Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers
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In this paper we provide an O(m (loglog n)^O(1)log(1/ϵ))-expected time algorithm for solving Laplacian systems on n-node m-edge graphs, improving improving upon the previous best expected runtime of O(m √(log n) (loglog n)^O(1)log(1/ϵ)) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of ℓ_p-stretch graph approximations with improved stretch and sparsity bounds. Additionally, as motivation for this work, we show that for every set of vectors in ℝ^d (not just those induced by graphs) and all k > 0 there exist an ultra-sparsifiers with d-1 + O(d/k) re-weighted vectors of relative condition number at most k^2. For small k, this improves upon the previous best known multiplicative factor of k ·Õ(log d), which is only known for the graph case.
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