This paper identifies the Stein-exponent of distributed detection when the sensor communicates to the decision center over a discrete memoryless channel (DMC) subject to one of three stringent communication constraints: 1) The number of channel uses of the DMC grows sublinearly in the number of source observations n; 2) The number of channel uses is n but a block-input cost constraint is imposed almost surely, which grows sublinearly in n; 3) The block-input constraint is imposed only on expectation. We identify a dichotomy in the Stein-exponent of all these setups depending on the structure of the DMC's transition law. Under any of these three constraints, when the DMC is partially-connected (i.e., some outputs cannot be induced by certain inputs) the Stein-exponent matches the exponent identified by Han and Kobayashi and by Shalaby and Papamarcou for the scenario where communication is of zero-rate but over a noiseless link. We prove our result by adapting Han's zero-rate coding strategy to partially-connected DMCs. In contrast, for fully-connected DMCs, in our scenarios 1) and 2) the Stein-exponent collapses to that of a local test at the decision center, rendering the remote sensor and communication useless. %To prove this result, we propose new converse proofs relying on change of measure arguments. In scenario 3), the sensor remains beneficial even for fully-connected DMCs, however also collapses compared to the case of a partially-connected DMC. Moreover, the Stein-exponent is larger when the expectation constraint is imposed only under the null hypothesis compared to when it is imposed under both hypotheses. To prove these results, we propose both new coding strategies and new converse proofs.

翻译：本文研究了当传感器通过离散无记忆信道（DMC）向决策中心通信，并面临以下三种严格通信约束之一时的分布式检测Stein指数：1) DMC的信道使用次数随源观测数n呈次线性增长；2) 信道使用次数为n，但几乎必然施加一个随n次线性增长的块输入成本约束；3) 块输入约束仅在期望意义上施加。我们发现，根据DMC转移律的结构，所有这些设置下的Stein指数存在二分性。在上述任一约束下，当DMC为部分连接（即某些输入无法产生某些输出）时，其Stein指数与Han和Kobayashi以及Shalaby和Papamarcou针对通信为零速率但通过无噪声链路场景所确定的指数一致。我们通过将Han的零速率编码策略适配至部分连接的DMC来证明此结果。相比之下，对于全连接DMC，在我们的场景1)和2)中，Stein指数退化为决策中心本地检验的指数，使得远程传感器和通信变得无用。%为证明此结果，我们提出了基于测度变换论证的新逆定理证明。在场景3)中，即使对于全连接DMC，传感器仍然有益，但与部分连接DMC的情况相比，其指数同样有所退化。此外，与在两种假设下均施加期望约束相比，仅在零假设下施加期望约束时，Stein指数更大。为证明这些结果，我们提出了新的编码策略和新的逆定理证明。