code: def cov_nearest_factor_homog(cov, rank):
      """
      Approximate an arbitrary square matrix with a factor-structured
      matrix of the form k*I + XX'.
  
      Parameters
      ----------
      cov : array_like
          The input array, must be square but need not be positive
          semidefinite
      rank : int
          The rank of the fitted factor structure
  
      Returns
      -------
      A FactoredPSDMatrix instance containing the fitted matrix
  
      Notes
      -----
      This routine is useful if one has an estimated covariance matrix
      that is not SPD, and the ultimate goal is to estimate the inverse,
      square root, or inverse square root of the true covariance
      matrix. The factor structure allows these tasks to be performed
      without constructing any n x n matrices.
  
      The calculations use the fact that if k is known, then X can be
      determined from the eigen-decomposition of cov - k*I, which can
      in turn be easily obtained form the eigen-decomposition of `cov`.
      Thus the problem can be reduced to a 1-dimensional search for k
      that does not require repeated eigen-decompositions.
  
      If the input matrix is sparse, then cov - k*I is also sparse, so
      the eigen-decomposition can be done efficiently using sparse
      routines.
  
      The one-dimensional search for the optimal value of k is not
      convex, so a local minimum could be obtained.
  
      Examples
      --------
      Hard thresholding a covariance matrix may result in a matrix that
      is not positive semidefinite.  We can approximate a hard
      thresholded covariance matrix with a PSD matrix as follows:
  
      >>> import numpy as np
      >>> np.random.seed(1234)
      >>> b = 1.5 - np.random.rand(10, 1)
      >>> x = np.random.randn(100,1).dot(b.T) + np.random.randn(100,10)
      >>> cov = np.cov(x)
      >>> cov = cov * (np.abs(cov) >= 0.3)
      >>> rslt = cov_nearest_factor_homog(cov, 3)
      """
  
      m, n = cov.shape
  
      Q, Lambda, _ = svds(cov, rank)
  
      if sparse.issparse(cov):
          QSQ = np.dot(Q.T, cov.dot(Q))
          ts = cov.diagonal().sum()
          tss = cov.dot(cov).diagonal().sum()
      else:
          QSQ = np.dot(Q.T, np.dot(cov, Q))
          ts = np.trace(cov)
          tss = np.trace(np.dot(cov, cov))
  
      def fun(k):
          Lambda_t = Lambda - k
          v = tss + m * (k**2) + np.sum(Lambda_t**2) - 2 * k * ts
          v += 2 * k * np.sum(Lambda_t) - 2 * np.sum(
              np.diag(QSQ) * Lambda_t
          )
          return v
  
      # Get the optimal decomposition
      k_opt = fminbound(fun, 0, 1e5)
      Lambda_opt = Lambda - k_opt
      fac_opt = Q * np.sqrt(Lambda_opt)
  
      diag = k_opt * np.ones(
          m, dtype=np.float64
      )  # - (fac_opt**2).sum(1)
      return FactoredPSDMatrix(diag, fac_opt)
  
imports: import numpy
  import scipy.sparse
  from scipy.sparse.linalg import svds
  from scipy.optimize import fminbound
  import warnings
  from statsmodels.tools.tools import Bunch
  from statsmodels.tools.sm_exceptions import IterationLimitWarning, iteration_limit_doc
label_desc: Use image node_id 9 for calling a global function with example usage: cov_nearest_factor_homog(cov, rank) and returns: FactoredPSDMatrix, v