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### The Gram Dimension of a Graph [chapter]

Monique Laurent, Antonios Varvitsiotis
2012 Lecture Notes in Computer Science
The Gram dimension gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in R k  ...  We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 6.  ...

### The Gram dimension of a graph [article]

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
The Gram dimension (G) of a graph is the smallest integer k > 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ^k, having  ...  We show that a graph has Gram dimension at most 4 if and only if it does not have K_5 and K_2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ...

### Page 491 of Science Education Vol. 16, Issue 6 [page]

1932 Science Education
Plot a graph and continue the line until it reads at least 1000 grams along the gram- axis. Now read the value in pounds.  ...  : (a) Two dimensions to obtain areas, in each system.  ...

### Gram Spectrahedra of Ternary Quartics [article]

Julian Vill
2021 arXiv   pre-print
We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1.  ...  The Gram spectrahedron of a real form f∈ℝ[x]_2d parametrizes all sum of squares representations of f.  ...  We consider the 3-dimensional affine slice of Gram(f ) spanned by the four rank 3 Gram tensors in one K 4 of the Steiner graph of f .  ...

### On Representations of Graphs as Two-Distance Sets [article]

A. Y. Alfakih
2018 arXiv   pre-print
, the adjacency matrix of the complement graph G̅.  ...  A Euclidean representation of a simple graph G in R^r is a mapping of the nodes of G into points in R^r such that the squared Euclidean distance between any two points is a if the corresponding nodes are  ...  Let D be an n × n EDM of embedding dimension r ≤ n − 2 and let X be the projected Gram matrix of D. Further, let U be the matrix whose columns form an orthonormal basis of the null space of X.  ...

### The Algorithm of Automatic Text Summarization Based on Network Representation Learning [chapter]

Xinghao Song, Chunming Yang, Hui Zhang, Xujian Zhao
2018 Lecture Notes in Computer Science
The sentences graph was construed by TF-IDF, and controlled the number of edges by a threshold. The Node2Vec is used to embedding the graph, and the sentences were clustered by k-means.  ...  Finally, the Modularity is used to control the number of clusters, and generating a brief summary of the document.  ...  This work is supported by the Ministry of education of Humanities and Social Science project (17YJCZH260), the Next Generation Internet Technology Innovation Project (NGII20170901), the Fund of Fundamental  ...

### N-Gram Graph: Simple Unsupervised Representation for Graphs, with Applications to Molecules [article]

Shengchao Liu, Mehmet Furkan Demirel, Yingyu Liang
2019 arXiv   pre-print
This paper introduces the N-gram graph, a simple unsupervised representation for molecules. The method first embeds the vertices in the molecule graph.  ...  It then constructs a compact representation for the graph by assembling the vertex embeddings in short walks in the graph, which we show is equivalent to a simple graph neural network that needs no training  ...  The authors would also like to acknowledge computing resources from the University of Wisconsin-Madison Center for High Throughput Computing and support provided by the University of Wisconsin-Madison  ...

### Unsupervised Learning of Image Manifolds by Semidefinite Programming

Kilian Q. Weinberger, Lawrence K. Saul
2006 International Journal of Computer Vision
Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition.  ...  In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be used to analyze high dimensional data that lies on or near a low dimensional manifold.  ...  rotation by the elements of the Gram matrix) lie in a subspace of dimensionality d < D, or at least approximately lie in such a subspace?  ...

### A new graph parameter related to bounded rank positive semidefinite matrix completions

Monique Laurent, Antonios Varvitsiotis
2013 Mathematical programming
The Gram dimension gd(G) of a graph G is the smallest integer k ≥ 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding  ...  For any fixed k the class of graphs satisfying gd(G) ≤ k is minor closed, hence it can characterized by a finite list of forbidden minors.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ...

### A new graph parameter related to bounded rank positive semidefinite matrix completions [article]

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
The Gram dimension (G) of a graph G is the smallest integer k> 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding  ...  For any fixed k the class of graphs satisfying (G) < k is minor closed, hence it can characterized by a finite list of forbidden minors.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ...

### Uniqueness of Low-Rank Matrix Completion by Rigidity Theory [article]

Amit Singer, Mihai Cucuringu
2009 arXiv   pre-print
A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances.  ...  The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative  ...  It is a pleasure to thank Haim Avron and Sivan Toledo for many valuable suggestions and interesting discussions on finding the null space of sparse matrices, sparse factorizations and iterative methods  ...

### Exploring Function Call Graph Vectorization and File Statistical Features in Malicious PE File Classification

Yipin Zhang, Xiaolin Chang, Yuzhou Lin, Jelena Misic, Vojislav B. Misic
2020 IEEE Access
For the purpose of resisting a large scale of malware variants, machine learning (ML) classifiers for malicious Portable Executable (PE) files have been proposed to achieve automated classification.  ...  This paper aims to further improve the classification accuracy of FCGV-based ML model by applying both graph and non-graph features.  ...  Hence, it can effectively reduce the dimensionality of high-dimension data.  ...

### Fast and Accurate Network Embeddings via Very Sparse Random Projection [article]

Haochen Chen, Syed Fahad Sultan, Yingtao Tian, Muhao Chen, Steven Skiena
2019 arXiv   pre-print
We show that the success of these methods should be attributed to the proper construction of this similarity matrix, rather than the dimension reduction method employed.  ...  We present FastRP, a scalable and performant algorithm for learning distributed node representations in a graph.  ...  Notation and Task Definition We consider the problem of embedding a network: given an undirected graph, the goal is to learn a low-dimensional latent representation for each node in the graph 2 .  ...

### Uniqueness of Low-Rank Matrix Completion by Rigidity Theory

Amit Singer, Mihai Cucuringu
2010 SIAM Journal on Matrix Analysis and Applications
A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances.  ...  The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative  ...  It is a pleasure to thank Haim Avron and Sivan Toledo for many valuable suggestions and interesting discussions on finding the null space of sparse matrices, sparse factorizations and iterative methods  ...

### KONG: Kernels for ordered-neighborhood graphs [article]

Moez Draief, Konstantin Kutzkov, Kevin Scaman, Milan Vojnovic
2018 arXiv   pre-print
For the special case of general graphs, i.e. graphs without ordered neighborhoods, the new graph kernels yield efficient and simple algorithms for the comparison of label distributions between graphs.  ...  Graphs with ordered neighborhoods are a natural data representation for evolving graphs where edges are created over time, which induces an order.  ...  Using the key property of the incremental string generation approach and a sketching algorithm, which is a linear projection of the original data onto a lower-dimensional space, we will show how to sketch  ...
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