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

TL;DR

This paper proves a universal phase transition phenomenon called 'all-or-nothing' for noiseless discrete channels, including models like Bernoulli group testing, extending previous limited results to all signals with sublinear sparsity.

Contribution

It introduces a novel technique showing that for noiseless discrete channels, proving the 'all' part implies the 'nothing' part, enabling broader application of the all-or-nothing phenomenon.

Findings

Establishes the all-or-nothing phase transition for all signals with sublinear sparsity.

Shows the equivalence of 'all' and 'nothing' phases in noiseless discrete channels.

Provides a general method to prove the phenomenon using the first-moment method.

Abstract

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.