Laplacian Graphs Research Articles

Multi-view clustering via double spaces structure learning and adaptive multiple projection regression learning

Multi-view clustering aims to group objects with high similarity into one group according to the heterogeneous features of different views. The graph-based clustering methods have obtained excellent results. However, there remain a few common drawbacks. For example, some methods do not consider graphs' high-order structure information. Thus, fuller data information cannot be obtained. In addition, some methods remove noise, outliers, and redundant information in the graph learning phase, resulting in the loss of graph information. Furthermore, using predefined graphs cannot exploit complementary information between views. A triple strategy-based multi-view clustering method is presented to solve the above issues. First, Laplacian graphs are used for fusion learning, and the underlying first-order and second-order structure information among views are explored simultaneously. Then, a label fusion scheme is designed to eliminate noise, outliers, and redundant information and to mine the intrinsic characteristics of data labels. Besides, the consistent label matrix in adaptive regression learning is used to explore complementary information between views in a mutually guided learning way. Finally, the objective function is solved by using an efficient iterative method. Six types of experiments are conducted on eleven real-world multi-view datasets, and the conclusions that can be drawn are: (1) the proposed algorithm achieves the best results in terms of clustering accuracy on ten datasets with an average accuracy improvement of 5.11% compared to other algorithms. Specifically, the accuracy improved by 9.05% on dataset HW and 10.95% on dataset Reuters compared to the second results; (2) The ablation experiments confirm that the different learning strategies included in the proposed algorithm allow it to achieve better clustering performance.

Golden Laplacian Graphs

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n≤10. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.

Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I

Abstract The set S i , n = { 0 , 1 , 2 , … , n − 1 , n } \ { i } {S}_{i,n}=\left\{0,1,2,\ldots ,n-1,n\right\}\setminus \left\{i\right\} , 1 ⩽ i ⩽ n 1\leqslant i\leqslant n , is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is S i , n {S}_{i,n} . The existence of such graphs was established by Fallat et al. (On graphs whose Laplacian matrices have distinct integer eigenvalues, J. Graph Theory 50 (2005), 162–174). In this article, we consider graphs whose Laplacian spectra have the form S { i , j } n m = { 0 , 1 , 2 , … , m − 1 , m , m , m + 1 , … , n − 1 , n } \ { i , j } , 0 < i < j ⩽ n , {S}_{{\left\{i,j\right\}}_{n}^{m}}=\left\{0,1,2,\ldots ,m-1,m,m,m+1,\ldots ,n-1,n\right\}\setminus \left\{i,j\right\},\hspace{1.0em}0\lt i\lt j\leqslant n, and completely describe those with m = n − 1 m=n-1 and m = n m=n . We also show close relations between graphs realizing S i , n {S}_{i,n} and S { i , j } n m {S}_{{\left\{i,j\right\}}_{n}^{m}} and discuss the so-called S n , n {S}_{n,n} -conjecture and the corresponding conjecture for S { i , n } n m {S}_{{\left\{i,n\right\}}_{n}^{m}} .

Structure learning with consensus label information for multi-view unsupervised feature selection

Structure learning based feature selection has attracted increasing attention for selecting these features which can preserve the learned structures. However, existing methods fail to effectively explore the heterogeneous and homogeneous information from multiple views, which leads to the suboptimal results. To solve this problem, we propose Structure Learning with Consensus Label Information for Multi-View Feature Selection (SCMvFS). Noting the heterogeneity of views, the graph of each view should be a perturbation of the intrinsic graph yet the clustering structure are shared across views. In light of this, we generate a unique clustering indicator through the spectral analysis of multiple Laplacian graphs for the structure learning based feature selection. Therefore, SCMvFS considers both the graph heterogeneity and indicator consistency to effectively explore the heterogeneous and homogeneous information for facilitating the feature selection task. Further, we carefully design an efficient algorithm to solve the resulting optimization problem. Extensive experimental results demonstrate that the proposed method outperforms the state-of-the-art methods on seven benchmark datasets with respect to two indicators. In particular, SCMvFS achieves an ACC of 61.87 (55.94) on the Outdoor Scene (Yale) dataset, which is an up to 43% (15%) performance improvement compared with the latest structure learning based method TLR. The code and datasets are available at https://github.com/HdTgon/2023-ESWA-SCMvFS.

Autoweighted Multiview Feature Selection With Graph Optimization.

In this article, we focus on the unsupervised multiview feature selection, which tries to handle high-dimensional data in the field of multiview learning. Although some graph-based methods have achieved satisfactory performance, they ignore the underlying data structure across different views. Besides, their predefined Laplacian graphs are sensitive to the noises in the original data space and fail to obtain the optimal neighbor assignment. To address the above problems, we propose a novel unsupervised multiview feature selection model based on graph learning, and the contributions are three-fold: 1) during the feature selection procedure, the consensus similarity graph shared by different views is learned. Therefore, the proposed model can reveal the data relationship from the feature subset; 2) a reasonable rank constraint is added to optimize the similarity matrix to obtain more accurate information; and 3) an autoweighted framework is presented to assign view weights adaptively, and an effective alternative iterative algorithm is proposed to optimize the problem. Experiments on various datasets demonstrate the superiority of the proposed method compared to the state-of-the-art methods.

Tensorial Multi-View Clustering via Low-Rank Constrained High-Order Graph Learning

Multi-view clustering aims to partition multi-view data into different categories by optimally exploring the consistency and complementary information from multiple sources. However, most existing multi-view clustering algorithms heavily rely on the similarity graphs from respective views and fail to comprehend multiple views holistically. Moreover, due to the noise and redundancy maintained in the original data, the original errors of multiple similarity graphs will continue to accumulate in the process of constructing consistent graphs. These situations always lead to the limitation to effective fuse the essential information from multiple views, which always influences the clustering performance and cries out for reliable solutions. Based on the above considerations, we propose a novel method termed Tensorial Multi-view Clustering (TMvC), which learns high-order graph by low-rank tensor constraint to uncover the essential information stored in multiple views. TMvC first learns the Laplacian graphs of all views and stacks them into a tensor which can be viewed as a high-order graph. With the high-order graph, consistency and complementary information from different views can be propagated smoothly across all views. Then, based on low-rank constraint, high-order graph is constrained in the horizontal and vertical directions to better uncover the inter-view and inter-class correlations between multi-view data, which is of vital importance for multi-view clustering. Extensive experiments on document and image datasets demonstrate that TMvC can achieve the state-of-the-art performance for multi-view clustering.

MLapSVM-LBS: Predicting DNA-binding proteins via a multiple Laplacian regularized support vector machine with local behavior similarity

DNA-binding proteins (DBPs) are of great significance in many basic cellular processes. Experiment-based methods for identifying DBPs are costly and time-consuming. To deal with large-scale DBP identification tasks, a variety of computation-based methods have been developed. Inspired by previous work, we propose a multiple Laplacian regularized support vector machine with local behavior similarity (MLapSVM-LBS) to predict DBP. We serially combine three features that are extracted from protein sequences (including PsePSSM, GE, NMBAC) and feed them into MLapSVM-LBS. Based on human behavior learning theory, MLapSVM-LBS can better represent the relationship between samples through local behavior similarity. We introduce a new edge weight calculation method that takes label information into consideration. In addition, a local distribution parameter reflecting the underlying probability distribution of a sample’s neighborhood is also employed. To further improve the robustness of the model, we utilize multiple Laplacian regularization to build a multigraph model in which five Laplacian graphs are constructed with local behavior similarity by changing the neighborhood size. To appraise the performance of our model, MLapSVM-LBS is trained and tested on the PDB186, PDB1075, PDB2272 and PDB14189 datasets. On two independent testing sets (PDB186 and PDB2272), our method reaches the accuracies of 0.887 and 0.712, respectively. The good results on both datasets demonstrate the reliable performance of our model.

Laplacian integral graphs with a given degreee sequence constraint

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Microbiome Data Analysis by Symmetric Non-negative Matrix Factorization With Local and Global Regularization.

A network is an efficient tool to organize complicated data. The Laplacian graph has attracted more and more attention for its good properties and has been applied to many tasks including clustering, feature selection, and so on. Recently, studies have indicated that though the Laplacian graph can capture the global information of data, it lacks the power to capture fine-grained structure inherent in network. In contrast, a Vicus matrix can make full use of local topological information from the data. Given this consideration, in this paper we simultaneously introduce Laplacian and Vicus graphs into a symmetric non-negative matrix factorization framework (LVSNMF) to seek and exploit the global and local structure patterns that inherent in the original data. Extensive experiments are conducted on three real datasets (cancer, cell populations, and microbiome data). The experimental results show the proposed LVSNMF algorithm significantly outperforms other competing algorithms, suggesting its potential in biological data analysis.

Una revisión a las propiedades y familia de expansores en grafos de Cayley

In this paper some concepts used in graph theory are introduced, such as directed, undirected, connected, tree, regular or gradient, divergent or Laplacian graphs, and relationships between the diameter of the graph, or the largest second proper value of its adjacency matrix, with respect to the Cheeger constant to identify expander graphs k-regular. With these guidelines defined, some properties are introduced in Cayley graphs, with illustrative examples, and methodologies to identify if the corresponding graph is k-regular or a directed tree. Finally, Cayley expander graphs are related to their diameter or the second largest eigenvalue.

Regularized graph-embedded covariance discriminative learning for image set classification

Riemannian manifold has attracted an increasing amount of attention for visual classification tasks, especially for video or image set classification. Covariance matrices are the natural second-order statistics of image sets. However, nonsingular covariance matrices, known as symmetric positive defined (SPD) matrices, lie on the non-Euclidean Riemannian manifold (SPD manifold). Covariance discriminative learning (CDL) is an effective discriminative learning method that employs the Riemannian manifold in the SPD kernel space. However, in practice, the discriminative learning of CDL often suffers from the problems of poor generalization and overfitting caused by a finite number of training samples and noise corruption. Hence, we propose to address these problems by importing eigenspectrum regularization and graph-embedded frameworks. Discriminative learning with SPD manifold is generalized by the graph-embedded framework, which combines with eigenspectrum regularization in the SPD kernel space. Three local Laplacian graphs of graph-embedded framework and two eigenspectrum regularized models are incorporated to the proposed method. Comprehensive mathematical deduction of the proposed method is depicted with the “kernel tricks.” Experimental results on set-based face recognition and object categorization tasks reveal the effectiveness of the proposed method.

Layered Multitask Tracker via Spatial–Temporal Laplacian Graph

Most multitask trackers define the trace of each candidate as one task, and assume all tasks are equally related. Multitask learning is only evaluated on the current frame. In fact, these assumptions are limited, and ignore the multitask relationship in consecutive frames. In this letter, we propose a discriminative layered multitask tracker via spatial–temporal Laplacian graphs, which defines the layered tasks from a novel view, and naturally incorporates the global and local target information into reverse multitask tracking process. The spatial–temporal Laplacian graphs not only exploit the sequential consistent information of the target, but also make full use of the geometric structure corresponding to the tasks among the adjacent frames. Besides, $l_{0}$ norm constraint and labeling information are used to improve the tracking robustness. Encouraging experimental results on challenging sequences justify that the proposed method performs well both in accuracy and robustness against some related trackers.

On the distance spectra of coronas

In this article, we give a complete description of the eigenvalues and the corresponding eigenvectors of the distance matrix and the distance signless Laplacian matrix of corona of two graphs and (denoted by ), when is connected transmission regular and is regular. Further, when is connected transmission regular, we give a complete description of the eigenvalues and the eigenvectors of the distance Laplacian matrix of for any graph . As an application, we find families of distance, distance Laplacian and distance signless Laplacian cospectral nonisomorphic graphs.

Normalized Laplacian spectrum of some subdivision-coronas of two regular graphs

In this paper, we determine the full normalized Laplacian spectrum of the subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighbourhood corona and subdivision-edge neighbourhood corona of a connected regular graph with an arbitrary regular graph in terms of their normalized Laplacian eigenvalues. Moreover, applying these results, we find some non-regular normalized Laplacian cospectral graphs.

Multi-source adaptation joint kernel sparse representation for visual classification

Most of the existing domain adaptation learning (DAL) methods relies on a single source domain to learn a classifier with well-generalized performance for the target domain of interest, which may lead to the so-called negative transfer problem. To this end, many multi-source adaptation methods have been proposed. While the advantages of using multi-source domains of information for establishing an adaptation model have been widely recognized, how to boost the robustness of the computational model for multi-source adaptation learning has only recently received attention. To address this issue for achieving enhanced performance, we propose in this paper a novel algorithm called multi-source Adaptation Regularization Joint Kernel Sparse Representation (ARJKSR) for robust visual classification problems. Specifically, ARJKSR jointly represents target dataset by a sparse linear combination of training data of each source domain in some optimal Reproduced Kernel Hilbert Space (RKHS), recovered by simultaneously minimizing the inter-domain distribution discrepancy and maximizing the local consistency, whilst constraining the observations from both target and source domains to share their sparse representations. The optimization problem of ARJKSR can be solved using an efficient alternative direction method. Under the framework ARJKSR, we further learn a robust label prediction matrix for the unlabeled instances of target domain based on the classical graph-based semi-supervised learning (GSSL) diagram, into which multiple Laplacian graphs constructed with the ARJKSR are incorporated. The validity of our method is examined by several visual classification problems. Results demonstrate the superiority of our method in comparison to several state-of-the-arts.

The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups

In this paper, a formula of the Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups is given. As applications of our main result, special formulae of Laplacian and signless Laplacian spectrum are also given for two classes of semi-Cayley graphs (one matching bi-Cayley graphs and the join of two Cayley graphs over isomorphic abelian groups). In particular, a method to construct Laplacian and signless Laplacian integral semi-Cayley graphs is obtained.

Adaptive multi-view feature selection for human motion retrieval

Human motion retrieval plays an important role in many motion data based applications. In the past, many researchers tended to use a single type of visual feature as data representation. Because different visual feature describes different aspects about motion data, and they have dissimilar discriminative power with respect to one particular class of human motion, it led to poor retrieval performance. Thus, it would be beneficial to combine multiple visual features together for motion data representation. In this article, we present an Adaptive Multi-view Feature Selection (AMFS) method for human motion retrieval. Specifically, we first use a local linear regression model to automatically learn multiple view-based Laplacian graphs for preserving the local geometric structure of motion data. Then, these graphs are combined together with a non-negative view-weight vector to exploit the complementary information between different features. Finally, in order to discard the redundant and irrelevant feature components from the original high-dimensional feature representation, we formulate the objective function of AMFS as a general trace ratio optimization problem, and design an effective algorithm to solve the corresponding optimization problem. Extensive experiments on two public human motion database, i.e., HDM05 and MSR Action3D, demonstrate the effectiveness of the proposed AMFS over the state-of-art methods for motion data retrieval. The scalability with large motion dataset, and insensitivity with the algorithm parameters, make our method can be widely used in real-world applications.

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.

The New Class of Laplacian Integral Graphs

Graph is Laplacian integral, if all the eigenvalues of its Laplacian matrix are integral. In this paper, we obtain that the Laplacian characteristic polynomials of graphs by calculation. Characterizes the new class of Laplacian integral graphs .

A characterization of spectral integral variation in two places for Laplacian matrices

We describe the graphs having the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing by 1 and the other Laplacian eigenvalues remaining fixed. For a certain subclass of graphs, we also characterize the Laplacian integral graphs with that property. Finally, we investigate a situation in which the algebraic connectivity is one of the eigenvalues that increases by 1.