np.sqrt(
squared_norm(rho *
(U_consensus)) + squared_norm(rho *
(Y_consensus)) +
squared_norm(rho * (X_consensus)) + squared_norm(rho * (X))))
Z_consensus_old = Z_consensus.copy()
W_consensus_old = W_consensus.copy()
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
else:
warnings.warn("Objective did not converge.")
return_list = [K, L, emp_cov]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
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 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 enet_kernel_learning_admm(
K, y, lamda=0.01, beta=0.01, rho=1., max_iter=100, verbose=0, rtol=1e-4,
tol=1e-4, return_n_iter=True, update_rho_options=None):
"""Elastic Net kernel learning.
Solve the following problem via ADMM:
min sum_{i=1}^p 1/2 ||alpha_i * w * K_i - y_i||^2 + lamda ||w||_1 +
+ beta||w||_2^2
"""
n_patients = len(K)
n_kernels = len(K[0])
coef = np.ones(n_kernels)
u_1 = np.zeros(n_kernels)
u_2 = np.zeros(n_kernels)
w_1 = np.zeros(n_kernels)
w_2 = np.zeros(n_kernels)
w_1_old = w_1.copy()
w_2_old = w_2.copy()
checks = []
for iteration_ in range(max_iter):
# update alpha
# solve (AtA + 2I)^-1 (Aty) with A = wK
A = [K[j].T.dot(coef) for j in range(n_patients)]
KK = [A[j].dot(A[j].T) for j in range(n_patients)]
yy = [y[j].dot(A[j]) for j in range(n_patients)]
alpha = [_solve_cholesky_kernel(
KK[j], yy[j][..., None], 2).ravel() for j in range(n_patients)]
# alpha = [_solve_cholesky_kernel(
# K_dot_coef[j], y[j][..., None], 0).ravel() for j in range(n_patients)]
w_1 = soft_thresholding(coef + u_1, lamda / rho)
w_2 = prox_laplacian(coef + u_2, beta / rho)
# equivalent to alpha_dot_K
# solve (sum(AtA) + 2*rho I)^-1 (sum(Aty) + rho(w1+w2-u1-u2))
# with A = K * alpha
A = [K[j].dot(alpha[j]) for j in range(n_patients)]
KK = sum(A[j].dot(A[j].T) for j in range(n_patients))
yy = sum(y[j].dot(A[j].T) for j in range(n_patients))
yy += rho * (w_1 + w_2 - u_1 - u_2)
coef = _solve_cholesky_kernel(KK, yy[..., None], 2 * rho).ravel()
# update residuals
u_1 += coef - w_1
u_2 += coef - w_2
# diagnostics, reporting, termination checks
rnorm = np.sqrt(squared_norm(coef - w_1) + squared_norm(coef - w_2))
snorm = rho * np.sqrt(
squared_norm(w_1 - w_1_old) + squared_norm(w_2 - w_2_old))
obj = objective_admm(K, y, alpha, lamda, beta, coef, w_1, w_2)
check = convergence(
obj=obj, rnorm=rnorm, snorm=snorm,
e_pri=np.sqrt(2 * coef.size) * tol + rtol * max(
np.sqrt(squared_norm(coef) + squared_norm(coef)),
np.sqrt(squared_norm(w_1) + squared_norm(w_2))),
e_dual=np.sqrt(2 * coef.size) * tol + rtol * rho * (
np.sqrt(squared_norm(u_1) + squared_norm(u_2))))
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)
checks.append(check)
if check.rnorm <= check.e_pri and check.snorm <= check.e_dual and iteration_ > 1:
break
rho_new = update_rho(rho, rnorm, snorm, iteration=iteration_,
**(update_rho_options or {}))
# scaled dual variables should be also rescaled
u_1 *= rho / rho_new
u_2 *= rho / rho_new
rho = rho_new
else:
warnings.warn("Objective did not converge.")
return_list = [alpha, coef]
if return_n_iter:
return_list.append(iteration_)
return return_list
def latent_time_matrix_decomposition(emp_cov,
alpha=0.01,
tau=1.,
rho=1.,
beta=1.,
eta=1.,
max_iter=100,
verbose=False,
psi='laplacian',
phi='laplacian',
mode='admm',
tol=1e-4,
rtol=1e-4,
assume_centered=False,
return_history=False,
return_n_iter=True,
update_rho_options=None,
compute_objective=True):
r"""Latent variable time-varying matrix decomposition solver.
Solves the following problem via ADMM:
min sum_{i=1}^T || S_i-(K_i-L_i)||^2 + 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 matrix to decompose.
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Matrix to decompose.
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)
Z_0 = np.zeros_like(emp_cov)
Z_1 = np.zeros_like(Z_0)[:-1]
Z_2 = np.zeros_like(Z_0)[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
checks = []
for iteration_ in range(max_iter):
# update R
A = Z_0 - W_0 - X_0
R = (rho * A + 2 * emp_cov) / (2 + rho)
# 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, 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)
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.")
return_list = [Z_0, W_0]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_)
return return_list
def tgl_forward_backward(
emp_cov, alpha=0.01, beta=1., max_iter=100, n_samples=None, verbose=False,
tol=1e-4, delta=1e-4, gamma=1., lamda=1., eps=0.5, debug=False,
return_history=False, return_n_iter=True, choose='gamma',
lamda_criterion='b', time_norm=1, compute_objective=True,
return_n_linesearch=False, vareps=1e-5, stop_at=None, stop_when=1e-4,
laplacian_penalty=False, init='empirical'):
"""Time-varying graphical lasso solver with forward-backward splitting.
Solves the following problem via FBS:
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_times, n_features, n_features)
Empirical covariance of data.
alpha, beta : float, optional
Regularisation parameters.
max_iter : int, optional
Maximum number of iterations.
n_samples : ndarray
Number of samples available for each time point.
verbose : bool, default False
Print info at each iteration.
tol : float, optional
Absolute tolerance for convergence.
delta, gamma, lamda, eps : float, optional
FBS parameters.
debug : bool, default False
Run in debug mode.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
choose : ('gamma', 'lambda', 'fixed', 'both)
Search iteratively gamma / lambda / none / both.
lamda_criterion : ('a', 'b', 'c')
Criterion to choose lamda. See ref for details.
time_norm : float, optional
Choose the temporal norm between points.
compute_objective : bool, default True
Choose to compute the objective value.
return_n_linesearch : bool, optional
Return the number of line-search iterations before convergence.
vareps : float, optional
Jitter for the loss.
stop_at, stop_when : float, optional
Other convergence criteria, as used in the paper.
laplacian_penalty : bool, default False
Use Laplacian penalty.
init : {'empirical', 'zero', ndarray}
Choose how to initialize the precision matrix, with the inverse
empirical covariance, zero matrix or precomputed.
Returns
-------
K, covariance : 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.
"""
available_choose = ('gamma', 'lamda', 'fixed', 'both')
if choose not in available_choose:
raise ValueError(
"`choose` parameter must be one of %s." % available_choose)
n_times, _, n_features = emp_cov.shape
K = init_precision(emp_cov, mode=init)
if laplacian_penalty:
obj_partial = partial(
objective_laplacian, n_samples=n_samples, emp_cov=emp_cov,
alpha=alpha, beta=beta, vareps=vareps)
function_f = partial(
loss_laplacian, beta=beta, n_samples=n_samples, S=emp_cov,
vareps=vareps)
gradient_f = partial(
grad_loss_laplacian, emp_cov=emp_cov, beta=beta,
n_samples=n_samples, vareps=vareps)
function_g = partial(penalty_laplacian, alpha=alpha)
else:
psi = partial(vector_p_norm, p=time_norm)
obj_partial = partial(
objective, n_samples=n_samples, emp_cov=emp_cov, alpha=alpha,
beta=beta, psi=psi, vareps=vareps)
function_f = partial(
loss, n_samples=n_samples, S=emp_cov, vareps=vareps)
gradient_f = partial(
grad_loss, emp_cov=emp_cov, n_samples=n_samples, vareps=vareps)
function_g = partial(penalty, alpha=alpha, beta=beta, psi=psi)
max_residual = -np.inf
n_linesearch = 0
checks = [convergence(obj=obj_partial(precision=K))]
for iteration_ in range(max_iter):
k_previous = K.copy()
x_inv = np.array([linalg.pinvh(x) for x in K])
grad = gradient_f(K, x_inv=x_inv)
if choose in ['gamma', 'both']:
gamma, y = choose_gamma(
gamma / eps if iteration_ > 0 else gamma, K,
function_f=function_f, beta=beta, alpha=alpha, lamda=lamda,
grad=grad, delta=delta, eps=eps, max_iter=200, p=time_norm,
x_inv=x_inv, choose=choose,
laplacian_penalty=laplacian_penalty)
x_hat = K - gamma * grad
if choose not in ['gamma', 'both']:
if laplacian_penalty:
y = soft_thresholding_od(x_hat, alpha * gamma)
else:
y = prox_FL(
x_hat, beta * gamma, alpha * gamma, p=time_norm,
symmetric=True)
if choose in ('lamda', 'both'):
lamda, n_ls = choose_lamda(
min(lamda / eps if iteration_ > 0 else lamda,
1), K, function_f=function_f, objective_f=obj_partial,
gradient_f=gradient_f, function_g=function_g, gamma=gamma,
delta=delta, eps=eps, criterion=lamda_criterion, max_iter=200,
p=time_norm, grad=grad, prox=y, vareps=vareps)
n_linesearch += n_ls
K = K + min(max(lamda, 0), 1) * (y - K)
# K, t = fista_step(Y, Y - Y_old, t)
check = convergence(
obj=obj_partial(precision=K),
rnorm=np.linalg.norm(upper_diag_3d(K) - upper_diag_3d(k_previous)),
snorm=np.linalg.norm(
obj_partial(precision=K) - obj_partial(precision=k_previous)),
e_pri=np.sqrt(upper_diag_3d(K).size) * tol + tol * max(
np.linalg.norm(upper_diag_3d(K)),
np.linalg.norm(upper_diag_3d(k_previous))), e_dual=tol)
if verbose and iteration_ % (50 if verbose < 2 else 1) == 0:
print(
"obj: %.4f, rnorm: %.7f, snorm: %.4f,"
"eps_pri: %.4f, eps_dual: %.4f" % check[:5])
if return_history:
checks.append(check)
if np.isnan(check.rnorm) or np.isnan(check.snorm):
warnings.warn("precision is not positive definite.")
if stop_at is not None:
if abs(check.obj - stop_at) / abs(stop_at) < stop_when:
break
else:
# use this convergence criterion
subgrad = (x_hat - K) / gamma
if 0:
if laplacian_penalty:
grad = grad_loss_laplacian(
K, emp_cov, n_samples, vareps=vareps)
else:
grad = grad_loss(K, emp_cov, n_samples, vareps=vareps)
res_norm = np.linalg.norm(grad + subgrad)
if iteration_ == 0:
normalizer = res_norm + 1e-6
max_residual = max(
np.linalg.norm(grad), np.linalg.norm(subgrad)) + 1e-6
else:
res_norm = np.linalg.norm(K - k_previous) / gamma
max_residual = max(max_residual, res_norm)
normalizer = max(
np.linalg.norm(grad), np.linalg.norm(subgrad)) + 1e-6
r_rel = res_norm / max_residual
r_norm = res_norm / normalizer
if not debug and (r_rel <= tol
or r_norm <= tol) and iteration_ > 0: # or (
# check.rnorm <= check.e_pri and iteration_ > 0):
break
else:
warnings.warn("Objective did not converge.")
covariance_ = np.array([linalg.pinvh(k) for k in K])
return_list = [K, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_ + 1)
if return_n_linesearch:
return_list.append(n_linesearch)
return return_list
def equality_time_graphical_lasso(
S,
K_init,
max_iter,
loss,
C,
rho, # n_samples=None,
psi,
gamma,
tol,
rtol,
verbose,
return_history,
return_n_iter,
mode,
compute_objective,
stop_at,
stop_when,
update_rho_options,
init):
"""Equality constrained time-varying graphical LASSO solver.
Solves the following problem via ADMM:
min sum_{i=1}^T ||K_i||_{od,1} + beta sum_{i=2}^T Psi(K_i - K_{i-1})
s.t. objective = c_i for i = 1, ..., T
where S_i = (1/n_i) X_i^T 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.
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.
gamma: float, optional
Kernel parameter when psi is chosen to be 'kernel'.
constrained_to: float or ndarray, shape (time steps)
Log likelihood constraints for K_i
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)
psi_name = psi.__name__
if loss == 'LL':
loss_function = neg_logl
else:
loss_function = dtrace
K = K_init
Z_0 = K.copy()
Z_1 = K.copy()[:-1]
Z_2 = K.copy()[1:]
u = np.zeros((S.shape[0]))
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)
I = np.eye(S.shape[1])
checks = [
convergence(
obj=equality_objective(loss_function, S, K, C, Z_0, Z_1, Z_2, psi))
]
for iteration_ in range(max_iter):
# update K
A_K = U_0 - Z_0
A_K[:-1] += Z_1 - U_1
A_K[1:] += Z_2 - U_2
A_K += A_K.transpose(0, 2, 1)
A_K /= 2.
K = soft_thresholding_od(A_K, lamda=1. / rho)
# update Z_0
residual_loss_constraint_u = loss_gen(loss_function, S, Z_0) - C + u
A_Z = K + U_0
A_Z += A_Z.transpose(0, 2, 1)
A_Z /= 2.
if loss_function == neg_logl:
A_Z -= residual_loss_constraint_u[:, None, None] * S
Z_0 = np.array([
prox_logdet_constrained(_A, _a, I)
for _A, _a in zip(A_Z, residual_loss_constraint_u)
])
elif loss_function == dtrace:
Z_0 = np.array([
prox_dtrace_constrained(_A, _S, _a, I)
for _A, _S, _a in zip(A_Z, S, residual_loss_constraint_u)
])
# 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. / 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 / rho,
rho=rho,
tol=tol,
rtol=rtol,
max_iter=max_iter)
# update residuals
residual_loss_constraint = loss_gen(loss_function, S, Z_0) - C
u += residual_loss_constraint
U_0 += K - Z_0
U_1 += K[:-1] - Z_1
U_2 += K[1:] - Z_2
print(residual_loss_constraint)
# diagnostics, reporting, termination checks
rnorm = np.sqrt(
np.sum(residual_loss_constraint**2) + 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 = equality_objective(loss_function, S, K, C, Z_0, Z_1, Z_2,
psi) if compute_objective else np.nan
check = convergence(
obj=obj,
rnorm=rnorm,
snorm=snorm,
e_pri=np.sqrt(Z_0.size + 2 * Z_1.size + S.shape[0]) * tol +
rtol * max(
np.sqrt(
np.sum(C**2) + squared_norm(Z_0) + squared_norm(Z_1) +
squared_norm(Z_2)),
np.sqrt(
np.sum(
(residual_loss_constraint + C)**2) + squared_norm(K) +
squared_norm(K[:-1]) + squared_norm(K[1:]))),
e_dual=np.sqrt(Z_0.size + 2 * Z_1.size) * tol + rtol * rho *
np.sqrt(squared_norm(U_0) + squared_norm(U_1) + squared_norm(U_2)),
)
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 *= rho / rho_new
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 K])
return_list = [K, covariance_]
if return_history:
return_list.append(checks)
if return_n_iter:
return_list.append(iteration_ + 1)
return return_list
