## 1 Introduction
The spectral unmixing approach tries to separate these responses, i.e., to recover the underlying pure material spectra (called endmembers) and their relative proportions (called abundances). Since LR-HSI and HR-MSI capture the same scene, the underlying pure matter members are identical. Therefore, it can extract a spectral dictionary from one image to interpret the other, thus achieving high spectral image super-resolution. Based on linear spectral unmixing, the literature [19] has proposed a coupled spectral unmixing method, which solves the super-resolution problem by alternating the spectral unmixing of HR-MSI and LR-HSI simultaneously while updating spectral and abundance. However, the fusion quality of this method is closely related to the estimation of the number of underlying endmembers. The requirement for estimating the underlying endmembers is high, and it is not easy to achieve an accurate estimation. Therefore, in the literature [20], a sparse constraint is added to the fusion. As a result, a hyperspectral, multispectral image fusion algorithm based on sparse representation is proposed (NSSR). Through updating the number of the underlying endmembers by an effective non-negative dictionary learning algorithm, NSSR effectively improves the fusion quality. However, this method demands more parameters to be determined, and the fusion quality fluctuates wildly. In recent years, matrix factorization-based fusion methods have started to emerge. The matrix factorization-based fusion methods assume HR-HSI as the product of the spectral basis matrix and a coefficient matrix. They estimate the spectral basis from LR-HSI and coefficient matrix from HR-MSI, respectively, to achieve hyperspectral super-resolution. The authors of [21] propose the coupled non-negative matrix factorization (CNMF) technology, which extracts the spectral basis and coefficient matrix by performing non-negative matrix factorization (NMF) on LR-HSI and HR-MSI, respectively. However, since the NMF is usually not unique, the results produced in [21] are not always good. In [13], a standard linear inverse problem model for LR-HSI and HR-MSI fusion was developed to solve the optimized spectral basis and coefficient matrices by introducing full variational regularization terms in the pairwise matrix equations. However, this approach usually requires splitting of variables requiring relatively more computational time as well. Further studies began to add a priori information to the fusion to improve the quality of the fusion. For example, in [22], by adding spatial similarity to the fusion, and in [23], by using the nonlocal coefficient representation method and the self-similarity constraint to impose a priori constraints on the fusion, the quality of image fusion can be improved to some extent. However, the fusion process becomes more complicated due to the added constraints, which leads to a further increase in the fusion time and makes the efficiency for fusion decrease. Thus, for some application scenarios that require real-time response, this fusion method is not suitable. Unlike matrix factorization-based methods, tensor factorization-based methods can preserve the three-dimensional characteristics of image data. This has led to the continuous application of tensor factorization-based methods to image processing work, such as multi-frame data denoising [24], completion [25], compression perception [26], and classification [27]. Tensor factorization-based methods for solving hyperspectral image fusion have also been proposed. Based on the Tucker decomposition, [28] views HR-HSI as the product of a kernel tensor and three-factor matrices and estimates the kernel tensor and factor matrices from HR-MSI and LR-HSI, respectively. The autocorrelation of hyperspectral images is cited in [29], and the method of nonlocal tensor factorization is proposed to effectively use the nonlocal similarity information to realize hyperspectral image fusion.