657 lines
23 KiB
Python
657 lines
23 KiB
Python
|
# -*- coding: utf8
|
||
|
|
"""Random Projection transformers.
|
||
|
|
|
||
|
|
Random Projections are a simple and computationally efficient way to
|
||
|
|
reduce the dimensionality of the data by trading a controlled amount
|
||
|
|
of accuracy (as additional variance) for faster processing times and
|
||
|
|
smaller model sizes.
|
||
|
|
|
||
|
|
The dimensions and distribution of Random Projections matrices are
|
||
|
|
controlled so as to preserve the pairwise distances between any two
|
||
|
|
samples of the dataset.
|
||
|
|
|
||
|
|
The main theoretical result behind the efficiency of random projection is the
|
||
|
|
`Johnson-Lindenstrauss lemma (quoting Wikipedia)
|
||
|
|
<https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma>`_:
|
||
|
|
|
||
|
|
In mathematics, the Johnson-Lindenstrauss lemma is a result
|
||
|
|
concerning low-distortion embeddings of points from high-dimensional
|
||
|
|
into low-dimensional Euclidean space. The lemma states that a small set
|
||
|
|
of points in a high-dimensional space can be embedded into a space of
|
||
|
|
much lower dimension in such a way that distances between the points are
|
||
|
|
nearly preserved. The map used for the embedding is at least Lipschitz,
|
||
|
|
and can even be taken to be an orthogonal projection.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# Authors: Olivier Grisel <olivier.grisel@ensta.org>,
|
||
|
|
# Arnaud Joly <a.joly@ulg.ac.be>
|
||
|
|
# License: BSD 3 clause
|
||
|
|
|
||
|
|
import warnings
|
||
|
|
from abc import ABCMeta, abstractmethod
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
import scipy.sparse as sp
|
||
|
|
|
||
|
|
from .base import BaseEstimator, TransformerMixin
|
||
|
|
|
||
|
|
from .utils import check_random_state
|
||
|
|
from .utils.extmath import safe_sparse_dot
|
||
|
|
from .utils.random import sample_without_replacement
|
||
|
|
from .utils.validation import check_array, check_is_fitted
|
||
|
|
from .utils.validation import _deprecate_positional_args
|
||
|
|
from .exceptions import DataDimensionalityWarning
|
||
|
|
|
||
|
|
|
||
|
|
__all__ = ["SparseRandomProjection",
|
||
|
|
"GaussianRandomProjection",
|
||
|
|
"johnson_lindenstrauss_min_dim"]
|
||
|
|
|
||
|
|
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def johnson_lindenstrauss_min_dim(n_samples, *, eps=0.1):
|
||
|
|
"""Find a 'safe' number of components to randomly project to.
|
||
|
|
|
||
|
|
The distortion introduced by a random projection `p` only changes the
|
||
|
|
distance between two points by a factor (1 +- eps) in an euclidean space
|
||
|
|
with good probability. The projection `p` is an eps-embedding as defined
|
||
|
|
by:
|
||
|
|
|
||
|
|
(1 - eps) ||u - v||^2 < ||p(u) - p(v)||^2 < (1 + eps) ||u - v||^2
|
||
|
|
|
||
|
|
Where u and v are any rows taken from a dataset of shape (n_samples,
|
||
|
|
n_features), eps is in ]0, 1[ and p is a projection by a random Gaussian
|
||
|
|
N(0, 1) matrix of shape (n_components, n_features) (or a sparse
|
||
|
|
Achlioptas matrix).
|
||
|
|
|
||
|
|
The minimum number of components to guarantee the eps-embedding is
|
||
|
|
given by:
|
||
|
|
|
||
|
|
n_components >= 4 log(n_samples) / (eps^2 / 2 - eps^3 / 3)
|
||
|
|
|
||
|
|
Note that the number of dimensions is independent of the original
|
||
|
|
number of features but instead depends on the size of the dataset:
|
||
|
|
the larger the dataset, the higher is the minimal dimensionality of
|
||
|
|
an eps-embedding.
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <johnson_lindenstrauss>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_samples : int or array-like of int
|
||
|
|
Number of samples that should be a integer greater than 0. If an array
|
||
|
|
is given, it will compute a safe number of components array-wise.
|
||
|
|
|
||
|
|
eps : float or ndarray of shape (n_components,), dtype=float, \
|
||
|
|
default=0.1
|
||
|
|
Maximum distortion rate in the range (0,1 ) as defined by the
|
||
|
|
Johnson-Lindenstrauss lemma. If an array is given, it will compute a
|
||
|
|
safe number of components array-wise.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
n_components : int or ndarray of int
|
||
|
|
The minimal number of components to guarantee with good probability
|
||
|
|
an eps-embedding with n_samples.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
|
||
|
|
>>> johnson_lindenstrauss_min_dim(1e6, eps=0.5)
|
||
|
|
663
|
||
|
|
|
||
|
|
>>> johnson_lindenstrauss_min_dim(1e6, eps=[0.5, 0.1, 0.01])
|
||
|
|
array([ 663, 11841, 1112658])
|
||
|
|
|
||
|
|
>>> johnson_lindenstrauss_min_dim([1e4, 1e5, 1e6], eps=0.1)
|
||
|
|
array([ 7894, 9868, 11841])
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma
|
||
|
|
|
||
|
|
.. [2] Sanjoy Dasgupta and Anupam Gupta, 1999,
|
||
|
|
"An elementary proof of the Johnson-Lindenstrauss Lemma."
|
||
|
|
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.3654
|
||
|
|
|
||
|
|
"""
|
||
|
|
eps = np.asarray(eps)
|
||
|
|
n_samples = np.asarray(n_samples)
|
||
|
|
|
||
|
|
if np.any(eps <= 0.0) or np.any(eps >= 1):
|
||
|
|
raise ValueError(
|
||
|
|
"The JL bound is defined for eps in ]0, 1[, got %r" % eps)
|
||
|
|
|
||
|
|
if np.any(n_samples) <= 0:
|
||
|
|
raise ValueError(
|
||
|
|
"The JL bound is defined for n_samples greater than zero, got %r"
|
||
|
|
% n_samples)
|
||
|
|
|
||
|
|
denominator = (eps ** 2 / 2) - (eps ** 3 / 3)
|
||
|
|
return (4 * np.log(n_samples) / denominator).astype(int)
|
||
|
|
|
||
|
|
|
||
|
|
def _check_density(density, n_features):
|
||
|
|
"""Factorize density check according to Li et al."""
|
||
|
|
if density == 'auto':
|
||
|
|
density = 1 / np.sqrt(n_features)
|
||
|
|
|
||
|
|
elif density <= 0 or density > 1:
|
||
|
|
raise ValueError("Expected density in range ]0, 1], got: %r"
|
||
|
|
% density)
|
||
|
|
return density
|
||
|
|
|
||
|
|
|
||
|
|
def _check_input_size(n_components, n_features):
|
||
|
|
"""Factorize argument checking for random matrix generation."""
|
||
|
|
if n_components <= 0:
|
||
|
|
raise ValueError("n_components must be strictly positive, got %d" %
|
||
|
|
n_components)
|
||
|
|
if n_features <= 0:
|
||
|
|
raise ValueError("n_features must be strictly positive, got %d" %
|
||
|
|
n_features)
|
||
|
|
|
||
|
|
|
||
|
|
def _gaussian_random_matrix(n_components, n_features, random_state=None):
|
||
|
|
"""Generate a dense Gaussian random matrix.
|
||
|
|
|
||
|
|
The components of the random matrix are drawn from
|
||
|
|
|
||
|
|
N(0, 1.0 / n_components).
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <gaussian_random_matrix>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int,
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_features : int,
|
||
|
|
Dimensionality of the original source space.
|
||
|
|
|
||
|
|
random_state : int, RandomState instance or None, default=None
|
||
|
|
Controls the pseudo random number generator used to generate the matrix
|
||
|
|
at fit time.
|
||
|
|
Pass an int for reproducible output across multiple function calls.
|
||
|
|
See :term:`Glossary <random_state>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
components : ndarray of shape (n_components, n_features)
|
||
|
|
The generated Gaussian random matrix.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
GaussianRandomProjection
|
||
|
|
"""
|
||
|
|
_check_input_size(n_components, n_features)
|
||
|
|
rng = check_random_state(random_state)
|
||
|
|
components = rng.normal(loc=0.0,
|
||
|
|
scale=1.0 / np.sqrt(n_components),
|
||
|
|
size=(n_components, n_features))
|
||
|
|
return components
|
||
|
|
|
||
|
|
|
||
|
|
def _sparse_random_matrix(n_components, n_features, density='auto',
|
||
|
|
random_state=None):
|
||
|
|
"""Generalized Achlioptas random sparse matrix for random projection.
|
||
|
|
|
||
|
|
Setting density to 1 / 3 will yield the original matrix by Dimitris
|
||
|
|
Achlioptas while setting a lower value will yield the generalization
|
||
|
|
by Ping Li et al.
|
||
|
|
|
||
|
|
If we note :math:`s = 1 / density`, the components of the random matrix are
|
||
|
|
drawn from:
|
||
|
|
|
||
|
|
- -sqrt(s) / sqrt(n_components) with probability 1 / 2s
|
||
|
|
- 0 with probability 1 - 1 / s
|
||
|
|
- +sqrt(s) / sqrt(n_components) with probability 1 / 2s
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <sparse_random_matrix>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int,
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_features : int,
|
||
|
|
Dimensionality of the original source space.
|
||
|
|
|
||
|
|
density : float or 'auto', default='auto'
|
||
|
|
Ratio of non-zero component in the random projection matrix in the
|
||
|
|
range `(0, 1]`
|
||
|
|
|
||
|
|
If density = 'auto', the value is set to the minimum density
|
||
|
|
as recommended by Ping Li et al.: 1 / sqrt(n_features).
|
||
|
|
|
||
|
|
Use density = 1 / 3.0 if you want to reproduce the results from
|
||
|
|
Achlioptas, 2001.
|
||
|
|
|
||
|
|
random_state : int, RandomState instance or None, default=None
|
||
|
|
Controls the pseudo random number generator used to generate the matrix
|
||
|
|
at fit time.
|
||
|
|
Pass an int for reproducible output across multiple function calls.
|
||
|
|
See :term:`Glossary <random_state>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
components : {ndarray, sparse matrix} of shape (n_components, n_features)
|
||
|
|
The generated Gaussian random matrix. Sparse matrix will be of CSR
|
||
|
|
format.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
SparseRandomProjection
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] Ping Li, T. Hastie and K. W. Church, 2006,
|
||
|
|
"Very Sparse Random Projections".
|
||
|
|
https://web.stanford.edu/~hastie/Papers/Ping/KDD06_rp.pdf
|
||
|
|
|
||
|
|
.. [2] D. Achlioptas, 2001, "Database-friendly random projections",
|
||
|
|
http://www.cs.ucsc.edu/~optas/papers/jl.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_input_size(n_components, n_features)
|
||
|
|
density = _check_density(density, n_features)
|
||
|
|
rng = check_random_state(random_state)
|
||
|
|
|
||
|
|
if density == 1:
|
||
|
|
# skip index generation if totally dense
|
||
|
|
components = rng.binomial(1, 0.5, (n_components, n_features)) * 2 - 1
|
||
|
|
return 1 / np.sqrt(n_components) * components
|
||
|
|
|
||
|
|
else:
|
||
|
|
# Generate location of non zero elements
|
||
|
|
indices = []
|
||
|
|
offset = 0
|
||
|
|
indptr = [offset]
|
||
|
|
for _ in range(n_components):
|
||
|
|
# find the indices of the non-zero components for row i
|
||
|
|
n_nonzero_i = rng.binomial(n_features, density)
|
||
|
|
indices_i = sample_without_replacement(n_features, n_nonzero_i,
|
||
|
|
random_state=rng)
|
||
|
|
indices.append(indices_i)
|
||
|
|
offset += n_nonzero_i
|
||
|
|
indptr.append(offset)
|
||
|
|
|
||
|
|
indices = np.concatenate(indices)
|
||
|
|
|
||
|
|
# Among non zero components the probability of the sign is 50%/50%
|
||
|
|
data = rng.binomial(1, 0.5, size=np.size(indices)) * 2 - 1
|
||
|
|
|
||
|
|
# build the CSR structure by concatenating the rows
|
||
|
|
components = sp.csr_matrix((data, indices, indptr),
|
||
|
|
shape=(n_components, n_features))
|
||
|
|
|
||
|
|
return np.sqrt(1 / density) / np.sqrt(n_components) * components
|
||
|
|
|
||
|
|
|
||
|
|
class BaseRandomProjection(TransformerMixin, BaseEstimator, metaclass=ABCMeta):
|
||
|
|
"""Base class for random projections.
|
||
|
|
|
||
|
|
Warning: This class should not be used directly.
|
||
|
|
Use derived classes instead.
|
||
|
|
"""
|
||
|
|
|
||
|
|
@abstractmethod
|
||
|
|
def __init__(self, n_components='auto', *, eps=0.1, dense_output=False,
|
||
|
|
random_state=None):
|
||
|
|
self.n_components = n_components
|
||
|
|
self.eps = eps
|
||
|
|
self.dense_output = dense_output
|
||
|
|
self.random_state = random_state
|
||
|
|
|
||
|
|
@abstractmethod
|
||
|
|
def _make_random_matrix(self, n_components, n_features):
|
||
|
|
"""Generate the random projection matrix.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int,
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_features : int,
|
||
|
|
Dimensionality of the original source space.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
components : {ndarray, sparse matrix} of shape \
|
||
|
|
(n_components, n_features)
|
||
|
|
The generated random matrix. Sparse matrix will be of CSR format.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Generate a sparse random projection matrix.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
||
|
|
Training set: only the shape is used to find optimal random
|
||
|
|
matrix dimensions based on the theory referenced in the
|
||
|
|
afore mentioned papers.
|
||
|
|
|
||
|
|
y
|
||
|
|
Ignored
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self
|
||
|
|
|
||
|
|
"""
|
||
|
|
X = self._validate_data(X, accept_sparse=['csr', 'csc'])
|
||
|
|
|
||
|
|
n_samples, n_features = X.shape
|
||
|
|
|
||
|
|
if self.n_components == 'auto':
|
||
|
|
self.n_components_ = johnson_lindenstrauss_min_dim(
|
||
|
|
n_samples=n_samples, eps=self.eps)
|
||
|
|
|
||
|
|
if self.n_components_ <= 0:
|
||
|
|
raise ValueError(
|
||
|
|
'eps=%f and n_samples=%d lead to a target dimension of '
|
||
|
|
'%d which is invalid' % (
|
||
|
|
self.eps, n_samples, self.n_components_))
|
||
|
|
|
||
|
|
elif self.n_components_ > n_features:
|
||
|
|
raise ValueError(
|
||
|
|
'eps=%f and n_samples=%d lead to a target dimension of '
|
||
|
|
'%d which is larger than the original space with '
|
||
|
|
'n_features=%d' % (self.eps, n_samples, self.n_components_,
|
||
|
|
n_features))
|
||
|
|
else:
|
||
|
|
if self.n_components <= 0:
|
||
|
|
raise ValueError("n_components must be greater than 0, got %s"
|
||
|
|
% self.n_components)
|
||
|
|
|
||
|
|
elif self.n_components > n_features:
|
||
|
|
warnings.warn(
|
||
|
|
"The number of components is higher than the number of"
|
||
|
|
" features: n_features < n_components (%s < %s)."
|
||
|
|
"The dimensionality of the problem will not be reduced."
|
||
|
|
% (n_features, self.n_components),
|
||
|
|
DataDimensionalityWarning)
|
||
|
|
|
||
|
|
self.n_components_ = self.n_components
|
||
|
|
|
||
|
|
# Generate a projection matrix of size [n_components, n_features]
|
||
|
|
self.components_ = self._make_random_matrix(self.n_components_,
|
||
|
|
n_features)
|
||
|
|
|
||
|
|
# Check contract
|
||
|
|
assert self.components_.shape == (self.n_components_, n_features), (
|
||
|
|
'An error has occurred the self.components_ matrix has '
|
||
|
|
' not the proper shape.')
|
||
|
|
|
||
|
|
return self
|
||
|
|
|
||
|
|
def transform(self, X):
|
||
|
|
"""Project the data by using matrix product with the random matrix
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
||
|
|
The input data to project into a smaller dimensional space.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
X_new : {ndarray, sparse matrix} of shape (n_samples, n_components)
|
||
|
|
Projected array.
|
||
|
|
"""
|
||
|
|
X = check_array(X, accept_sparse=['csr', 'csc'])
|
||
|
|
|
||
|
|
check_is_fitted(self)
|
||
|
|
|
||
|
|
if X.shape[1] != self.components_.shape[1]:
|
||
|
|
raise ValueError(
|
||
|
|
'Impossible to perform projection:'
|
||
|
|
'X at fit stage had a different number of features. '
|
||
|
|
'(%s != %s)' % (X.shape[1], self.components_.shape[1]))
|
||
|
|
|
||
|
|
X_new = safe_sparse_dot(X, self.components_.T,
|
||
|
|
dense_output=self.dense_output)
|
||
|
|
return X_new
|
||
|
|
|
||
|
|
|
||
|
|
class GaussianRandomProjection(BaseRandomProjection):
|
||
|
|
"""Reduce dimensionality through Gaussian random projection.
|
||
|
|
|
||
|
|
The components of the random matrix are drawn from N(0, 1 / n_components).
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <gaussian_random_matrix>`.
|
||
|
|
|
||
|
|
.. versionadded:: 0.13
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int or 'auto', default='auto'
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_components can be automatically adjusted according to the
|
||
|
|
number of samples in the dataset and the bound given by the
|
||
|
|
Johnson-Lindenstrauss lemma. In that case the quality of the
|
||
|
|
embedding is controlled by the ``eps`` parameter.
|
||
|
|
|
||
|
|
It should be noted that Johnson-Lindenstrauss lemma can yield
|
||
|
|
very conservative estimated of the required number of components
|
||
|
|
as it makes no assumption on the structure of the dataset.
|
||
|
|
|
||
|
|
eps : float, default=0.1
|
||
|
|
Parameter to control the quality of the embedding according to
|
||
|
|
the Johnson-Lindenstrauss lemma when `n_components` is set to
|
||
|
|
'auto'. The value should be strictly positive.
|
||
|
|
|
||
|
|
Smaller values lead to better embedding and higher number of
|
||
|
|
dimensions (n_components) in the target projection space.
|
||
|
|
|
||
|
|
random_state : int, RandomState instance or None, default=None
|
||
|
|
Controls the pseudo random number generator used to generate the
|
||
|
|
projection matrix at fit time.
|
||
|
|
Pass an int for reproducible output across multiple function calls.
|
||
|
|
See :term:`Glossary <random_state>`.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_components_ : int
|
||
|
|
Concrete number of components computed when n_components="auto".
|
||
|
|
|
||
|
|
components_ : ndarray of shape (n_components, n_features)
|
||
|
|
Random matrix used for the projection.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from sklearn.random_projection import GaussianRandomProjection
|
||
|
|
>>> rng = np.random.RandomState(42)
|
||
|
|
>>> X = rng.rand(100, 10000)
|
||
|
|
>>> transformer = GaussianRandomProjection(random_state=rng)
|
||
|
|
>>> X_new = transformer.fit_transform(X)
|
||
|
|
>>> X_new.shape
|
||
|
|
(100, 3947)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
SparseRandomProjection
|
||
|
|
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def __init__(self, n_components='auto', *, eps=0.1, random_state=None):
|
||
|
|
super().__init__(
|
||
|
|
n_components=n_components,
|
||
|
|
eps=eps,
|
||
|
|
dense_output=True,
|
||
|
|
random_state=random_state)
|
||
|
|
|
||
|
|
def _make_random_matrix(self, n_components, n_features):
|
||
|
|
""" Generate the random projection matrix.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int,
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_features : int,
|
||
|
|
Dimensionality of the original source space.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
components : {ndarray, sparse matrix} of shape \
|
||
|
|
(n_components, n_features)
|
||
|
|
The generated random matrix. Sparse matrix will be of CSR format.
|
||
|
|
|
||
|
|
"""
|
||
|
|
random_state = check_random_state(self.random_state)
|
||
|
|
return _gaussian_random_matrix(n_components,
|
||
|
|
n_features,
|
||
|
|
random_state=random_state)
|
||
|
|
|
||
|
|
|
||
|
|
class SparseRandomProjection(BaseRandomProjection):
|
||
|
|
"""Reduce dimensionality through sparse random projection.
|
||
|
|
|
||
|
|
Sparse random matrix is an alternative to dense random
|
||
|
|
projection matrix that guarantees similar embedding quality while being
|
||
|
|
much more memory efficient and allowing faster computation of the
|
||
|
|
projected data.
|
||
|
|
|
||
|
|
If we note `s = 1 / density` the components of the random matrix are
|
||
|
|
drawn from:
|
||
|
|
|
||
|
|
- -sqrt(s) / sqrt(n_components) with probability 1 / 2s
|
||
|
|
- 0 with probability 1 - 1 / s
|
||
|
|
- +sqrt(s) / sqrt(n_components) with probability 1 / 2s
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <sparse_random_matrix>`.
|
||
|
|
|
||
|
|
.. versionadded:: 0.13
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int or 'auto', default='auto'
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_components can be automatically adjusted according to the
|
||
|
|
number of samples in the dataset and the bound given by the
|
||
|
|
Johnson-Lindenstrauss lemma. In that case the quality of the
|
||
|
|
embedding is controlled by the ``eps`` parameter.
|
||
|
|
|
||
|
|
It should be noted that Johnson-Lindenstrauss lemma can yield
|
||
|
|
very conservative estimated of the required number of components
|
||
|
|
as it makes no assumption on the structure of the dataset.
|
||
|
|
|
||
|
|
density : float or 'auto', default='auto'
|
||
|
|
Ratio in the range (0, 1] of non-zero component in the random
|
||
|
|
projection matrix.
|
||
|
|
|
||
|
|
If density = 'auto', the value is set to the minimum density
|
||
|
|
as recommended by Ping Li et al.: 1 / sqrt(n_features).
|
||
|
|
|
||
|
|
Use density = 1 / 3.0 if you want to reproduce the results from
|
||
|
|
Achlioptas, 2001.
|
||
|
|
|
||
|
|
eps : float, default=0.1
|
||
|
|
Parameter to control the quality of the embedding according to
|
||
|
|
the Johnson-Lindenstrauss lemma when n_components is set to
|
||
|
|
'auto'. This value should be strictly positive.
|
||
|
|
|
||
|
|
Smaller values lead to better embedding and higher number of
|
||
|
|
dimensions (n_components) in the target projection space.
|
||
|
|
|
||
|
|
dense_output : bool, default=False
|
||
|
|
If True, ensure that the output of the random projection is a
|
||
|
|
dense numpy array even if the input and random projection matrix
|
||
|
|
are both sparse. In practice, if the number of components is
|
||
|
|
small the number of zero components in the projected data will
|
||
|
|
be very small and it will be more CPU and memory efficient to
|
||
|
|
use a dense representation.
|
||
|
|
|
||
|
|
If False, the projected data uses a sparse representation if
|
||
|
|
the input is sparse.
|
||
|
|
|
||
|
|
random_state : int, RandomState instance or None, default=None
|
||
|
|
Controls the pseudo random number generator used to generate the
|
||
|
|
projection matrix at fit time.
|
||
|
|
Pass an int for reproducible output across multiple function calls.
|
||
|
|
See :term:`Glossary <random_state>`.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_components_ : int
|
||
|
|
Concrete number of components computed when n_components="auto".
|
||
|
|
|
||
|
|
components_ : sparse matrix of shape (n_components, n_features)
|
||
|
|
Random matrix used for the projection. Sparse matrix will be of CSR
|
||
|
|
format.
|
||
|
|
|
||
|
|
density_ : float in range 0.0 - 1.0
|
||
|
|
Concrete density computed from when density = "auto".
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from sklearn.random_projection import SparseRandomProjection
|
||
|
|
>>> rng = np.random.RandomState(42)
|
||
|
|
>>> X = rng.rand(100, 10000)
|
||
|
|
>>> transformer = SparseRandomProjection(random_state=rng)
|
||
|
|
>>> X_new = transformer.fit_transform(X)
|
||
|
|
>>> X_new.shape
|
||
|
|
(100, 3947)
|
||
|
|
>>> # very few components are non-zero
|
||
|
|
>>> np.mean(transformer.components_ != 0)
|
||
|
|
0.0100...
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
GaussianRandomProjection
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] Ping Li, T. Hastie and K. W. Church, 2006,
|
||
|
|
"Very Sparse Random Projections".
|
||
|
|
https://web.stanford.edu/~hastie/Papers/Ping/KDD06_rp.pdf
|
||
|
|
|
||
|
|
.. [2] D. Achlioptas, 2001, "Database-friendly random projections",
|
||
|
|
https://users.soe.ucsc.edu/~optas/papers/jl.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def __init__(self, n_components='auto', *, density='auto', eps=0.1,
|
||
|
|
dense_output=False, random_state=None):
|
||
|
|
super().__init__(
|
||
|
|
n_components=n_components,
|
||
|
|
eps=eps,
|
||
|
|
dense_output=dense_output,
|
||
|
|
random_state=random_state)
|
||
|
|
|
||
|
|
self.density = density
|
||
|
|
|
||
|
|
def _make_random_matrix(self, n_components, n_features):
|
||
|
|
""" Generate the random projection matrix
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_components : int
|
||
|
|
Dimensionality of the target projection space.
|
||
|
|
|
||
|
|
n_features : int
|
||
|
|
Dimensionality of the original source space.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
components : {ndarray, sparse matrix} of shape \
|
||
|
|
(n_components, n_features)
|
||
|
|
The generated random matrix. Sparse matrix will be of CSR format.
|
||
|
|
|
||
|
|
"""
|
||
|
|
random_state = check_random_state(self.random_state)
|
||
|
|
self.density_ = _check_density(self.density, n_features)
|
||
|
|
return _sparse_random_matrix(n_components,
|
||
|
|
n_features,
|
||
|
|
density=self.density_,
|
||
|
|
random_state=random_state)
|