_fastica.py

"""
Python implementation of the fast ICA algorithms.

Reference: Tables 8.3 and 8.4 page 196 in the book:
Independent Component Analysis, by  Hyvarinen et al.
"""

# Authors: Pierre Lafaye de Micheaux, Stefan van der Walt, Gael Varoquaux,
#          Bertrand Thirion, Alexandre Gramfort, Denis A. Engemann
# License: BSD 3 clause

import warnings

import numpy as np
from scipy import linalg

from ..base import BaseEstimator, TransformerMixin, _ClassNamePrefixFeaturesOutMixin
from ..exceptions import ConvergenceWarning

from ..utils import check_array, as_float_array, check_random_state
from ..utils.validation import check_is_fitted

__all__ = ["fastica", "FastICA"]


def _gs_decorrelation(w, W, j):
    """
    Orthonormalize w wrt the first j rows of W.

    Parameters
    ----------
    w : ndarray of shape (n,)
        Array to be orthogonalized

    W : ndarray of shape (p, n)
        Null space definition

    j : int < p
        The no of (from the first) rows of Null space W wrt which w is
        orthogonalized.

    Notes
    -----
    Assumes that W is orthogonal
    w changed in place
    """
    w -= np.linalg.multi_dot([w, W[:j].T, W[:j]])
    return w


def _sym_decorrelation(W):
    """Symmetric decorrelation
    i.e. W <- (W * W.T) ^{-1/2} * W
    """
    s, u = linalg.eigh(np.dot(W, W.T))
    # Avoid sqrt of negative values because of rounding errors. Note that
    # np.sqrt(tiny) is larger than tiny and therefore this clipping also
    # prevents division by zero in the next step.
    s = np.clip(s, a_min=np.finfo(W.dtype).tiny, a_max=None)

    # u (resp. s) contains the eigenvectors (resp. square roots of
    # the eigenvalues) of W * W.T
    return np.linalg.multi_dot([u * (1.0 / np.sqrt(s)), u.T, W])


def _ica_def(X, tol, g, fun_args, max_iter, w_init):
    """Deflationary FastICA using fun approx to neg-entropy function

    Used internally by FastICA.
    """

    n_components = w_init.shape[0]
    W = np.zeros((n_components, n_components), dtype=X.dtype)
    n_iter = []

    # j is the index of the extracted component
    for j in range(n_components):
        w = w_init[j, :].copy()
        w /= np.sqrt((w**2).sum())

        for i in range(max_iter):
            gwtx, g_wtx = g(np.dot(w.T, X), fun_args)

            w1 = (X * gwtx).mean(axis=1) - g_wtx.mean() * w

            _gs_decorrelation(w1, W, j)

            w1 /= np.sqrt((w1**2).sum())

            lim = np.abs(np.abs((w1 * w).sum()) - 1)
            w = w1
            if lim < tol:
                break

        n_iter.append(i + 1)
        W[j, :] = w

    return W, max(n_iter)


def _ica_par(X, tol, g, fun_args, max_iter, w_init):
    """Parallel FastICA.

    Used internally by FastICA --main loop

    """
    W = _sym_decorrelation(w_init)
    del w_init
    p_ = float(X.shape[1])
    for ii in range(max_iter):
        gwtx, g_wtx = g(np.dot(W, X), fun_args)
        W1 = _sym_decorrelation(np.dot(gwtx, X.T) / p_ - g_wtx[:, np.newaxis] * W)
        del gwtx, g_wtx
        # builtin max, abs are faster than numpy counter parts.
        lim = max(abs(abs(np.diag(np.dot(W1, W.T))) - 1))
        W = W1
        if lim < tol:
            break
    else:
        warnings.warn(
            "FastICA did not converge. Consider increasing "
            "tolerance or the maximum number of iterations.",
            ConvergenceWarning,
        )

    return W, ii + 1


# Some standard non-linear functions.
# XXX: these should be optimized, as they can be a bottleneck.
def _logcosh(x, fun_args=None):
    alpha = fun_args.get("alpha", 1.0)  # comment it out?

    x *= alpha
    gx = np.tanh(x, x)  # apply the tanh inplace
    g_x = np.empty(x.shape[0], dtype=x.dtype)
    # XXX compute in chunks to avoid extra allocation
    for i, gx_i in enumerate(gx):  # please don't vectorize.
        g_x[i] = (alpha * (1 - gx_i**2)).mean()
    return gx, g_x


def _exp(x, fun_args):
    exp = np.exp(-(x**2) / 2)
    gx = x * exp
    g_x = (1 - x**2) * exp
    return gx, g_x.mean(axis=-1)


def _cube(x, fun_args):
    return x**3, (3 * x**2).mean(axis=-1)


def fastica(
    X,
    n_components=None,
    *,
    algorithm="parallel",
    whiten="warn",
    fun="logcosh",
    fun_args=None,
    max_iter=200,
    tol=1e-04,
    w_init=None,
    random_state=None,
    return_X_mean=False,
    compute_sources=True,
    return_n_iter=False,
):
    """Perform Fast Independent Component Analysis.

    The implementation is based on [1]_.

    Read more in the :ref:`User Guide <ICA>`.

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

    n_components : int, default=None
        Number of components to extract. If None no dimension reduction
        is performed.

    algorithm : {'parallel', 'deflation'}, default='parallel'
        Apply a parallel or deflational FASTICA algorithm.

    whiten : str or bool, default="warn"
        Specify the whitening strategy to use.
        If 'arbitrary-variance'  (default), a whitening with variance arbitrary is used.
        If 'unit-variance', the whitening matrix is rescaled to ensure that each
        recovered source has unit variance.
        If False, the data is already considered to be whitened, and no
        whitening is performed.

        .. deprecated:: 1.1
            From version 1.3, `whiten='unit-variance'` will be used by default.
            `whiten=True` is deprecated from 1.1 and will raise ValueError in 1.3.
            Use `whiten=arbitrary-variance` instead.

    fun : {'logcosh', 'exp', 'cube'} or callable, default='logcosh'
        The functional form of the G function used in the
        approximation to neg-entropy. Could be either 'logcosh', 'exp',
        or 'cube'.
        You can also provide your own function. It should return a tuple
        containing the value of the function, and of its derivative, in the
        point. The derivative should be averaged along its last dimension.
        Example:

        def my_g(x):
            return x ** 3, np.mean(3 * x ** 2, axis=-1)

    fun_args : dict, default=None
        Arguments to send to the functional form.
        If empty or None and if fun='logcosh', fun_args will take value
        {'alpha' : 1.0}.

    max_iter : int, default=200
        Maximum number of iterations to perform.

    tol : float, default=1e-04
        A positive scalar giving the tolerance at which the
        un-mixing matrix is considered to have converged.

    w_init : ndarray of shape (n_components, n_components), default=None
        Initial un-mixing array of dimension (n.comp,n.comp).
        If None (default) then an array of normal r.v.'s is used.

    random_state : int, RandomState instance or None, default=None
        Used to initialize ``w_init`` when not specified, with a
        normal distribution. Pass an int, for reproducible results
        across multiple function calls.
        See :term:`Glossary <random_state>`.

    return_X_mean : bool, default=False
        If True, X_mean is returned too.

    compute_sources : bool, default=True
        If False, sources are not computed, but only the rotation matrix.
        This can save memory when working with big data. Defaults to True.

    return_n_iter : bool, default=False
        Whether or not to return the number of iterations.

    Returns
    -------
    K : ndarray of shape (n_components, n_features) or None
        If whiten is 'True', K is the pre-whitening matrix that projects data
        onto the first n_components principal components. If whiten is 'False',
        K is 'None'.

    W : ndarray of shape (n_components, n_components)
        The square matrix that unmixes the data after whitening.
        The mixing matrix is the pseudo-inverse of matrix ``W K``
        if K is not None, else it is the inverse of W.

    S : ndarray of shape (n_samples, n_components) or None
        Estimated source matrix.

    X_mean : ndarray of shape (n_features,)
        The mean over features. Returned only if return_X_mean is True.

    n_iter : int
        If the algorithm is "deflation", n_iter is the
        maximum number of iterations run across all components. Else
        they are just the number of iterations taken to converge. This is
        returned only when return_n_iter is set to `True`.

    Notes
    -----
    The data matrix X is considered to be a linear combination of
    non-Gaussian (independent) components i.e. X = AS where columns of S
    contain the independent components and A is a linear mixing
    matrix. In short ICA attempts to `un-mix' the data by estimating an
    un-mixing matrix W where ``S = W K X.``
    While FastICA was proposed to estimate as many sources
    as features, it is possible to estimate less by setting
    n_components < n_features. It this case K is not a square matrix
    and the estimated A is the pseudo-inverse of ``W K``.

    This implementation was originally made for data of shape
    [n_features, n_samples]. Now the input is transposed
    before the algorithm is applied. This makes it slightly
    faster for Fortran-ordered input.

    References
    ----------
    .. [1] A. Hyvarinen and E. Oja, "Fast Independent Component Analysis",
           Algorithms and Applications, Neural Networks, 13(4-5), 2000,
           pp. 411-430.
    """
    est = FastICA(
        n_components=n_components,
        algorithm=algorithm,
        whiten=whiten,
        fun=fun,
        fun_args=fun_args,
        max_iter=max_iter,
        tol=tol,
        w_init=w_init,
        random_state=random_state,
    )
    S = est._fit(X, compute_sources=compute_sources)

    if est._whiten in ["unit-variance", "arbitrary-variance"]:
        K = est.whitening_
        X_mean = est.mean_
    else:
        K = None
        X_mean = None

    returned_values = [K, est._unmixing, S]
    if return_X_mean:
        returned_values.append(X_mean)
    if return_n_iter:
        returned_values.append(est.n_iter_)

    return returned_values


class FastICA(_ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
    """FastICA: a fast algorithm for Independent Component Analysis.

    The implementation is based on [1]_.

    Read more in the :ref:`User Guide <ICA>`.

    Parameters
    ----------
    n_components : int, default=None
        Number of components to use. If None is passed, all are used.

    algorithm : {'parallel', 'deflation'}, default='parallel'
        Apply parallel or deflational algorithm for FastICA.

    whiten : str or bool, default="warn"
        Specify the whitening strategy to use.
        If 'arbitrary-variance' (default), a whitening with variance arbitrary is used.
        If 'unit-variance', the whitening matrix is rescaled to ensure that each
        recovered source has unit variance.
        If False, the data is already considered to be whitened, and no
        whitening is performed.

        .. deprecated:: 1.1
            From version 1.3 whiten='unit-variance' will be used by default.
            `whiten=True` is deprecated from 1.1 and will raise ValueError in 1.3.
            Use `whiten=arbitrary-variance` instead.

    fun : {'logcosh', 'exp', 'cube'} or callable, default='logcosh'
        The functional form of the G function used in the
        approximation to neg-entropy. Could be either 'logcosh', 'exp',
        or 'cube'.
        You can also provide your own function. It should return a tuple
        containing the value of the function, and of its derivative, in the
        point. Example::

            def my_g(x):
                return x ** 3, (3 * x ** 2).mean(axis=-1)

    fun_args : dict, default=None
        Arguments to send to the functional form.
        If empty and if fun='logcosh', fun_args will take value
        {'alpha' : 1.0}.

    max_iter : int, default=200
        Maximum number of iterations during fit.

    tol : float, default=1e-4
        Tolerance on update at each iteration.

    w_init : ndarray of shape (n_components, n_components), default=None
        The mixing matrix to be used to initialize the algorithm.

    random_state : int, RandomState instance or None, default=None
        Used to initialize ``w_init`` when not specified, with a
        normal distribution. Pass an int, for reproducible results
        across multiple function calls.
        See :term:`Glossary <random_state>`.

    Attributes
    ----------
    components_ : ndarray of shape (n_components, n_features)
        The linear operator to apply to the data to get the independent
        sources. This is equal to the unmixing matrix when ``whiten`` is
        False, and equal to ``np.dot(unmixing_matrix, self.whitening_)`` when
        ``whiten`` is True.

    mixing_ : ndarray of shape (n_features, n_components)
        The pseudo-inverse of ``components_``. It is the linear operator
        that maps independent sources to the data.

    mean_ : ndarray of shape(n_features,)
        The mean over features. Only set if `self.whiten` is True.

    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

    n_iter_ : int
        If the algorithm is "deflation", n_iter is the
        maximum number of iterations run across all components. Else
        they are just the number of iterations taken to converge.

    whitening_ : ndarray of shape (n_components, n_features)
        Only set if whiten is 'True'. This is the pre-whitening matrix
        that projects data onto the first `n_components` principal components.

    See Also
    --------
    PCA : Principal component analysis (PCA).
    IncrementalPCA : Incremental principal components analysis (IPCA).
    KernelPCA : Kernel Principal component analysis (KPCA).
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.
    SparsePCA : Sparse Principal Components Analysis (SparsePCA).

    References
    ----------
    .. [1] A. Hyvarinen and E. Oja, Independent Component Analysis:
           Algorithms and Applications, Neural Networks, 13(4-5), 2000,
           pp. 411-430.

    Examples
    --------
    >>> from sklearn.datasets import load_digits
    >>> from sklearn.decomposition import FastICA
    >>> X, _ = load_digits(return_X_y=True)
    >>> transformer = FastICA(n_components=7,
    ...         random_state=0,
    ...         whiten='unit-variance')
    >>> X_transformed = transformer.fit_transform(X)
    >>> X_transformed.shape
    (1797, 7)
    """

    def __init__(
        self,
        n_components=None,
        *,
        algorithm="parallel",
        whiten="warn",
        fun="logcosh",
        fun_args=None,
        max_iter=200,
        tol=1e-4,
        w_init=None,
        random_state=None,
    ):
        super().__init__()
        self.n_components = n_components
        self.algorithm = algorithm
        self.whiten = whiten
        self.fun = fun
        self.fun_args = fun_args
        self.max_iter = max_iter
        self.tol = tol
        self.w_init = w_init
        self.random_state = random_state

    def _fit(self, X, compute_sources=False):
        """Fit the model.

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

        compute_sources : bool, default=False
            If False, sources are not computes but only the rotation matrix.
            This can save memory when working with big data. Defaults to False.

        Returns
        -------
        S : ndarray of shape (n_samples, n_components) or None
            Sources matrix. `None` if `compute_sources` is `False`.
        """
        self._whiten = self.whiten

        if self._whiten == "warn":
            warnings.warn(
                "From version 1.3 whiten='unit-variance' will be used by default.",
                FutureWarning,
            )
            self._whiten = "arbitrary-variance"

        if self._whiten is True:
            warnings.warn(
                "From version 1.3 whiten=True should be specified as "
                "whiten='arbitrary-variance' (its current behaviour). This "
                "behavior is deprecated in 1.1 and will raise ValueError in 1.3.",
                FutureWarning,
                stacklevel=2,
            )
            self._whiten = "arbitrary-variance"

        XT = self._validate_data(
            X, copy=self._whiten, dtype=[np.float64, np.float32], ensure_min_samples=2
        ).T
        fun_args = {} if self.fun_args is None else self.fun_args
        random_state = check_random_state(self.random_state)

        alpha = fun_args.get("alpha", 1.0)
        if not 1 <= alpha <= 2:
            raise ValueError("alpha must be in [1,2]")

        if self.fun == "logcosh":
            g = _logcosh
        elif self.fun == "exp":
            g = _exp
        elif self.fun == "cube":
            g = _cube
        elif callable(self.fun):

            def g(x, fun_args):
                return self.fun(x, **fun_args)

        else:
            exc = ValueError if isinstance(self.fun, str) else TypeError
            raise exc(
                "Unknown function %r;"
                " should be one of 'logcosh', 'exp', 'cube' or callable"
                % self.fun
            )

        n_features, n_samples = XT.shape

        n_components = self.n_components
        if not self._whiten and n_components is not None:
            n_components = None
            warnings.warn("Ignoring n_components with whiten=False.")

        if n_components is None:
            n_components = min(n_samples, n_features)
        if n_components > min(n_samples, n_features):
            n_components = min(n_samples, n_features)
            warnings.warn(
                "n_components is too large: it will be set to %s" % n_components
            )

        if self._whiten:
            # Centering the features of X
            X_mean = XT.mean(axis=-1)
            XT -= X_mean[:, np.newaxis]

            # Whitening and preprocessing by PCA
            u, d, _ = linalg.svd(XT, full_matrices=False, check_finite=False)

            del _
            K = (u / d).T[:n_components]  # see (6.33) p.140
            del u, d
            X1 = np.dot(K, XT)
            # see (13.6) p.267 Here X1 is white and data
            # in X has been projected onto a subspace by PCA
            X1 *= np.sqrt(n_samples)
        else:
            # X must be casted to floats to avoid typing issues with numpy
            # 2.0 and the line below
            X1 = as_float_array(XT, copy=False)  # copy has been taken care of

        w_init = self.w_init
        if w_init is None:
            w_init = np.asarray(
                random_state.normal(size=(n_components, n_components)), dtype=X1.dtype
            )

        else:
            w_init = np.asarray(w_init)
            if w_init.shape != (n_components, n_components):
                raise ValueError(
                    "w_init has invalid shape -- should be %(shape)s"
                    % {"shape": (n_components, n_components)}
                )

        if self.max_iter < 1:
            raise ValueError(
                "max_iter should be greater than 1, got (max_iter={})".format(
                    self.max_iter
                )
            )

        kwargs = {
            "tol": self.tol,
            "g": g,
            "fun_args": fun_args,
            "max_iter": self.max_iter,
            "w_init": w_init,
        }

        if self.algorithm == "parallel":
            W, n_iter = _ica_par(X1, **kwargs)
        elif self.algorithm == "deflation":
            W, n_iter = _ica_def(X1, **kwargs)
        else:
            raise ValueError(
                "Invalid algorithm: must be either `parallel` or `deflation`."
            )
        del X1

        self.n_iter_ = n_iter

        if compute_sources:
            if self._whiten:
                S = np.linalg.multi_dot([W, K, XT]).T
            else:
                S = np.dot(W, XT).T
        else:
            S = None

        if self._whiten:
            if self._whiten == "unit-variance":
                if not compute_sources:
                    S = np.linalg.multi_dot([W, K, XT]).T
                S_std = np.std(S, axis=0, keepdims=True)
                S /= S_std
                W /= S_std.T

            self.components_ = np.dot(W, K)
            self.mean_ = X_mean
            self.whitening_ = K
        else:
            self.components_ = W

        self.mixing_ = linalg.pinv(self.components_, check_finite=False)
        self._unmixing = W

        return S

    def fit_transform(self, X, y=None):
        """Fit the model and recover the sources from X.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data, 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
        -------
        X_new : ndarray of shape (n_samples, n_components)
            Estimated sources obtained by transforming the data with the
            estimated unmixing matrix.
        """
        return self._fit(X, compute_sources=True)

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

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data, 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.
        """
        self._fit(X, compute_sources=False)
        return self

    def transform(self, X, copy=True):
        """Recover the sources from X (apply the unmixing matrix).

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Data to transform, where `n_samples` is the number of samples
            and `n_features` is the number of features.

        copy : bool, default=True
            If False, data passed to fit can be overwritten. Defaults to True.

        Returns
        -------
        X_new : ndarray of shape (n_samples, n_components)
            Estimated sources obtained by transforming the data with the
            estimated unmixing matrix.
        """
        check_is_fitted(self)

        X = self._validate_data(
            X, copy=(copy and self._whiten), dtype=[np.float64, np.float32], reset=False
        )
        if self._whiten:
            X -= self.mean_

        return np.dot(X, self.components_.T)

    def inverse_transform(self, X, copy=True):
        """Transform the sources back to the mixed data (apply mixing matrix).

        Parameters
        ----------
        X : array-like of shape (n_samples, n_components)
            Sources, where `n_samples` is the number of samples
            and `n_components` is the number of components.
        copy : bool, default=True
            If False, data passed to fit are overwritten. Defaults to True.

        Returns
        -------
        X_new : ndarray of shape (n_samples, n_features)
            Reconstructed data obtained with the mixing matrix.
        """
        check_is_fitted(self)

        X = check_array(X, copy=(copy and self._whiten), dtype=[np.float64, np.float32])
        X = np.dot(X, self.mixing_.T)
        if self._whiten:
            X += self.mean_

        return X

    @property
    def _n_features_out(self):
        """Number of transformed output features."""
        return self.components_.shape[0]

    def _more_tags(self):
        return {"preserves_dtype": [np.float32, np.float64]}