Adaptive and flexible \(\ell _1\)-norm graph embedding for unsupervised feature selection

Unsupervised feature selection (UFS) is a fundamental and indispensable dimension reduction method for large amount of high-dimensional unlabeled data samples. Without label information, the manifold learning technique is leveraged to compensate for the lack of discrimination with the selected features. However, it is still a challenging problem to capture the geometrical structure for practical data, which are often contaminated by noises and outliers. Additionally, the predetermined graph embedded UFS models suffer from the parameter tuning problem and the separated model optimization procedures. To generate more compact and discriminative feature subsets, we propose a Robust UFS model with Adaptive and Flexible \(\varvec{\ell }_\textbf{1}\) ${\mathit{\ell }}_{\mathbf{1}}$ -norm Graph (RAFG) embedding. Specifically, the \(\varvec{\ell }_\textbf{2,1}\) ${\mathit{\ell }}_{\mathbf{2}},\mathbf{1}$ -norm is imposed on the flexible regression term to alleviate the adverse effects of both noisy features and outliers, and \(\varvec{\ell }_\textbf{2,p}\) ${\mathit{\ell }}_{\mathbf{2}},\mathbf{p}$ -norm regularization term is incorporated to ensure that the selected transformation matrix is sufficiently sparse. Moreover, the adaptive \(\varvec{\ell }_\textbf{1}\) ${\mathit{\ell }}_{\mathbf{1}}$ -norm graph learning characterize the clustering distribution via consistent embeddings, which avoids time-consuming distance computations in a high-dimensional feature space. To solve the challenging problem, we propose an efficient alternative updating algorithm with an iterative reweighted strategy, together with the necessary convergence and complexity analyses. Finally, experimental results on two synthetic data and eight benchmark datasets illustrate the effectiveness and superiority of the proposed RAFG method compared with state-of-the-art methods.