基于最优传输的任意维后验匹配的构造与分析
Construction and Analysis of Posterior Matching in Arbitrary Dimensions via Optimal Transport
摘要：
后验匹配方案，为反馈编码的消息点躺在单位间隔上的无记忆信道，最大化互信息为任意数量的信道使用。然而，一般来说，它并不总是达到任何积极的比率;迄今为止，需要进行详细的分析，以表明它达到的任何积极的比率低于能力。最近的努力已经将编码器和解码器共享的随机“抖动”引入到问题公式中，以简化分析并保证随机化方案实现低于容量的任何速率。受应用程序（例如人机接口）的启发，其中（a）由编码器和解码器共享的公共随机性可能是不可行的，（b）消息点位于高维空间中，我们在这里专注于没有公共随机性的原始公式，并使用最优传输理论来概括高维空间中消息点的方案。通过定义一个更严格的，几乎肯定的，消息解码的概念，我们使用经典的概率技术（例如测量和鞅收敛的变化）建立简洁的必要和充分条件时，消息点可以从无限的观察：Birkhoff遍历随机过程顺序产生的编码器。我们还展示了一个令人惊讶的“全有或全无”的结果：相同的遍历条件是必要的，足以实现任何低于容量的速率。我们提供了这个消息点框架在人机界面和多天线通信中的应用。
英文摘要：
The posterior matching scheme, for feedback encoding of a message point lying on the unit interval over memoryless channels, maximizes mutual information for an arbitrary number of channel uses. However, it in general does not always achieve any positive rate; so far, elaborate analyses have been required to show that it achieves any positive rate below capacity. More recent efforts have introduced a random "dither" shared by the encoder and decoder to the problem formulation, to simplify analyses and guarantee that the randomized scheme achieves any rate below capacity. Motivated by applications (e.g. human-computer interfaces) where (a) common randomness shared by the encoder and decoder may not be feasible and (b) the message point lies in a higher dimensional space, we focus here on the original formulation without common randomness, and use optimal transport theory to generalize the scheme for a message point in a higher dimensional space. By defining a stricter, almost sure, notion of message decoding, we use classical probabilistic techniques (e.g. change of measure and martingale convergence) to establish succinct necessary and sufficient conditions on when the message point can be recovered from infinite observations: Birkhoff ergodicity of a random process sequentially generated by the encoder. We also show a surprising "all or nothing" result: the same ergodicity condition is necessary and sufficient to achieve any rate below capacity. We provide applications of this message point framework in human-computer interfaces and multi-antenna communications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
