A Class of Subadditive Information Measures and their Applications

TL;DR

This paper introduces a new two-parameter family of divergence measures called $(G,f)$-divergences, explores their subadditivity properties, and applies these findings to problems in channel coding, hypothesis testing, and sphere-packing bounds.

Contribution

It defines the $(G,f)$-divergences, establishes subadditivity conditions, and demonstrates their applications in information theory problems.

Findings

Derived tractable conditions for subadditivity of $(G,f)$-divergences.

Showed subadditivity reduces to binary alphabet cases for broad classes of $G$.

Extended classical bounds and exponents using the new divergence measures.

Abstract

We introduce a two-parameter family of discrepancy measures, termed \emph{$(G,f)$-divergences}, obtained by applying a non-decreasing function $G$ to an $f$-divergence $D_f$. Building on Csisz\'ar's formulation of mutual $f$-information, we define a corresponding $(G,f)$-information measure $ I_{G,f}(X;Y)$. A central theme of the paper is subadditivity over product distributions and product channels. We develop reduction principles showing that, for broad classes of $G$, it suffices to verify divergence subadditivity on binary alphabets. Specializing to the functions $G(x)\in\{x,\log(1+x),-\log(1-x)\}$, we derive tractable sufficient conditions on $f$ that guarantee subadditivity, covering many standard $f$-divergences. Finally, we present applications to finite-blocklength converses for channel coding, bounds in binary hypothesis testing, and an extension of the Shannon--Gallager--Berlekamp sphere-packing exponent framework to subadditive $(G,f)$-divergences.