Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality

TL;DR

This paper develops extended Kraft inequalities to refine and generalize the Shannon lower bound for various rate-distortion coding scenarios, including sharper bounds and individual-sequence versions.

Contribution

It introduces new Kraft inequality extensions that lead to sharper and more general Shannon lower bounds for lossy compression.

Findings

Sharper bounds for one-to-one codes and D-semifaithful codes

A Shannon lower bound for sliding-window distortion measures

An individual-sequence version of the Shannon lower bound

Abstract

We derive a few extended versions of the Kraft inequality for lossy compression, which pave the way to the derivation of several refinements and extensions of the well known Shannon lower bound in a variety of instances of rate-distortion coding. These refinements and extensions include sharper bounds for one-to-one codes and $D$-semifaithful codes, a Shannon lower bound for distortion measures based on sliding-window functions, and an individual-sequence counterpart of the Shannon lower bound.