Efficient O(n/ε) Spectral Sketches for the Laplacian and its Pseudoinverse
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In this paper we consider the problem of efficiently computing ϵ-sketches for the Laplacian and its pseudoinverse. Given a Laplacian and an error tolerance ϵ, we seek to construct a function f such that for any vector x (chosen obliviously from f), with high probability (1-ϵ) x^ A x ≤ f(x) ≤ (1 + ϵ) x^ A x where A is either the Laplacian or its pseudoinverse. Our goal is to construct such a sketch f efficiently and to store it in the least space possible. We provide nearly-linear time algorithms that, when given a Laplacian matrix L∈R^n × n and an error tolerance ϵ, produce Õ(n/ϵ)-size sketches of both L and its pseudoinverse. Our algorithms improve upon the previous best sketch size of O(n / ϵ^1.6) for sketching the Laplacian form by Andoni et al (2015) and O(n / ϵ^2) for sketching the Laplacian pseudoinverse by Batson, Spielman, and Srivastava (2008). Furthermore we show how to compute all-pairs effective resistances from O(n/ϵ) size sketch in O(n^2/ϵ) time. This improves upon the previous best running time of O(n^2/ϵ^2) by Spielman and Srivastava (2008).
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