Vertex Representations Research Articles

Several realizations of Fock modules for toroidal $\ddot {U}_{q,d}(\mathfrak {sl}_{n})$

In this paper, we relate the well-known Fock representations of the quantum toroidal algebra of sl(n) to the vertex, shuffle, and L-operator representations of the same algebra. These identifications generalize those for the quantum toroidal algebra of gl(1), which were recently established by Feigin-Jimbo-Miwa-Mukhin.

Deep Dynamic Network Embedding for Link Prediction

Network embedding task aims at learning low-dimension latent representations of vertices while preserving the structure of a network simultaneously. Most existing network embedding methods mainly focus on static networks, which extract and condense the network information without temporal information. However, in the real world, networks keep evolving, where the linkage states between the same vertex pairs at consequential timestamps have very close correlations. In this paper, we propose to study the network embedding problem and focus on modeling the linkage evolution in the dynamic network setting. To address this problem, we propose a deep dynamic network embedding method. More specifically, the method utilizes the historical information obtained from the network snapshots at past timestamps to learn latent representations of the future network. In the proposed embedding method, the objective function is carefully designed to incorporate both the network internal and network dynamic transition structures. Extensive empirical experiments prove the effectiveness of the proposed model on various categories of real-world networks, including a human contact network, a bibliographic network, and e-mail networks. Furthermore, the experimental results also demonstrate the significant advantages of the method compared with both the state-of-the-art embedding techniques and several existing baseline methods.

Structural Representations Learning of Social Influence in Heterogeneous Social Networks

The research of structural representations of vertexes is critical in classifying nodes, detecting communities, and predicting social links. The existing studies focus on capturing and preserving the structure of social relations, embedding the structure information into low-dimensional vector spaces. However, current approaches cannot fully consider the diversity of influence relations patterns and rich information of vertexes. To better reveal the latent patterns and preserve local and global influence structure, this paper proposes a time sequence-based semi-supervised deep synergistic method, MR-iNE (Multi-Relationship Influence Network Embedding), which mines the relationships of multi-typed of entities to capture latent influence structure, and exploits self-fusion eigenmaps reinforcement representations for vertexes in heterogeneous networks. In MR-iNE, we learn the structural representation for relationships of entities by preserving first-order, second-order, and high-order proximity influence structure and mapping them to a low-dimensional space, where fuse information from the aligned semi-supervised component that maximize the likelihood of preserving both the global and local structure of the influence relations of vertexes to overcome data sparsity. Our experiment results show the effectiveness of MR-iNE outperforms the state-of-the-art algorithms for learning rich representations on multi-label classification in real datasets from diverse domains.

Network representation learning: an overview

Networks are important ways of representing objects and their relationships. A key problem in the study of networks is how to represent the network information properly. With the developments in machine learning, feature learning of network vertices has become an important area of study. Network representation learning algorithms turn network information into dense, low-dimensional real-valued vectors that can be used as inputs for existing machine learning algorithms. For example, the representation of vertices can be fed to a classifier such as a Support Vector Machine (SVM) for vertex classification. In addition, the representations can be used for visualization by taking the representations as points in a Euclidean space. The study of network representation learning has attracted the attention of many researchers. In this article, recent works on network representation learning are introduced and summarized.

Boundary modeling in model-based calibration for automotive engines via the vertex representation of the convex hulls

When using the convex hull approach in the boundary modeling process, Model-Based Calibration (MBC) software suites – such as Model-Based Calibration Toolbox from MathWorks – can be computationally intensive depending on the amount of data modeled. The reason for this is that the half-space representation of the convex hull is used. We discuss here another representation of the convex hull, the vertex representation, which proves capable to reduce the computational cost. Numerical comparisons in this article are executed in MATLAB by using MBC Toolbox commands, and show that for certain conditions, the vertex representation outperforms the half-space representation.

H-Representation of the Kimura-3 Polytope for the $m$-Claw Tree

Given a group-based Markov model on a tree, one can compute the vertex representation of a polytope describing a toric variety associated with the algebraic statistical model. In the cases of ${\mathbb Z}_2$ and ${\mathbb Z}_2\times{\mathbb Z}_2$, these polytopes have applications in the field of phylogenetics. We provide a half-space representation of the polytope for the $m$-claw tree where $G={\mathbb Z}_2\times{\mathbb Z}_2$. This choice of group corresponds to the Kimura-3 model of evolution.

Direct point-based registration for precise non-rigid surface matching using Student’s-t mixture model

Abstract One of the main challenges in the non-rigid surface matching is to match complex surfaces with absence of salient landmarks (marker-less) and salient structures (structure-less). We propose an accurate non-rigid surface registration method, called DSMM, to match complex surfaces based on a dense point-to-point correspondence alignment. The key idea of our approach is to model the correspondences on surfaces by using Student’s-t mixture model and represent local spatial structures via Dirichlet distribution and the directional springs. Firstly, we formulate the problem of alignment of two point sets as a probability density estimate, modeling one set as Student’s-t mixture model centroids, and the other one as observation data. We subsequently incorporate spatial representations of vertices on the surfaces into the prior probability of the finite Student’s-t mixture model by exploiting the Dirichlet distribution and Dirichlet law. We later explicitly add an additional structure regularization to get an approximate isometric and near-conformal transformation. Finally, we obtain closed-form solutions of registration parameters using Expectation Maximization (EM) framework, leading to a computationally efficient registration method. We compare DSMM with other state-of-the-art direct point-based non-rigid surface matching methods based on finite mixture models on artificial shapes with large deformation and real complex shapes from various segmented brain structures. DSMM demonstrates its statistical accuracy and robustness, outperforming the competing

Bloom Filter Trie: an alignment-free and reference-free data structure for pan-genome storage.

BackgroundHigh throughput sequencing technologies have become fast and cheap in the past years. As a result, large-scale projects started to sequence tens to several thousands of genomes per species, producing a high number of sequences sampled from each genome. Such a highly redundant collection of very similar sequences is called a pan-genome. It can be transformed into a set of sequences “colored” by the genomes to which they belong. A colored de Bruijn graph (C-DBG) extracts from the sequences all colored k-mers, strings of length k, and stores them in vertices.ResultsIn this paper, we present an alignment-free, reference-free and incremental data structure for storing a pan-genome as a C-DBG: the bloom filter trie (BFT). The data structure allows to store and compress a set of colored k-mers, and also to efficiently traverse the graph. Bloom filter trie was used to index and query different pangenome datasets. Compared to another state-of-the-art data structure, BFT was up to two times faster to build while using about the same amount of main memory. For querying k-mers, BFT was about 52–66 times faster while using about 5.5–14.3 times less memory.ConclusionWe present a novel succinct data structure called the Bloom Filter Trie for indexing a pan-genome as a colored de Bruijn graph. The trie stores k-mers and their colors based on a new representation of vertices that compress and index shared substrings. Vertices use basic data structures for lightweight substrings storage as well as Bloom filters for efficient trie and graph traversals. Experimental results prove better performance compared to another state-of-the-art data structure.Availabilityhttps://www.github.com/GuillaumeHolley/BloomFilterTrie.

Using 2-Opt based evolution strategy for travelling salesman problem

Harmony search algorithm that matches the (µ+1) evolution strategy, is a heuristic method simulated by the process of music improvisation. In this paper, a harmony search algorithm is directly used for the travelling salesman problem. Instead of conventional selection operators such as roulette wheel, the transformation of real number values of harmony search algorithm to order index of vertex representation and improvement of solutions are obtained by using the 2-Opt local search algorithm. Then, the obtained algorithm is tested on two different parameter groups of TSPLIB. The proposed method is compared with classical 2-Opt which randomly started at each step and best known solutions of test instances from TSPLIB. It is seen that the proposed algorithm offers valuable solutions.

Constrained zonotopes: A new tool for set-based estimation and fault detection

This article introduces a new class of sets, called constrained zonotopes, that can be used to enclose sets of interest for estimation and control. The numerical representation of these sets is sufficient to describe arbitrary convex polytopes when the complexity of the representation is not limited. At the same time, this representation permits the computation of exact projections, intersections, and Minkowski sums using very simple identities. Efficient and accurate methods for computing an enclosure of one constrained zonotope by another of lower complexity are provided. The advantages and disadvantages of these sets are discussed in comparison to ellipsoids, parallelotopes, zonotopes, and convex polytopes in halfspace and vertex representations. Moreover, extensive numerical comparisons demonstrate significant advantages over other classes of sets in the context of set-based state estimation and fault detection.

Deep Neural Networks for Learning Graph Representations

In this paper, we propose a novel model for learning graph representations, which generates a low-dimensional vector representation for each vertex by capturing the graph structural information. Different from other previous research efforts, we adopt a random surfing model to capture graph structural information directly, instead of using the sampling-based method for generating linear sequences proposed by Perozzi et al. (2014). The advantages of our approach will be illustrated from both theorical and empirical perspectives. We also give a new perspective for the matrix factorization method proposed by Levy and Goldberg (2014), in which the pointwise mutual information (PMI) matrix is considered as an analytical solution to the objective function of the skip-gram model with negative sampling proposed by Mikolov et al. (2013). Unlike their approach which involves the use of the SVD for finding the low-dimensitonal projections from the PMI matrix, however, the stacked denoising autoencoder is introduced in our model to extract complex features and model non-linearities. To demonstrate the effectiveness of our model, we conduct experiments on clustering and visualization tasks, employing the learned vertex representations as features. Empirical results on datasets of varying sizes show that our model outperforms other stat-of-the-art models in such tasks.

Learning of Multimodal Representations With Random Walks on the Click Graph.

In multimedia information retrieval, most classic approaches tend to represent different modalities of media in the same feature space. With the click data collected from the users' searching behavior, existing approaches take either one-to-one paired data (text-image pairs) or ranking examples (text-query-image and/or image-query-text ranking lists) as training examples, which do not make full use of the click data, particularly the implicit connections among the data objects. In this paper, we treat the click data as a large click graph, in which vertices are images/text queries and edges indicate the clicks between an image and a query. We consider learning a multimodal representation from the perspective of encoding the explicit/implicit relevance relationship between the vertices in the click graph. By minimizing both the truncated random walk loss as well as the distance between the learned representation of vertices and their corresponding deep neural network output, the proposed model which is named multimodal random walk neural network (MRW-NN) can be applied to not only learn robust representation of the existing multimodal data in the click graph, but also deal with the unseen queries and images to support cross-modal retrieval. We evaluate the latent representation learned by MRW-NN on a public large-scale click log data set Clickture and further show that MRW-NN achieves much better cross-modal retrieval performance on the unseen queries/images than the other state-of-the-art methods.

Efficient edge-skeleton computation for polytopes defined by oracles

In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify problems and polytope representations for which total polynomial-time algorithms can be obtained. We offer the first total polynomial-time algorithm for computing the edge-skeleton—including vertex enumeration—of a polytope given by an optimization or separation oracle, where we are also given a superset of its edge directions. We also offer a space-efficient variant of our algorithm by employing reverse search. All complexity bounds refer to the (oracle) Turing machine model. There is a number of polytope classes naturally defined by oracles; for some of them neither vertex nor facet representation is obvious. We consider two main applications, where we obtain (weakly) total polynomial-time algorithms: Signed Minkowski sums of convex polytopes, where polytopes can be subtracted provided the signed sum is a convex polytope, and computation of secondary, resultant, and discriminant polytopes. Further applications include convex combinatorial optimization and convex integer programming, where we offer a new approach, thus removing the complexity's exponential dependence in the dimension.

Vertex representations of toroidal special linear Lie superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, free field representation of toroidal Lie superalgebras of type $A(m, n)$ is constructed using both vertex operators and bosonic fields.

Two-parameter quantum affine algebra of type G2(1), Drinfeld realization and vertex representation

In this paper, we define the two-parameter quantum affine algebra for type G2(1) and give the (r, s)-Drinfeld realization of Ur,s(G2(1)), as well as establish and prove its Drinfeld isomorphism. We construct and verify explicitly the level-one vertex representation of two-parameter quantum affine algebra Ur,s(G2(1)), which also supports an evidence in nontwisted type G2(1) for the uniform defining approach via the two-parameter τ-invariant generating functions proposed in Hu and Zhang [Generating functions with τ-invariance and vertex representations of two-parameter quantum affine algebras Ur,s(ĝ): Simply laced cases e-print arXiv:1401.4925]

Vertex Operator Representations of Type C l (1) and Product-Sum Identities

We construct a class of homogeneous vertex representations of C (1) , l ≥ 2, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory.

Beyond rainbow-ladder in bound state equations

In this work we devise a new method to study quark-anti-quark interactions beyond simple ladder-exchange that yield massless pions in the chiral limit. The method is based on the requirement to have a representation of the quark-gluon vertex that is explicitly given in terms of quark dressings functions. We outline a general procedure to generate the Bethe-Salpeter kernel for a given vertex representation. Our method allows not only the identification of the mesons' masses but also the extraction of their Bethe-Salpeter wave functions exposing their internal structure. We exemplify our method with vertex models that are of phenomenological interest.

Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.

Bounding bubbles: The vertex representation of 3d group field theory and the suppression of pseudomanifolds

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the group field theory (GFT) fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudomanifold configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.

On the stability of Archimedean tilings formed by patchy particles

We have investigated the possibility of decorating, using a bottom-up strategy, patchy particles in such a way that they self-assemble in (two-dimensional) Archimedean tilings. Except for the trihexagonal tiling, we have identified conditions under which this is indeed possible. The more compact tilings, i.e., the elongated triangular and the snub square tilings (which are built up by triangles and squares only) are found to be stable up to intermediate pressure values in the vertex representation, i.e., where the tiling is decorated with particles at its vertices. The other tilings, which are built up by rather large hexagons, octagons and dodecagons, are stable over a relatively large pressure range in the centre representation where the particles occupy the centres of the polygonal units.