Soft Subspace Research Articles

Soft nodec spaces

<abstract><p>Following van Douwen, we call a soft topological space soft nodec if every soft nowhere dense subset of it is soft closed. This paper considers soft nodec spaces, which contain soft submaximal and soft door spaces. We investigate the basic properties and characterizations of soft nodec spaces. More precisely, we show that a soft nodec space can be written as a union of two disjoint soft closed soft dense (or soft open) soft nodec subspaces. Then, we study the behavior of soft nodec spaces under various operations, including the following: taking soft subspaces, soft products, soft topological sums, and images under specific soft functions with the support of appropriate counterexamples. Additionally, we show that the Krull dimension of a soft nodec soft $ T_{0} $-space is less than or equal to one. After that, we present some connections among soft nodec, soft strong nodec, and soft compact spaces. Finally, we successfully determine a condition under which the soft one-point compactification of a soft space is soft nodec if and only if the soft space is soft strong nodec.</p></abstract>

Three new soft separation axioms in soft topological spaces

<abstract><p>Soft $ \omega $-almost-regularity, soft $ \omega $ -semi-regularity, and soft $ \omega $-$ T_{2\frac{1}{2}} $ as three novel soft separation axioms are introduced. It is demonstrated that soft $ \omega $ -almost-regularity is strictly between "soft regularity" and "soft almost-regularity"; soft $ \omega $-$ T_{2\frac{1}{2}} $ is strictly between "soft $ T_{2\frac{1}{2}} $" and "soft $ T_{2} $", and soft $ \omega $ -semi-regularity is a weaker form of both "soft semi-regularity" and "soft $ \omega $-regularity". Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft $ \omega $-almost-regularity are obtained, and a decomposition theorem of soft regularity by means of "soft $ \omega $ -semi-regularity" and "soft $ \omega $-almost-regularity" is obtained. Furthermore, it is shown that soft $ \omega $-almost-regularity is heritable for specific kinds of soft subspaces. It is also proved that the soft product of two soft $ \omega $-almost regular soft topological spaces is soft $ \omega $-almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed.</p></abstract>

Between Soft Complete Continuity and Soft Somewhat-Continuity

This paper introduces two novel concepts of mappings over soft topological spaces: “soft somewhat-r-continuity” and “soft somewhat-r-openness”. We provide characterizations and discuss soft composition and soft subspaces. With the use of examples, we offer numerous connections between these two notions and their accompanying concepts. We also offer extension theorems for them. Finally, we investigated a symmetry between our new concepts with their topological analogs.

SEPARATION AXIOMS ON NEAR SOFT TOPOLOGICAL SPACES

Near soft sets are a very successful mathematical model that has been used in order to express the decision-making process for uncertainty in a more ideal way, especially in recent years. The purpose of this paper is to contribute to the theoretical studies on near soft topological spaces. In addition, it presents basic concepts and constructs that will form the basis for a near theoretical set-up of near soft topological spaces. These concepts and structures include sub near soft set, near soft subspaces of a near soft topological space and near soft $T_i$-spaces for $0\leq i\leq 4$. The important aspects of the paper are discussed, especially by examining the definitions and properties given.

An Improved Soft Subspace Clustering Algorithm Based on Particle Swarm Optimization for MR Image Segmentation.

Soft subspace clustering (SSC), which analyzes high-dimensional data and applies various weights to each cluster class to assess the membership degree of each cluster to the space, has shown promising results in recent years. This method of clustering assigns distinct weights to each cluster class. By introducing spatial information, enhanced SSC algorithms improve the degree to which intraclass compactness and interclass separation are achieved. However, these algorithms are sensitive to noisy data and have a tendency to fall into local optima. In addition, the segmentation accuracy is poor because of the influence of noisy data. In this study, an SSC approach that is based on particle swarm optimization is suggested with the intention of reducing the interference caused by noisy data. The particle swarm optimization method is used to locate the best possible clustering center. Second, increasing the amount of geographical membership makes it possible to utilize the spatial information to quantify the link between different clusters in a more precise manner. In conclusion, the extended noise clustering method is implemented in order to maximize the weight. Additionally, the constraint condition of the weight is changed from the equality constraint to the boundary constraint in order to reduce the impact of noise. The methodology presented in this research works to reduce the amount of sensitivity the SSC algorithm has to noisy data. It is possible to demonstrate the efficacy of this algorithm by using photos with noise already present or by introducing noise to existing photographs. The revised SSC approach based on particle swarm optimization (PSO) is demonstrated to have superior segmentation accuracy through a number of trials; as a result, this work gives a novel method for the segmentation of noisy images.

Soft Regular Generalized ω-Closed Sets and Soft ω-T1/2 Spaces

Soft rgω-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft rg-closed sets and soft gω-closed sets. Furthermore, the behavior of soft rgω-closed sets with respect to soft unions, soft intersections, and soft subspaces, as well as induced soft topologies are investigated. Moreover, soft ω-T1/2 spaces which is a weaker form soft T1/2 spaces is defined and investigated. In addition to these, the characterizations of soft rg-T1/2 spaces and soft rgω-T1/2 spaces are discussed. The work also looks at the relationship between our novel notions in soft topological spaces and their analogs in topological spaces.

On soft submaximal spaces

In the present paper, detailed properties of soft submaximal topological spaces have been discussed. It starts by examining how well soft submaximal spaces behave when subjected to some operations, such as: taking soft subspaces, soft products, soft topological sum and images/preimages under certain soft mappings. Furthermore, several characterizations of soft submaximal spaces are obtained with respect to various types of soft subsets of soft topological spaces and their simple extensions.

Design of a modular neural network based on an improved soft subspace clustering algorithm

Being a commonly used way for task decomposition in modular neural network (MNN), clustering analysis is employed to decompose the complex task into several simple subtasks for learning. Recent studies mainly focus on hard clustering, but the clusters might be not sufficiently represented when the cluster boundary is ambiguous, which may degenerate the learning performance of subnetworks in MNN. To solve this problem, we design a modular neural network based on an improved soft subspace clustering (IESSC-MNN) algorithm in this study. Firstly, we propose an improved soft subspace clustering algorithm for task decomposition in MNN, which divides the original space into several interactive feature subspaces and allocates a weight item to each subspace to describe the contribution of the subtasks at the same time. Secondly, each RBF subnetwork is adaptively constructed using a structure growing strategy, and all subnetworks learning the corresponding subtask in parallel. Finally, all subnetworks’ outputs are integrated by weighted summation using the contribution weight of subnetworks. The simulation results of the proposed model on five benchmark data and a practical dataset indicate that IESSC-MNN improves the modeling accuracy and generalization performance with a simple structure when compared with other MNNs.

Spherical Fuzzy Soft Topology and Its Application in Group Decision-Making Problems

The spherical fuzzy soft set is a generalized soft set model, which is more realistic, practical, and accurate. It is an extended version of existing fuzzy soft set models that can be used to describe imprecise data in real-world scenarios. The paper seeks to introduce the new concept of spherical fuzzy soft topology defined on spherical fuzzy soft sets. In this work, we define some basic concepts including spherical fuzzy soft basis, spherical fuzzy soft subspace, spherical fuzzy soft interior, spherical fuzzy soft closure, and spherical fuzzy soft boundary. The properties of these defined set are also discussed and explained with an appropriate examples. Also, we establish certain important results and introduce spherical fuzzy soft separation axioms, spherical fuzzy soft regular space, and spherical fuzzy soft normal space. Furthermore, as an application, a group decision-making algorithm is presented based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) method for solving the decision-making problems. The applicability of the proposed method is demonstrated through a numerical example. The comprehensive advantages of the proposed work have been stated over the existing methods.

On Soft Generalized ω-Closed Sets and Soft T1/2 Spaces in Soft Topological Spaces

In this paper, we define a soft generalized ω-closed set, which is a generalization of both the soft ω-closed set and the soft generalized closed set. We show that the classes of generalized closed sets and generalized ω-closed sets coincide in soft anti-locally countable soft topological spaces. Additionally, in soft locally countable soft topological spaces, we show that every soft set is a soft generalized ω-closed set. Furthermore, we prove that the classes of soft generalized closed sets and soft generalized ω-closed sets coincide in the soft topological space (X,τω,A). In addition to these, we determine the behavior of soft generalized ω-closed sets relative to soft unions, soft intersections, soft subspaces, and generated soft topologies. Furthermore, we investigate soft images and soft inverse images of soft generalized closed sets and soft generalized ω-closed sets under soft continuous, soft closed soft transformations. Finally, we continue the study of soft T1/2 spaces, in which we obtain two characterizations of these soft spaces, and investigate their behavior with respect to soft subspaces, soft transformations, and generated soft topologies.

Fuzzy clustering algorithms with distance metric learning and entropy regularization

Clustering has been used in various fields, such as image processing, data mining, pattern recognition, and statistical analysis. Generally, clustering algorithms consider all variables equally relevant or not correlated. Nevertheless, the pattern of data samples in the multidimensional space can be geometrically complicated, e.g., clusters may exist in different subsets of features. In this regard, new soft subspace clustering algorithms have been proposed, in which the correlation and relevance of variables are considered to improve their performance. Since regularization-based methods are robust for initializations, the approaches proposed introduce an entropy regularization term for controlling the membership degree of the objects. Such regularizations are popular due to high performance in large-scale data clustering and low computational complexity. These three-step iterative algorithms provide a fuzzy partition, a representative for each cluster, and the relevance weight of the variables or their correlation by minimizing a suitable objective function. Several experiments on synthetic and real datasets, including their application to the segmentation of noisy image textures, demonstrate the usefulness of the proposed clustering methods.

Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

Soft subspace clustering of interval-valued data with regularizations

Data analysis plays an indispensable role in understanding different phenomena. One of the vital means of handling these data is to group them into a set of clusters given a measure of similarity. Usually, clustering methods deal with objects described by single-valued variables. Nevertheless, this representation is too restrictive for representing complex data, such as lists, histograms, or even intervals. Furthermore, in some problems, many dimensions are irrelevant and can mask existing clusters. In this regard, new interval-valued data clustering methods with regularizations and adaptive distances are proposed. These approaches consider that the boundaries of the interval-valued variables have the same and different importance for the clustering process. The algorithms optimize an objective function alternating three steps for obtaining the representatives of each group, a fuzzy partition, and the relevance weights of the variables. Experiments on synthetic and real data sets corroborate the robustness and usefulness of the proposed methods.

Multi-objective soft subspace clustering in the composite kernel space

Conventional subspace clustering algorithms group the data samples by optimizing the objective function which aggregates different clustering criteria using the linear combination. However, the performance is sensitive to the user-defined coefficients. Besides, the widely used Euclidean distance metric falls short of handling the linear indivisible problems. Some composite kernel metrics are proposed to overcome this drawback, but it is still difficult to determine the proper weight of base kernels. To address these problems, a novel multi-objective soft subspace clustering model is proposed. The novel model simultaneously optimizes three clustering criteria without setting coefficients. The distance between data points is measured in a composite kernel space. The weight of base kernels is optimized by a multi-objective evolutionary algorithm. A decomposition-based local search strategy is developed to enhance the performance of the proposed algorithm. The experimental results indicate that the proposed algorithm can achieve better solutions.

Some Structures on Neutrosophic Topological Spaces

Abstract In this paper, we define boundary of neutrosophic soft set, neutrosophic soft dense set, neutrosophic soft basis and neutrosophic soft subspace topology on neutrosophic soft topological spaces. Furthermore, some important theorems are proved and interesting examples are given.

On characterizations of fuzzy supra soft connected spaces

In this work, we initiate and study notion of connectedness for fuzzy supra soft topological spaces and present fundamentals properties. Moreover, we discuss its properties as well as with respect to fuzzy supra soft subspaces and fuzzy supra irresolute soft functions. Furthermore, we generalize C_i-fuzzy soft connectedness (i=1,2,3,4), which plays an important role in fuzzy soft topological spaces, to fuzzy supra soft topological spaces. The relationship between the classes FSSCi-connected sets is discussed. Moreover, we introduce counterexamples to clarify that the reverse implications aren’t satisfied.

Subspace data stream clustering with global and local weighting models

Subspace clustering discovers clusters embedded in multiple, overlapping subspaces of high dimensional data. It has been successfully applied in many domains. Data streams are ordered and potentially infinite sequences of data points created by a typically non-stationary data generating process. Clustering this type of data requires some restrictions in time and memory. In this paper, we propose the S2G-Stream algorithm based on growing neural gas and soft subspace clustering. We introduce two types of entropy weighting for both features and blocks, and also two weighting models (local and global). Experiments on public datasets demonstrated the ability of S2G-Stream to detect relevant features and blocks and to provide the best partitioning of the data.

Self-organizing subspace clustering for high-dimensional and multi-view data

A surge in the availability of data from multiple sources and modalities is correlated with advances in how to obtain, compress, store, transfer, and process large amounts of complex high-dimensional data. The clustering challenge increases with the growth of data dimensionality which decreases the discriminate power of the distance metrics. Subspace clustering aims to group data drawn from a union of subspaces. In such a way, there is a large number of state-of-the-art approaches and we divide them into families regarding the method used in the clustering. We introduce a soft subspace clustering algorithm, a Self-organizing Map (SOM) with a time-varying structure, to cluster data without any prior knowledge of the number of categories or of the neural network topology, both determined during the training process. The model also assigns proper relevancies (weights) to different dimensions, capturing from the learning process the influence of each dimension on uncovering clusters. We employ a number of real-world datasets to validate the model. This algorithm presents a competitive performance in a diverse range of contexts among them data mining, gene expression, multi-view, computer vision and text clustering problems which include high-dimensional data. Extensive experiments suggest that our method very often outperforms the state-of-the-art approaches in all types of problems considered.

Evaluation and Analysis of Electronic Information Industry Clustering Level Based on Soft Subspace Clustering Algorithm

Based on the soft subspace clustering, this paper constructs an industry clustering index which can be used to test the significance, and then calculates and analyzes the clustering level of China’s electronic information industry. The results show that the clustering phenomenon of China’s electronic information industry is very obvious, and the clustering index shows an inverted U-shaped trend of rapid rise and slow decline in the whole research range, In terms of the distribution of the gathering areas, they are mainly concentrated in the eastern coastal areas such as Guangdong, Jiangsu and Shanghai, and the concentration degree is constantly increasing.

Adaptive soft subspace clustering combining within-cluster and between-cluster information

Adaptive soft subspace clustering combining within-cluster and between-cluster information