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.

中文翻译：

---

对于“什么都没有”，这是“全部”：无噪声离散通道的尖锐相变

我们为无噪声离散通道建立了一个称为“全有或全无”现象的相变。此类模型包括伯努利小组测试模型和已植入的高斯感知器模型。以前，仅在有限的参数范围内知道此类模型的“全有或全无”现象的存在。我们的工作将结果扩展到具有任意亚线性稀疏性的所有信号。在过去的几年中，由于两种看似不相交的结果，在各种模型中都建立了全有或全无的现象：一个肯定的结果建立了全有或全部的“全部”一半，而一个不可能的结果建立了全有或全部的一半。 “一无所有”一半。我们在当前工作中的主要技术是表明，对于无噪声的离散通道，“全部”一半表示“