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Local atomic environment descriptors (LAEDs) are used in the materials science and chemistry communities, for example, for the development of machine learning interatomic potentials. Despite the fact that LAEDs have been extensively studied and benchmarked for various applications, global structure descriptors (GSDs), i.e., descriptors for entire molecules or crystal structures, have been mostly developed independently based on other approaches. Here, we propose a systematically improvable methodology for constructing a space of representations of GSDs from LAEDs by incorporating statistical information and information about chemical elements. We apply the method to construct GSDs of varying complexity for lithium thiophosphate structures that are of interest as solid electrolytes and use an information-theoretic approach to obtain an optimally compressed GSD. Finally, we report the performance of the compressed GSD for energy prediction tasks.