_truncated_svd.py
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"""Truncated SVD for sparse matrices, aka latent semantic analysis (LSA).
"""
# Author: Lars Buitinck
# Olivier Grisel <olivier.grisel@ensta.org>
# Michael Becker <mike@beckerfuffle.com>
# 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.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 <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 (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, default=None
Used during randomized svd. Pass an int for reproducible results across
multiple function calls.
See :term:`Glossary <random_state>`.
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_ : array, shape (n_components,)
The variance of the training samples transformed by a projection to
each component.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
singular_values_ : array, 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
>>> from sklearn.random_projection import sparse_random_matrix
>>> 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 LSI model on training data X.
Parameters
----------
X : {array-like, sparse matrix}, 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 LSI model to X and perform dimensionality reduction on X.
Parameters
----------
X : {array-like, sparse matrix}, shape (n_samples, n_features)
Training data.
y : Ignored
Returns
-------
X_new : array, 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":
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 = U * 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
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}, 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', 'csc'])
check_is_fitted(self)
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_)