Search

Attribute reduction is a crucial issue of rough set theory and has been applied to various fields. It aims to remove useless or redundant features from data and extract precise rules. Traditional attribute reduction methods have obvious limitations, that is, it can not guarantee the generalization ability of reduced decision rules to classification but can only guarantee the classification performance to visible objects. To overcome this issue and enhance the model classification generalization ability without increasing rules' complexity based attribute reduction, we propose a novel attribute method by employing the Structural Risk Minimization principle, which is a classic method in machine learning to balance model complexity and performance. The improved attribute reduction method with rough set tries to get a trade-off between the number of features and empirical error, wherein the former is defined by rule confidence, and the latter is explored by the mutual information between the condition attribute subset and the decision attribute. In other words, model accuracy can be regarded as an empirical error term to measure the model's classification or prediction performance, and the model complexity as a penalty term to characterize the size of the reduction subset. To implement this approach, genetic algorithm is employed as a heuristic search technique, to obtain the optimal reduction subset. Several comparative experiments are performed on ten UCI datasets to evaluate the model's classification accuracy and the size of the reduction subset. The experimental results indicate the method proposed in this paper with better generalization ability compared to other traditional algorithms under the condition of equal reducted set length. • We propose a new attribute reduction method by considering the model accuracy and complexity simultaneously. • We use rule confidence and mutual information to define the model's accuracy and complexity, respectively. • We employ the genetic algorithm as a heuristic search technique to obtain the optimal reducted subset. [ABSTRACT FROM AUTHOR]

• A new regularizer is proposed based on a linear projection. • Two iterative update procedures are developed for minimizing the new objective function. • Various experiments verify the superiority of the proposed algorithm. Nonnegative Matrix Factorization (NMF) produces interpretable solutions for many applications including collaborative filtering. Typically, regularization is needed to address issues such as overfitting and interpretability, especially for collaborative filtering where the rating matrices are sparse. However, the existing regularizers are typically constructed from the factorization results instead of the rating matrices. Intuitively, we regard these existing regularizers as representing either user factors or item factors and anticipate that a more holistic regularizer could improve the effectiveness of NMF. To this end, we propose a graph regularizer based on a linear projection of the rating matrix, and call the resulting method: Linear Projection and Graph Regularized Nonnegative Matrix Factorization (LPGNMF). We develop two iterative methods to minimize the cost function and derive two update rules named LPGNMF and F-LPGNMF. Additionally, we prove the value of the objective function decreases with LPGNMF and converges to a fixed point with F-LPGNMF. Finally, we test these methods against a number of NMF algorithms on different data sets and show both LPGNMF and F-LPGNMF always achieve smaller errors based on two different error measures. [ABSTRACT FROM AUTHOR]

• A new regularizer is proposed based on a linear projection. • Two iterative update procedures are developed for minimizing the new objective function. • Various experiments verify the superiority of the proposed algorithm. Nonnegative Matrix Factorization (NMF) produces interpretable solutions for many applications including collaborative filtering. Typically, regularization is needed to address issues such as overfitting and interpretability, especially for collaborative filtering where the rating matrices are sparse. However, the existing regularizers are typically constructed from the factorization results instead of the rating matrices. Intuitively, we regard these existing regularizers as representing either user factors or item factors and anticipate that a more holistic regularizer could improve the effectiveness of NMF. To this end, we propose a graph regularizer based on a linear projection of the rating matrix, and call the resulting method: Linear Projection and Graph Regularized Nonnegative Matrix Factorization (LPGNMF). We develop two iterative methods to minimize the cost function and derive two update rules named LPGNMF and F-LPGNMF. Additionally, we prove the value of the objective function decreases with LPGNMF and converges to a fixed point with F-LPGNMF. Finally, we test these methods against a number of NMF algorithms on different data sets and show both LPGNMF and F-LPGNMF always achieve smaller errors based on two different error measures. [ABSTRACT FROM AUTHOR]

Although graph neural networks (GNNs) work well on graph data, they are black-box models that lack of reliable explanations for their predictions. We propose a multi-granularity subgraph aggregation method based on graph topology to explain GNNs. Specifically, given a trained GNN model and an input graph, our method constructs a subgraph by heuristics from fine-grained to coarse-grained and sorts the subgraph nodes to obtain subgraph and node-level explain. Furthermore, we propose an improved Shapley value as a heuristic function for the search algorithm, which strikes a balance between the time complexity and accuracy. Finally, experimental results on both synthetic and real datasets demonstrate that our method achieves best performance on seven datasets, quantifying the influence of individual nodes on prediction results and providing more reliable explanations. • We propose a multi-granularity subgraph aggregation method with graph's topology. • We propose an improved Shapley value for balancing time complexity and performance. • We propose a node ranking and verification method utilizing perturbation. • We propose a new metric to evaluate the explainability algorithm. [ABSTRACT FROM AUTHOR]

Extracting fundamental behavior patterns or governing equations from data can deepen our understanding and insights into physical systems, it will lead to the better control and application of these systems in science and engineering. Currently, most existing methods in extracting governing equations require a candidate function term library in advance, which results in the limitations of those learned equations. To overcome this problem in this paper we propose a new method for data-driven discovery of parsimonious partial differential equations (PDEs) by utilizing symbolic regression based on multi-objective optimization and reinforcement learning, we call the MORL4PDEs in short. Specifically, neural network agent aims to generate the pre-order traversal sequence of a binary tree, and through which we can obtain the expression for each PDE. Then the resulting individuals can be used as the initial population in the multi-objective genetic algorithm to ensure the accuracy and parsimony of the equations, whose plausibility is guaranteed according to the constraints generated from the rules of PDEs. Meanwhile, the neural network is optimized through reinforcement learning with the final expression of each PDE as a reward. Finally, several experiments are conduct to demonstrate the effectiveness of the proposed method, and the results show MORL4PDEs can identify governing equations in different dynamic systems, including those PDEs with complex forms and high-order derivatives. [ABSTRACT FROM AUTHOR]

Uncertainty learning is an important research direction of rough set theory, wherein the most popular one is rough set-based prediction, whose goal is to extract decision rules from decision systems and then assign the corresponding decision labels for new samples in terms of the decision rules. To design efficient prediction algorithms, it is necessary and meaningful to measure the uncertainty of rough set-based prediction, especially the stability and generalization performance. In this paper, we analyze the generalization performance of rough set-based prediction algorithms in terms of algorithmic stability analysis and give the generalization error bounds. Firstly, we propose a general rough set-based prediction algorithm to predict the labels for new samples, and then define a scoring function and the corresponding loss function. Secondly, we define two kinds of algorithmic stability for this prediction algorithm in terms of their loss functions, by which two general generalization error bounds are obtained according to two different kinds of stability: strong stability and pointwise hypothesis stability. The bounds numerically imply the performance of the proposed rough set-based prediction algorithm is related to the number of samples and stability parameter. Thirdly, we adopt the confidence and max confidence, min support algorithms as the specific scoring functions instead of general scoring functions. The results show the prediction performance of the confidence algorithm is related to the number of samples and stability parameter, as well as that of max confidence, min support algorithm is associated with the number of samples and minimum support threshold. Based on these discussions, a general framework of stability and generalization error bounds analysis for rough set-based prediction is established. Finally, several experiments are performed to test the previous conclusions. [ABSTRACT FROM AUTHOR]

Dominance rough sets generalize classical rough sets by replacing the equivalence relation with a dominance relation. However, the existing dominance relations are still too restrictive to convenient for its practical applications because they are always considering a strict decreasing or increasing order for each attribute. In fact, in many real-world problems, we only need to employ the decreasing or increasing order in terms of partial attributes rather than all attributes or even just focus on overall evaluations of objects. Based on this phenomenon, in this paper we define two new dominance relations and obtain two generalized dominance rough set models according to define the overall evaluations and add particular requirements for some individual attributes. Meanwhile, the attribute reductions of dominance intuitionistic fuzzy decision information systems are also examined with these two models. Firstly, we define a generalized dominance relation and obtain a generalized dominance rough set model by using the intuitionistic fuzzy additive operator to aggregate individual attribute value of each object into a overall evaluation. The attribute reduction of intuitionistic fuzzy information systems with generalized dominance rough set are also explored. Secondly, we introduce another dominance relation named as generalized β -dominance relation and the generalized β -dominance rough set model by adding a parameter β ∈ [0, 1] in generalized dominance relation in order to control the number of attributes which satisfy dominance relations, from which we can induce all “at least” and “at most” decision rules. An algorithm is developed to compute the lower and upper β -dominance approximations in order to get all the lower and upper reducts of intuitionistic fuzzy decision systems. Some numerical examples are employed to substantiate the conceptual arguments. [ABSTRACT FROM AUTHOR]

Covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings. The existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems. Less efforts have been made in measurements of covering rough sets. To enrich this field, in this paper we investigate the measurements of covering rough sets based on granules and evidence theory, which can be used to solve more complex practical problems. Firstly, it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non-dual covering approximation operators according to definitions based on element, granule, and subsystem. Secondly, we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework. The underlying uncertainty of covering rough sets results from the roughness of granulation of coverings. Finally, we investigate the relationships between the covering approximation operators and evidence theory. It shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions, while others do not share this property. The sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored, and the corresponding counterexamples are given to illustrate those that cannot be measured. Based on above discussions, we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory. [ABSTRACT FROM AUTHOR]