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For many application scenarios where raw and even multidomain training data can be easily collected, and at the same time, validation data (as ground-truth data) are available, it becomes naturally desirable for us to perform an enhanced classification/prediction on only validation data with the appropriate leverage of training data. In this article, a novel ensemble framework EP-TSK-FK of Takagi–Sugeno–Kang (TSK) fuzzy subclassifiers, is proposed to achieve the following distinctive characteristics: 1) each interpretable TSK fuzzy subclassifier on each training subset can be quickly built in parallel such that its outputs provide the values of the corresponding augmented features of the original validation data space; 2) as a novel ensemble method of fuzzy subclassifiers, EP-TSK-FK trains all the interpretable TSK fuzzy subclassifiers only once and does not explicitly reuse them while predicting a testing sample, which thereby reduces the computational complexity of the ensemble process for prediction; 3) after running the proposed iterative fuzzy c-means clustering algorithm iterative fuzzy C-means clustering (IFCM) on the augmented validation data to obtain the representative centroids, the fast classification/prediction of EP-TSK-FK on the testing samples is realized by using the $k$ -nearest neighbor (KNN) method on the representative centroids with the original features; and 4) enhanced classification performance by the IFCM & KNN method is theoretically revealed, and the experimental results on the benchmarking datasets indicate the effectiveness of EP-TSK-FK and its parallel learning method in the sense of enhanced classification performance, running time, and interpretability. [ABSTRACT FROM AUTHOR]

Although high-order Takagi–Sugeno–Kang (TSK) fuzzy systems have demonstrated their computational advantages and simultaneously circumvent the weakness that the number of rules with the number of input variables and membership functions grows exponentially in both zero-order and first-order TSK fuzzy systems for complex modeling tasks, they still face two serious issues: incapability for a changing environment and no interpretability of the coefficients in high-order polynomial used in the consequent part of each fuzzy rule. In order to circumvent these two challenges, a novel stacked architecture of an interpretable deep higher order TSK fuzzy classifier called DHO-TSK and its deep learning method are proposed by proving the equivalence between a high-order TSK fuzzy classifier and a deep ensemble of interpretable zero-order TSK fuzzy classifiers in this article. DHO-TSK can be built by assembling interpretable zero-order TSK fuzzy classifiers in a special stacked way. Each zero-order TSK fuzzy classifier can be learnt by randomly selecting input features, randomly assigning an antecedent fuzzy subset from a fixed fuzzy partition to each of the selected input features, and then multiplying the output of each TSK fuzzy classifier by a randomly selected feature. Except for the abovementioned solid theoretical equivalence, DHO-TSK is featured in the following aspects: first, the consequent part of each fuzzy rule in DHO-TSK becomes interpretable and the output expression of each layer in DHO-TSK becomes comprehensible due to the adopted stacked ensemble; second, its enhanced classification performance can be achieved in a stacked deep learning way; third, DHO-TSK has its adoptability for changing environments owing to random selection of both features and fuzzy membership functions. Our experimental results on the benchmarking UCI and KEEL datasets and a real dataset indicate the effectiveness of DHO-TSK and its learning method in the sense of both classification performance and interpretability. [ABSTRACT FROM AUTHOR]

As the first attempt, this article considers how to provide a design methodology for style clustering on stylistic data, where each cluster depends on both the similarities between data samples and its latently or apparently distinguishable style. By taking our previous fuzzy k plane clustering algorithm as the basic framework, a fuzzy style k-plane clustering (S-KPC) algorithm is proposed to have its distinctive merits: First, the nuances between styles of clusters can be well identified by using the proposed twofold data representation. That is to say, style matrices are used to express the structure, hence style information of each cluster, whereas the augmentation of the original features of data with enhanced nodes is taken as an abstract representation so as to move the manifold structure of data apart. Such a twofold data representation can make us realize S-KPC readily in an incremental way. Second, by means of alternating optimization strategy, the objective function of S-KPC can be optimized such that each discriminant function of each cluster shares the advantages of both simple regression models and functional-link neural networks. Extensive experiments on synthetic and real-world datasets demonstrate that S-KPC has comparable clustering performance with several compared methods on the adopted ordinary datasets, and yet it obviously outperforms them on stylistic datasets. [ABSTRACT FROM AUTHOR]

There may exist more than one Pareto optimal solution with the same objective vector to a multimodal multiobjective optimization problem (MMOP). The difficulties in finding such solutions can be different. Although a number of evolutionary multimodal multiobjective algorithms (EMMAs) have been proposed, they are unable to solve such an MMOP due to their convergence-first selection criteria. They quickly converge to the Pareto optimal solutions which are easy to find and therefore lose diversity in the decision space. That is, such an MMOP features an imbalance between achieving convergence and preserving diversity in the decision space. In this article, we first present a set of imbalanced distance minimization benchmark problems. Then we propose an evolutionary algorithm using a convergence-penalized density method (CPDEA). In CPDEA, the distances among solutions in the decision space are transformed based on their local convergence quality. Their density values are estimated based on the transformed distances and used as the selection criterion. We compare CPDEA with five state-of-the-art EMMAs on the proposed benchmarks. Our experimental results show that CPDEA is clearly superior in solving these problems. [ABSTRACT FROM AUTHOR]

The performance of decomposition-based multiobjective evolutionary algorithms (MOEAs) often deteriorates clearly when solving multiobjective optimization problems with irregular Pareto fronts (PFs). The main reason is the improper settings of reference vectors and scalarizing functions. In this paper, we propose a decomposition-based MOEA guided by a growing neural gas network, which learns the topological structure of the PF. Both reference vectors and scalarizing functions are adapted based on the topological structure to enhance the evolutionary algorithm’s search ability. The proposed algorithm is compared with eight state-of-the-art optimizers on 34 test problems. The experimental results demonstrate that the proposed method is competitive in handling irregular PFs. [ABSTRACT FROM AUTHOR]

Many practical scenarios have demanded that we should classify unlabeled data more accurately based on both physical features (e.g., color, distance, or similarity) and implicit style features of data. As most extant classification algorithms classify unlabeled data based only on their physical features, they become weak in achieving expected classification results for many scenarios. To work around this drawback in this paper, a novel classification method (FuCM) from the perspective of fuzzy social network based on both physical and implicit style features of data is proposed. Based on the proposed fuzzy social network and its dynamics about fuzzy influences of nodes, FuCM comprises two stages. In its training stage, after the fuzzy social network has been built, it learns the topological structure, reflecting physical features and implicit style features of data by carrying out fuzzy influence dynamics in the built network. In its prediction stage, both physical and implicit style features of data are effectively integrated to yield the double structure efficiency characterized by fuzzy influences of nodes. FuCM classifies unlabeled data according to the strongest connection measure based on the proposed double structure efficiency. FuCM does not assume that both data distribution and the classification by physical features or by both physical and implicit style features of data must be known in advance. Thus, it is a novel unified classification framework in this sense. In contrast to all the nine comparative methods, FuCM experimentally demonstrates its comparable classification performance on most synthetic, UCI and KEEL datasets, which can be well classified based only on physical features of data. Furthermore, it displays distinctive superiority on five case studies where satisfactory classification certainly depends on both physical and implicit style features. [ABSTRACT FROM AUTHOR]

The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First, we explain difficulties of fair comparison in the following two cases: one is the use of all nondominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto front but also from outside the Pareto front. [ABSTRACT FROM AUTHOR]