"""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_in_place(self, K):

"""Fit's using kernel K"""

# center kernel in place

K = self._centerer.fit(K).transform(K, copy=False)

# 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)

# When kernel="precomputed", K is X but it's safe to perform in place operations

# on K because a copy was made before if requested by copy_X.

self._fit_transform_in_place(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."""