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arXiv · 2401.14596

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

Xin Jiang·Edward Duc Hien Nguyen·César A. Uribe·Bicheng Ying
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Abstract

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.

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Xin Jiang, Edward Duc Hien Nguyen, César A. Uribe, Bicheng Ying. 2024-01-26. Sparse factorization of the square all-ones matrix of arbitrary order. https://arxiv.org/abs/2401.14596

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