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The Chow-Liu algorithm (IEEE Trans.~Inform.~Theory, 1968) has been a mainstayfor the learning of tree-structured graphical models from i.i.d.\ sampled datavectors. Its theoretical properties have been well-studied and arewell-understood. In this paper, we focus on the class of trees that arearguably even more fundamental, namely {\em homogeneous} trees in which eachpair of nodes that forms an edge has the same correlation $\rho$. We askwhether we are able to further reduce the error probability of learning thestructure of the homogeneous tree model when {\em active learning} or {\emactive sampling of nodes or variables} is allowed. Our figure of merit is the{\em error exponent}, which quantifies the exponential rate of decay of theerror probability with an increasing number of data samples. At first sight, animprovement in the error exponent seems impossible, as all the edges arestatistically identical. We design and analyze an algorithm Active LearningAlgorithm for Trees with Homogeneous Edge (Active-LATHE), which surprisinglyboosts the error exponent by at least 40\% when $\rho$ is at least $0.8$. Forall other values of $\rho$, we also observe commensurate, but more modest,improvements in the error exponent. Our analysis hinges on judiciouslyexploiting the minute but detectable statistical variation of the samples toallocate more data to parts of the graph in which we are less confident ofbeing correct.

Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees