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sklearn/decomposition/_kernel_pca.py
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Sota Goto
DOC Fix broken link to Kernel PCA paper (#32314)
02 окт 2025, 11:19
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"""Kernel Principal Components Analysis.""" # Authors: The scikit-learn developers # SPDX-License-Identifier: 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 sklearn.base import ( BaseEstimator, ClassNamePrefixFeaturesOutMixin, TransformerMixin, _fit_context, ) from sklearn.exceptions import NotFittedError from sklearn.metrics.pairwise import pairwise_kernels from sklearn.preprocessing import KernelCenterer from sklearn.utils._arpack import _init_arpack_v0 from sklearn.utils._param_validation import Interval, StrOptions from sklearn.utils.extmath import _randomized_eigsh, svd_flip from sklearn.utils.validation import ( _check_psd_eigenvalues, check_is_fitted, validate_data, ) 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://graphics.stanford.edu/courses/cs233-25-spring/ReferencedPapers/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 = validate_data(self, 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) Transformed values. """ 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) Projection of X in the first principal components, where `n_samples` is the number of samples and `n_components` is the number of the components. """ check_is_fitted(self) X = validate_data(self, 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_original : ndarray of shape (n_samples, n_features) Original data, where `n_samples` is the number of samples and `n_features` is the number of features. 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 __sklearn_tags__(self): tags = super().__sklearn_tags__() tags.input_tags.sparse = True tags.transformer_tags.preserves_dtype = ["float64", "float32"] tags.input_tags.pairwise = self.kernel == "precomputed" return tags @property def _n_features_out(self): """Number of transformed output features.""" return self.eigenvalues_.shape[0]