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The Gram Dimension of a Graph
[chapter]

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]

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]

2021
*
arXiv
*
pre-print

We show that

arXiv:2112.10533v1
fatcat:2qmd73kn2vfbbn6s6k5xeflnoq
*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]

2018
*
arXiv
*
pre-print

*A̅*,

*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]

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]

2019
*
arXiv
*
pre-print

This paper introduces

arXiv:1806.09206v2
fatcat:yhvq2tlh2bdaxkgwgd3bycagaa
*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

2006
*
International Journal of Computer Vision
*

Can we detect low

doi:10.1007/s11263-005-4939-z
fatcat:jaefu7usv5g6pbl54jehhqofpy
*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

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]

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]

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

2020
*
IEEE Access
*

For

doi:10.1109/access.2020.2978335
fatcat:biiyb5qrijcsfce3cowfxnlsom
*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]

2019
*
arXiv
*
pre-print

We show that

arXiv:1908.11512v1
fatcat:j4dblin5wfabnffadp42dmt7bm
*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

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]

2018
*
arXiv
*
pre-print

For

arXiv:1805.10014v2
fatcat:k4s3bfox7resjilnmvhkvm3csq
*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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