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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 spectral_clustering(geom, K, eigen_solver = 'dense', random_state = None, solver_kwds = None, renormalize = True, stabalize = True, additional_vectors = 0):
 """ Spectral clustering for find K clusters by using the eigenvectors of a matrix which is derived from a set of similarities S. Parameters S: array-like,shape(n_sample,n_sample) similarity matrix K: integer number of K clusters eigen_solver : {'auto', 'dense', 'arpack', 'lobpcg', or 'amg'} 'auto' : algorithm will attempt to choose the best method for input data 'dense' : use standard dense matrix operations for the eigenvalue decomposition. For this method, M must be an array or matrix type. This method should be avoided for large problems. 'arpack' : use arnoldi iteration in shift-invert mode. For this method, M may be a dense matrix, sparse matrix, or general linear operator. Warning: ARPACK can be unstable for some problems. It is best to try several random seeds in order to check results. 'lobpcg' : Locally Optimal Block Preconditioned Conjugate Gradient Method. A preconditioned eigensolver for large symmetric positive definite (SPD) generalized eigenproblems. 'amg' : AMG requires pyamg to be installed. It can be faster on very large, sparse problems, but may also lead to instabilities. random_state : numpy.RandomState or int, optional The generator or seed used to determine the starting vector for arpack iterations. Defaults to numpy.random.RandomState solver_kwds : any additional keyword arguments to pass to the selected eigen_solver renormalize : (bool) whether or not to set the rows of the eigenvectors to have norm 1 this can improve label quality stabalize : (bool) whether or not to compute the (more stable) eigenvectors of L = D^-1/2*S*D^-1/2 instead of P = D^-1*S additional_vectors : (int) compute additional eigen vectors when computing eigen decomposition. When eigen_solver = 'amg' or 'lopcg' often if a small number of eigen values is sought the largest eigenvalue returned is *not* equal to 1 (it should be). This can usually be fixed by requesting more than K eigenvalues until the first eigenvalue is close to 1 and then omitted. The remaining K-1 eigenvectors should be informative. Returns ------- labels: array-like, shape (1,n_samples) """
 
    # Step 1: get similarity matrix
    if geom.affinity_matrix is None:
        S = geom.compute_affinity_matrix()
    else:
        S = geom.affinity_matrix
        
    # Check for stability method, symmetric solvers require this
    if eigen_solver in ['lobpcg', 'amg']:
        stabalize = True
    if stabalize:
        geom.laplacian_type = 'symmetricnormalized'
        return_lapsym = True
    else:
        geom.laplacian_type = 'randomwalk'
        return_lapsym = False
    
    # Step 2: get the Laplacian matrix
    P = geom.compute_laplacian_matrix(return_lapsym = return_lapsym)
    # by default the Laplacian is subtracted from the Identify matrix (this step may not be needed)
    P += identity(P.shape[0])        
    
    # Step 3: Compute the top K eigenvectors and drop the first 
    if eigen_solver in ['auto', 'amg', 'lobpcg']:
        n_components = 2*int(np.log(P.shape[0]))*K + 1
        n_components += int(additional_vectors)
    else:
        n_components = K
    n_components = min(n_components, P.shape[0])
    (lambdas, eigen_vectors) = eigen_decomposition(P, n_components=n_components, eigen_solver=eigen_solver, 
                                                   random_state=random_state, drop_first = True,
                                                   solver_kwds=solver_kwds)
    # the first vector is usually uninformative 
    if eigen_solver in ['auto', 'lobpcg', 'amg']:
        if np.abs(lambdas[0] - 1) > 1e-4:
            warnings.warn("largest eigenvalue not equal to 1. Results may be poor. Try increasing additional_vectors parameter")
    eigen_vectors = eigen_vectors[:, 1:K]
    lambdas = lambdas[1:K]
    
    # If stability method chosen, adjust eigenvectors
    if stabalize:
        w = np.array(geom.laplacian_weights)
        eigen_vectors /= np.sqrt(w[:,np.newaxis])
        eigen_vectors /= np.linalg.norm(eigen_vectors, axis = 0)    
    
    # If renormalize: set each data point to unit length
    if renormalize:
        norms = np.linalg.norm(eigen_vectors, axis=1)
        eigen_vectors /= norms[:,np.newaxis]
    
    # Step 4: run k-means clustering
    labels =  k_means_clustering(eigen_vectors,K)    
    return labels, eigen_vectors, P