  Molecular clouds show complex structures reflecting their non-linear
dynamics. Many studies, investigating the bridge between their morphology and
physical properties, have shown the interest provided by non-Gaussian
higher-order statistics to grasp physical information. Yet, as this bridge is
usually characterized in the supervised world of simulations, transferring it
onto observations can be hazardous, especially when the discrepancy between
simulations and observations remains unknown. In this paper, we aim at
identifying relevant summary statistics directly from the observation data. To
do so, we develop a test that compares the informative power of two sets of
summary statistics for a given dataset. Contrary to supervised approaches, this
test does not require the knowledge of any data label or parameter, but focuses
instead on comparing the degeneracy levels of these descriptors, relying on a
notion of statistical compatibility. We apply this test to column density maps
of 14 nearby molecular clouds observed by Herschel, and iteratively compare
different sets of usual summary statistics. We show that a standard Gaussian
description of these clouds is highly degenerate but can be substantially
improved when being estimated on the logarithm of the maps. This illustrates
that low-order statistics, properly used, remain a very powerful tool. We then
further show that such descriptions still exhibit a small quantity of
degeneracies, some of which are lifted by the higher order statistics provided
by reduced wavelet scattering transforms. This property of observations
quantitatively differs from state-of-the-art simulations of dense molecular
cloud collapse and is not reproduced by logfBm models. Finally we show how the
summary statistics identified can be cooperatively used to build a
morphological distance, which is evaluated visually, and gives very
satisfactory results.
