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