Some real-world problems are partially decomposable, in that they can be decomposed into a set of coupled subproblems, that are each relatively easy to solve. However, when these sub-problem share some common variables, it is not sufficient to simply solve each sub-problem in isolation. We develop a technology for such problems, and use it to address the challenge of finding the concentrations of the chemicals that appear in a complex mixture, based on its one-dimensional ~1H Nuclear Magnetic Resonance (NMR) spectrum. As each chemical involves clusters of spatially localized peaks, this requires finding the shifts for the clusters and the concentrations of the chemicals, that collectively produce the best match to the observed NMR spectrum. Here, each sub-problem requires finding the chemical concentrations and cluster shifts that can appear within a limited spectrum range; these are coupled as these limited regions can share many chemicals, and so must agree on the concentrations and cluster shifts of the common chemicals. This task motivates CEED: a novel extension to the Cross-Entropy stochastic optimization method constructed to address such partially decomposable problems. Our experimental results in the NMR task show that our CEED system is superior to other well-known optimization methods, and indeed produces the best-known results in this important, real-world application.
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