Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds
Abstract
We study the following generalized matrix rank estimation problem: given an $n \times n$ matrix and a constant $c \geq 0$, estimate the number of eigenvalues that are greater than $c$. In the distributed setting, the matrix of interest is the sum of $m$ matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate $\Omega(n^2)$ bits, which is orderequivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only $\widetilde O(n)$ bits. The upper bound is matched by an $\Omega(n)$ lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.01403
 Bibcode:
 2015arXiv150201403Z
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Statistics  Machine Learning
 EPrint:
 23 pages, 5 figures