It was all for nothing: sharp phase transitions for noiseless discrete channels

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.