  Quantum many-body systems realise many different phases of matter
characterised by their exotic emergent phenomena. While some simple versions of
these properties can occur in systems of free fermions, their occurrence
generally implies that the physics is dictated by an interacting Hamiltonian.
The interaction distance has been successfully used to quantify the effect of
interactions in a variety of states of matter via the entanglement spectrum
[Nat. Commun. 8, 14926 (2017), arXiv:1705.09983]. The computation of the
interaction distance reduces to a global optimisation problem whose goal is to
search for the free-fermion entanglement spectrum closest to the given
entanglement spectrum. In this work, we employ techniques from machine learning
in order to perform this same task. In a supervised learning setting, we use
labelled data obtained by computing the interaction distance and predict its
value via linear regression. Moving to a semi-supervised setting, we train an
auto-encoder to estimate an alternative measure to the interaction distance,
and we show that it behaves in a similar manner.
