Implementation:Scikit learn Scikit learn KernelPCA

Knowledge Sources	Scikit_learn Scikit-learn Docs
Domains	Dimensionality Reduction, Kernel Methods
Last Updated	2026-02-08 15:00 GMT

Overview

Concrete tool for non-linear dimensionality reduction through kernel principal component analysis provided by scikit-learn.

Description

KernelPCA performs non-linear dimensionality reduction by applying PCA in a high-dimensional feature space induced by a kernel function. It uses the LAPACK implementation of full SVD or the ARPACK implementation of truncated SVD, depending on the input data shape and number of components to extract. It also supports a randomized truncated SVD solver. Multiple kernel functions are available including linear, polynomial, RBF, sigmoid, and cosine, as well as precomputed kernel matrices.

Usage

Use KernelPCA when you need to capture non-linear relationships in your data that standard PCA would miss. It is ideal for non-linear feature extraction, manifold learning, and denoising applications where the underlying data structure is not well represented by linear projections.

Code Reference

Source Location

Repository: scikit-learn
File: sklearn/decomposition/_kernel_pca.py

Signature

class KernelPCA(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
    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,
    ):

Import

from sklearn.decomposition import KernelPCA

I/O Contract

Inputs

Name	Type	Required	Description
n_components	int	No	Number of components. If None, all non-zero components are kept.
kernel	str or callable	No	Kernel function: 'linear', 'poly', 'rbf', 'sigmoid', 'cosine', 'precomputed', or callable (default='linear').
gamma	float	No	Kernel coefficient for rbf, poly, and sigmoid kernels. Defaults to 1/n_features.
degree	float	No	Degree for poly kernels (default=3).
coef0	float	No	Independent term in poly and sigmoid kernels (default=1).
kernel_params	dict	No	Parameters for callable kernel.
alpha	float	No	Ridge regression hyperparameter for inverse transform (default=1.0).
fit_inverse_transform	bool	No	Whether to learn the inverse transform (default=False).
eigen_solver	str	No	Solver: 'auto', 'dense', 'arpack', or 'randomized' (default='auto').
tol	float	No	Convergence tolerance for arpack solver (default=0).
max_iter	int	No	Maximum number of iterations for arpack solver.
random_state	int or RandomState	No	Random state for reproducibility.

Outputs

Name	Type	Description
eigenvalues_	ndarray of shape (n_components,)	Eigenvalues of the centered kernel matrix in decreasing order.
eigenvectors_	ndarray of shape (n_samples, n_components)	Eigenvectors of the centered kernel matrix.
dual_coef_	ndarray of shape (n_samples, n_features)	Inverse transform matrix (only available when fit_inverse_transform=True).
X_fit_	ndarray of shape (n_samples, n_features)	The data used to fit the model.
n_features_in_	int	Number of features seen during fit.

Usage Examples

Basic Usage

import numpy as np
from sklearn.decomposition import KernelPCA

X = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype=float)
kpca = KernelPCA(n_components=2, kernel='rbf', gamma=0.1)
X_transformed = kpca.fit_transform(X)
print(X_transformed.shape)  # (4, 2)
print(kpca.eigenvalues_)

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