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.