  Machine learning-inspired techniques have emerged as a new paradigm for
analysis of phase transitions in quantum matter. In this work, we introduce a
supervised learning algorithm for studying critical phenomena from measurement
data, which is based on iteratively training convolutional networks of
increasing complexity, and test it on the transverse field Ising chain and q=6
Potts model. At the continuous Ising transition, we identify scaling behavior
in the classification accuracy, from which we infer a characteristic
classification length scale. It displays a power-law divergence at the critical
point, with a scaling exponent that matches with the diverging correlation
length. Our algorithm correctly identifies the thermodynamic phase of the
system and extracts scaling behavior from projective measurements,
independently of the basis in which the measurements are performed.
Furthermore, we show the classification length scale is absent for the $q=6$
Potts model, which has a first order transition and thus lacks a divergent
correlation length. The main intuition underlying our finding is that, for
measurement patches of sizes smaller than the correlation length, the system
appears to be at the critical point, and therefore the algorithm cannot
identify the phase from which the data was drawn.
