code: def _group_sparse_covariance(
      emp_covs,
      n_samples,
      alpha,
      max_iter=10,
      tol=1e-3,
      precisions_init=None,
      probe_function=None,
      verbose=0,
      debug=False,
  ):
      """Implement an internal version of group_sparse_covariance.
  
      See its docstring for details.
  
      """
      if tol == -1:
          tol = None
      if not isinstance(alpha, (int, float)) or alpha < 0:
          raise ValueError(
              "Regularization parameter alpha must be a "
              "positive number.\n"
              f"You provided: {alpha}"
          )
  
      n_subjects = emp_covs.shape[-1]
      n_features = emp_covs[0].shape[0]
      n_samples = np.asarray(n_samples)
      n_samples /= n_samples.sum()  # essential for numerical stability
  
      # Check diagonal normalization.
      ones = np.ones(emp_covs.shape[0])
      for k in range(n_subjects):
          if (
              abs(
                  emp_covs[..., k].flat[:: emp_covs.shape[0] + 1] - ones
              )
              > 0.1
          ).any():
              warnings.warn(
                  "input signals do not all have unit variance. This "
                  "can lead to numerical instability."
              )
              break
  
      if precisions_init is None:
          # Fortran order make omega[..., k] contiguous, which is often useful.
          omega = np.ndarray(
              shape=emp_covs.shape, dtype=np.float64, order="F"
          )
          for k in range(n_subjects):
              # Values on main diagonals are far from zero, because they
              # are timeseries energy.
              omega[..., k] = np.diag(1.0 / np.diag(emp_covs[..., k]))
      else:
          omega = precisions_init.copy()
  
      # Preallocate arrays
      y = np.ndarray(
          shape=(n_subjects, n_features - 1), dtype=np.float64
      )
      u = np.ndarray(
          shape=(n_subjects, n_features - 1), dtype=np.float64
      )
      y_1 = np.ndarray(
          shape=(n_subjects, n_features - 2), dtype=np.float64
      )
      h_12 = np.ndarray(
          shape=(n_subjects, n_features - 2), dtype=np.float64
      )
      q = np.ndarray(shape=(n_subjects,), dtype=np.float64)
      aq = np.ndarray(
          shape=(n_subjects,), dtype=np.float64
      )  # temp. array
      c = np.ndarray(shape=(n_subjects,), dtype=np.float64)
      W = np.ndarray(
          shape=(
              omega.shape[0] - 1,
              omega.shape[1] - 1,
              omega.shape[2],
          ),
          dtype=np.float64,
          order="F",
      )
      W_inv = np.ndarray(shape=W.shape, dtype=np.float64, order="F")
  
      # Auxiliary arrays.
      v = np.ndarray((omega.shape[0] - 1,), dtype=np.float64)
      h = np.ndarray((omega.shape[1] - 1,), dtype=np.float64)
  
      # Optional.
      tolerance_reached = False
      max_norm = None
  
      omega_old = np.empty_like(omega)
      if probe_function is not None:
          # iteration number -1 means called before iteration loop.
          probe_function(
              emp_covs, n_samples, alpha, max_iter, tol, -1, omega, None
          )
      probe_interrupted = False
  
      # Start optimization loop. Variables are named following (mostly) the
      # Honorio-Samaras paper notations.
  
      # Used in the innermost loop. Computed here to save some computation.
      alpha2 = alpha**2
  
      for n in range(max_iter):
          if max_norm is not None:
              suffix = f" variation (max norm): {max_norm:.3e} "
          else:
              suffix = ""
          if verbose > 1:
              logger.log(
                  f"* iteration {n:d} "
                  f"({100.0 * n / max_iter:.0f} %){suffix} ...",
                  verbose=verbose,
              )
  
          omega_old[...] = omega
          for p in range(n_features):
              if p == 0:
                  # Initial state: remove first col/row
                  W = omega[1:, 1:, :].copy()  # stack of W(k)
                  W_inv = np.ndarray(shape=W.shape, dtype=np.float64)
                  for k in range(W.shape[2]):
                      # stack of W^-1(k)
                      W_inv[..., k] = scipy.linalg.inv(W[..., k])
                      if debug:
                          np.testing.assert_almost_equal(
                              np.dot(W_inv[..., k], W[..., k]),
                              np.eye(W_inv[..., k].shape[0]),
                              decimal=10,
                          )
                          _assert_submatrix(omega[..., k], W[..., k], p)
                          assert is_spd(W_inv[..., k])
              else:
                  # Update W and W_inv
                  if debug:
                      omega_orig = omega.copy()
  
                  for k in range(n_subjects):
                      _update_submatrix(
                          omega[..., k],
                          W[..., k],
                          W_inv[..., k],
                          p,
                          h,
                          v,
                      )
  
                      if debug:
                          _assert_submatrix(omega[..., k], W[..., k], p)
                          assert is_spd(W_inv[..., k], decimal=14)
                          np.testing.assert_almost_equal(
                              np.dot(W[..., k], W_inv[..., k]),
                              np.eye(W_inv[..., k].shape[0]),
                              decimal=10,
                          )
                  if debug:
                      # Check that omega has not been modified.
                      np.testing.assert_almost_equal(omega_orig, omega)
  
              # In the following lines, implicit loop on k (subjects)
              # Extract y and u
              y[:, :p] = omega[:p, p, :].T
              y[:, p:] = omega[p + 1 :, p, :].T
  
              u[:, :p] = emp_covs[:p, p, :].T
              u[:, p:] = emp_covs[p + 1 :, p, :].T
  
              for m in range(n_features - 1):
                  # Coordinate descent on y
  
                  # T(k) -> n_samples[k]
                  # v(k) -> emp_covs[p, p, k]
                  # h_22(k) -> W_inv[m, m, k]
                  # h_12(k) -> W_inv[:m, m, k],  W_inv[m+1:, m, k]
                  # y_1(k) -> y[k, :m], y[k, m+1:]
                  # u_2(k) -> u[k, m]
                  h_12[:, :m] = W_inv[:m, m, :].T
                  h_12[:, m:] = W_inv[m + 1 :, m, :].T
                  y_1[:, :m] = y[:, :m]
                  y_1[:, m:] = y[:, m + 1 :]
  
                  c[:] = -n_samples * (
                      emp_covs[p, p, :] * (h_12 * y_1).sum(axis=1)
                      + u[:, m]
                  )
                  c2 = np.sqrt(np.dot(c, c))
  
                  # x -> y[:][m]
                  if c2 <= alpha:
                      y[:, m] = 0  # x* = 0
                  else:
                      # q(k) -> T(k) * v(k) * h_22(k)
                      # \lambda -> gamma   (lambda is a Python keyword)
                      q[:] = (
                          n_samples * emp_covs[p, p, :] * W_inv[m, m, :]
                      )
                      if debug:
                          assert np.all(q > 0)
                      # x* = \lambda* diag(1 + \lambda q)^{-1} c
  
                      # Newton-Raphson loop. Loosely based on Scipy's.
                      # Tolerance does not seem to be important for numerical
                      # stability (tolerance of 1e-2 works) but has an effect on
                      # overall convergence rate (the tighter the better.)
  
                      gamma = 0.0  # initial value
                      # Precompute some quantities
                      cc = c * c
                      two_ccq = 2.0 * cc * q
                      for _ in itertools.repeat(None, 100):
                          # Function whose zero must be determined (fval) and
                          # its derivative (fder).
                          # Written inplace to save some function calls.
                          aq = 1.0 + gamma * q
                          aq2 = aq * aq
                          fder = (two_ccq / (aq2 * aq)).sum()
  
                          if fder == 0:
                              msg = "derivative was zero."
                              warnings.warn(msg, RuntimeWarning)
                              break
                          fval = -(alpha2 - (cc / aq2).sum()) / fder
                          gamma = fval + gamma
                          if abs(fval) < 1.5e-8:
                              break
  
                      if abs(fval) > 0.1:
                          warnings.warn(
                              "Newton-Raphson step did not converge.\n"
                              "This may indicate a badly conditioned "
                              "system."
                          )
  
                      if debug:
                          assert gamma >= 0.0, gamma
                      y[:, m] = (gamma * c) / aq  # x*
  
              # Copy back y in omega (column and row)
              omega[:p, p, :] = y[:, :p].T
              omega[p + 1 :, p, :] = y[:, p:].T
              omega[p, :p, :] = y[:, :p].T
              omega[p, p + 1 :, :] = y[:, p:].T
  
              for k in range(n_subjects):
                  omega[p, p, k] = 1.0 / emp_covs[p, p, k] + np.dot(
                      np.dot(y[k, :], W_inv[..., k]), y[k, :]
                  )
  
                  if debug:
                      assert is_spd(omega[..., k])
  
          if probe_function is not None:
              if probe_function(
                  emp_covs,
                  n_samples,
                  alpha,
                  max_iter,
                  tol,
                  n,
                  omega,
                  omega_old,
              ):
                  probe_interrupted = True
                  logger.log(
                      "probe_function interrupted loop",
                      verbose=verbose,
                      msg_level=2,
                  )
                  break
  
          # Compute max of variation
          omega_old -= omega
          omega_old = abs(omega_old)
          max_norm = omega_old.max()
  
          if tol is not None and max_norm < tol:
              logger.log(
                  f"tolerance reached at iteration number {n + 1:d}: "
                  f"{max_norm:.3e}",
                  verbose=verbose,
              )
              tolerance_reached = True
              break
  
      if (
          tol is not None
          and not tolerance_reached
          and not probe_interrupted
      ):
          warnings.warn(
              "Maximum number of iterations reached without getting "
              "to the requested tolerance level."
          )
  
      return omega
  
imports: import collections.abc
  import itertools
  import operator
  import warnings
  import numpy
  import scipy.linalg
  from joblib import Memory, Parallel, delayed
  from sklearn.base import BaseEstimator
  from sklearn.covariance import empirical_covariance
  from sklearn.model_selection import check_cv
  from sklearn.utils.extmath import fast_logdet
  from .._utils import CacheMixin, logger
  from .._utils.extmath import is_spd
label_desc: Use image node_id 5 for calling a global function with example usage: _group_sparse_covariance(emp_covs, n_samples, alpha, max_iter, tol, precisions_init, probe_function, verbose, debug) and returns: omega