Circular backpropagation networks for classification

The class of mapping networks is a general family of tools to perform a wide variety of tasks. This paper presents a standardized, uniform representation for this class of networks, and introduces a simple modification of the multilayer perceptron with interesting practical properties, especially well suited to cope with pattern classification tasks. The proposed model unifies the two main representation paradigms found in the class of mapping networks for classification, namely, the surface-based and the prototype-based schemes, while retaining the advantage of being trainable by backpropagation. The enhancement in the representation properties and the generalization performance are assessed through results about the worst-case requirement in terms of hidden units and about the Vapnik-Chervonenkis dimension and cover capacity. The theoretical properties of the network also suggest that the proposed modification to the multilayer perceptron is in many senses optimal. A number of experimental verifications also confirm theoretical results about the model's increased performances, as compared with the multilayer perceptron and the Gaussian radial basis functions network.     

A parametrization of piecewise linear Lyapunov functions via linear programming

In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stabiilty analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta et al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.     

Some extensions of a new method to analyze complete stability of neural networks

In a recent work, a new method has been introduced to analyze complete stability of the standard symmetric cellular neural networks (CNNs), which are characterized by local interconnections and neuron activations modeled by a three-segment piecewise-linear (PWL) function. By complete stability it is meant that each trajectory of the neural network converges toward an equilibrium point. The goal of this paper is to extend that method in order to address complete stability of the much wider class of symmetric neural networks with an additive interconnecting structure where the neuron activations are general PWL functions with an arbitrary number of straight segments. The main result obtained is that complete stability holds for any choice of the parameters within the class of symmetric additive neural networks with PWL neuron activations, i.e., such a class of neural networks enjoys the important property of absolute stability of global pattern formation. It is worth pointing out that complete stability is proved for generic situations where the neural network has finitely many (isolated) equilibrium points, as well as for degenerate situations where there are infinite (nonisolated) equilibrium points. The extension in this paper is of practical importance since it includes neural networks useful to solve significant signal processing tasks (e.g., neural networks with multilevel neuron activations). It is of theoretical interest too, due to the possibility of approximating any continuous function (e.g., a sigmoidal function), using PWL functions. The results in this paper confirm the advantages of the method of Forti and Tesi, with respect to LaSalle approach, to address complete stability of PWL neural networks.     

多媒体信号处理的数学理论前沿进展

深度学习模型广泛应用于多媒体信号处理领域,通过引入非线性能够极大地提升性能,但是其黑箱结构无法解析地给出最优点和优化条件。因此如何利用传统信号处理理论,基于变换/基映射模型逼近深度学习模型,解析优化问题,成为当前研究的前沿问题。本文从信号处理的基础理论出发,分析了当前针对高维非线性非规则结构方法的数学模型和理论边界,主要包括：结构化稀疏表示模型、基于框架理论的深度网络模型、多层卷积稀疏编码模型以及图信号处理理论。详细描述了基于组稀疏性和层次化稀疏性的表示模型和优化方法,分析基于半离散框架和卷积稀疏编码构建深度/多层网络模型,进一步在非欧氏空间上扩展形成图信号处理模型,并对国内外关于记忆网络的研究进展进行了比较。最后,展望了多媒体信号处理的理论模型发展,认为图信号处理通过解析谱图模型的数学性质,解释其中的关联性,为建立广义的大规模非规则多媒体信号处理模型提供理论基础,是未来研究的重要领域之一。     

A geometrical representation of McCulloch-Pitts neural model andits applications

In this paper, a geometrical representation of McCulloch-Pitts neural model (1943) is presented, From the representation, a clear visual picture and interpretation of the model can be seen. Two interesting applications based on the interpretation are discussed. They are 1) a new design principle of feedforward neural networks and 2) a new proof of mapping abilities of three-layer feedforward neural networks.     

An iterative method for training multilayer networks with thresholdfunctions

Concerns the problem of finding weights for feed-forward networks in which threshold functions replace the more common logistic node output function. The advantage of such weights is that the complexity of the hardware implementation of such networks is greatly reduced. If the task to be learned does not change over time, it may be sufficient to find the correct weights for a threshold function network off-line and to transfer these weights to the hardware implementation. This paper provides a mathematical foundation for training a network with standard logistic function nodes and gradually altering the function to allow a mapping to a threshold unit network. The procedure is analogous to taking the limit of the logistic function as the gain parameter goes to infinity. It is demonstrated that, if the error in a trained network is small, a small change in the gain parameter will cause a small change in the network error. The result is that a network that must be implemented with threshold functions can first be trained using a traditional back propagation network using gradient descent, and further trained with progressively steeper logistic functions. In theory, this process could require many repetitions. In simulations, however, the weights have be successfully mapped to a true threshold network after a modest number of slope changes. It is important to emphasize that this method is only applicable to situations for which off-line learning is appropriate.     

Analysis and synthesis of feedforward neural networks usingdiscrete affine wavelet transformations

A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined.     

On the generalized Sylvester mapping and matrix equations

General parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the so-called generalized Sylvester mapping which has some elegant properties. The solution consists of some polynomial matrices satisfying certain conditions and a parametric matrix representing the degree of freedom in the solution. The results provide great convenience to the computation and analysis of the solutions to this family of equations, and can perform important functions in many analysis and design problems in linear system theory. It is also expected that this so-called generalized Sylvester mapping tool may have some other applications in control system theory.     

An MLP-based representation of neural tensor networks for the RDF data models

In this paper, a new representation of neural tensor networks is presented. Recently, state-of-the-art neural tensor networks have been introduced to complete RDF knowledge bases. However, mathematical model representation of these networks is still a challenging problem, due to tensor parameters. To solve this problem, it is proposed that these networks can be represented as two-layer perceptron network. To complete the network topology, the traditional gradient based learning rule is then developed. It should be mentioned that for tensor networks there have been developed some learning rules which are complex in nature due to the complexity of the objective function used. Indeed, this paper is aimed to show that the tensor network can be viewed and represented by the two-layer feedforward neural network in its traditional form. The simulation results presented in the paper easily verify this claim.

     

A generalized scheme for mapping parallel algorithms

A generalized mapping strategy that uses a combination of graph theory, mathematical programming, and heuristics is proposed. The authors use the knowledge from the given algorithm and the architecture to guide the mapping. The approach begins with a graphical representation of the parallel algorithm (problem graph) and the parallel computer (host graph). Using these representations, the authors generate a new graphical representation (extended host graph) on which the problem graph is mapped. An accurate characterization of the communication overhead is used in the objective functions to evaluate the optimality of the mapping. An efficient mapping scheme is developed which uses two levels of optimization procedures. The objective functions include minimizing the communication overhead and minimizing the total execution time which includes both computation and communication times. The mapping scheme is tested by simulation and further confirmed by mapping a real world application onto actual distributed environments     

Qualitative behavior rules for the cold rolling process extracted from trained ANN via the FCANN method

Nowadays, artificial neural networks (ANN) are being widely used in the representation of different systems and physics processes. In this paper, a neural representation of the cold rolling process will be considered. In general, once trained, the networks are capable of dealing with operational conditions not seen during the training process, keeping acceptable errors in their responses. However, humans cannot assimilate the knowledge kept by those networks, since such knowledge is implicit and difficult to be extracted. For this reason, the neural networks are considered a “black-box”.In this work, the FCANN method based on formal concept analysis (FCA) is being used in order to extract and represent knowledge from previously trained ANN. The new FCANN approach permits to obtain a non-redundant canonical base with minimum implications, which qualitatively describes the process. The approach can be used to understand the relationship among the process parameters through implication rules in different operational conditions on the load-curve of the cold rolling process. Metrics for evaluation of the rules extraction process are also proposed, which permit a better analysis of the results obtained.     

P2P网络中基于模糊理论的综合信任模型研究

在分析P2P网络传统信任模型的基础上,提出一种模糊理论和传统数学理论相结合的综合信任模型。该模型将模糊理论应用于网络节点间主观信任的计算,能够更好地处理复杂的网络因素对信任的影响;利用传统的数学模型来计算推荐信任不但可以得到准确的信任值,还可以将网络的负担降到最低。同时模型中引入了等级反馈机制和时间因子,使得模型的信任度评估更加准确和灵活。仿真结果表明,该模型较传统模型有一定的改进。     

M-P神经元模型的几何意义及其应用

给出M-P神经元模型的几何意义,这个几何的铨释,给神经元一个非常直观的理解,利用这个直观的理解,给出两个颇为有趣的应用:(1)用此法给出三层前向神经网络的学习能力的基本定理的新的证明;(2)给出前向网络的拓扑结构设计的新方法.     

Bio-PEPA: An Extension of the Process Algebra PEPA for Biochemical Networks

In this work we introduce Bio-PEPA, a process algebra for the modelling and the analysis of biochemical networks. It is a modification of PEPA to deal with some features of biological models, such as stoichiometry and the use of generic kinetic laws. Bio-PEPA may be seen as an intermediate, formal, compositional representation of biological systems, on which different kinds of analysis can be carried out. Finally, we show a representation of a model, concerning a simple genetic network, in the new language.     

Vertex-Edge Degree Based Indices of Honey Comb Derived Network

Chemical graph theory is a branch of mathematics which combines graph theory and chemistry. Chemical reaction network theory is a territory of applied mathematics that endeavors to display the conduct of genuine compound frameworks. It pulled the research community due to its applications in theoretical and organic chemistry since 1960. Additionally, it also increases the interest the mathematicians due to the interesting mathematical structures and problems are involved. The structure of an interconnection network can be represented by a graph. In the network, vertices represent the processor nodes and edges represent the links between the processor nodes. Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks. In this paper, we determined the newly introduced topological indices namely, first -degree Zagreb index, first -degree Zagreb index, second -degree Zagreb index, -degree Randic index, -degree atom-bond connectivity index, -degree geometric-arithmetic index, -degree harmonic index and -degree sum-connectivity index for honey comb derived network. In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structureactivity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. Also, we give the numerical and graphical representation of our outcomes.     

Belief trees and networks for biometric applications

This paper aims to introduce the novel approach to the design of a class of decision-making tools based on belief networks for biometric applications. The problem is formulated as mapping the belief networks into the homogeneous computing network, in order to meet the requirements of real-time computing, in particular, the biometric-based physical access control system. The feasible approach to this problem is the accelerating of software computing using hardware. Our experiments show that the straightforward utilization of the hardware tools may not satisfy real-time applications, since the belief networks may not be mapped directly into the hardware. We propose generating the belief network based on mapping of the belief trees into the linear networks with further fusion, so that the obtained structures can be mapped into homogeneous computing arrays.     

A metabolism–repair theory of by-products and side-effects

The plurality of process outputs is a genericity of Nature. In this paper, Natural Law receives a new mathematical formulation founded on two axioms: ‘Everything is a set.’ and ‘Every process is a set-valued mapping.’ I present a brief introduction to the algebraic theory of set-valued mappings, which culminates in two particular morphisms: the metabolism bundle and the imminence mapping. These are relations defined on the collection of processes of a natural system, and serve to characterize material entailment and functional entailment. Generalized metabolism is material entailment of (by-)products, and generalized repair is functional entailment of (side-)effects. Metabolism–Repair networks, hence equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems.     

多阶段建模过程中可重用数学模型表示方法

为解决多阶段建模过程中数学模型重用性问题,分析多阶段建模过程中数学模型表示现状及MDA(model driven architecture,模型驱动体系结构)对数学模型的重用性和平台无关性需求,提出作战训练仿真数学模型广义定义和面向重用的数学模型表示方法;通过建立可重用变量转换关系网络组织、管理和描述作战训练仿真数学模型中的变量转换关系(variable conversion relation,VCR),扩展MathML(math markup language,数学标记语言)语义并基于扩展的MathML语义对变量转换数学表达式进行表示。最后基于变量转换关系网络实现作战训练仿真数学模型动态逻辑行为模型生成。     

A basic mathematical framework for conceptual graphs

Based on the original idea of Sowa on conceptual graph and a recent formalism by Corbett on ontology, this paper presents a rigorous mathematization of basic concepts encountered in the conceptual structure theory, including canon, ontology, conceptual graph, projection, and canonical formation operations, with the aim of deriving their mathematical properties and applying them to future research and development on knowledge representation. Our proposed formalism enhances the conceptual structure theory and enables it to compare favorably with other alternative methods such as the formal concept analysis theory.