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It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements., Comment: To appear in Discrete & Computational Geometry. arXiv admin note: substantial text overlap with arXiv:1707.08449, arXiv:1703.09402

Press-pack IGBTs are now widely used in HVDC and FACTS since its extraordinary reliability and high power density. A method to estimate the reliability of press-pack IGBT based on FEM is proposed. A circuit model of MMC was simulated by Simplorer in order to obtain the power loss model of press-pack IGBT. The coupled steady-state simulation of thermal, electrical and mechanical model was analyzed by Workbench. Furthermore, the transient junction swing of IGBT was estimated by a power loss with half sine profile. LESIT lifetime prediction model was used to analyze the lifetime and reliability of press-pack IGBT in MMC system. Press-pack IGBT has high reliability in MMC in long time scale since the low junction temperature of it.

For the widely used multilayer perceptrons (MLPs), the existed singularities in the parameter space have seriously affected the learning dynamics of MLPs, which cause several singular learning behaviors. Since the Fisher information matrix (FIM) plays a significant role in analyzing the singular learning dyanmics of MLPs, it is very worthy to obtain the analytical form of FIM to do further investigation. In this paper, by choosing the bipolar error function as the activation function, the analytical form of FIM are obtained, where the validity of the obtained results are verified by taking three experiments.

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first combinatorial formula and asked to give a combinatorial interpretation. In this article we prove that the Falk invariant of an arrangement associated with signed graph G without loops is double of the sum of $$k_3$$ , $$k_4$$ and $$k^{\pm }_3,$$ where $$k_l$$ is the number of subgraphs of G that are switching equivalent to the cliques with l vertices, $$k_3^{\pm }$$ is that of subgraphs which have 3 vertices and each two vertices are connected by double edges with different signs. This formula modifies the one given by Schenck and Suciu, and answers partially Falk’s question in the case of signed graphic arrangements.

For the bipolar-activation-function multilayer perceptrons (MLPs), there exist opposite singularities in the parameter space. The Fisher information matrix degenerates on the opposite singularity which causes strange learning behaviors. As the stability is the fundamental to analyze the properties of the opposite singularity, this paper concerns the stability analysis of the opposite singularity in MLPs. The analytical form of the best approximation on the opposite singularity is obtained at first, then the concrete expression of Hessian matrix can be obtained. By analyzing the eigenvalues of Hessian matrix on the opposite singularity, the stability of the opposite singularity is investigated. Finally, two experiments are taken to verify the obtained results.

The existing overlap singularities in the parameter space significantly affect the learning dynamics of the multilayer perceptrons. From the obtained theoretical learning trajectories near overlap singularity, when the learning process has been affected by the overlap singularity, the influence area of the overlap singularity is just the line space where the two hidden units equal to each other. However, in the practical applications, different case has been observed and the influence area of such singularity may be larger. By analyzing the generalization error of multilayer perceptrons, we find that the error surface is much flatter near overlap singularity and the singularity would have much larger influence area. Finally, the validity of the obtained results are verified by taking an artificial experiment and two real-data experiments.

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an interesting and complicated invariant. The third rank of successive quotients in the lower central series of the fundamental group was called the global invariant of the arrangement by Falk. Falk gave a general formula to compute the global invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the number of cliques with three or four vertices in the graph with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements. While in the case of signed graphic arrangements, we obtained a similar combinatorial formula. In this paper, we give a direct and simple proof for this combinatorial formula.

The multilayer perceptrons (MLPs) are widely used in many fields, however, singularities in the parameter space may seriously influence the learning dynamics of MLPs and cause strange learning behaviors. Given that the singularities are the subspaces of the parameter space where the Fisher information matrix (FIM) degenerates, the FIM plays a key role in the study of the singular learning dynamics of the MLPs. In this paper, we obtain the analytical form of the FIM for unipolar activation function-based MLPs where the input subjects to the Gaussian distribution with general covariance matrix and the unipolar error function is chosen as the activation function. Then three simulation experiments are taken to verify the validity of the obtained results.

The multilayer perceptrons (MLPs) have been widely used in many communication applications, however, the learning process of the multilayer perceptrons often becomes very slow, which is due to the existence of the singularities in the parameter space. As the singularities significantly affect the learning dynamics of MLPs, the standard gradient descent method is not Fisher efficient. In order to overcome this problem, natural gradient method and adaptive natural gradient method were proposed to accelerate the learning process. As is well known, step size in each iteration plays a key role in the performance of algorithms. In this paper, the modified adaptive natural gradient method is proposed where the step size in each iteration is adaptive modified. The aim of the proposed algorithm is to accelerate the convergence speed and increase the performance of MLPs. The simulation results verify the validity of the analytical results.

Feature selection is a significant preprocessing procedure for many high-dimensional forecasting problems. In multivariate time series forecasting, the purpose of feature selection is to select a relevant feature subset from the original time series. One of the most common strategies for feature selection is mutual information (MI) criterion. K-nearest neighbors (k-NN) is a powerful means which has been widely used to estimate the MI between two high-dimensional variables directly from data set. In order to improve the effect of feature selection, a novel procedure is proposed in this paper to use MI and k-NN to perform feature selection in multivariate time series forecasting. This feature selection procedure not only reduces the irrelevance between the inputs and outputs variables, but also removes the redundant input variables. The procedure of this method is elaborated on a synthetic data set as well as on a real-world example.

To avoid unpredictable losses because of network failure, the reliability of the network needs to be evaluated in some application scenarios. This paper start the network failure prediction research upon 14 months' network alarm logs we collected. The logs are of one Metropolitan area network. The research method is shown as below: firstly, construct features to represent network characteristics by the means of the feature construction method which is based on two levels time windows; secondly, select optimal parameter combination to create the feature files through multiple experiments; thirdly, design and build adaptive failure prediction model according to classification learning methods. Numbers of experiments show that accuracy of predicting whether the network failure takes place in 6 hours is up to 70%, is better than the prediction result of Weibull distribution model obviously; the results of classification prediction for network equipment failure are slightly better than the prediction method on the basis of Weibull distribution. Preliminary research results show that most network failures can be predicted through analyzing previous network running logs and the method proposed in this paper is verified to be with good prediction effect. This method can detect failures in practical application on early stage and reduce unnecessary economic losses.