scikit-learn/sklearn/linear_model/_omp.py

"""Orthogonal matching pursuit algorithms
"""

# Author: Vlad Niculae
#
# License: BSD 3 clause

import warnings
from math import sqrt

import numpy as np
from scipy import linalg
from scipy.linalg.lapack import get_lapack_funcs
from joblib import Parallel

from ._base import LinearModel, _pre_fit
from ..base import RegressorMixin, MultiOutputMixin
from ..utils import as_float_array, check_array
from ..utils.validation import _deprecate_positional_args
from ..utils.fixes import delayed
from ..model_selection import check_cv

premature = """ Orthogonal matching pursuit ended prematurely due to linear
dependence in the dictionary. The requested precision might not have been met.
"""


def _cholesky_omp(X, y, n_nonzero_coefs, tol=None, copy_X=True,
                  return_path=False):
    """Orthogonal Matching Pursuit step using the Cholesky decomposition.

    Parameters
    ----------
    X : ndarray of shape (n_samples, n_features)
        Input dictionary. Columns are assumed to have unit norm.

    y : ndarray of shape (n_samples,)
        Input targets.

    n_nonzero_coefs : int
        Targeted number of non-zero elements.

    tol : float, default=None
        Targeted squared error, if not None overrides n_nonzero_coefs.

    copy_X : bool, default=True
        Whether the design matrix X must be copied by the algorithm. A false
        value is only helpful if X is already Fortran-ordered, otherwise a
        copy is made anyway.

    return_path : bool, default=False
        Whether to return every value of the nonzero coefficients along the
        forward path. Useful for cross-validation.

    Returns
    -------
    gamma : ndarray of shape (n_nonzero_coefs,)
        Non-zero elements of the solution.

    idx : ndarray of shape (n_nonzero_coefs,)
        Indices of the positions of the elements in gamma within the solution
        vector.

    coef : ndarray of shape (n_features, n_nonzero_coefs)
        The first k values of column k correspond to the coefficient value
        for the active features at that step. The lower left triangle contains
        garbage. Only returned if ``return_path=True``.

    n_active : int
        Number of active features at convergence.
    """
    if copy_X:
        X = X.copy('F')
    else:  # even if we are allowed to overwrite, still copy it if bad order
        X = np.asfortranarray(X)

    min_float = np.finfo(X.dtype).eps
    nrm2, swap = linalg.get_blas_funcs(('nrm2', 'swap'), (X,))
    potrs, = get_lapack_funcs(('potrs',), (X,))

    alpha = np.dot(X.T, y)
    residual = y
    gamma = np.empty(0)
    n_active = 0
    indices = np.arange(X.shape[1])  # keeping track of swapping

    max_features = X.shape[1] if tol is not None else n_nonzero_coefs

    L = np.empty((max_features, max_features), dtype=X.dtype)

    if return_path:
        coefs = np.empty_like(L)

    while True:
        lam = np.argmax(np.abs(np.dot(X.T, residual)))
        if lam < n_active or alpha[lam] ** 2 < min_float:
            # atom already selected or inner product too small
            warnings.warn(premature, RuntimeWarning, stacklevel=2)
            break

        if n_active > 0:
            # Updates the Cholesky decomposition of X' X
            L[n_active, :n_active] = np.dot(X[:, :n_active].T, X[:, lam])
            linalg.solve_triangular(L[:n_active, :n_active],
                                    L[n_active, :n_active],
                                    trans=0, lower=1,
                                    overwrite_b=True,
                                    check_finite=False)
            v = nrm2(L[n_active, :n_active]) ** 2
            Lkk = linalg.norm(X[:, lam]) ** 2 - v
            if Lkk <= min_float:  # selected atoms are dependent
                warnings.warn(premature, RuntimeWarning, stacklevel=2)
                break
            L[n_active, n_active] = sqrt(Lkk)
        else:
            L[0, 0] = linalg.norm(X[:, lam])

        X.T[n_active], X.T[lam] = swap(X.T[n_active], X.T[lam])
        alpha[n_active], alpha[lam] = alpha[lam], alpha[n_active]
        indices[n_active], indices[lam] = indices[lam], indices[n_active]
        n_active += 1

        # solves LL'x = X'y as a composition of two triangular systems
        gamma, _ = potrs(L[:n_active, :n_active], alpha[:n_active], lower=True,
                         overwrite_b=False)

        if return_path:
            coefs[:n_active, n_active - 1] = gamma
        residual = y - np.dot(X[:, :n_active], gamma)
        if tol is not None and nrm2(residual) ** 2 <= tol:
            break
        elif n_active == max_features:
            break

    if return_path:
        return gamma, indices[:n_active], coefs[:, :n_active], n_active
    else:
        return gamma, indices[:n_active], n_active


def _gram_omp(Gram, Xy, n_nonzero_coefs, tol_0=None, tol=None,
              copy_Gram=True, copy_Xy=True, return_path=False):
    """Orthogonal Matching Pursuit step on a precomputed Gram matrix.

    This function uses the Cholesky decomposition method.

    Parameters
    ----------
    Gram : ndarray of shape (n_features, n_features)
        Gram matrix of the input data matrix.

    Xy : ndarray of shape (n_features,)
        Input targets.

    n_nonzero_coefs : int
        Targeted number of non-zero elements.

    tol_0 : float, default=None
        Squared norm of y, required if tol is not None.

    tol : float, default=None
        Targeted squared error, if not None overrides n_nonzero_coefs.

    copy_Gram : bool, default=True
        Whether the gram matrix must be copied by the algorithm. A false
        value is only helpful if it is already Fortran-ordered, otherwise a
        copy is made anyway.

    copy_Xy : bool, default=True
        Whether the covariance vector Xy must be copied by the algorithm.
        If False, it may be overwritten.

    return_path : bool, default=False
        Whether to return every value of the nonzero coefficients along the
        forward path. Useful for cross-validation.

    Returns
    -------
    gamma : ndarray of shape (n_nonzero_coefs,)
        Non-zero elements of the solution.

    idx : ndarray of shape (n_nonzero_coefs,)
        Indices of the positions of the elements in gamma within the solution
        vector.

    coefs : ndarray of shape (n_features, n_nonzero_coefs)
        The first k values of column k correspond to the coefficient value
        for the active features at that step. The lower left triangle contains
        garbage. Only returned if ``return_path=True``.

    n_active : int
        Number of active features at convergence.
    """
    Gram = Gram.copy('F') if copy_Gram else np.asfortranarray(Gram)

    if copy_Xy or not Xy.flags.writeable:
        Xy = Xy.copy()

    min_float = np.finfo(Gram.dtype).eps
    nrm2, swap = linalg.get_blas_funcs(('nrm2', 'swap'), (Gram,))
    potrs, = get_lapack_funcs(('potrs',), (Gram,))

    indices = np.arange(len(Gram))  # keeping track of swapping
    alpha = Xy
    tol_curr = tol_0
    delta = 0
    gamma = np.empty(0)
    n_active = 0

    max_features = len(Gram) if tol is not None else n_nonzero_coefs

    L = np.empty((max_features, max_features), dtype=Gram.dtype)

    L[0, 0] = 1.
    if return_path:
        coefs = np.empty_like(L)

    while True:
        lam = np.argmax(np.abs(alpha))
        if lam < n_active or alpha[lam] ** 2 < min_float:
            # selected same atom twice, or inner product too small
            warnings.warn(premature, RuntimeWarning, stacklevel=3)
            break
        if n_active > 0:
            L[n_active, :n_active] = Gram[lam, :n_active]
            linalg.solve_triangular(L[:n_active, :n_active],
                                    L[n_active, :n_active],
                                    trans=0, lower=1,
                                    overwrite_b=True,
                                    check_finite=False)
            v = nrm2(L[n_active, :n_active]) ** 2
            Lkk = Gram[lam, lam] - v
            if Lkk <= min_float:  # selected atoms are dependent
                warnings.warn(premature, RuntimeWarning, stacklevel=3)
                break
            L[n_active, n_active] = sqrt(Lkk)
        else:
            L[0, 0] = sqrt(Gram[lam, lam])

        Gram[n_active], Gram[lam] = swap(Gram[n_active], Gram[lam])
        Gram.T[n_active], Gram.T[lam] = swap(Gram.T[n_active], Gram.T[lam])
        indices[n_active], indices[lam] = indices[lam], indices[n_active]
        Xy[n_active], Xy[lam] = Xy[lam], Xy[n_active]
        n_active += 1
        # solves LL'x = X'y as a composition of two triangular systems
        gamma, _ = potrs(L[:n_active, :n_active], Xy[:n_active], lower=True,
                         overwrite_b=False)
        if return_path:
            coefs[:n_active, n_active - 1] = gamma
        beta = np.dot(Gram[:, :n_active], gamma)
        alpha = Xy - beta
        if tol is not None:
            tol_curr += delta
            delta = np.inner(gamma, beta[:n_active])
            tol_curr -= delta
            if abs(tol_curr) <= tol:
                break
        elif n_active == max_features:
            break

    if return_path:
        return gamma, indices[:n_active], coefs[:, :n_active], n_active
    else:
        return gamma, indices[:n_active], n_active


@_deprecate_positional_args
def orthogonal_mp(X, y, *, n_nonzero_coefs=None, tol=None, precompute=False,
                  copy_X=True, return_path=False,
                  return_n_iter=False):
    r"""Orthogonal Matching Pursuit (OMP).

    Solves n_targets Orthogonal Matching Pursuit problems.
    An instance of the problem has the form:

    When parametrized by the number of non-zero coefficients using
    `n_nonzero_coefs`:
    argmin ||y - X\gamma||^2 subject to ||\gamma||_0 <= n_{nonzero coefs}

    When parametrized by error using the parameter `tol`:
    argmin ||\gamma||_0 subject to ||y - X\gamma||^2 <= tol

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

    Parameters
    ----------
    X : ndarray of shape (n_samples, n_features)
        Input data. Columns are assumed to have unit norm.

    y : ndarray of shape (n_samples,) or (n_samples, n_targets)
        Input targets.

    n_nonzero_coefs : int, default=None
        Desired number of non-zero entries in the solution. If None (by
        default) this value is set to 10% of n_features.

    tol : float, default=None
        Maximum norm of the residual. If not None, overrides n_nonzero_coefs.

    precompute : 'auto' or bool, default=False
        Whether to perform precomputations. Improves performance when n_targets
        or n_samples is very large.

    copy_X : bool, default=True
        Whether the design matrix X must be copied by the algorithm. A false
        value is only helpful if X is already Fortran-ordered, otherwise a
        copy is made anyway.

    return_path : bool, default=False
        Whether to return every value of the nonzero coefficients along the
        forward path. Useful for cross-validation.

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

    Returns
    -------
    coef : ndarray of shape (n_features,) or (n_features, n_targets)
        Coefficients of the OMP solution. If `return_path=True`, this contains
        the whole coefficient path. In this case its shape is
        (n_features, n_features) or (n_features, n_targets, n_features) and
        iterating over the last axis yields coefficients in increasing order
        of active features.

    n_iters : array-like or int
        Number of active features across every target. Returned only if
        `return_n_iter` is set to True.

    See Also
    --------
    OrthogonalMatchingPursuit
    orthogonal_mp_gram
    lars_path
    sklearn.decomposition.sparse_encode

    Notes
    -----
    Orthogonal matching pursuit was introduced in S. Mallat, Z. Zhang,
    Matching pursuits with time-frequency dictionaries, IEEE Transactions on
    Signal Processing, Vol. 41, No. 12. (December 1993), pp. 3397-3415.
    (http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf)

    This implementation is based on Rubinstein, R., Zibulevsky, M. and Elad,
    M., Efficient Implementation of the K-SVD Algorithm using Batch Orthogonal
    Matching Pursuit Technical Report - CS Technion, April 2008.
    https://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf

    """
    X = check_array(X, order='F', copy=copy_X)
    copy_X = False
    if y.ndim == 1:
        y = y.reshape(-1, 1)
    y = check_array(y)
    if y.shape[1] > 1:  # subsequent targets will be affected
        copy_X = True
    if n_nonzero_coefs is None and tol is None:
        # default for n_nonzero_coefs is 0.1 * n_features
        # but at least one.
        n_nonzero_coefs = max(int(0.1 * X.shape[1]), 1)
    if tol is not None and tol < 0:
        raise ValueError("Epsilon cannot be negative")
    if tol is None and n_nonzero_coefs <= 0:
        raise ValueError("The number of atoms must be positive")
    if tol is None and n_nonzero_coefs > X.shape[1]:
        raise ValueError("The number of atoms cannot be more than the number "
                         "of features")
    if precompute == 'auto':
        precompute = X.shape[0] > X.shape[1]
    if precompute:
        G = np.dot(X.T, X)
        G = np.asfortranarray(G)
        Xy = np.dot(X.T, y)
        if tol is not None:
            norms_squared = np.sum((y ** 2), axis=0)
        else:
            norms_squared = None
        return orthogonal_mp_gram(G, Xy, n_nonzero_coefs=n_nonzero_coefs,
                                  tol=tol, norms_squared=norms_squared,
                                  copy_Gram=copy_X, copy_Xy=False,
                                  return_path=return_path)

    if return_path:
        coef = np.zeros((X.shape[1], y.shape[1], X.shape[1]))
    else:
        coef = np.zeros((X.shape[1], y.shape[1]))
    n_iters = []

    for k in range(y.shape[1]):
        out = _cholesky_omp(
            X, y[:, k], n_nonzero_coefs, tol,
            copy_X=copy_X, return_path=return_path)
        if return_path:
            _, idx, coefs, n_iter = out
            coef = coef[:, :, :len(idx)]
            for n_active, x in enumerate(coefs.T):
                coef[idx[:n_active + 1], k, n_active] = x[:n_active + 1]
        else:
            x, idx, n_iter = out
            coef[idx, k] = x
        n_iters.append(n_iter)

    if y.shape[1] == 1:
        n_iters = n_iters[0]

    if return_n_iter:
        return np.squeeze(coef), n_iters
    else:
        return np.squeeze(coef)


@_deprecate_positional_args
def orthogonal_mp_gram(Gram, Xy, *, n_nonzero_coefs=None, tol=None,
                       norms_squared=None, copy_Gram=True,
                       copy_Xy=True, return_path=False,
                       return_n_iter=False):
    """Gram Orthogonal Matching Pursuit (OMP).

    Solves n_targets Orthogonal Matching Pursuit problems using only
    the Gram matrix X.T * X and the product X.T * y.

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

    Parameters
    ----------
    Gram : ndarray of shape (n_features, n_features)
        Gram matrix of the input data: X.T * X.

    Xy : ndarray of shape (n_features,) or (n_features, n_targets)
        Input targets multiplied by X: X.T * y.

    n_nonzero_coefs : int, default=None
        Desired number of non-zero entries in the solution. If None (by
        default) this value is set to 10% of n_features.

    tol : float, default=None
        Maximum norm of the residual. If not None, overrides n_nonzero_coefs.

    norms_squared : array-like of shape (n_targets,), default=None
        Squared L2 norms of the lines of y. Required if tol is not None.

    copy_Gram : bool, default=True
        Whether the gram matrix must be copied by the algorithm. A false
        value is only helpful if it is already Fortran-ordered, otherwise a
        copy is made anyway.

    copy_Xy : bool, default=True
        Whether the covariance vector Xy must be copied by the algorithm.
        If False, it may be overwritten.

    return_path : bool, default=False
        Whether to return every value of the nonzero coefficients along the
        forward path. Useful for cross-validation.

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

    Returns
    -------
    coef : ndarray of shape (n_features,) or (n_features, n_targets)
        Coefficients of the OMP solution. If `return_path=True`, this contains
        the whole coefficient path. In this case its shape is
        (n_features, n_features) or (n_features, n_targets, n_features) and
        iterating over the last axis yields coefficients in increasing order
        of active features.

    n_iters : array-like or int
        Number of active features across every target. Returned only if
        `return_n_iter` is set to True.

    See Also
    --------
    OrthogonalMatchingPursuit
    orthogonal_mp
    lars_path
    sklearn.decomposition.sparse_encode

    Notes
    -----
    Orthogonal matching pursuit was introduced in G. Mallat, Z. Zhang,
    Matching pursuits with time-frequency dictionaries, IEEE Transactions on
    Signal Processing, Vol. 41, No. 12. (December 1993), pp. 3397-3415.
    (http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf)

    This implementation is based on Rubinstein, R., Zibulevsky, M. and Elad,
    M., Efficient Implementation of the K-SVD Algorithm using Batch Orthogonal
    Matching Pursuit Technical Report - CS Technion, April 2008.
    https://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf

    """
    Gram = check_array(Gram, order='F', copy=copy_Gram)
    Xy = np.asarray(Xy)
    if Xy.ndim > 1 and Xy.shape[1] > 1:
        # or subsequent target will be affected
        copy_Gram = True
    if Xy.ndim == 1:
        Xy = Xy[:, np.newaxis]
        if tol is not None:
            norms_squared = [norms_squared]
    if copy_Xy or not Xy.flags.writeable:
        # Make the copy once instead of many times in _gram_omp itself.
        Xy = Xy.copy()

    if n_nonzero_coefs is None and tol is None:
        n_nonzero_coefs = int(0.1 * len(Gram))
    if tol is not None and norms_squared is None:
        raise ValueError('Gram OMP needs the precomputed norms in order '
                         'to evaluate the error sum of squares.')
    if tol is not None and tol < 0:
        raise ValueError("Epsilon cannot be negative")
    if tol is None and n_nonzero_coefs <= 0:
        raise ValueError("The number of atoms must be positive")
    if tol is None and n_nonzero_coefs > len(Gram):
        raise ValueError("The number of atoms cannot be more than the number "
                         "of features")

    if return_path:
        coef = np.zeros((len(Gram), Xy.shape[1], len(Gram)))
    else:
        coef = np.zeros((len(Gram), Xy.shape[1]))

    n_iters = []
    for k in range(Xy.shape[1]):
        out = _gram_omp(
            Gram, Xy[:, k], n_nonzero_coefs,
            norms_squared[k] if tol is not None else None, tol,
            copy_Gram=copy_Gram, copy_Xy=False,
            return_path=return_path)
        if return_path:
            _, idx, coefs, n_iter = out
            coef = coef[:, :, :len(idx)]
            for n_active, x in enumerate(coefs.T):
                coef[idx[:n_active + 1], k, n_active] = x[:n_active + 1]
        else:
            x, idx, n_iter = out
            coef[idx, k] = x
        n_iters.append(n_iter)

    if Xy.shape[1] == 1:
        n_iters = n_iters[0]

    if return_n_iter:
        return np.squeeze(coef), n_iters
    else:
        return np.squeeze(coef)


class OrthogonalMatchingPursuit(MultiOutputMixin, RegressorMixin, LinearModel):
    """Orthogonal Matching Pursuit model (OMP).

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

    Parameters
    ----------
    n_nonzero_coefs : int, default=None
        Desired number of non-zero entries in the solution. If None (by
        default) this value is set to 10% of n_features.

    tol : float, default=None
        Maximum norm of the residual. If not None, overrides n_nonzero_coefs.

    fit_intercept : bool, default=True
        whether to calculate the intercept for this model. If set
        to false, no intercept will be used in calculations
        (i.e. data is expected to be centered).

    normalize : bool, default=True
        This parameter is ignored when ``fit_intercept`` is set to False.
        If True, the regressors X will be normalized before regression by
        subtracting the mean and dividing by the l2-norm.
        If you wish to standardize, please use
        :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
        on an estimator with ``normalize=False``.

    precompute : 'auto' or bool, default='auto'
        Whether to use a precomputed Gram and Xy matrix to speed up
        calculations. Improves performance when :term:`n_targets` or
        :term:`n_samples` is very large. Note that if you already have such
        matrices, you can pass them directly to the fit method.

    Attributes
    ----------
    coef_ : ndarray of shape (n_features,) or (n_targets, n_features)
        Parameter vector (w in the formula).

    intercept_ : float or ndarray of shape (n_targets,)
        Independent term in decision function.

    n_iter_ : int or array-like
        Number of active features across every target.

    Examples
    --------
    >>> from sklearn.linear_model import OrthogonalMatchingPursuit
    >>> from sklearn.datasets import make_regression
    >>> X, y = make_regression(noise=4, random_state=0)
    >>> reg = OrthogonalMatchingPursuit().fit(X, y)
    >>> reg.score(X, y)
    0.9991...
    >>> reg.predict(X[:1,])
    array([-78.3854...])

    Notes
    -----
    Orthogonal matching pursuit was introduced in G. Mallat, Z. Zhang,
    Matching pursuits with time-frequency dictionaries, IEEE Transactions on
    Signal Processing, Vol. 41, No. 12. (December 1993), pp. 3397-3415.
    (http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf)

    This implementation is based on Rubinstein, R., Zibulevsky, M. and Elad,
    M., Efficient Implementation of the K-SVD Algorithm using Batch Orthogonal
    Matching Pursuit Technical Report - CS Technion, April 2008.
    https://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf

    See Also
    --------
    orthogonal_mp
    orthogonal_mp_gram
    lars_path
    Lars
    LassoLars
    sklearn.decomposition.sparse_encode
    OrthogonalMatchingPursuitCV
    """
    @_deprecate_positional_args
    def __init__(self, *, n_nonzero_coefs=None, tol=None, fit_intercept=True,
                 normalize=True, precompute='auto'):
        self.n_nonzero_coefs = n_nonzero_coefs
        self.tol = tol
        self.fit_intercept = fit_intercept
        self.normalize = normalize
        self.precompute = precompute

    def fit(self, X, y):
        """Fit the model using X, y as training data.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data.

        y : array-like of shape (n_samples,) or (n_samples, n_targets)
            Target values. Will be cast to X's dtype if necessary


        Returns
        -------
        self : object
            returns an instance of self.
        """
        X, y = self._validate_data(X, y, multi_output=True, y_numeric=True)
        n_features = X.shape[1]

        X, y, X_offset, y_offset, X_scale, Gram, Xy = \
            _pre_fit(X, y, None, self.precompute, self.normalize,
                     self.fit_intercept, copy=True)

        if y.ndim == 1:
            y = y[:, np.newaxis]

        if self.n_nonzero_coefs is None and self.tol is None:
            # default for n_nonzero_coefs is 0.1 * n_features
            # but at least one.
            self.n_nonzero_coefs_ = max(int(0.1 * n_features), 1)
        else:
            self.n_nonzero_coefs_ = self.n_nonzero_coefs

        if Gram is False:
            coef_, self.n_iter_ = orthogonal_mp(
                X, y, n_nonzero_coefs=self.n_nonzero_coefs_, tol=self.tol,
                precompute=False, copy_X=True,
                return_n_iter=True)
        else:
            norms_sq = np.sum(y ** 2, axis=0) if self.tol is not None else None

            coef_, self.n_iter_ = orthogonal_mp_gram(
                Gram, Xy=Xy, n_nonzero_coefs=self.n_nonzero_coefs_,
                tol=self.tol, norms_squared=norms_sq,
                copy_Gram=True, copy_Xy=True,
                return_n_iter=True)
        self.coef_ = coef_.T
        self._set_intercept(X_offset, y_offset, X_scale)
        return self


def _omp_path_residues(X_train, y_train, X_test, y_test, copy=True,
                       fit_intercept=True, normalize=True, max_iter=100):
    """Compute the residues on left-out data for a full LARS path.

    Parameters
    ----------
    X_train : ndarray of shape (n_samples, n_features)
        The data to fit the LARS on.

    y_train : ndarray of shape (n_samples)
        The target variable to fit LARS on.

    X_test : ndarray of shape (n_samples, n_features)
        The data to compute the residues on.

    y_test : ndarray of shape (n_samples)
        The target variable to compute the residues on.

    copy : bool, default=True
        Whether X_train, X_test, y_train and y_test should be copied.  If
        False, they may be overwritten.

    fit_intercept : bool, default=True
        Whether to calculate the intercept for this model. If set
        to false, no intercept will be used in calculations
        (i.e. data is expected to be centered).

    normalize : bool, default=True
        This parameter is ignored when ``fit_intercept`` is set to False.
        If True, the regressors X will be normalized before regression by
        subtracting the mean and dividing by the l2-norm.
        If you wish to standardize, please use
        :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
        on an estimator with ``normalize=False``.

    max_iter : int, default=100
        Maximum numbers of iterations to perform, therefore maximum features
        to include. 100 by default.

    Returns
    -------
    residues : ndarray of shape (n_samples, max_features)
        Residues of the prediction on the test data.
    """

    if copy:
        X_train = X_train.copy()
        y_train = y_train.copy()
        X_test = X_test.copy()
        y_test = y_test.copy()

    if fit_intercept:
        X_mean = X_train.mean(axis=0)
        X_train -= X_mean
        X_test -= X_mean
        y_mean = y_train.mean(axis=0)
        y_train = as_float_array(y_train, copy=False)
        y_train -= y_mean
        y_test = as_float_array(y_test, copy=False)
        y_test -= y_mean

    if normalize:
        norms = np.sqrt(np.sum(X_train ** 2, axis=0))
        nonzeros = np.flatnonzero(norms)
        X_train[:, nonzeros] /= norms[nonzeros]

    coefs = orthogonal_mp(X_train, y_train, n_nonzero_coefs=max_iter, tol=None,
                          precompute=False, copy_X=False,
                          return_path=True)
    if coefs.ndim == 1:
        coefs = coefs[:, np.newaxis]
    if normalize:
        coefs[nonzeros] /= norms[nonzeros][:, np.newaxis]

    return np.dot(coefs.T, X_test.T) - y_test


class OrthogonalMatchingPursuitCV(RegressorMixin, LinearModel):
    """Cross-validated Orthogonal Matching Pursuit model (OMP).

    See glossary entry for :term:`cross-validation estimator`.

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

    Parameters
    ----------
    copy : bool, default=True
        Whether the design matrix X must be copied by the algorithm. A false
        value is only helpful if X is already Fortran-ordered, otherwise a
        copy is made anyway.

    fit_intercept : bool, default=True
        whether to calculate the intercept for this model. If set
        to false, no intercept will be used in calculations
        (i.e. data is expected to be centered).

    normalize : bool, default=True
        This parameter is ignored when ``fit_intercept`` is set to False.
        If True, the regressors X will be normalized before regression by
        subtracting the mean and dividing by the l2-norm.
        If you wish to standardize, please use
        :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
        on an estimator with ``normalize=False``.

    max_iter : int, default=None
        Maximum numbers of iterations to perform, therefore maximum features
        to include. 10% of ``n_features`` but at least 5 if available.

    cv : int, cross-validation generator or iterable, default=None
        Determines the cross-validation splitting strategy.
        Possible inputs for cv are:

        - None, to use the default 5-fold cross-validation,
        - integer, to specify the number of folds.
        - :term:`CV splitter`,
        - An iterable yielding (train, test) splits as arrays of indices.

        For integer/None inputs, :class:`KFold` is used.

        Refer :ref:`User Guide <cross_validation>` for the various
        cross-validation strategies that can be used here.

        .. versionchanged:: 0.22
            ``cv`` default value if None changed from 3-fold to 5-fold.

    n_jobs : int, default=None
        Number of CPUs to use during the cross validation.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    verbose : bool or int, default=False
        Sets the verbosity amount.

    Attributes
    ----------
    intercept_ : float or ndarray of shape (n_targets,)
        Independent term in decision function.

    coef_ : ndarray of shape (n_features,) or (n_targets, n_features)
        Parameter vector (w in the problem formulation).

    n_nonzero_coefs_ : int
        Estimated number of non-zero coefficients giving the best mean squared
        error over the cross-validation folds.

    n_iter_ : int or array-like
        Number of active features across every target for the model refit with
        the best hyperparameters got by cross-validating across all folds.

    Examples
    --------
    >>> from sklearn.linear_model import OrthogonalMatchingPursuitCV
    >>> from sklearn.datasets import make_regression
    >>> X, y = make_regression(n_features=100, n_informative=10,
    ...                        noise=4, random_state=0)
    >>> reg = OrthogonalMatchingPursuitCV(cv=5).fit(X, y)
    >>> reg.score(X, y)
    0.9991...
    >>> reg.n_nonzero_coefs_
    10
    >>> reg.predict(X[:1,])
    array([-78.3854...])

    See Also
    --------
    orthogonal_mp
    orthogonal_mp_gram
    lars_path
    Lars
    LassoLars
    OrthogonalMatchingPursuit
    LarsCV
    LassoLarsCV
    sklearn.decomposition.sparse_encode

    """
    @_deprecate_positional_args
    def __init__(self, *, copy=True, fit_intercept=True, normalize=True,
                 max_iter=None, cv=None, n_jobs=None, verbose=False):
        self.copy = copy
        self.fit_intercept = fit_intercept
        self.normalize = normalize
        self.max_iter = max_iter
        self.cv = cv
        self.n_jobs = n_jobs
        self.verbose = verbose

    def fit(self, X, y):
        """Fit the model using X, y as training data.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data.

        y : array-like of shape (n_samples,)
            Target values. Will be cast to X's dtype if necessary.

        Returns
        -------
        self : object
            returns an instance of self.
        """
        X, y = self._validate_data(X, y, y_numeric=True, ensure_min_features=2,
                                   estimator=self)
        X = as_float_array(X, copy=False, force_all_finite=False)
        cv = check_cv(self.cv, classifier=False)
        max_iter = (min(max(int(0.1 * X.shape[1]), 5), X.shape[1])
                    if not self.max_iter
                    else self.max_iter)
        cv_paths = Parallel(n_jobs=self.n_jobs, verbose=self.verbose)(
            delayed(_omp_path_residues)(
                X[train], y[train], X[test], y[test], self.copy,
                self.fit_intercept, self.normalize, max_iter)
            for train, test in cv.split(X))

        min_early_stop = min(fold.shape[0] for fold in cv_paths)
        mse_folds = np.array([(fold[:min_early_stop] ** 2).mean(axis=1)
                              for fold in cv_paths])
        best_n_nonzero_coefs = np.argmin(mse_folds.mean(axis=0)) + 1
        self.n_nonzero_coefs_ = best_n_nonzero_coefs
        omp = OrthogonalMatchingPursuit(n_nonzero_coefs=best_n_nonzero_coefs,
                                        fit_intercept=self.fit_intercept,
                                        normalize=self.normalize)
        omp.fit(X, y)
        self.coef_ = omp.coef_
        self.intercept_ = omp.intercept_
        self.n_iter_ = omp.n_iter_
        return self