Sparse factorization of the square all-ones matrix of arbitrary order

In this paper, we study sparse factorization of the (scaled) square all-ones matrix $J$ of arbitrary order. We introduce the concept of hierarchically banded matrices and propose two types of hierarchically banded factorization of $J$: the reduced hierarchically banded (RHB) factorization and the doubly stochastic hierarchically banded (DSHB) factorization. Based on the DSHB factorization, we propose the sequential doubly stochastic (SDS) factorization, in which~$J$ is decomposed as a product of sparse, doubly stochastic matrices. Finally, we discuss the application of the proposed sparse factorizations to the decentralized average consensus problem and decentralized optimization.