IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Heterogeneous Regularizations Based Tensor Subspace Clustering for Hyperspectral Band Selection Shaoguang Huang, Member, IEEE, Hongyan Zhang, Senior Member, IEEE, Jize Xue, Student Member, IEEE, and Aleksandra Piˇ zurica, Senior Member, IEEE Abstract—Band selection (BS) reduces effectively the spectral dimension of a hyperspectral image (HSI) by selecting relatively few representative bands, which allows efficient processing in subsequent tasks. Existing unsupervised BS methods based on subspace clustering are built on matrix-based models, where each band is reshaped as a vector. They encode the correlation of data only in spectral-mode (dimension) and neglect strong correlations between different modes, i.e., spatial modes and spectral mode. Another issue is that the subspace representa- tion of bands is performed in the raw data space, where the dimension is often excessively high, resulting in a less efficient and less robust performance. To address these issues, in this paper we propose a tensor based subspace clustering model for hyperspectral band selection. Our model is developed on the well-known Tucker decomposition. The three factor matrices and a core tensor in our model encodes jointly the multi-mode correlations of HSI, avoiding effectively to destroy the tensor structure and information loss. In addition, we propose well- motivated heterogeneous regularizations on the factor matrices by taking into account the important local and global property of HSI along three dimensions, which facilitates the learning of the intrinsic cluster structure of bands in the low-dimensional subspaces. Instead of learning the correlations of bands in the original domain, a common way for the matrix-based models, our model learns naturally the band correlations in a low-dimensional latent feature space, which is derived by the projections of two factor matrices associated with spatial dimensions, leading to a computationally efficient model. More importantly, the latent feature space is learned in a unified framework. We also develop an efficient algorithm to solve the resulting model. Experimental results on benchmark data sets demonstrate that our model yields improved performance compared to the state-of-the-art. Index Terms—Band selection, hyperspectral image, remote sensing, tensor, subspace clustering. I. I NTRODUCTION This work was supported in part by the Flanders AI Research Programme under Grant 174B09119, in part by the Bijzonder Onderzoeksfonds (BOF) under Grant BOF.24Y.2021.0049.01 and in part by the National Natural Science Foundation of China under Grant 61871298 and Grant 42071322. (Corresponding author: Hongyan Zhang.) S. Huang and A. Piˇ zurica are with the Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium (e-mail: shaoguang.huang@ugent.be; Aleksandra.Pizurica@ugent.be). H. Zhang is with the State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Collaborative Innovation Center of Geospatial Technology, Wuhan University, Wuhan 430079, China (e-mail: zhanghongyan@whu.edu.cn). J. Xue is with the Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China and also with the Image Processing and Interpretation, IMEC Research Group, Ghent University, Ghent, Belgium(e-mail: xuejize900507@mail.nwpu.edu.cn) R ECENT advances on hyperspectral sensors significantly increase the spectral resolution of hyperspectral images (HSIs) [1]. While providing richer spectral information than multi-spectral data, which enables improved discriminations between different materials, HSIs raise in the meantime the challenges of dealing with high-dimensional data [2]. On the one hand, high-dimensional data surges the burdens in the data acquisition, storage and transmission. On the other hand, it leads to the curse of dimensionality problem, which deteriorates the performance of the related post-processing algorithms [3, 4]. Both issues limit the real applications of HSIs. Band selection (BS), as an effective dimension reduction method, selects the most relevant spectral bands from HSIs for post-processing such as classification, which reduces ef- fectively the dimension of HSIs while obtaining comparable or even better performance in the related task [5, 6]. De- pending on how supervised information is involved in the band selection, band selection methods can be categorized into supervised methods [7–9], semi-supervised methods [10– 12] and unsupervised methods [13–17]. A recent overview for BS can be found in [18]. We here mainly focus on the unsupervised BS method given the fact that data labelling is expensive, leading to often a scarce of labeled data in practice [19, 20]. Common unsupervised BS methods include ranking-based [14, 21–26], searching-based [5, 27–31] and clustering-based methods [6, 16, 17, 32–38]. Ranking-based methods select the top-ranked bands as representatives according to the scores of bands measured by a given criterion such as band variation [21], mutual information [23] and band correlation [22, 24]. Searching-based methods select desired bands by greedy algorithm or evolutionary algorithms, including im- mune clone [30], firefly algorithms [31] and particle swarm [27]. Clustering-based methods cluster the bands of HSIs into different groups by well-developed clustering algorithms, including spectral clustering [39], hierarchical clustering [6], subspace clustering [37, 40] and probabilistic clustering [41]. As each of the resulting clusters contains spectral bands of high similarity, one can select a representative band from each cluster to obtain the optimal subset of bands. In general, spectral clustering deals well with non-spherical cluster structure of bands by using graph spectral analysis. However, the performance of spectral clustering is sensitive to the neighborhood size and similarity measurement between