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Quantum simulators offer powerful means to investigate strongly correlated quantum matter. However, interpreting measurement outcomes in such systems poses significant challenges. Here, we present a theoretical framework for information extraction in synthetic quantum matter, illustrated for the case of a quantum quench in a spinor Bose-Einstein condensate experiment. Employing non-parametric unsupervised learning tools that provide different measures of information content, we demonstrate a system-agnostic approach to identify dominant degrees of freedom. This enables us to rank operators according to their relevance, akin to effective field theory. To characterize the corresponding effective description, we then explore the intrinsic dimension of data sets as a measure of the complexity of the dynamics. This reveals a simplification of the data structure, which correlates with the emergence of time-dependent universal behavior in the studied system. Our assumption-free approach can be immediately applied in a variety of experimental platforms.