  The mid-infrared (MIR) spectra observed with the \textit{Spitzer} Infrared
Spectrograph (IRS) provide a valuable dataset for untangling the physical
processes and conditions within galaxies.
  This paper presents the first attempt to blindly learn fundamental spectral
components of MIR galaxy spectra, using non-negative matrix factorisation
(NMF). NMF is a recently developed multivariate technique shown to be
successful in blind source separation problems. Unlike the more popular
multivariate analysis technique, principal component analysis, NMF imposes the
condition that weights and spectral components are non-negative. This more
closely resembles the physical process of emission in the mid-infrared,
resulting in physically intuitive components. By applying NMF to galaxy spectra
in the Cornell Atlas of Spitzer/IRS sources (CASSIS), we find similar
components amongst different NMF sets. These similar components include two for
AGN emission and one for star formation. [... ABBREVIATED...] We show an NMF
set with seven components can reconstruct the general spectral shape of a wide
variety of objects, though struggle to fit the varying strength of emission
lines. We also show that the seven components can be used to separate out
different types of objects. We model this separation with Gaussian Mixtures
modelling and use the result to provide a classification tool.
  We also show the NMF components can be used to separate out the emission from
AGN and star formation regions and define a new star formation/AGN diagnostic
which is consistent with all mid-infrared diagnostics already in use but has
the advantage that it can be applied to mid-infrared spectra with low signal to
noise or with limited spectral range. The 7 NMF components and code for
classification are made public on arxiv and are available at:
\url{https://github.com/pdh21/NMF_software/}
