The Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS

Abstract

In this paper, we prove that for the doubly symmetric binary distribution, the lower increasing envelope and the upper envelope of the minimum-relative-entropy region are respectively convex and concave. We also prove that another function induced the minimum-relative-entropy region is concave. These two envelopes and this function were previously used to characterize the optimal exponents in strong small-set expansion problems and strong Brascamp-Lieb inequalities. The results in this paper, combined with the strong small-set expansion theorem derived by Yu, Anantharam, and Chen (2021), and the strong Brascamp-Lieb inequality derived by Yu ( 2021), confirm positively Ordentlich-Polyanskiy-Shayevitz's conjecture on the strong small-set expansion (2019) and Polyanskiy's conjecture on the strong Brascamp-Lieb inequality (2016). The proofs in this paper are based on the equivalence between the convexity of a function and the convexity of the set of minimizers of its Lagrangian dual.

Common Information, Noise Stability, and Their Extensions

“…In particular, he showed that Υ LD is convex and Υ LD is concave. Combining this result with the strong SSE theorem (Theorem 8.5.1) allows us to conclude that the OPS conjecture is unconditionally true and that Hamming balls or spheres (without time-sharing) are optimal in the LD regime [193]. That is, for the DSBS and α, β ∈ (0, 1),…”

“…As stated in [97], Conjecture 10.1 for all q > 1 was proved by Polyanskiy in an unpublished work [137]. It was also proven independently by the first author of this monograph in [192], [193]. In particular, it was shown that Conjecture 10.1 holds even for p = 1.…”

“…To prove the OPS conjecture, we need to remove the operations of taking the lower convex and upper concave envelopes in the strong SSE theorem. This was done by the first author of this monograph [193]. In particular, he showed that Υ LD is convex and Υ LD is concave.…”

“…It has been shown in [193] that for q ≥ 1, Υ q,LD is convex, and for q ∈ (−∞, 1)\{0}, Υ q,LD is concave. Combining this result with the strong q-stability theorem (Theorem 9.5.2) tells us that the lower convex envelope and upper concave envelope operations in (9.62) can be removed and Hamming balls or spheres are optimal in the LD regime.…”

“…Indeed, the asymptotically sharp bound on q(t) such that T ⊗n t f q(t) ≤ f p holds for any f : {0, 1} n → [0, ∞) with 1 n Ent p (f ) ≥ α can be characterized by the strong BL inequality derived in the same references. Readers may refer to [192], [193] for the details.…”