Finite-Time In-Network Computation of Linear Transforms

TL;DR

This paper develops a systematic algebraic geometric approach to enable finite-time distributed computation of general linear transforms over directed networks, extending prior work on simpler cases.

Contribution

It introduces a novel algebraic geometric framework for scalable finite-time in-network linear transform computation, including directed graphs and heterogeneous agent objectives.

Findings

Almost all linear transforms on networks with Hamiltonian cycles can be computed in at most N iterations.

The approach applies to directed graphs, broadening previous undirected graph results.

New insights into sparsity-constrained matrix factorizations are provided.

Abstract

This paper focuses on finite-time in-network computation of linear transforms of distributed graph data. Finite-time transform computation problems are of interest in graph-based computing and signal processing applications in which the objective is to compute, by means of distributed iterative methods, various (linear) transforms of the data distributed at the agents or nodes of the graph. While finite-time computation of consensus-type or more generally rank-one transforms have been studied, systematic approaches toward scalable computing of general linear transforms, specifically in the case of heterogeneous agent objectives in which each agent is interested in obtaining a different linear combination of the network data, are relatively less explored. In this paper, by employing ideas from algebraic geometry, we develop a systematic characterization of linear transforms that are amenable to distributed in-network computation in finite-time using linear iterations. Further, we consider the general case of directed inter-agent communication graphs. Specifically, it is shown that \emph{almost all} linear transformations of data distributed on the nodes of a digraph containing a Hamiltonian cycle may be computed using at most $N$ linear distributed iterations. Finally, by studying an associated matrix factorization based reformulation of the transform computation problem, we obtain, as a by-product, certain results and characterizations on sparsity-constrained matrix factorization that are of independent interest.