Embedding complex networks in a low dimensional Euclidean space based on vertex dissimilarities

2012 ◽  
Vol 391 (20) ◽  
pp. 5018-5030 ◽  
Author(s):  
Chang-Yong Lee
Keyword(s):  
Euclidean Space ◽  
Complex Networks ◽  
Dimensional Euclidean Space ◽  
Low Dimensional

scholarly journals The Observable Representation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 310
Author(s):  
L. Schulman
Keyword(s):  
Phase Transition ◽  
Euclidean Space ◽  
Complex Systems ◽  
Random Walks ◽  
Stochastic Dynamics ◽  
Dimensional Euclidean Space ◽  
Significant Feature ◽  
Original Space ◽  
Definition Of ◽  
Low Dimensional

The observable representation (OR) is an embedding of the space on which a stochastic dynamics is taking place into a low dimensional Euclidean space. The most significant feature of the OR is that it respects the dynamics. Examples are given in several areas: the definition of a phase transition (including metastable phases), random walks in which the OR recovers the original space, complex systems, systems in which the number of extrema exceed convenient viewing capacity, and systems in which successful features are displayed, but without the support of known theorems.

scholarly journals Semantic Visualization for Short Texts with Word Embeddings

2017 ◽  
Author(s):  
Tuan M. V. Le ◽  
Hady W. Lauw
Keyword(s):  
Euclidean Space ◽  
Topic Modeling ◽  
Topic Models ◽  
Auxiliary Information ◽  
Dimensional Euclidean Space ◽  
Word Embeddings ◽  
Topic Distribution ◽  
Low Dimensional ◽  
News Headlines ◽  
Semantic Visualization

Semantic visualization integrates topic modeling and visualization, such that every document is associated with a topic distribution as well as visualization coordinates on a low-dimensional Euclidean space. We address the problem of semantic visualization for short texts. Such documents are increasingly common, including tweets, search snippets, news headlines, or status updates. Due to their short lengths, it is difficult to model semantics as the word co-occurrences in such a corpus are very sparse. Our approach is to incorporate auxiliary information, such as word embeddings from a larger corpus, to supplement the lack of co-occurrences. This requires the development of a novel semantic visualization model that seamlessly integrates visualization coordinates, topic distributions, and word vectors. We propose a model called GaussianSV, which outperforms pipelined baselines that derive topic models and visualization coordinates as disjoint steps, as well as semantic visualization baselines that do not consider word embeddings.

scholarly journals On Metric DBSCAN with Low Doubling Dimension

2020 ◽  
Author(s):  
Hu Ding ◽  
Fan Yang ◽  
Mingyue Wang
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Spatial Clustering ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Index Structures ◽  
Doubling Dimension ◽  
Density Based Clustering ◽  
Speed Up ◽  
Low Dimensional

The density based clustering method Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is a popular method for outlier recognition and has received tremendous attention from many different areas. A major issue of the original DBSCAN is that the time complexity could be as large as quadratic. Most of existing DBSCAN algorithms focus on developing efficient index structures to speed up the procedure in low-dimensional Euclidean space. However, the research of DBSCAN in high-dimensional Euclidean space or general metric spaces is still quite limited, to the best of our knowledge. In this paper, we consider the metric DBSCAN problem under the assumption that the inliers (excluding the outliers) have a low doubling dimension. We apply a novel randomized k-center clustering idea to reduce the complexity of range query, which is the most time consuming step in the whole DBSCAN procedure. Our proposed algorithms do not need to build any complicated data structures and are easy to implement in practice. The experimental results show that our algorithms can significantly outperform the existing DBSCAN algorithms in terms of running time.

scholarly journals HOMOTOPY GROUPS OF MODULI SPACES OF STABLE QUIVER REPRESENTATIONS

2010 ◽  
Vol 21 (09) ◽  
pp. 1219-1238
Author(s):  
GRAEME WILKIN
Keyword(s):  
Euclidean Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Homotopy Groups ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Fixed Dimension ◽  
Low Dimensional

The purpose of this paper is to describe a method for computing homotopy groups of the space of α-stable representations of a quiver with fixed dimension vector and stability parameter α. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low dimensional homotopy groups of the moduli space of α-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

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