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Recently, various high-order methods have been developed to solve the convex optimization problem. The auxiliary problem of these methods shares the general form that is the same as the high-order proximal operator proposed by Nesterov. In this paper, we present a linearly convergent method to solve the high-order proximal operator based on the classical proximal operator. In addition, some experiments are performed to demonstrate the performance of the proposed method.

The proximal point algorithm (PPA) has been developed to solve the monotone variational inequality problem. It provides a theoretical foundation for some methods, such as the augmented Lagrangian method (ALM) and the alternating direction method of multipliers (ADMM). This paper generalizes the PPA to the $p$th-order ($p\geq 1$) and proves its convergence rate $O \left(1/k^{p/2}\right)$ . Additionally, the $p$th-order ALM is proposed based on the $p$th-order PPA. Some numerical experiments are presented to demonstrate the performance of the $p$th-order ALM.

The concept of tensor eigenpairs has received more researches in past decades. Recent works have paid attentions to a special class of symmetric tensors termed regular simplex tensors, which is constructed by equiangular tight frame of n + 1 vectors in n-dimensional space, and the robustness of eigenpairs was investigated. In the end of the literature, a conjecture was claimed that the robust eigenvectors of a regular simplex tensor are precisely the vectors in the frame. One later work theoretically proved that the case of n = 2 was true. In this paper, we proceed further and complete the proof for the above conjecture. Some promising directions are discussed in the end for future works., Comment: arXiv admin note: text overlap with arXiv:2303.00274

In this paper, we focus on a special class of symmetric tensors, which can be orthogonally diagonalizable, and investigate their Z-eigenpairs problem. We show that the eigenpairs can be uniformly expressed using several basic eigenpairs, and the number of all the eigenpairs is uniquely determined by the order and rank of the symmetric tensor. In addition, we exploit the local optimality of each eigenpair by checking the second-order necessary condition.

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a third-order generalization of principal component analysis (PCA), its solution of searching for the locally maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, it can prevent the new eigenpairs from converging to the same maxima that has been determined before. However, the eigenvectors of the supersymmetric tensor are not inherently orthogonal in general, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to solve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous one, instead of its orthogonal complement space. We give a detailed theoretical proof to illustrate why the new strategy can result in the more accurate eigenpairs. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

Our research has revealed a hidden relationship among several basic components, which leads to the best target detection result. Further, we have proved that the matched filter (MF) is always superior to the constrained energy minimization (CEM) operator, both of which were originally of parallel importance in the field of target detection for remotely sensed image., Comment: 8 pages, 8 figures

The objective of this paper is to evaluate the potential of Gaofen-2 (GF-2) high resolution multispectral sensor (MS) and panchromatic (PAN) imagery on water mapping. Difficulties of water mapping on high resolution data includes: 1) misclassification between water and shadows or other low-reflectance ground objects, which is mostly caused by the spectral similarity within the given band range; 2) small water bodies with size smaller than the spatial resolution of MS image. To solve the confusion between water and low-reflectance objects, the Landsat 8 time series with two shortwave infrared (SWIR) bands is added because water has extremely strong absorption in SWIR. In order to integrate the three multi-sensor, multi-resolution data sets, the probabilistic graphical model (PGM) is utilized here with conditional probability distribution defined mainly based on the size of each object. For comparison, results from the SVM classifier on the PCA fused and MS data, thresholding method on the PAN image, and water index method on the Landsat data are computed. The confusion matrices are calculated for all the methods. The results demonstrate that the PGM method can achieve the best performance with the highest overall accuracy. Moreover, small rivers can also be extracted by adding weight on the PAN result in PGM. Finally, the post-classification procedure is applied on the PGM result to further exclude misclassification in shadow and water-land boundary regions. Accordingly, the producer's, user's and overall accuracy are all increased, indicating the effectiveness of our method., Comment: 17 pages, 9 figures, 6 tables

The constrained energy minimization (CEM) and matched filter (MF) are two most frequently used target detection algorithms in the remotely sensed community. In this paper, we first introduce an augmented CEM (ACEM) by adding an all-one band. According to a recently published conclusion that CEM can always achieve a better performance by adding any linearly independent bands, ACEM is better than CEM. Further, we prove that ACEM is mathematically equivalent to MF. As a result, we can conclude that the classical matched filter (MF) is always superior to the CEM operator., Comment: 4 pages

The determination of cluster centers generally depends on the scale that we use to analyze the data to be clustered. Inappropriate scale usually leads to unreasonable cluster centers and thus unreasonable results. In this study, we first consider the similarity of elements in the data as the connectivity of nodes in an undirected graph, then present the concept of a connection center and regard it as the cluster center of the data. Based on this definition, the determination of cluster centers and the assignment of class are very simple, natural and effective. One more crucial finding is that the cluster centers of different scales can be obtained easily by the different powers of a similarity matrix and the change of power from small to large leads to the dynamic evolution of cluster centers from local (microscopic) to global (microscopic). Further, in this process of evolution, the number of categories changes discontinuously, which means that the presented method can automatically skip the unreasonable number of clusters, suggest appropriate observation scales and provide corresponding cluster results.

For most hyperspectral remote sensing applications, removing bad bands, such as water absorption bands, is a required preprocessing step. Currently, the commonly applied method is by visual inspection, which is very time-consuming and it is easy to overlook some noisy bands. In this study, we find an inherent connection between target detection algorithms and the corrupted band removal. As an example, for the matched filter (MF), which is the most widely used target detection method for hyperspectral data, we present an automatic MF-based algorithm for bad band identification. The MF detector is a filter vector, and the resulting filter output is the sum of all bands weighted by the MF coefficients. Therefore, we can identify bad bands only by using the MF filter vector itself, the absolute value of whose entry accounts for the importance of each band for the target detection. For a specific target of interest, the bands with small MF weights correspond to the noisy or bad ones. Based on this fact, we develop an automatic bad band preremoval algorithm by utilizing the average absolute value of MF weights for multiple targets within a scene. Experiments with three well known hyperspectral datasets show that our method can always identify the water absorption and other low signal-to-noise (SNR) bands that are usually chosen as bad bands manually., Comment: 17 pages, 8 figures