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/}
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