Unsupervised feature selection is challenging in machine learning, pattern recognition, and data mining. The crucial difficulty is to learn a moderate subspace that preserves the intrinsic structure and to find uncorrelated or independent features simultaneously. The most common solution is first to project the original data into a lower dimensional space and then force them to preserve the similar intrinsic structure under linear uncorrelation constraint. However, there are three shortcomings. First, the final graph generated by the iterative learning process differs significantly from the initial graph in which the original intrinsic structure is embedded. Second, it requires prior knowledge about a moderate dimension of subspace. Third, it is inefficient when dealing with high-dimensional datasets. The first shortcoming, which is longstanding and undiscovered, makes the previous methods fail to achieve their expected results. The last two ones increase the difficulty of applying in different fields. Therefore, two unsupervised feature selection methods are proposed based on controllable adaptive graph learning and uncorrelated/independent feature learning (CAG-U and CAG-I) to address the abovementioned issues. In the proposed methods, the final graph that preserves intrinsic structure can be adaptively learned while the difference between the two graphs can be well controlled. Besides, relatively uncorrelated/independent features can be selected using a discrete projection matrix. The experimental results on 12 datasets in different fields show the superiority of CAG-U and CAG-I.