650 research outputs found
A Metric for genus-zero surfaces
We present a new method to compare the shapes of genus-zero surfaces. We
introduce a measure of mutual stretching, the symmetric distortion energy, and
establish the existence of a conformal diffeomorphism between any two
genus-zero surfaces that minimizes this energy. We then prove that the energies
of the minimizing diffeomorphisms give a metric on the space of genus-zero
Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms
are shown to have properties that are highly desirable for applications.Comment: 33 pages, 8 figure
Making Thin Data Thick: User Behavior Analysis with Minimum Information
abstract: With the rise of social media, user-generated content has become available at an unprecedented scale. On Twitter, 1 billion tweets are posted every 5 days and on Facebook, 20 million links are shared every 20 minutes. These massive collections of user-generated content have introduced the human behavior's big-data.
This big data has brought about countless opportunities for analyzing human behavior at scale. However, is this data enough? Unfortunately, the data available at the individual-level is limited for most users. This limited individual-level data is often referred to as thin data. Hence, researchers face a big-data paradox, where this big-data is a large collection of mostly limited individual-level information. Researchers are often constrained to derive meaningful insights regarding online user behavior with this limited information. Simply put, they have to make thin data thick.
In this dissertation, how human behavior's thin data can be made thick is investigated. The chief objective of this dissertation is to demonstrate how traces of human behavior can be efficiently gleaned from the, often limited, individual-level information; hence, introducing an all-inclusive user behavior analysis methodology that considers social media users with different levels of information availability. To that end, the absolute minimum information in terms of both link or content data that is available for any social media user is determined. Utilizing only minimum information in different applications on social media such as prediction or recommendation tasks allows for solutions that are (1) generalizable to all social media users and that are (2) easy to implement. However, are applications that employ only minimum information as effective or comparable to applications that use more information?
In this dissertation, it is shown that common research challenges such as detecting malicious users or friend recommendation (i.e., link prediction) can be effectively performed using only minimum information. More importantly, it is demonstrated that unique user identification can be achieved using minimum information. Theoretical boundaries of unique user identification are obtained by introducing social signatures. Social signatures allow for user identification in any large-scale network on social media. The results on single-site user identification are generalized to multiple sites and it is shown how the same user can be uniquely identified across multiple sites using only minimum link or content information.
The findings in this dissertation allows finding the same user across multiple sites, which in turn has multiple implications. In particular, by identifying the same users across sites, (1) patterns that users exhibit across sites are identified, (2) how user behavior varies across sites is determined, and (3) activities that are observed only across sites are identified and studied.Dissertation/ThesisDoctoral Dissertation Computer Science 201
Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces
Knowledge graph embedding (KGE) is a increasingly popular technique that aims
to represent entities and relations of knowledge graphs into low-dimensional
semantic spaces for a wide spectrum of applications such as link prediction,
knowledge reasoning and knowledge completion. In this paper, we provide a
systematic review of existing KGE techniques based on representation spaces.
Particularly, we build a fine-grained classification to categorise the models
based on three mathematical perspectives of the representation spaces: (1)
Algebraic perspective, (2) Geometric perspective, and (3) Analytical
perspective. We introduce the rigorous definitions of fundamental mathematical
spaces before diving into KGE models and their mathematical properties. We
further discuss different KGE methods over the three categories, as well as
summarise how spatial advantages work over different embedding needs. By
collating the experimental results from downstream tasks, we also explore the
advantages of mathematical space in different scenarios and the reasons behind
them. We further state some promising research directions from a representation
space perspective, with which we hope to inspire researchers to design their
KGE models as well as their related applications with more consideration of
their mathematical space properties.Comment: 32 pages, 6 figure
Universal Factorization of Symbols of the First and Second Kinds for SU(2) Group and Their Direct and Exact Calculation and Tabulation
We show that general symbols of the first kind and the second
kind for the group SU(2) can be reformulated in terms of binomial coefficients.
The proof is based on the graphical technique established by Yutsis, et al. and
through a definition of a reduced symbol. The resulting symbols
thereby take a combinatorial form which is simply the product of two factors.
The one is an integer or polynomial which is the single sum over the products
of reduced symbols. They are in the form of summing over the products of
binomial coefficients. The other is a multiplication of all the triangle
relations appearing in the symbols, which can also be rewritten using binomial
coefficients. The new formulation indicates that the intrinsic structure for
the general recoupling coefficients is much nicer and simpler, which might
serves as a bridge for the study with other fields. Along with our newly
developed algorithms, this also provides a basis for a direct, exact and
efficient calculation or tabulation of all the symbols of the SU(2)
group for all range of quantum angular momentum arguments. As an illustration,
we present teh results for the symbols of the first kind.Comment: Add tables and reference
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
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