Representation learning serves as a critical bridge between human cognition and the data world, constituting an essential component of machine learning architectures where comprehensiveness and flexibility are paramount. However, existing multi-view feature selection methods are constrained by the raw-scale representations, neglecting comprehensive depth-breadth integration and flexible adaptability. This paper presents multi-scale fuzzy fusion-based heterogeneous granular-ball flexible representation learning for multi-view feature selection (MFHGBFR). A comprehensive multi-view, multi-scale analytical foundation is established to promote depth-breadth representation learning, where views represent breadth and scales represent depth in human cognition. Heterogeneous granular ball-based flexible representations are adaptively developed via multi-scale fuzzy fusion, effectively capturing intricate data manifolds and implicit fuzzy patterns across multi-granularity spaces. Intricate data manifolds are fitted with heterogeneous rather than conventional homogeneous structures to improve the adaptability of representations' multi-granularity. Implicit fuzzy patterns are extracted with fuzzy approximation operators to mitigate fuzziness and uncertainty in a multi-granularity space. For the first time, granular-ball representation learning and feature selection are adaptively optimized in a unified one-step framework, rather than the conventional two-step frameworks, for flexible adaptability. An effective optimization algorithm with proven convergence is derived. Through the learned high-quality representations, the MFHGBFR manifests superior performance, robustness, and efficiency. Comparative experiments against contemporary state-of-the-art algorithms substantiate these advantages quantitatively and qualitatively.

Multi-view unsupervised feature selection (MV-UFS) has recently gained significant attention as an effective technique for handling high-dimensional multi-view data. Despite recent progress, three flaws are present in most existing MV-UFS methods. First, they predominantly pay attention to the feature dimensionality while neglecting sample size considerations, typically resulting in cubic time complexity concerning sample size (i.e.$\mathcal {O}(n^{3})$) that hinders them in real-world applications. Second, existing methods primarily focus on the relationships between pairwise views, failing to leverage the complex interactions among multiple views effectively. Third, they generally overlook crucial graph semantic information during graph fusion. An ingenious MV-UFS method is presented in this paper, namedJoint Information Interaction and Semantic Fusion for Multi-View Unsupervised Feature Selection (JIISF), to overcome these limitations. Specifically, JIISF devises a multi-view double reduction learning framework that projects original data into a low-dimensional embedding space and dynamically picks out representative anchors, eliminating redundant information while constructing compact view-specific anchor graphs. Meanwhile, JIISF presents a cross-view information interaction scheme, which enables comprehensive information interaction among multiple views rather than being limited to pairwise views. A semantic alignment fusion strategy is presented to learn a diverse, consistent anchor graph with an explicit semantic alignment module, which can preserve diversity while maintaining consistency. Finally, an alternating optimization algorithm is devised to resolve the proposed JIISF, and experimental results on real-life datasets validate its advantage compared with some state-of-the-art UFS and MV-UFS methods.

Multiview data possess different discriminability in different views, which is challenging to catch but crucial for a feature selection model. Multiscale information, which represents vertical exploration in each view, is vital for further mining traits implied in multiview data. However, most existing studies neglect these beneficial multi-granulation characteristics. This study first embeds the multiscale information into the sparse learning framework for multiview feature selection. A class-specific discriminability and multiscale information-based multiview feature selection (CDMIMFS) method is proposed. It explores the fuzzy and uncertain class-specific discriminability which is inherently discrepant in different views by the fuzzy rough set theory. It relaxes the over strict requirement for complete consistency in multiscale information systems to make a trade-off, which further enhances discriminative feature selection. An effective iteration algorithm is proposed to solve the optimization. Both the theoretical proof and experimental demonstration of convergence are provided. Comprehensive experiments are conducted on the CDMIMFS compared with state-of-the-art algorithms. Results on different evaluation metrics exhibit the advantages of the proposed method.