赋权图匹配问题的一种新的松弛模型

图匹配是一个NP难(NP-hard)问题. 基于置换矩阵是非负正交矩阵这一经典结论, 提出赋权图匹配(Weighted graph matching, WGM)的双向松弛障碍规划, 理论上证明新模型的解与原模型的解是一致的. 该规划是一个二元连续规划, 它是正交矩阵上的线性优化问题, 同时也是非负矩阵上的凸二次优化问题. 故设计求解新模型的交替迭代算法, 并证明算法的局部收敛性. 数值实验表明, 在匹配精度方面, 新方法强于线性规划方法和特征值分解方法.     

平衡图匹配方法在多视角图像配准中的研究

在多视角图像处理中,多点匹配是一个基本的问题。针对有较多制约因素的多点匹配问题,对现有的光谱匹配技术进行两点改进:首先提出一种新的光谱松弛技术用以求解匹配函数的近似解,其次将现有的相容性矩阵重构为二分图边矩阵并进行双随机归一化,减小噪声对结果的影响。在基于CAVIAR与PETS2009数据集上的实验证明,所提出的算法可以很好地完成图匹配,并且有较高的精度与较好的鲁棒性。     

基于非精确图匹配的CAD模型搜索方法

为了弥补现有的三维CAD模型搜索方法难以搜索到不同近似程度的相似模型的缺陷,提出一种基于面属性化邻接图非精确匹配的CAD模型搜索方法.首先提取CAD模型中的B-rep信息将CAD模型转化为面属性化邻接图;然后计算目标模型与被搜索模型的面属性化邻接图之间的顶点相容程度矩阵和边相容程度矩阵,并由此建立2个模型相似程度的度量作为选择不同顶点匹配矩阵M的优化目标函数;在对匹配矩阵M进行连续化松弛后,运用Sinkhorn行列交替规范化方法求解匹配优化问题.实验结果表明,采用该方法能够搜索到不同近似程度的相似模型;并且由于避免了具有NP复杂性的精确图匹配过程,检索效率也能满足实际要求.     

图松弛优化聚类的快速近似提升方法

基于图松弛优化为非近似迭代方法提供了有效的分析解决方案,且实现简单。然而,由于矩阵的逆在计算时需要多项式时间,则在运行速度方面不是很理想,当面对较大规模数据时此方法将变得不可行。提出了对基于图松弛优化聚类进行快速近似提升的两种方法:一个是基于k均值聚类,另一个是基于随机投影树。广泛实验表明,这些算法在运算速度方面表现较优,聚类精度变化非常小。具体来讲,该算法在运算大规模数据时精度优于k均值算法,并且在保证精度的情况下运行速度远快于基于图松弛优化聚类算法。值得注意的是,该算法可以使得单个机器在数分钟内对具有数百万样本的数据集进行聚类。     

基于遗传算法的有向无环图画图算法*

在研究了现有画有向无环图的主要方法的基础上提出一种基于遗传算法的有向无环图画图算法,将一般有向无环图的画图问题转换为函数优化问题,用遗传算法求目标函数最优解的近似值。实验表明此算法具有算法统一、方法简单、容易实现、易于修改,并且具有自适应、自学习和易于并行化的特点。     

一种应用峰值特征匹配的SAR图象自动目标识别方法

针对合成孔径雷达（SAR）图象自动目标识别问题，在对SAR图象的特征提取问题进行分析的基础上，提出了一种特征点匹配算法，该算法根据Birkhoff-von Neumann定量，首先将广义置换矩阵约束松驰为广义双随机矩阵约束；然后利用拉格朗日乘子和障碍函数法，把约束加到目标函数中，从而将点集匹配问题转化为非线性最优化问题；最后利用确定性退火和软分配技术求解该问题，将得到的匹配代价用特征点数目的比值进行修正后，用于目标的识别。实验结果表明，该算法非常有效。     

耳廓点云形状特征匹配的路径跟随算法

路径跟随算法结合凸松弛方法与凹松弛方法,通过跟随凸凹问题的解路径,近似地求解图匹配问题,具有较高的匹配精度.将路径跟随算法用于耳廓特征图的匹配问题:首先,基于PCA方法构造耳廓点云的显著性关键点集合;然后,采用乘积型参数域上的单值二次曲面方法拟合关键点邻域内的点集,并将曲面的局部形状特征定义为耳廓的局部形状相似测度;第三,对关键点集合进行Delaunay三角剖分,得到关键点集合在三维空间内的拓扑结构图,并定义关键点图的整体结构差异测度;最后,记耳廓关键点图的组合差异测度为关键点图的整体结构差异测度与关键点上的局部形状相似测度的线性组合,并基于路径跟随算法快速求解关键点图之间的精确匹配.相关实验结果表明:与其他相关算法相比,该算法具有较高的匹配效率和匹配精度.     

基于文档属性单元松弛的XML近似查询方法

为解决普通用户对XML文档的近似查询问题,提出了一种基于文档属性单元松弛的XML近似查询方法.该方法将XML文档中的叶子结点和属性结点作为属性单元处理,基于一致集的概念导出最大集,生成最小非平凡函数依赖集,从而找出属性单元之间的近似函数依赖关系,进而求出近似候选码和近似关键字.在此基础上,根据属性单元支持度将属性单元按重要程度排列并据此对初始查询条件进行松弛,最不重要的属性单元最先松弛并且松弛程度最大.利用松弛后的查询条件对XML文档进行查询,可得到与初始查询条件近似的查询结果.实验结果和分析表明:提出的XML近似查询方法能够很好地满足用户的查询意图,具有较高的执行效率.     

非正交联合对角化的罚函数粒子群算法

针对多个矩阵近似联合对角化盲分离问题,提出一种新的非正交近似联合对角化算法.首先采用罚函数法将联合对角化的非线性约束优化模型转化为无约束优化模型;其次将粒子群优化算法引入无约束优化模型中实现目标函数的最优化,从而完成矩阵组的联合对角化.分析了惩罚因子的更新策略及算法的收敛性能,并设计仿真实验进行对比分析以检验算法解决实际盲分离问题的能力.     

模态试验传感器优化布置的松弛序列法

合理选择传感器的数目和位置是模态试验是否成功的关键步骤.将松弛思想融入传感器优化布置的序列优化方法,提出了模态试验传感器优化布置的松弛序列法.选取模态保证准则矩阵的最大非对角元素为目标函数,在积累序列法的基础上融入松弛的思想,使其求解的结果得到进一步优化.首先,积累序列法产生初始解集;然后,对解集的元素进行多次松弛操作,直至所有元素无法继续松弛.提出的松弛序列法被应用于某桥梁的模态试验传感器优化布置问题,传统的序列法作为对比算法.结果表明:松弛序列法优于传统的序列法.     

A Path Following Algorithm for the Graph Matching Problem

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We, therefore, construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.     

Graph Matching by Simplified Convex-Concave Relaxation Procedure

The convex and concave relaxation procedure (CCRP) was recently proposed and exhibited state-of-the-art performance on the graph matching problem. However, CCRP involves explicitly both convex and concave relaxations which typically are difficult to find, and thus greatly limit its practical applications. In this paper we propose a simplified CCRP scheme, which can be proved to realize exactly CCRP, but with a much simpler formulation without needing the concave relaxation in an explicit way, thus significantly simplifying the process of developing CCRP algorithms. The simplified CCRP can be generally applied to any optimizations over the partial permutation matrix, as long as the convex relaxation can be found. Based on two convex relaxations, we obtain two graph matching algorithms defined on adjacency matrix and affinity matrix, respectively. Extensive experimental results witness the simplicity as well as state-of-the-art performance of the two simplified CCRP graph matching algorithms.     

A dynamical systems approach to weighted graph matching

Graph matching is a fundamental problem that arises frequently in the areas of distributed control, computer vision, and facility allocation. In this paper, we consider the optimal graph matching problem for weighted graphs, which is computationally challenging due the combinatorial nature of the set of permutations. Contrary to optimization-based relaxations to this problem, in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices. In particular, since permutation matrices are orthogonal matrices with nonnegative elements, we define two gradient flows in the space of orthogonal matrices. The first minimizes the cost of weighted graph matching over orthogonal matrices, whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations. The combination of the two dynamical systems converges to a permutation matrix, which provides a suboptimal solution to the weighted graph matching problem. Finally, our approach is shown to be promising by illustrating it on nontrivial problems.     

Maximal Biclique Subgraphs and Closed Pattern Pairs of the Adjacency Matrix: A One-to-One Correspondence and Mining Algorithms

Maximal biclique (also known as complete bipartite) subgraphs can model many applications in Web mining, business, and bioinformatics. Enumerating maximal biclique subgraphs from a graph is a computationally challenging problem, as the size of the output can become exponentially large with respect to the vertex number when the graph grows. In this paper, we efficiently enumerate them through the use of closed patterns of the adjacency matrix of the graph. For an undirected graph G without self-loops, we prove that 1) the number of closed patterns in the adjacency matrix of G is even, 2) the number of the closed patterns is precisely double the number of maximal biclique subgraphs of G, and 3) for every maximal biclique subgraph, there always exists a unique pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, the problem of enumerating maximal bicliques can be solved by using efficient algorithms for mining closed patterns, which are algorithms extensively studied in the data mining field. However, this direct use of existing algorithms causes a duplicated enumeration. To achieve high efficiency, we propose an O(mn) time delay algorithm for a nonduplicated enumeration, in particular, for enumerating those maximal bicliques with a large size, where m and n. are the number of edges and vertices of the graph, respectively. We evaluate the high efficiency of our algorithm by comparing it to state- of-the-art algorithms on three categories of graphs: randomly generated graphs, benchmarks, and a real-life protein interaction network. In this paper, we also prove that if self-loops are allowed in a graph, then the number of closed patterns in the adjacency matrix is not necessarily even, but the maximal bicliques are exactly the same as those of the graph after removing all the self-loops.     

A Lagrangian relaxation network for graph matching

A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions". Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetry-breaking self-amplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and self-amplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variables. Simulation results are shown on 100 node random graphs and for a wide range of connectivities.     

超网络能量研究与应用

图能量是图论研究的重要内容,图能量及其变种已在无向图、有向图、混合图等其他多种类型的图中得到很多成功的应用。超网络是一类较传统意义上的复杂网络更为复杂的网络。大多数图能量均是基于矩阵特征值计算得到的,无法推广应用到超网络中,应用范围受限。基于网络维数的网络能量已先后应用于无向图、有向图等多种类型图的分析研究中,并与无向图的图能量及有向图的斜能量等其他类似能量之间存在密切关联。基于超网络的超网络维数,结合网络能量,将超网络维数应用于超网络中,提出了超网络的超网络能量,给出了超网络能量的若干上下限,论证了超网络的超网络能量与图的网络能量之间的内在关联,最后分析了超网络的超网络能量若干重要性质。     

Semi-local Model of Computations on Graphs to Break the Local Symmetry: Work in Progress

We consider finite connected undirected graphs without self-loops as a model of computer networks. The nodes of the graph represent computers or processors, while the edges of the graph correspond to the links between them. We present a model of distributed computations, called semi-local. This extension of the classical local model breaks the local symmetry. As a result, many useful tasks become deterministically solvable in every network assuming a very small initial knowledge about its graph representation. One of these tasks is a creation of a token in an arbitrary anonymous ring – an example of election of a leader. A semi-local solution to this problem is presented.     

基于图匹配的分层布局算法

针对节点数目较大并且度数比较平均的无向图,根据分层扩展的思想,提出一种基于图匹配的分层布局算法（Graph Matching Hierarchy,GMH）。基于图匹配思想对大图进行递归化简,然后应用FR算法对最粗化图进行布局,最后利用质心布局算法对图进行扩展。实验结果表明,GMH算法能够提高可视化效率,改善布局效果,且分层布局的结果更易于理解。      

Average Convergence for Directed & Undirected Graphs in Distributed Systems

Consensus control of multi-agent systems is an innovative paradigm for the development of intelligent distributed systems. This has fascinated numerous scientific groups for their promising applications as they have the freedom to achieve their local and global goals and make their own decisions. Network communication topologies based on graph and matrix theory are widely used in a various real-time applications ranging from software agents to robotics. Therefore, while sustaining the significance of both directed and undirected graphs, this research emphases on the demonstration of a distributed average consensus algorithm. It uses the harmonic mean in the domain of multi-agent systems with directed and undirected graphs under static topologies based on a control input scheme. The proposed agreement protocol focuses on achieving a constant consensus on directional and undirected graphs using the exchange of information between neighbors to update their status values and to be able to calculate the total number of agents that contribute to the communication network at the same time. The proposed method is implemented for the identical networks that are considered under the directional and non-directional communication links. Two different scenarios are simulated and it is concluded that the undirected approach has an advantage over directed graph communication in terms of processing time and the total number of iterations required to achieve convergence. The same network parameters are introduced for both orientations of the communication graphs. In addition, the results of the simulation and the calculation of various matrices are provided at the end to validate the effectiveness of the proposed algorithm to achieve consensus.