dict_learning.py

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

import time
import sys
import itertools

from math import ceil

import numpy as np
from scipy import linalg
from joblib import Parallel, delayed, effective_n_jobs

from ..base import BaseEstimator, TransformerMixin
from ..utils import (check_array, check_random_state, gen_even_slices,
                     gen_batches)
from ..utils.extmath import randomized_svd, row_norms
from ..utils.validation import check_is_fitted
from ..linear_model import Lasso, orthogonal_mp_gram, LassoLars, Lars


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(X, dictionary, gram, cov=None, algorithm='lasso_lars',
                   regularization=None, copy_cov=True,
                   init=None, max_iter=1000, check_input=True, verbose=0,
                   positive=False):
    """Generic sparse coding

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

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

    dictionary : array 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 : None | array, shape=(n_components, n_components)
        Precomputed Gram matrix, dictionary * dictionary'
        gram can be None if method is 'threshold'.

    cov : array, shape=(n_components, n_samples)
        Precomputed covariance, dictionary * X'

    algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', 'threshold'}
        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 | float
        The regularization parameter. It corresponds to alpha when
        algorithm is 'lasso_lars', 'lasso_cd' or 'threshold'.
        Otherwise it corresponds to n_nonzero_coefs.

    init : array of shape (n_samples, n_components)
        Initialization value of the sparse code. Only used if
        `algorithm='lasso_cd'`.

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

    copy_cov : boolean, optional
        Whether to copy the precomputed covariance matrix; if False, it may be
        overwritten.

    check_input : boolean, optional
        If False, the input arrays X and dictionary will not be checked.

    verbose : int
        Controls the verbosity; the higher, the more messages. Defaults to 0.

    positive: boolean
        Whether to enforce a positivity constraint on the sparse code.

        .. versionadded:: 0.20

    Returns
    -------
    code : array of shape (n_components, n_features)
        The sparse codes

    See also
    --------
    sklearn.linear_model.lars_path
    sklearn.linear_model.orthogonal_mp
    sklearn.linear_model.Lasso
    SparseCoder
    """
    if X.ndim == 1:
        X = X[:, np.newaxis]
    n_samples, n_features = X.shape
    n_components = dictionary.shape[0]
    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))
    if cov is None and algorithm != 'lasso_cd':
        # overwriting cov is safe
        copy_cov = False
        cov = np.dot(dictionary, X.T)

    _check_positive_coding(algorithm, positive)

    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, normalize=False,
                                   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, normalize=False,
                    precompute=gram, max_iter=max_iter, warm_start=True,
                    positive=positive)

        if init is not None:
            clf.coef_ = init

        clf.fit(dictionary.T, X.T, check_input=check_input)
        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, normalize=False,
                        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
    else:
        raise ValueError('Sparse coding method must be "lasso_lars" '
                         '"lasso_cd", "lasso", "threshold" or "omp", got %s.'
                         % algorithm)
    if new_code.ndim != 2:
        return new_code.reshape(n_samples, n_components)
    return new_code


# 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 of shape (n_samples, n_features)
        Data matrix

    dictionary : array 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, shape=(n_components, n_components)
        Precomputed Gram matrix, dictionary * dictionary'

    cov : array, shape=(n_components, n_samples)
        Precomputed covariance, dictionary' * X

    algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', 'threshold'}
        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'

    n_nonzero_coefs : int, 0.1 * n_features by default
        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.

    alpha : float, 1. by default
        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`.

    copy_cov : boolean, optional
        Whether to copy the precomputed covariance matrix; if False, it may be
        overwritten.

    init : array of shape (n_samples, n_components)
        Initialization value of the sparse codes. Only used if
        `algorithm='lasso_cd'`.

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

    n_jobs : int or None, optional (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 : boolean, optional
        If False, the input arrays X and dictionary will not be checked.

    verbose : int, optional
        Controls the verbosity; the higher, the more messages. Defaults to 0.

    positive : boolean, optional
        Whether to enforce positivity when finding the encoding.

        .. versionadded:: 0.20

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

    See also
    --------
    sklearn.linear_model.lars_path
    sklearn.linear_model.orthogonal_mp
    sklearn.linear_model.Lasso
    SparseCoder
    """
    if check_input:
        if algorithm == 'lasso_cd':
            dictionary = check_array(dictionary, order='C', dtype='float64')
            X = check_array(X, order='C', dtype='float64')
        else:
            dictionary = check_array(dictionary)
            X = check_array(X)

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

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

    # Enter parallel code block
    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)(
            X[this_slice], dictionary, gram,
            cov[:, this_slice] if cov is not None else None,
            algorithm,
            regularization=regularization, copy_cov=copy_cov,
            init=init[this_slice] if init is not None else None,
            max_iter=max_iter,
            check_input=False,
            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, verbose=False, return_r2=False,
                 random_state=None, positive=False):
    """Update the dense dictionary factor in place.

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

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

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

    verbose:
        Degree of output the procedure will print.

    return_r2 : bool
        Whether to compute and return the residual sum of squares corresponding
        to the computed solution.

    random_state : int, RandomState instance or None, optional (default=None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.

    positive : boolean, optional
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    Returns
    -------
    dictionary : array of shape (n_features, n_components)
        Updated dictionary.

    """
    n_components = len(code)
    n_features = Y.shape[0]
    random_state = check_random_state(random_state)
    # Get BLAS functions
    gemm, = linalg.get_blas_funcs(('gemm',), (dictionary, code, Y))
    ger, = linalg.get_blas_funcs(('ger',), (dictionary, code))
    nrm2, = linalg.get_blas_funcs(('nrm2',), (dictionary,))
    # Residuals, computed with BLAS for speed and efficiency
    # R <- -1.0 * U * V^T + 1.0 * Y
    # Outputs R as Fortran array for efficiency
    R = gemm(-1.0, dictionary, code, 1.0, Y)
    for k in range(n_components):
        # R <- 1.0 * U_k * V_k^T + R
        R = ger(1.0, dictionary[:, k], code[k, :], a=R, overwrite_a=True)
        dictionary[:, k] = np.dot(R, code[k, :])
        if positive:
            np.clip(dictionary[:, k], 0, None, out=dictionary[:, k])
        # Scale k'th atom
        # (U_k * U_k) ** 0.5
        atom_norm = nrm2(dictionary[:, k])
        if atom_norm < 1e-10:
            if verbose == 1:
                sys.stdout.write("+")
                sys.stdout.flush()
            elif verbose:
                print("Adding new random atom")
            dictionary[:, k] = random_state.randn(n_features)
            if positive:
                np.clip(dictionary[:, k], 0, None, out=dictionary[:, k])
            # Setting corresponding coefs to 0
            code[k, :] = 0.0
            # (U_k * U_k) ** 0.5
            atom_norm = nrm2(dictionary[:, k])
            dictionary[:, k] /= atom_norm
        else:
            dictionary[:, k] /= atom_norm
            # R <- -1.0 * U_k * V_k^T + R
            R = ger(-1.0, dictionary[:, k], code[k, :], a=R, overwrite_a=True)
    if return_r2:
        R = nrm2(R) ** 2.0
        return dictionary, R
    return dictionary


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):
    """Solves 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 ||_2^2 + alpha * || U ||_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.

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

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

    n_components : int,
        Number of dictionary atoms to extract.

    alpha : int,
        Sparsity controlling parameter.

    max_iter : int,
        Maximum number of iterations to perform.

    tol : float,
        Tolerance for the stopping condition.

    method : {'lars', 'cd'}
        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 or None, optional (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 : array of shape (n_components, n_features),
        Initial value for the dictionary for warm restart scenarios.

    code_init : array of shape (n_samples, n_components),
        Initial value for the sparse code for warm restart scenarios.

    callback : callable or None, optional (default: None)
        Callable that gets invoked every five iterations

    verbose : bool, optional (default: False)
        To control the verbosity of the procedure.

    random_state : int, RandomState instance or None, optional (default=None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.

    return_n_iter : bool
        Whether or not to return the number of iterations.

    positive_dict : bool
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    positive_code : bool
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

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

        .. versionadded:: 0.22

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

    dictionary : array 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
    DictionaryLearning
    MiniBatchDictionaryLearning
    SparsePCA
    MiniBatchSparsePCA
    """
    if method not in ('lars', 'cd'):
        raise ValueError('Coding method %r not supported as a fit algorithm.'
                         % method)

    _check_positive_coding(method, positive_code)

    method = 'lasso_' + method

    t0 = time.time()
    # Avoid integer division problems
    alpha = float(alpha)
    random_state = check_random_state(random_state)

    # 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)
        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, as we are going to access its row vectors
    dictionary = np.array(dictionary, order='F')

    residuals = 0

    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
        dictionary, residuals = _update_dict(dictionary.T, X.T, code.T,
                                             verbose=verbose, return_r2=True,
                                             random_state=random_state,
                                             positive=positive_dict)
        dictionary = dictionary.T

        # Cost function
        current_cost = 0.5 * residuals + 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


def dict_learning_online(X, n_components=2, alpha=1, n_iter=100,
                         return_code=True, dict_init=None, callback=None,
                         batch_size=3, verbose=False, shuffle=True,
                         n_jobs=None, method='lars', iter_offset=0,
                         random_state=None, return_inner_stats=False,
                         inner_stats=None, return_n_iter=False,
                         positive_dict=False, positive_code=False,
                         method_max_iter=1000):
    """Solves 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 ||_2^2 + alpha * || U ||_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. 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 of shape (n_samples, n_features)
        Data matrix.

    n_components : int,
        Number of dictionary atoms to extract.

    alpha : float,
        Sparsity controlling parameter.

    n_iter : int,
        Number of mini-batch iterations to perform.

    return_code : boolean,
        Whether to also return the code U or just the dictionary V.

    dict_init : array of shape (n_components, n_features),
        Initial value for the dictionary for warm restart scenarios.

    callback : callable or None, optional (default: None)
        callable that gets invoked every five iterations

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

    verbose : bool, optional (default: False)
        To control the verbosity of the procedure.

    shuffle : boolean,
        Whether to shuffle the data before splitting it in batches.

    n_jobs : int or None, optional (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'}
        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.

    iter_offset : int, default 0
        Number of previous iterations completed on the dictionary used for
        initialization.

    random_state : int, RandomState instance or None, optional (default=None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.

    return_inner_stats : boolean, optional
        Return the inner statistics A (dictionary covariance) and B
        (data approximation). Useful to restart the algorithm in an
        online setting. If return_inner_stats is True, return_code is
        ignored

    inner_stats : tuple of (A, B) ndarrays
        Inner sufficient statistics that are kept by the algorithm.
        Passing them at initialization is useful in online settings, to
        avoid loosing the history of the evolution.
        A (n_components, n_components) is the dictionary covariance matrix.
        B (n_features, n_components) is the data approximation matrix

    return_n_iter : bool
        Whether or not to return the number of iterations.

    positive_dict : bool
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

    positive_code : bool
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

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

        .. versionadded:: 0.22

    Returns
    -------
    code : array of shape (n_samples, n_components),
        the sparse code (only returned if `return_code=True`)

    dictionary : array 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
    DictionaryLearning
    MiniBatchDictionaryLearning
    SparsePCA
    MiniBatchSparsePCA

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

    if method not in ('lars', 'cd'):
        raise ValueError('Coding method not supported as a fit algorithm.')

    _check_positive_coding(method, positive_code)

    method = 'lasso_' + method

    t0 = time.time()
    n_samples, n_features = X.shape
    # Avoid integer division problems
    alpha = float(alpha)
    random_state = check_random_state(random_state)

    # Init V with SVD of X
    if dict_init is not None:
        dictionary = dict_init
    else:
        _, S, dictionary = randomized_svd(X, n_components,
                                          random_state=random_state)
        dictionary = S[:, np.newaxis] * dictionary
    r = len(dictionary)
    if n_components <= r:
        dictionary = dictionary[:n_components, :]
    else:
        dictionary = np.r_[dictionary,
                           np.zeros((n_components - r, dictionary.shape[1]))]

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

    if shuffle:
        X_train = X.copy()
        random_state.shuffle(X_train)
    else:
        X_train = X

    dictionary = check_array(dictionary.T, order='F', dtype=np.float64,
                             copy=False)
    dictionary = np.require(dictionary, requirements='W')

    X_train = check_array(X_train, order='C', dtype=np.float64, copy=False)

    batches = gen_batches(n_samples, batch_size)
    batches = itertools.cycle(batches)

    # The covariance of the dictionary
    if inner_stats is None:
        A = np.zeros((n_components, n_components))
        # The data approximation
        B = np.zeros((n_features, n_components))
    else:
        A = inner_stats[0].copy()
        B = inner_stats[1].copy()

    # If n_iter is zero, we need to return zero.
    ii = iter_offset - 1

    for ii, batch in zip(range(iter_offset, iter_offset + n_iter), batches):
        this_X = X_train[batch]
        dt = (time.time() - t0)
        if verbose == 1:
            sys.stdout.write(".")
            sys.stdout.flush()
        elif verbose:
            if verbose > 10 or ii % ceil(100. / verbose) == 0:
                print("Iteration % 3i (elapsed time: % 3is, % 4.1fmn)"
                      % (ii, dt, dt / 60))

        this_code = sparse_encode(this_X, dictionary.T, algorithm=method,
                                  alpha=alpha, n_jobs=n_jobs,
                                  check_input=False,
                                  positive=positive_code,
                                  max_iter=method_max_iter, verbose=verbose).T

        # Update the auxiliary variables
        if ii < batch_size - 1:
            theta = float((ii + 1) * batch_size)
        else:
            theta = float(batch_size ** 2 + ii + 1 - batch_size)
        beta = (theta + 1 - batch_size) / (theta + 1)

        A *= beta
        A += np.dot(this_code, this_code.T)
        B *= beta
        B += np.dot(this_X.T, this_code.T)

        # Update dictionary
        dictionary = _update_dict(dictionary, B, A, verbose=verbose,
                                  random_state=random_state,
                                  positive=positive_dict)
        # XXX: Can the residuals be of any use?

        # Maybe we need a stopping criteria based on the amount of
        # modification in the dictionary
        if callback is not None:
            callback(locals())

    if return_inner_stats:
        if return_n_iter:
            return dictionary.T, (A, B), ii - iter_offset + 1
        else:
            return dictionary.T, (A, B)
    if return_code:
        if verbose > 1:
            print('Learning code...', end=' ')
        elif verbose == 1:
            print('|', end=' ')
        code = sparse_encode(X, dictionary.T, algorithm=method, alpha=alpha,
                             n_jobs=n_jobs, check_input=False,
                             positive=positive_code, max_iter=method_max_iter,
                             verbose=verbose)
        if verbose > 1:
            dt = (time.time() - t0)
            print('done (total time: % 3is, % 4.1fmn)' % (dt, dt / 60))
        if return_n_iter:
            return code, dictionary.T, ii - iter_offset + 1
        else:
            return code, dictionary.T

    if return_n_iter:
        return dictionary.T, ii - iter_offset + 1
    else:
        return dictionary.T


class SparseCodingMixin(TransformerMixin):
    """Sparse coding mixin"""

    def _set_sparse_coding_params(self, n_components,
                                  transform_algorithm='omp',
                                  transform_n_nonzero_coefs=None,
                                  transform_alpha=None, split_sign=False,
                                  n_jobs=None, positive_code=False,
                                  transform_max_iter=1000):
        self.n_components = n_components
        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):
        """Encode the data as a sparse combination of the dictionary atoms.

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

        Parameters
        ----------
        X : array 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 : array, shape (n_samples, n_components)
            Transformed data

        """
        check_is_fitted(self)

        X = check_array(X)

        code = sparse_encode(
            X, self.components_, algorithm=self.transform_algorithm,
            n_nonzero_coefs=self.transform_n_nonzero_coefs,
            alpha=self.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


class SparseCoder(SparseCodingMixin, 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 : array, [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'}
        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, ``0.1 * n_features`` by default
        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.

    transform_alpha : float, 1. by default
        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`.

    split_sign : bool, False by default
        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 or None, optional (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
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

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

        .. versionadded:: 0.22

    Attributes
    ----------
    components_ : array, [n_components, n_features]
        The unchanged dictionary atoms

    See also
    --------
    DictionaryLearning
    MiniBatchDictionaryLearning
    SparsePCA
    MiniBatchSparsePCA
    sparse_encode
    """
    _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):
        self._set_sparse_coding_params(dictionary.shape[0],
                                       transform_algorithm,
                                       transform_n_nonzero_coefs,
                                       transform_alpha, split_sign, n_jobs,
                                       positive_code, transform_max_iter)
        self.components_ = 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

        y : Ignored

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


class DictionaryLearning(SparseCodingMixin, BaseEstimator):
    """Dictionary learning

    Finds a dictionary (a set of atoms) that can best be used to represent data
    using a sparse code.

    Solves the optimization problem::

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

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

    Parameters
    ----------
    n_components : int,
        number of dictionary elements to extract

    alpha : float,
        sparsity controlling parameter

    max_iter : int,
        maximum number of iterations to perform

    tol : float,
        tolerance for numerical error

    fit_algorithm : {'lars', 'cd'}
        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.

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

    transform_algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', \
    'threshold'}
        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'``

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

    transform_n_nonzero_coefs : int, ``0.1 * n_features`` by default
        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.

    transform_alpha : float, 1. by default
        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`.

    n_jobs : int or None, optional (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 : array of shape (n_samples, n_components),
        initial value for the code, for warm restart

    dict_init : array of shape (n_components, n_features),
        initial values for the dictionary, for warm restart

    verbose : bool, optional (default: False)
        To control the verbosity of the procedure.

    split_sign : bool, False by default
        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, optional (default=None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.

    positive_code : bool
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    positive_dict : bool
        Whether to enforce positivity when finding the dictionary

        .. versionadded:: 0.20

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

        .. versionadded:: 0.22

    Attributes
    ----------
    components_ : array, [n_components, n_features]
        dictionary atoms extracted from the data

    error_ : array
        vector of errors at each iteration

    n_iter_ : int
        Number of iterations run.

    Notes
    -----
    **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)

    See also
    --------
    SparseCoder
    MiniBatchDictionaryLearning
    SparsePCA
    MiniBatchSparsePCA
    """
    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, verbose=False,
                 split_sign=False, random_state=None, positive_code=False,
                 positive_dict=False, transform_max_iter=1000):

        self._set_sparse_coding_params(n_components, transform_algorithm,
                                       transform_n_nonzero_coefs,
                                       transform_alpha, split_sign, n_jobs,
                                       positive_code, transform_max_iter)
        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.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, shape (n_samples, n_features)
            Training vector, where n_samples in the number of samples
            and n_features is the number of features.

        y : Ignored

        Returns
        -------
        self : object
            Returns the object itself
        """
        random_state = check_random_state(self.random_state)
        X = check_array(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, self.alpha,
            tol=self.tol, max_iter=self.max_iter,
            method=self.fit_algorithm,
            method_max_iter=self.transform_max_iter,
            n_jobs=self.n_jobs,
            code_init=self.code_init,
            dict_init=self.dict_init,
            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 self


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

    Finds a dictionary (a set of atoms) that can best be used to represent data
    using a sparse code.

    Solves the optimization problem::

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

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

    Parameters
    ----------
    n_components : int,
        number of dictionary elements to extract

    alpha : float,
        sparsity controlling parameter

    n_iter : int,
        total number of iterations to perform

    fit_algorithm : {'lars', 'cd'}
        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 or None, optional (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,
        number of samples in each mini-batch

    shuffle : bool,
        whether to shuffle the samples before forming batches

    dict_init : array of shape (n_components, n_features),
        initial value of the dictionary for warm restart scenarios

    transform_algorithm : {'lasso_lars', 'lasso_cd', 'lars', 'omp', \
    'threshold'}
        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, ``0.1 * n_features`` by default
        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.

    transform_alpha : float, 1. by default
        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`.

    verbose : bool, optional (default: False)
        To control the verbosity of the procedure.

    split_sign : bool, False by default
        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, optional (default=None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.

    positive_code : bool
        Whether to enforce positivity when finding the code.

        .. versionadded:: 0.20

    positive_dict : bool
        Whether to enforce positivity when finding the dictionary.

        .. versionadded:: 0.20

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

        .. versionadded:: 0.22

    Attributes
    ----------
    components_ : array, [n_components, n_features]
        components extracted from the data

    inner_stats_ : tuple of (A, B) ndarrays
        Internal sufficient statistics that are kept by the algorithm.
        Keeping them is useful in online settings, to avoid loosing the
        history of the evolution, but they shouldn't have any use for the
        end user.
        A (n_components, n_components) is the dictionary covariance matrix.
        B (n_features, n_components) is the data approximation matrix

    n_iter_ : int
        Number of iterations run.

    iter_offset_ : int
        The number of iteration on data batches that has been
        performed before.

    random_state_ : RandomState
        RandomState instance that is generated either from a seed, the random
        number generattor or by `np.random`.

    Notes
    -----
    **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)

    See also
    --------
    SparseCoder
    DictionaryLearning
    SparsePCA
    MiniBatchSparsePCA

    """
    def __init__(self, n_components=None, alpha=1, n_iter=1000,
                 fit_algorithm='lars', n_jobs=None, batch_size=3, 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):

        self._set_sparse_coding_params(n_components, transform_algorithm,
                                       transform_n_nonzero_coefs,
                                       transform_alpha, split_sign, n_jobs,
                                       positive_code, transform_max_iter)
        self.alpha = alpha
        self.n_iter = n_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

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

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

        y : Ignored

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        random_state = check_random_state(self.random_state)
        X = check_array(X)

        U, (A, B), self.n_iter_ = dict_learning_online(
            X, self.n_components, self.alpha,
            n_iter=self.n_iter, return_code=False,
            method=self.fit_algorithm,
            method_max_iter=self.transform_max_iter,
            n_jobs=self.n_jobs, dict_init=self.dict_init,
            batch_size=self.batch_size, shuffle=self.shuffle,
            verbose=self.verbose, random_state=random_state,
            return_inner_stats=True,
            return_n_iter=True,
            positive_dict=self.positive_dict,
            positive_code=self.positive_code)
        self.components_ = U
        # Keep track of the state of the algorithm to be able to do
        # some online fitting (partial_fit)
        self.inner_stats_ = (A, B)
        self.iter_offset_ = self.n_iter
        return self

    def partial_fit(self, X, y=None, iter_offset=None):
        """Updates the model using the data in X as a mini-batch.

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

        y : Ignored

        iter_offset : integer, optional
            The number of iteration on data batches that has been
            performed before this call to partial_fit. This is optional:
            if no number is passed, the memory of the object is
            used.

        Returns
        -------
        self : object
            Returns the instance itself.
        """
        if not hasattr(self, 'random_state_'):
            self.random_state_ = check_random_state(self.random_state)
        X = check_array(X)
        if hasattr(self, 'components_'):
            dict_init = self.components_
        else:
            dict_init = self.dict_init
        inner_stats = getattr(self, 'inner_stats_', None)
        if iter_offset is None:
            iter_offset = getattr(self, 'iter_offset_', 0)
        U, (A, B) = dict_learning_online(
            X, self.n_components, self.alpha,
            n_iter=self.n_iter, method=self.fit_algorithm,
            method_max_iter=self.transform_max_iter,
            n_jobs=self.n_jobs, dict_init=dict_init,
            batch_size=len(X), shuffle=False,
            verbose=self.verbose, return_code=False,
            iter_offset=iter_offset, random_state=self.random_state_,
            return_inner_stats=True, inner_stats=inner_stats,
            positive_dict=self.positive_dict,
            positive_code=self.positive_code)
        self.components_ = U

        # Keep track of the state of the algorithm to be able to do
        # some online fitting (partial_fit)
        self.inner_stats_ = (A, B)
        self.iter_offset_ = iter_offset + self.n_iter
        return self