287 research outputs found
Embedding Graphs under Centrality Constraints for Network Visualization
Visual rendering of graphs is a key task in the mapping of complex network
data. Although most graph drawing algorithms emphasize aesthetic appeal,
certain applications such as travel-time maps place more importance on
visualization of structural network properties. The present paper advocates two
graph embedding approaches with centrality considerations to comply with node
hierarchy. The problem is formulated first as one of constrained
multi-dimensional scaling (MDS), and it is solved via block coordinate descent
iterations with successive approximations and guaranteed convergence to a KKT
point. In addition, a regularization term enforcing graph smoothness is
incorporated with the goal of reducing edge crossings. A second approach
leverages the locally-linear embedding (LLE) algorithm which assumes that the
graph encodes data sampled from a low-dimensional manifold. Closed-form
solutions to the resulting centrality-constrained optimization problems are
determined yielding meaningful embeddings. Experimental results demonstrate the
efficacy of both approaches, especially for visualizing large networks on the
order of thousands of nodes.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphic
Fast Graph Laplacian regularized kernel learning via semidefinite-quadratic-linear programming.
Wu, Xiaoming.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 30-34).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.ivChapter 1 --- Introduction --- p.1Chapter 2 --- Preliminaries --- p.4Chapter 2.1 --- Kernel Learning Theory --- p.4Chapter 2.1.1 --- Positive Semidefinite Kernel --- p.4Chapter 2.1.2 --- The Reproducing Kernel Map --- p.6Chapter 2.1.3 --- Kernel Tricks --- p.7Chapter 2.2 --- Spectral Graph Theory --- p.8Chapter 2.2.1 --- Graph Laplacian --- p.8Chapter 2.2.2 --- Eigenvectors of Graph Laplacian --- p.9Chapter 2.3 --- Convex Optimization --- p.10Chapter 2.3.1 --- From Linear to Conic Programming --- p.11Chapter 2.3.2 --- Second-Order Cone Programming --- p.12Chapter 2.3.3 --- Semidefinite Programming --- p.12Chapter 3 --- Fast Graph Laplacian Regularized Kernel Learning --- p.14Chapter 3.1 --- The Problems --- p.14Chapter 3.1.1 --- MVU --- p.16Chapter 3.1.2 --- PCP --- p.17Chapter 3.1.3 --- Low-Rank Approximation: from SDP to QSDP --- p.18Chapter 3.2 --- Previous Approach: from QSDP to SDP --- p.20Chapter 3.3 --- Our Formulation: from QSDP to SQLP --- p.21Chapter 3.4 --- Experimental Results --- p.23Chapter 3.4.1 --- The Results --- p.25Chapter 4 --- Conclusion --- p.28Bibliography --- p.3
Modeling outcomes of soccer matches
We compare various extensions of the Bradley-Terry model and a hierarchical
Poisson log-linear model in terms of their performance in predicting the
outcome of soccer matches (win, draw, or loss). The parameters of the
Bradley-Terry extensions are estimated by maximizing the log-likelihood, or an
appropriately penalized version of it, while the posterior densities of the
parameters of the hierarchical Poisson log-linear model are approximated using
integrated nested Laplace approximations. The prediction performance of the
various modeling approaches is assessed using a novel, context-specific
framework for temporal validation that is found to deliver accurate estimates
of the test error. The direct modeling of outcomes via the various
Bradley-Terry extensions and the modeling of match scores using the
hierarchical Poisson log-linear model demonstrate similar behavior in terms of
predictive performance
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Graph Embedding and Nonlinear Dimensionality Reduction
Traditionally, spectral methods such as principal component analysis (PCA) have been applied to many graph embedding and dimensionality reduction tasks. These methods aim to find low-dimensional representations of data that preserve its inherent structure. However, these methods often perform poorly when applied to data which does not lie exactly near a linear manifold. In this thesis, I present a set of novel graph embedding algorithms which extend spectral methods, allowing graph representations of high-dimensional data or networks to be accurately embedded in a low-dimensional space. I first propose minimum volume embedding (MVE) which, like other leading dimensionality reduction algorithms, first encodes the high-dimensional data as a nearest-neighbor graph, where the edge weights between neighbors correspond to kernel values between points, and then embeds this graph in a low-dimensional space. Next I present structure preserving embedding (SPE), an algorithm for embedding unweighted graphs where similarity between nodes is not known. SPE finds low-dimensional embeddings which explicitly preserve graph topology, meaning a connectivity algorithm, such as k-nearest neighbors, will recover the edges of the input graph from only the coordinates of the nodes after embedding. I further explore preserving graph structure during embedding, and find the concept applicable to dimensionality reduction, large-scale network visualization, and metric learning for link prediction. This thesis posits that simply preserving pairwise distances, as with many spectral methods, is insufficient for capturing the structure of many datasets and that preserving both local distances and graph topology is crucial for producing accurate low-dimensional representations of networks and high-dimensional data
Data-Driven Shape Analysis and Processing
Data-driven methods play an increasingly important role in discovering
geometric, structural, and semantic relationships between 3D shapes in
collections, and applying this analysis to support intelligent modeling,
editing, and visualization of geometric data. In contrast to traditional
approaches, a key feature of data-driven approaches is that they aggregate
information from a collection of shapes to improve the analysis and processing
of individual shapes. In addition, they are able to learn models that reason
about properties and relationships of shapes without relying on hard-coded
rules or explicitly programmed instructions. We provide an overview of the main
concepts and components of these techniques, and discuss their application to
shape classification, segmentation, matching, reconstruction, modeling and
exploration, as well as scene analysis and synthesis, through reviewing the
literature and relating the existing works with both qualitative and numerical
comparisons. We conclude our report with ideas that can inspire future research
in data-driven shape analysis and processing.Comment: 10 pages, 19 figure
kLog: A Language for Logical and Relational Learning with Kernels
We introduce kLog, a novel approach to statistical relational learning.
Unlike standard approaches, kLog does not represent a probability distribution
directly. It is rather a language to perform kernel-based learning on
expressive logical and relational representations. kLog allows users to specify
learning problems declaratively. It builds on simple but powerful concepts:
learning from interpretations, entity/relationship data modeling, logic
programming, and deductive databases. Access by the kernel to the rich
representation is mediated by a technique we call graphicalization: the
relational representation is first transformed into a graph --- in particular,
a grounded entity/relationship diagram. Subsequently, a choice of graph kernel
defines the feature space. kLog supports mixed numerical and symbolic data, as
well as background knowledge in the form of Prolog or Datalog programs as in
inductive logic programming systems. The kLog framework can be applied to
tackle the same range of tasks that has made statistical relational learning so
popular, including classification, regression, multitask learning, and
collective classification. We also report about empirical comparisons, showing
that kLog can be either more accurate, or much faster at the same level of
accuracy, than Tilde and Alchemy. kLog is GPLv3 licensed and is available at
http://klog.dinfo.unifi.it along with tutorials
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