  All-sky observations of the Milky Way show both Galactic and non-Galactic
diffuse emission, for example from interstellar matter or the cosmic microwave
background (CMB). The different emitters are partly superimposed in the
measurements, partly they obscure each other, and sometimes they dominate
within a certain spectral range. The decomposition of the underlying radiative
components from spectral data is a signal reconstruction problem and often
associated with detailed physical modeling and substantial computational
effort. We aim to build an effective and self-instructing algorithm detecting
the essential spectral information contained Galactic all-sky data covering
spectral bands from $\gamma$-ray to radio waves. Utilizing principles from
information theory, we develop a state-of-the-art variational autoencoder
specialized on the adaption to Gaussian noise statistics. We first derive a
generic generative process that leads from a low-dimensional set of emission
features to the observed high-dimensional data. We formulate a posterior
distribution of these features using Bayesian methods and approximate this
posterior with variational inference. The algorithm efficiently encodes the
information of 35 Galactic emission data sets in ten latent feature maps. These
contain the essential information required to reconstruct the initial data with
high fidelity and are ranked by the algorithm according to their significance
for data regeneration. The three most significant feature maps encode
astrophysical components: (1) The dense interstellar medium (ISM), (2) the hot
and dilute regions of the ISM and (3) the CMB. The machine-assisted and
data-driven dimensionality reduction of spectral data is able to uncover the
physical features encoding the input data. Our algorithm is able to extract the
dense and dilute Galactic regions, as well as the CMB, from the sky brightness
values only.
