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

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.

💡 Research Summary

This paper presents a unified framework for refining and generalizing the well-known Shannon Lower Bound (SLB) in rate-distortion theory through novel extensions of the Kraft inequality. The central idea is to derive lower bounds on the rate-distortion performance by upper-bounding a generalized Kraft sum/integral that incorporates both codeword length and distortion.

The authors begin by reviewing the significance of the SLB as a simple, explicit, and often tight lower bound to the rate-distortion function, especially useful in low-distortion regimes and where exact calculation is intractable. They then introduce the key technical tool: the extended Kraft integral Z_n(α, β), defined as the integral (or sum for discrete alphabets) over the source space of exp2{-αL