_kernel_pca.py

"""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) [1]_.

    Non-linear dimensionality reduction through the use of kernels (see
    :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`.

    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 : int, 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(Integral, 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(
            self.eigenvectors_, np.zeros_like(self.eigenvectors_).T
        )

        # 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]