Compute the Wasserstein barycenter of distributions A

     The function solves the following optimization problem [16]:

    .. math::
       \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{1}(\mathbf{a},\mathbf{a}_i)

    where :

    - :math:`W_1(\cdot,\cdot)` is the Wasserstein distance (see ot.emd.sinkhorn)
    - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}`

    The linear program is solved using the interior point solver from scipy.optimize.
    If cvxopt solver if installed it can use cvxopt

    Note that this problem do not scale well (both in memory and computational time).

    Parameters
    ----------
    A : np.ndarray (d,n)
        n training distributions a_i of size d
    M : np.ndarray (d,d)
        loss matrix   for OT
    reg : float
        Regularization term >0
    weights : np.ndarray (n,)
        Weights of each histogram a_i on the simplex (barycentric coodinates)
    verbose : bool, optional
        Print information along iterations
    log : bool, optional
        record log if True
    solver : string, optional
        the solver used, default 'interior-point' use the lp solver from
        scipy.optimize. None, or 'glpk' or 'mosek' use the solver from cvxopt.

    Returns
    -------
    a : (d,) ndarray
        Wasserstein barycenter
    log : dict
        log dictionary return only if log==True in parameters


    References
    ----------

    .. [16] Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.

def barycenter(A, M, weights=None, verbose=False, log=False, solver='interior-point'):
    """Compute the Wasserstein barycenter of distributions A

     The function solves the following optimization problem [16]:

    .. math::
       \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{1}(\mathbf{a},\mathbf{a}_i)

    where :

    - :math:`W_1(\cdot,\cdot)` is the Wasserstein distance (see ot.emd.sinkhorn)
    - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}`

    The linear program is solved using the interior point solver from scipy.optimize.
    If cvxopt solver if installed it can use cvxopt

    Note that this problem do not scale well (both in memory and computational time).

    Parameters
    ----------
    A : np.ndarray (d,n)
        n training distributions a_i of size d
    M : np.ndarray (d,d)
        loss matrix   for OT
    reg : float
        Regularization term >0
    weights : np.ndarray (n,)
        Weights of each histogram a_i on the simplex (barycentric coodinates)
    verbose : bool, optional
        Print information along iterations
    log : bool, optional
        record log if True
    solver : string, optional
        the solver used, default 'interior-point' use the lp solver from
        scipy.optimize. None, or 'glpk' or 'mosek' use the solver from cvxopt.

    Returns
    -------
    a : (d,) ndarray
        Wasserstein barycenter
    log : dict
        log dictionary return only if log==True in parameters


    References
    ----------

    .. [16] Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.



    """

    if weights is None:
        weights = np.ones(A.shape[1]) / A.shape[1]
    else:
        assert(len(weights) == A.shape[1])

    n_distributions = A.shape[1]
    n = A.shape[0]

    n2 = n * n
    c = np.zeros((0))
    b_eq1 = np.zeros((0))
    for i in range(n_distributions):
        c = np.concatenate((c, M.ravel() * weights[i]))
        b_eq1 = np.concatenate((b_eq1, A[:, i]))
    c = np.concatenate((c, np.zeros(n)))

    lst_idiag1 = [sps.kron(sps.eye(n), np.ones((1, n))) for i in range(n_distributions)]
    #  row constraints
    A_eq1 = sps.hstack((sps.block_diag(lst_idiag1), sps.coo_matrix((n_distributions * n, n))))

    # columns constraints
    lst_idiag2 = []
    lst_eye = []
    for i in range(n_distributions):
        if i == 0:
            lst_idiag2.append(sps.kron(np.ones((1, n)), sps.eye(n)))
            lst_eye.append(-sps.eye(n))
        else:
            lst_idiag2.append(sps.kron(np.ones((1, n)), sps.eye(n - 1, n)))
            lst_eye.append(-sps.eye(n - 1, n))

    A_eq2 = sps.hstack((sps.block_diag(lst_idiag2), sps.vstack(lst_eye)))
    b_eq2 = np.zeros((A_eq2.shape[0]))

    # full problem
    A_eq = sps.vstack((A_eq1, A_eq2))
    b_eq = np.concatenate((b_eq1, b_eq2))

    if not cvxopt or solver in ['interior-point']:
        # cvxopt not installed or interior point

        if solver is None:
            solver = 'interior-point'

        options = {'sparse': True, 'disp': verbose}
        sol = sp.optimize.linprog(c, A_eq=A_eq, b_eq=b_eq, method=solver,
                                  options=options)
        x = sol.x
        b = x[-n:]

    else:

        h = np.zeros((n_distributions * n2 + n))
        G = -sps.eye(n_distributions * n2 + n)

        sol = solvers.lp(matrix(c), scipy_sparse_to_spmatrix(G), matrix(h),
                         A=scipy_sparse_to_spmatrix(A_eq), b=matrix(b_eq),
                         solver=solver)

        x = np.array(sol['x'])
        b = x[-n:].ravel()

    if log:
        return b, sol
    else:
        return b