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We extend Ziv and Lempel's model of finite-state encoders to the realm oflossy compression of individual sequences. In particular, the model of theencoder includes a finite-state reconstruction codebook followed by aninformation lossless finite-state encoder that compresses the reconstructioncodeword with no additional distortion. We first derive two different lowerbounds to the compression ratio that depend on the number of states of thelossless encoder. Both bounds are asymptotically achievable by conceptuallysimple coding schemes. We then show that when the number of states of thelossless encoder is large enough in terms of the reconstruction block-length,the performance can be improved, sometimes significantly so. In particular, theimproved performance is achievable using a random-coding ensemble that isuniversal, not only in terms of the source sequence, but also in terms of thedistortion measure.

Lossy Compression of Individual Sequences Revisited: Fundamental Limits of Finite-State Encoders