code: def mean_ale(
      X=None, tol=10e-7, maxiter=50, sample_weight=None, covmats=None
  ):
      """AJD-based log-Euclidean (ALE) mean of SPD matrices.
  
      Return the mean of a set of SPD matrices using the approximate joint
      diagonalization (AJD) based log-Euclidean (ALE) mean [1]_.
  
      Parameters
      ----------
      X : ndarray, shape (n_matrices, n, n)
          Set of SPD matrices.
      tol : float, default=10e-7
          The tolerance to stop the gradient descent.
      maxiter : int, default=50
          The maximum number of iterations.
      sample_weight : None | ndarray, shape (n_matrices,), default=None
          Weights for each matrix. If None, it uses equal weights.
  
      Returns
      -------
      M : ndarray, shape (n, n)
          ALE mean.
  
      Notes
      -----
      .. versionadded:: 0.2.4
  
      See Also
      --------
      mean_covariance
  
      References
      ----------
      .. [1] `Approximate Joint Diagonalization and Geometric Mean of Symmetric
          Positive Definite Matrices
          <https://arxiv.org/abs/1505.07343>`_
          M. Congedo, B. Afsari, A. Barachant, M. Moakher. PLOS ONE, 2015
      """
      X = _deprecate_covmats(covmats, X)
      n_matrices, n, _ = X.shape
      sample_weight = check_weights(sample_weight, n_matrices)
  
      # init with AJD
      B = ajd_pham(X)[0]
  
      eye_n = np.eye(n)
      crit = np.inf
      for _ in range(maxiter):
          J = np.einsum(
              "a,abc->bc", sample_weight, logm(B @ X @ B.conj().T)
          )
          delta = np.real(np.diag(expm(J)))
          B = (np.abs(delta) ** -0.5)[:, np.newaxis] * B
  
          crit = distance_riemann(eye_n, np.diag(delta))
          if crit <= tol:
              break
      else:
          warnings.warn("Convergence not reached")
  
      J = np.einsum(
          "a,abc->bc", sample_weight, logm(B @ X @ B.conj().T)
      )
      A = np.linalg.inv(B)
      M = A @ expm(J) @ A.conj().T
      return M
  
imports: from copy import deepcopy
  import numpy
  import warnings
  from .ajd import ajd_pham
  from .base import sqrtm, invsqrtm, logm, expm, powm
  from .distance import distance_riemann
  from .geodesic import geodesic_riemann
  from .utils import check_weights
label_desc: Use image node_id 2 for calling a global function with example usage: mean_ale(X, tol, maxiter, sample_weight, covmats) and returns: M