650 research outputs found

    A Metric for genus-zero surfaces

    Full text link
    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

    Get PDF
    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

    Full text link
    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 3n−j(j>2)3n-j (j > 2) Symbols of the First and Second Kinds for SU(2) Group and Their Direct and Exact Calculation and Tabulation

    Full text link
    We show that general 3n−j(n>2)3n-j (n>2) 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 6−j6-j symbol. The resulting 3n−j3n-j 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 6−j6-j 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 3n−j3n-j symbols of the SU(2) group for all range of quantum angular momentum arguments. As an illustration, we present teh results for the 12−j12-j symbols of the first kind.Comment: Add tables and reference

    An extensive English language bibliography on graph theory and its applications

    Get PDF
    Bibliography on graph theory and its application
    • …
    corecore