Hierarchical Guessing with a Fidelity Criterion

In an earlier paper, we studied the problem of guessing a random vector X within distortion D, and characterized the best attainable exponent E(D; ) of the th moment of the number of required guesses G(X) until the guessing error falls below D. In this paper, we extend these results to a multi-stage, hierarchical guessing model, which allows for a faster search of a codeword vector at the encoder of a rate-distortion codebook. In the two-stage case of this model, if the target distortion level is D2, the guesser rst makes guesses w.r.t. (a higher) distortion level D1, and then, upon his/her rst success, directs the subsequent guesses to distortion D2. As in the above mentioned earlier paper, we provide a single-letter characterization of the best attainable guessing exponent, which relies heavily on well-known results on the successive re nement problem. We also relate this guessing exponent function to the source coding error exponent function of the two-step coding process.