A matrix hyperbolic cosine algorithm and applications
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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In this thesis we prove the following two basic statements in linear algebra. Let B be an arbitrary n × m in matrix where m ≥ n and suppose 0 (1) Spectral Sparsification . There is a diagonal matrix Sm×m⪰0 with at most &ceill0; n/ε2 &ceilr0; nonzero entries for which 1-e2BB T⪯BSBT⪯1+e 2BBT . Thus the spectral behavior of BBT is captured by a weighted subset of the columns of B, of size proportional to its rank n. (2) Restricted Invertibility. There is a diagonal matrix Sm×m with at least 1-e2 B2 FB 22 nonzero entries, all equal to 1, for which BSBT⪰e2 B2F mI. Thus there is a large coordinate restriction of B (i.e., a submatrix of its columns, given by S), of size proportional to its numerical rank B 2FB 22 , which is well-invertible. This significantly improves a theorem of Bourgain and Tzafriri [14]. We give deterministic polynomial time algorithms for constructing the promised diagonal matrices S in time O(mn3/ε 2) and O((1 – ε)2 mn3), respectively.By applying (1) to the class of Laplacian matrices of graphs, we show that every graph on n vertices can be spectrally approximated by a weighted graph with O(n) edges, thus generalizing the concept of expander graphs, which are constant-degree approximations of the complete graph. We then present a second graph sparsification algorithm based on random sampling, which produces weaker sparsifiers with O(n log n) edges but runs in nearly-linear time. We also prove a refinement of (1) for the special case of B arising from John's decompositions of convex bodies, which allows us to show that every convex body can be approximated by one which has very few contact points with its minimum volume ellipsoid.