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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 _sparsify(X, remove_mean=False, modify_data=False, sparse_mode='auto', sparse_tol=0.0):
 """ Determines the sparsity of X and returns a selected sub-matrix Only conducts sparsification if the number of constant columns is at least max(a N - b, min_const_col_number), Parameters X : ndarray data matrix remove_mean : bool True: remove column mean from the data, False: don't remove mean. modify_data : bool If remove_mean=True, the mean will be removed in the data matrix X, without creating an independent copy. This option is faster but might lead to surprises because your input array is changed. sparse_mode : str one of: * 'dense' : always use dense mode * 'sparse' : always use sparse mode if possible * 'auto' : automatic Returns ------- X0 : ndarray (view of X) Either X itself (if not sufficiently sparse), or a sliced view of X, containing only the variable columns mask : ndarray(N, dtype=bool) or None Bool selection array that indicates which columns of X were selected for X0, i.e. X0 = X[:, mask]. mask is None if no sparse selection was made. xconst : ndarray(N) Constant column values that are outside the sparse selection, i.e. X[i, ~mask] = xconst for any row i. xconst=0 if no sparse selection was made. """

    if sparse_mode.lower() == 'sparse':
        min_const_col_number = 0  # enforce sparsity. A single constant column will lead to sparse treatment
    elif sparse_mode.lower() == 'dense':
        min_const_col_number = X.shape[1] + 1  # never use sparsity
    else:
        if remove_mean and not modify_data:  # in this case we have to copy the data anyway, and can be permissive
            min_const_col_number = max(0.1 * X.shape[1], 50)
        else:
            # This is a rough heuristic to choose a minimum column number for which sparsity may pay off.
            # This heuristic is good for large number of samples, i.e. it may be inadequate for small matrices X.
            if X.shape[1] < 250:
                min_const_col_number = X.shape[1] - 0.25 * X.shape[1]
            elif X.shape[1] < 1000:
                min_const_col_number = X.shape[1] - (0.5 * X.shape[1] - 100)
            else:
                min_const_col_number = X.shape[1] - (0.8 * X.shape[1] - 400)
        # ensure we have an integer again.
        min_const_col_number = int(min_const_col_number)

    if X.shape[1] > min_const_col_number:
        mask = covartools.variable_cols(X, tol=sparse_tol, min_constant=min_const_col_number)  # bool vector
        nconst = len(np.where(~mask)[0])
        if nconst > min_const_col_number:
            xconst = X[0, ~mask]
            X = X[:, mask]  # sparsify
        else:
            xconst = None
            mask = None
    else:
        xconst = None
        mask = None

    return X, mask, xconst