kernel_pca.py

"""Kernel Principal Components Analysis"""

# Author: Mathieu Blondel <mathieu@mblondel.org>
# License: BSD 3 clause

import numpy as np
from scipy import linalg

from ..utils.arpack import eigsh
from ..base import BaseEstimator, TransformerMixin
from ..preprocessing import KernelCenterer
from ..metrics.pairwise import pairwise_kernels


class KernelPCA(BaseEstimator, TransformerMixin):
    """Kernel Principal component analysis (KPCA)

    Non-linear dimensionality reduction through the use of kernels.

    Parameters
    ----------
    n_components: int or None
        Number of components. If None, all non-zero components are kept.

    kernel: "linear" | "poly" | "rbf" | "sigmoid" | "cosine" | "precomputed"
        Kernel.
        Default: "linear"

    degree : int, default=3
        Degree for poly, rbf and sigmoid kernels. Ignored by other kernels.

    gamma : float, optional
        Kernel coefficient for rbf and poly kernels. Default: 1/n_features.
        Ignored by other kernels.

    coef0 : float, optional
        Independent term in poly and sigmoid kernels.
        Ignored by other kernels.

    kernel_params : mapping of string to any, optional
        Parameters (keyword arguments) and values for kernel passed as
        callable object. Ignored by other kernels.

    alpha: int
        Hyperparameter of the ridge regression that learns the
        inverse transform (when fit_inverse_transform=True).
        Default: 1.0

    fit_inverse_transform: bool
        Learn the inverse transform for non-precomputed kernels.
        (i.e. learn to find the pre-image of a point)
        Default: False

    eigen_solver: string ['auto'|'dense'|'arpack']
        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.

    tol: float
        convergence tolerance for arpack.
        Default: 0 (optimal value will be chosen by arpack)

    max_iter : int
        maximum number of iterations for arpack
        Default: None (optimal value will be chosen by arpack)

    remove_zero_eig : boolean, default=True
        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.

    Attributes
    ----------

    `lambdas_`, `alphas_`:
        Eigenvalues and eigenvectors of the centered kernel matrix

    `dual_coef_`:
        Inverse transform matrix

    `X_transformed_fit_`:
        Projection of the fitted data on the kernel principal components

    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.
    """

    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, remove_zero_eig=False):
        if fit_inverse_transform and kernel == 'precomputed':
            raise ValueError(
                "Cannot fit_inverse_transform with a precomputed kernel.")
        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.remove_zero_eig = remove_zero_eig
        self.tol = tol
        self.max_iter = max_iter
        self._centerer = KernelCenterer()

    @property
    def _pairwise(self):
        return self.kernel == "precomputed"

    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, **params)

    def _fit_transform(self, K):
        """ Fit's using kernel K"""
        # center kernel
        K = self._centerer.fit_transform(K)

        if self.n_components is None:
            n_components = K.shape[0]
        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':
            self.lambdas_, self.alphas_ = linalg.eigh(
                K, eigvals=(K.shape[0] - n_components, K.shape[0] - 1))
        elif eigen_solver == 'arpack':
            self.lambdas_, self.alphas_ = eigsh(K, n_components,
                                                which="LA",
                                                tol=self.tol,
                                                maxiter=self.max_iter)

        # sort eigenvectors in descending order
        indices = self.lambdas_.argsort()[::-1]
        self.lambdas_ = self.lambdas_[indices]
        self.alphas_ = self.alphas_[:, indices]

        # remove eigenvectors with a zero eigenvalue
        if self.remove_zero_eig or self.n_components is None:
            self.alphas_ = self.alphas_[:, self.lambdas_ > 0]
            self.lambdas_ = self.lambdas_[self.lambdas_ > 0]

        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, sym_pos=True, overwrite_a=True)
        self.X_transformed_fit_ = X_transformed

    def fit(self, X, y=None):
        """Fit the model from data in X.

        Parameters
        ----------
        X: array-like, shape (n_samples, n_features)
            Training vector, where n_samples in the number of samples
            and n_features is the number of features.

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        K = self._get_kernel(X)
        self._fit_transform(K)

        if self.fit_inverse_transform:
            sqrt_lambdas = np.diag(np.sqrt(self.lambdas_))
            X_transformed = np.dot(self.alphas_, sqrt_lambdas)
            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, 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_new: array-like, shape (n_samples, n_components)
        """
        self.fit(X, **params)

        X_transformed = self.alphas_ * np.sqrt(self.lambdas_)

        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, shape (n_samples, n_features)

        Returns
        -------
        X_new: array-like, shape (n_samples, n_components)
        """
        K = self._centerer.transform(self._get_kernel(X, self.X_fit_))
        return np.dot(K, self.alphas_ / np.sqrt(self.lambdas_))

    def inverse_transform(self, X):
        """Transform X back to original space.

        Parameters
        ----------
        X: array-like, shape (n_samples, n_components)

        Returns
        -------
        X_new: array-like, shape (n_samples, n_features)

        References
        ----------
        "Learning to Find Pre-Images", G BakIr et al, 2004.
        """
        if not self.fit_inverse_transform:
            raise ValueError("Inverse transform was not fitted!")

        K = self._get_kernel(X, self.X_transformed_fit_)

        return np.dot(K, self.dual_coef_)