Hierarchical guessing with a fidelity criterion

Arikan and Merhav (1998) studied the problem of guessing a random vector X within distortion D, and characterized the best attainable exponent E(D,ρ) of the ρth moment of the number of required guesses G(X) until the guessing error falls below D. We extend these results to a multistage, hierarchical guessing model, which allows for a faster search for 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 first makes guesses with respect to (a higher) distortion level D₁, and then, upon his/her first 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 refinement problem. We also relate this guessing exponent function to the source-coding error exponent function of the two-step coding process