There is a growing recognition that electronic band structure is a local property of materials and devices, and there is steep growth in capabilities to collect the relevant data. New photon sources, from small-laboratory-based lasers to free electron lasers, together with focussing beam optics and advanced electron spectrometers, are beginning to enable angle-resolved photoemission spectroscopy (ARPES) in scanning mode with a spatial resolution of near to and below microns, two- to three orders of magnitude smaller than what has been typical for ARPES hitherto. The results are vast data sets inhabiting a five-dimensional subspace of the ten-dimensional space spanned by two scanning dimensions of real space, three of reciprocal space, three of spin-space, time, and energy. In this work, we demonstrate that recent developments in representational learning (self-supervised learning) combined with k-means clustering can help automate the labeling and spatial mapping of dispersion cuts, thus saving precious time relative to manual analysis, albeit with low performance. Finally, we introduce a few-shot learning (k-nearest neighbor or kNN) in representational space where we selectively choose one (k=1) image reference for each known label and subsequently label the rest of the data with respect to the nearest reference image. This last approach demonstrates the strength of self-supervised learning to automate image analysis in ARPES in particular and can be generalized to any scientific image analysis.