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7.2.3 Tensor problems Tensors are generalizations of indexed data, e.g. a matrix, with more than two indices. Modern tensor theory and applications of tensor analysis are quickly evolving fields, see Kolda and Bader [2009] for a recent survey. One application synthesizing tensors and PageRank is the sports ranking problem from Govan et al. [2008]. They form a linear combination of multiple frames of a tensor: (I−α1P1 −α2P2 −⋯−αkPk)x=(1−∑αj)v. Using ideas from Constantine [2009], we could analyze this equation as a multi-variate parameterized matrix problem. Similar techniques will apply to other types of “tensor-frame” problems. We would have liked to experiment with these ideas, but there is only so much time. Perhaps we will be able to explore these techniques in the future. 7.2.4 gaimc performance We have not discussed the software chapter at all in the current chapter. That is not to imply that it is not important, but rather the conclusions of the thesis are about PageRank and not software. We would like to conclude that using the Matlab code in gaimc is nearly as fast as using the complicated MatlabBGL package. Our current perfor- mance results indicate that this occurs for a few of the functions, but not for others. The next step on the way to this conclusion is to track down these perfor- mance discrepancies and understand why they are unavoidable or produce a solution. 7.2.5 Inner-Outer extensions We ended the chapter on the inner-outer iteration with a conjecture about its asymptotic performance. Namely, the asymptotic convergence rate of the inner-outer algorithm seems to be αλ2, where λ2 is the first eigenvalue with ∣λ2∣ < 1, even though it runs iterations of the power method, which 3 0000 converge at rate α. As written, the conjecture is false. For many large graphs, it appears to be a close approximation of what happens. We hope to formalize this conjecture and establish when and why it holds. Rigorously proving such a conjecture is incredibly important because we are not aware of any other iterative algorithm that always computes PageRank faster than the power method with similar memory requirements. With the inner-outer iteration, it seems possible. 3 Using P = [ 1 0 0 1 ] produces a coun- 0100 7.2 ⋅ future work 149 terexample. 0 0 1 0PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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