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"""Kernel Principal Components Analysis."""
# Author: Mathieu Blondel <mathieu@mblondel.org>
# Sylvain Marie <sylvain.marie@schneider-electric.com>
# License: BSD 3 clause
from numbers import Integral, Real
import numpy as np
from scipy import linalg
from scipy.linalg import eigh
from scipy.sparse.linalg import eigsh
from ..base import (
BaseEstimator,
ClassNamePrefixFeaturesOutMixin,
TransformerMixin,
_fit_context,
)
from ..exceptions import NotFittedError
from ..metrics.pairwise import pairwise_kernels
from ..preprocessing import KernelCenterer
from ..utils._arpack import _init_arpack_v0
from ..utils._param_validation import Interval, StrOptions
from ..utils.extmath import _randomized_eigsh, svd_flip
from ..utils.validation import (
_check_psd_eigenvalues,
check_is_fitted,
)
class KernelPCA(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
"""Kernel Principal component analysis (KPCA).
Non-linear dimensionality reduction through the use of kernels [1]_, see also
:ref:`metrics`.
It uses the :func:`scipy.linalg.eigh` LAPACK implementation of the full SVD
or the :func:`scipy.sparse.linalg.eigsh` ARPACK implementation of the
truncated SVD, depending on the shape of the input data and the number of
components to extract. It can also use a randomized truncated SVD by the
method proposed in [3]_, see `eigen_solver`.
For a usage example and comparison between
Principal Components Analysis (PCA) and its kernelized version (KPCA), see
:ref:`sphx_glr_auto_examples_decomposition_plot_kernel_pca.py`.
For a usage example in denoising images using KPCA, see
:ref:`sphx_glr_auto_examples_applications_plot_digits_denoising.py`.
Read more in the :ref:`User Guide <kernel_PCA>`.
Parameters
----------
n_components : int, default=None
Number of components. If None, all non-zero components are kept.
kernel : {'linear', 'poly', 'rbf', 'sigmoid', 'cosine', 'precomputed'} \
or callable, default='linear'
Kernel used for PCA.
gamma : float, default=None
Kernel coefficient for rbf, poly and sigmoid kernels. Ignored by other
kernels. If ``gamma`` is ``None``, then it is set to ``1/n_features``.
degree : float, default=3
Degree for poly kernels. Ignored by other kernels.
coef0 : float, default=1
Independent term in poly and sigmoid kernels.
Ignored by other kernels.
kernel_params : dict, default=None
Parameters (keyword arguments) and
values for kernel passed as callable object.
Ignored by other kernels.
alpha : float, default=1.0
Hyperparameter of the ridge regression that learns the
inverse transform (when fit_inverse_transform=True).
fit_inverse_transform : bool, default=False
Learn the inverse transform for non-precomputed kernels
(i.e. learn to find the pre-image of a point). This method is based
on [2]_.
eigen_solver : {'auto', 'dense', 'arpack', 'randomized'}, \
default='auto'
Select eigensolver to use. If `n_components` is much
less than the number of training samples, randomized (or arpack to a
smaller extent) may be more efficient than the dense eigensolver.
Randomized SVD is performed according to the method of Halko et al
[3]_.
auto :
the solver is selected by a default policy based on n_samples
(the number of training samples) and `n_components`:
if the number of components to extract is less than 10 (strict) and
the number of samples is more than 200 (strict), the 'arpack'
method is enabled. Otherwise the exact full eigenvalue
decomposition is computed and optionally truncated afterwards
('dense' method).
dense :
run exact full eigenvalue decomposition calling the standard
LAPACK solver via `scipy.linalg.eigh`, and select the components
by postprocessing
arpack :
run SVD truncated to n_components calling ARPACK solver using
`scipy.sparse.linalg.eigsh`. It requires strictly
0 < n_components < n_samples
randomized :
run randomized SVD by the method of Halko et al. [3]_. The current
implementation selects eigenvalues based on their module; therefore
using this method can lead to unexpected results if the kernel is
not positive semi-definite. See also [4]_.
.. versionchanged:: 1.0
`'randomized'` was added.
tol : float, default=0
Convergence tolerance for arpack.
If 0, optimal value will be chosen by arpack.
max_iter : int, default=None
Maximum number of iterations for arpack.
If None, optimal value will be chosen by arpack.
iterated_power : int >= 0, or 'auto', default='auto'
Number of iterations for the power method computed by
svd_solver == 'randomized'. When 'auto', it is set to 7 when
`n_components < 0.1 * min(X.shape)`, other it is set to 4.
.. versionadded:: 1.0
remove_zero_eig : bool, default=False
If True, then all components with zero eigenvalues are removed, so
that the number of components in the output may be < n_components
(and sometimes even zero due to numerical instability).
When n_components is None, this parameter is ignored and components
with zero eigenvalues are removed regardless.
random_state : int, RandomState instance or None, default=None
Used when ``eigen_solver`` == 'arpack' or 'randomized'. Pass an int
for reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
.. versionadded:: 0.18
copy_X : bool, default=True
If True, input X is copied and stored by the model in the `X_fit_`
attribute. If no further changes will be done to X, setting
`copy_X=False` saves memory by storing a reference.
.. versionadded:: 0.18
n_jobs : int, default=None
The number of parallel jobs to run.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
.. versionadded:: 0.18
Attributes
----------
eigenvalues_ : ndarray of shape (n_components,)
Eigenvalues of the centered kernel matrix in decreasing order.
If `n_components` and `remove_zero_eig` are not set,
then all values are stored.
eigenvectors_ : ndarray of shape (n_samples, n_components)
Eigenvectors of the centered kernel matrix. If `n_components` and
`remove_zero_eig` are not set, then all components are stored.
dual_coef_ : ndarray of shape (n_samples, n_features)
Inverse transform matrix. Only available when
``fit_inverse_transform`` is True.
X_transformed_fit_ : ndarray of shape (n_samples, n_components)
Projection of the fitted data on the kernel principal components.
Only available when ``fit_inverse_transform`` is True.
X_fit_ : ndarray of shape (n_samples, n_features)
The data used to fit the model. If `copy_X=False`, then `X_fit_` is
a reference. This attribute is used for the calls to transform.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
gamma_ : float
Kernel coefficient for rbf, poly and sigmoid kernels. When `gamma`
is explicitly provided, this is just the same as `gamma`. When `gamma`
is `None`, this is the actual value of kernel coefficient.
.. versionadded:: 1.3
See Also
--------
FastICA : A fast algorithm for Independent Component Analysis.
IncrementalPCA : Incremental Principal Component Analysis.
NMF : Non-Negative Matrix Factorization.
PCA : Principal Component Analysis.
SparsePCA : Sparse Principal Component Analysis.
TruncatedSVD : Dimensionality reduction using truncated SVD.
References
----------
.. [1] `Schölkopf, Bernhard, Alexander Smola, and Klaus-Robert Müller.
"Kernel principal component analysis."
International conference on artificial neural networks.
Springer, Berlin, Heidelberg, 1997.
<https://people.eecs.berkeley.edu/~wainwrig/stat241b/scholkopf_kernel.pdf>`_
.. [2] `Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
"Learning to find pre-images."
Advances in neural information processing systems 16 (2004): 449-456.
<https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf>`_
.. [3] :arxiv:`Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp.
"Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions."
SIAM review 53.2 (2011): 217-288. <0909.4061>`
.. [4] `Martinsson, Per-Gunnar, Vladimir Rokhlin, and Mark Tygert.
"A randomized algorithm for the decomposition of matrices."
Applied and Computational Harmonic Analysis 30.1 (2011): 47-68.
<https://www.sciencedirect.com/science/article/pii/S1063520310000242>`_
Examples
--------
>>> from sklearn.datasets import load_digits
>>> from sklearn.decomposition import KernelPCA
>>> X, _ = load_digits(return_X_y=True)
>>> transformer = KernelPCA(n_components=7, kernel='linear')
>>> X_transformed = transformer.fit_transform(X)
>>> X_transformed.shape
(1797, 7)
"""
_parameter_constraints: dict = {
"n_components": [
Interval(Integral, 1, None, closed="left"),
None,
],
"kernel": [
StrOptions({"linear", "poly", "rbf", "sigmoid", "cosine", "precomputed"}),
callable,
],
"gamma": [
Interval(Real, 0, None, closed="left"),
None,
],
"degree": [Interval(Real, 0, None, closed="left")],
"coef0": [Interval(Real, None, None, closed="neither")],
"kernel_params": [dict, None],
"alpha": [Interval(Real, 0, None, closed="left")],
"fit_inverse_transform": ["boolean"],
"eigen_solver": [StrOptions({"auto", "dense", "arpack", "randomized"})],
"tol": [Interval(Real, 0, None, closed="left")],
"max_iter": [
Interval(Integral, 1, None, closed="left"),
None,
],
"iterated_power": [
Interval(Integral, 0, None, closed="left"),
StrOptions({"auto"}),
],
"remove_zero_eig": ["boolean"],
"random_state": ["random_state"],
"copy_X": ["boolean"],
"n_jobs": [None, Integral],
}
def __init__(
self,
n_components=None,
*,
kernel="linear",
gamma=None,
degree=3,
coef0=1,
kernel_params=None,
alpha=1.0,
fit_inverse_transform=False,
eigen_solver="auto",
tol=0,
max_iter=None,
iterated_power="auto",
remove_zero_eig=False,
random_state=None,
copy_X=True,
n_jobs=None,
):
self.n_components = n_components
self.kernel = kernel
self.kernel_params = kernel_params
self.gamma = gamma
self.degree = degree
self.coef0 = coef0
self.alpha = alpha
self.fit_inverse_transform = fit_inverse_transform
self.eigen_solver = eigen_solver
self.tol = tol
self.max_iter = max_iter
self.iterated_power = iterated_power
self.remove_zero_eig = remove_zero_eig
self.random_state = random_state
self.n_jobs = n_jobs
self.copy_X = copy_X
def _get_kernel(self, X, Y=None):
if callable(self.kernel):
params = self.kernel_params or {}
else:
params = {"gamma": self.gamma_, "degree": self.degree, "coef0": self.coef0}
return pairwise_kernels(
X, Y, metric=self.kernel, filter_params=True, n_jobs=self.n_jobs, **params
)
def _fit_transform(self, K):
"""Fit's using kernel K"""
# center kernel
K = self._centerer.fit_transform(K)
# adjust n_components according to user inputs
if self.n_components is None:
n_components = K.shape[0] # use all dimensions
else:
n_components = min(K.shape[0], self.n_components)
# compute eigenvectors
if self.eigen_solver == "auto":
if K.shape[0] > 200 and n_components < 10:
eigen_solver = "arpack"
else:
eigen_solver = "dense"
else:
eigen_solver = self.eigen_solver
if eigen_solver == "dense":
# Note: subset_by_index specifies the indices of smallest/largest to return
self.eigenvalues_, self.eigenvectors_ = eigh(
K, subset_by_index=(K.shape[0] - n_components, K.shape[0] - 1)
)
elif eigen_solver == "arpack":
v0 = _init_arpack_v0(K.shape[0], self.random_state)
self.eigenvalues_, self.eigenvectors_ = eigsh(
K, n_components, which="LA", tol=self.tol, maxiter=self.max_iter, v0=v0
)
elif eigen_solver == "randomized":
self.eigenvalues_, self.eigenvectors_ = _randomized_eigsh(
K,
n_components=n_components,
n_iter=self.iterated_power,
random_state=self.random_state,
selection="module",
)
# make sure that the eigenvalues are ok and fix numerical issues
self.eigenvalues_ = _check_psd_eigenvalues(
self.eigenvalues_, enable_warnings=False
)
# flip eigenvectors' sign to enforce deterministic output
self.eigenvectors_, _ = svd_flip(u=self.eigenvectors_, v=None)
# sort eigenvectors in descending order
indices = self.eigenvalues_.argsort()[::-1]
self.eigenvalues_ = self.eigenvalues_[indices]
self.eigenvectors_ = self.eigenvectors_[:, indices]
# remove eigenvectors with a zero eigenvalue (null space) if required
if self.remove_zero_eig or self.n_components is None:
self.eigenvectors_ = self.eigenvectors_[:, self.eigenvalues_ > 0]
self.eigenvalues_ = self.eigenvalues_[self.eigenvalues_ > 0]
# Maintenance note on Eigenvectors normalization
# ----------------------------------------------
# there is a link between
# the eigenvectors of K=Phi(X)'Phi(X) and the ones of Phi(X)Phi(X)'
# if v is an eigenvector of K
# then Phi(X)v is an eigenvector of Phi(X)Phi(X)'
# if u is an eigenvector of Phi(X)Phi(X)'
# then Phi(X)'u is an eigenvector of Phi(X)'Phi(X)
#
# At this stage our self.eigenvectors_ (the v) have norm 1, we need to scale
# them so that eigenvectors in kernel feature space (the u) have norm=1
# instead
#
# We COULD scale them here:
# self.eigenvectors_ = self.eigenvectors_ / np.sqrt(self.eigenvalues_)
#
# But choose to perform that LATER when needed, in `fit()` and in
# `transform()`.
return K
def _fit_inverse_transform(self, X_transformed, X):
if hasattr(X, "tocsr"):
raise NotImplementedError(
"Inverse transform not implemented for sparse matrices!"
)
n_samples = X_transformed.shape[0]
K = self._get_kernel(X_transformed)
K.flat[:: n_samples + 1] += self.alpha
self.dual_coef_ = linalg.solve(K, X, assume_a="pos", overwrite_a=True)
self.X_transformed_fit_ = X_transformed
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit the model from data in X.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : Ignored
Not used, present for API consistency by convention.
Returns
-------
self : object
Returns the instance itself.
"""
if self.fit_inverse_transform and self.kernel == "precomputed":
raise ValueError("Cannot fit_inverse_transform with a precomputed kernel.")
X = self._validate_data(X, accept_sparse="csr", copy=self.copy_X)
self.gamma_ = 1 / X.shape[1] if self.gamma is None else self.gamma
self._centerer = KernelCenterer().set_output(transform="default")
K = self._get_kernel(X)
self._fit_transform(K)
if self.fit_inverse_transform:
# no need to use the kernel to transform X, use shortcut expression
X_transformed = self.eigenvectors_ * np.sqrt(self.eigenvalues_)
self._fit_inverse_transform(X_transformed, X)
self.X_fit_ = X
return self
def fit_transform(self, X, y=None, **params):
"""Fit the model from data in X and transform X.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : Ignored
Not used, present for API consistency by convention.
**params : kwargs
Parameters (keyword arguments) and values passed to
the fit_transform instance.
Returns
-------
X_new : ndarray of shape (n_samples, n_components)
Returns the instance itself.
"""
self.fit(X, **params)
# no need to use the kernel to transform X, use shortcut expression
X_transformed = self.eigenvectors_ * np.sqrt(self.eigenvalues_)
if self.fit_inverse_transform:
self._fit_inverse_transform(X_transformed, X)
return X_transformed
def transform(self, X):
"""Transform X.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
X_new : ndarray of shape (n_samples, n_components)
Returns the instance itself.
"""
check_is_fitted(self)
X = self._validate_data(X, accept_sparse="csr", reset=False)
# Compute centered gram matrix between X and training data X_fit_
K = self._centerer.transform(self._get_kernel(X, self.X_fit_))
# scale eigenvectors (properly account for null-space for dot product)
non_zeros = np.flatnonzero(self.eigenvalues_)
scaled_alphas = np.zeros_like(self.eigenvectors_)
scaled_alphas[:, non_zeros] = self.eigenvectors_[:, non_zeros] / np.sqrt(
self.eigenvalues_[non_zeros]
)
# Project with a scalar product between K and the scaled eigenvectors
return np.dot(K, scaled_alphas)
def inverse_transform(self, X):
"""Transform X back to original space.
``inverse_transform`` approximates the inverse transformation using
a learned pre-image. The pre-image is learned by kernel ridge
regression of the original data on their low-dimensional representation
vectors.
.. note:
:meth:`~sklearn.decomposition.fit` internally uses a centered
kernel. As the centered kernel no longer contains the information
of the mean of kernel features, such information is not taken into
account in reconstruction.
.. note::
When users want to compute inverse transformation for 'linear'
kernel, it is recommended that they use
:class:`~sklearn.decomposition.PCA` instead. Unlike
:class:`~sklearn.decomposition.PCA`,
:class:`~sklearn.decomposition.KernelPCA`'s ``inverse_transform``
does not reconstruct the mean of data when 'linear' kernel is used
due to the use of centered kernel.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_components)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
X_new : ndarray of shape (n_samples, n_features)
Returns the instance itself.
References
----------
`Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
"Learning to find pre-images."
Advances in neural information processing systems 16 (2004): 449-456.
<https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf>`_
"""
if not self.fit_inverse_transform:
raise NotFittedError(
"The fit_inverse_transform parameter was not"
" set to True when instantiating and hence "
"the inverse transform is not available."
)
K = self._get_kernel(X, self.X_transformed_fit_)
return np.dot(K, self.dual_coef_)
def _more_tags(self):
return {
"preserves_dtype": [np.float64, np.float32],
"pairwise": self.kernel == "precomputed",
}
@property
def _n_features_out(self):
"""Number of transformed output features."""
return self.eigenvalues_.shape[0]