基于拟Laplacian谱和点对拓扑特征的点模式匹配算法

针对谱匹配方法对噪声和出格点的鲁棒性较差的问题,提出了一种基于拟Laplacian谱和点对拓扑特征的点模式匹配算法。首先,用赋权图的最小生成树构造无符号Laplacian矩阵,通过对矩阵谱分解得到的特征值和特征向量表示点的特征,进而计算点的初始匹配概率;其次,利用点对拓扑特征的相似性测度来定义点对间的局部相容性,然后借助概率松弛的方法更新由拟Laplacian谱得到的匹配概率,得出匹配结果。对比实验结果表明,该方法在处理存在噪声和出格点的点集匹配上具有较高的鲁棒性。     

一种VLSI剖分系统的研究与实现

基于多水平方法,设计并实现了一种VLSI剖分系统（Multilevel-based VLSI Partitioner,MVP）。介绍了MVP系统的结构框图、处理流程及模块功能。MVP系统的多水平剖分程序引入图核到粗化阶段,谱图论到初始剖分阶段,群智能到投影优化阶段,得到了无向赋权图更优的剖分。MVP系统特点体现在VLSI线网到无向赋权图的转换,避免了剖分算法直接在VLSI线网上进行剖分,提高了VLSI剖分的效率。实验及分析表明MVP系统的多水平剖分程序能找到更优的图剖分,以及MVP系统找到比现有技术更优的VLSI剖分,提高了VLSI剖分的性能。     

基于均值的谱聚类特征向量选择算法

在数据聚类当中,谱聚类是最流行的方法之一,其性能取决于所选取相关图的拉普拉斯(Laplacian)矩阵的特征向量。对于一个K类问题,Ng-Jordan-Weiss(NJW)谱聚类算法通常采用Laplacian矩阵的前K个最大特征值对应的特征向量作为数据的一种表示。然而,对于某些分类问题,这K个特征向量不一定能够很好地体现原始数据的信息。本文提出一种基于均值的谱聚类特征向量选择算法。该算法首先得出图的Laplacian矩阵的前3K个最大特征值的均值,然后选取K个离均值最近的特征值所对应的特征向量。相比传统谱聚类算法,该算法在UCI数据集上获得了较好的聚类性能。     

PEBI网格节点编号优化方法研究

基于PEBI网格的油藏数值模拟能够更准确地模拟地下油藏流动,模拟过程中主要是求解以PEBI网格为差分单元的有限差分方程。提出采用谱算法优化PEBI网格节点的编号来减少差分方程中系数矩阵的带宽,以节约计算时间和数据存储量。首先计算网格按初始编号所形成的邻接矩阵及其Laplacian矩阵,然后通过计算Laplacian矩阵的特征值和特征向量得到Fiedler特征向量,最后对Fiedler特征向量进行排序,并根据排序后的向量对PEBI重新编号。最后通过实验验证了谱算法在PEBI网格编号优化中的有效应用。     

微分方程在复杂网络社团发现中的研究

理解复杂网络的关键在于迅速精确地发现网络中的社团结构。基于图理论的谱聚类算法是一种有效并全局收敛的优秀社团发现算法,其计算量集中于特征值和特征向量的计算。结合常系数线性常微分方程的解与系数矩阵特征值的关系,提出了基于微分方程的谱聚类社团发现算法(AMCF和LMCF);这两种算法避免了矩阵的特征值和特征向量的复杂计算过程,为社团发现算法提供了新的思路。理论分析和实验验证了算法的有效性。     

一种计算矩阵特征值特征向量的神经网络方法

当把Oja学习规则描述的连续型全反馈神经网络(Oja-N)用于求解矩阵特征值特征向量时,网络初始向量需位于单位超球面上,这给应用带来不便.由此,提出一种求解矩阵特征值特征向量的神经网络(1yNN)方法.在lyNN解析解基础上得到了以下结果:初始向量属于任意特征值对应特征向量张成的子空间,则网络平衡向量也将属于该空间;分析了lyNN收敛于矩阵最大特征值对应特征向量的初始向量取值条件;明确了lyNN收敛于矩阵不同特征值的特征子空间时,网络初始向量的最大取值空间;网络初始向量与已知特征向量垂直,则lyNN平衡解向量将垂直于该特征向量;证明了平衡解向量位于由非零初始向量确定的超球面上的结论.基于以上分析,设计了用lyNN求矩阵特征值特征向量的具体算法,实例演算验证了该算法的有效性.1yNN不出现有限溢,而基于Oja-N的方法在矩阵负定、初始向量位于单位超球面外时必出现有限溢,算法失效.与基于优化的方法相比,lyNN实现容易,计算量较小.     

谱聚类算法对输入数据顺序的敏感性*

结合矩阵分析知识,还原了实施谱聚类算法过程中的矩阵表示.发现了不同数据输入顺序使得相应的Affinity矩阵及Laplacian矩阵是相似的.这样,Laplacian矩阵的特征向量生成的矩阵Y也是相似的;而以Y的行向量作为输入数据的K-平均算法依赖于初始的k个对象的选择.由此给出了导致谱聚类算法对数据输入顺序敏感的原因.     

点云模型的谱聚类分割

为了实现点云模型的有意义分割,提出一种基于谱聚类的分割算法.首先用图G表示点云模型,将分割问题转化为图切割问题;然后根据归一化的非对称Laplacian矩阵构造谱聚类空间;最后通过移除掉多余的特征向量,在一个更低维的空间中找到了分割问题的松弛解.文中还给出了该算法相关定理的证明,并通过实验验证了算法的正确性和有效性.     

二维ESPRIT参数配对及FPGA实现

与MUSIC算法相比,二维ESPRIT算法不需要谱峰搜索、运算量小,但存在参数配对问题。基于此,提出一种易于实现的参数配对方法,基本原理是2个表出矩阵特征值的和差等于表出矩阵和差的特征值,不需求解表出矩阵的特征向量。给出基于CORDIC算法和脉动阵的参数配对并行化实现方案,整个系统只有移位相加运算。使用ISE软件和ModelSim软件得到的仿真结果验证了该方案的正确性。     

多特征-谱聚类的SAR图像溢油分割

经典的K聚类算法,并不适合实现任意形状的聚类,而且有容易陷入局部最小值的不足.提出基于多个纹理特征的谱聚类算法,该方法用灰度共生矩阵(GLCM)提取合成孔径雷达 (SAR)图像的多个特征值,构建谱聚类的特征矩阵,并依据规范切准则,用K均值聚类的方法对拉普拉斯矩阵的第二小的特征值对应的特征向量进行聚类,实现基于SAR图像的溢油的分割.新方法与传统的K聚类方法比较,可以减少相干斑噪声对分割结果的影响,较好的保持图像边缘.仿真结果显示,该算法对于相干斑噪声影响较大的图像具有较强的鲁棒性.     

基于Ncut准则的图分割的新算法

针对有权图分割时不能很好解决子图内部耦合度不高的问题,使用可以同时优化子图内部顶点耦合度和子图之间顶点耦合度的Ncut准则,提出了一种新的基于迭代改善策略的RNK分割算法。算法通过不断交换可以改善Ncut值的顶点对优化现有分割。与传统分割算法相比,可以同时保证子图内最大耦合度和子图间最小的耦合度。并提出一种散列技术,提高查找最优交换顶点对的效率。当图为稠密矩阵时,改善效果尤为明显。通过对随机图分割的实验结果表明,该算法较传统的KL算法可以得到更理想的分割结果。     

Graph spectral image smoothing using the heat kernel

A new method for smoothing both gray-scale and color images is presented that relies on the heat diffusion equation on a graph. We represent the image pixel lattice using a weighted undirected graph. The edge weights of the graph are determined by the Gaussian weighted distances between local neighboring windows. We then compute the associated Laplacian matrix (the degree matrix minus the adjacency matrix). Anisotropic diffusion across this weighted graph-structure with time is captured by the heat equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigensystem with time. Image smoothing is accomplished by convolving the heat kernel with the image, and its numerical implementation is realized by using the Krylov subspace technique. The method has the effect of smoothing within regions, but does not blur region boundaries. We also demonstrate the relationship between our method, standard diffusion-based PDEs, Fourier domain signal processing and spectral clustering. Experiments and comparisons on standard images illustrate the effectiveness of the method.     

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation

This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the "length" of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commute-time distance. This graph PCA provides a nice interpretation to the "Fiedler vector," widely used for graph partitioning. The model is evaluated on a collaborative-recommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities perform well in comparison with other methods. The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a relational database     

An optimal algorithm for solving the searchlight guarding problem on weighted two-terminal series-parallel graphs

In this paper, we consider a graph problem on a connected weighted undirected graph, called the searchlight guarding problem. Our problem is an extension of so-called graph searching/guarding problem by considering the time slot parameter in addition to the traditional building cost. Suppose that there is a fugitive who moves along the edges of the graph at any speed. We want to place a set of searchlights at the vertices to search the edges of the graph and capture the fugitive. It costs some building cost to place a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total costs of the vertices in S is minimized. If there is more than one set of searchlights with the minimum building cost, then find the one with the minimum searching time, that is, the time slots needed to capture the fugitive is the minimum. The problem is known to be NP-hard on weighted bipartite graphs, split graphs, and chordal graphs; and it is linear time solvable on weighted trees and interval graphs. In this paper, an algorithm is designed to solve the problem on weighted two-terminal series-parallel graphs. It works on the parsing tree structure of the given two-terminal series-parallel graph. The algorithm is divided into two phases. In the phase one, we first extract some useful properties of optimal solutions. Employing these properties, an algorithm is designed to find the set of searchlights with the minimum guarding cost and to assign the searching directions of all edges by the dynamic programming strategy. In the phase two, the searched time slots of all edges are determined by the breadth-first-search from the root of the parsing tree. The time complexities of both phases are linear. Thus, our algorithm is time optimal. Received: 12 March 1996 / 27 May 1997     

异构计算中一种图的非均衡划分算法

现有的图的划分算法大多是均衡划分，要求划分块的权值相等，划分块之间的连接代价尽量最小。但是在异构计算环境中，不同的处理机的计算能力不尽相同，从而在并行任务调度时所分配的计算任务量也应随之不同。所以为了适应更广泛意义上的异构负栽均衡，本文提出了异构计算中的一种任务图的非均衡划分算法。该算法根据任意给定的需求，使得划分好的各个子集权值不均等。其中划分子集的个数等于异构环境中处理机的个数，各子集的大小比例于不同处理机的计算能力。算法包括3步：粗化阶段、非均衡划分阶段以及精化还原阶段。本文通过用格林威治大学提供的系列开放图来测试该算法，实验结果表明算法是准确有效的。     

Semidefinite spectral clustering

Multi-way partitioning of an undirected weighted graph where pairwise similarities are assigned as edge weights, provides an important tool for data clustering, but is an NP-hard problem. Spectral relaxation is a popular way of relaxation, leading to spectral clustering where the clustering is performed by the eigen-decomposition of the (normalized) graph Laplacian. On the other hand, semidefinite relaxation, is an alternative way of relaxing a combinatorial optimization, leading to a convex optimization. In this paper we employ a semidefinite programming (SDP) approach to the graph equipartitioning for clustering, where sufficient conditions for strong duality hold. The method is referred to as semidefinite spectral clustering, where the clustering is based on the eigen-decomposition of the optimal feasible matrix computed by SDP. Numerical experiments with several data sets, demonstrate the useful behavior of our semidefinite spectral clustering, compared to existing spectral clustering methods.     

An optimal algorithm for solving the searchlight guarding problem on weighted interval graphs

In this paper, a graph problem on connected, weighted, undirected graphs, called the searchlight guarding problem, is considered. Assume that there is a fugitive who moves along the edges of the graph at a random speed. The task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive. Suppose that placing a searchlight at some vertex incurs some building cost. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total cost of the vertices in S is minimized. If there is more than one set of searchlights, each with a minimum building cost, then identify the set with the minimum search time, that is, where the time slots needed to spot the fugitive is the minimum. As is well established, the problem is NP-hard on weighted bipartite graphs but is linear-time solvable on weighted trees. In this paper, the design of a linear-time optimal algorithm for the searchlight guarding problem on weighted interval graphs is presented. It entails two phases. In the first phase, a set of searchlights with minimum guarding cost is identified and the search directions of all edges are assigned. To achieve this task, a new problem, called the edge-direction assignment problem, is first defined and the problem on weighted complete-split graphs is solved by the greedy strategy. Based on this computational result, the problem of finding the set of searchlights with minimum guarding cost and assigning the search directions of all edges is solved by the dynamic programming strategy. Then, in the second phase, the search time slots of each edge are determined on the basis of the results of the first phase and on some properties of interval graphs.     

知识图谱划分算法研究综述

知识图谱是人工智能的重要基石,因其包含丰富的图结构和属性信息而受到广泛关注.知识图谱可以精确语义描述现实世界中的各种实体及其联系,其中顶点表示实体,边表示实体间的联系.知识图谱划分是大规模知识图谱分布式处理的首要工作,对知识图谱分布式存储、查询、推理和挖掘起基础支撑作用.随着知识图谱数据规模及分布式处理需求的不断增长,如何对其进行划分已成为目前知识图谱研究的热点问题.从知识图谱和图划分的定义出发,系统性地介绍当前知识图谱数据划分的各类算法,包括基本、多级、流式、分布式和其他类型图划分算法.首先,介绍4种基本图划分算法:谱划分算法、几何划分算法、分支定界算法、KL及其衍生算法,这类算法通常用于小规模图数据或作为其他划分算法的一部分;然后,介绍多级图划分算法,这类算法对图粗糙化后进行划分再投射回原始图,根据粗糙化过程分为基于匹配的算法和基于聚合的算法;其次,描述3种流式图划分算法,这类算法将顶点或边加载为序列后进行划分,包括Hash算法、贪心算法、Fennel算法,以及这3种算法的衍生算法;再次,介绍以KaPPa、JA-BE-JA和轻量级重划分为代表的分布式图划分算法及它们的衍生算法;同时,在其他类型图划分算法中,介绍近年来新兴的2种图划分算法:标签传播算法和基于查询负载的算法.通过在合成与真实知识图谱数据集上的丰富实验,比较了5类知识图谱代表性划分算法在划分效果、查询处理与图数据挖掘方面的性能差异,分析实验结果并推广到推理层面,获得了基于实验的知识图谱划分算法性能评价结论.最后,在对已有方法分析和比较的基础上,总结目前知识图谱数据划分面临的主要挑战,提出相应的研究问题,并展望未来的研究方向.