Feature selection is an important preprocessing and interpretable method in the fields where big data plays an essential role. In this paper, we first reformulate and analyze some representative information theoretic feature selection methods from the perspective of approximations of feature inner correlations, and indicate that many of these methods cannot guarantee any theoretical bounds of feature inner correlations. We thus introduce two lower bounds that have very simple forms for feature redundancy and complementarity, and verify that they are closer to the optima than the existing lower bounds applied by some state-of-the-art information theoretic methods. A simple and effective feature selection method based on the proposed lower bounds is then proposed and empirically verified with a wide scope of real-world datasets. The experimental results show that the proposed method achieves promising improvement on feature selection, indicating the effectiveness of the feature criterion consisting of the proposed lower bounds of redundancy and complementarity.