The paper presents the exact capacity characterization for the 3-user linear computation broadcast (LCBC) problem. The LCBC problem involves a central server with d-dimensional data over a finite field Fq, and 3 users, each of whom has some linear function of the data as side-information and wants to retrieve a different linear function of the data.
The key insights and highlights of the paper are:
The capacity is expressed in two equivalent forms - a closed-form expression (Theorem 1) and a linear programming formulation (Theorem 2). The linear programming formulation provides constructive insights into the optimal coding scheme.
The converse (impossibility) proof shows that the entropic formulation used for the 2-user LCBC is insufficient for the 3-user case, and instead relies on functional submodularity.
The achievability scheme utilizes a subspace decomposition that parallels previous results on degrees of freedom in wireless MIMO broadcast channels. This decomposition is essential for identifying the appropriate dimensions to broadcast.
The optimal scheme involves a non-trivial tradeoff between the number of dimensions broadcast from different subspaces, leading to a constrained waterfilling solution.
Remarkably, non-linear schemes are not needed, and the optimal scheme can be achieved using vector linear coding over the finite field Fq.
The paper provides a comprehensive capacity characterization for the 3-user LCBC problem, shedding light on the technical challenges that arise when moving from 2 to 3 users in computation broadcast networks.
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by Yuhang Yao,S... at arxiv.org 05-07-2024
https://arxiv.org/pdf/2206.10049.pdfDeeper Inquiries