Feature selection (FS) is essential in machine learning and data mining as it makes handling high-dimensional data more efficient and reliable. More attention has been paid to unsupervised feature selection (UFS) due to the extra resources required to obtain labels for data in the real world. Most of the existing embedded UFS utilize a sparse projection matrix for FS. However, this may introduce additional regularization terms, and it is difficult to control the sparsity of the projection matrix well. Moreover, such methods may seriously destroy the original feature structure in the embedding space. Instead, avoiding projecting the original data into the low-dimensional embedding space and identifying features directly from the raw features that perform well in the process of making the data show a distinct cluster structure is a feasible solution. Inspired by this, this paper proposes a model called A General Adaptive Unsupervised Feature Selection with Auto-weighting (GAWFS), which utilizes two techniques, non-negative matrix factorization, and adaptive graph learning, to simulate the process of dividing data into clusters, and identifies the features that are most discriminative in the clustering process by a feature weighting matrix Θ. Since the weighting matrix is sparse, it also plays the role of FS or a filter. Finally, experiments comparing GAWFS with several state-of-the-art UFS methods on synthetic datasets and real-world datasets are conducted, and the results demonstrate the superiority of the GAWFS.

The inherent characteristics involved in data can be mined from multi-scale information systems by extracting information from different value levels of features. In real applications, noise data and irrelevant or redundant features affect the generality of learning models. Therefore, keeping meaningful features and avoiding the effect of noise is essential for feature selection in a multi-scale information system. In bireduct, multi-scale granulation can be used to characterize the importance and correlation of features at different scales. However, little work has taken the distribution of multi-scale data into account when granulating it. In addition, these approaches focus on solving the task of multi-scale data reduction only from the dimension perspective. To this end, a fuzzy-rough bireduct with supervised multi-scale granulation (FrBSmg) is proposed. First, the supervised multi-scale fuzzy granulation based on data distribution is constructed. Then, scaled uncertainty measures are defined to describe the fuzzy relevance of each feature. Furthermore, the global and local distributions of a sample are characterized simultaneously based on the positive region, which can reflect the degree of a sample belonging to some class, and the supervised fuzzy similarity relation can describe the degree of a sample belonging to its class. A strategy of Feature-Correlated Selection and Sample-Noisy Removal is devised for bireduct. Finally, the experimental results on twenty-one public datasets show the effectiveness of FrBSmg.