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, p) of the pth moment of the number of required guesses G(X) until the guessing error falls below D. In this correspondence, 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 D 2, the guesser first makes guesses with respect to (a higher) distortion level D 1, and then, upon his/her first success, directs the subsequent guesses to distortion DI. As in the abovementioned 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.