content: """Truncated SVD for sparse matrices, aka latent semantic analysis (LSA).
  """
  
  # Author: Lars Buitinck <L.J.Buitinck@uva.nl>
  #         Olivier Grisel <olivier.grisel@ensta.org>
  #         Michael Becker <mike@beckerfuffle.com>
  # License: 3-clause BSD.
  
  import numpy as np
  import scipy.sparse as sp
  
  try:
      from scipy.sparse.linalg import svds
  except ImportError:
      from ..utils.arpack import svds
  
  from ..base import BaseEstimator, TransformerMixin
  from ..utils import check_array, as_float_array, check_random_state
  from ..utils.extmath import randomized_svd, safe_sparse_dot, svd_flip
  from ..utils.sparsefuncs import mean_variance_axis
  
  __all__ = ["TruncatedSVD"]
  
  
  class TruncatedSVD(BaseEstimator, TransformerMixin):
      """Dimensionality reduction using truncated SVD (aka LSA).
  
      This transformer performs linear dimensionality reduction by means of
      truncated singular value decomposition (SVD). It is very similar to PCA,
      but operates on sample vectors directly, instead of on a covariance matrix.
      This means it can work with scipy.sparse matrices efficiently.
  
      In particular, truncated SVD works on term count/tf-idf matrices as
      returned by the vectorizers in sklearn.feature_extraction.text. In that
      context, it is known as latent semantic analysis (LSA).
  
      This estimator supports two algorithm: a fast randomized SVD solver, and
      a "naive" algorithm that uses ARPACK as an eigensolver on (X * X.T) or
      (X.T * X), whichever is more efficient.
  
      Read more in the :ref:`User Guide <LSA>`.
  
      Parameters
      ----------
      n_components : int, default = 2
          Desired dimensionality of output data.
          Must be strictly less than the number of features.
          The default value is useful for visualisation. For LSA, a value of
          100 is recommended.
  
      algorithm : string, default = "randomized"
          SVD solver to use. Either "arpack" for the ARPACK wrapper in SciPy
          (scipy.sparse.linalg.svds), or "randomized" for the randomized
          algorithm due to Halko (2009).
  
      n_iter : int, optional
          Number of iterations for randomized SVD solver. Not used by ARPACK.
  
      random_state : int or RandomState, optional
          (Seed for) pseudo-random number generator. If not given, the
          numpy.random singleton is used.
  
      tol : float, optional
          Tolerance for ARPACK. 0 means machine precision. Ignored by randomized
          SVD solver.
  
      Attributes
      ----------
      components_ : array, shape (n_components, n_features)
  
      explained_variance_ratio_ : array, [n_components]
          Percentage of variance explained by each of the selected components.
  
      explained_variance_ : array, [n_components]
          The variance of the training samples transformed by a projection to
          each component.
  
      Examples
      --------
      >>> from sklearn.decomposition import TruncatedSVD
      >>> from sklearn.random_projection import sparse_random_matrix
      >>> X = sparse_random_matrix(100, 100, density=0.01, random_state=42)
      >>> svd = TruncatedSVD(n_components=5, random_state=42)
      >>> svd.fit(X) # doctest: +NORMALIZE_WHITESPACE
      TruncatedSVD(algorithm='randomized', n_components=5, n_iter=5,
              random_state=42, tol=0.0)
      >>> print(svd.explained_variance_ratio_) # doctest: +ELLIPSIS
      [ 0.07825... 0.05528... 0.05445... 0.04997... 0.04134...]
      >>> print(svd.explained_variance_ratio_.sum()) # doctest: +ELLIPSIS
      0.27930...
  
      See also
      --------
      PCA
      RandomizedPCA
  
      References
      ----------
      Finding structure with randomness: Stochastic algorithms for constructing
      approximate matrix decompositions
      Halko, et al., 2009 (arXiv:909) http://arxiv.org/pdf/0909.4061
  
      Notes
      -----
      SVD suffers from a problem called "sign indeterminancy", which means the
      sign of the ``components_`` and the output from transform depend on the
      algorithm and random state. To work around this, fit instances of this
      class to data once, then keep the instance around to do transformations.
  
      """
      def __init__(self, n_components=2, algorithm="randomized", n_iter=5,
                   random_state=None, tol=0.):
          self.algorithm = algorithm
          self.n_components = n_components
          self.n_iter = n_iter
          self.random_state = random_state
          self.tol = tol
  
      def fit(self, X, y=None):
          """Fit LSI model on training data X.
  
          Parameters
          ----------
          X : {array-like, sparse matrix}, shape (n_samples, n_features)
              Training data.
  
          Returns
          -------
          self : object
              Returns the transformer object.
          """
          self.fit_transform(X)
          return self
  
      def fit_transform(self, X, y=None):
          """Fit LSI model to X and perform dimensionality reduction on X.
  
          Parameters
          ----------
          X : {array-like, sparse matrix}, shape (n_samples, n_features)
              Training data.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Reduced version of X. This will always be a dense array.
          """
          X = as_float_array(X, copy=False)
          random_state = check_random_state(self.random_state)
  
          # If sparse and not csr or csc, convert to csr
          if sp.issparse(X) and X.getformat() not in ["csr", "csc"]:
              X = X.tocsr()
  
          if self.algorithm == "arpack":
              U, Sigma, VT = svds(X, k=self.n_components, tol=self.tol)
              # svds doesn't abide by scipy.linalg.svd/randomized_svd
              # conventions, so reverse its outputs.
              Sigma = Sigma[::-1]
              U, VT = svd_flip(U[:, ::-1], VT[::-1])
  
          elif self.algorithm == "randomized":
              k = self.n_components
              n_features = X.shape[1]
              if k >= n_features:
                  raise ValueError("n_components must be < n_features;"
                                   " got %d >= %d" % (k, n_features))
              U, Sigma, VT = randomized_svd(X, self.n_components,
                                            n_iter=self.n_iter,
                                            random_state=random_state)
          else:
              raise ValueError("unknown algorithm %r" % self.algorithm)
  
          self.components_ = VT
  
          # Calculate explained variance & explained variance ratio
          X_transformed = np.dot(U, np.diag(Sigma))
          self.explained_variance_ = exp_var = np.var(X_transformed, axis=0)
          if sp.issparse(X):
              _, full_var = mean_variance_axis(X, axis=0)
              full_var = full_var.sum()
          else:
              full_var = np.var(X, axis=0).sum()
          self.explained_variance_ratio_ = exp_var / full_var
          return X_transformed
  
      def transform(self, X):
          """Perform dimensionality reduction on X.
  
          Parameters
          ----------
          X : {array-like, sparse matrix}, shape (n_samples, n_features)
              New data.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Reduced version of X. This will always be a dense array.
          """
          X = check_array(X, accept_sparse='csr')
          return safe_sparse_dot(X, self.components_.T)
  
      def inverse_transform(self, X):
          """Transform X back to its original space.
  
          Returns an array X_original whose transform would be X.
  
          Parameters
          ----------
          X : array-like, shape (n_samples, n_components)
              New data.
  
          Returns
          -------
          X_original : array, shape (n_samples, n_features)
              Note that this is always a dense array.
          """
          X = check_array(X)
          return np.dot(X, self.components_)
  