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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def smooth_ot_dual(a, b, M, reg, reg_type='l2', method="L-BFGS-B", stopThr=1e-9, numItermax=500, verbose=False, log=False):
 r""" Solve the regularized OT problem in the dual and return the OT matrix The function solves the smooth relaxed dual formulation (7) in [17]_ : .. math:: \max_{\alpha,\beta}\quad a^T\alpha+b^T\beta-\sum_j\delta_\Omega(\alpha+\beta_j-\mathbf{m}_j) where : - :math:`\mathbf{m}_j` is the jth column of the cost matrix - :math:`\delta_\Omega` is the convex conjugate of the regularization term :math:`\Omega` - a and b are source and target weights (sum to 1) The OT matrix can is reconstructed from the gradient of :math:`\delta_\Omega` (See [17]_ Proposition 1). The optimization algorithm is using gradient decent (L-BFGS by default). Parameters a : np.ndarray (ns,) samples weights in the source domain b : np.ndarray (nt,) or np.ndarray (nt,nbb) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) M : np.ndarray (ns,nt) loss matrix reg : float Regularization term >0 reg_type : str Regularization type, can be the following (default ='l2'):
 - 'kl' : Kullback Leibler (~ Neg-entropy used in sinkhorn [2]_) - 'l2' : Squared Euclidean regularization method : str Solver to use for scipy.optimize.minimize numItermax : int, optional Max number of iterations stopThr : float, optional Stop threshol on error (>0) verbose : bool, optional Print information along iterations log : bool, optional record log if True Returns ------- gamma : (ns x nt) ndarray Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters References .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013 .. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS). See Also -------- ot.lp.emd : Unregularized OT ot.sinhorn : Entropic regularized OT ot.optim.cg : General regularized OT """


    if reg_type.lower() in ['l2', 'squaredl2']:
        regul = SquaredL2(gamma=reg)
    elif reg_type.lower() in ['entropic', 'negentropy', 'kl']:
        regul = NegEntropy(gamma=reg)
    else:
        raise NotImplementedError('Unknown regularization')

    # solve dual
    alpha, beta, res = solve_dual(a, b, M, regul, max_iter=numItermax,
                                  tol=stopThr, verbose=verbose)

    # reconstruct transport matrix
    G = get_plan_from_dual(alpha, beta, M, regul)

    if log:
        log = {'alpha': alpha, 'beta': beta, 'res': res}
        return G, log
    else:
        return G