4,000 research outputs found
Second order accurate distributed eigenvector computation for extremely large matrices
We propose a second-order accurate method to estimate the eigenvectors of
extremely large matrices thereby addressing a problem of relevance to
statisticians working in the analysis of very large datasets. More
specifically, we show that averaging eigenvectors of randomly subsampled
matrices efficiently approximates the true eigenvectors of the original matrix
under certain conditions on the incoherence of the spectral decomposition. This
incoherence assumption is typically milder than those made in matrix completion
and allows eigenvectors to be sparse. We discuss applications to spectral
methods in dimensionality reduction and information retrieval.Comment: Complete proofs are included on averaging performanc
On Imprisoned Curves and b-length in General Relativity
This paper is concerned with two themes: imprisoned curves and the b-length
functional. In an earlier paper by the author, it was claimed that an endless
incomplete curve partially imprisoned in a compact set admits an endless null
geodesic cluster curve. Unfortunately, the proof was flawed. We give an outline
of the problem and remedy the situation by providing a proof by different
methods. Next, we obtain some results concerning the structure of b-length
neighbourhoods, which gives a clue to how the geometry of a spacetime is
encoded in the pseudo-orthonormal frame bundle equipped with the b-metric. We
also show that a previous result by the author, proving total degeneracy of a
b-boundary fibre in some cases, does not apply to imprisoned curves. Finally,
we correct some results in the literature linking the b-lengths of general
curves in the frame bundle with the b-length of the corresponding horizontal
curves.Comment: 26 pages, 7 figures, LaTeX 2e with AMSLaTeX 1.2 and AMSFonts,
submitted to J. Math. Phy
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