The spatial resolution of cavitation maps generated from passive recordings of cavitation emissions is compromised by the bandlimited nature of the recordings. Deconvolution based on the assumption that cavitation consists of a sparse series of discrete events allows the recovery of frequency components that are not only outside the frequency band of the receivers, but may also have been attenuated by the medium before being detectable. In the current work, two sparse deconvolution techniques, matching pursuit and basis pursuit, were applied to simulated and experimental cavitation recordings before they were beamformed to provide passive maps of cavitation activity. Matching pursuit was shown to reduce the maximal diameter of the point spread function by almost a third, at the cost of greater susceptibility to inter-source interference. In contrast, although basis pursuit causes an almost 20% increase in the maximal diameter of the point spread function, its application to experimental data appears to enhance the ability of passive mapping to resolve multiple sources.