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, iterated_power='auto', remove_zero_eig=False, random_state=None, copy_X=True, n_jobs=None)[source]¶
Kernel Principal component analysis (KPCA).
Non-linear dimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels).
It uses the
scipy.linalg.eigh
LAPACK implementation of the full SVD or thescipy.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 of Halko et al. 2009, seeeigen_solver
.Read more in the User Guide.
- Parameters
- n_componentsint, default=None
Number of components. If None, all non-zero 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 non-precomputed kernels. (i.e. learn to find the pre-image of a point)
- 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 extend) may be more efficient than the dense eigensolver. Randomized SVD is performed according to the method of Halko et al.- 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. 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.
Changed in version 1.0:
'randomized'
was added.- 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.
- iterated_powerint >= 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.New in version 1.0.
- 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’ or ‘randomized’. 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.- n_features_in_int
Number of features seen during fit.
New in version 0.24.
References
- Kernel PCA was introduced in:
Bernhard Schoelkopf, Alexander J. Smola, and Klaus-Robert Mueller. 1999. Kernel principal component analysis. In Advances in kernel methods, MIT Press, Cambridge, MA, USA 327-352.
For eigen_solver == ‘arpack’, refer to
scipy.sparse.linalg.eigsh
.- For eigen_solver == ‘randomized’, see:
Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions Halko, et al., 2009 (arXiv:909) A randomized algorithm for the decomposition of matrices Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
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{array-like, 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{array-like, 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 pre-image. The pre-image is learned by kernel ridge regression of the original data on their low-dimensional 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{array-like, sparse matrix} of shape (n_samples, n_components)
- Returns
- X_newndarray of shape (n_samples, n_features)
References
“Learning to Find Pre-Images”, 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.