## 1 Introduction
As is well known, HSIs and MSIs are both cubic data, which can be considered three-dimensional tensors. Recently, another category of state-of-the-art method has been developed on basis of tensor decomposition, and includes canonical polyadic decomposition (CPD) [18; 19] and Tucker decomposition [20; 21; 22], which can retain the cubic structure information of remote sensing images. For example, [PERSON] et al. [20] proposed a fusion algorithm based on coupled sparse tensor factorization (CSTF), which was reformulated as the iterations of a core tensor and dictionaries of the three modes. [PERSON] et al. [19] brought forward a CPD-based fusion framework, in which the identifiability of the reconstructed image was discussed under mild conditions. [PERSON] et al. [22] proposed a tensor-based approach to handle the problem of HSI spatial super-resolution by incorporating the regularizers of nonlocal low-rank tensor approximation and total variation. However, as the rank-1 version of Tucker decomposition [23], CPD is usually used to describe linear local features in the image, rather than complex features. In NMF-based hyperspectral unmixing, the data can be decomposed into the product of two matrices, which represent the endmember signature and fractional abundance, respectively. However, the physical meaning of decomposed tensor is not the same as the aforementioned matrices, so that the spectral signature (i.e., [PERSON]'s belief [24], etc.) cannot be fully explored. Different from a matrix, the rank of a tensor is determined by a non-deterministicpolynomial (NP) problem. In summary, the fusion methods based on tensor decomposition usually suffer from the uncertainty of accurate rank and the non-uniqueness of decomposition [18], especially when the remote sensing images are contaminated by serious noise. Besides the tensor-based methods, [PERSON]'s belief in hyperspectral unmixing can effectively solve the problem of data structure loss. Another popular fusion approach is based on spectral unmixing, whose representative work is known as coupled non-negative matrix factorization (CNMF) [25]. According to the observation model, CNMF also amounts to an ill-posed inverse problem that can usually be solved by regularization methods. Different from the tensor-based approach, the matrix decomposition-based algorithm (called matrixing method) requires reshaping the cubic data into a two-dimensional form, resulting in the loss of three-dimensional structure information. To lower the effect of data dimension reduction, spatial signature regularizers can be incorporated into the original fusion method, such as spatial smoothing [26; 27] and sparse coding [28; 29]. However, the sparsity-promoting regularizer itself may not be sufficient to yield high-quality fused data [28; 30; 31]; meanwhile, joint spatial and spectral regularization can perform well [32; 33]. [PERSON] et al. proposed an NMF-based fusion method with several physical constraints by jointly unmixing the HIS and MSI data into pure reflectance spectra of the observed materials for hyperspectral super-resolution. A regularized CNMF-based method was proposed [34] by introducing the volume of signature vectors' simplex regularizer, yet the heuristic algorithm did not improve the performance substantially. [PERSON] et al. [32] proposed a convex optimization-based CNMF based on the sum-of-squared distances (SSD) between all the endmembers, in which the sparsity and SSD-based regularizers were employed to bring significant improvements in fusion performance. However, this algorithm was not suitable for a high noise environment, yielding poor anti-noise performance. A spectral-total-variation image fusion framework was proposed by [PERSON] et al. [35], which established a model for decomposing components by tailed \(\alpha\)-stable-based random variable distribution. [PERSON] et al. [36] employed total variation and signature-based regularization, in which the horizontal and vertical difference matrices were constructed separately for spatial smoothing.