392 lines
15 KiB
Python
392 lines
15 KiB
Python
"""Kernel Principal Components Analysis."""
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# Author: Mathieu Blondel <mathieu@mblondel.org>
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# License: BSD 3 clause
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import numpy as np
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from scipy import linalg
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from scipy.sparse.linalg import eigsh
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from ..utils._arpack import _init_arpack_v0
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from ..utils.extmath import svd_flip
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from ..utils.validation import check_is_fitted, _check_psd_eigenvalues
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from ..utils.deprecation import deprecated
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from ..exceptions import NotFittedError
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from ..base import BaseEstimator, TransformerMixin
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from ..preprocessing import KernelCenterer
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from ..metrics.pairwise import pairwise_kernels
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from ..utils.validation import _deprecate_positional_args
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class KernelPCA(TransformerMixin, BaseEstimator):
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"""Kernel Principal component analysis (KPCA).
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Non-linear dimensionality reduction through the use of kernels (see
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:ref:`metrics`).
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Read more in the :ref:`User Guide <kernel_PCA>`.
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Parameters
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----------
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n_components : int, default=None
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Number of components. If None, all non-zero components are kept.
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kernel : {'linear', 'poly', \
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'rbf', 'sigmoid', 'cosine', 'precomputed'}, default='linear'
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Kernel used for PCA.
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gamma : float, default=None
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Kernel coefficient for rbf, poly and sigmoid kernels. Ignored by other
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kernels. If ``gamma`` is ``None``, then it is set to ``1/n_features``.
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degree : int, default=3
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Degree for poly kernels. Ignored by other kernels.
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coef0 : float, default=1
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Independent term in poly and sigmoid kernels.
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Ignored by other kernels.
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kernel_params : dict, default=None
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Parameters (keyword arguments) and
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values for kernel passed as callable object.
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Ignored by other kernels.
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alpha : float, default=1.0
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Hyperparameter of the ridge regression that learns the
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inverse transform (when fit_inverse_transform=True).
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fit_inverse_transform : bool, default=False
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Learn the inverse transform for non-precomputed kernels.
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(i.e. learn to find the pre-image of a point)
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eigen_solver : {'auto', 'dense', 'arpack'}, default='auto'
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Select eigensolver to use. If n_components is much less than
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the number of training samples, arpack may be more efficient
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than the dense eigensolver.
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tol : float, default=0
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Convergence tolerance for arpack.
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If 0, optimal value will be chosen by arpack.
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max_iter : int, default=None
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Maximum number of iterations for arpack.
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If None, optimal value will be chosen by arpack.
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remove_zero_eig : bool, default=False
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If True, then all components with zero eigenvalues are removed, so
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that the number of components in the output may be < n_components
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(and sometimes even zero due to numerical instability).
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When n_components is None, this parameter is ignored and components
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with zero eigenvalues are removed regardless.
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random_state : int, RandomState instance or None, default=None
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Used when ``eigen_solver`` == 'arpack'. Pass an int for reproducible
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results across multiple function calls.
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See :term:`Glossary <random_state>`.
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.. versionadded:: 0.18
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copy_X : bool, default=True
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If True, input X is copied and stored by the model in the `X_fit_`
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attribute. If no further changes will be done to X, setting
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`copy_X=False` saves memory by storing a reference.
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.. versionadded:: 0.18
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n_jobs : int, default=None
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The number of parallel jobs to run.
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``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
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``-1`` means using all processors. See :term:`Glossary <n_jobs>`
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for more details.
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.. versionadded:: 0.18
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Attributes
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----------
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lambdas_ : ndarray of shape (n_components,)
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Eigenvalues of the centered kernel matrix in decreasing order.
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If `n_components` and `remove_zero_eig` are not set,
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then all values are stored.
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alphas_ : ndarray of shape (n_samples, n_components)
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Eigenvectors of the centered kernel matrix. If `n_components` and
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`remove_zero_eig` are not set, then all components are stored.
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dual_coef_ : ndarray of shape (n_samples, n_features)
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Inverse transform matrix. Only available when
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``fit_inverse_transform`` is True.
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X_transformed_fit_ : ndarray of shape (n_samples, n_components)
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Projection of the fitted data on the kernel principal components.
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Only available when ``fit_inverse_transform`` is True.
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X_fit_ : ndarray of shape (n_samples, n_features)
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The data used to fit the model. If `copy_X=False`, then `X_fit_` is
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a reference. This attribute is used for the calls to transform.
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Examples
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--------
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>>> from sklearn.datasets import load_digits
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>>> from sklearn.decomposition import KernelPCA
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>>> X, _ = load_digits(return_X_y=True)
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>>> transformer = KernelPCA(n_components=7, kernel='linear')
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>>> X_transformed = transformer.fit_transform(X)
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>>> X_transformed.shape
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(1797, 7)
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References
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----------
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Kernel PCA was introduced in:
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Bernhard Schoelkopf, Alexander J. Smola,
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and Klaus-Robert Mueller. 1999. Kernel principal
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component analysis. In Advances in kernel methods,
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MIT Press, Cambridge, MA, USA 327-352.
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"""
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@_deprecate_positional_args
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def __init__(self, n_components=None, *, kernel="linear",
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gamma=None, degree=3, coef0=1, kernel_params=None,
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alpha=1.0, fit_inverse_transform=False, eigen_solver='auto',
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tol=0, max_iter=None, remove_zero_eig=False,
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random_state=None, copy_X=True, n_jobs=None):
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if fit_inverse_transform and kernel == 'precomputed':
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raise ValueError(
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"Cannot fit_inverse_transform with a precomputed kernel.")
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self.n_components = n_components
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self.kernel = kernel
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self.kernel_params = kernel_params
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self.gamma = gamma
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self.degree = degree
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self.coef0 = coef0
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self.alpha = alpha
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self.fit_inverse_transform = fit_inverse_transform
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self.eigen_solver = eigen_solver
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self.remove_zero_eig = remove_zero_eig
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self.tol = tol
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self.max_iter = max_iter
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self.random_state = random_state
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self.n_jobs = n_jobs
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self.copy_X = copy_X
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# TODO: Remove in 1.1
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# mypy error: Decorated property not supported
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@deprecated("Attribute _pairwise was deprecated in " # type: ignore
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"version 0.24 and will be removed in 1.1 (renaming of 0.26).")
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@property
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def _pairwise(self):
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return self.kernel == "precomputed"
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def _get_kernel(self, X, Y=None):
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if callable(self.kernel):
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params = self.kernel_params or {}
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else:
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params = {"gamma": self.gamma,
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"degree": self.degree,
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"coef0": self.coef0}
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return pairwise_kernels(X, Y, metric=self.kernel,
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filter_params=True, n_jobs=self.n_jobs,
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**params)
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def _fit_transform(self, K):
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""" Fit's using kernel K"""
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# center kernel
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K = self._centerer.fit_transform(K)
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if self.n_components is None:
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n_components = K.shape[0]
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else:
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n_components = min(K.shape[0], self.n_components)
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# compute eigenvectors
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if self.eigen_solver == 'auto':
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if K.shape[0] > 200 and n_components < 10:
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eigen_solver = 'arpack'
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else:
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eigen_solver = 'dense'
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else:
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eigen_solver = self.eigen_solver
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if eigen_solver == 'dense':
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self.lambdas_, self.alphas_ = linalg.eigh(
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K, eigvals=(K.shape[0] - n_components, K.shape[0] - 1))
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elif eigen_solver == 'arpack':
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v0 = _init_arpack_v0(K.shape[0], self.random_state)
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self.lambdas_, self.alphas_ = eigsh(K, n_components,
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which="LA",
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tol=self.tol,
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maxiter=self.max_iter,
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v0=v0)
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# make sure that the eigenvalues are ok and fix numerical issues
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self.lambdas_ = _check_psd_eigenvalues(self.lambdas_,
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enable_warnings=False)
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# flip eigenvectors' sign to enforce deterministic output
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self.alphas_, _ = svd_flip(self.alphas_,
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np.zeros_like(self.alphas_).T)
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# sort eigenvectors in descending order
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indices = self.lambdas_.argsort()[::-1]
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self.lambdas_ = self.lambdas_[indices]
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self.alphas_ = self.alphas_[:, indices]
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# remove eigenvectors with a zero eigenvalue (null space) if required
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if self.remove_zero_eig or self.n_components is None:
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self.alphas_ = self.alphas_[:, self.lambdas_ > 0]
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self.lambdas_ = self.lambdas_[self.lambdas_ > 0]
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# Maintenance note on Eigenvectors normalization
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# ----------------------------------------------
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# there is a link between
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# the eigenvectors of K=Phi(X)'Phi(X) and the ones of Phi(X)Phi(X)'
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# if v is an eigenvector of K
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# then Phi(X)v is an eigenvector of Phi(X)Phi(X)'
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# if u is an eigenvector of Phi(X)Phi(X)'
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# then Phi(X)'u is an eigenvector of Phi(X)'Phi(X)
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#
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# At this stage our self.alphas_ (the v) have norm 1, we need to scale
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# them so that eigenvectors in kernel feature space (the u) have norm=1
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# instead
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#
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# We COULD scale them here:
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# self.alphas_ = self.alphas_ / np.sqrt(self.lambdas_)
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#
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# But choose to perform that LATER when needed, in `fit()` and in
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# `transform()`.
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return K
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def _fit_inverse_transform(self, X_transformed, X):
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if hasattr(X, "tocsr"):
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raise NotImplementedError("Inverse transform not implemented for "
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"sparse matrices!")
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n_samples = X_transformed.shape[0]
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K = self._get_kernel(X_transformed)
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K.flat[::n_samples + 1] += self.alpha
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self.dual_coef_ = linalg.solve(K, X, sym_pos=True, overwrite_a=True)
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self.X_transformed_fit_ = X_transformed
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def fit(self, X, y=None):
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"""Fit the model from data in X.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training vector, where n_samples in the number of samples
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and n_features is the number of features.
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Returns
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-------
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self : object
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Returns the instance itself.
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"""
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X = self._validate_data(X, accept_sparse='csr', copy=self.copy_X)
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self._centerer = KernelCenterer()
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K = self._get_kernel(X)
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self._fit_transform(K)
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if self.fit_inverse_transform:
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# no need to use the kernel to transform X, use shortcut expression
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X_transformed = self.alphas_ * np.sqrt(self.lambdas_)
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self._fit_inverse_transform(X_transformed, X)
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self.X_fit_ = X
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return self
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def fit_transform(self, X, y=None, **params):
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"""Fit the model from data in X and transform X.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training vector, where n_samples in the number of samples
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and n_features is the number of features.
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Returns
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-------
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X_new : ndarray of shape (n_samples, n_components)
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"""
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self.fit(X, **params)
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# no need to use the kernel to transform X, use shortcut expression
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X_transformed = self.alphas_ * np.sqrt(self.lambdas_)
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if self.fit_inverse_transform:
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self._fit_inverse_transform(X_transformed, X)
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return X_transformed
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def transform(self, X):
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"""Transform X.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Returns
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-------
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X_new : ndarray of shape (n_samples, n_components)
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"""
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check_is_fitted(self)
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X = self._validate_data(X, accept_sparse='csr', reset=False)
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# Compute centered gram matrix between X and training data X_fit_
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K = self._centerer.transform(self._get_kernel(X, self.X_fit_))
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# scale eigenvectors (properly account for null-space for dot product)
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non_zeros = np.flatnonzero(self.lambdas_)
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scaled_alphas = np.zeros_like(self.alphas_)
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scaled_alphas[:, non_zeros] = (self.alphas_[:, non_zeros]
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/ np.sqrt(self.lambdas_[non_zeros]))
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# Project with a scalar product between K and the scaled eigenvectors
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return np.dot(K, scaled_alphas)
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def inverse_transform(self, X):
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"""Transform X back to original space.
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``inverse_transform`` approximates the inverse transformation using
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a learned pre-image. The pre-image is learned by kernel ridge
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regression of the original data on their low-dimensional representation
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vectors.
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.. note:
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:meth:`~sklearn.decomposition.fit` internally uses a centered
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kernel. As the centered kernel no longer contains the information
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of the mean of kernel features, such information is not taken into
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account in reconstruction.
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.. note::
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When users want to compute inverse transformation for 'linear'
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kernel, it is recommended that they use
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:class:`~sklearn.decomposition.PCA` instead. Unlike
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:class:`~sklearn.decomposition.PCA`,
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:class:`~sklearn.decomposition.KernelPCA`'s ``inverse_transform``
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does not reconstruct the mean of data when 'linear' kernel is used
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due to the use of centered kernel.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_components)
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Returns
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-------
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X_new : ndarray of shape (n_samples, n_features)
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References
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----------
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"Learning to Find Pre-Images", G BakIr et al, 2004.
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"""
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if not self.fit_inverse_transform:
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raise NotFittedError("The fit_inverse_transform parameter was not"
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" set to True when instantiating and hence "
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"the inverse transform is not available.")
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K = self._get_kernel(X, self.X_transformed_fit_)
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return np.dot(K, self.dual_coef_)
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def _more_tags(self):
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return {'preserves_dtype': [np.float64, np.float32],
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'pairwise': self.kernel == 'precomputed'}
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