code: def graph_lasso(
      emp_cov,
      alpha,
      cov_init=None,
      mode="cd",
      tol=1e-4,
      enet_tol=1e-4,
      max_iter=100,
      verbose=False,
      return_costs=False,
      eps=np.finfo(np.float64).eps,
      return_n_iter=False,
  ):
      """l1-penalized covariance estimator
  
      Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
  
      Parameters
      ----------
      emp_cov : 2D ndarray, shape (n_features, n_features)
          Empirical covariance from which to compute the covariance estimate.
  
      alpha : positive float
          The regularization parameter: the higher alpha, the more
          regularization, the sparser the inverse covariance.
  
      cov_init : 2D array (n_features, n_features), optional
          The initial guess for the covariance.
  
      mode : {'cd', 'lars'}
          The Lasso solver to use: coordinate descent or LARS. Use LARS for
          very sparse underlying graphs, where p > n. Elsewhere prefer cd
          which is more numerically stable.
  
      tol : positive float, optional
          The tolerance to declare convergence: if the dual gap goes below
          this value, iterations are stopped.
  
      enet_tol : positive float, optional
          The tolerance for the elastic net solver used to calculate the descent
          direction. This parameter controls the accuracy of the search direction
          for a given column update, not of the overall parameter estimate. Only
          used for mode='cd'.
  
      max_iter : integer, optional
          The maximum number of iterations.
  
      verbose : boolean, optional
          If verbose is True, the objective function and dual gap are
          printed at each iteration.
  
      return_costs : boolean, optional
          If return_costs is True, the objective function and dual gap
          at each iteration are returned.
  
      eps : float, optional
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems.
  
      return_n_iter : bool, optional
          Whether or not to return the number of iterations.
  
      Returns
      -------
      covariance : 2D ndarray, shape (n_features, n_features)
          The estimated covariance matrix.
  
      precision : 2D ndarray, shape (n_features, n_features)
          The estimated (sparse) precision matrix.
  
      costs : list of (objective, dual_gap) pairs
          The list of values of the objective function and the dual gap at
          each iteration. Returned only if return_costs is True.
  
      n_iter : int
          Number of iterations. Returned only if `return_n_iter` is set to True.
  
      See Also
      --------
      GraphLasso, GraphLassoCV
  
      Notes
      -----
      The algorithm employed to solve this problem is the GLasso algorithm,
      from the Friedman 2008 Biostatistics paper. It is the same algorithm
      as in the R `glasso` package.
  
      One possible difference with the `glasso` R package is that the
      diagonal coefficients are not penalized.
  
      """
      _, n_features = emp_cov.shape
      if alpha == 0:
          if return_costs:
              precision_ = linalg.inv(emp_cov)
              cost = -2.0 * log_likelihood(emp_cov, precision_)
              cost += n_features * np.log(2 * np.pi)
              d_gap = np.sum(emp_cov * precision_) - n_features
              if return_n_iter:
                  return emp_cov, precision_, (cost, d_gap), 0
              else:
                  return emp_cov, precision_, (cost, d_gap)
          else:
              if return_n_iter:
                  return emp_cov, linalg.inv(emp_cov), 0
              else:
                  return emp_cov, linalg.inv(emp_cov)
      if cov_init is None:
          covariance_ = emp_cov.copy()
      else:
          covariance_ = cov_init.copy()
      # As a trivial regularization (Tikhonov like), we scale down the
      # off-diagonal coefficients of our starting point: This is needed, as
      # in the cross-validation the cov_init can easily be
      # ill-conditioned, and the CV loop blows. Beside, this takes
      # conservative stand-point on the initial conditions, and it tends to
      # make the convergence go faster.
      covariance_ *= 0.95
      diagonal = emp_cov.flat[:: n_features + 1]
      covariance_.flat[:: n_features + 1] = diagonal
      precision_ = pinvh(covariance_)
  
      indices = np.arange(n_features)
      costs = list()
      # The different l1 regression solver have different numerical errors
      if mode == "cd":
          errors = dict(over="raise", invalid="ignore")
      else:
          errors = dict(invalid="raise")
      try:
          # be robust to the max_iter=0 edge case, see:
          # https://github.com/scikit-learn/scikit-learn/issues/4134
          d_gap = np.inf
          for i in range(max_iter):
              for idx in range(n_features):
                  sub_covariance = np.ascontiguousarray(
                      covariance_[indices != idx].T[indices != idx]
                  )
                  row = emp_cov[idx, indices != idx]
                  with np.errstate(**errors):
                      if mode == "cd":
                          # Use coordinate descent
                          coefs = -(
                              precision_[indices != idx, idx]
                              / (precision_[idx, idx] + 1000 * eps)
                          )
                          (
                              coefs,
                              _,
                              _,
                              _,
                          ) = cd_fast.enet_coordinate_descent_gram(
                              coefs,
                              alpha,
                              0,
                              sub_covariance,
                              row,
                              row,
                              max_iter,
                              enet_tol,
                              check_random_state(None),
                              False,
                          )
                      else:
                          # Use LARS
                          _, _, coefs = lars_path(
                              sub_covariance,
                              row,
                              Xy=row,
                              Gram=sub_covariance,
                              alpha_min=alpha / (n_features - 1),
                              copy_Gram=True,
                              method="lars",
                              return_path=False,
                          )
                  # Update the precision matrix
                  precision_[idx, idx] = 1.0 / (
                      covariance_[idx, idx]
                      - np.dot(covariance_[indices != idx, idx], coefs)
                  )
                  precision_[indices != idx, idx] = (
                      -precision_[idx, idx] * coefs
                  )
                  precision_[idx, indices != idx] = (
                      -precision_[idx, idx] * coefs
                  )
                  coefs = np.dot(sub_covariance, coefs)
                  covariance_[idx, indices != idx] = coefs
                  covariance_[indices != idx, idx] = coefs
              d_gap = _dual_gap(emp_cov, precision_, alpha)
              cost = _objective(emp_cov, precision_, alpha)
              if verbose:
                  print(
                      "[graph_lasso] Iteration % 3i, cost % 3.2e, dual gap %.3e"
                      % (i, cost, d_gap)
                  )
              if return_costs:
                  costs.append((cost, d_gap))
              if np.abs(d_gap) < tol:
                  break
              if not np.isfinite(cost) and i > 0:
                  raise FloatingPointError(
                      "Non SPD result: the system is "
                      "too ill-conditioned for this solver"
                  )
          else:
              warnings.warn(
                  "graph_lasso: did not converge after %i iteration:"
                  " dual gap: %.3e" % (max_iter, d_gap),
                  ConvergenceWarning,
              )
      except FloatingPointError as e:
          e.args = (
              e.args[0]
              + ". The system is too ill-conditioned for this solver",
          )
          raise e
  
      if return_costs:
          if return_n_iter:
              return covariance_, precision_, costs, i + 1
          else:
              return covariance_, precision_, costs
      else:
          if return_n_iter:
              return covariance_, precision_, i + 1
          else:
              return covariance_, precision_
  
imports: import warnings
  import operator
  import sys
  import time
  import numpy
  from scipy import linalg
  from .empirical_covariance_ import empirical_covariance, EmpiricalCovariance, log_likelihood
  from ..exceptions import ConvergenceWarning
  from ..utils.extmath import pinvh
  from ..utils.validation import check_random_state, check_array
  from ..linear_model import lars_path
  from ..linear_model import cd_fast
  from ..model_selection import check_cv, cross_val_score
  from ..externals.joblib import Parallel, delayed
  import collections
label_desc: Use image node_id 4 for calling a global function with example usage: graph_lasso(emp_cov, alpha, cov_init, mode, tol, enet_tol, max_iter, verbose, return_costs, eps, return_n_iter) and returns: covariance_,  precision_,  costs,  unknown, covariance_,  precision_,  costs, covariance_,  precision_,  unknown, covariance_,  precision_, emp_cov,  precision_,  cost,  d_gap,  int, emp_cov,  precision_,  cost,  d_gap, emp_cov,  linalg,  int, emp_cov,  linalg