_dict_learning.py

"""Dictionary learning."""

# Author: Vlad Niculae, Gael Varoquaux, Alexandre Gramfort
# License: BSD 3 clause

import itertools
import sys
import time
from numbers import Integral, Real
from warnings import warn

import numpy as np
from joblib import effective_n_jobs
from scipy import linalg

from ..base import (
    BaseEstimator,
    ClassNamePrefixFeaturesOutMixin,
    TransformerMixin,
    _fit_context,
)
from ..linear_model import Lars, Lasso, LassoLars, orthogonal_mp_gram
from ..utils import check_array, check_random_state, gen_batches, gen_even_slices
from ..utils._param_validation import Hidden, Interval, StrOptions, validate_params
from ..utils.extmath import randomized_svd, row_norms, svd_flip
from ..utils.parallel import Parallel, delayed
from ..utils.validation import check_is_fitted


def _check_positive_coding(method, positive):
    if positive and method in ["omp", "lars"]:
        raise ValueError(
            "Positive constraint not supported for '{}' coding method.".format(method)
        )


def _sparse_encode_precomputed(
    X,
    dictionary,
    *,
    gram=None,
    cov=None,
    algorithm="lasso_lars",
    regularization=None,
    copy_cov=True,
    init=None,
    max_iter=1000,
    verbose=0,
    positive=False,
):
    """Generic sparse coding with precomputed Gram and/or covariance matrices.

    Each row of the result is the solution to a Lasso problem.

    Parameters
    ----------
    X : ndarray of shape (n_samples, n_features)
        Data matrix.

    dictionary : ndarray of shape (n_components, n_features)
        The dictionary matrix against which to solve the sparse coding of
        the data. Some of the algorithms assume normalized rows.

    gram : ndarray of shape (n_components, n_components), default=None
        Precomputed Gram matrix, `dictionary * dictionary'`
        gram can be `None` if method is 'threshold'.

    cov : ndarray of shape (n_components, n_samples), default=None
        Precomputed covariance, `dictionary * X'`.

    algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', 'threshold'}, \
            default='lasso_lars'
        The algorithm used:

        * `'lars'`: uses the least angle regression method
          (`linear_model.lars_path`);
        * `'lasso_lars'`: uses Lars to compute the Lasso solution;
        * `'lasso_cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). lasso_lars will be faster if
          the estimated components are sparse;
        * `'omp'`: uses orthogonal matching pursuit to estimate the sparse
          solution;
        * `'threshold'`: squashes to zero all coefficients less than
          regularization from the projection `dictionary * data'`.

    regularization : int or float, default=None
        The regularization parameter. It corresponds to alpha when
        algorithm is `'lasso_lars'`, `'lasso_cd'` or `'threshold'`.
        Otherwise it corresponds to `n_nonzero_coefs`.

    init : ndarray of shape (n_samples, n_components), default=None
        Initialization value of the sparse code. Only used if
        `algorithm='lasso_cd'`.

    max_iter : int, default=1000
        Maximum number of iterations to perform if `algorithm='lasso_cd'` or
        `'lasso_lars'`.

    copy_cov : bool, default=True
        Whether to copy the precomputed covariance matrix; if `False`, it may
        be overwritten.

    verbose : int, default=0
        Controls the verbosity; the higher, the more messages.

    positive: bool, default=False
        Whether to enforce a positivity constraint on the sparse code.

        .. versionadded:: 0.20

    Returns
    -------
    code : ndarray of shape (n_components, n_features)
        The sparse codes.
    """
    n_samples, n_features = X.shape
    n_components = dictionary.shape[0]

    if algorithm == "lasso_lars":
        alpha = float(regularization) / n_features  # account for scaling
        try:
            err_mgt = np.seterr(all="ignore")

            # Not passing in verbose=max(0, verbose-1) because Lars.fit already
            # corrects the verbosity level.
            lasso_lars = LassoLars(
                alpha=alpha,
                fit_intercept=False,
                verbose=verbose,
                precompute=gram,
                fit_path=False,
                positive=positive,
                max_iter=max_iter,
            )
            lasso_lars.fit(dictionary.T, X.T, Xy=cov)
            new_code = lasso_lars.coef_
        finally:
            np.seterr(**err_mgt)

    elif algorithm == "lasso_cd":
        alpha = float(regularization) / n_features  # account for scaling

        # TODO: Make verbosity argument for Lasso?
        # sklearn.linear_model.coordinate_descent.enet_path has a verbosity
        # argument that we could pass in from Lasso.
        clf = Lasso(
            alpha=alpha,
            fit_intercept=False,
            precompute=gram,
            max_iter=max_iter,
            warm_start=True,
            positive=positive,
        )

        if init is not None:
            # In some workflows using coordinate descent algorithms:
            #  - users might provide NumPy arrays with read-only buffers
            #  - `joblib` might memmap arrays making their buffer read-only
            # TODO: move this handling (which is currently too broad)
            # closer to the actual private function which need buffers to be writable.
            if not init.flags["WRITEABLE"]:
                init = np.array(init)
            clf.coef_ = init

        clf.fit(dictionary.T, X.T, check_input=False)
        new_code = clf.coef_

    elif algorithm == "lars":
        try:
            err_mgt = np.seterr(all="ignore")

            # Not passing in verbose=max(0, verbose-1) because Lars.fit already
            # corrects the verbosity level.
            lars = Lars(
                fit_intercept=False,
                verbose=verbose,
                precompute=gram,
                n_nonzero_coefs=int(regularization),
                fit_path=False,
            )
            lars.fit(dictionary.T, X.T, Xy=cov)
            new_code = lars.coef_
        finally:
            np.seterr(**err_mgt)

    elif algorithm == "threshold":
        new_code = (np.sign(cov) * np.maximum(np.abs(cov) - regularization, 0)).T
        if positive:
            np.clip(new_code, 0, None, out=new_code)

    elif algorithm == "omp":
        new_code = orthogonal_mp_gram(
            Gram=gram,
            Xy=cov,
            n_nonzero_coefs=int(regularization),
            tol=None,
            norms_squared=row_norms(X, squared=True),
            copy_Xy=copy_cov,
        ).T

    return new_code.reshape(n_samples, n_components)


@validate_params(
    {
        "X": ["array-like"],
        "dictionary": ["array-like"],
        "gram": ["array-like", None],
        "cov": ["array-like", None],
        "algorithm": [
            StrOptions({"lasso_lars", "lasso_cd", "lars", "omp", "threshold"})
        ],
        "n_nonzero_coefs": [Interval(Integral, 1, None, closed="left"), None],
        "alpha": [Interval(Real, 0, None, closed="left"), None],
        "copy_cov": ["boolean"],
        "init": ["array-like", None],
        "max_iter": [Interval(Integral, 0, None, closed="left")],
        "n_jobs": [Integral, None],
        "check_input": ["boolean"],
        "verbose": ["verbose"],
        "positive": ["boolean"],
    },
    prefer_skip_nested_validation=True,
)
# XXX : could be moved to the linear_model module
def sparse_encode(
    X,
    dictionary,
    *,
    gram=None,
    cov=None,
    algorithm="lasso_lars",
    n_nonzero_coefs=None,
    alpha=None,
    copy_cov=True,
    init=None,
    max_iter=1000,
    n_jobs=None,
    check_input=True,
    verbose=0,
    positive=False,
):
    """Sparse coding.

    Each row of the result is the solution to a sparse coding problem.
    The goal is to find a sparse array `code` such that::

        X ~= code * dictionary

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Data matrix.

    dictionary : array-like of shape (n_components, n_features)
        The dictionary matrix against which to solve the sparse coding of
        the data. Some of the algorithms assume normalized rows for meaningful
        output.

    gram : array-like of shape (n_components, n_components), default=None
        Precomputed Gram matrix, `dictionary * dictionary'`.

    cov : array-like of shape (n_components, n_samples), default=None
        Precomputed covariance, `dictionary' * X`.

    algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', 'threshold'}, \
            default='lasso_lars'
        The algorithm used:

        * `'lars'`: uses the least angle regression method
          (`linear_model.lars_path`);
        * `'lasso_lars'`: uses Lars to compute the Lasso solution;
        * `'lasso_cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). lasso_lars will be faster if
          the estimated components are sparse;
        * `'omp'`: uses orthogonal matching pursuit to estimate the sparse
          solution;
        * `'threshold'`: squashes to zero all coefficients less than
          regularization from the projection `dictionary * data'`.

    n_nonzero_coefs : int, default=None
        Number of nonzero coefficients to target in each column of the
        solution. This is only used by `algorithm='lars'` and `algorithm='omp'`
        and is overridden by `alpha` in the `omp` case. If `None`, then
        `n_nonzero_coefs=int(n_features / 10)`.

    alpha : float, default=None
        If `algorithm='lasso_lars'` or `algorithm='lasso_cd'`, `alpha` is the
        penalty applied to the L1 norm.
        If `algorithm='threshold'`, `alpha` is the absolute value of the
        threshold below which coefficients will be squashed to zero.
        If `algorithm='omp'`, `alpha` is the tolerance parameter: the value of
        the reconstruction error targeted. In this case, it overrides
        `n_nonzero_coefs`.
        If `None`, default to 1.

    copy_cov : bool, default=True
        Whether to copy the precomputed covariance matrix; if `False`, it may
        be overwritten.

    init : ndarray of shape (n_samples, n_components), default=None
        Initialization value of the sparse codes. Only used if
        `algorithm='lasso_cd'`.

    max_iter : int, default=1000
        Maximum number of iterations to perform if `algorithm='lasso_cd'` or
        `'lasso_lars'`.

    n_jobs : int, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    check_input : bool, default=True
        If `False`, the input arrays X and dictionary will not be checked.

    verbose : int, default=0
        Controls the verbosity; the higher, the more messages.

    positive : bool, default=False
        Whether to enforce positivity when finding the encoding.

        .. versionadded:: 0.20

    Returns
    -------
    code : ndarray of shape (n_samples, n_components)
        The sparse codes.

    See Also
    --------
    sklearn.linear_model.lars_path : Compute Least Angle Regression or Lasso
        path using LARS algorithm.
    sklearn.linear_model.orthogonal_mp : Solves Orthogonal Matching Pursuit problems.
    sklearn.linear_model.Lasso : Train Linear Model with L1 prior as regularizer.
    SparseCoder : Find a sparse representation of data from a fixed precomputed
        dictionary.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.decomposition import sparse_encode
    >>> X = np.array([[-1, -1, -1], [0, 0, 3]])
    >>> dictionary = np.array(
    ...     [[0, 1, 0],
    ...      [-1, -1, 2],
    ...      [1, 1, 1],
    ...      [0, 1, 1],
    ...      [0, 2, 1]],
    ...    dtype=np.float64
    ... )
    >>> sparse_encode(X, dictionary, alpha=1e-10)
    array([[ 0.,  0., -1.,  0.,  0.],
           [ 0.,  1.,  1.,  0.,  0.]])
    """
    if check_input:
        if algorithm == "lasso_cd":
            dictionary = check_array(
                dictionary, order="C", dtype=[np.float64, np.float32]
            )
            X = check_array(X, order="C", dtype=[np.float64, np.float32])
        else:
            dictionary = check_array(dictionary)
            X = check_array(X)

    if dictionary.shape[1] != X.shape[1]:
        raise ValueError(
            "Dictionary and X have different numbers of features:"
            "dictionary.shape: {} X.shape{}".format(dictionary.shape, X.shape)
        )

    _check_positive_coding(algorithm, positive)

    return _sparse_encode(
        X,
        dictionary,
        gram=gram,
        cov=cov,
        algorithm=algorithm,
        n_nonzero_coefs=n_nonzero_coefs,
        alpha=alpha,
        copy_cov=copy_cov,
        init=init,
        max_iter=max_iter,
        n_jobs=n_jobs,
        verbose=verbose,
        positive=positive,
    )


def _sparse_encode(
    X,
    dictionary,
    *,
    gram=None,
    cov=None,
    algorithm="lasso_lars",
    n_nonzero_coefs=None,
    alpha=None,
    copy_cov=True,
    init=None,
    max_iter=1000,
    n_jobs=None,
    verbose=0,
    positive=False,
):
    """Sparse coding without input/parameter validation."""

    n_samples, n_features = X.shape
    n_components = dictionary.shape[0]

    if algorithm in ("lars", "omp"):
        regularization = n_nonzero_coefs
        if regularization is None:
            regularization = min(max(n_features / 10, 1), n_components)
    else:
        regularization = alpha
        if regularization is None:
            regularization = 1.0

    if gram is None and algorithm != "threshold":
        gram = np.dot(dictionary, dictionary.T)

    if cov is None and algorithm != "lasso_cd":
        copy_cov = False
        cov = np.dot(dictionary, X.T)

    if effective_n_jobs(n_jobs) == 1 or algorithm == "threshold":
        code = _sparse_encode_precomputed(
            X,
            dictionary,
            gram=gram,
            cov=cov,
            algorithm=algorithm,
            regularization=regularization,
            copy_cov=copy_cov,
            init=init,
            max_iter=max_iter,
            verbose=verbose,
            positive=positive,
        )
        return code

    # Enter parallel code block
    n_samples = X.shape[0]
    n_components = dictionary.shape[0]
    code = np.empty((n_samples, n_components))
    slices = list(gen_even_slices(n_samples, effective_n_jobs(n_jobs)))

    code_views = Parallel(n_jobs=n_jobs, verbose=verbose)(
        delayed(_sparse_encode_precomputed)(
            X[this_slice],
            dictionary,
            gram=gram,
            cov=cov[:, this_slice] if cov is not None else None,
            algorithm=algorithm,
            regularization=regularization,
            copy_cov=copy_cov,
            init=init[this_slice] if init is not None else None,
            max_iter=max_iter,
            verbose=verbose,
            positive=positive,
        )
        for this_slice in slices
    )
    for this_slice, this_view in zip(slices, code_views):
        code[this_slice] = this_view
    return code


def _update_dict(
    dictionary,
    Y,
    code,
    A=None,
    B=None,
    verbose=False,
    random_state=None,
    positive=False,
):
    """Update the dense dictionary factor in place.

    Parameters
    ----------
    dictionary : ndarray of shape (n_components, n_features)
        Value of the dictionary at the previous iteration.

    Y : ndarray of shape (n_samples, n_features)
        Data matrix.

    code : ndarray of shape (n_samples, n_components)
        Sparse coding of the data against which to optimize the dictionary.

    A : ndarray of shape (n_components, n_components), default=None
        Together with `B`, sufficient stats of the online model to update the
        dictionary.

    B : ndarray of shape (n_features, n_components), default=None
        Together with `A`, sufficient stats of the online model to update the
        dictionary.

    verbose: bool, default=False
        Degree of output the procedure will print.

    random_state : int, RandomState instance or None, default=None
        Used for randomly initializing the dictionary. Pass an int for
        reproducible results across multiple function calls.
        See :term:`Glossary <random_state>`.

    positive : bool, default=False
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20
    """
    n_samples, n_components = code.shape
    random_state = check_random_state(random_state)

    if A is None:
        A = code.T @ code
    if B is None:
        B = Y.T @ code

    n_unused = 0

    for k in range(n_components):
        if A[k, k] > 1e-6:
            # 1e-6 is arbitrary but consistent with the spams implementation
            dictionary[k] += (B[:, k] - A[k] @ dictionary) / A[k, k]
        else:
            # kth atom is almost never used -> sample a new one from the data
            newd = Y[random_state.choice(n_samples)]

            # add small noise to avoid making the sparse coding ill conditioned
            noise_level = 0.01 * (newd.std() or 1)  # avoid 0 std
            noise = random_state.normal(0, noise_level, size=len(newd))

            dictionary[k] = newd + noise
            code[:, k] = 0
            n_unused += 1

        if positive:
            np.clip(dictionary[k], 0, None, out=dictionary[k])

        # Projection on the constraint set ||V_k|| <= 1
        dictionary[k] /= max(linalg.norm(dictionary[k]), 1)

    if verbose and n_unused > 0:
        print(f"{n_unused} unused atoms resampled.")


def _dict_learning(
    X,
    n_components,
    *,
    alpha,
    max_iter,
    tol,
    method,
    n_jobs,
    dict_init,
    code_init,
    callback,
    verbose,
    random_state,
    return_n_iter,
    positive_dict,
    positive_code,
    method_max_iter,
):
    """Main dictionary learning algorithm"""
    t0 = time.time()
    # Init the code and the dictionary with SVD of Y
    if code_init is not None and dict_init is not None:
        code = np.array(code_init, order="F")
        # Don't copy V, it will happen below
        dictionary = dict_init
    else:
        code, S, dictionary = linalg.svd(X, full_matrices=False)
        # flip the initial code's sign to enforce deterministic output
        code, dictionary = svd_flip(code, dictionary)
        dictionary = S[:, np.newaxis] * dictionary
    r = len(dictionary)
    if n_components <= r:  # True even if n_components=None
        code = code[:, :n_components]
        dictionary = dictionary[:n_components, :]
    else:
        code = np.c_[code, np.zeros((len(code), n_components - r))]
        dictionary = np.r_[
            dictionary, np.zeros((n_components - r, dictionary.shape[1]))
        ]

    # Fortran-order dict better suited for the sparse coding which is the
    # bottleneck of this algorithm.
    dictionary = np.asfortranarray(dictionary)

    errors = []
    current_cost = np.nan

    if verbose == 1:
        print("[dict_learning]", end=" ")

    # If max_iter is 0, number of iterations returned should be zero
    ii = -1

    for ii in range(max_iter):
        dt = time.time() - t0
        if verbose == 1:
            sys.stdout.write(".")
            sys.stdout.flush()
        elif verbose:
            print(
                "Iteration % 3i (elapsed time: % 3is, % 4.1fmn, current cost % 7.3f)"
                % (ii, dt, dt / 60, current_cost)
            )

        # Update code
        code = sparse_encode(
            X,
            dictionary,
            algorithm=method,
            alpha=alpha,
            init=code,
            n_jobs=n_jobs,
            positive=positive_code,
            max_iter=method_max_iter,
            verbose=verbose,
        )

        # Update dictionary in place
        _update_dict(
            dictionary,
            X,
            code,
            verbose=verbose,
            random_state=random_state,
            positive=positive_dict,
        )

        # Cost function
        current_cost = 0.5 * np.sum((X - code @ dictionary) ** 2) + alpha * np.sum(
            np.abs(code)
        )
        errors.append(current_cost)

        if ii > 0:
            dE = errors[-2] - errors[-1]
            # assert(dE >= -tol * errors[-1])
            if dE < tol * errors[-1]:
                if verbose == 1:
                    # A line return
                    print("")
                elif verbose:
                    print("--- Convergence reached after %d iterations" % ii)
                break
        if ii % 5 == 0 and callback is not None:
            callback(locals())

    if return_n_iter:
        return code, dictionary, errors, ii + 1
    else:
        return code, dictionary, errors


@validate_params(
    {
        "X": ["array-like"],
        "return_code": ["boolean"],
        "method": [StrOptions({"cd", "lars"})],
        "method_max_iter": [Interval(Integral, 0, None, closed="left")],
    },
    prefer_skip_nested_validation=False,
)
def dict_learning_online(
    X,
    n_components=2,
    *,
    alpha=1,
    max_iter=100,
    return_code=True,
    dict_init=None,
    callback=None,
    batch_size=256,
    verbose=False,
    shuffle=True,
    n_jobs=None,
    method="lars",
    random_state=None,
    positive_dict=False,
    positive_code=False,
    method_max_iter=1000,
    tol=1e-3,
    max_no_improvement=10,
):
    """Solve a dictionary learning matrix factorization problem online.

    Finds the best dictionary and the corresponding sparse code for
    approximating the data matrix X by solving::

        (U^*, V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                     (U,V)
                     with || V_k ||_2 = 1 for all  0 <= k < n_components

    where V is the dictionary and U is the sparse code. ||.||_Fro stands for
    the Frobenius norm and ||.||_1,1 stands for the entry-wise matrix norm
    which is the sum of the absolute values of all the entries in the matrix.
    This is accomplished by repeatedly iterating over mini-batches by slicing
    the input data.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Data matrix.

    n_components : int or None, default=2
        Number of dictionary atoms to extract. If None, then ``n_components``
        is set to ``n_features``.

    alpha : float, default=1
        Sparsity controlling parameter.

    max_iter : int, default=100
        Maximum number of iterations over the complete dataset before
        stopping independently of any early stopping criterion heuristics.

        .. versionadded:: 1.1

        .. deprecated:: 1.4
           `max_iter=None` is deprecated in 1.4 and will be removed in 1.6.
           Use the default value (i.e. `100`) instead.

    return_code : bool, default=True
        Whether to also return the code U or just the dictionary `V`.

    dict_init : ndarray of shape (n_components, n_features), default=None
        Initial values for the dictionary for warm restart scenarios.
        If `None`, the initial values for the dictionary are created
        with an SVD decomposition of the data via
        :func:`~sklearn.utils.extmath.randomized_svd`.

    callback : callable, default=None
        A callable that gets invoked at the end of each iteration.

    batch_size : int, default=256
        The number of samples to take in each batch.

        .. versionchanged:: 1.3
           The default value of `batch_size` changed from 3 to 256 in version 1.3.

    verbose : bool, default=False
        To control the verbosity of the procedure.

    shuffle : bool, default=True
        Whether to shuffle the data before splitting it in batches.

    n_jobs : int, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    method : {'lars', 'cd'}, default='lars'
        * `'lars'`: uses the least angle regression method to solve the lasso
          problem (`linear_model.lars_path`);
        * `'cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). Lars will be faster if
          the estimated components are sparse.

    random_state : int, RandomState instance or None, default=None
        Used for initializing the dictionary when ``dict_init`` is not
        specified, randomly shuffling the data when ``shuffle`` is set to
        ``True``, and updating the dictionary. Pass an int for reproducible
        results across multiple function calls.
        See :term:`Glossary <random_state>`.

    positive_dict : bool, default=False
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    positive_code : bool, default=False
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    method_max_iter : int, default=1000
        Maximum number of iterations to perform when solving the lasso problem.

        .. versionadded:: 0.22

    tol : float, default=1e-3
        Control early stopping based on the norm of the differences in the
        dictionary between 2 steps.

        To disable early stopping based on changes in the dictionary, set
        `tol` to 0.0.

        .. versionadded:: 1.1

    max_no_improvement : int, default=10
        Control early stopping based on the consecutive number of mini batches
        that does not yield an improvement on the smoothed cost function.

        To disable convergence detection based on cost function, set
        `max_no_improvement` to None.

        .. versionadded:: 1.1

    Returns
    -------
    code : ndarray of shape (n_samples, n_components),
        The sparse code (only returned if `return_code=True`).

    dictionary : ndarray of shape (n_components, n_features),
        The solutions to the dictionary learning problem.

    n_iter : int
        Number of iterations run. Returned only if `return_n_iter` is
        set to `True`.

    See Also
    --------
    dict_learning : Solve a dictionary learning matrix factorization problem.
    DictionaryLearning : Find a dictionary that sparsely encodes data.
    MiniBatchDictionaryLearning : A faster, less accurate, version of the dictionary
        learning algorithm.
    SparsePCA : Sparse Principal Components Analysis.
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.datasets import make_sparse_coded_signal
    >>> from sklearn.decomposition import dict_learning_online
    >>> X, _, _ = make_sparse_coded_signal(
    ...     n_samples=30, n_components=15, n_features=20, n_nonzero_coefs=10,
    ...     random_state=42,
    ... )
    >>> U, V = dict_learning_online(
    ...     X, n_components=15, alpha=0.2, max_iter=20, batch_size=3, random_state=42
    ... )

    We can check the level of sparsity of `U`:

    >>> np.mean(U == 0)
    0.53...

    We can compare the average squared euclidean norm of the reconstruction
    error of the sparse coded signal relative to the squared euclidean norm of
    the original signal:

    >>> X_hat = U @ V
    >>> np.mean(np.sum((X_hat - X) ** 2, axis=1) / np.sum(X ** 2, axis=1))
    0.05...
    """
    # TODO(1.6): remove in 1.6
    if max_iter is None:
        warn(
            (
                "`max_iter=None` is deprecated in version 1.4 and will be removed in "
                "version 1.6. Use the default value (i.e. `100`) instead."
            ),
            FutureWarning,
        )
        max_iter = 100

    transform_algorithm = "lasso_" + method

    est = MiniBatchDictionaryLearning(
        n_components=n_components,
        alpha=alpha,
        max_iter=max_iter,
        n_jobs=n_jobs,
        fit_algorithm=method,
        batch_size=batch_size,
        shuffle=shuffle,
        dict_init=dict_init,
        random_state=random_state,
        transform_algorithm=transform_algorithm,
        transform_alpha=alpha,
        positive_code=positive_code,
        positive_dict=positive_dict,
        transform_max_iter=method_max_iter,
        verbose=verbose,
        callback=callback,
        tol=tol,
        max_no_improvement=max_no_improvement,
    ).fit(X)

    if not return_code:
        return est.components_
    else:
        code = est.transform(X)
        return code, est.components_


@validate_params(
    {
        "X": ["array-like"],
        "method": [StrOptions({"lars", "cd"})],
        "return_n_iter": ["boolean"],
        "method_max_iter": [Interval(Integral, 0, None, closed="left")],
    },
    prefer_skip_nested_validation=False,
)
def dict_learning(
    X,
    n_components,
    *,
    alpha,
    max_iter=100,
    tol=1e-8,
    method="lars",
    n_jobs=None,
    dict_init=None,
    code_init=None,
    callback=None,
    verbose=False,
    random_state=None,
    return_n_iter=False,
    positive_dict=False,
    positive_code=False,
    method_max_iter=1000,
):
    """Solve a dictionary learning matrix factorization problem.

    Finds the best dictionary and the corresponding sparse code for
    approximating the data matrix X by solving::

        (U^*, V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                     (U,V)
                    with || V_k ||_2 = 1 for all  0 <= k < n_components

    where V is the dictionary and U is the sparse code. ||.||_Fro stands for
    the Frobenius norm and ||.||_1,1 stands for the entry-wise matrix norm
    which is the sum of the absolute values of all the entries in the matrix.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Data matrix.

    n_components : int
        Number of dictionary atoms to extract.

    alpha : int or float
        Sparsity controlling parameter.

    max_iter : int, default=100
        Maximum number of iterations to perform.

    tol : float, default=1e-8
        Tolerance for the stopping condition.

    method : {'lars', 'cd'}, default='lars'
        The method used:

        * `'lars'`: uses the least angle regression method to solve the lasso
           problem (`linear_model.lars_path`);
        * `'cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). Lars will be faster if
          the estimated components are sparse.

    n_jobs : int, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    dict_init : ndarray of shape (n_components, n_features), default=None
        Initial value for the dictionary for warm restart scenarios. Only used
        if `code_init` and `dict_init` are not None.

    code_init : ndarray of shape (n_samples, n_components), default=None
        Initial value for the sparse code for warm restart scenarios. Only used
        if `code_init` and `dict_init` are not None.

    callback : callable, default=None
        Callable that gets invoked every five iterations.

    verbose : bool, default=False
        To control the verbosity of the procedure.

    random_state : int, RandomState instance or None, default=None
        Used for randomly initializing the dictionary. Pass an int for
        reproducible results across multiple function calls.
        See :term:`Glossary <random_state>`.

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

    positive_dict : bool, default=False
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    positive_code : bool, default=False
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    method_max_iter : int, default=1000
        Maximum number of iterations to perform.

        .. versionadded:: 0.22

    Returns
    -------
    code : ndarray of shape (n_samples, n_components)
        The sparse code factor in the matrix factorization.

    dictionary : ndarray of shape (n_components, n_features),
        The dictionary factor in the matrix factorization.

    errors : array
        Vector of errors at each iteration.

    n_iter : int
        Number of iterations run. Returned only if `return_n_iter` is
        set to True.

    See Also
    --------
    dict_learning_online : Solve a dictionary learning matrix factorization
        problem online.
    DictionaryLearning : Find a dictionary that sparsely encodes data.
    MiniBatchDictionaryLearning : A faster, less accurate version
        of the dictionary learning algorithm.
    SparsePCA : Sparse Principal Components Analysis.
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.datasets import make_sparse_coded_signal
    >>> from sklearn.decomposition import dict_learning
    >>> X, _, _ = make_sparse_coded_signal(
    ...     n_samples=30, n_components=15, n_features=20, n_nonzero_coefs=10,
    ...     random_state=42,
    ... )
    >>> U, V, errors = dict_learning(X, n_components=15, alpha=0.1, random_state=42)

    We can check the level of sparsity of `U`:

    >>> np.mean(U == 0)
    0.6...

    We can compare the average squared euclidean norm of the reconstruction
    error of the sparse coded signal relative to the squared euclidean norm of
    the original signal:

    >>> X_hat = U @ V
    >>> np.mean(np.sum((X_hat - X) ** 2, axis=1) / np.sum(X ** 2, axis=1))
    0.01...
    """
    estimator = DictionaryLearning(
        n_components=n_components,
        alpha=alpha,
        max_iter=max_iter,
        tol=tol,
        fit_algorithm=method,
        n_jobs=n_jobs,
        dict_init=dict_init,
        callback=callback,
        code_init=code_init,
        verbose=verbose,
        random_state=random_state,
        positive_code=positive_code,
        positive_dict=positive_dict,
        transform_max_iter=method_max_iter,
    ).set_output(transform="default")
    code = estimator.fit_transform(X)
    if return_n_iter:
        return (
            code,
            estimator.components_,
            estimator.error_,
            estimator.n_iter_,
        )
    return code, estimator.components_, estimator.error_


class _BaseSparseCoding(ClassNamePrefixFeaturesOutMixin, TransformerMixin):
    """Base class from SparseCoder and DictionaryLearning algorithms."""

    def __init__(
        self,
        transform_algorithm,
        transform_n_nonzero_coefs,
        transform_alpha,
        split_sign,
        n_jobs,
        positive_code,
        transform_max_iter,
    ):
        self.transform_algorithm = transform_algorithm
        self.transform_n_nonzero_coefs = transform_n_nonzero_coefs
        self.transform_alpha = transform_alpha
        self.transform_max_iter = transform_max_iter
        self.split_sign = split_sign
        self.n_jobs = n_jobs
        self.positive_code = positive_code

    def _transform(self, X, dictionary):
        """Private method allowing to accommodate both DictionaryLearning and
        SparseCoder."""
        X = self._validate_data(X, reset=False)

        if hasattr(self, "alpha") and self.transform_alpha is None:
            transform_alpha = self.alpha
        else:
            transform_alpha = self.transform_alpha

        code = sparse_encode(
            X,
            dictionary,
            algorithm=self.transform_algorithm,
            n_nonzero_coefs=self.transform_n_nonzero_coefs,
            alpha=transform_alpha,
            max_iter=self.transform_max_iter,
            n_jobs=self.n_jobs,
            positive=self.positive_code,
        )

        if self.split_sign:
            # feature vector is split into a positive and negative side
            n_samples, n_features = code.shape
            split_code = np.empty((n_samples, 2 * n_features))
            split_code[:, :n_features] = np.maximum(code, 0)
            split_code[:, n_features:] = -np.minimum(code, 0)
            code = split_code

        return code

    def transform(self, X):
        """Encode the data as a sparse combination of the dictionary atoms.

        Coding method is determined by the object parameter
        `transform_algorithm`.

        Parameters
        ----------
        X : ndarray of shape (n_samples, n_features)
            Test data to be transformed, must have the same number of
            features as the data used to train the model.

        Returns
        -------
        X_new : ndarray of shape (n_samples, n_components)
            Transformed data.
        """
        check_is_fitted(self)
        return self._transform(X, self.components_)


class SparseCoder(_BaseSparseCoding, BaseEstimator):
    """Sparse coding.

    Finds a sparse representation of data against a fixed, precomputed
    dictionary.

    Each row of the result is the solution to a sparse coding problem.
    The goal is to find a sparse array `code` such that::

        X ~= code * dictionary

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

    Parameters
    ----------
    dictionary : ndarray of shape (n_components, n_features)
        The dictionary atoms used for sparse coding. Lines are assumed to be
        normalized to unit norm.

    transform_algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', \
            'threshold'}, default='omp'
        Algorithm used to transform the data:

        - `'lars'`: uses the least angle regression method
          (`linear_model.lars_path`);
        - `'lasso_lars'`: uses Lars to compute the Lasso solution;
        - `'lasso_cd'`: uses the coordinate descent method to compute the
          Lasso solution (linear_model.Lasso). `'lasso_lars'` will be faster if
          the estimated components are sparse;
        - `'omp'`: uses orthogonal matching pursuit to estimate the sparse
          solution;
        - `'threshold'`: squashes to zero all coefficients less than alpha from
          the projection ``dictionary * X'``.

    transform_n_nonzero_coefs : int, default=None
        Number of nonzero coefficients to target in each column of the
        solution. This is only used by `algorithm='lars'` and `algorithm='omp'`
        and is overridden by `alpha` in the `omp` case. If `None`, then
        `transform_n_nonzero_coefs=int(n_features / 10)`.

    transform_alpha : float, default=None
        If `algorithm='lasso_lars'` or `algorithm='lasso_cd'`, `alpha` is the
        penalty applied to the L1 norm.
        If `algorithm='threshold'`, `alpha` is the absolute value of the
        threshold below which coefficients will be squashed to zero.
        If `algorithm='omp'`, `alpha` is the tolerance parameter: the value of
        the reconstruction error targeted. In this case, it overrides
        `n_nonzero_coefs`.
        If `None`, default to 1.

    split_sign : bool, default=False
        Whether to split the sparse feature vector into the concatenation of
        its negative part and its positive part. This can improve the
        performance of downstream classifiers.

    n_jobs : int, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    positive_code : bool, default=False
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    transform_max_iter : int, default=1000
        Maximum number of iterations to perform if `algorithm='lasso_cd'` or
        `lasso_lars`.

        .. versionadded:: 0.22

    Attributes
    ----------
    n_components_ : int
        Number of atoms.

    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

    See Also
    --------
    DictionaryLearning : Find a dictionary that sparsely encodes data.
    MiniBatchDictionaryLearning : A faster, less accurate, version of the
        dictionary learning algorithm.
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.
    SparsePCA : Sparse Principal Components Analysis.
    sparse_encode : Sparse coding where each row of the result is the solution
        to a sparse coding problem.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.decomposition import SparseCoder
    >>> X = np.array([[-1, -1, -1], [0, 0, 3]])
    >>> dictionary = np.array(
    ...     [[0, 1, 0],
    ...      [-1, -1, 2],
    ...      [1, 1, 1],
    ...      [0, 1, 1],
    ...      [0, 2, 1]],
    ...    dtype=np.float64
    ... )
    >>> coder = SparseCoder(
    ...     dictionary=dictionary, transform_algorithm='lasso_lars',
    ...     transform_alpha=1e-10,
    ... )
    >>> coder.transform(X)
    array([[ 0.,  0., -1.,  0.,  0.],
           [ 0.,  1.,  1.,  0.,  0.]])
    """

    _required_parameters = ["dictionary"]

    def __init__(
        self,
        dictionary,
        *,
        transform_algorithm="omp",
        transform_n_nonzero_coefs=None,
        transform_alpha=None,
        split_sign=False,
        n_jobs=None,
        positive_code=False,
        transform_max_iter=1000,
    ):
        super().__init__(
            transform_algorithm,
            transform_n_nonzero_coefs,
            transform_alpha,
            split_sign,
            n_jobs,
            positive_code,
            transform_max_iter,
        )
        self.dictionary = dictionary

    def fit(self, X, y=None):
        """Do nothing and return the estimator unchanged.

        This method is just there to implement the usual API and hence
        work in pipelines.

        Parameters
        ----------
        X : Ignored
            Not used, present for API consistency by convention.

        y : Ignored
            Not used, present for API consistency by convention.

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        return self

    def transform(self, X, y=None):
        """Encode the data as a sparse combination of the dictionary atoms.

        Coding method is determined by the object parameter
        `transform_algorithm`.

        Parameters
        ----------
        X : ndarray 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
        -------
        X_new : ndarray of shape (n_samples, n_components)
            Transformed data.
        """
        return super()._transform(X, self.dictionary)

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

    @property
    def n_components_(self):
        """Number of atoms."""
        return self.dictionary.shape[0]

    @property
    def n_features_in_(self):
        """Number of features seen during `fit`."""
        return self.dictionary.shape[1]

    @property
    def _n_features_out(self):
        """Number of transformed output features."""
        return self.n_components_


class DictionaryLearning(_BaseSparseCoding, BaseEstimator):
    """Dictionary learning.

    Finds a dictionary (a set of atoms) that performs well at sparsely
    encoding the fitted data.

    Solves the optimization problem::

        (U^*,V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                    (U,V)
                    with || V_k ||_2 <= 1 for all  0 <= k < n_components

    ||.||_Fro stands for the Frobenius norm and ||.||_1,1 stands for
    the entry-wise matrix norm which is the sum of the absolute values
    of all the entries in the matrix.

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

    Parameters
    ----------
    n_components : int, default=None
        Number of dictionary elements to extract. If None, then ``n_components``
        is set to ``n_features``.

    alpha : float, default=1.0
        Sparsity controlling parameter.

    max_iter : int, default=1000
        Maximum number of iterations to perform.

    tol : float, default=1e-8
        Tolerance for numerical error.

    fit_algorithm : {'lars', 'cd'}, default='lars'
        * `'lars'`: uses the least angle regression method to solve the lasso
          problem (:func:`~sklearn.linear_model.lars_path`);
        * `'cd'`: uses the coordinate descent method to compute the
          Lasso solution (:class:`~sklearn.linear_model.Lasso`). Lars will be
          faster if the estimated components are sparse.

        .. versionadded:: 0.17
           *cd* coordinate descent method to improve speed.

    transform_algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', \
            'threshold'}, default='omp'
        Algorithm used to transform the data:

        - `'lars'`: uses the least angle regression method
          (:func:`~sklearn.linear_model.lars_path`);
        - `'lasso_lars'`: uses Lars to compute the Lasso solution.
        - `'lasso_cd'`: uses the coordinate descent method to compute the
          Lasso solution (:class:`~sklearn.linear_model.Lasso`). `'lasso_lars'`
          will be faster if the estimated components are sparse.
        - `'omp'`: uses orthogonal matching pursuit to estimate the sparse
          solution.
        - `'threshold'`: squashes to zero all coefficients less than alpha from
          the projection ``dictionary * X'``.

        .. versionadded:: 0.17
           *lasso_cd* coordinate descent method to improve speed.

    transform_n_nonzero_coefs : int, default=None
        Number of nonzero coefficients to target in each column of the
        solution. This is only used by `algorithm='lars'` and
        `algorithm='omp'`. If `None`, then
        `transform_n_nonzero_coefs=int(n_features / 10)`.

    transform_alpha : float, default=None
        If `algorithm='lasso_lars'` or `algorithm='lasso_cd'`, `alpha` is the
        penalty applied to the L1 norm.
        If `algorithm='threshold'`, `alpha` is the absolute value of the
        threshold below which coefficients will be squashed to zero.
        If `None`, defaults to `alpha`.

        .. versionchanged:: 1.2
            When None, default value changed from 1.0 to `alpha`.

    n_jobs : int or None, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    code_init : ndarray of shape (n_samples, n_components), default=None
        Initial value for the code, for warm restart. Only used if `code_init`
        and `dict_init` are not None.

    dict_init : ndarray of shape (n_components, n_features), default=None
        Initial values for the dictionary, for warm restart. Only used if
        `code_init` and `dict_init` are not None.

    callback : callable, default=None
        Callable that gets invoked every five iterations.

        .. versionadded:: 1.3

    verbose : bool, default=False
        To control the verbosity of the procedure.

    split_sign : bool, default=False
        Whether to split the sparse feature vector into the concatenation of
        its negative part and its positive part. This can improve the
        performance of downstream classifiers.

    random_state : int, RandomState instance or None, default=None
        Used for initializing the dictionary when ``dict_init`` is not
        specified, randomly shuffling the data when ``shuffle`` is set to
        ``True``, and updating the dictionary. Pass an int for reproducible
        results across multiple function calls.
        See :term:`Glossary <random_state>`.

    positive_code : bool, default=False
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    positive_dict : bool, default=False
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    transform_max_iter : int, default=1000
        Maximum number of iterations to perform if `algorithm='lasso_cd'` or
        `'lasso_lars'`.

        .. versionadded:: 0.22

    Attributes
    ----------
    components_ : ndarray of shape (n_components, n_features)
        dictionary atoms extracted from the data

    error_ : array
        vector of errors at each iteration

    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
        Number of iterations run.

    See Also
    --------
    MiniBatchDictionaryLearning: A faster, less accurate, version of the
        dictionary learning algorithm.
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.
    SparseCoder : Find a sparse representation of data from a fixed,
        precomputed dictionary.
    SparsePCA : Sparse Principal Components Analysis.

    References
    ----------

    J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009: Online dictionary learning
    for sparse coding (https://www.di.ens.fr/sierra/pdfs/icml09.pdf)

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.datasets import make_sparse_coded_signal
    >>> from sklearn.decomposition import DictionaryLearning
    >>> X, dictionary, code = make_sparse_coded_signal(
    ...     n_samples=30, n_components=15, n_features=20, n_nonzero_coefs=10,
    ...     random_state=42,
    ... )
    >>> dict_learner = DictionaryLearning(
    ...     n_components=15, transform_algorithm='lasso_lars', transform_alpha=0.1,
    ...     random_state=42,
    ... )
    >>> X_transformed = dict_learner.fit(X).transform(X)

    We can check the level of sparsity of `X_transformed`:

    >>> np.mean(X_transformed == 0)
    0.52...

    We can compare the average squared euclidean norm of the reconstruction
    error of the sparse coded signal relative to the squared euclidean norm of
    the original signal:

    >>> X_hat = X_transformed @ dict_learner.components_
    >>> np.mean(np.sum((X_hat - X) ** 2, axis=1) / np.sum(X ** 2, axis=1))
    0.05...
    """

    _parameter_constraints: dict = {
        "n_components": [Interval(Integral, 1, None, closed="left"), None],
        "alpha": [Interval(Real, 0, None, closed="left")],
        "max_iter": [Interval(Integral, 0, None, closed="left")],
        "tol": [Interval(Real, 0, None, closed="left")],
        "fit_algorithm": [StrOptions({"lars", "cd"})],
        "transform_algorithm": [
            StrOptions({"lasso_lars", "lasso_cd", "lars", "omp", "threshold"})
        ],
        "transform_n_nonzero_coefs": [Interval(Integral, 1, None, closed="left"), None],
        "transform_alpha": [Interval(Real, 0, None, closed="left"), None],
        "n_jobs": [Integral, None],
        "code_init": [np.ndarray, None],
        "dict_init": [np.ndarray, None],
        "callback": [callable, None],
        "verbose": ["verbose"],
        "split_sign": ["boolean"],
        "random_state": ["random_state"],
        "positive_code": ["boolean"],
        "positive_dict": ["boolean"],
        "transform_max_iter": [Interval(Integral, 0, None, closed="left")],
    }

    def __init__(
        self,
        n_components=None,
        *,
        alpha=1,
        max_iter=1000,
        tol=1e-8,
        fit_algorithm="lars",
        transform_algorithm="omp",
        transform_n_nonzero_coefs=None,
        transform_alpha=None,
        n_jobs=None,
        code_init=None,
        dict_init=None,
        callback=None,
        verbose=False,
        split_sign=False,
        random_state=None,
        positive_code=False,
        positive_dict=False,
        transform_max_iter=1000,
    ):
        super().__init__(
            transform_algorithm,
            transform_n_nonzero_coefs,
            transform_alpha,
            split_sign,
            n_jobs,
            positive_code,
            transform_max_iter,
        )
        self.n_components = n_components
        self.alpha = alpha
        self.max_iter = max_iter
        self.tol = tol
        self.fit_algorithm = fit_algorithm
        self.code_init = code_init
        self.dict_init = dict_init
        self.callback = callback
        self.verbose = verbose
        self.random_state = random_state
        self.positive_dict = positive_dict

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

        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.

        y : Ignored
            Not used, present for API consistency by convention.

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        self.fit_transform(X)
        return self

    @_fit_context(prefer_skip_nested_validation=True)
    def fit_transform(self, X, y=None):
        """Fit the model from data in X and return the transformed data.

        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.

        y : Ignored
            Not used, present for API consistency by convention.

        Returns
        -------
        V : ndarray of shape (n_samples, n_components)
            Transformed data.
        """
        _check_positive_coding(method=self.fit_algorithm, positive=self.positive_code)

        method = "lasso_" + self.fit_algorithm

        random_state = check_random_state(self.random_state)
        X = self._validate_data(X)

        if self.n_components is None:
            n_components = X.shape[1]
        else:
            n_components = self.n_components

        V, U, E, self.n_iter_ = _dict_learning(
            X,
            n_components,
            alpha=self.alpha,
            tol=self.tol,
            max_iter=self.max_iter,
            method=method,
            method_max_iter=self.transform_max_iter,
            n_jobs=self.n_jobs,
            code_init=self.code_init,
            dict_init=self.dict_init,
            callback=self.callback,
            verbose=self.verbose,
            random_state=random_state,
            return_n_iter=True,
            positive_dict=self.positive_dict,
            positive_code=self.positive_code,
        )
        self.components_ = U
        self.error_ = E

        return V

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

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


class MiniBatchDictionaryLearning(_BaseSparseCoding, BaseEstimator):
    """Mini-batch dictionary learning.

    Finds a dictionary (a set of atoms) that performs well at sparsely
    encoding the fitted data.

    Solves the optimization problem::

       (U^*,V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                    (U,V)
                    with || V_k ||_2 <= 1 for all  0 <= k < n_components

    ||.||_Fro stands for the Frobenius norm and ||.||_1,1 stands for
    the entry-wise matrix norm which is the sum of the absolute values
    of all the entries in the matrix.

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

    Parameters
    ----------
    n_components : int, default=None
        Number of dictionary elements to extract.

    alpha : float, default=1
        Sparsity controlling parameter.

    max_iter : int, default=1_000
        Maximum number of iterations over the complete dataset before
        stopping independently of any early stopping criterion heuristics.

        .. versionadded:: 1.1

        .. deprecated:: 1.4
           `max_iter=None` is deprecated in 1.4 and will be removed in 1.6.
           Use the default value (i.e. `1_000`) instead.

    fit_algorithm : {'lars', 'cd'}, default='lars'
        The algorithm used:

        - `'lars'`: uses the least angle regression method to solve the lasso
          problem (`linear_model.lars_path`)
        - `'cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). Lars will be faster if
          the estimated components are sparse.

    n_jobs : int, default=None
        Number of parallel jobs to run.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
        for more details.

    batch_size : int, default=256
        Number of samples in each mini-batch.

        .. versionchanged:: 1.3
           The default value of `batch_size` changed from 3 to 256 in version 1.3.

    shuffle : bool, default=True
        Whether to shuffle the samples before forming batches.

    dict_init : ndarray of shape (n_components, n_features), default=None
        Initial value of the dictionary for warm restart scenarios.

    transform_algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', \
            'threshold'}, default='omp'
        Algorithm used to transform the data:

        - `'lars'`: uses the least angle regression method
          (`linear_model.lars_path`);
        - `'lasso_lars'`: uses Lars to compute the Lasso solution.
        - `'lasso_cd'`: uses the coordinate descent method to compute the
          Lasso solution (`linear_model.Lasso`). `'lasso_lars'` will be faster
          if the estimated components are sparse.
        - `'omp'`: uses orthogonal matching pursuit to estimate the sparse
          solution.
        - `'threshold'`: squashes to zero all coefficients less than alpha from
          the projection ``dictionary * X'``.

    transform_n_nonzero_coefs : int, default=None
        Number of nonzero coefficients to target in each column of the
        solution. This is only used by `algorithm='lars'` and
        `algorithm='omp'`. If `None`, then
        `transform_n_nonzero_coefs=int(n_features / 10)`.

    transform_alpha : float, default=None
        If `algorithm='lasso_lars'` or `algorithm='lasso_cd'`, `alpha` is the
        penalty applied to the L1 norm.
        If `algorithm='threshold'`, `alpha` is the absolute value of the
        threshold below which coefficients will be squashed to zero.
        If `None`, defaults to `alpha`.

        .. versionchanged:: 1.2
            When None, default value changed from 1.0 to `alpha`.

    verbose : bool or int, default=False
        To control the verbosity of the procedure.

    split_sign : bool, default=False
        Whether to split the sparse feature vector into the concatenation of
        its negative part and its positive part. This can improve the
        performance of downstream classifiers.

    random_state : int, RandomState instance or None, default=None
        Used for initializing the dictionary when ``dict_init`` is not
        specified, randomly shuffling the data when ``shuffle`` is set to
        ``True``, and updating the dictionary. Pass an int for reproducible
        results across multiple function calls.
        See :term:`Glossary <random_state>`.

    positive_code : bool, default=False
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    positive_dict : bool, default=False
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    transform_max_iter : int, default=1000
        Maximum number of iterations to perform if `algorithm='lasso_cd'` or
        `'lasso_lars'`.

        .. versionadded:: 0.22

    callback : callable, default=None
        A callable that gets invoked at the end of each iteration.

        .. versionadded:: 1.1

    tol : float, default=1e-3
        Control early stopping based on the norm of the differences in the
        dictionary between 2 steps.

        To disable early stopping based on changes in the dictionary, set
        `tol` to 0.0.

        .. versionadded:: 1.1

    max_no_improvement : int, default=10
        Control early stopping based on the consecutive number of mini batches
        that does not yield an improvement on the smoothed cost function.

        To disable convergence detection based on cost function, set
        `max_no_improvement` to None.

        .. versionadded:: 1.1

    Attributes
    ----------
    components_ : ndarray of shape (n_components, n_features)
        Components extracted from the data.

    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
        Number of iterations over the full dataset.

    n_steps_ : int
        Number of mini-batches processed.

        .. versionadded:: 1.1

    See Also
    --------
    DictionaryLearning : Find a dictionary that sparsely encodes data.
    MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.
    SparseCoder : Find a sparse representation of data from a fixed,
        precomputed dictionary.
    SparsePCA : Sparse Principal Components Analysis.

    References
    ----------

    J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009: Online dictionary learning
    for sparse coding (https://www.di.ens.fr/sierra/pdfs/icml09.pdf)

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.datasets import make_sparse_coded_signal
    >>> from sklearn.decomposition import MiniBatchDictionaryLearning
    >>> X, dictionary, code = make_sparse_coded_signal(
    ...     n_samples=30, n_components=15, n_features=20, n_nonzero_coefs=10,
    ...     random_state=42)
    >>> dict_learner = MiniBatchDictionaryLearning(
    ...     n_components=15, batch_size=3, transform_algorithm='lasso_lars',
    ...     transform_alpha=0.1, max_iter=20, random_state=42)
    >>> X_transformed = dict_learner.fit_transform(X)

    We can check the level of sparsity of `X_transformed`:

    >>> np.mean(X_transformed == 0) > 0.5
    True

    We can compare the average squared euclidean norm of the reconstruction
    error of the sparse coded signal relative to the squared euclidean norm of
    the original signal:

    >>> X_hat = X_transformed @ dict_learner.components_
    >>> np.mean(np.sum((X_hat - X) ** 2, axis=1) / np.sum(X ** 2, axis=1))
    0.052...
    """

    _parameter_constraints: dict = {
        "n_components": [Interval(Integral, 1, None, closed="left"), None],
        "alpha": [Interval(Real, 0, None, closed="left")],
        "max_iter": [Interval(Integral, 0, None, closed="left"), Hidden(None)],
        "fit_algorithm": [StrOptions({"cd", "lars"})],
        "n_jobs": [None, Integral],
        "batch_size": [Interval(Integral, 1, None, closed="left")],
        "shuffle": ["boolean"],
        "dict_init": [None, np.ndarray],
        "transform_algorithm": [
            StrOptions({"lasso_lars", "lasso_cd", "lars", "omp", "threshold"})
        ],
        "transform_n_nonzero_coefs": [Interval(Integral, 1, None, closed="left"), None],
        "transform_alpha": [Interval(Real, 0, None, closed="left"), None],
        "verbose": ["verbose"],
        "split_sign": ["boolean"],
        "random_state": ["random_state"],
        "positive_code": ["boolean"],
        "positive_dict": ["boolean"],
        "transform_max_iter": [Interval(Integral, 0, None, closed="left")],
        "callback": [None, callable],
        "tol": [Interval(Real, 0, None, closed="left")],
        "max_no_improvement": [Interval(Integral, 0, None, closed="left"), None],
    }

    def __init__(
        self,
        n_components=None,
        *,
        alpha=1,
        max_iter=1_000,
        fit_algorithm="lars",
        n_jobs=None,
        batch_size=256,
        shuffle=True,
        dict_init=None,
        transform_algorithm="omp",
        transform_n_nonzero_coefs=None,
        transform_alpha=None,
        verbose=False,
        split_sign=False,
        random_state=None,
        positive_code=False,
        positive_dict=False,
        transform_max_iter=1000,
        callback=None,
        tol=1e-3,
        max_no_improvement=10,
    ):
        super().__init__(
            transform_algorithm,
            transform_n_nonzero_coefs,
            transform_alpha,
            split_sign,
            n_jobs,
            positive_code,
            transform_max_iter,
        )
        self.n_components = n_components
        self.alpha = alpha
        self.max_iter = max_iter
        self.fit_algorithm = fit_algorithm
        self.dict_init = dict_init
        self.verbose = verbose
        self.shuffle = shuffle
        self.batch_size = batch_size
        self.split_sign = split_sign
        self.random_state = random_state
        self.positive_dict = positive_dict
        self.callback = callback
        self.max_no_improvement = max_no_improvement
        self.tol = tol

    def _check_params(self, X):
        # n_components
        self._n_components = self.n_components
        if self._n_components is None:
            self._n_components = X.shape[1]

        # fit_algorithm
        _check_positive_coding(self.fit_algorithm, self.positive_code)
        self._fit_algorithm = "lasso_" + self.fit_algorithm

        # batch_size
        self._batch_size = min(self.batch_size, X.shape[0])

    def _initialize_dict(self, X, random_state):
        """Initialization of the dictionary."""
        if self.dict_init is not None:
            dictionary = self.dict_init
        else:
            # Init V with SVD of X
            _, S, dictionary = randomized_svd(
                X, self._n_components, random_state=random_state
            )
            dictionary = S[:, np.newaxis] * dictionary

        if self._n_components <= len(dictionary):
            dictionary = dictionary[: self._n_components, :]
        else:
            dictionary = np.concatenate(
                (
                    dictionary,
                    np.zeros(
                        (self._n_components - len(dictionary), dictionary.shape[1]),
                        dtype=dictionary.dtype,
                    ),
                )
            )

        dictionary = check_array(dictionary, order="F", dtype=X.dtype, copy=False)
        dictionary = np.require(dictionary, requirements="W")

        return dictionary

    def _update_inner_stats(self, X, code, batch_size, step):
        """Update the inner stats inplace."""
        if step < batch_size - 1:
            theta = (step + 1) * batch_size
        else:
            theta = batch_size**2 + step + 1 - batch_size
        beta = (theta + 1 - batch_size) / (theta + 1)

        self._A *= beta
        self._A += code.T @ code / batch_size
        self._B *= beta
        self._B += X.T @ code / batch_size

    def _minibatch_step(self, X, dictionary, random_state, step):
        """Perform the update on the dictionary for one minibatch."""
        batch_size = X.shape[0]

        # Compute code for this batch
        code = _sparse_encode(
            X,
            dictionary,
            algorithm=self._fit_algorithm,
            alpha=self.alpha,
            n_jobs=self.n_jobs,
            positive=self.positive_code,
            max_iter=self.transform_max_iter,
            verbose=self.verbose,
        )

        batch_cost = (
            0.5 * ((X - code @ dictionary) ** 2).sum()
            + self.alpha * np.sum(np.abs(code))
        ) / batch_size

        # Update inner stats
        self._update_inner_stats(X, code, batch_size, step)

        # Update dictionary
        _update_dict(
            dictionary,
            X,
            code,
            self._A,
            self._B,
            verbose=self.verbose,
            random_state=random_state,
            positive=self.positive_dict,
        )

        return batch_cost

    def _check_convergence(
        self, X, batch_cost, new_dict, old_dict, n_samples, step, n_steps
    ):
        """Helper function to encapsulate the early stopping logic.

        Early stopping is based on two factors:
        - A small change of the dictionary between two minibatch updates. This is
          controlled by the tol parameter.
        - No more improvement on a smoothed estimate of the objective function for a
          a certain number of consecutive minibatch updates. This is controlled by
          the max_no_improvement parameter.
        """
        batch_size = X.shape[0]

        # counts steps starting from 1 for user friendly verbose mode.
        step = step + 1

        # Ignore 100 first steps or 1 epoch to avoid initializing the ewa_cost with a
        # too bad value
        if step <= min(100, n_samples / batch_size):
            if self.verbose:
                print(f"Minibatch step {step}/{n_steps}: mean batch cost: {batch_cost}")
            return False

        # Compute an Exponentially Weighted Average of the cost function to
        # monitor the convergence while discarding minibatch-local stochastic
        # variability: https://en.wikipedia.org/wiki/Moving_average
        if self._ewa_cost is None:
            self._ewa_cost = batch_cost
        else:
            alpha = batch_size / (n_samples + 1)
            alpha = min(alpha, 1)
            self._ewa_cost = self._ewa_cost * (1 - alpha) + batch_cost * alpha

        if self.verbose:
            print(
                f"Minibatch step {step}/{n_steps}: mean batch cost: "
                f"{batch_cost}, ewa cost: {self._ewa_cost}"
            )

        # Early stopping based on change of dictionary
        dict_diff = linalg.norm(new_dict - old_dict) / self._n_components
        if self.tol > 0 and dict_diff <= self.tol:
            if self.verbose:
                print(f"Converged (small dictionary change) at step {step}/{n_steps}")
            return True

        # Early stopping heuristic due to lack of improvement on smoothed
        # cost function
        if self._ewa_cost_min is None or self._ewa_cost < self._ewa_cost_min:
            self._no_improvement = 0
            self._ewa_cost_min = self._ewa_cost
        else:
            self._no_improvement += 1

        if (
            self.max_no_improvement is not None
            and self._no_improvement >= self.max_no_improvement
        ):
            if self.verbose:
                print(
                    "Converged (lack of improvement in objective function) "
                    f"at step {step}/{n_steps}"
                )
            return True

        return False

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

        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.

        y : Ignored
            Not used, present for API consistency by convention.

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        X = self._validate_data(
            X, dtype=[np.float64, np.float32], order="C", copy=False
        )

        self._check_params(X)
        self._random_state = check_random_state(self.random_state)

        dictionary = self._initialize_dict(X, self._random_state)
        old_dict = dictionary.copy()

        if self.shuffle:
            X_train = X.copy()
            self._random_state.shuffle(X_train)
        else:
            X_train = X

        n_samples, n_features = X_train.shape

        if self.verbose:
            print("[dict_learning]")

        # Inner stats
        self._A = np.zeros(
            (self._n_components, self._n_components), dtype=X_train.dtype
        )
        self._B = np.zeros((n_features, self._n_components), dtype=X_train.dtype)

        # TODO(1.6): remove in 1.6
        if self.max_iter is None:
            warn(
                (
                    "`max_iter=None` is deprecated in version 1.4 and will be removed"
                    " in version 1.6. Use the default value (i.e. `1_000`) instead."
                ),
                FutureWarning,
            )
            max_iter = 1_000
        else:
            max_iter = self.max_iter

        # Attributes to monitor the convergence
        self._ewa_cost = None
        self._ewa_cost_min = None
        self._no_improvement = 0

        batches = gen_batches(n_samples, self._batch_size)
        batches = itertools.cycle(batches)
        n_steps_per_iter = int(np.ceil(n_samples / self._batch_size))
        n_steps = max_iter * n_steps_per_iter

        i = -1  # to allow max_iter = 0

        for i, batch in zip(range(n_steps), batches):
            X_batch = X_train[batch]

            batch_cost = self._minibatch_step(
                X_batch, dictionary, self._random_state, i
            )

            if self._check_convergence(
                X_batch, batch_cost, dictionary, old_dict, n_samples, i, n_steps
            ):
                break

            # XXX callback param added for backward compat in #18975 but a common
            # unified callback API should be preferred
            if self.callback is not None:
                self.callback(locals())

            old_dict[:] = dictionary

        self.n_steps_ = i + 1
        self.n_iter_ = np.ceil(self.n_steps_ / n_steps_per_iter)
        self.components_ = dictionary

        return self

    @_fit_context(prefer_skip_nested_validation=True)
    def partial_fit(self, X, y=None):
        """Update the model using the data in X as a mini-batch.

        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.

        y : Ignored
            Not used, present for API consistency by convention.

        Returns
        -------
        self : object
            Return the instance itself.
        """
        has_components = hasattr(self, "components_")

        X = self._validate_data(
            X, dtype=[np.float64, np.float32], order="C", reset=not has_components
        )

        if not has_components:
            # This instance has not been fitted yet (fit or partial_fit)
            self._check_params(X)
            self._random_state = check_random_state(self.random_state)

            dictionary = self._initialize_dict(X, self._random_state)

            self.n_steps_ = 0

            self._A = np.zeros((self._n_components, self._n_components), dtype=X.dtype)
            self._B = np.zeros((X.shape[1], self._n_components), dtype=X.dtype)
        else:
            dictionary = self.components_

        self._minibatch_step(X, dictionary, self._random_state, self.n_steps_)

        self.components_ = dictionary
        self.n_steps_ += 1

        return self

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

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