它“全”是为了“无”：无噪声离散信道的尖锐相变
It was "all" for "nothing": sharp phase transitions for noiseless discrete channels
AI总结：
本文为无噪声离散信道建立了“全或无”相变现象，将结果扩展至任意次线性稀疏信号，并证明“全”部分蕴含“无”部分，提供了一种通用证明方法。
摘要：
我们为无噪声离散信道建立了一种被称为“全或无”现象的相变。这类模型包括 Bernoulli 群测试模型和植入高斯感知器模型。此前，此类模型中“全或无”现象的存在性仅在有限的参数范围内已知。我们的工作将结果扩展到了具有任意次线性稀疏性的所有信号。在过去的几年里，“全或无”现象已在各种模型中被确立，作为两个看似不相关联的结果的产物：一个确立“全或无”中“全”部分的正面结果，以及一个确立“无”部分的不可能性结果。我们在本工作中的主要技术是证明，对于无噪声离散信道，“全”部分蕴含“无”部分，即对“全”的证明可以转化为对“无”的证明。由于“全”部分通常可以通过直接的方法证明——例如，通过一阶矩方法——我们的等价性为在其他背景下确立该现象的存在提供了一种强大且通用的方法。
英文摘要：
We establish a phase transition known as the "all-or-nothing" phenomenon for noiseless discrete channels. This class of models includes the Bernoulli group testing model and the planted Gaussian perceptron model. Previously, the existence of the all-or-nothing phenomenon for such models was only known in a limited range of parameters. Our work extends the results to all signals with arbitrary sublinear sparsity. Over the past several years, the all-or-nothing phenomenon has been established in various models as an outcome of two seemingly disjoint results: one positive result establishing the "all" half of all-or-nothing, and one impossibility result establishing the "nothing" half. Our main technique in the present work is to show that for noiseless discrete channels, the "all" half implies the "nothing" half, that is a proof of "all" can be turned into a proof of "nothing." Since the "all" half can often be proven by straightforward means -- for instance, by the first-moment method -- our equivalence gives a powerful and general approach towards establishing the existence of this phenomenon in other contexts.
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。
