Quantile Search: A Distance-Penalized Active Learning Algorithm for Spatial Sampling

Adaptive sampling theory has shown that, with proper assumptions on the signal class, algorithms exist to reconstruct a signal in $\mathbb{R}^{d}$ with an optimal number of samples. We generalize this problem to when the cost of sampling is not only the number of samples but also the distance traveled between samples. This is motivated by our work studying regions of low oxygen concentration in the Great Lakes. We show that for one-dimensional threshold classifiers, a tradeoff between number of samples and distance traveled can be achieved using a generalization of binary search, which we refer to as quantile search. We derive the expected total sampling time for noiseless measurements and the expected number of samples for an extension to the noisy case. We illustrate our results in simulations relevant to our sampling application.