On limited-delay lossy coding and filtering of individual sequences

We continue the study of adaptive schemes for the sequential lossy coding of individual sequences which was initiated by Linder and Lugosi (see IEEE Trans. Inform. Theory. (See also Proc. 2000 IEEE ISIT, Sorento, Italy, p.125, 2000). Specifically, we consider fixed-rate lossy coding systems of fixed (or zero) delay where the encoder (allowed to use randomization) and the decoder are connected via a noiseless channel of a given capacity. It is shown that for any finite set of such coding schemes of a given rate there exists a source code (adhering to the same structural and delay limitations) with the same rate whose distortion is very likely to be almost as small as that of the best scheme in that set, uniformly for all individual sequences. This result can be applied to reference classes of special interest. Finally, we consider the case where the individual sequence is corrupted by noise prior to reaching the coding system, whose goal now is to reconstruct a sequence with small distortion relative to the clean individual sequence. It is shown that for the case of a finite alphabet and an invertible channel matrix, for any finite set of sliding window schemes of a given rate there exists a source code (allowed to use randomization yet adhering to the same delay constraints) which is very likely to do essentially as well as the best scheme in the class, for all individual sequences.