Efficient Õ (n/ϵ) spectral sketches for the Laplacian and its pseudo inverse

Abstract

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)xAx ϵ f(x) ϵ (1+ ϵ)xAx 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 2 Rn ϵn 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 eO (n/ϵ1:6) for sketching the Laplacian form by [1] and O(n/ϵ2) for sketching the Laplacian pseudoinverse by [2]. Furthermore we show how to compute all-pairs effective resistances from our O(n/ϵ) size sketch in eO (n2/ϵ) time. This improves upon the previous best running time of O (n2/ϵ2) by [3].