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# `sklearn.decomposition`.KernelPCA

class sklearn.decomposition.KernelPCA(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)

  Kernel Principal component analysis (KPCA) [1].

  Non-linear dimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels).

  It uses the `scipy.linalg.eigh` LAPACK implementation of the full SVD or the `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 User Guide.

  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].

      Changed in version 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.

      New in version 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 Glossary.

      New in version 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.

      New in version 0.18.

    n_jobs int, default=None

      The number of parallel jobs to run. `None` means 1 unless in a `joblib.parallel_backend` context. `-1` means using all processors. See Glossary for more details.

      New in version 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 fit.

      New in version 0.24.

    feature_names_in_ndarray of shape (`n_features_in_`,)

      Names of features seen during fit. Defined only when `X` has feature names that are all strings.

      New in version 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.

      New in version 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.

  [2]

  Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf. “Learning to find pre-images.” Advances in neural information processing systems 16 (2004): 449-456.

  [3] (1,2,3)

  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.

  [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.

  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)

  Methods

  `fit`(X[, y]) | Fit the model from data in X.
  `fit_transform`(X[, y]) | Fit the model from data in X and transform X.
  `get_feature_names_out`([input_features]) | Get output feature names for transformation.
  `get_metadata_routing`() | Get metadata routing of this object.
  `get_params`([deep]) | Get parameters for this estimator.
  `inverse_transform`(X) | Transform X back to original space.
  `set_output`(*[, transform]) | Set output container.
  `set_params`(**params) | Set the parameters of this estimator.
  `transform`(X) | Transform X.

  fit(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.

  fit_transform(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.

  get_feature_names_out(input_features=None)

    Get output feature names for transformation.

    The feature names out will prefixed by the lowercased class name. For example, if the transformer outputs 3 features, then the feature names out are: `["class_name0", "class_name1", "class_name2"]`.

    Parameters:
      input_features array-like of str or None, default=None

        Only used to validate feature names with the names seen in `fit`.

    Returns:
      feature_names_out ndarray of str objects

        Transformed feature names.

  get_metadata_routing()

    Get metadata routing of this object.

    Please check User Guide on how the routing mechanism works.

    Returns:
      routing MetadataRequest

        A `MetadataRequest` encapsulating routing information.

  get_params(deep=True)

    Get parameters for this estimator.

    Parameters:
      deep bool, default=True

        If True, will return the parameters for this estimator and contained subobjects that are estimators.

    Returns:
      params dict

        Parameter names mapped to their values.

  inverse_transform(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

    When users want to compute inverse transformation for ‘linear’ kernel, it is recommended that they use `PCA` instead. Unlike `PCA`, `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.

  set_output(*, transform=None)

    Set output container.

    See Introducing the set_output API for an example on how to use the API.

    Parameters:
      transform{“default”, “pandas”}, default=None

        Configure output of `transform` and `fit_transform`.

        - `"default"`: Default output format of a transformer

        - `"pandas"`: DataFrame output

        - `None`: Transform configuration is unchanged

    Returns:
      self estimator instance

        Estimator instance.

  set_params(**params)

    Set the parameters of this estimator.

    The method works on simple estimators as well as on nested objects (such as `Pipeline`). The latter have parameters of the form `<component>__<parameter>` so that it’s possible to update each component of a nested object.

    Parameters:
      **params dict

        Estimator parameters.

    Returns:
      self estimator instance

        Estimator instance.

  transform(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.

## Examples using `sklearn.decomposition.KernelPCA`

Kernel PCA

Kernel PCA

Image denoising using kernel PCA

Image denoising using kernel PCA