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.

翻译：我们针对无噪离散信道建立了被称为"全有或全无"现象的相变。这类模型包括伯努利分组检测模型和植入高斯感知机模型。此前，此类模型中"全有或全无"现象的存在性仅在有限参数范围内得到确认。我们的工作将结论推广至任意具有次线性稀疏性的信号。过去数年间，"全有或全无"现象已在多种模型中被证明，其印证过程看似由两个独立结果构成：一个建立"全有或全无"中"全部"部分的肯定性结论，另一个揭示"一无所获"部分的不可行性结论。本文的核心技术在于证明：对于无噪离散信道，"全部"部分蕴含"一无所获"部分——即对"全部"的证明可转化为对"一无所获"的证明。由于"全部"部分通常可通过直接方法（例如一阶矩方法）得到证明，本文提出的等价性为在其他场景下建立该现象的存在性提供了强大而普适的途径。