New Non-Asymptotic Bounds on Numbers of Codewords for the Fixed-Length Lossy Compression

Abstract

In this paper, we deal with the fixed-length lossy compression, where a fixed-length sequence emitted from the information source is encoded into a codeword, and the source sequence is reproduced from the codeword with a certain distortion. We give lower and upper bounds on the minimum number of codewords such that the probability of exceeding a given distortion level is less than a given probability. These bounds are characterized by using the α-mutual information of order infinity. Further, for i.i.d. binary sources, we provide numerical examples of tight upper bounds which are computable in polynomial time in the blocklength.

Non-Asymptotic Bounds and a General Formula for the Rate-Distortion Region of the Successive Refinement Problem

“…In Section 3, we give several lemmas for an inner bound on pairs of numbers of codewords and the rate-distortion region. These lemmas are extended versions of our previous results [12,Lemma 2] and [13,Lemma 1]. In Section 4, we give outer and inner bounds using the smooth max Rényi divergence on pairs of numbers of codewords.…”

“…The inner bound is derived by using an extended version of the previous lemma [12,Lemma 2]. We give this lemma as a special case of an extended version of the previous generalized covering lemma [13,Lemma 1]. The outer bound is derived by using an extended version of the previous converse bound [12,Lemma 4].…”