1,304 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard
A study of the quantum triangular billiard requires consideration of a
boundary value problem for the Green's function of the Laplacian on a trianglar
domain. Our main result is a reformulation of this problem in terms of coupled
non--singular integral equations. A non--singular formulation, via Fredholm's
theory, guarantees uniqueness and provides a mathematically firm foundation for
both numerical and analytic studies. We compare and contrast our reformulation,
based on the exact solution for the wedge, with the standard singular integral
equations using numerical discretisation techniques. We consider in detail the
(integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our
non--singular formulation produces results which are well behaved
mathematically. In contrast, while resolving the eigenvalues very well, the
standard approach displays various behaviours demonstrating the need for some
sort of ``renormalisation''. The non-singular formulation provides a
mathematically firm basis for the generation and analysis of periodic orbit
expansions. We discuss their convergence paying particular emphasis to the
computational effort required in comparision with Einstein--Brillouin--Keller
quantisation and the standard discretisation, which is analogous to the method
of Bogomolny. We also discuss the generalisation of our technique to smooth,
chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and
subfigure. 17 figures. To appear Annals of Physics, southern sprin
Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems
Development of robust dynamical systems and networks such as autonomous
aircraft systems capable of accomplishing complex missions faces challenges due
to the dynamically evolving uncertainties coming from model uncertainties,
necessity to operate in a hostile cluttered urban environment, and the
distributed and dynamic nature of the communication and computation resources.
Model-based robust design is difficult because of the complexity of the hybrid
dynamic models including continuous vehicle dynamics, the discrete models of
computations and communications, and the size of the problem. We will overview
recent advances in methodology and tools to model, analyze, and design robust
autonomous aerospace systems operating in uncertain environment, with stress on
efficient uncertainty quantification and robust design using the case studies
of the mission including model-based target tracking and search, and trajectory
planning in uncertain urban environment. To show that the methodology is
generally applicable to uncertain dynamical systems, we will also show examples
of application of the new methods to efficient uncertainty quantification of
energy usage in buildings, and stability assessment of interconnected power
networks
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