Sublinear estimation of a single element in sparse linear systems

Abstract

We present a fast bidirectional algorithm for estimating a single element of the product of a matrix power and vector. This is an important primitive in many applications; in particular, we describe how it can be used to estimate a single element in the solution of a linear system Ax = b, with sublinear average-case running time guarantees for sparse systems. Our work combines the von Neumann-Ulam MCMC scheme for matrix multiplication with recent developments in bidirectional algorithms for estimating random-walk metrics. In particular, given a target additive-error threshold, we show how to combine a reverse local-variational technique with forward MCMC sampling, such that the resulting algorithm is order-wise faster than each individual approach.