参考下列代码，如何在(N,N,M)维张量的cp分解过程中约束(N,N)维的两个因子矩阵相同？：import numpy as np
import warnings

import tensorly as tl
from ..random import random_cp
from ._base_decomposition import DecompositionMixin
from ..base import unfold
from ..cp_tensor import CPTensor, cp_norm, validate_cp_rank
from ..tenalg.proximal import admm, proximal_operator, validate_constraints
from ..tenalg.svd import svd_interface
from ..tenalg import unfolding_dot_khatri_rao


# Author: Jean Kossaifi
#         Jeremy Cohen <<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>>
#         Caglayan Tuna <<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>>

# License: BSD 3 clause


def initialize_constrained_parafac(
    tensor,
    rank,
    init="svd",
    svd="truncated_svd",
    random_state=None,
    non_negative=None,
    l1_reg=None,
    l2_reg=None,
    l2_square_reg=None,
    unimodality=None,
    normalize=None,
    simplex=None,
    normalized_sparsity=None,
    soft_sparsity=None,
    smoothness=None,
    monotonicity=None,
    hard_sparsity=None,
):
    r"""Initialize factors used in `constrained_parafac`.

    Parameters
    ----------

    The type of initialization is set using `init`. If `init == 'random'` then
    initialize factor matrices with uniform distribution using `random_state`. If `init == 'svd'` then
    initialize the `m`th factor matrix using the `rank` left singular vectors
    of the `m`th unfolding of the input tensor. If init is a previously initialized `cp tensor`, all
    the weights are pulled in the last factor and then the weights are set to "1" for the output tensor.
    Lastly, factors are updated with proximal operator according to the selected constraint(s), so that they satisfy the
    imposed constraints (does not apply to cptensor initialization).

    Parameters
    ----------
    tensor : ndarray
    rank : int
    random_state : {None, int, np.random.RandomState}
    init : {'svd', 'random', cptensor}, optional
    svd : str, default is 'truncated_svd'
        function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
    non_negative : bool or dictionary
        This constraint is clipping negative values to '0'.
        If it is True, non-negative constraint is applied to all modes.
    l1_reg : float or list or dictionary, optional
        Penalizes the factor with the l1 norm using the input value as regularization parameter.
    l2_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 norm using the input value as regularization parameter.
    l2_square_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 square norm using the input value as regularization parameter.
    unimodality : bool or dictionary, optional
        If it is True, unimodality constraint is applied to all modes.
        Applied to each column seperately.
    normalize : bool or dictionary, optional
        This constraint divides all the values by maximum value of the input array.
        If it is True, normalize constraint is applied to all modes.
    simplex : float or list or dictionary, optional
        Projects on the simplex with the given parameter
        Applied to each column seperately.
    normalized_sparsity : float or list or dictionary, optional
        Normalizes with the norm after hard thresholding
    soft_sparsity : float or list or dictionary, optional
        Impose that the columns of factors have L1 norm bounded by a user-defined threshold.
    smoothness : float or list or dictionary, optional
        Optimizes the factors by solving a banded system
    monotonicity : bool or dictionary, optional
        Projects columns to monotonically decreasing distrbution
        Applied to each column seperately.
        If it is True, monotonicity constraint is applied to all modes.
    hard_sparsity : float or list or dictionary, optional
        Hard thresholding with the given threshold
    Returns
    -------
    factors : CPTensor
        An initial cp tensor.
    """
    n_modes = tl.ndim(tensor)
    rng = tl.check_random_state(random_state)

    if init == "random":
        weights, factors = random_cp(
            tl.shape(tensor), rank, normalise_factors=False, **tl.context(tensor)
        )

    elif init == "svd":
        factors = []
        for mode in range(tl.ndim(tensor)):
            U, S, _ = svd_interface(unfold(tensor, mode), n_eigenvecs=rank, method=svd)

            # Put SVD initialization on the same scaling as the tensor in case normalize_factors=False
            if mode == 0:
                idx = min(rank, tl.shape(S)[0])
                U = tl.index_update(U, tl.index[:, :idx], U[:, :idx] * S[:idx])

            if tensor.shape[mode] < rank:
                random_part = tl.tensor(
                    rng.random_sample((U.shape[0], rank - tl.shape(tensor)[mode])),
                    **tl.context(tensor),
                )
                U = tl.concatenate([U, random_part], axis=1)

            factors.append(U[:, :rank])

    elif isinstance(init, (tuple, list, CPTensor)):
        try:
            weights, factors = CPTensor(init)

            if tl.all(weights == 1):
                weights, factors = CPTensor((None, factors))
            else:
                weights_avg = tl.prod(weights) ** (1.0 / tl.shape(weights)[0])
                for i in range(len(factors)):
                    factors[i] = factors[i] * weights_avg
            kt = CPTensor((None, factors))
            return kt
        except ValueError:
            raise ValueError(
                "If initialization method is a mapping, then it must "
                "be possible to convert it to a CPTensor instance"
            )
    else:
        raise ValueError(f'Initialization method "{init}" not recognized')

    for i in range(n_modes):
        factors[i] = proximal_operator(
            factors[i],
            non_negative=non_negative,
            l1_reg=l1_reg,
            l2_reg=l2_reg,
            l2_square_reg=l2_square_reg,
            unimodality=unimodality,
            normalize=normalize,
            simplex=simplex,
            normalized_sparsity=normalized_sparsity,
            soft_sparsity=soft_sparsity,
            smoothness=smoothness,
            monotonicity=monotonicity,
            hard_sparsity=hard_sparsity,
            n_const=n_modes,
            order=i,
        )
    kt = CPTensor((None, factors))
    return kt


def constrained_parafac(
    tensor,
    rank,
    n_iter_max=100,
    n_iter_max_inner=10,
    init="svd",
    svd="truncated_svd",
    tol_outer=1e-8,
    tol_inner=1e-6,
    random_state=None,
    verbose=0,
    return_errors=False,
    cvg_criterion="abs_rec_error",
    fixed_modes=None,
    non_negative=None,
    l1_reg=None,
    l2_reg=None,
    l2_square_reg=None,
    unimodality=None,
    normalize=None,
    simplex=None,
    normalized_sparsity=None,
    soft_sparsity=None,
    smoothness=None,
    monotonicity=None,
    hard_sparsity=None,
):
    """CANDECOMP/PARAFAC decomposition via alternating optimization of
    alternating direction method of multipliers (AO-ADMM):

    Computes a rank-`rank` decomposition of `tensor` [1]_ such that::

        tensor = [|weights; factors[0], ..., factors[-1] |],

    where factors are either penalized or constrained according to the user-defined constraint.

    In order to compute the factors efficiently, the ADMM algorithm
    introduces an auxilliary factor which is called factor_aux in the function.

    Parameters
    ----------
    tensor : ndarray
    rank  : int
        Number of components.
    n_iter_max : int
        Maximum number of iteration for outer loop
    n_iter_max_inner : int
        Number of iteration for inner loop
    init : {'svd', 'random', cptensor}, optional
        Type of factor matrix initialization. See `initialize_factors`.
    svd : str, default is 'truncated_svd'
        function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
    tol_outer : float, optional
        (Default: 1e-8) Relative reconstruction error tolerance for outer loop. The
        algorithm is considered to have found a local minimum when the
        reconstruction error is less than `tol_outer`.
    tol_inner : float, optional
        (Default: 1e-6) Absolute reconstruction error tolerance for factor update during inner loop, i.e. ADMM optimization.
    random_state : {None, int, np.random.RandomState}
    verbose : int, optional
        Level of verbosity
    return_errors : bool, optional
        Activate return of iteration errors
    non_negative : bool or dictionary
        This constraint is clipping negative values to '0'.
        If it is True, non-negative constraint is applied to all modes.
    l1_reg : float or list or dictionary, optional
        Penalizes the factor with the l1 norm using the input value as regularization parameter.
    l2_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 norm using the input value as regularization parameter.
    l2_square_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 square norm using the input value as regularization parameter.
    unimodality : bool or dictionary, optional
        If it is True, unimodality constraint is applied to all modes.
        Applied to each column seperately.
    normalize : bool or dictionary, optional
        This constraint divides all the values by maximum value of the input array.
        If it is True, normalize constraint is applied to all modes.
    simplex : float or list or dictionary, optional
        Projects on the simplex with the given parameter
        Applied to each column seperately.
    normalized_sparsity : float or list or dictionary, optional
        Normalizes with the norm after hard thresholding
    soft_sparsity : float or list or dictionary, optional
        Impose that the columns of factors have L1 norm bounded by a user-defined threshold.
    smoothness : float or list or dictionary, optional
        Optimizes the factors by solving a banded system
    monotonicity : bool or dictionary, optional
        Projects columns to monotonically decreasing distrbution
        Applied to each column seperately.
        If it is True, monotonicity constraint is applied to all modes.
    hard_sparsity : float or list or dictionary, optional
        Hard thresholding with the given threshold
    cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
       Stopping criterion if `tol` is not None.
       If 'rec_error',  algorithm stops at current iteration if ``(previous rec_error - current rec_error) < tol``.
       If 'abs_rec_error', algorithm terminates when `|previous rec_error - current rec_error| < tol`.
    fixed_modes : list, default is None
        A list of modes for which the initial value is not modified.
        The last mode cannot be fixed due to error computation.

    Returns
    -------
    CPTensor : (weight, factors)
        * weights : 1D array of shape (rank, )
        * factors : List of factors of the CP decomposition element `i` is of shape ``(tensor.shape[i], rank)``
    errors : list
        A list of reconstruction errors at each iteration of the algorithms.

    References
    ----------
    .. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications", SIAM
           REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
    .. [2] Huang, Kejun, Nicholas D. Sidiropoulos, and Athanasios P. Liavas.
           "A flexible and efficient algorithmic framework for constrained matrix and tensor factorization." IEEE
           Transactions on Signal Processing 64.19 (2016): 5052-5065.
    """
    rank = validate_cp_rank(tl.shape(tensor), rank=rank)
    _, _ = validate_constraints(
        non_negative=non_negative,
        l1_reg=l1_reg,
        l2_reg=l2_reg,
        l2_square_reg=l2_square_reg,
        unimodality=unimodality,
        normalize=normalize,
        simplex=simplex,
        normalized_sparsity=normalized_sparsity,
        soft_sparsity=soft_sparsity,
        smoothness=smoothness,
        monotonicity=monotonicity,
        hard_sparsity=hard_sparsity,
        n_const=tl.ndim(tensor),
    )

    weights, factors = initialize_constrained_parafac(
        tensor,
        rank,
        init=init,
        svd=svd,
        random_state=random_state,
        non_negative=non_negative,
        l1_reg=l1_reg,
        l2_reg=l2_reg,
        l2_square_reg=l2_square_reg,
        unimodality=unimodality,
        normalize=normalize,
        simplex=simplex,
        normalized_sparsity=normalized_sparsity,
        soft_sparsity=soft_sparsity,
        smoothness=smoothness,
        monotonicity=monotonicity,
        hard_sparsity=hard_sparsity,
    )

    rec_errors = []
    norm_tensor = tl.norm(tensor, 2)

    if fixed_modes is None:
        fixed_modes = []

    if tl.ndim(tensor) - 1 in fixed_modes:
        warnings.warn(
            "You asked for fixing the last mode, which is not supported.\n "
            "The last mode will not be fixed. Consider using tl.moveaxis()"
        )
        fixed_modes.remove(tl.ndim(tensor) - 1)
    modes_list = [mode for mode in range(tl.ndim(tensor)) if mode not in fixed_modes]

    # ADMM inits
    dual_variables = []
    factors_aux = []
    for i in range(len(factors)):
        dual_variables.append(tl.zeros(tl.shape(factors[i])))
        factors_aux.append(tl.transpose(tl.zeros(tl.shape(factors[i]))))

    for iteration in range(n_iter_max):
        if verbose > 1:
            print("Starting iteration", iteration + 1)
        for mode in modes_list:
            if verbose > 1:
                print("Mode", mode, "of", tl.ndim(tensor))

            pseudo_inverse = tl.tensor(np.ones((rank, rank)), **tl.context(tensor))
            for i, factor in enumerate(factors):
                if i != mode:
                    pseudo_inverse = pseudo_inverse * tl.dot(
                        tl.transpose(factor), factor
                    )

            mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)

            factors[mode], factors_aux[mode], dual_variables[mode] = admm(
                mttkrp,
                pseudo_inverse,
                factors[mode],
                dual_variables[mode],
                n_iter_max=n_iter_max_inner,
                n_const=tl.ndim(tensor),
                order=mode,
                non_negative=non_negative,
                l1_reg=l1_reg,
                l2_reg=l2_reg,
                l2_square_reg=l2_square_reg,
                unimodality=unimodality,
                normalize=normalize,
                simplex=simplex,
                normalized_sparsity=normalized_sparsity,
                soft_sparsity=soft_sparsity,
                smoothness=smoothness,
                monotonicity=monotonicity,
                hard_sparsity=hard_sparsity,
                tol=tol_inner,
            )

        factors_norm = cp_norm((weights, factors))
        iprod = tl.sum(tl.sum(mttkrp * factors[-1], axis=0) * weights)
        rec_error = (
            tl.sqrt(tl.abs(norm_tensor**2 + factors_norm**2 - 2 * iprod)) / norm_tensor
        )
        rec_errors.append(rec_error)
        constraint_error = 0
        for mode in modes_list:
            constraint_error += tl.norm(
                factors[mode] - tl.transpose(factors_aux[mode])
            ) / tl.norm(factors[mode])
        if tol_outer:
            if iteration >= 1:
                rec_error_decrease = rec_errors[-2] - rec_errors[-1]

                if verbose:
                    print(
                        f"iteration {iteration}, reconstruction error: {rec_error}, decrease = {rec_error_decrease}"
                    )

                if constraint_error < tol_outer:
                    break
                if cvg_criterion == "abs_rec_error":
                    stop_flag = tl.abs(rec_error_decrease) < tol_outer
                elif cvg_criterion == "rec_error":
                    stop_flag = rec_error_decrease < tol_outer
                else:
                    raise TypeError("Unknown convergence criterion")

                if stop_flag:
                    if verbose:
                        print(f"PARAFAC converged after {iteration} iterations")
                    break

            else:
                if verbose:
                    print(f"reconstruction error={rec_errors[-1]}")

    cp_tensor = CPTensor((weights, factors))

    if return_errors:
        return cp_tensor, rec_errors
    else:
        return cp_tensor


class ConstrainedCP(DecompositionMixin):
    """CANDECOMP/PARAFAC decomposition via alternating optimization of
    alternating direction method of multipliers (AO-ADMM):

    Computes a rank-`rank` decomposition of `tensor` [1]_ such that::

        tensor = [|weights; factors[0], ..., factors[-1] |],

    where factors are either penalized or constrained according to the user-defined constraint.

    In order to compute the factors efficiently, the ADMM algorithm
    introduces an auxilliary factor which is called factor_aux in the function.

    Parameters
    ----------
    tensor : ndarray
    rank  : int
        Number of components.
    n_iter_max : int
        Maximum number of iteration for outer loop
    n_iter_max_inner : int
        Number of iteration for inner loop
    init : {'svd', 'random', cptensor}, optional
        Type of factor matrix initialization. See `initialize_factors`.
    svd : str, default is 'truncated_svd'
        function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
    tol_outer : float, optional
        (Default: 1e-8) Relative reconstruction error tolerance for outer loop. The
        algorithm is considered to have found a local minimum when the
        reconstruction error is less than `tol_outer`.
    tol_inner : float, optional
        (Default: 1e-6) Absolute reconstruction error tolerance for factor update during inner loop, i.e. ADMM optimization.
    random_state : {None, int, np.random.RandomState}
    verbose : int, optional
        Level of verbosity
    return_errors : bool, optional
        Activate return of iteration errors
    non_negative : bool or dictionary
        This constraint is clipping negative values to '0'.
        If it is True, non-negative constraint is applied to all modes.
    l1_reg : float or list or dictionary, optional
        Penalizes the factor with the l1 norm using the input value as regularization parameter.
    l2_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 norm using the input value as regularization parameter.
    l2_square_reg : float or list or dictionary, optional
        Penalizes the factor with the l2 square norm using the input value as regularization parameter.
    unimodality : bool or dictionary, optional
        If it is True, unimodality constraint is applied to all modes.
        Applied to each column seperately.
    normalize : bool or dictionary, optional
        This constraint divides all the values by maximum value of the input array.
        If it is True, normalize constraint is applied to all modes.
    simplex : float or list or dictionary, optional
        Projects on the simplex with the given parameter
        Applied to each column seperately.
    normalized_sparsity : float or list or dictionary, optional
        Normalizes with the norm after hard thresholding
    soft_sparsity : float or list or dictionary, optional
        Impose that the columns of factors have L1 norm bounded by a user-defined threshold.
    smoothness : float or list or dictionary, optional
        Optimizes the factors by solving a banded system
    monotonicity : bool or dictionary, optional
        Projects columns to monotonically decreasing distrbution
        Applied to each column seperately.
        If it is True, monotonicity constraint is applied to all modes.
    hard_sparsity : float or list or dictionary, optional
        Hard thresholding with the given threshold
    cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
       Stopping criterion if `tol` is not None.
       If 'rec_error',  algorithm stops at current iteration if ``(previous rec_error - current rec_error) < tol``.
       If 'abs_rec_error', algorithm terminates when `|previous rec_error - current rec_error| < tol`.
    fixed_modes : list, default is None
        A list of modes for which the initial value is not modified.
        The last mode cannot be fixed due to error computation.

    Returns
    -------
    CPTensor : (weight, factors)
        * weights : 1D array of shape (rank, )
        * factors : List of factors of the CP decomposition element `i` is of shape ``(tensor.shape[i], rank)``
    errors : list
        A list of reconstruction errors at each iteration of the algorithms.

    References
    ----------
    .. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications", SIAM
           REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
    .. [2] Huang, Kejun, Nicholas D. Sidiropoulos, and Athanasios P. Liavas.
           "A flexible and efficient algorithmic framework for constrained matrix and tensor factorization." IEEE
           Transactions on Signal Processing 64.19 (2016): 5052-5065.
    """

    def __init__(
        self,
        rank,
        n_iter_max=100,
        n_iter_max_inner=10,
        init="svd",
        svd="truncated_svd",
        tol_outer=1e-8,
        tol_inner=1e-6,
        random_state=None,
        verbose=0,
        return_errors=False,
        cvg_criterion="abs_rec_error",
        fixed_modes=None,
        non_negative=None,
        l1_reg=None,
        l2_reg=None,
        l2_square_reg=None,
        unimodality=None,
        normalize=None,
        simplex=None,
        normalized_sparsity=None,
        soft_sparsity=None,
        smoothness=None,
        monotonicity=None,
        hard_sparsity=None,
    ):
        self.rank = rank
        self.n_iter_max = n_iter_max
        self.n_iter_max_inner = n_iter_max_inner
        self.init = init
        self.svd = svd
        self.tol_outer = tol_outer
        self.tol_inner = tol_inner
        self.random_state = random_state
        self.verbose = verbose
        self.return_errors = return_errors
        self.cvg_criterion = cvg_criterion
        self.fixed_modes = fixed_modes
        self.non_negative = non_negative
        self.l1_reg = l1_reg
        self.l2_reg = l2_reg
        self.l2_square_reg = l2_square_reg
        self.unimodality = unimodality
        self.normalize = normalize
        self.simplex = simplex
        self.normalized_sparsity = normalized_sparsity
        self.soft_sparsity = soft_sparsity
        self.smoothness = smoothness
        self.monotonicity = monotonicity
        self.hard_sparsity = hard_sparsity

    def fit_transform(self, tensor):
        """Decompose an input tensor

        Parameters
        ----------
        tensor : tensorly tensor
            input tensor to decompose

        Returns
        -------
        CPTensor
            decomposed tensor
        """
        cp_tensor, errors = constrained_parafac(
            tensor,
            rank=self.rank,
            n_iter_max=self.n_iter_max,
            n_iter_max_inner=self.n_iter_max_inner,
            init=self.init,
            svd=self.svd,
            tol_outer=self.tol_outer,
            tol_inner=self.tol_inner,
            random_state=self.random_state,
            verbose=self.verbose,
            cvg_criterion=self.cvg_criterion,
            fixed_modes=self.fixed_modes,
            non_negative=self.non_negative,
            l1_reg=self.l1_reg,
            l2_reg=self.l2_reg,
            l2_square_reg=self.l2_square_reg,
            unimodality=self.unimodality,
            normalize=self.normalize,
            simplex=self.simplex,
            normalized_sparsity=self.normalized_sparsity,
            soft_sparsity=self.soft_sparsity,
            smoothness=self.smoothness,
            monotonicity=self.monotonicity,
            hard_sparsity=self.hard_sparsity,
            return_errors=True,
        )
        self.decomposition_ = cp_tensor
        self.errors_ = errors
        return self.decomposition_