Rényi Resolvability, Noise Stability, and Anti-contractivity

As indicated by the title, this paper investigates three closely related topics – Rényi resolvability, noise stability, and anti-contractivity. The Rényi resolvability problem refers to approximating a target output distribution of a given channel in the Rényi divergence when the input is set to a function of a given uniform random variable. This problem for the Rényi parameter in [0,2]∪{∞} was studied by the present author and Tan in 2019. In the present paper, we provide a complete solution to this problem for the Rényi parameter in the entire range ℝ∪{±∞}. We then connect the Rényi resolvability problem to the noise stability problem, by observing that the q-stability of a set can be expressed in terms of the Rényi divergence between the true output distribution and the target distribution in a variant of the Rényi resolvability problem. By such a connection, we provide sharp dimension-free bounds on the q-stability. We lastly relate the noise stability problem to the anti-contractivity of a Markov operator (i.e., conditional expectation operator), where anti-contractivity introduced by us refers to as the opposite property of the well-known contractivity/hyercontractivity. We derive sharp dimension-free anti-contractivity inequalities. All of the results in this paper are evaluated for binary distributions. Our proofs in this paper are mainly based on the method of types, especially a strengthened version of packing-covering lemma.