sklearn.decomposition
.KernelPCA¶

class
sklearn.decomposition.
KernelPCA
(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, remove_zero_eig=False, random_state=None, copy_X=True, n_jobs=None)[source]¶ Kernel Principal component analysis (KPCA).
Nonlinear dimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels).
Read more in the User Guide.
 Parameters
 n_componentsint, default=None
Number of components. If None, all nonzero components are kept.
 kernel{‘linear’, ‘poly’, ‘rbf’, ‘sigmoid’, ‘cosine’, ‘precomputed’}, default=’linear’
Kernel used for PCA.
 gammafloat, default=None
Kernel coefficient for rbf, poly and sigmoid kernels. Ignored by other kernels. If
gamma
isNone
, then it is set to1/n_features
. degreeint, default=3
Degree for poly kernels. Ignored by other kernels.
 coef0float, default=1
Independent term in poly and sigmoid kernels. Ignored by other kernels.
 kernel_paramsdict, default=None
Parameters (keyword arguments) and values for kernel passed as callable object. Ignored by other kernels.
 alphafloat, default=1.0
Hyperparameter of the ridge regression that learns the inverse transform (when fit_inverse_transform=True).
 fit_inverse_transformbool, default=False
Learn the inverse transform for nonprecomputed kernels. (i.e. learn to find the preimage of a point)
 eigen_solver{‘auto’, ‘dense’, ‘arpack’}, default=’auto’
Select eigensolver to use. If n_components is much less than the number of training samples, arpack may be more efficient than the dense eigensolver.
 tolfloat, default=0
Convergence tolerance for arpack. If 0, optimal value will be chosen by arpack.
 max_iterint, default=None
Maximum number of iterations for arpack. If None, optimal value will be chosen by arpack.
 remove_zero_eigbool, 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_stateint, RandomState instance or None, default=None
Used when
eigen_solver
== ‘arpack’. Pass an int for reproducible results across multiple function calls. See Glossary.New in version 0.18.
 copy_Xbool, 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, settingcopy_X=False
saves memory by storing a reference.New in version 0.18.
 n_jobsint, default=None
The number of parallel jobs to run.
None
means 1 unless in ajoblib.parallel_backend
context.1
means using all processors. See Glossary for more details.New in version 0.18.
 Attributes
 lambdas_ndarray of shape (n_components,)
Eigenvalues of the centered kernel matrix in decreasing order. If
n_components
andremove_zero_eig
are not set, then all values are stored. alphas_ndarray of shape (n_samples, n_components)
Eigenvectors of the centered kernel matrix. If
n_components
andremove_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
, thenX_fit_
is a reference. This attribute is used for the calls to transform.
References
 Kernel PCA was introduced in:
Bernhard Schoelkopf, Alexander J. Smola, and KlausRobert Mueller. 1999. Kernel principal component analysis. In Advances in kernel methods, MIT Press, Cambridge, MA, USA 327352.
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)
Methods
fit
(X[, y])Fit the model from data in X.
fit_transform
(X[, y])Fit the model from data in X and transform X.
get_params
([deep])Get parameters for this estimator.
Transform X back to original space.
set_params
(**params)Set the parameters of this estimator.
transform
(X)Transform X.

fit
(X, y=None)[source]¶ Fit the model from data in X.
 Parameters
 X{arraylike, sparse matrix} of shape (n_samples, n_features)
Training vector, where n_samples in the number of samples and n_features is the number of features.
 Returns
 selfobject
Returns the instance itself.

fit_transform
(X, y=None, **params)[source]¶ Fit the model from data in X and transform X.
 Parameters
 X{arraylike, sparse matrix} of shape (n_samples, n_features)
Training vector, where n_samples in the number of samples and n_features is the number of features.
 Returns
 X_newndarray of shape (n_samples, n_components)

get_params
(deep=True)[source]¶ Get parameters for this estimator.
 Parameters
 deepbool, default=True
If True, will return the parameters for this estimator and contained subobjects that are estimators.
 Returns
 paramsdict
Parameter names mapped to their values.

inverse_transform
(X)[source]¶ Transform X back to original space.
inverse_transform
approximates the inverse transformation using a learned preimage. The preimage is learned by kernel ridge regression of the original data on their lowdimensional representation vectors.Note
When users want to compute inverse transformation for ‘linear’ kernel, it is recommended that they use
PCA
instead. UnlikePCA
,KernelPCA
’sinverse_transform
does not reconstruct the mean of data when ‘linear’ kernel is used due to the use of centered kernel. Parameters
 X{arraylike, sparse matrix} of shape (n_samples, n_components)
 Returns
 X_newndarray of shape (n_samples, n_features)
References
“Learning to Find PreImages”, G BakIr et al, 2004.

set_params
(**params)[source]¶ Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as
Pipeline
). The latter have parameters of the form<component>__<parameter>
so that it’s possible to update each component of a nested object. Parameters
 **paramsdict
Estimator parameters.
 Returns
 selfestimator instance
Estimator instance.