[Paper Review] Efficient sphere-covering and converse measure concentration via generalized coding theorems

Ioannis Kontoyiannis|ArXiv.org|Oct 12, 1999

Wireless Communication Security Techniques15 references3 citations

TL;DR

This paper introduces a generalized framework for analyzing efficient sphere-covering and converse measure concentration in discrete probability spaces using coding-theoretic principles. By characterizing optimal covering sets via a mass function and product measures, it derives tight bounds on covering efficiency, unifying results such as Shannon's data compression theorem and Stein's lemma, and establishes new converse inequalities for measure concentration on finite spaces with minimal assumptions.

ABSTRACT

Suppose A is a finite set equipped with a probability measure P and let M be a ``mass'' function on A. We give a probabilistic characterization of the most efficient way in which A^n can be almost-covered using spheres of a fixed radius. An almost-covering is a subset C_n of A^n, such that the union of the spheres centered at the points of C_n has probability close to one with respect to the product measure P^n. An efficient covering is one with small mass M^n(C_n); n is typically large. With different choices for M and the geometry on A our results give various corollaries as special cases, including Shannon's data compression theorem, a version of Stein's lemma (in hypothesis testing), and a new converse to some measure concentration inequalities on discrete spaces. Under mild conditions, we generalize our results to abstract spaces and non-product measures.

Motivation & Objective

• To develop a probabilistic framework for characterizing the most efficient sphere-covering of product probability spaces.
• To generalize classical results in information theory and probability, such as Shannon's theorem and Stein's lemma, within a unified coding-theoretic structure.
• To establish converse measure concentration inequalities for discrete spaces using minimal assumptions on the underlying probability measure.
• To extend the results beyond product measures to abstract spaces with general measures.

Proposed method

• The framework uses a finite set A equipped with a probability measure P and a mass function M to quantify the efficiency of covering sets in A^n.
• It defines an almost-covering as a subset C_n ⊂ A^n such that the union of M-spheres around points in C_n captures probability close to one under P^n.
• The efficiency is measured by minimizing M^n(C_n), leading to a variational characterization of optimal covering sets.
• The approach leverages generalized coding theorems to derive asymptotic bounds on the minimal mass required for such coverings.
• It applies large deviations and entropy-based arguments to analyze the trade-off between covering probability and mass cost.
• The method generalizes to abstract spaces and non-product measures under mild regularity conditions.

Experimental results

Research questions

• RQ1 What is the minimal mass required to almost-cover A^n with M-spheres under the product measure P^n, and how is this related to entropy and divergence?
• RQ2 How can classical results like Shannon's data compression theorem and Stein's lemma be derived as special cases of a unified coding-theoretic principle?
• RQ3 What converse inequalities for measure concentration on discrete spaces can be obtained using this generalized framework?
• RQ4 In what sense is the sphere-covering construction optimal, and how does the choice of mass function M affect efficiency?
• RQ5 To what extent can the results be extended beyond product measures and finite spaces?

Key findings

• The paper establishes a precise characterization of the minimal mass M^n(C_n) required for an almost-covering of A^n, showing it is asymptotically determined by a variational expression involving relative entropy.
• Shannon's data compression theorem emerges as a special case when M is chosen as the uniform measure and the covering radius corresponds to typical sequences.
• A version of Stein's lemma in hypothesis testing is recovered by interpreting the covering problem as a test between two measures with a divergence threshold.
• The framework yields new converse bounds for measure concentration on discrete spaces, showing that concentration cannot be stronger than predicted by the entropy-based rate function.
• The results extend to abstract spaces and non-product measures under mild conditions, demonstrating robustness of the generalized coding-theoretic approach.
• The derived bounds are tight and provide a unifying perspective on sphere-covering, compression, and concentration in probability and information theory.