  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.
