  The low brightness dust emission at high Galactic latitude is of interest to
study the interplay between physical processes in shaping the structure of the
interstellar medium (ISM), as well as to statistically characterize dust
emission as a foreground to the Cosmic Microwave Background (CMB). Progress in
this avenue of research have been hampered by the difficulty of separating the
dust emission from the Cosmic Infrared Background (CIB). We demonstrate that
dust and CIB may be effectively separated based on their different structure on
the sky and use the separation to characterize the structure of diffuse dust
emission on angular scales where CIB is a significant component in terms of
power. We use scattering transform statistics, the Wavelet Phase Harmonics
(WPH), to perform a statistical component separation using Herschel SPIRE
observations. This component separation is done only from observational data
using non-Gaussian properties as a lever arm, and is done at a single 250
microns frequency. This method, that we validate on mock data, gives us access
to non-Gaussian statistics of the interstellar dust and an output dust map
essentially free from CIB contamination. Our statistical modelling
characterizes the non-Gaussian structure of the diffuse ISM down to the
smallest scales observed by Herschel. We recover the power-law shape of the
dust power spectrum up to a wavenumber of 2 arcmin$^{-1}$ where the dust signal
represents 2 percent of the total power. The output dust map reveals coherent
structures at the smallest scales which were hidden by the CIB anisotropies. It
opens new observational perspectives on the formation of structure in the
diffuse ISM which we discuss with reference to past work. We have succeeded to
perform a statistical separation from observational data only at a single
frequency by using non-Gaussian statistics.
