"""Truncated SVD for sparse matrices, aka latent semantic analysis (LSA). """ # Author: Lars Buitinck # Olivier Grisel # Michael Becker # License: 3-clause BSD. import numpy as np import scipy.sparse as sp from scipy.sparse.linalg import svds from ..base import BaseEstimator, TransformerMixin from ..utils import check_array, check_random_state from ..utils._arpack import _init_arpack_v0 from ..utils.extmath import randomized_svd, safe_sparse_dot, svd_flip from ..utils.sparsefuncs import mean_variance_axis from ..utils.validation import _deprecate_positional_args from ..utils.validation import check_is_fitted __all__ = ["TruncatedSVD"] class TruncatedSVD(TransformerMixin, BaseEstimator): """Dimensionality reduction using truncated SVD (aka LSA). This transformer performs linear dimensionality reduction by means of truncated singular value decomposition (SVD). Contrary to PCA, this estimator does not center the data before computing the singular value decomposition. This means it can work with sparse matrices efficiently. In particular, truncated SVD works on term count/tf-idf matrices as returned by the vectorizers in :mod:`sklearn.feature_extraction.text`. In that context, it is known as latent semantic analysis (LSA). This estimator supports two algorithms: 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 `. 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 : {'arpack', 'randomized'}, 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, default=5 Number of iterations for randomized SVD solver. Not used by ARPACK. The default is larger than the default in :func:`~sklearn.utils.extmath.randomized_svd` to handle sparse matrices that may have large slowly decaying spectrum. random_state : int, RandomState instance or None, default=None Used during randomized svd. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. tol : float, default=0. Tolerance for ARPACK. 0 means machine precision. Ignored by randomized SVD solver. Attributes ---------- components_ : ndarray of shape (n_components, n_features) explained_variance_ : ndarray of shape (n_components,) The variance of the training samples transformed by a projection to each component. explained_variance_ratio_ : ndarray of shape (n_components,) Percentage of variance explained by each of the selected components. singular_values_ : ndarray od shape (n_components,) The singular values corresponding to each of the selected components. The singular values are equal to the 2-norms of the ``n_components`` variables in the lower-dimensional space. Examples -------- >>> from sklearn.decomposition import TruncatedSVD >>> from scipy.sparse import random as sparse_random >>> X = sparse_random(100, 100, density=0.01, format='csr', ... random_state=42) >>> svd = TruncatedSVD(n_components=5, n_iter=7, random_state=42) >>> svd.fit(X) TruncatedSVD(n_components=5, n_iter=7, random_state=42) >>> print(svd.explained_variance_ratio_) [0.0646... 0.0633... 0.0639... 0.0535... 0.0406...] >>> print(svd.explained_variance_ratio_.sum()) 0.286... >>> print(svd.singular_values_) [1.553... 1.512... 1.510... 1.370... 1.199...] See Also -------- PCA References ---------- Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions Halko, et al., 2009 (arXiv:909) https://arxiv.org/pdf/0909.4061.pdf Notes ----- SVD suffers from a problem called "sign indeterminacy", 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. """ @_deprecate_positional_args 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 model on training data X. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) Training data. y : Ignored Returns ------- self : object Returns the transformer object. """ self.fit_transform(X) return self def fit_transform(self, X, y=None): """Fit model to X and perform dimensionality reduction on X. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) Training data. y : Ignored Returns ------- X_new : ndarray of shape (n_samples, n_components) Reduced version of X. This will always be a dense array. """ X = self._validate_data(X, accept_sparse=['csr', 'csc'], ensure_min_features=2) random_state = check_random_state(self.random_state) if self.algorithm == "arpack": v0 = _init_arpack_v0(min(X.shape), random_state) U, Sigma, VT = svds(X, k=self.n_components, tol=self.tol, v0=v0) # 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 # As a result of the SVD approximation error on X ~ U @ Sigma @ V.T, # X @ V is not the same as U @ Sigma if self.algorithm == "randomized" or \ (self.algorithm == "arpack" and self.tol > 0): X_transformed = safe_sparse_dot(X, self.components_.T) else: X_transformed = U * Sigma # Calculate explained variance & explained variance ratio 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 self.singular_values_ = Sigma # Store the singular values. return X_transformed def transform(self, X): """Perform dimensionality reduction on X. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) New data. Returns ------- X_new : ndarray of shape (n_samples, n_components) Reduced version of X. This will always be a dense array. """ check_is_fitted(self) X = self._validate_data(X, accept_sparse=['csr', 'csc'], reset=False) 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 of shape (n_samples, n_components) New data. Returns ------- X_original : ndarray of shape (n_samples, n_features) Note that this is always a dense array. """ X = check_array(X) return np.dot(X, self.components_) def _more_tags(self): return {'preserves_dtype': [np.float64, np.float32]}