High-rate vector quantization for detection

We investigate high-rate quantization for various detection and reconstruction loss criteria. A new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance. The distortion criterion is related to the area under the receiver-operating-characteristic (ROC) curve. Specifically, motivated by Sanov´s theorem, we define a performance curve as the trajectory of the pair of optimal asymptotic Type I and Type II error rates of the most powerful Neyman-Pearson test of the hypotheses. The distortion measure is then defined as the difference between the area-under-the-curve (AUC) of the optimal pre-encoded hypothesis test and the AUC of the optimal post-encoded hypothesis test. As compared to many previously introduced distortion measures for decision making, this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses, which are generally difficult to specify in the code design process. A high-resolution Zador-Gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high-rate vector quantizer. The analysis applies to a restricted class of high-rate quantizers that have bounded cells with vanishing volumes. The optimal point density is used to specify a Lloyd-type algorithm which allocates its finest resolution to regions where the gradient of the pre-encoded likelihood ratio has greatest magnitude.