Uninorms, as important generalizations of triangular norms and conorms, let the identity [Formula: see text] exist anywhere on a bounded lattice. In this paper, we focus on new characterizations of uninorms allowed to act on more general bounded lattices. In particular, we present several necessary and sufficient conditions to verify the construction approaches introduced by (Çaylı and Karaçal, Kybernetika 53 (2017) 394–417) and (Çaylı, Fuzzy Sets Syst. 395 (2020) 107–129), yielding a uninorm on bounded lattices.

Based on the closed operational laws in picture fuzzy numbers and strict triangular norms, we extend the Bonferroni mean (BM) operator under the picture fuzzy environment to propose the picture fuzzy interactional Bonferroni mean (PFIBM), picture fuzzy interactional weighted Bonferroni mean (PFIWBM), and picture fuzzy interactional normalized weighted Bonferroni mean (PFINWBM) operators. We prove the monotonicity, idempotency, boundedness, and commutativity for the PFIBM and PFINWBM operators. We also establish a novel multicriteria decision making (MCDM) method under the picture fuzzy environment by applying the PFINWBM operator. Furthermore, we apply our MCDM method to the enterprise resource planning (ERP) systems selection. The comparative results for our MCDM method induced by six classes of well-known triangular norms ensure that the best selection is always the same ERP system. Therefore, our MCDM method is effective for dealing with the picture fuzzy MCDM problems.

All intuitionistic fuzzy TOPSIS methods contain two key elements: (1) the order structure, which can affect the choices of positive and negative ideal-points, and construction of admissible distance/similarity measures; (2) the distance/similarity measure, which is closely related to the values of the relative closeness degrees and determines the accuracy and rationality of decision-making. For the order structure, many efforts are devoted to constructing some score functions, which can strictly distinguish different intuitionistic fuzzy values (IFVs) and preserve the natural partial order for IFVs. This paper proves that such a score function does not exist. For the distance or similarity measure, some examples are given to show that classical similarity measures based on the Euclidean distance and Minkowski distance do not meet the axiomatic definition of IF similarity measures. Moreover, some illustrative examples are given to show that classical intuitionistic fuzzy TOPSIS methods do not ensure the monotonicity with the natural partial order or linear orders, which may yield some counter-intuitive results. To overcome the limitation of non-monotonicity, we propose a novel intuitionistic fuzzy TOPSIS method, using three new admissible distances with the linear orders measured by a score degree/similarity function and accuracy degree, or two aggregation functions, and prove that the proposed TOPSIS method is monotonous under these three linear orders. This is the first result with a strict mathematical proof on the monotonicity with the linear orders for the intuitionistic fuzzy TOPSIS method. Finally, we show two practical examples to illustrate the efficiency of the developed TOPSIS.

Graph neural networks have been widely used for a variety of learning tasks. Link prediction is a relatively under-studied graph learning task, with current state-of-the-art models based on one- or two-layer shallow graph auto-encoder (GAE) architectures. In this paper, we overcome the limitation of current methods for link prediction of non-Euclidean network data, which can only use shallow GAEs and variational GAEs. Our proposed methods innovatively incorporate standard auto-encoders (AEs) into the architectures of GAEs to capitalize on the intimate coupling of node and edge information in complex network data. Empirically, extensive experiments on various datasets demonstrate the competitive performance of our proposed approach. Theoretically, we prove that our deep extensions can inclusively express multiple polynomial filters with different orders. The codes of this paper are available at https://github.com/xinxingwu-uk/DGAE.

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this article for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt,ordinal sums of triangular norms on bounded lattices, Inf. Sci., vol. 517, (2020) 198–216.

In this paper, we propose a novel nonconvex approach to robust principal component analysis for HSI denoising, which focuses on simultaneously developing more accurate approximations to both rank and column-wise sparsity for the low-rank and sparse components, respectively. In particular, the new method adopts the log-determinant rank approximation and a novell2,lognorm, to restrict the local low-rank or column-wisely sparse properties for the component matrices, respectively. For thel2,log-regularized shrinkage problem, we develop an efficient, closed-form solution, which is namedl2,log-shrinkage operator. The new regularization and the corresponding operator can be generally used in other problems that require column-wise sparsity. Moreover, we impose the spatial-spectral total variation regularization in the log-based nonconvex RPCA model, which enhances the global piece-wise smoothness and spectral consistency from the spatial and spectral views in the recovered HSI. Extensive experiments on both simulated and real HSIs demonstrate the effectiveness of the proposed method in denoising HSIs.

Feature selection reduces the dimensionality of data by identifying a subset of the most informative features. In this paper, we propose an innovative framework for unsupervised feature selection, called fractal autoencoders (FAE). It trains a neural network to pinpoint informative features for global exploring of representability and for local excavating of diversity. Architecturally, FAE extends autoencoders by adding a one-to-one scoring layer and a small sub-neural network for feature selection in an unsupervised fashion. With such a concise architecture, FAE achieves state-of-the-art performances; extensive experimental results on fourteen datasets, including very high-dimensional data, have demonstrated the superiority of FAE over existing contemporary methods for unsupervised feature selection. In particular, FAE exhibits substantial advantages on gene expression data exploration, reducing measurement cost by about 15% over the widely used L1000 landmark genes. Further, we show that the FAE framework is easily extensible with an application.

Feature selection, as a vital dimension reduction technique, reduces data dimension by identifying an essential subset of input features, which can facilitate interpretable insights into learning and inference processes. Algorithmic stability is a key characteristic of an algorithm regarding its sensitivity to perturbations of input samples. In this paper, we propose an innovative unsupervised feature selection algorithm attaining this stability with provable guarantees. The architecture of our algorithm consists of a feature scorer and a feature selector. The scorer trains a neural network (NN) to globally score all the features, and the selector adopts a dependent sub-NN to locally evaluate the representation abilities for selecting features. Further, we present algorithmic stability analysis and show that our algorithm has a performance guarantee via a generalization error bound. Extensive experimental results on real-world datasets demonstrate superior generalization performance of our proposed algorithm to strong baseline methods. Also, the properties revealed by our theoretical analysis and the stability of our algorithm-selected features are empirically confirmed.

As one of the efficient approaches to deal with big data, divide-and-conquer distributed algorithms, such as the distributed kernel regression, bootstrap, structured perception training algorithms, and so on, are proposed and broadly used in learning systems. Some learning theories have been built to analyze the feasibility, approximation, and convergence bounds of these distributed learning algorithms. However, less work has been studied on the stability of these distributed learning algorithms. In this paper, we discuss the generalization bounds of distributed learning algorithms from the view of algorithmic stability. First, we introduce a definition of uniform distributed stability for distributed algorithms and study the distributed algorithms' generalization risk bounds. Then, we analyze the stability properties and generalization risk bounds of a kind of regularization-based distributed algorithms. Two generalization distributed risks obtained show that the generalization distributed risk bounds for the difference between their generalization distributed and empirical distributed/leave-one-computer-out risks are closely related to the size of samples n and the amount of working computers m as O(m/n1/2) . Furthermore, the results in this paper indicate that, for a good generalization regularized distributed kernel algorithm, the regularization parameter λ should be adjusted with the change of the term m/n1/2. These theoretic discoveries provide the useful guidance when deploying the distributed algorithms on practical big data platforms. We explore our theoretic analyses through two simulation experiments. Finally, we discuss some problems about the sufficient amount of working computers, nonequivalence, and generalization for distributed learning. We show that the rules for the computation on one single computer may not always hold for distributed learning.