# Sublinear estimation of a single element in sparse linear systems

Nitin Shyamkumar, Siddhartha Banerjee, Peter Lofgren

October 2016
### 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.

Publication

*54th Annual Allerton Conference on Communication, Control, and Computing, 2016 *

###### Associate Professor

Sid Banerjee is an associate professor in the School of Operations Research at Cornell, working on topics at the intersection of data-driven decision-making, market design, and algorithms for large-scale networks.