发表机构
EURECOM（欧洲通信学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非线性函数计算广播问题，提出基于广播图和Körner特征图的可达方案及多字母响应轮廓等逆定理，并证明非线性编码可严格优于线性方案。
AI 中文摘要
本文研究非线性函数计算广播（NFCB）问题，其中发送方拥有 $N$ 个数据集 $(X_1,\dots,X_N)$，并向 $K$ 个用户广播一条公共消息，每个用户拥有边信息并请求数据集的某个函数。目标是使所有需求实现渐近无损恢复所需的最小速率。我们引入一个广播图，该图联合刻画了源分布、边信息和所请求的函数。利用 Körner 的特征图框架，我们为任意 $K$、一般源分布以及一般有限域需求（包括线性、不可分离和非线性函数）提出了一种可实现方案，且不限制编码或解码操作为线性。我们还提出了一种边信息辅助的独立集方案，并刻画了兼容函数下基于图的最优可达速率。对于逆定理，我们推导了一个多字母响应轮廓界，该界加强了基本的边信息逆定理、基于操作块广播图的零误差和渐近无损团熵界，以及一个精灵辅助下界。对于 $N=K$ 且用户 $i$ 拥有边信息 $X_i$ 的二元 NFCB，我们在若干特殊情形下刻画了最优速率，并给出了 $K=3$ 和 $K=4$ 时提出的可达速率与精灵辅助逆定理之间的最坏情况和平均加性间隙。最后，具有布尔和线性需求的三用户示例说明了所提出的界，并表明非线性编码可以严格优于最佳的标量和向量线性方案。
英文摘要
This work studies non-linear function computation broadcast (NFCB), in which a sender with access to $N$ datasets $(X_1,\dots,X_N)$ broadcasts a common message to $K$ users, each possessing side information and requesting a function of the datasets. The goal is to minimize the rate required for asymptotically lossless recovery of all demands. We introduce a broadcast graph that jointly captures the source distribution, side information, and demanded functions. Using Körner's characteristic-graph framework, we develop an achievable scheme for arbitrary $K$, general source distributions, and general finite-field demands, including linear, non-separable, and non-linear functions, without restricting the encoding or decoding operations to be linear. We also present a side-information-assisted independent-set scheme and characterize the optimal graph-based achievable rate for compatible functions. For the converse, we derive a multi-letter response-profile bound that strengthens a basic side-information converse, zero-error and asymptotically lossless clique-entropy bounds based on the operational block broadcast graph, and a genie-aided lower bound. For binary NFCB with $N=K$ and side information $X_i$ at user $i$, we characterize the optimal rate in several special cases and bound the worst-case and average additive gaps between the proposed achievable rate and the genie-aided converse for $K=3$ and $K=4$. Finally, three-user examples with Boolean and linear demands illustrate the proposed bounds and show that non-linear encoding can strictly outperform the best scalar and vector linear schemes.
CommentsThis work extends the conference version presented at IEEE ISIT 2025, available at https://arxiv.org/abs/2502.13688. Example 2, concerning three-user linear computation broadcast, has been corrected. Contact author: Derya Malak (malak@eurecom.fr)
