Identifying the relevant degrees of freedom is key to developing an effective theory of a complex system. “Relevance” is surprisingly well defined in physics and forms the cornerstone of the renormalisation group (RG) program, but its practical execution in unfamiliar systems is difficult. Machine learning approaches, on the other hand, though promising, often lack formal interpretability: it is unclear what relation the architecture- and training-dependent learned "relevant features” bear to objects of physical theory.

I will discuss how the above gap can be bridged, paving a path to automated discovery of mathematically formal and interpretable physical theories, from raw data. To this end we show that the field-theoretic “relevance” is in fact equivalent* to the notion of “relevant information” defined in the Information Bottleneck (IB) formalism of compression theory. Employing recent tools of ML-based estimation of information-theoretic quantities we then construct an unsupervised algorithm whose inputs are raw configurations of a statistical system, and whose outputs are neural nets parametrising formal objects such as “order parameters”, or more generally “scaling operators” characterising the system. The information about the phase diagram, correlations and symmetries (also emergent) can be obtained, which we validate on the example of the interacting dimer model. I will discuss applications to quasiperiodic and disordered systems.