  Despite the widespread use and success of machine-learning techniques for
detecting phase transitions from data, their working principle and fundamental
limits remain elusive. Here, we explain the inner workings and identify
potential failure modes of these techniques by rooting popular machine-learning
indicators of phase transitions in information-theoretic concepts. Using tools
from information geometry, we prove that several machine-learning indicators of
phase transitions approximate the square root of the system's (quantum) Fisher
information from below -- a quantity that is known to indicate phase
transitions but is often difficult to compute from data. We numerically
demonstrate the quality of these bounds for phase transitions in classical and
quantum systems.
