Graph Laplacian Matrix Learning from Smooth Time-Vertex Signal

In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesian product graph of the time-and vertex-graphs.By assuming the signals follow a Gaussian prior distribution on the joint graph,a meaningful representation that promotes the smoothness property of the joint graph signal is derived.Furthermore,by decoupling the joint graph,the graph learning framework is formulated as a joint optimization problem which includes signal denoising,timeand vertex-graphs learning together.Specifically,two algorithms are proposed to solve the optimization problem,where the discrete second-order difference operator with reversed sign(DSODO)in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm,and the time-graph,as well as the vertex-graph,is estimated by the other algorithm.Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time-and vertex-graphs from noisy and incomplete data.

A Note on the Spectral Radius of Weighted Signless Laplacian Matrix

A weighted graph is a graph that has a numeric label associated with each edge, called the weight of edge. In many applications, the edge weights are usually represented by nonnegative integers or square matrices. The weighted signless Laplacian matrix of a weighted graph is defined as the sum of adjacency matrix and degree matrix of same weighted graph. In this paper, a brief overview of the notation and concepts of weighted graphs that will be used throughout this study is given. In Section 2, the weighted signless Laplacian matrix of simple connected weighted graphs is considered, some upper bounds for the spectral radius of the weighted signless Laplacian matrix are obtained and some results on weighted and unweighted graphs are found.

On the Largest Eigenvalue of Signless Laplacian Matrix of a Graph

The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.

On the Eigenvalues of General Sum-Connectivity Laplacian Matrix

The connectivity index was introduced by Randi´c(J.Am.Chem.Soc.97(23):6609–6615,1975)and was generalized by Bollobás and Erdös(Ars Comb.50:225–233,1998).It studies the branching property of graphs,and has been applied to studying network structures.In this paper we focus on the general sum-connectivity index which is a variant of the connectivity index.We characterize the tight upper and lower bounds of the largest eigenvalue of the general sum-connectivity matrix,as well as its spectral diameter.We show the corresponding extremal graphs.In addition,we show that the general sum-connectivity index is determined by the eigenvalues of the general sum-connectivity Laplacian matrix.

ON THE κ-th LARGEST EIGENVALUE OF THE LAPLACIAN MATRIX OF A GRAPH

In this paper,we give the upper bound and lower bound of k-th largest eigenvalue λ_κ of the Laplacian matrix of a graph G in terms of the edge number of G and the number of spanning trees of G.

多通道Laplacian矩阵融合的超图直推学习模型

超图直推学习模型是机器学习领域研究热点.超图模型的性能取决于构造的超图结构及其Laplacian矩阵的质量.现有超图模型基于单一超图结构,信息表达能力有限.本文提出超图结构扩张法,将异构超图的关联矩阵和权重矩阵拼接,融合更多的顶点间全局高阶信息,增加Markov随机游走的扩散范围.但这会导致矩阵维度高,计算开销大.因此进一步提出多通道Laplacian矩阵融合法,用多个通道计算异构超图结构各自的Laplacian矩阵,再加权累加.在4个数据集上的实验表明,两种方法都能提高超图直推学习模型的分类性能,且Laplacian矩阵融合法比结构扩张法平均节约40%左右时间成本,F1指标最高提升8.4%.

RGBD Salient Object Detection by Structured Low-Rank Matrix Recovery and Laplacian Constraint

A structured low-rank matrix recovery model for RGBD salient object detection is proposed. Firstly, the problem is described by a low-rank matrix recovery, and the hierarchical structure of RGB image is added to the sparsity term. Secondly, the depth information is fused into the model by a Laplacian regularization term to ensure that the image regions which share similar depth value will be allocated to similar saliency value. Thirdly, a variation of alternating direction method is proposed to solve the proposed model. Finally, both quantitative and qualitative experimental results on NLPR1000 and NJU400 show the advantage of the proposed RGBD salient object detection model. © 2017, Tianjin University and Springer-Verlag Berlin Heidelberg.

非平衡符号双圈图的拉普拉斯谱半径的排序

研究了非平衡符号双圈图的第一到第六大的拉普拉斯特征值的分布规律,完善了现有结论中一些不准确的情况,推广了现有的结果,并给出了取得极值情况的图例。

二部图性质的谱刻画

为了刻画二部图的性质,研究了图的邻接矩阵、邻接特征多项式、线图、关联矩阵、拉普拉斯矩阵、无符号拉普拉斯矩阵等。用图的谱性质刻画了二部图的特征,并得到了以下结论:二部图G的奇数阶谱矩为0,邻接谱在实数轴上关于原点对称,–2是线图l(G)的重数为m−n+1的特征值,拉普拉斯矩阵和无符号拉普拉斯矩阵有相同的谱,最小无符号拉普拉斯特征值等于0,最大拉普拉斯特征值等于最大无符号拉普拉斯特征值。

The Bounds for Eigenvalues of Normalized Laplacian Matrices and Signless Laplacian Matrices

In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian matrices by using their traces. In addition, we found the bounds for k-th eigenvalues of normalized Laplacian matrix and signless Laplacian matrix.

一类单圈图的Laplacian谱刻画

针对哪些图可由它们的谱刻画这一问题,在lollipop图和图H(n;q,n1,n2)的基础上定义了一类新的图类,符号表示为H(n;q,n1,n2,n3),它是通过在圈Cq的同一个顶点上连接3条悬挂路Pn1、Pn2、Pn3而得到的顶点数为n的单圈图.首先,证明了此图类中,如果2个图形不同构,那么它们必定具有不同的Laplacian谱.在此结论的基础上,证明了图H(n;q,n1,n2,n3)可由它的Laplacian谱刻画.

一种基于Laplacian矩阵的图像匹配算法

文章提出了一种基于Laplacian矩阵的图像特征匹配算法。首先分别构造两幅图像特征点集的Laplacian矩阵,并对这两个矩阵进行奇异值分解(SVD),然后利用分解的结果构造出一个反应特征点之间匹配程度的关系矩阵,最后根据关系矩阵实现两幅图像的特征点匹配。大量实验结果表明,该文所提出的算法具有较高的匹配精度。

基于图的自适应加权多视图聚类

针对现有的基于图的多视图聚类算法没有考虑不同视图的权重和视图数据存在噪声的问题,提出一种基于图的自适应加权多视图聚类算法。通过自适应邻域学习从原始数据中构造多个关系图,引入视图权重调节参数,减少噪声的影响;通过自适应学习将各个关系图融合成统一关系图,通过秩约束优化使数据点自动划分成所需的簇,从而得到聚类结果。在多视图数据集上的实验结果表明了该算法的有效性。

联合Laplacian正则项和特征自适应的数据聚类算法

在信息爆炸时代,大数据处理已成为当前国内外热点研究方向之一.谱分析型算法因其特有的性能而获得了广泛的应用,然而受维数灾难影响,主流的谱分析法对高维数据的处理仍是一个极具挑战的问题.提出一种兼顾维数特征优选和图Laplacian约束的聚类模型,即联合拉普拉斯正则项和自适应特征学习(joint Laplacian regularization and adaptive feature learning,简称LRAFL)的数据聚类算法.基于自适应近邻进行图拉普拉斯学习,并将低维嵌入、特征选择和子空间聚类纳入同一框架,替换传统谱聚类算法先图Laplacian构建、后谱分析求解的两级操作.通过添加非负加和约束以及低秩约束,LRAFL能获得稀疏的特征权值向量并具有块对角结构的Laplacian矩阵.此外,提出一种有效的求解方法用于模型参数优化,并对算法的收敛性、复杂度以及平衡参数设定进行了理论分析.在合成数据和多个公开数据集上的实验结果表明,LRAFL在效果效率及实现便捷性等指标上均优于现有的其他数据聚类算法.

单圈图的Laplacian谱(英文)

G是一个图,A(G),D(G)分别是G的邻接矩阵和顶点度序列对角矩阵,则矩阵L(G)=D(G)-A(G)称为G的Laplacian矩阵。作者考察了单圈图的Laplacian矩阵的谱性质,并着重讨论了单圈图的代数连通度。

给定阶与边独立数的树和单圈图的Laplacian矩阵的最大特征值

得到了给定顶点数和边独立数的树与单圈图的Laplacian矩阵的最大特征值的精确上界 。

给定点连通度的图的补图的无符号拉普拉斯谱半径

假设G是一个具有点集V(G)={v_(1),v_(2),…,v_(n)}和边集E(G)的连通简单图,矩阵Q(G)=D(G)+A(G)被称为图G的无符号拉普拉斯矩阵,其中D(G)和A(G)分别是图G的度对角矩阵和邻接矩阵。称矩阵Q(G)的最大特征值为图G的无符号拉普拉斯谱半径。图G的补图记为G^(c)=(V(G^(c))),E(G^(c)),这里V(G^(c))=V(G)和E(G^(c))={xy|x,y∈V(G),xy∉E(G)}.文章在给定点连通度且直径大于3的图的所有补图中,确定了无符号拉普拉斯谱半径达到最小时的唯一图。

合成图的Laplacian特征值(英文)

给出了任意两个图的合成图的Laplacian特征值和特征向量 ,同时得到了合成图的生成树的数目 .

结合K均值与Laplacian的聚类集成算法

聚类集成可以有效提高传统聚类算法的精度,其关键问题在于如何根据聚类成员提供的信息获得更加优越的聚类结果。设计一种聚类集成算法,它结合K均值算法与基于拉普拉斯矩阵的谱聚类算法,充分利用聚类成员提供的属性信息与关系信息。为了降低算法计算复杂度,通过代数变换方法有效避免了大规模矩阵的特征值分解问题。在多组真实数据集上的实验结果表明,提出的算法优于其他聚类集成算法。

两类冠图的Laplacian谱

图的谱蕴含着图的许多信息。冠图是一种比较复杂的图,冠图的谱更加难以计算。文中定义了两类冠图,分别是:图G1和G2的剖分图的冠点图G1◇G2和剖分图的冠边图G1☆G2。应用分块矩阵、矩阵的coronal、克罗内克积证明了两类冠图的Laplacian谱可以表示为原图G1和G2的Laplacian谱;并给出了两类冠图的生成树数目以及Kirchhoff指数。