This paper considers a massive connectivity setting in which a base-station (BS) aims to communicate sources $(X_1,\cdots,X_k)$ to a randomly activated subset of $k$ users, among a large pool of $n$ users, via a common downlink message. Although the identities of the $k$ active users are assumed to be known at the BS, each active user only knows whether itself is active and does not know the identities of the other active users. A naive coding strategy is to transmit the sources alongside the identities of the users for which the source information is intended, which would require $H(X_1,\cdots,X_k) + k\log(n)$ bits, because the cost of specifying the identity of a user is $\log(n)$ bits. For large $n$, this overhead can be significant. This paper shows that it is possible to develop coding techniques that eliminate the dependency of the overhead on $n$, if the source distribution follows certain symmetry. Specifically, if the source distribution is independent and identically distributed (i.i.d.) then the overhead can be reduced to at most $O(\log(k))$ bits, and in case of uniform i.i.d. sources, the overhead can be further reduced to $O(1)$ bits. For sources that follow a more general exchangeable distribution, the overhead is at most $O(k)$ bits, and in case of finite-alphabet exchangeable sources, the overhead can be further reduced to $O(\log(k))$ bits. The downlink massive random access problem is closely connected to the study of finite exchangeable sequences. The proposed coding strategy allows bounds on the relative entropy distance between finite exchangeable distributions and i.i.d. mixture distributions to be developed, and gives a new relative entropy version of the finite de Finetti theorem which is scaling optimal.

翻译：本文研究一种大规模连接场景：基站（BS）通过公共下行链路消息，向一个包含n个用户的大集合中随机激活的k个用户子集传输信源$(X_1,\cdots,X_k)$。尽管BS已知这k个活跃用户的身份，但每个活跃用户仅知晓自身是否被激活，而无法获知其他活跃用户的身份。一种朴素的编码策略是将信源与接收用户身份信息一同传输，这需要$H(X_1,\cdots,X_k) + k\log(n)$比特，因为指定单个用户身份的成本为$\log(n)$比特。当n很大时，该开销可能非常显著。本文证明，若信源分布满足特定对称性，则可开发出消除开销对n依赖性的编码技术。具体而言：若信源分布为独立同分布（i.i.d.），则开销可降至最多$O(\log(k))$比特；对于均匀i.i.d.信源，开销可进一步降至$O(1)$比特。对于更一般的可交换分布信源，开销不超过$O(k)$比特；而对于有限字母表可交换信源，开销可进一步降至$O(\log(k))$比特。下行链路海量随机接入问题与有限可交换序列的研究密切相关。本文提出的编码策略可推导出有限可交换分布与i.i.d.混合分布之间相对熵距离的界，并给出一种新的关于相对熵的有限de Finetti定理，该定理在标度意义下是最优的。