  The upcoming facilities like the Vera C. Rubin Observatory will provide
extremely deep photometry of thousands of star clusters to the edge of the
Galaxy and beyond, which will require adequate tools for automatic analysis,
capable of performing tasks such as the characterization of a star cluster
through the analysis of color-magnitude diagrams (CMDs). The latter are
essentially point clouds in N-dimensional space, with the number of dimensions
corresponding to the photometric bands employed. In this context, machine
learning techniques suitable for tabular data are not immediately applicable to
CMDs because the number of stars included in a given CMD is variable, and
equivariance for permutations is required. To address this issue without
introducing ad-hoc manipulations that would require human oversight, here we
present a new CMD featurization procedure that summarizes a CMD by means of a
quadtree-like structure through iterative partitions of the color-magnitude
plane, extracting a fixed number of meaningful features of the relevant
subregion from any given CMD. The present approach is robust to photometric
noise and contamination and it shows that a simple linear regression on our
features predicts distance modulus (metallicity) with a scatter of 0.33 dex
(0.16 dex) in cross-validation.
