Improved bounds on Gaussian MAC and sparse regression via Gaussian inequalities

We consider the Gaussian multiple-access channel with two critical departures from the classical asymptotics: a) number of users proportional to block-length and b) each user sends a fixed number of data bits. We provide improved bounds on the trade-off between the user density and the energy-per-bit. Interestingly, in this information-theoretic problem we rely on Gordon's lemma from Gaussian process theory. From the engineering standpoint, we discover a surprising new effect: good coded-access schemes can achieve perfect multi-user interference cancellation at low user density. In addition, by a similar method we analyze the limits of false-discovery in binary sparse regression problem in the asymptotic regime of number of measurements going to infinity at fixed ratios with problem dimension, sparsity and noise level. Our rigorous bound matches the formal replica-method prediction for some range of parameters with imperceptible numerical precision.