American Mathematical Society

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Eigenvectors in Graph Theory and Related Problems in Numerical Linear Algebra

Month: May 2014

Date: May 5--9

Name: Eigenvectors in Graph Theory and Related Problems in Numerical Linear Algebra

Location: Brown University, Providence, Rhode Island.


The analysis of problems modeled by large graphs is greatly hampered by a lack of efficient computational tools. The purpose of the workshop is to explore possibilities for designing appropriate computational methods that draw on recent advances in numerical methods and scientific computation. Specifically, the questions of how to form the matrices representing graph Laplacians, and how to compute the leading eigenvectors of such matrices will be addressed. It seems likely that these problems will be amenable to algorithms based on randomized projections that dramatically reduce the effective dimensionality of the underlying problems. Such techniques have recently proven highly effective for the related problems of how to find approximate lists of nearest neighbors for clouds of points in high dimensional spaces, and for constructing approximate low-rank factorizations of large matrices. In both cases, a key observation is that the problem of distortions of distances that is inherent to randomized projection techniques can be overcome by using the randomized projections only as pre-conditioners; they inform the algorithm of where to look, and then highly accurate deterministic techniques are used to compute the actual output. The resulting algorithms scale extra-ordinarily well on modern parallel and multicore architectures. To successfully address the enormous problems arising in the analysis of graphs, it is expected that additional machinery will be needed, such as the use of multi-resolution data structures, and more efficient scalable randomized projections.